About Rationally Speaking


Rationally Speaking is a blog maintained by Prof. Massimo Pigliucci, a philosopher at the City University of New York. The blog reflects the Enlightenment figure Marquis de Condorcet's idea of what a public intellectual (yes, we know, that's such a bad word) ought to be: someone who devotes himself to "the tracking down of prejudices in the hiding places where priests, the schools, the government, and all long-established institutions had gathered and protected them." You're welcome. Please notice that the contents of this blog can be reprinted under the standard Creative Commons license.

Wednesday, December 11, 2013

Mathematical Universe? I ain’t convinced

by Massimo Pigliucci

So the other day Julia Galef and I had the pleasure of interviewing mathematical cosmologist Max Tegmark for the Rationally Speaking podcast. The episode will come out in late January, close to the release of Max’s book, presenting his Mathematical Universe Hypothesis (MUH). We had a lively and interesting conversation, but in the end, I’m not convinced (and I doubt Julia was either).

The basic idea is that the ultimate structure of reality is, well, a mathematical one. Please understand this well, because it is the crux of the discussion: Tegmark isn’t saying anything as mundane as that the world is best described by mathematics; he is saying that the ultimate nature of reality is mathematics.

This is actually not at all a new thesis, though Max is advancing it in new form and based on different reasoning then before. Indeed, the idea has a long philosophical history, and can fruitfully be thought of as based on two distinct philosophical positions: Pythagoreanism, or mathematical Platonism; and Mathematical monism.

Mathematical Platonism is the idea that mathematical structures are real in a mind-independent fashion. They are not “real” in the same sense as, say, chairs and electrons, but they do have an ontological status independent of the human (or any other) mind. As readers of this blog know, I’m actually sympathetic to (though not necessarily completely on board with) mathematical Platonism. The best point in its favor is the so-called “no miracles” argument, the idea that mathematics is too unreasonably effective (at predicting things about the world) for it to be just a human invention, rather than somehow part of the inherent fabric of the world. (Interestingly, this argument is equivalent to one by the same name advanced by scientific realists to claim that science really does describe — approximately — how the world is, as opposed to the antirealist position that the only thing we can say about science is that it is empirically adequate.)

Mathematical monism is the stronger doctrine that not only are mathematical structures real, but they are the only real thing out there (or, more precisely, everywhere).

The combination of Platonism and monism yields a class of theories about the ultimate nature of reality, of which Tegmark’s MUH is one example. We have seen another one several times in the past, in the form of Ladyman and Ross’ ontic structural realism, the notion that there are no “objects” or “things” at the bottom, just (mathematical) relations.

While I have commented positively on ontic structural realism (again, without necessarily buying into it), and more generally on the idea of a “naturalistic” metaphysics (i.e., a metaphysics that takes seriously the best known physics), my conversation with Max Tegmark actually generated more doubts than illumination.
One obvious problem is posed by what it would mean for the world to be “made of” mathematical structures. The notion of mathematical structure is well developed, so that’s not the issue. A structure, strictly speaking, is a property or a group of mathematical objects that attach themselves to a given set. For instance, the set of real numbers has a number of structures, including an order (with any given number being either less or more than another number), a metric (measuring the distance between points in the set), an algebraic structure (the operations of addition and multiplication), and so on.

The problem is in what sense, if any, can a mathematical structure, so defined, actually be the fundamental constituent of the physical world, i.e. being the substance of which chairs, electrons, and so on, are made.

Of course, both Julia and I asked Max that very question, and we were both very unconvinced by his answer. When Tegmark said that fundamental particles, like electrons, are, ultimately mathematical in nature, Julia suggested that perhaps what he meant was that their properties are described by mathematical quantities. But Max was adamant, mentioning, for instance, the spin (which in the case of the electron has magnitude 1/2). Now, the spin of a particle, although normally described as its angular momentum, is an exquisitely quantum mechanical property (i.e., with no counterpart in classical mechanics), and it is highly misleading to think of it as anything like the angular momentum of a macroscopic object. Nevertheless, Julia and I insisted, it is a physical property described by a mathematical quantity, the latter is not the same as the former.

Could it be that theories like MUH are actually based on a category mistake? Obviously, I’m not suggesting that people like Tegmark make the elementary mistake of confusing the normal meaning of words like “objects” and “properties,” or of “physical” and “mathematical.” But perhaps they are making precisely that mistake in a metaphysical sense?

There are other problems with MUH. For one, several critics of Tegmark’s ideas have pointed out that they run afoul of the seemingly omnipresent (and much misunderstood) Gödel’s incompleteness theorems. Mark Alford, specifically, during a debate with Tegmark and Piet Hut has suggested that the idea that mathematics is “out there” is incompatible with the idea that it consists of formal systems. To which Tegmark replied that perhaps only Gödel-complete mathematical structures have physical existence (something referred to as the Computable Universe Hypothesis, CUH).

This, apparently, results in serious problems for Max’s theory, since it excludes much of the landscape of mathematical structures, not to mention that pretty much every successful physical theory so far would violate CUH. Oops.

Prompted by the above, I also asked Max about Gödel, and his response was that Gödel-related problems appear only in the case of infinite quantities, and he professed himself to be an infinity-skeptic. That took me by surprise, what do you mean you don’t believe in infinity? I thought this was a pretty darn well established concept in mathematics, at least since the work of Georg Cantor in the 19th century! But of course Tegmark was referring to the existence of physical, not mathematical, infinities. As is well known, there are certain calculations in physics that do generate infinities, for instance the singularity that shows up in the description of black holes, or the infinite quantities that are postulated in standard descriptions of phase transitions. The question of whether there really are infinities in physical systems is open, so surely Max is entitled to his skepticism. But it did seem a bit too convenient a position, in light of the above mentioned Gödel-related problems.

Another issue that didn’t convince either Julia or me during our conversation with Max is a crucial one: testability. I’m okay with philosophical speculations (and I use the term in a positive fashion!) about modal realism or the principle of plenitude, but if we are claiming to be doing science (as Tegmark surely is), then our speculations better make contact with empirical reality. Jim Baggott, in his Farewell to Reality: How Modern Physics Has Betrayed the Search for Scientific Truth, is already accusing physicists of losing touch with what it means to do science. Is Tegmark the latest example of the trend?

When we asked, he claimed that the MUH does make empirical predictions, but when pressed on the details the answer becomes far less satisfying than one would hope. For instance, Max said that one prediction is that physics will continue to uncover mathematical regularities in nature. Well, probably, but one surely doesn’t need to postulate MUH to account for that. He also has stated in the past that — assuming we live in an average universe (within the multiverse of mathematical structures) — then we “start testing multiverse predictions by assessing how typical our universe is.” But how would we carry out such tests, if we have no access to the other parts of the multiverse?

Max went on to say that his hypothesis has “zero free parameters” and is therefore favored by Occam’s razor. But if you check his paper at arxiv.org he says: “If this theory is correct, then since it has no free parameters, all properties of all parallel universes … could in principle be derived by an infinitely intelligent mathematician. … Finally, the ultimate ensemble of the Level IV multiverse would require 0 bits to specify, since it has no free parameters.” There are a couple of obvious problems here. One is the dearth of infinitely intelligent mathematicians, the second the fact that the above mentioned Level IV multiverse is precisely what gets dramatically (and unrealistically) shrunk as a result of Gödel-imposed limitations. And let’s not forget that Occam’s razor is just a useful heuristic, it should never be used as the final arbiter to decide which theory is to be favored, especially when we are talking about such highly speculative and empirically next to impossible (or even downright impossible) ideas to test.


In Many Worlds in One: The Search for Other Universes, critic Alex Vilenkin says that “the number of mathematical structures [in the multiverse] increases with increasing complexity, suggesting that ‘typical’ structures should be horrendously large and cumbersome. This seems to be in conflict with the beauty and simplicity of the theories describing our world.” In order to get around that problem, Tegmark assigns lower weights to more complex structures, but since this is done without a priori justification, it is an ad hoc move, which of course violates Occam’s razor. So, as much as I enjoyed our conversation with Max, for the time being I remain skeptical of the MUH and related hypotheses. Maybe we just need to wait for the appearance of an infinitely intelligent mathematician.

_______

[This just in from Max Tegmark himself!]

Thanks Massimo for the fun conversation during the interview and for raising these important questions! They are excellent ones, and a key reason why I spent three years writing this book is because I wanted to make sure to finally answer them all properly. Needless to say, I couldn't do justice to them in our short interview, so I'm very much look forward to hear what you think about my detailed answers in chapters 6, 10, 11 and 12. I think you'll find that our viewpoints are closer than your post suggest - for example, your statement "Tegmark assigns lower weights to more complex structures" is not something you'll find in the book. Rather, I describe how the measure problem is a terrible embarrassment for modern cosmology (regardless of whether the MUH is true or not) that we need to solve, and that our untested assumption that truly infinite things exist in nature are my prime suspect: we've never measured anything to better than 17 decimal places, have only 10^89 particles in our universe, and manage to do all our publishable physics simulations with computers that have finite resources, so even though my physics courses at MIT use infinity as a convenient tool, I respectfully object to your "OPS" argument that we somehow have experimental evidence for infinity in physics. Without infinity, there are, as you say, no Gödel issues in our physics.

I look forward to continuing this interesting conversation! ;-)

190 comments:

  1. > That took me by surprise, what do you mean you don’t believe in infinity? I thought this was a pretty darn well established concept in mathematics, at least since the work of Georg Cantor in the 19th century!

    I consider it obvious that we don't "really" use infinity; instead, mathematicians use symbolic representations of infinity to reason about infinite sets and processes. I think of it as akin to reasoning about a computer program. You can have an infinite loop in a program that would run forever if executed, but it's possible for a finite program to analyze the loop and decide if it will stop. This isn't possible in the general case (Halting Problem), but easy in practical cases (static analysis, program optimization).

    When mathematicians use infinity, they generally use limits or infinite maps, either concretely or abstractly. Limits and maps of these kinds are shorthand for, "this process would continue forever, if possible". Theorems are developed by reasoning about the symbolic representation (lim x -> 0, x:N |-> x+1) and the rules defined for those symbols, but those proofs are found via finite means.

    For a different take, see the work of Feng Ye in trying to place mathematics on a naturalistic and strictly finitistic position: http://www.academia.edu/4737368/Introduction_to_a_Naturalistic_Philosophy_of_Mathematics

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  2. My understanding is the the existence of an infinite set cannot be deduced from the other axioms of ZFC - it has to be stated as an axiom, and it's negation is completely consistent with the other axioms. But I'm not a set theorist.

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    1. This is true. But proving this consistency result requires postulating an infinite amount of (finite) sets (or requires you to make an assumption about an infinite ordinal). So while the ZFC-without-infinity axioms are formally consistent, it requires a kind of cognitive dissonance to believe them but to not believe in infinity. By analogy, the claim that ZFC is consistent cannot be deduced from the ZFC axioms, but it would require a kind of cognitive dissonance to believe all the axioms of ZFC are true but *not* believe that ZFC is consistent (unless you believe in true contradictions).

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  3. @ Massimo

    > When Tegmark said that fundamental particles, like electrons, are, ultimately mathematical in nature, Julia suggested that perhaps what he meant was that their properties are described by mathematical quantities. But Max was adamant, mentioning, for instance, the spin (which in the case of the electron has magnitude 1/2). Now, the spin of a particle, although normally described as its angular momentum, is an exquisitely quantum mechanical property (i.e., with no counterpart in classical mechanics), and it is highly misleading to think of it as anything like the angular momentum of a macroscopic object <

    Quantum mechanics holds that nature is fundamentally dualistic - the wave/particle duality. (In Aristotelian terms, this is known as "hylomorphism" - the duality of "form" ("morphe" in Greek) and "matter" ("hyle" in Greek)).

    > Max went on to say that his hypothesis has “zero free parameters” and is therefore favored by Occam’s razor. But if you check his paper at arxiv.org he says: “If this theory is correct, then since it has no free parameters, all properties of all parallel universes … could in principle be derived by an infinitely intelligent mathematician. <

    The term "infinitely intelligent mathematician" sounds like a euphemism for God. Of course, God is the most parsimonious explanation why there is something rather than nothing. (Just FYI. William of Ockham (credited with for "Occam's razor") was a Christian monk.)

    Question: Is professor Tegmark engaging in sound metaphysics or pseudoscience?

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  4. We have mathematical expressions for the current cutting-edge views of physics, and we don't have convincing physical interpretations for many of them. I can understand why that would affect him this way, although the way I tend to think about things, I doubt the theory until I comprehend a coherent physical interpretation.

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  5. Great article Massimo.

    Always a fan of bringing philosophy of mathematics and realism/platonism into play on this blog. I've been reading into Tegmark's proposal and the MUH for years, and although I'm accepting of his position, or at least one that's very similar, I think that his defense of it is lacking when arguments against it spring up. As a result, I'll do my best to mount a defense of my own (one that I am sure I have made elsewhere on this blog. The work towards objective truth is never over I suppose, and repetition is key).

    First, the idea that Platonism is somehow incomprehensible and the "What would a mathematical structure even be?" questions that constantly spring up against it are somewhat laughable when one considers what "physical object" really entails. In the article, the statement that captures this sentiment is:

    "The problem is in what sense, if any, can a mathematical structure, so defined, actually be the fundamental constituent of the physical world, i.e. being the substance of which chairs, electrons, and so on, are made."

    This problem is not one that is unique to the idea that the world is literally mathematics. Physicalism, a doctrine that virtually all scientists and a large portion of philosophers subscribe to, also runs into the same difficulties. I have always considered myself a physicalist as well, but after really contemplating what that means I start to revert to simply considering myself a naturalist, which contrary to many opinions can embrace abstract objects without a problem. When one really begins to delve into the nature of the "real" objects that are out there, their decidedly ephemeral nature is exposed. For one, objects in the universe are made almost entirely of empty space. The atoms that compose physical entities are something like 99.999% empty space, with a very tiny nucleaus and some extremely small electrons that occupy probability clouds around it. The nucleus itself is composed of even tinier quarks and gluons, the true nature of which (along with every other elementary particle in existence) is extremely hard to pin down. Maybe they're "vibrating strands of energy" as String Theory posits, or “knots” in the fabric of space-time as postulated by Loop Quantum Gravity.

    The point is, what we think about as being physical is really no such thing at all. At the bottom, it really seems to become mathematical equations and relations. And at that point, a la Massimo's invocation of no miracles and the idea of indispensability to our understanding of the world, you have a strong case for mathematical realism. The ironic part of the opposite position, that of nominalism with regard to mathematics, where something must be physically instantiated in space-time if it’s to be considered real (something which makes mathematics nonexistent), is its inherent assumption of physicality that is largely misunderstood and nonexistent in and of itself.

    Second, the invocation of the incompleteness theorems in arguments for/against platonism really has to stop, at least until someone comes up with a cogent argument that gets some sort of substantial utility from them. I had the pleasure of reading the arXiv article mentioned in the post several years ago, and the "incompatibility" of Platonism with Formalism is never defended and largely untrue. The article simply states that they cannot coexist because incompleteness means we will never have a complete and consistent axiomatic system that reflects the entirety of the mathematical universe. This is true, but so what? Mathematical truths can exist out there, and they can be forever out of reach, leading us to add axioms as we gain evidence/intuitive comfort with their truth, lighting more of the undiscovered world of mathematics as we go. Some truths will likely be forever out of reach, but this in no way means Platonism is out the window. At all.

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    1. I think that the problems physicalism runs into when you begin peeling back the deepest layers of reality, not to mention the unreasonable effectiveness arguments, makes mathematical realism much more intuitive, dare I say inevitable. People have to get over their hatred of all things Platonism just because of preconceived notions that it deals only with a mystical Theory of Forms and has supernatural undertones. If that’s not the case and they genuinely want to avoid adding any unnecessary ontology and sticking to what they think is real and physical, perhaps they should sit down and honestly inspect their own “common sense” notions of reality a bit further.

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  6. "the idea that mathematics is too unreasonably effective (at predicting things about the world) for it to be just a human invention, rather than somehow part of the inherent fabric of the world"
    I've never understood this concept. We invented mathematics in order to describe precisely the regularities we see in the world. And then we turn around and wonder why mathematics is so effective at describing the world? Isn't that what we invented it for?
    We can describe the world using English as well - it just wouldn't be as precise a description - but we don't wonder why the universe is inherently amenable to English description.
    I must be missing something here.

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    1. Well, with respect, I think you are! We're not talking here about mathematics "invented ... in order to describe precisely the regularities we see in the world". We're talking about a great mass of abstract mathematics which is developed purely for its own sake, totally within the bounds of the mathematical domain, which later get picked up by physicists because it happens to be useful for what they're investigating. And not only do the physicists find that the topic is of relevance to their work, but that other aspects of that mathematical domain hint at new possibilities in the physical world, which the physicist then goes on to find by experiment. This is the reverse of what you described, and has happened often in widely differing fields. A good example of a mathematical concept that was developed without any reference to physics is that of imaginary numbers, which only came into their full real-world significance much later on in quantum mechanics.

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  7. >The best point in its favor is the so-called “no miracles” argument, the idea that mathematics is too unreasonably effective (at predicting things about the world) for it to be just a human invention<

    This "unreasonable effectiveness" of mathematics is too often cited (even in pop culture) without good examples. I suspect the relevant thinking is akin to thinking that humans couldn't have invented language given its uncanny correspondence to reality. We have the word 'duck' and lo and behold there are ducks. Chills. If anyone here can give an example of this "weird match" between math and reality, it would be appreciated.

    That this "weird match" is the best point in favor of mathematical Platonism is just a nice way of saying that it just gets loonier from there. Let's consider the (real) Platonist "problem" of too may 7s for example. If the number 7 were not a transcendent entity, there would just be the 7s in each of our heads; but wait, that's too many 7s!
    :-\

    >I’m okay with philosophical speculations (and I use the term in a positive fashion!) about modal realism or the principle of plenitude<

    'Speculation' is an interesting word choice here. It's Platonistic in that it suggests the existence of a non-empirical, language-independent reality that we can only speculate about. Yes, are possible worlds in the realist sense are out there? If so, what are they like? (Are realist possible worlds possible?)

    But I look forward to the podcast. I'll be wearing my purple tennis shoes ;)

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    1. p.s. My counter-view to Platonism is that numbers are artificial entities like fictional characters (though not fictional - they do not pretend to be real - qua fictional character Sherlock Holmes is not fictional - fictional characters as such are not fictional). Platonism arises by not reading Wittgenstein and asking misguided questions about numbers, such as that of how many number 7s there are.

      We can make the same sort of mistake with Sherlock Holmes by asking how much he actually weighs. Well he's a fictional character so he doesn't weigh anything. Thus, Sherlock Holmes is weightless, which might help him in solving case.

      Artificial entities have only the properties we give them, and those that follow logically from their definitions. Platonism begs the question in that it assumes that e.g. numbers are entities of a kind that certain questions can asked about.

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  8. It's good to hear that Max's thinking is close to mine.

    But I like the term PUH (Programmatic Universe Hypothesis) instead: the universe/multiverse consists of programs. There is the issue of defining the nature (e.g. quantum) of the programming language(s). (I don't know if hypercomputation is involved.)

    And it's a theorem (by Jan Mycielski in J. of Symbolic Logic) that says The Axiom of Infinity becomes a statement about finite sets when its quantifiers are relativized.

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  9. - Evan (and Paul)

    I'm afraid you're missing something pretty big bud. The reason mathematics is entirely different from English when it comes to describing the world is because our mathematical models make unique predictions that can be empirically verified in nature. English (or any other language aside from mathematics/logic) doesn't do this at all, especially when you think of it in terms of giving descriptive names to certain objects. Now the very interesting thing is that looks highly inaccurate to say that we've tailored our mathematics to describe the regularities of nature. People always say this and I don't quite understand why. Mathematics was likely born from primitive counting, and many advances in the field since those days were totally unexpected and in many cases defied common sense. For example, would anyone have expected that many types of polynomial equations would not have general algebraic solutions after a certain degree, let alone predict the birth of Galois Theory and its beautiful solution to this problem? This doesn't even touch on many surprising results in calculus, combinatorics, or number theory that were totally unexpected. The discovery of calculus was indeed born from a need to model continuous and dynamical physical situations, but many of the details almost seem forced on us, rather than meticulously invented by mathematicians of old. Mathematical models make unique predictions, and when the level of abstraction reaches new heights (i.e. things like Quantum Field Theory), mathematical predictions that are verified in experiments makes the case for the universe being "mathematical" in some sense very hard to neglect.

    - Paul

    As for specific examples, there are several that have come about over the last few decades in the area of theoretical physics. For one, mathematical models that were developed in the mid 20th Century actually predicted particles that were later discovered in particle accelerators. This was an amazing example of mathematics predicting something in the real world, and the models that these physicists were esoteric structures of pure mathematics that had been studied by mathematicians with absolutely no interest in the "real" world and physics. These just happened to predict real particles many years later. An excellent article that highlights this mysterious connection is here:

    http://blogs.discovermagazine.com/crux/2012/07/30/the-mathematical-magic-behind-the-mysterious-higgs-boson/#.UqjmxeJWous

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    1. Thanks Pete. The example you give is a good one, because it involves the prediction of an entirely new entity, whereas I was thinking mostly of predicting the behaviour of systems.
      It may be difficult for me to speak intelligibly about this, as I'm not a physicist or mathematician. But in the case of discovering the Higgs Boson, for example, suppose we discovered empirical evidence for the particle first, and only later came up with the maths that describes it. Would we then still say that the maths *predicts* the particle, or just that we happened to find a way to describe it mathematically? We found a previously unobserved regularity in nature (ie. certain results from particle accelerator experiments) and precisely because they were regular they were amenable to mathematical description.
      And presumably there were other mathematical theories that would have predicted different physical realities. It just so happens that we found the right one before we had empirical evidence for it, rather than the other way 'round.

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    2. Pete - Thanks for the examples, the link, and the attempt at explanation.

      I read the article and it seems the fact to be explained is that pure math structures have been developed that later turn out to describe structures in nature. Note that to say that such math structures 'predict' natural structures is question begging in that it assumes that the special relation between math in nature in question exists. If I had claimed a few years ago that my Tarot cards predict that the Higgs Boson will be found, I would have had had to hold a certain view about the relation between Tarot and physics.

      The article gives no explanation of the special relation. It just assumes it exists through use of the word 'predict'. One possible explanation is that the cases of "prediction" are lucky cases among many ignored unlucky cases. How often do pure mathematical structures fail to predict natural structure? Sometimes even Tarot cards get it right. There's also the possibility that the sense that a special connection exists is bolstered by the fact that most measurable structures in nature can be mathematically modeled.

      But let's suppose the special relation exists. What does it consist in. It appears Tegmark takes it to be identity. I look forward to an explanation but presently I don't see how pure math relations can add up to solidity. It's a brute fact that there is stuff in the universe. That physics disappears down into "pure" mathematical relations contradicts this only at the risk of physics being wrong. It's obvious that chairs are not made of nothing, even if it seems so at a basic level.

      It is not weirdness of Platonism that bothers me. I like weirdness as it portents a possible revolution in understanding. To me Platonism just seems like another mystical idea that people like too much to drop. Perhaps philosophy of math needs a Richard Dawkins to come in and say: "Math is just made up, but that doesn't diminish its beauty. In fact it makes it more beautiful because it is true!" ;)

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    3. Great replies. I'll do my best to try and give my answer to these concerns.

      Eric -

      I agree that if the particles were indeed discovered first and then some sort of mathematical structure was utilized to explain the patterns, it might be hard to advocate the situation as evidence in favor of mathematical realism. That being said, this case is one of several I have come across that shows that mathematical structure seems to be inherent in physical reality.

      Now you're right to point out that other mathematical theories might give rise to different physics, and I think this is where the MUH (and modal realism for that matter) come into play. One might argue that different mathematical theories do in fact have an existence and are instantiated in different physical worlds that exist beyond our observable universe. I realize that this idea is not amenable to empirical verification at the present (and perhaps never will be), but I think your point of regularity and structure gives us a direction to proceed in. The fact that our Universe can be described by mathematical structures should push us to accept that those mathematical structures are instantiated in the world. Many have argued that it is simply one language that is used to understand the world, but that others might exist. I find this very hard to believe. There are thousands of languages that human beings have developed, and it seems that only one has the universal quality of being able to describe the world so effectively and on top of that predict other phenomena that are born out in experiment.

      To further my point, there is something in quantum mechanics known as the Aharonov-Bohm effect, which can be understood by taking the idea of a "mathematical potential" as physically real. Now from physics class you may remember the ideas of fields and potentials (think potential energy). For centuries, fields were considered to be physically real whereas potentials were considered mathematical artifacts useful for solving problems but not necessarily real in and of themselves. The Aharanov-Bohm effect, however, forced people to take the idea of a "potential" as having an ontological existence of its own.

      For a quick synopsis of this effect, you can check the Wiki article and look over this link (page 21 mentions the reality of the potential):

      http://www.ece.rice.edu/~kono/ELEC563/2005/AharonovBohm.pdf

      Paul -

      Even though chairs and other objects seem to you very solid and "real," I'm afraid modern physics shows that they have very little "physical" existence to them. It is the electromagnetic force and the fact that certain wavelengths of light are reflected off the surface that gives them the appearance of solidity. Again, common sense notions sometimes have to be dispensed with upon closer examination.

      As far as what this special relation exists of, I honestly do not know. But I also don't know what the true nature of quantum mechanics is (its safe to say no one does). On top of that, the foundation of quantum mechanics, the wavefuction, exists in an infinite dimensional Hilbert space, a mathematical structure that one might be hard pressed to describe as "real" at first blush. The fact that it is the most accurate theory in the history of Mankind (to something like 12 decimal places at the present time), should act as strong evidence in favor of it being the correct (or at least on the right track) description of nature. If we're talking mysticism, QM would be right up there on the list. But there is no need for mysticism. Abstract objects, though bothersome, in no way entails Gods, angels, or other supernatural entities that have been associated with Platonism at times. Platonism has never needed those supernatural elements.

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    4. As far as what this special relation exists of, I honestly do not know. But I also don't know what the true nature of quantum mechanics is (its safe to say no one does). On top of that, the foundation of quantum mechanics, the wavefuction, exists in an infinite dimensional Hilbert space, a mathematical structure that one might be hard pressed to describe as "real" at first blush. The fact that it is the most accurate theory in the history of Mankind (to something like 12 decimal places at the present time), should act as strong evidence in favor of it being the correct (or at least on the right track) description of nature. If we're talking mysticism, QM would be right up there on the list. But there is no need for mysticism. Abstract objects, though bothersome, in no way entails Gods, angels, or other supernatural entities that have been associated with Platonism at times. Platonism has never needed those supernatural elements.

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    5. Pete,

      >Again, common sense notions sometimes have to be dispensed with upon closer examination. <

      Yes, once we let go of the commonsense notion that the universe exists, everything falls into place ;) More seriously, it cannot be coherently argued that solidity is an illusion. Things cannot seem solid to the touch without being so. What would it mean for it to be an illusion that a piece of granite is solid, i.e., that it does not give when pressed upon?

      >The fact that it is the most accurate theory in the history of Mankind (to something like 12 decimal places at the present time)<

      If math fits the world so magically well why do we need decimal places?
      ;)

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    6. Hahaha come on now Paul the reason we have decimal places is because we are cursed with never being able to attain infinite precision. As of now our measurements are accurate enough to many significant figures, and will only get more accurate as time goes on.

      And I don't in any way support any form of idealism, so I definitely understand where you're coming from with the solidity comment. What I mean is, and this was considered to be the case by many before modern atomic theory, the idea of continously "solid" objects does not exist. These objects are amalgamations of smaller fundamental constituents (atoms) that are mostly empty. The non-empty part consists of objects that are heavily described by mathematical concepts and relations. You see where I'm going with this...

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  10. What is the "OPS" argument referred to?

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  11. ...the only thing we can say about science is that it is empirically adequate.

    Well, it is grounded in human experience. What's more real than that?

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  12. The mathematically equation for Nature's reality is =.
    The proof or truth is self-evident.
    =

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  13. Hi Massimo,


    Oh my!

    Very excited to see you writing about this. This is my number one pet topic in philosophy, and in fact the reason I got interested in philosophy in the first place. I came up with the MUH independently a couple of years ago and started doing research to find out if the idea was already out there, which is when I came across Max Tegmark.

    I was considering writing a book about this myself, but now that Tegmark has done so my efforts would be superfluous.

    I am absolutely convinced Tegmark is right, but there are a few points where I would differ with him.

    1. It's metaphysics,not philosophy. This is not science.
    2. I don't see the incompleteness theorems as a problem. The retreat to the CUH is unnecessary.
    3. The unreasonable effectiveness of mathematics is not the best argument for the MUH. The best argument for the MUH is that it is necessarily true that the universe must exist as a mathematical object if mathematical Platonism is true. If so, then there is no need to posit a "physical" universe, and in fact it becomes clear that the concept of "physical" as distinct from "mathematical" is in fact incoherent, and in some sense magic.
    4. I'm not an infinity skeptic. At least, even if there is no infinite quantity in this universe, I see no problem with there being infinite quantities in other universes.

    Pete has already explained much of this, and I agree with what he has said.

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    Replies
    1. @ Disagreeable Me

      > 1. It's metaphysics,not philosophy. <

      Metaphysics is a branch of philosophy.

      > The best argument for the MUH is that it is necessarily true that the universe must exist as a mathematical object if mathematical Platonism is true <

      That's a tautological statement, not a compelling argument.

      > If so, then there is no need to posit a "physical" universe, and in fact it becomes clear that the concept of "physical" as distinct from "mathematical" is in fact incoherent, and in some sense magic. <

      Mathematical abstractions are nonphysical. And you would have to invoke magic to explain how an immaterial mathematical abstraction manages to materialiize itself. (The idea is completely unintelligible.)

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    2. Hi Alastair,

      > 1. It's metaphysics,not philosophy. <

      I agree, this is mistaken. I was writing my post in a hurry while about to board a flight and I didn't proofread.

      I meant to say "It's metaphyics, not physics".

      Sorry about that.

      >That's a tautological statement, not a compelling argument.<

      It ought to be a compelling argument if you entertain mathematical Platonism.

      >Mathematical abstractions are nonphysical. <

      The whole point of the MUH is that what is physical is a matter of perspective. To you, a chair is physical, because it exists within the same mathematical structure that you do (our universe). From the point of view of someone in another universe, it is not physical, just as they are not physical from your point of view.

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    3. @ Disagreeable Me

      > It ought to be a compelling argument if you entertain mathematical Platonism. <

      I will retract me previous statement. It's not a tautology. Mathematical Platonism and the MUH are different. Mathematical Platonism (unlike the MUH) does not hold that everything is nothing more than mathematical abstractions. It simply holds that mathematical abstractions have an independent existence. (It doesn't necessarily deny matter/energy, consciousness). That being said, I subscribe to a form of Platonism. But I do not limit my eternal Platonic forms to simply mathematical abstractions. There are other eternal objects (e.g. truth, beauty, and goodness). Also, I believe it is nonsense to talk about perfect forms and/or ideals as having some kind of existence independent of any subjectivity whatsoever. (They require an eternal, subjective locus - namely, the mind of God.)

      > The whole point of the MUH is that what is physical is a matter of perspective. <

      The whole point of the MUH is that only mathematical abstractions exist, independently of any perspective whatsoever. It's completely unintelligible. How exactly do nonphysical mathematical abstractions bring about the physical world we experience? Moreover, how exactly do nonphysical mathematical abstractions bring about any kind of experience whatsoever?

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    4. Hi Alastair,

      >How exactly do nonphysical mathematical abstractions bring about the physical world we experience?<

      It follows from three premises.

      (1) The computational theory of mind - or more specifically that certain logical/computational/mathematical structures are responsible for consciousness and perception and not any particular "stuff"
      (2) Mathematical Platonism
      (3) Naturalism - our universe is a universe governed by mathematical laws (the laws of physics)

      Given 3, there exists some mathematical structure that is isomorphic to this universe. Simulation of this structure, if such a thing were possible, would let us see how the lives of beings isomorphic to you and I play out. This very conversation could be found somewhere within the structure just as a particular wrinkled curlicue might be found in some obscure corner of the Mandelbrot set.

      But whether we simulate it or not, this structure and the beings within it exist abstractly, given (2).

      The beings within this structure are genuinely conscious, given (1).

      These beings therefore have precisely the same conscious experience as we do, so there is no way we can be sure we are not such beings.

      >Moreover, how exactly do nonphysical mathematical abstractions bring about any kind of experience whatsoever?<

      As I was saying to Massimo, it helps to visualise it if you imagine that the universe is a computer simulation being run by an infinitely powerful computer. For this to work, you need to accept the computational theory of mind. If you don't, adopt it for the sake of the analogy if only to understand how I can actually believe something so bizarre.

      However mathematical Platonism means you don't actually need the computer because what happens within the simulation is a mathematical structure which exists regardless, just as the shape of the Mandelbrot set has always existed and was not created when computers started to visualise it.

      So though I believe in the computational theory of mind, I don't actually think that you need computers for consciousness or that computers actually can be conscious. Rather, I think certain algorithms are conscious.

      But this is a subtle point and easily misunderstood. I probably don't mean it in quite the way that you think I do, as I find it really hard to concisely express what it is I do mean.

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    5. @ Disagreeable Me

      > But this is a subtle point and easily misunderstood. I probably don't mean it in quite the way that you think I do, as I find it really hard to concisely express what it is I do mean. <

      The reason that you are having a difficult time expressing your worldview is because it is incoherent. (I'm not deliberately trying to be flippant here. I'm simply calling it like I see it.)

      > As I was saying to Massimo, it helps to visualise it if you imagine that the universe is a computer simulation being run by an infinitely powerful computer. <

      What you're not intellectually grasping here is that it is not possible to visualize something that is truly abstract. For example, you cannot make an image of a perfect circle, not even in your mind's eye.

      > However mathematical Platonism means you don't actually need the computer because what happens within the simulation is a mathematical structure which exists regardless, just as the shape of the Mandelbrot set has always existed and was not created when computers started to visualise it. <

      That's the point. There is no computer simulation taking place, because there is no hardware for the software to run on. (That's what you have completely failed to explain. How exactly does nonphysical software generate physical hardware? How exactly does nonphysical information process itself when there is no physical processor for it to process on?)

      What you're describing is more suggestive of a "thought-process" than a "computer simulation" - something more suggestive of "idealism" (the true from of "immaterialism", not your mathematical monism (a faux version of it)).

      Incidentally, the "laws of physics" are not literally governing the universe. They may be descriptive of some natural process, but they're not the causal agents of it. (You apparently believe that mathematical abstractions are causally efficacious.)

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    6. >The reason that you are having a difficult time expressing your worldview is because it is incoherent.<

      I beg to differ.

      Ok, if I must attempt to explain this...

      I don't think it makes sense to talk of physical hardware being conscious because consciousness is not a property of a physical object. One computer could host any number of conscious minds, so it's not the computer that has the mind.

      And indeed a mind could be hosted across several distributed computers. In fact two computers could perform the same calculations in parallel to host the same mind even with no communication between them.

      So when I say an algorithm is conscious, I only mean algorithms that could be thought of as simulations of temporal environments within which conscious minds exist and process information. I think that no matter how many times you run this simulation on the computer, or on what hardware you run it, it's the same mind each time, just as each time you read a novel it's the same story that plays out, and it doesn't matter who is reading it. The characters (within the fiction of the story) are conscious and perceive their world to be real.

      I think mathematical structures which host minds are like fictional stories. I think we are like characters in the stories. We feel like we are conscious and feel that the world exists, but we are objectively no more real than entities within mathematical structures we discover in the abstract.

      But that's not to say that we are not real, because what I am saying is that mathematical structure IS reality, and the fact that we perceive a difference between the physical and the abstract is a subjective property of our perspective as observers within one particular mathematical universe.

      >What you're not intellectually grasping here is that it is not possible to visualize something that is truly abstract.<

      You're taking me too literally. I'm not asking you to draw a literal picture. I'm asking you to conduct a thought experiment imagining the universe is a simulation. Whether or not I can picture a perfect circle has nothing to do with what I'm saying.

      >How exactly does nonphysical software generate physical hardware? <

      It doesn't, because there is no physical hardware. The physical stuff in this universe only seems physical to us because we can interact with it, the way a sentient character in a virtual environment in a computer simulation would perceive objects within that environment to be physical. This idea is perfectly straightforwards, I feel.

      >How exactly does nonphysical information process itself when there is no physical processor for it to process on?<

      Because it's not processed, it just exists Platonicaly. Like the Mandelbrot set. How did the Mandelbrot set exist before Benoit Mandelbrot discovered it and started processing it? Processing has nothing to do with existence. Mathematical objects do not need to be created or sustained. They just are.

      >Incidentally, the "laws of physics" are not literally governing the universe.<

      I disagree.

      >You apparently believe that mathematical abstractions are causally efficacious.<

      They are within the context of those abstractions. The rules of Conway's Game of Life are causally efficacious to the evolution of the grid within that mathematical object.

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    7. @ Disagreeable Me

      > In fact two computers could perform the same calculations in parallel to host the same mind even with no communication between them. <

      I'll probably have some regrets later for asking this question now: But how exactly do "two minds" (or "two computers") act as one if they're not in communication with each other?

      > I think that no matter how many times you run this simulation on the computer, or on what hardware you run it, it's the same mind each time, just as each time you read a novel it's the same story that plays out, and it doesn't matter who is reading it. <

      What you don't seem to be intellectually grasping here is that the "reader" itself must be conscious in order for any subjective experiences to take place. Also, you are presupposing some kind of hardware for your software to run on when your worldview specifically precludes the existence of any hardware.

      > We feel like we are conscious and feel that the world exists, but we are objectively no more real than entities within mathematical structures we discover in the abstract. <

      Consciousness is axiomatic ("self evident"). Any attempt to deny it, presupposes it. (And it definitely appears that you are attempting to deny the reality of your own subjectivity. You're entire mental life may be an illusion, but the illusion itself presupposes consciousness.)

      > But that's not to say that we are not real. <

      You just denied consciousness in your previous paragraph. You are now contradicting yourself in this one.

      I don't "feel like I'm conscious." I am conscious. Consciousness is no illusion. Illusions themselves presuppose consciousness.

      > You're taking me too literally. I'm not asking you to draw a literal picture. I'm asking you to conduct a thought experiment imagining the universe is a simulation. Whether or not I can picture a perfect circle has nothing to do with what I'm saying <

      You have argued that all physical phenomena (as well as mental phenomena) reduce to mathematical abstractions. Did you or did you not mean this literally? If you did not mean this literally, then there is no point to continue this debate. If you meant this literally, then I cannot imagine that all phenomena are the simulation of mathematical abstractions. Why? Because I cannot LITERALLY make an image of a mathematical abstraction. I can't. You can't. No one can't. That's the point!

      > It doesn't, because there is no physical hardware. The physical stuff in this universe only seems physical to us because we can interact with it, the way a sentient character in a virtual environment in a computer simulation would perceive objects within that environment to be physical. This idea is perfectly straightforwards, I feel. <

      If there is no physical hardware, then your software cannot run. Computers require hardware, software, and information. Therefore, your analogy (if that is the right word) falls apart.

      > Because it's not processed, it just exists Platonicaly. Like the Mandelbrot set... Processing has nothing to do with existence. Mathematical objects do not need to be created or sustained. They just are <

      Our subjective experiences are clearly undergoing change. That cannot be denied. So, unless you can account for change, we can summarily disregard your metaphysical theory. It doesn't speak to the empirical facts. (Apparently, your "simulation" is a still-frame picture, not a motion picture.)

      > I disagree <

      If you believe the "laws of physics" are LITERALLY governing the world, then you have ascribed conscious agency to the laws of physics.

      > They are within the context of those abstractions. <

      Mathematical abstractions are not causally efficacious. No amount of idolizing mathematical abstractions will change this fact.

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    8. >how exactly do ... "two computers" act as one if they're not in communication with each other?<

      If I run a program to simulate the Mandelbrot set here and you do the same where you are, we're both simulating the same thing. As long as both computers do the same computations, then in my view whatever is computed is the same, whether that be a run of the mill number crunching exercise or the simulation of a mind.

      >What you don't seem to be intellectually grasping here is that the "reader" itself must be conscious<

      What you don't seem to be grasping is that I'm not talking about the reader at all. I'm talking about the experience of the characters in the story.

      Now, I agree that they don't actually have a subjective experience, because their minds are not mathematical structures but scripted by an author. But if they're entities within a mathematical construct, that's no longer true, and fictional characters are in my view good familiar analogies to such entities even if they are not conscious.

      >Also, you are presupposing some kind of hardware for your software to run on<

      No I'm not. I'm inviting you to imagine that the universe is a simulation as only a first step on the road to grasping what it might mean for the universe to be non-physical.

      >You just denied consciousness in your previous paragraph.<

      I never said we were not conscious. I said we feel like we are conscious. That's not a denial of consciousness.

      >You have argued that all physical phenomena (as well as mental phenomena) reduce to mathematical abstractions. Did you or did you not mean this literally?<

      I did. That's not where you seemed to be taking me too literally. I asked you to visualise that the universe was a simulation and your rejoinder was that I can't visualise a perfect circle. You seemed to be putting rather too much emphasis on the word "visualise".

      I don't get how you move from the proposition that no one can imagine a perfect circle to the proposition that it is inconceivable that the universe is mathematical. As far as I can see it's a complete non sequitur.

      >If there is no physical hardware, then your software cannot run. Computers require hardware, software, and information.<

      There is no hardware. The universe is pure mathematics. The analogy to a computer simulation is a crutch to get you to visualise the universe as something other than physical. It's an analogy, not the MUH itself, so of course it's not precisely the same.

      >So, unless you can account for change, we can summarily disregard your metaphysical theory.<

      There is no change from an objective point of view. There is only change from the point of view of an observer who can only see the state of the universe at one particular point in time. This idea is well established in philosophy and is known as the B theory of time.

      >Apparently, your "simulation" is a still-frame picture, not a motion picture.<

      No, it's like a motion picture. A DVD sitting on the shelf is not changing, yet the characters within the movie perceive their world to be changing.

      >If you believe the "laws of physics" are LITERALLY governing the world, then you have ascribed conscious agency to the laws of physics. <

      Only if you take a very particular interpretation of "governing". I could say "determining" or "causing all events" instead if you prefer. I certainly don't mean that they are conscious.

      >Mathematical abstractions are not causally efficacious. No amount of idolizing mathematical abstractions will change this fact.<

      You've ignored my point. Take Conway's Game of Life for example. Why do you think it is so wrong for me to say that the laws of that toy universe are causally efficacious in the evolution of that universe? It seems to me there's no problem with mathematical abstractions being causally efficacious within a mathematical abstraction.

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    9. @ Disagreeable Me

      > There is no change from an objective point of view. There is only change from the point of view of an observer who can only see the state of the universe at one particular point in time. This idea is well established in philosophy and is known as the B theory of time. <

      It's really ironic. You presuppose a God's-eye view of the world-process (the changeless, "still-frame," timeless, objective point of view). And yet you reject the very "eye" that perceives that view and endows the world with its very existence.

      "esse est percipi, (to be is to be perceived)" - Bishop George Berkeley

      Philosopher John McTaggart coined the terms "A-series" and "B-series" in order to argue for the unreality of time. He argued for the unreality of time in order to make an argument for philosophical idealism. (Plato was an idealist.)

      "Idealism is the group of philosophies which assert that reality, or reality as we can know it, is fundamentally mental, mentally constructed, or otherwise immaterial." (source: Wikipdia: Idealism)

      I have already argued in another thread that, if time is a subjective illusion, then this implies that subjectivity (consciousness) itself is eternal (timeless).

      > It seems to me there's no problem with mathematical abstractions being causally efficacious within a mathematical abstraction. <

      It's problematic for me.

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  14. I also have some comments and questions for you Massimo.

    If you think Godel is a problem for the MUH, please explain the argument. As Pete has already pointed out, this argument was never really spelled out. It seems that it simply doesn't sit well with some people that the universe could be made of mathematics if Godel has shown there are limits to what mathematics can prove. I don't however see any link between the two concepts.

    2) You don't get how it could be that the universe is made of math. I agree, it's unintuitive, but I think it actually works if you think about it and it makes more sense than that it's made out of physical "stuff", which is a concept that while intuitive doesn't really bear up under scrutiny.

    To start with, you can imagine that the universe is a simulation running on an infinitely powerful computer. This computer simulates the laws of physics perfectly, and we would have no way of knowing that we were in a simulation because from our point of view everything would be real.

    But if mathematical Platonism were true, this computational structure would exist even if no actual computer is ever built. Our universe is defined and our life stories play out within this structure just the same way that the arc of a parabola plays out within the structure of a quadratic equation, whether or not any particular mathematician or graphing program plots it.

    As such, our universe only feels physical to us because of our local, limited perspective.

    If mathematical Platonism is true, then we and our universe must exist in this way, whether or not the physical universe or any physical universe exists. The physical universe hypothesis thus adds nothing of value and is essentially meaningless.

    Does this in any way answer your question about what it would mean for the universe to be made of math?

    The apparent simplicity of this universe is the single best argument against the MUH. I agree, it seems we should expect the universe to be infinitely complex if the MUH is true. There are counter-arguments against this, however.

    Perhaps the universe IS infinitely complex. Perhaps there are an infinite number of physical laws, but most of them are pretty inconsequential or are too feeble to make their effects felt. It may be, for example, that quarks and gluons etc are made up of more fundamental particles, which are made up of more fundamental particles, etc ad infinitum.

    Also, I would suggest that life could only exist in a universe that was reasonably regular. Life could not evolve if the laws of the universe were so chaotic and complex that organisms could not evolve to exploit predictable regularities. As such, there may be an anthropic selection effect to explain why our universe is so relatively neat.

    Finally, I think we need to bring the computational theory of mind into the argument. If you doubt the CUH, then you think there's something more than mathematical/computational structures responsible for consciousness. There's something about the "stuff" in our minds that makes us conscious. As such, if you buy the Chinese Room argument and its ilk, that's an additional reason for doubting the MUH.

    Of course, as has been discussed on previous posts, I don't think any such arguments hold up.

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    1. My beef with the "made of math" or "made of comp-math" hypothesis is that it lacks some beef. I ask: What is the underlying (programming) language? Could be the lambda calculus ("can be called the smallest universal programming language of the world"), the binary lambda calculus, a quantum lambda calculus, and so on up (or down).

      http://www.inf.fu-berlin.de/lehre/WS03/alpi/lambda.pdf
      http://en.wikipedia.org/wiki/Binary_lambda_calculus
      http://www.het.brown.edu/people/andre/qlambda/

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    2. Hi Philip,
      I don't think there is an underlying programming language.

      I've explained this to you before. I don't see that the language used to represent a concept matters in the slightest. A programming language is just a way to express an algorithm. It's essentially just a form of notation. Whether we use Pi or some other symbol to represent the ratio of the radius to the circumference is of just as little consequence.

      The actual concept is distinct from the language.

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    3. Also, due to typing in a rush, the sentence in the second last paragraph from my post above should read:

      "If you doubt the CTM [[The Computational Theory of Mind, not the CUH as I originally said]], then you think there's something more than mathematical/computational structures responsible for consciousness. "

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    4. Disagreeable Me: "I don't see that the language used to represent a concept matters in the slightest. ... The actual concept is distinct from the language."

      Now THAT is what I call PLATONISM!

      Yuk. :)

      Delete
  15. Hi Massimo,

    > not to mention that pretty much every successful physical theory so far would violate CUH.<

    Please explain this, as I think you may be wrong.

    In particular, quantum mechanics is often thought to be uncomputable but this is incorrect. It is computable, but the difficulty of computation grows horribly as the system becomes more complex. It may be for all practical purposes impossible to simulate complex quantum systems on real computers, but the physics itself is still computable in the relevant sense.

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  16. Max (Tegmark),

    thanks for taking the time to write your brief response to the article!

    > your statement "Tegmark assigns lower weights to more complex structures" is not something you'll find in the book. <

    Well, the book is not out yet, but I trust you! The problem to which that quote (from one of your writings, the 2003 chapter on parallel universes) refers, however, remains, and was brought up by one of your critics, Alexander Vilenkin, in his 2006 book.

    > our untested assumption that truly infinite things exist in nature are my prime suspect <

    I’m not sure how the problem of infinities is related to Vilenkin’s objection, but that may simply be my limitation (I’m not a physicist!).

    > I respectfully object to your "OPS" argument that we somehow have experimental evidence for infinity in physics. <

    The oops did not refer to the problem of infinities, but to the apparent exclusion of much of the mathematical landscape, and (as I understand it) of most successful physical theories, imposed by Godel-limitations. I understand you think Godel-limitations apply only to infinite systems, but that doesn’t seem to be the position of your critics. Do you think your critics accept infinities in physics, hence their problem with the theory?

    Alastair,

    > Quantum mechanics holds that nature is fundamentally dualistic - the wave/particle duality. <

    I think you are misusing the term “dualism” in this context.

    > Of course, God is the most parsimonious explanation why there is something rather than nothing <

    Wrong, and irrelevant to the discussion at hand.

    > Is professor Tegmark engaging in sound metaphysics or pseudoscience? <

    Neither. It’s speculative science bordering on metaphysics.

    Pete,

    > I start to revert to simply considering myself a naturalist, which contrary to many opinions can embrace abstract objects without a problem. <

    I agree, I consider myself a naturalist, and I have no problem with some versions of mathematical Platonism. I do, however, have a problem with the idea that the physical world *is* a mathematical structure (i.e., I have a problem with monism, not with Platonism per se).

    > objects in the universe are made almost entirely of empty space. The atoms that compose physical entities are something like 99.999% empty space <

    Indeed, but that *almost* makes all the difference…

    > The article simply states that they cannot coexist because incompleteness means we will never have a complete and consistent axiomatic system that reflects the entirety of the mathematical universe. This is true, but so what? <

    I share your puzzlement on that particular point, but again my problem is with monism. The objection that Godel’s theorems would limit the physical multiverse to computable ones, and that most accepted physical theories would violate that restriction seems compelling to me. But I’m no expert on Godel…

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    1. @ Massimo

      > I think you are misusing the term “dualism” in this context. <

      No, I am not misusing the term. The "wave-particle duality" holds that nature has a nonphysical aspect (the "wave" aspect) as well as a physical aspect (the "particle" aspect). (This qualifies as hylomorphic dualism - the Aristotelian duality of form and matter.)

      "The much-vaunted wave–particle duality of quantum mechanics conceals a subtlety concerning the meaning of the terms. Particle talk refers to hardware: physical stuff such as electrons. By contrast, the wave function that attaches to an electron encodes what we know about the system. The wave is NOT a wave of ‘stuff,’ it is an information wave. Since information and ‘stuff’ refer to two different conceptual levels, quantum mechanics seems to imply a DUALITY of levels akin to mind-brain duality." (soure: pg. 8, "The Physics of Downward Causation" by physicist Paul Davies...reprinted on pp. 44-45, "The Re-Emergence of Emergence" edited by Philip Clayton and Paul Davies)

      > Wrong, and irrelevant to the discussion at hand. <

      It's quite relevant seeing how you broached the topics of "Occam's razor," "cosmology," and an "infinitely intelligent mathematician" - all three in the same context. And God is the most parsimonious explanation for why there is anything all. The idea that something spontaneously emerged from nothing is a magical explanation. And anyone who would argue otherwise is simply smuggling in magic without acknowledging it as such. It's intellectually dishonest.

      > Neither. It’s speculative science bordering on metaphysics. <

      That's about as clear as mud.

      If an individual is engaging in metaphysics and selling it as science, then he/she is engaging in pseudoscience. (It's seems to me that you're holding a double standard in regards to pseudoscience.)

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    2. Alastair, why is god the most parsimonious explanation for why there is something rather than nothing? I've never understood why that isn't just a form of a god of the gaps argument. For example, what would have prevented me a few hundred years ago from saying 'God is the most parsimonious answer for why there are humans on earth. The idea that they just spontaneously emerged is a magical explanation.' I'm assuming you'd state those are different, but how? To me, we don't know something, and that's the end of the story.

      Also, absent good reasons to believe in god, would I be justified in withholding belief that god is, in fact, the most parsimonious explanation? Thanks.

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    3. @ Stewy0013

      > Alastair, why is god the most parsimonious explanation for why there is something rather than nothing? <

      Do you have a more parsimonious explanation?

      > I've never understood why that isn't just a form of a god of the gaps argument. <

      Something from nothing is a pretty big gap to fill.

      > For example, what would have prevented me a few hundred years ago from saying 'God is the most parsimonious answer for why there are humans on earth. The idea that they just spontaneously emerged is a magical explanation.' I'm assuming you'd state those are different, but how? <

      If you would have argued that they just spontaneously emerged, then you would be invoking a magical explanation. (You're making my point. Today we have physicsts (e.g. Hawkings, Krauss, and (apparently) Tegmark) making the very same argument...that the universe just spontaneously emerged out of nothing.)

      > To me, we don't know something, and that's the end of the story. <

      If you don't believe we can engage in a rational discussion of metaphysics, then that is certainly your prerogative. But by so doing, you bar yourself from the conversation at hand. (This particular blog post is dealing with a metaphysical issue, just in case you weren't aware of that fact.)

      > Also, absent good reasons to believe in god, would I be justified in withholding belief that god is, in fact, the most parsimonious explanation? Thanks <

      That there is something rather than nothing is sufficient evidence to posit the existence of God. (No explanation (your position) does not qualify as any explanation whatsoever.)

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    4. Alastair,

      Thanks. I too think Krauss and Hawking are making nonsense claims but I still don't understand why the fact that there is something rather than nothing is sufficient to posit the existence of God. By the same reasoning, would I not be able to posit the existence of some other type of creator with any characteristics I want? What are the characteristics of a God (to you) and how do you know? Thanks.

      Delete
    5. @ Stewy0013

      > Thanks. I too think Krauss and Hawking are making nonsense claims but I still don't understand why the fact that there is something rather than nothing is sufficient to posit the existence of God. <

      Because if there were nothing, then there would be nothing to explain. But the fact is that there is something, and that something is not self-explanatory.

      > By the same reasoning, would I not be able to posit the existence of some other type of creator with any characteristics I want? <

      The way I see it there are only two "gap" theories to explain why there is something rather than nothing - the "God of the gaps" theory or the "materialism of the gaps" theory (the gap argument slices both ways).

      > What are the characteristics of a God (to you) and how do you know? Thanks. <

      If you want to engage in a metaphysical debate, then you're going to have to show your cards. Because I'm not going to play only defense while you have the luxury of playing only offense. Previously, I asked you if you had a more parsimonious explanation. No explanation was forthcoming.

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    6. You misunderstand me. I'm not trying to engage in a metaphysical debate. I'm just asking what your conception is of god so that I can make some sense of your parsimonious concept. I don't believe in god. I don't know how the universe was created.

      Delete
    7. @ Stewy0013

      > You misunderstand me. I'm not trying to engage in a metaphysical debate. I'm just asking what your conception is of god so that I can make some sense of your parsimonious concept. I don't believe in god. I don't know how the universe was created <

      If you're not attempting to engage in a metaphysical debate, then there is no need for you to ask that question.

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  17. Evan,

    > I've never understood this concept. We invented mathematics in order to describe precisely the regularities we see in the world. And then we turn around and wonder why mathematics is so effective at describing the world? <

    This is a common objection to the no-miracles argument, but I think it has no bite. The issue isn’t that mathematics does its job, it’s that it does it in so many complex and completely unexpected ways that its effectiveness seems unreasonable. We didn’t go for decades into high dimensional topology, say, to solve practical problems. But it then turns out that, somehow, topology has lots of application to physical theories. Why?

    > We can describe the world using English as well - it just wouldn't be as precise a description - but we don't wonder why the universe is inherently amenable to English description. <

    Good analogy, but ultimately unconvincing, I think. As others have pointed out, the ability of English descriptions of the world to discover new things about the world is extremely limited when compared to that of mathematics. Completely different order of magnitude, and that difference does require an explanation. Whether that explanation is to be found in Platonism is another thing, of course.

    Paul,

    > This "unreasonable effectiveness" of mathematics is too often cited (even in pop culture) without good examples. <

    I just gave one above, and there are plenty of examples that mathematicians and philosophers of mathematics have provided. It’s a real issue that any philosopher of mathematics has to deal with, again regardless of whether the answer turns out to be Platonism or not.

    > We have the word 'duck' and lo and behold there are ducks. Chills. <

    As I mentioned above, not even close.

    mufi,

    > “...the only thing we can say about science is that it is empirically adequate.”

    Well, it is grounded in human experience. What's more real than that? <

    Yes, but that wasn’t the point. A theory can be empirically adequate and yet fundamentally incorrect, like Newtonian mechanics (it works very well for certain sets of problems, but gets the description of space-time entirely wrong). The debate btw realists and antirealists in philosophy of science is about whether scientific theories are all like Newton’s (antirealists), or whether there is a sense in which we can say that we are getting closer and closer to a true description of the world (realists).

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    1. Understood. Although I dislike the term "antirealism" (as suggested in my earlier statement re: the realism of human experience), I dislike dogmatism and arrogance much more, and I strongly detect those in the "realism" of the debate to which you refer.

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  18. DM,

    > It's metaphysics, not philosophy. This is not science. <

    I agree, except of course that metaphysics *is* (a branch of) philosophy.

    > The unreasonable effectiveness of mathematics is not the best argument for the MUH. The best argument for the MUH is that it is necessarily true that the universe must exist as a mathematical object if mathematical Platonism is true. <

    But we need some reason to think that Platonism is true, and the no-miracles argument is the best (only?) argument out there.

    > it becomes clear that the concept of "physical" as distinct from "mathematical" is in fact incoherent, and in some sense magic. <

    Funny, I had the exact opposite reaction. Whenever I think “4!” I get a very different feeling from whenever I think “pasta!”

    > If you think Godel is a problem for the MUH, please explain the argument. <

    Just citing others. And Max agrees that it is a problem, though apparently only if one accepts the idea of physical infinities.

    > To start with, you can imagine that the universe is a simulation running on an infinitely powerful computer. <

    As you know, that’s a no starter for me, considering my position in philosophy of mind (biological naturalism).

    > If mathematical Platonism is true, then we and our universe must exist in this way, whether or not the physical universe or any physical universe exists. <

    Not at all. Even if one accepts Platonism one needs the addition of monism — which is logically independent from it — to get a mathematical universe. I am sympathetic to Platonism, but I reject monism as incoherent (at my current stage of understanding of things).

    > Perhaps the universe IS infinitely complex. Perhaps there are an infinite number of physical laws, but most of them are pretty inconsequential or are too feeble to make their effects felt. <

    Perhaps, but from where I stand this sounds *extremely* ad hoc.

    > It may be, for example, that quarks and gluons etc are made up of more fundamental particles, which are made up of more fundamental particles, etc ad infinitum. <

    Which is pretty much contrary to currently accepted physical theories…

    > I would suggest that life could only exist in a universe that was reasonably regular. <

    Sure, but that’s an unrelated issue (though of course some people do see a connection between the mathematical universe and the anthropic principle problem — but the former isn’t the only way to get around the latter).

    > I think we need to bring the computational theory of mind into the argument. <

    Of course we do. And you know very well by now what I think of it…

    > “not to mention that pretty much every successful physical theory so far would violate CUH.”

    Please explain this, as I think you may be wrong. <

    See my response above to Max, and the two works referred to there.

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    1. @ Massimo

      > As you know, that’s a no starter for me, considering my position in philosophy of mind (biological naturalism). <

      Biological naturalism actually qualifies as a form of property dualism (Searle's objections to the contrary notwithstanding).

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    2. Hi Massimo,

      Firstly, D'oh!

      >I agree, except of course that metaphysics *is* (a branch of) philosophy.

      Yes, I meant to write that it was metaphysics, not physics. Sorry. Rushing for a plane. Of course metaphysics is philosophy.

      >But we need some reason to think that Platonism is true, and the no-miracles argument is the best (only?) argument out there.<

      It's not the best or the only argument in my view. I've mentioned before that I think any denial of Platonism is simply a misunderstanding of what we mean when we say "exists". Mathematical objects have to exist in some sense. I've written about my thoughts on this on my blog.

      I would see the no miracles argument as supporting the MUH specifically rather than Platonism in general. I don't see why the utility of some mathematical objects in describing the world should argue for the view that all mathematical objects are real.

      But since you're sympathetic to Platonism and yet not to the MUH, can we not just assume for the sake of argument that Platonism is true?

      >Funny, I had the exact opposite reaction. Whenever I think “4!” I get a very different feeling from whenever I think “pasta!”<

      For two reasons.

      Firstly, you're going with your instincts, and the MUH is after all profoundly weird.

      Secondly, I do agree that there is a fundamental difference between "pizza" and "four", but this difference comes down to your point of view. To you, a physical object is an object which is present within your universe. Four is not.

      What is incoherent is the idea that this pizza is objectively physical from the point of view of an observer outside this universe, or that this universe is objectively physical or real at all in a sense that other mathematical universes or objects are not.

      Pizza is physical to you, but "Four" is physical to "Five" (if Five were an observer). A better example might be that Han Solo is a physical entity from the point of view of Luke Skywalker, but you are not.

      >Just citing others.<

      And the others you cite don't really have any argument to back this up. I've read that original debate, and there is no argument there. Just some rhetoric about Godel being problematic for Platonism in general (and not the MUH in particular).

      >And Max agrees that it is a problem<

      And I wish he would explain why, because I really don't think it is. I'm on board with the MUH, but not always with Max's approach.

      >As you know, that’s a no starter for me, considering my position in philosophy of mind (biological naturalism).<

      Please suspend your doubt for now so I can explain my views on the MUH. I agree with you that I can't convince you of the MUH if I can't first convince you of the CTM. But since you have said before that you are not 100% sure the CTM is false, please entertain for now the possibility that it is true.
      In any case, what I'm trying to do is explain what Max, Pete and I mean when we say the universe is made of math. Imagining the universe to be a computer simulation is for me the easiest way to explain it. Even if you doubt the CTM, it ought to give you an intuition as to how it's not simply a category error to say that an electron *really* is a mathematical object.

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    3. >Even if one accepts Platonism one needs the addition of monism — which is logically independent from it — to get a mathematical universe.<

      You've missed my point. If Platonism is true, there is an abstract mathematical object which perfectly describes this universe, and within it exist entities perfectly isomorphic to you and I. That's without positing monism, because the universe we are in is not necessarily that abstract universe. We could still be in the "real" or physical universe.

      Monism is simply what logically follows if we apply Occam's razor to this. Why posit two identical universes, one physical, one mathematical, which are indistinguishable from the point of view of observers within, when only one universe is needed?
      Even without Occam's razor, if no observer exists who can tell the difference, then what does it really mean to say that one universe is physical anyway? Doesn't it make the concept of "physical" into magical nonsense? It certainly seems to for me.

      >Perhaps, but from where I stand this sounds *extremely* ad hoc.<

      My point is we have no idea how complex the ultimate physical laws are. If the universe were randomly selected from the population of all possible universes supporting life, we might indeed expect there to be wide discrepencies between the significance of different physical laws. It seems to me to be entirely reasonable to suppose that perhaps we have only grasped the broad strokes. Science has a history of uncovering more and more detail. Why are you so confident that this process will not continue?

      [On quarks and gluons being composed of more basic stuff]

      >Which is pretty much contrary to currently accepted physical theories…<

      Well, not really, because current physical theories just posit that quarks and gluons exist. They don't say anything about what they may be made of. In the same way, the discovery that atoms were composed of smaller parts was not contrary to the existing atomic theory but an elaboration of it.

      >Sure, but that’s an unrelated issue <

      I think the anthropic principle is absolutely crucial if you want to discuss why the universe is so regular and relatively simple-looking. If the apparent simplicity of physical law is a problem you have with the MUH, then I think you need to address the anthropic argument carefully. What would life look like in a universe that was infinitely complex, with infinitely many laws of nature each of approximately equal significance? I'm saying it would be impossible for such a chaotic universe to support life, and if I'm right then this seems to me to defeat the argument.

      >See my response above to Max, and the two works referred to there.<

      Sorry, I can find no answer to my question there. I think you may have been answering a different question.

      You have stated that every successful physical theory would violate the Computable Universe Hypothesis. I interpreted this to mean that all successful theories have been shown to be uncomputable -- that is they could not be simulated by an infinitely powerful Turing machine.

      If that is now what you meant, please explain what you did. If it is what you meant, could you give an example?

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  19. Massimo,

    You attribute the following claims to Tegmark:

    "To which Tegmark replied that perhaps only Gödel-complete mathematical structures have PHYSICAL existence"

    "But of course Tegmark was referring to the existence of PHYSICAL, not mathematical, infinities...The question of whether there really are infinities in PHYSICAL systems is open, so surely Max is entitled to his skepticism."

    How can Tegmark distinguish between what "physically" exists and what "mathematically" exists, if he thinks that the physical world is made up of mathematical objects? I imagine he has to say that some mathematical structures are "physically instantiated" while others are not. But it's not clear to me what it means for a mathematical structure to be "physically instantiated" if that doesn't involve postulating the existence of some "physical stuff" that can be described by the mathematical structure.

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    1. >But it's not clear to me what it means for a mathematical structure to be "physically instantiated" if that doesn't involve postulating the existence of some "physical stuff" that can be described by the mathematical structure.<

      I think I can explain.

      There is no such thing as physical stuff.

      For something to be perceived to be physical, there must be a conscious observer who can interact with it in some way.

      We are conscious observers in this mathematical universe who perceive objects in this universe to physically exist, while mathematical objects not in this universe seem to us to be abstract.

      So all other universes are abstract to us, but ours is physical. Meanwhile, from the point of view of an observer in another universe, we are abstract but stuff in her universe is physical.

      From a broader, more objective perspective not limited to our universe, we could say that something is physically instantiated if there exists any observer who would perceive it as physical.

      Since most mathematical objects do not directly interact with such observers, most mathematical objects are not physically instantiated.

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    2. Disagreeable,

      The existence of physical stuff simply cannot be denied. By the meanings of 'physical stuff' and 'exists' plus observation, it is certain that physical stuff exists.

      I think what you mean to say is that some of our beliefs about physical stuff are false, such as that it is mind-independent. Precisely which beliefs you think are false is what is of philosophical interest regarding your view about physical stuff.

      Regarding an earlier comments of yours, language is not only important in philosophical theorizing, but everything. Generally, the development of knowledge is the development of language. Math is language, despite the lingering Pythagorean mysticism. For all intents and purposes, our thoughts are only as good as our ability to put them into language, whether in poems or mathematical formulas.


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    3. What do you mean when you say that an object can be in one "mathematical universe" or another?

      When someone like David Lewis talks about multiple worlds, for example, he can say that two objects don't exist in the same world if there is no spacetime path between them But you can't individuate mathematical universes by spacetime relations, since mathematical objects don't exist in spacetime.

      Maybe you're thinking of mathematical universes simply as sets, and the objects existing in them as their members. So you might say that X and Y don't belong to the same universe if they don't belong to the same set. But for any object X and Y, there is the set {X, Y}. So by this criterion, there aren't any two objects X and Y that don't belong to the same universe.

      So you must mean something else when you talk about distinct mathematical universes, but I don't know what you mean.

      (Joel Hamkins writes on what he calls the "set-theoretic multiverse", but I think this has serious problems).

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    4. Hi Paul,

      >The existence of physical stuff simply cannot be denied. By the meanings of 'physical stuff' and 'exists' plus observation, it is certain that physical stuff exists. <

      It is certainly true that there are objects which we perceive to be physical and others which we perceive to be abstract. My contention is that the distinction, at least from an objective viewpoint, is an illusion.

      However if we take "physical" to have meaning only from a subjective stance then I would agree that it is a coherent concept.

      Here's an example of how I think our intuitions about physicality leads us astray.

      Let's say I make up a universe that runs according to very different rules than our own. Let's say it's something like Conway's Game of Life.

      Let us furthermore assume that this universe is of the sort that can allow not just gliders, oscillators etc, but also reactive, intelligent structures. If we take the CTM to be true, we might even suppose that these structures could be conscious.

      Now, let's ask the question. Does this made-up universe correspond to an actual physical universe somewhere else in the multiverse?

      Even forgetting the MUH for a moment, there are reasons to believe that every possible universe might exist. Firstly, why not? Secondly, it would allow us to explain fine tuning of not just the constants but the laws themselves.

      So it doesn't seem unreasonable to think that this toy universe physically exists. However, thinking deeply about this question reveals some problems.

      What does "physically exist" mean? I think it means "is present within the universe". I can think of no other coherent non-circular definition, but I would welcome any attempt you might offer.

      But this other universe does not exist within ours, so by definition it does not physically exist.

      But that's not very satisfying, because by the same token, our universe does not physically exist from the point of view of an observer in that universe. The concept of physical existence simply doesn't work when applied to universes. It's a category mistake. Physical existence is a property of objects within a universe as perceived by observers within than universe.

      So the idea that a universe does or does not physically exist is simply not a coherent proposition.

      >language is not only important in philosophical theorizing, but everything.<

      I agree that language is crucial in communicating ideas and as a tool in forming them. But if you write an idea in one language or another, as long as the idea is expressed faithfully, it's the same concept.

      "One" is the same as "uno" is the same as "un" is the same as "yi" is the same as 1. It doesn't make sense to me to ask what is the correct word or symbol because it's all just different ways of expressing the same concept.

      >Math is language<

      I disagree. I think math is the study of formal systems. Mathematical notation is language.

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    5. Hi C,

      >What do you mean when you say that an object can be in one "mathematical universe" or another? <

      You're pretty close to what I mean when you mention sets, but I see the point your making with the existence of supersets.

      I see the objects as being in two different universes if there is no causal path between them. By causal path, I mean you can trace no pattern of events forward or back in time such that there is any connection between them whatsoever.

      Ultimately, everything does get subsumed into one big set of all mathematical objects, but within this big set there will be smaller sets of objects which interact with each other but not with other objects. If you draw a line around all the objects that interact with each other but no others, you get distinct universes.

      An analogy from fiction is that Star Wars and Star Trek are distinct universes, even though they both belong to the same set of sci-fi franchises arising in popularity in the latter half of the 20th century and currently helmed by JJ Abrams. Entities within the Star Trek canon do not interact with entities from Star Wars and vice versa.

      It does get muddy. It could turn out that Star Wars is a fictional franchise that happens to exist in the Star Trek universe (in the way we see Sherlock Holmes does). Similarly, by simulating and talking about the Conway's Life universe, we are interacting with it, so arguably you could consider the two universes to be joined.

      And yet it is clear that we do not share the perspective of a person within a universe we simulate, even if we can interact with the universe and talk to the people inside it, so there remains a distinction. I'm not sure I can give a formal account of this distinction just now. For now, it just seems obvious to me.

      In the end, perhaps the drawing of imaginary lines around interacting objects and calling the enclosed sets "universes" is more a helpful taxonomy than an ontological fundamental.

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    6. Disagreeable,

      >I think math is the study of formal systems. Mathematical notation is language.<

      Well formal systems are a kind of linguistic system, so math in the activity sense, as opposed to the sense of being a language (an ambiguity shared with many disciplinary terms), is, on a higher level of generality, the study of certain linguistic systems. In short, 'math' means both a language and the development of that language.

      Regarding physicality, I see what you're saying generally but I don't have much to say within your framework because I reject mind-independent abstract entities altogether. One can set up all the Platonic puzzles one wants and I will still think it is more plausible that we're making some sort of mistake regarding language than that there are mind-independent entities that exist outside space-time that things in the world instantiate.

      Delete
    7. >Well formal systems are a kind of linguistic system,<

      Only if you stretch the meaning of linguistic system beyond where I would. For me, a language is a means of communicating and expressing ideas. I think that the objects under study in mathematics are not the means of communication or expression but the concepts themselves. For instance, 20% is another way of expressing 1/5, but in my view the objects being expressed are identical.

      Perhaps this difference between us is just the result of my Platonism and your resistance to it.

      > I reject mind-independent abstract entities altogether<

      Fair enough, but if you're not on board with Platonism you're certainly not going to be persuaded by the MUH.

      My own arguments in favour of Platonism are on my blog

      http://disagreeableme.blogspot.co.uk/search/label/mathematical%20platonism

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  20. Alastair,

    > No, I am not misusing the term. The "wave-particle duality" holds that nature has a nonphysical aspect (the "wave" aspect) as well as a physical aspect (the "particle" aspect). <

    Bullocks.

    > And God is the most parsimonious explanation for why there is anything all. <

    See above.

    > “Neither. It’s speculative science bordering on metaphysics.”
    That's about as clear as mud. <

    Only if you refuse to make an effort to understand what I write, which seems to be your MO.

    > Biological naturalism actually qualifies as a form of property dualism (Searle's objections to the contrary notwithstanding). <

    Bullocks, for reasons already explained ad nauseam on this blog.

    C,

    > How can Tegmark distinguish between what "physically" exists and what "mathematically" exists, if he thinks that the physical world is made up of mathematical objects? <

    Ah, that’s a good question! But I really don’t think he was denying mathematical infinities, how could he? Does he think that the set of natural numbers, say, is finite? I don’t think so. Perhaps the charitable interpretation is the one you suggest: he thinks only a sub-set of the mathematical landscape instantiates what we call the physical world, and that sub-set doesn’t include infinities. But, as I wrote, that seems arbitrary and a bit convenient, since it is precisely that subset that, apparently, doesn’t suffer from Godel-related issues. (And yes, I’m not sure how he would cash out the notion of “physically instantiated” without invoking stuff.)

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    1. @ Massimo

      > Bullocks, for reasons already explained ad nauseam on this blog. <

      Biological naturalism (John Searle's stepchild) qualifies as a form of "non-reductive physicalism." Searle holds that "mental states are NOT ontologically reducible to physical states." (source: Wikipedia: Property dualism: non-reductive physicalism: biological naturalism)

      The term "non-reductive physicalism" is clearly an oxymoron. If something doesn't reduce to the physical, then it isn't physical. Duh! (Searle's objections to the contrary or you screaming "bullshit" doesn't refute my bullet-proof argument.)

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    2. @ Alastair (December 13, 2013 10.57 AM)

      oh my, that's not even wrong

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    3. @ Dan

      > oh my, that's not even wrong <

      I don't see any counterargument here.

      By the way, you didn't use the phrase "not even wrong" in its proper context. It used to describe pseudoscience, not what you believe to be an incorrect philosophical argument.

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  21. DM,

    (having fun with your arguments on this, so take the following as a friendly give and take)

    > I've mentioned before that I think any denial of Platonism is simply a misunderstanding of what we mean when we say "exists". Mathematical objects have to exist in some sense. <

    Well, but that’s too weak for my taste. By the same reasoning unicorns exists as well.

    > I don't see why the utility of some mathematical objects in describing the world should argue for the view that all mathematical objects are real. <

    Because mathematics describes all possible coherent structures, some of which must be the ones describing the physical universe.

    > you're going with your instincts, and the MUH is after all profoundly weird. <

    Indeed, but weirdness per se isn’t an argument.

    > I do agree that there is a fundamental difference between "pizza" and "four", but this difference comes down to your point of view <

    No, it comes down to the fact that if I eat pizza I get fatter, while I can keep thinking about “4” all day along and retain my slim appearance…

    > Pizza is physical to you, but "Four" is physical to "Five" (if Five were an observer). <

    But “5” is not an observer, so…

    > A better example might be that Han Solo is a physical entity from the point of view of Luke Skywalker, but you are not. <

    See comment above about unicorns. I think you are going to get yourself into a lot of ontological trouble if you pursue this line of argument.

    > I've read that original debate, and there is no argument there. Just some rhetoric about Godel being problematic for Platonism in general <

    Well, Tegmark thinks there is enough meat to the argument that he felt compelled to respond.

    > Imagining the universe to be a computer simulation is for me the easiest way to explain it. Even if you doubt the CTM, it ought to give you an intuition as to how it's not simply a category error to say that an electron *really* is a mathematical object. <

    I understand what you are saying, but I’m still inclined to file the whole thing under category mistake, for the pizza vs “4” reason outlined above.

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    1. Hi Massimo,

      >(having fun with your arguments on this, so take the following as a friendly give and take)<

      Of course! I welcome your critical feedback, and I'm glad you're enjoying the conversation (almost?) as much as I am!

      >By the same reasoning unicorns exists as well.<

      Precisely. I do actually believe unicorns exist in some sense. They certainly exist as concepts. And that's what mathematical objects are - concepts. No stronger form of existence is required for Platonism.

      But I would also believe that unicorns physically exist, though not on this planet. As a concept unicorns can be pretty clearly defined. For example: they're like horses but they're white and they have horns growing out of their foreheads.

      If our own universe is infinite in extent, as it may be, then there are certainly creatures that match this description somewhere.

      Even if the universe is not infinite in extent, then such creatures must exist elsewhere in the multiverse if there are many universes.

      When you scoff at the idea that unicorns exist, you are taking it as evident that they do not exist as physical objects to be found on planet earth. And this is true. But nobody is claiming that abstract objects are physical terrestrial objects so I don't think your criticism works.

      >No, it comes down to the fact that if I eat pizza I get fatter, while I can keep thinking about “4” all day along and retain my slim appearance…<

      Will you get fatter if you just think about pizza? No, because you need to physically interact with it. You can't physically interact with the number four because it is not an object within your universe. That's the difference. Five can "eat" Four and become Nine because they are in the same "universe" of numbers. It can't interact with pizza because pizza is not a number. Four is real to Five but Pizza is not. Pizza is real to you but Four is not.

      (Obviously don't take this too seriously - it's only an illustrative analogy).

      >But “5” is not an observer, so…<

      ...so this is an analogy which doesn't work. Star Wars is a better example because it contains observers. There is no observer who regards 4 as a physical object, so in that sense there is a difference between 4 and pizza. So to have a less tortured, and misfitting analogy we should be instead be comparing physical pizza to virtual pizza in a simulation, or comparing George Lucas to Luke Skywalker.

      >I think you are going to get yourself into a lot of ontological trouble if you pursue this line of argument.<

      I'm willing to risk it. Bring it on!

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  22. DM,

    > If Platonism is true, there is an abstract mathematical object which perfectly describes this universe, and within it exist entities perfectly isomorphic to you and I. That's without positing monism, because the universe we are in is not necessarily that abstract universe. We could still be in the "real" or physical universe. <

    No, if you are a monist there is no real and physical universe, it’s all math. And the key word you used is “describe,” which plays a very different role from “is.”

    > Monism is simply what logically follows if we apply Occam's razor to this. Why posit two identical universes, one physical, one mathematical, which are indistinguishable from the point of view of observers within, when only one universe is needed? <

    Because there are both mathematical structures and stuff in the universe. Occam doesn’t get you out of that one, since it doesn’t say that the simplest explanation is true, only that you should not multiply *unnecessary* explanatory entities.

    > if no observer exists who can tell the difference, then what does it really mean to say that one universe is physical anyway? <

    I can tell the difference, as in the case of pizza vs. “4.”

    > My point is we have no idea how complex the ultimate physical laws are. <

    Shall we disregard all physics today, which seems to point to a relatively simple set of laws and forces?

    > Science has a history of uncovering more and more detail. Why are you so confident that this process will not continue? <

    Because the trend has been toward simplification (e.g., several forces shown to be aspects of the same thing), not the opposite. Remember, Tegmark is worried about this as well.

    > current physical theories just posit that quarks and gluons exist. They don't say anything about what they may be made of. <

    So you are invoking possible — currently entirely unforeseen — experimental discoveries to defend a theory that smells of magic to begin with? Not very parsimonious, is it?

    > I think the anthropic principle is absolutely crucial if you want to discuss why the universe is so regular and relatively simple-looking. <

    But there are other ways to deal with the anthropic problem, including a *physical* multiverse. Besides, I’m not sure it’s a good idea to explain a problem by invoking a highly fanciful view of the universe. Sort of like explaining a small miracles by invoking a bigger one…

    > You have stated that every successful physical theory would violate the Computable Universe Hypothesis. I interpreted this to mean that all successful theories have been shown to be uncomputable <

    I don’t know where you got that interpretation. But, again, I was simply reporting what Tegmark’s critics have pointed out, you need to go and check the original works.

    > For something to be perceived to be physical, there must be a conscious observer who can interact with it in some way. <

    And that consciousness is non-physical?

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    1. @ Massimo

      > Bullocks.<

      I cited a professor of physics (and an award-winning writer on the communication of science to the general public) to support my claim. If you don't understand that the wave aspect in the wave-particle duality refers to a probability wave (a nonphysical, mathematical abstraction, not a wave like a sound wave or a water wave that propagates through a physical medium), then you apparently don't have a layman's understanding of quantum mechanics.

      > See above. <

      I really don't seeing any difference between an explanation that invokes God to account for why there is a universe (or multiverse) and one that invokes an "infinitely intelligent mathematician" to derive a universe (or multiverse) from nonphysical, mathematical abstractions. Clearly, both explanations involve creation ex nihilo.

      > Only if you refuse to make an effort to understand what I write, which seems to be your MO. <

      If an individual is presenting a metaphysical theory as a scientific theory, then he/she is engaging in pseudoscience. The difference between a metaphysical theory and a scientific theory is that the latter makes a testable prediction (the former does not). What testable predictable does the MUH make? By your own admission, the MUH is "empirically next to impossible (or even downright impossible) ideas to test."

      "Pseudoscience is a claim, belief, or practice which is presented as scientific, but does not adhere to a valid scientific method, lacks supporting evidence or plausibility, canNOT be reliably TESTED, or otherwise lacks scientific status.[1]" (souce: Wikipedia: Pseudoscience)

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    2. >No, if you are a monist there is no real and physical universe, it’s all math. And the key word you used is “describe,” which plays a very different role from “is.”<

      You've missed the point again, I'm afraid.

      You deny monism. OK. I'll accept that monism is false for now for the sake of argument, if you accept Platonism and CTM. Now, where does that take us?

      My point is that it leaves us with both a physical and a mathematical universe. The two are identical, the mathematical universe perfectly describing the physical universe. The mathematical universe contains self-aware substructures (the analogues of ourselves). To them, their universe feels physical. We are left with no way to tell whether we are in the physical universe or the mathematical universe.

      So, per your own description of Occam's razor, we must not multiply unnecessary entities. Since the mathematical universe must exist, given Platonism, the physical universe is unnecessary. It could exist, but there's no reason to believe it does, and furthermore, since there is even in principle no way for someone to tell whether they are in a physical or mathematical universe, the concept of a physical universe seems incoherent to me given Platonism and CTM.

      >Shall we disregard all physics today, which seems to point to a relatively simple set of laws and forces?<

      I'm not sure your confidence is shared by all physicists (although Tegmark seems to agree with you). Personally I'm agnostic on the question. I really have no idea how complex the ultimate laws of physics are.

      >So you are invoking possible — currently entirely unforeseen — experimental discoveries to defend a theory that smells of magic to begin with? Not very parsimonious, is it?<

      Not at all. I'm just expressing entirely reasonable uncertainty about how close we have come to the ultimate laws of reality. My point is not specifically that quarks and gluons are made of more basic particles. That's just an example.

      >But there are other ways to deal with the anthropic problem, including a *physical* multiverse.<

      I think you're losing track of why I brought the anthropic principle up. I didn't bring it up to defend the MUH. I agree that the anthropic principle only argues for a multiverse of some kind. Rather, I am invoking the anthropic principle specifically to counter the argument that the universe is too simple to have been selected randomly from the set of all possible mathematical structures. Again, my argument is that only simple structures are likely to be conducive to life, so an apparently simple universe is consistent with the MUH.

      >I don’t know where you got that interpretation.<

      I remain in the dark as to what you meant then. What do you mean when you say every successful physical theory would violate the CUH? I'm wondering now if you mean they include infinities? If so, I still don't think that's a problem and you don't seem to be confident enough to discuss it so I guess we can drop it.

      >And that consciousness is non-physical?<

      Yes. I believe consciousness is not a physical property. It is a property of an abstract structure. This is consistent with the Virtual Minds response to the Chinese Room.

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  23. A computer can output a "4" via a laser printer as a glyph of toner particles deposited on a piece of paper.

    A computer can output a "pizza" via a 3D printer as layers of food particles deposited on a plate:
    popsci.com/article/technology/watch-3-d-printer-make-pizza

    What's the difference?

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    1. A printed symbol "4" is not the number four. If it were, then Massimo would regard 4 as a physical object much like a pizza.

      A printed pizza, on the other hand, is a pizza.

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    2. There is actually no "the number four" just like there is no "the pizza". You can have a bunch of fours just like you can have a punch of pizzas.

      And to the computer: It prints a 4 or it prints a pizza.

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    3. There's really no point in discussing the MUH if you don't accept Platonism, at least for the sake of argument.

      Obviously, I do think there is a "the number four", so there's not much point asserting the contrary without argument. This Platonic four is what Massimo was referring to, so your comparison to a physical object representing the shape of the symbol 4 doesn't really address the point he was making.

      If you want to discuss Platonism specifically, you are always welcome to comment on my blog.

      In particular here: http://disagreeableme.blogspot.co.uk/2013/10/mathematical-platonism-is-true-because.html

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  24. The "how can real things be *made of* math?" problem seems rather trivial to me. The apparent solidity of things would just be the rules of interaction of various aspects of the structure. You can't push your hand through a table any more 5 can move through 6 to stand next to 7, *because* of that aspect of the set of real numbers called "order."
    Physical solidity would just be a more complex version of the same kind of limitation on interactions, in that the structure imposes rules on the various ways its simplest facets can relate to one another, that, iterated out enough times, can add up to instantiate things like people and tables that seem solid to one another.

    Put another way, when I type a sentence like this one, and go back to insert a bunch of ************ in the middle, that "in the middle" gets pushed to the right as if by a solid object, but of course it's all just mathematics inside a computer, and all we see is the "phenomenology" of the monitor where words seem like solid entities you can highlight and move around.

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    1. I think the problem is that most people when initially encountering the idea that the universe is math literally have no idea what we are talking about. It seems like a nonsensical platitude. It's not just a question of how stuff can appear solid, or how things can affect other things, it's a question of how physical objects can be made of abstract objects when the two appear to be completely different categories. It just seems like nonsense until you've had time to think about it deeply.

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    2. The expression "it's all just mathematics inside a computer" is interesting.

      What is inside a computer (after code is loaded from a HDD or the internet) is electrons flowing through the CPU and RAM.

      So the "mathematics inside" is physical stuff.

      I think Tegmark's idea is basically right. It's jives with what I've always thought.

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    3. That's true, although you could make the same program out of any materials, say, legos or cans on strings, as long as the abstract relationships among the parts them were the same, so it still has nothing to do with the causal powers of the physical stuff itself, except in the way they have to be engineered to influence each other to embody those same abstract relationships.

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    4. Could you run (rather than emulate) a quantum program (with superposition and entanglement of qubits) on a conventional computer? There is a difference.

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    5. Emulation is the whole point. You couldn't run a program for an abacus on a conventional computer, either (no wires *or* beads).

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    6. The difference is time complexity. Finding the prime factors of a number N takes a super-polynomial (perhaps exponential) number of time steps (as a function of the number of bits representing N) whether it's done on an abacus or conventional computer, but a polynomial number of time steps on a quantum computer (Shor's algorithm).

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    7. Hi Philip,

      There's a difference in time and feasibility, sure, but no difference in principle as to what can be achieved.

      But in the context of the MUH it doesn't matter much because the whole idea is that there need be no physical computer and that the mathematical object (or program in your language) stands alone.

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  25. I've organised my thoughts on the MUH in an essay on my blog.

    I intend to return to the topic soon to discuss criticisms of the MUH. If anyone would care to comment, specifically if there are arguments against the MUH given platonism, computational theory of mind and naturalism, then please let me know, either by commenting here or on my blog.

    http://disagreeableme.blogspot.co.uk/2013/12/the-universe-is-made-of-mathematics.html

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  26. It's my first comment here - I'm not a professional philosopher, so I hesitate to take part in the discussions on this blog.

    However ... I don't understand the argument that mathematics is unreasonably effective. Is it? As a tool to do biology or philosophy? How many equations did Darwin use? Mathematics is very effective in physics, I agree - but even in physics this effectiveness is misunderstood, I think. In itself mathematics is perfectly useless. in physics. One can construct a zillion different mathematical models to describe something. Are they all effective? No, a limited number are and it's not mathematics that decide which models are useful, but experiment. The standard model in particle physics is described with a nice formula U(1) x SU(2) x SU(3) in which the U's and SU's refer to mathematical symmetry groups. Why those groups and not SU(8)? Mathematics can't tell you. From the point of view of mathematics SU(8) is just as good a group as SU(3), although it has different properties. From the point of view of mathematics, a non Archimedean ordered field is just as good as an Archimedean ordered field. But for some strange reason the real numbers - an Archimedean ordered field - are very useful do to classical mechanics. There's no strictly mathematical reason why this should be so. Or take supersymmetry. A mathematically attractive idea, that however fails to be seen at the LHC. Mathematically that's no problem. You just have to tinker a bit with the parameters and the models et voilà! supersymmetry is invisible at the LHC. Another example of the effectiveness of mathematics?

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    1. This makes sense.

      Also, physicists write programs based on theories and run them on computers. The programs can be run to see how they match to experimental data. One might as well talk about the surprising effectiveness of programs. (But there are a lot of programs that are not "effective".)

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    2. Patrick -

      Don't hesitate. It's always good to get your thoughts out there in the open. I'll try to answer some of your questions and provide some links for you if you want to delve a little bit deeper.

      First, as to your comments on the usefulness of mathematics in biology and philosophy, you're statement, and I mean this with the utmost respect, is way off the mark. Mathematical biology has been around for years, and it has drastically expanded as a discipline over the last few decades (this link is a place to start):

      http://en.wikipedia.org/wiki/Mathematical_and_theoretical_biology#Model_example:_the_cell_cycle

      Mathematics is being used more and more to shed light on the workings of our genetic code, the cells that form the foundation of all organisms, and even the behavior of animals and humans. Just today I found this article on Phys.org, which is an excellent site with daily articles on math, science, and health. The article talks about different behavioral characteristics of humans being amenable to mathematical description (these discoveries seem to come out every other week nowadays):

      http://phys.org/news/2013-12-simple-mathematical-formula-human-struggles.html

      You might need to learn a bit about decision theory and game theory as well, which are highly rigorous areas of mathematics that are also able to describe how humans behave when different payouts and expectations are present. Obviously, we can't predict what any individual human or animal will do at the moment (if we ever can), but mathematics is clearly making strong inroads in explaining biological phenomena and higher level behavior, and we can expect this to continue.

      As far as mathematics and philosophy, I hope you understand that logic is probably THE foundation of philosophical inquiry. Logic and mathematics are related in deep ways (math can be built up from second-order logic), and modern mathematical logic is considered a crowning achievement of 19th and 20th century philosophers.

      When you examine all the different GUT/Theories of Everything that are out there, it becomes apparent that many are related as well. With the different group structures being subgroups of groups at an even higher level, so these differences aren't as severe as you think. The fact that there are various models is, as I explained in a comment earlier in the thread, where something like Tegmark's MUH might come into play, where various mathematical structures are present in different universes. It should be said that, regardless of whether or not some mathematical structures don't seem to be instantiated in our universe, it is a horrible argument to extrapolate from this that mathematics is ineffective. One can think of mathematics as the science of structures, and if those structures are instantiated in the world then it doesn't take much of a jump to say that those structures have an ontological existence in and of themselves.

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    3. Regarding biology, we may not have the quantum program for a wormwhole, but there is a project whose goal is to find the program for a worm: OpenWorm. Very interesting.

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    4. Phil -

      You're exactly right. Programs are effective. They are also mathematically defined, and they happen to be real. The theory of computation uses mathematics and logic at its base.

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    5. The equation for everything, unification, GUT., TOE., justice, equality, the Universe, Oneness, God, self, democracy, independence, freedom, the equation that Einstein died searching for, the equation Lincoln, Gandhi, and King spent their lives for, the equation the Civil War was fought, the equation we protest for, we fight and die for, the equation Socrates could not find, the equation for Descartes' "I", the solution, the most simple, elegant, beautiful equation for absolute, for truth, is =.

      =

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    6. Pete,

      Thanks for your reply. I studied physics and mathematics, so I don't think mathematics in ineffective - I only think its effectiveness is overstated here. Interesting mathematical things are happening in biology, but large areas of biology are distinctly unmathematical. Logic is a good tool in philosophy, but only if you talk about things between which the relevant logical relations are true. The literature on the Slingshot Argument is a nice example; it shows how complicated things get when you start to analyze an essentially logical argument.

      And even in physics ... You seem to know a lot about it, so I would like to ask you: how "mathematical" was the development of renormalisation? It was full of mathematical howlers of the worst kind - howlers that turned out to be excellent physical intuition. In the end they found something that worked (after discarding all the approaches that didn't work) but if they would have done "mathematics" straight from the beginning, there's a fair chance we still would be staring a those mysterious infinities that crop up when we're doing calculations with perturbation theory. I think many people underestimate the pragmatic – mathematicians would say: cynical – way physicists use mathematics. Mathematics is great when it gives you results; unmathematical intuition and reasoning is better when it gives you better results. Physics is too difficult to let mathematics dictate what you can or can't do.

      I don't get your argument that certain groups are subgroups of other groups. Of course they are. All finite groups are subgroups of a permutation group, etc. But a QFT with a SU(6) gauge symmetry doesn't give you the same physics as a QFT with a SU(3) gauge symmetry.

      I have to repeat here what I wrote about supersymmetry. The mathematics of these models is so flexible that it can explain the fact that SS is seen at the LHC; but now that it isn't seen it can explain that too. It's a form of effectiveness, but it's not really what we usually mean when we use that expression.

      I don't want to sound strident, but I sometimes feel that someone who claims that mathematics is unreasonably effective has never solved a wave equation. When you work with wave equations, you sometimes routinely discard perfectly sound mathematical solutions because they are "unphysical". Just like with renormalization, it seems that mathematics is less effective than physical intuition. Or only effective if you couple it to a solid dose of physical intuition.

      You write “…it is a horrible argument to extrapolate from this that mathematics is ineffective”. Perhaps. But, perhaps, we can turn this argument on its head. Do we really have enough arguments to claim that mathematics is effective at describing all of nature? A couple of years ago Steven Weinberg wrote (in The New York Review of Books, if I remember correctly) that elementary particle physicists only study the simplest systems (more complex systems being too difficult to solve). Mathematics – in combination with a solid amount of physical intuition – is good at describing, predicting etc. the behavior of these systems. But is it fair to extrapolate this success to an “unreasonable effectiveness” of mathematics in other domains?

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    7. Patrick,

      Mathematics is great. (It must be: I have a Ph.D. in Applied Mathematics.)

      But I think we are in a transition from "traditional" math to code. There will still be research papers published on arXiv with mathematical "theory" (expressed by TeX math commands and translated into MathML or PDF), but there is a trend to the computational, with code repositories (in Python, C++, Haskell, ...) published on GitHub.

      Models in science in the future will be made more of code than traditional math.

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    8. Patrick -

      As far as thinking certain areas of mathematics are more "unmathematical" than other areas and might be overstated, the brute fact is that mathematical biology is moving in leaps and bounds as we speak. Whether or not one might think this is appropriate or that mathematics is overstepping its boundaries is beside the point. As long as mathematical analysis helps biologists and medical researchers discover new biological principles and further their understanding of complex organisms, it will only continue to expand.

      Again, speaking of group representations and their use in particle physics, you made it sound in the first comment that things are drastically different from each separate theory. The point is, regardless of new physics that would result from different groups (no one is denying that), if many of these different theories have groups that are related to the groups of other theories, then they're not as vastly different as one might think. All of the different Lie groups you mentioned will give rise to different physics, but isn't this, as I said earlier, what we're talking about here, the idea of every mathematical structure being instantiated in different universes (MUH)?

      Maybe I'm missing something, but in the case of discarding unphysical solutions to wave equations I don't think its actually an argument against Platonism. In fact, I think it can be used as an argument for Platonism. Think of PAM Dirac, getting solutions to the Klein-Gordon equation, a wave equation, that seemed unphysical. Only they weren't. They were actually discovered to be antiparticles. Dirac literally followed the mathematics and it led him to a physically real entity (this is probably the third or fourth instance of this I have now described in the comments). Any yea I can remember discarding "unphysical" solutions in mathematics and physics classes myself, but they were for reasons that are easily explained. Maybe we only wanted to look at solutions at t = 0 or greater (negative t would represent the past). Or the dynamics we are talking about aren't even amenable to negative number descriptions, but we model it on the cartesian plane and just focus on the equation in the positive quadrant because negative values might not even make sense.

      As far as what Gell-Mann says, he's right, but that doesn't mean complex systems aren't also going to be described more accurately by mathematics as time goes on. In fact, there was a very interesting article in Wired Science a few days ago about the same patterns emerging in very complex systems, so maybe this progress will happen sooner rather than later:

      http://www.wired.com/wiredscience/2013/02/math-and-nature-universality/

      Phil-

      Again, that code you are referring to is at its foundation a formal system based on mathematical logic.

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    9. Whereas classical computation may be charaterised as “theory (mathematics) first”, unconventional computation might be thought of as “hardware (physics) first”.

      Programming Unconventional Computers: Dynamics, Development, Self-Reference

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  27. Curiosity getting the better of me, I watched a talk on Youtube in which Tegmark presents his argument. It appears to be something like the following:

    Basic physical objects (particles, say) are fully described by mathematical properties.
    Objects of pure mathematical structures are fully described by mathematical properties.
    Thus, particles do not differ from objects of pure mathematical structures.
    Thus, the universe is a pure mathematical structure.

    The problem with this argument, it seems to me, relates to the distinction between analytic and synthetic propositions. Analytic propositions are true or false in virtue of word meanings or definitions while synthetic propositions are true or false in virtue of something other than word meanings or definitions, such as how the world is.

    While the mathematical values of a pure mathematical object are analytic in that they follow from initial definitions, those of particles are synthetic in that they are determined by how the world is.

    'Having its mathematical values in virtue of how the world is' is a property difference between particles and objects of pure math structures. The former has it, the latter does not.

    If particles were like objects of pure mathematical structures, we wouldn't need to do experiments. But we do need to because we need to let the world get in. The world and its mathematical values are not the same thing.

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  28. Hi Paul,

    >'Having its mathematical values in virtue of how the world is' is a property difference between particles and objects of pure math structures. The former has it, the latter does not.<

    Interesting argument. Here's how I would rebut it.

    Say we have two parabolas, A defined by the equation 2x^2 and B defined by a drawing on graph paper. You want to find out what the equation of the graphed parabola is, and by a little trial and error you determine that 2x^2 is as good an approximation as you can find, and it's possibly exactly right.

    But the parabola A was defined analytically while parabola B was defined synthetically by virtue of how the graph paper is. Does this mean that parabola A is fundamentally different from parabola B?

    I don't think so: they're both mathematical structures with similar properties, and probably precisely the same mathematical object. The difference in analytic and synthetic properties only accounts for how we come to know of the parabolas and does not reflect intrinsic properties in the parabolas themselves.

    Similarly, the mathematical electron and the physical electron may be defined differently but that doesn't mean they cannot be the same object.

    >If particles were like objects of pure mathematical structures, we wouldn't need to do experiments.<

    We still need to do experiments because otherwise we can't know what part of the mathematical multiverse we live in. Even putting this consideration aside, deriving everything from first principles is unfeasibly hard.

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    1. Another great way to describe a parabola is to throw something to a friend. And if they catch it you should both get an A in analytic geometry.

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    2. There is a theory (William Calvin, Univ. Of Washington) that says the evolution of early humans ability to throw accurately (compute parabolas) for hunting gave the brain the recursive computational structure needed for language.

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    3. Disagreeable,

      >The difference in analytic and synthetic properties only accounts for how we come to know of the parabolas and does not reflect intrinsic properties in the parabolas themselves.<

      Actually, the two parabolas differ a lot intrinsically. Parabola A has the intrinsic property of being determined by a function while parabola B does not, for instance. You seem to be assuming that the distinction between intrinsic and extrinsic properties corresponds to the distinction between mathematical and non-mathematical properties. It does not; there are extrinsic quantitative properties (such as weight) and intrinsic non-quantitative properties, such as being the output of a function.

      >We still need to do experiments because otherwise we can't know what part of the mathematical multiverse we live in.<

      If this could also be stated as that we need to do experiments to know which possible physical world we live in, a question is: what determines which world we live in? Might it be the world?

      However inelegant it may be mathematically, the world exists as something other than a mathematical abstraction ;)

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    4. Hi Paul,

      >Actually, the two parabolas differ a lot intrinsically. Parabola A has the intrinsic property of being determined by a function while parabola B does not, for instance. <

      I don't see how you make out that these parabolas have different intrinsic properties. Let's say I printed out the graph of parabola B by feeding the equation 2x^2 into a computer program.

      Now, to you parabola B is defined by the graph but to me it's defined by the equation.

      But it's the same parabola! Furthermore parabola A and parabola B are the same parabola.

      Thus the "property of being defined by X" is not intrinsic, it's in the eye of the beholder.

      >You seem to be assuming that the distinction between intrinsic and extrinsic properties corresponds to the distinction between mathematical and non-mathematical properties<

      I really don't think I am.

      >If this could also be stated as that we need to do experiments to know which possible physical world we live in, a question is: what determines which world we live in? Might it be the world?<

      Sure, but this is compatible with my view that the world is a mathematical object (at least if we interpret "physical" to mean worlds perceived as physical by their inhabitants), so it's not an argument against the MUH.

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    5. Philip, that is an interesting theory. I'll try to check it out. But animals not only make the same calculation, but they usually do it better. Like those spitting fish that can factor the refraction of light into their aim when spitting at insects above the water's surface.

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  29. It was interesting to hear the physicist Sean Carroll on this podcast say that he thinks (to probably mangle the paraphrase) at root the universe is a single quantum wave function. He did not consider this position to be mathematical Platonism. I don't understand either part of his assertion, but I think it's worth noting.

    In my mind there is a whole class of philosophical questions that would all be solved if any one of them is solved. I put them under the heading of "what is the ontological status of thoughts?" Clearly answering this question would solve:

    "why is there something instead of nothing?"
    My assertion here is that "nothing" is a counterfactual with dubious ontological status. This is a pseudo problem that is really asking the question: "What is the status of this unreal thing we call nothing?"

    "What is the ontological value of the various abstracta, logical, mathematical and philosophical?"
    I am fascinated that, in order to assert that we live in a mathematical universe, Tegmark has to jettison infinity as a real thing!!! So he still has the epistemological problem of explaining how we can have ideas that are impossible. In this case a well trod mathematical concept.

    "What is the nature of consciousness?"
    My hunch about this one is that a form of mundane pansychism will be the answer. And probably to all the other questions too. Abstraction and cognition and subjectivity are possibilities at all levels of reality, but in the same boring way that mass, center of gravity and phase shifts are possible at all levels. There is waterness in a single quark, but not in a mystical way, but in the sense of "if you get a buttload of these quarks together under the correct temperatures and pressure (thanks to a whole other buttload of quarks) you will get a nifty thing that the quarks that make up our mind cooperate in a meta statistical way to call "water'." All the elements are there in a single quark in the same way that a pile of sand is possible in a single grain.

    The same is true of mathematic relationships. We don't invent them, we discover them, yes. However, what we are discovering is not a fundamental truth about the universe. What we are discovering is a fundamental truth about how our minds map the universe. Our map of the universe is a discovery in a sense, but it is also subjective and idiosyncratic. An ant colony's map of the universe would probably be unintelligible to us but it might work great for the ants. But it would probably not be perfect even for the ants.

    If I remember correctly Steven Hawking has come up with a model dependent description of reality. The model is made out of the same world it describes. The model is by definition, a necessary abstraction. We will always be limited by the model status of our worldview, but that is the only kind of worldview that is possible. But it is incredibly important to note that different models are possible. It is perspectival!

    I think Tagemak is, in fact, making a categorical mistake. He is elevating the very constraints of our mapping to that of the universe fact. To be a little straw-man about it, it's like saying at root the world is made of paper, because our map is made of paper. Mathematics is a human invention to describe the relationship of the human mind to the universe. It is very powerful, but there is no good reason to assume that, just because we are limited to this type of model, then the universe is born of it.

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    1. >at root the universe is a single quantum wave function. He did not consider this position to be mathematical Platonism. I don't understand either part of his assertion, but I think it's worth noting.<

      The first part you can take to be similar to what Tegmark is saying, only he is only talking about this universe. It's not Platonism because he doesn't make any claims that all mathematical objects exsit.

      >I am fascinated that, in order to assert that we live in a mathematical universe, Tegmark has to jettison infinity as a real thing!!!<

      No he doesn't. He just prefers to answer arguments from Godel by saying "Ok, maybe infinity isn't a real thing" and propose the Computational Universe Hypothesis rather than take the approach that Godel is no problem. As far as I can see, he's agnostic on infinity, not denying it.

      >However, what we are discovering is not a fundamental truth about the universe. What we are discovering is a fundamental truth about how our minds map the universe.<

      It's neither. What we're discovering is fundamental truths about formal systems, that may or may not map onto the universe.

      >Our map of the universe is a discovery in a sense, but it is also subjective and idiosyncratic. An ant colony's map of the universe would probably be unintelligible to us but it might work great for the ants. But it would probably not be perfect even for the ants. <

      Ok, but mathematics is mathematics. There may be many branches of mathematics, but only one mathematics itself. What follows from one system of axioms is true no matter who proposes those axioms or analyses them. Aliens will have different notation and may have discovered different theorems, but if they have mathematics at all it would work just as well for us. Mathematical relativism is just not tenable.

      >different models are possible.<

      Sure. Especially if those models are only approximations. Different approximations may be useful for different tasks, e.g. Newtonian mechanics for catching a baseball, relativity for GPS.

      However Tegmark's claim is that ultimately the universe is a mathematical construct. In that case, there is only one absolutely correct model (although perhaps it could be expressed in a number of different isomorphic ways).

      While you may be right that there may be more than one model for practical purposes, that is not an argument that these are not simply approximations to one ultimate mathematical object.

      >I think Tagemak is, in fact, making a categorical mistake. He is elevating the very constraints of our mapping to that of the universe fact. To be a little straw-man about it, it's like saying at root the world is made of paper, because our map is made of paper.<

      I really think there's more to the argument than that. It's not just a category mistake, though I can see why you might think it is. There's a solid argument to be made for the MUH given Platonism, naturalism and the computational theory of mind. To say we're just mistaking the map for the world is to simply ignore this argument.

      My own argument is spelled out on my blog here:
      http://disagreeableme.blogspot.co.uk/2013/12/the-universe-is-made-of-mathematics.html

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    2. OneDayMore -

      Unfortunately I don't think you're argument (especially as it relates to ants and their mapping of the universe) has any bite. Mathematics is not a subjective discipline. Once something is proven within maths axiomatic framework, that truth is revealed to mathematicians and society (though a Platonist, such as myself, would indeed also say that the truth has always existed) and does not depend on ones beliefs or cultural values in any way.

      Again, as mentioned in earlier comments, why would one not consider that our minds ability to discover the mathematical relationships inherent in the universe is a product of evolutionary forces allowing us to recognize that those mathematical structures are inherent in reality?

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    3. Thanks Pete and DM, I will read more and get back to you. If you have the time, check out "Scientific Perspectivism" by Giere. I learned about him on this site. Massimo has been more interested in Ross&Ladyman and now Tegmark, but I am more partial to Giere. He's not the most lucid of writers, but I think he's getting at something important.

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  30. The best point in its favor is the so-called “no miracles” argument, the idea that mathematics is too unreasonably effective (at predicting things about the world) for it to be just a human invention

    Dr Pigliucci, does it really make sense to make "plausibility" arguments about what reality is "probably" composed of? How would we know? To use words like "plausible" or "likely" in this way is to invoke epistemic probability. But epistemic probabilities, by definition, can change upon new information (such as new measurements or theoretical activity by scientists). In other words if you say A is more likely than B,C,D, ... in this epistemic sense it doesn't actually mean anything about the underlying nature of Reality.


    But perhaps they are making precisely that mistake in a metaphysical sense?
    Or perhaps metaphysics is the art of making linguistic distinctions between things that are not really distinct.... (I would not be the first to make that critique)


    The problem is in what sense, if any, can a mathematical structure, so defined, actually be the fundamental constituent of the physical world, i.e. being the substance of which chairs, electrons, and so on, are made.

    Suppose there were no such thing as electrons but rather solutions of PDE's that look just like electrons. They would behave exactly the same—you could sit on a force field for example. In fact if the maths for physics were "done" then the maths would be extensionally equivalent to the physical stuff.

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    Replies
    1. I guess I could better summarise my comment this way: aren't we really talking about whether things that are extensionally the same can be different?

      How could we ever tell the difference between a universe composed of matter and a universe composed of maths?

      Delete
  31. Alastair,

    > I cited a professor of physics (and an award-winning writer on the communication of science to the general public) to support my claim. <

    Yes, Paul Davies is well known for his ideo-theological bias. And I find it interesting (and a bit insulting) that you choose to trust academic authority when is convenient for you, and yet insist in citing Wikipedia to a professional philosopher (yours truly) about philosophical matters.

    > I really don't seeing any difference between an explanation that invokes God to account for why there is a universe (or multiverse) and one that invokes an "infinitely intelligent mathematician” <

    But that your own, absolutely bizarre, construction of that sentence. Tegmark isn’t *invoking* an infinitely intelligent mathematician in *support* of his hypothesis, he is only making a (somewhat hyperbolic) claim of epistemic humility (as in “we may never be able to fully understand this because we are not infinitely intelligent mathematicians”).

    > If an individual is presenting a metaphysical theory as a scientific theory, then he/she is engaging in pseudoscience. <

    According to whom? No, he is engaging in metaphysically informed scientific speculation. Pseudoscience is an altogether different beast, which you should have learned to recognize by now. (Hint: I just edited a whole book about it.)

    > The term "non-reductive physicalism" is clearly an oxymoron. <

    Only if you are a reductionist.

    > If something doesn't reduce to the physical, then it isn't physical. Duh! <

    You clearly have no idea of what strong emergence means. Check out my series of posts about it.

    Studio,

    > The "how can real things be *made of* math?" problem seems rather trivial to me. <

    As a matter of general principle, I’m weary of anyone claiming that a problem that professional scientists and philosophers find intriguing is, in reality, trivial.

    > The apparent solidity of things would just be the rules of interaction of various aspects of the structure. <

    This may be trivial, but it makes no sense to me.

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    Replies
    1. @ Massimo

      > Yes, Paul Davies is well known for his ideo-theological bias. And I find it interesting (and a bit insulting) that you choose to trust academic authority when is convenient for you, and yet insist in citing Wikipedia to a professional philosopher (yours truly) about philosophical matters. <

      Several points:

      1) The "wave-particle duality" is a well-established component of quantum mechanics. Attempting to divert attention away from this fact by attacking Davies' so-called "ideo-theological" bias doesn't change it.

      2) Attacking Wikipedia is a losing strategy. As you will recall, I cited Wikipedia to demonstrate to you that you were confusing the "naturalistic fallacy" with the fallacious "appeal to nature." (Wikipedia was right on this matter, you were wrong - your status a professional philosopher notwithstanding.)

      3) Your hypocrisy is showing. You yourself, in this very blog post, cited Wikipedia as a link to "mathematical structure."

      > But that your own, absolutely bizarre, construction of that sentence. Tegmark isn’t *invoking* an infinitely intelligent mathematician in *support* of his hypothesis, he is only making a (somewhat hyperbolic) claim of epistemic humility (as in “we may never be able to fully understand this because we are not infinitely intelligent mathematicians”). <

      Okay. I stand corrected on this particular point and will retract my previous comment. However, I still stand by my claim that an "infinite intelligent mathematician" is simply a euphemism for God. Also, I still maintain that I have every intellectual right to invoke God as the most parsimonious explanation why there is a universe at all, given that the topic of this blog post is clearly a metaphysical one. (Tegmark is proposing that mathematical abstractions are the only "things" that exist. Moreover, he is apparently claiming that they are causally efficacious.)

      > According to whom? <

      According to Michael Shermer.

      "'"claims presented so that they appear [to be] scientific even though they lack supporting evidence and plausibility"(p. 33). In contrast, science is "a set of methods designed to describe and interpret observed and inferred phenomena, past or present, and aimed at building a TESTABLE body of knowledge open to rejection or confirmation"(p. 17)' Shermer 1997, (this was the definition adopted by the National Science Foundation).' (source: Wikipedia: Pseudoscience: definitional footnote)

      > No, he is engaging in metaphysically informed scientific speculation. <

      I see. He's not engaging in metaphysics, but in "metaphysically informed scientific speculation."

      > Pseudoscience is an altogether different beast, which you should have learned to recognize by now. (Hint: I just edited a whole book about it.) <

      I understand that you wrote a book on the subject. But it appears that you are struggling with the demarcation problem, because you are clearly having problems in distinguishing between physics and metaphysics.

      > You clearly have no idea of what strong emergence means. <

      Au contraire. "Strong emergence" (which is basically a euphemism for magic) is employed only in the philosophy of mind. (Science employs only "weak emergence," not strong.) The bottom line is that you have completely failed to explain how the mental qualifies as physical if it does not ontologically reduce to the physical. (Arguing that the mental is a strong emergent property simply makes my point that biological naturalism actually qualifies as a form of property dualism.)

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    2. Hi Alastair,

      Give Massimo a break.

      >The "wave-particle duality" is a well-established component of quantum mechanics<

      Everybody knows this and nobody's denying it. It is not necessary for you to keep asserting it with citations and references. You're preaching to the choir.

      What people have a problem with is the interpretation you infer from wave particle duality, equating it with other kinds of duality.

      >As you will recall, I cited Wikipedia to demonstrate to you that you were confusing the "naturalistic fallacy" with the fallacious "appeal to nature."<

      I think you were right on this, but Massimo did have a plausible answer. He made the point that increasingly the naturalistic fallacy is being used as a synonym to the appeal to nature in philosophical circles. This was not documented on Wikipedia, true, but it is presumptuous to be crowing to a working philosopher that Wikipedia has proven him wrong when he has a plausible defense.

      >According to Michael Shermer. <

      I agree that it is difficult to distinguish Tegmark from a pseudoscientist, but the distinction must be made because he is not making any claims that are inconsistent with the evidence or with science, nor is he falsifying results or pretending to have evidence he does not.

      Unlike homeopaths and cold fusion or perpetual motion enthusiasts, he is advocating a position supported by rational argument. If the technical definition for pseudoscience fails to distinguish between him and these others, then we need to refine the definition because they are not the same thing.

      But I do think the definition from Shermer might be enough to save Tegmark from the label of pseudoscience.

      >even though they lack supporting evidence and plausibility<

      Tegmark's view is not implausible, since it is consistent with the evidence, parsimonious (certainly more so than theism) and supported by logical argument.

      It's also (albeit weakly) supported by the evidence that the universe exists, is fine tuned and appears to be governed by mathematical laws of physics, which are precisely the facts it purports to explain.

      I do however agree with you that it is not science. But it is not psuedoscience, it's metaphysics.

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    3. @ Disagreeable Me

      > Hi Alastair, <

      I wish you would dispense with the 'pleasantries.' It comes across as being disingenuous.

      > What people have a problem with is the interpretation you infer from wave particle duality, equating it with other kinds of duality. <

      It is not my interpretation. It's the standard interpretation (a.k.a. the Copenhagen interpretation). The wave aspect in the "wave-particle duality" is the nonphysical aspect. It's not like a water wave or a sound wave which propagates itself through a physical medium. It's a probability wave (represented mathematically by the wave function). IOW, it's pure abstraction. And for someone like yourself who believes that mathematical abstractions are the only "things'" that exist, you are clearly in no position to take an opposing view. (To do so is to simply discredit your own view.)

      > I think you were right on this <

      I was right. (Massimo may need your help to fight his battles. But I do not.)

      > but Massimo did have a plausible answer. He made the point that increasingly the naturalistic fallacy is being used as a synonym to the appeal to nature in philosophical circles. <

      The argument that the "naturalistic fallacy" is increasingly (mis)used by academics as a synonym for the fallacious "appeal to nature" simply makes my point. (I was fully aware that the naturalistic fallacy was being misused in philosophical circles. That's one of the the reasons why I raised the issue.)

      > I agree that it is difficult to distinguish Tegmark from a pseudoscientist, but the distinction must be made because he is not making any claims that are inconsistent with the evidence or with science, nor is he falsifying results or pretending to have evidence he does not. <

      If Tegmark is presenting the mathematical universe hypothesis as a scientific hypothesis, then it is required to make testable predictions. If it doesn't make any testable predictions (and Massimo apparently doesn't think that it does), then it qualifies as pseudoscience (according to Shermer's definiton of the term).

      > Unlike homeopaths and cold fusion or perpetual motion enthusiasts, he is advocating a position supported by rational argument. If the technical definition for pseudoscience fails to distinguish between him and these others, then we need to refine the definition because they are not the same thing. <

      Both physics and metaphysics are based on rationalism. What distinguishes the two is that the former is also based on empiricism while the latter is not.

      > Tegmark's view is not implausible, since it is consistent with the evidence, parsimonious (certainly more so than theism) and supported by logical argument. <

      How exactly is the MUH a more parsimonious explanation than a theistic explanation?

      > It's also (albeit weakly) supported by the evidence that the universe exists, is fine tuned and appears to be governed by mathematical laws of physics, which are precisely the facts it purports to explain <

      As I see it, that evidence qualifies as evidence for theism. And I would also argue that the theistic explanation is a more plausible explanation than the MUH, because it is clearly more plausible that a conscious mind would have the capacity to act as a causal agent than a mathematical abstraction.

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    4. @Alastair

      >the standard interpretation<

      The standard interpretation is not the problem. You describe the wave part as non-physical. That is your interpretation. It is certainly not a mainstream view that it is a pure abstraction, although it would be mine, Tegmark's, Carrol's and that of some others, sure. Massimoi and many others would not agree.

      But adding citations to the effect that wave-particle duality itself is a thing is not necessary because nobody doubts that.

      >I was fully aware that the naturalistic fallacy was being misused in philosophical circles.<

      If you were aware of this then why did you misunderstand Massimo when it was perfectly obvious what he meant from context? And why do you insist he is wrong when the meaning of terms is determined by usage, not prescriptive definitions?

      It would have been better to make the point that communication is aided by using terms clearly rather than to insist that Massimo is flat out wrong.

      >If Tegmark is presenting the mathematical universe hypothesis as a scientific hypothesis, then it is required to make testable predictions<

      Agreed. That's why it's not science, it's metaphysics.

      >If it doesn't make any testable predictions (and Massimo apparently doesn't think that it does), then it qualifies as pseudoscience (according to Shermer's definiton of the term).<

      Only if it has no evidence or plausibility, both of which I argue it has.

      >Both physics and metaphysics are based on rationalism. What distinguishes the two is that the former is also based on empiricism while the latter is not. <

      I have already said I agree it is not physics but metaphysics.

      >How exactly is the MUH a more parsimonious explanation than a theistic explanation? <

      Because the MUH is necessarily true given Platonism, naturalism and the computational theory of mind, as I have argued on my blog. All of these are pretty defensible premises. You yourself seem to agree with Platonism and naturalism.

      A god is not much of an answer because there is no account of where it might come from, and there are numerous problems with the concept as usually presented which render it incoherent.

      But it's not any kind of explanation at all in any case, it's just a word until you start defining what you mean. Is your god good, intelligent, etc? Perfectly so or quite so? Has he existed forever, or does he exist outside of space and time? Is he still around?

      If you don't answer these questions and others, then "God" is just a synonym for "the reason the universe exists", and so "God did it" is an entirely empty statement.

      For some people, God is a bearded guy in the sky. For others, God is basically the universe. If your God is the Platonic ensemble of mathematical objects, then we are agreed.

      >As I see it, that evidence qualifies as evidence for theism.<

      Sure, although not so much that it is governed by mathematical laws, I would think. But yeah, I would count the other facts as (weak) evidence for a god. So theism (or more appropriately deism) is not pseudoscience but metaphysics/theology. However the God concept is still not plausible for the reasons outlined above (and below).

      >because it is clearly more plausible that a conscious mind would have the capacity to act as a causal agent than a mathematical abstraction.<

      Only if you have a plausible account of where the conscious mind came from and how it has the abilty to create universes on a whim. Also, on the MUH, the universe was not caused by a mathematical structure, it is a mathematical structure.

      Delete
    5. @ Disagreeable Me

      > The standard interpretation is not the problem. You describe the wave part as non-physical. That is your interpretation. <

      That's not my interpretation; that's the Copenhagen interpretation - the standard interpretation accepted by the mainstream physics community.

      "It is important to resist the temptation to regard electron waves as waves of some material substance, like sound waves or water waves. The correct interpretation, proposed by Max Born in the 1920s, is that the waves are a measure of probability...The fact that electron waves are WAVES OF PROBABILITY is a vital component of quantum mechanics and in the quantum nature of reality." (source: pg. 202, "The Myth Matter" by Paul Davies)

      "1 According to Bohr, the concept of a physical state independent of the conditions of its experimental observation does NOT have a well-defined meaning. According to Heisenberg the wavefunction represents a PROBABILITY, but NOT an objective reality itself in space and time." (source: Wikipedia: Interpretations of quantum mechanics: footnote 1 to the Copenhagen interpretation)

      > If you were aware of this then why did you misunderstand Massimo when it was perfectly obvious what he meant from context? <

      I was aware that academics misuse the naturalistic fallacy. When I explained to Massimo that he was misusing the naturalistic fallacy (and I backed up my claim with Wiki documentation), he then decided to play the "I am a professional philosopher" card - an evasive tactic that he has repeatedly deployed. When that tactic failed to work for him, he then decided to engage in some lame spin-doctoring tactic in a vain attempt to spare himself further embarrassment. The record clearly bears this all out. There's nothing more to discuss.

      > Agreed. That's why it's not science, it's metaphysics. <

      You're making my point. He's engaging in metaphysics, not physics.

      > Only if it has no evidence or plausibility, both of which I argue it has. <

      If you present a metaphysical theory as a scientific theory, then you are engaging in pseudoscience. (That meets the criteria for Shermer's definition of pseudoscience.)

      The following is in regards to mathematical Platonism (on which the MUH is predicated), theism, the principle of parsimony (a.k.a. "Occam's razor"), and plausibility.

      To begin with, Platonism is theistic. Secondly, the underlying philosophy for classical theism is Thomism ( which is a synthesis of Aristotelianism and Platonism). I myself personally subscribe to Whiteheadian metaphysics. (Whiteheadian process metaphysics is the underlying philosophy for neo-classical theism - a.k.a. process theology). But I am also conversant in Thomistic metaphysics,

      You stated in your previous post to me that the concept of God was incoherent. Well, if you believe it is incoherent, then you will have to explain what exactly is incoherent. But in order to do that, you better be fairly conversant in Thomistic metaphysics and/or Whiteheadian metaphysics. Because you will have to explain to me what exactly is incoherent in either of those metaphysical systems.

      I suspect that Tegmark's "mathematical universe" is already included in what Whitehead referred to as the "primordial nature of God" (the locus for the eternal objects - a.k.a. the Platonic forms). And here's the bottom line: I believe it is more intelligible to argue that an infinite mind has the capacity to exert some causal power rather than an infinite mathematical abstraction. In fact, I consider the notion that an infinite mathematical abstraction has the capacity to exert some kind of causal power to be completely absurd.

      Delete
    6. In the Copenhagen interpretation, it's a probability function, true. That doesn't mean it's not physical. Massimo can say there is a difference between the probability function of an actual physical quantity and a probability function on paper, and that's that the former is physically manifested.

      It's not the science that's the problem, it's the conclusions you draw.

      Your quote from Paul Davies is not much help because Massimo has already claimed that he has a theological agenda.

      >He's engaging in metaphysics, not physics.<

      We're agreed on that. We are not agreed that he is doing pseudoscience.

      >If you present a metaphysical theory as a scientific theory, then you are engaging in pseudoscience.<

      I'm not sure that's the same as Shermer's definition.

      The Copenhagen Interpretation is often presented as and indeed widely regarded as science, yet there is no evidence for it and it makes no predictions. Is it pseudoscience?

      >To begin with, Platonism is theistic.<

      No it most certainly is not. Not my Platonism or Tegmark's anyway.

      >You stated in your previous post to me that the concept of God was incoherent. <

      I said that the concept of God as commonly presented is incoherent. There are different versions of God which may not be. I'm not familiar enough with theology to have a clear vision of what you interpret God to be based on your description.

      I'll throw out a few statements (which I have no idea if you agree with) and explain what I find to be problematic.

      God is omniscient
      Incoherent because God could not know, for example, if there is anything he does not know.

      God is perfectly good
      Inconsistent with the world he created.

      God is omnipotent
      Possibly incoherent because of various paradoxes such as not being able to limit his own power. Admittedly you can get around this by just saying he is only as omnipotent as doesn't lead to paradoxes, but that seems like a bit of a dodge to me.

      God is an intelligent being existing outside of time
      Incoherent because intelligence is a process which takes place in time.

      God is self-created
      Enough said.

      God sees the future
      Inconsistent with the free will of both humans and God

      God is love
      Meaningless platitude.

      God is perfect
      Too vague. I'm not sure that "perfect" really makes sense as a concept on its own, only in specific contexts according to specified criteria.

      God is a non-physical spiritual being which thinks and plans
      He has no brain. What does he think with? According to what algorithm does he think? I reject as incoherent the concept of any thought process (or any process at all) not governed by rules at some level. How does it work if it's not governed by rules? If it is governed by rules, then it's just a powerful algorithm, in which case how can it be perfect or omniscient in the light of Godel's theorems?

      >it is more intelligible to argue that an infinite mind has the capacity to exert some causal power rather than an infinite mathematical abstraction<

      What is an infinite mind? What does that even mean? How does it work? How did it come to be? How does it exert causal power? How did it create the universe? Why did it create the universe?

      There are no good answers to these questions. That's why it's not parsimonious - it raises more questions and causes more problems than it solves.

      >In fact, I consider the notion that an infinite mathematical abstraction has the capacity to exert some kind of causal power to be completely absurd.<

      That's because you don't seem to understand. Maths didn't cause the universe to be, any more than maths caused the Mandelbrot set to be. On the MUH, the universe is maths just as the Mandelbrot set is maths. End of story. No mystical causal powers needed.

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    7. @ Alastair

      I think I see where the confusion about the particle/wave duality comes from.
      The fundamental object in QM is the wave function, a probability wave as you say.
      Such a wave function is the general solution to e.g. the Dirac equation, describing electrons.

      But that's not the same as the 'wave' from the particle/wave duality. The duality comes in later
      when you do a measurement. Both aspects of that duality are built-in intoe the wave function.

      Doing a double-slit experiment projects the wave function into a 'wave',
      showing wave-like properties such as interference.
      Closing one slit projects the wave funtion into a 'particle', showing particle-like properties
      such as being localised in space.

      In QM a 'particle' is just a wave packet, which is a sum of wave functions for which the probability is
      concentrated to a very narrow spatial region.

      Why is a QM 'particle' less abstract than a 'wave'? Both are equally abstract, mathematical concepts with only
      approximate correspondences in classical mechanics/everyday life, like a water wave, or a ball.
      For example, is the concept of an infinitely small point mass not pretty abstract?
      Many more abstract properties apply both to particle and wave, like spin (not at all the same as angular momentum),
      or Pauli exclusion (no correspondence in classical physics).

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    8. @ Alastair

      Are you going to respond to either

      DM December 16, 2013 8:03 PM or

      lalanguedesetoiles December 17, 2013 11:17 AM?

      Silence at this point sends a pretty strong message of contempt

      Delete
    9. @ Disagreeable Me

      This part one of a two part response.

      > Your quote from Paul Davies is not much help because Massimo has already claimed that he has a theological agenda. <

      I also cited Wikipedia's article on the "interpretations of QM." It corroborated what Paul Davies had written. I'm afraid you don't have the luxury of simply disregarding that fact.

      By the way, I failed to give proper attribution to "The Myth Matter." It was also co-authored by John Gribbin. So, you are not only casting aspersions on Davies' integrity, but also Gribbin's.

      > I'm not sure that's the same as Shermer's definition. <

      "Pseudoscience is a claim, belief, or practice which is presented as scientific, but does not adhere to a valid scientific method, lacks supporting evidence or plausibility, cannot be reliably TESTED, or otherwise lacks scientific status.[1]" (source: Wikipedia: Pseudoscience)

      > The Copenhagen Interpretation is often presented as and indeed widely regarded as science, yet there is no evidence for it and it makes no predictions. Is it pseudoscience? <

      It's presented as an interpretation of what the scientific theory means. (It's not presented as a scientific theory, in and of itself.)

      > No it most certainly is not. Not my Platonism or Tegmark's anyway <

      I said "Platonism" was theistic, not "mathematical Platonism." But mathematical Platonism has basically co-opted Plato's "theory of forms" (hence the name, "mathematical Platonism).

      Moving on, I'll address some of your problems with the divine attributes.

      "Omiscience" means "all-knowing." God knows everything that is logically possible to know, both what is possible and what is actual. I don't see anything incoherent about that.

      "Omnipotence" means "all-powerful." This doesn't mean that God has the power to do that which is logically impossible to do. It means that God has the power to create ex nihilo.

      "Omnibenevolence" means "all-benevolent" (or "all-good"). God is all-good or (the supreme good). You believe that this is inconsistent with the world as it is, because of the so-called "problem of evil." But evil is relative, not absolute.

      "This is the part of the infinite goodness of God, that He should allow evil to exist, and out of it produce good." - St. Thomas Aquinas (ST I.2.3)

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    10. @ Disagreeable Me

      This is part two.

      > God is an intelligent being existing outside of time
      Incoherent because intelligence is a process which takes place in time. <

      If a computer had infinite computing power, how long would it take to process an infinite amount of data? Answer: No time. (An infinite mind with infinite power does not take any time to think or act. This might be mind-boggling (after all we are talking about the infinite), but it is not incoherent.)

      "All actions ever truly ascribed to God are just manifestations in time of one atemporal act." (source: pg. 132, "Our Idea of God" by Thomas V. Morris)

      > God is self-created <

      Classical theism does not hold that God is self-created. God simply is.

      > God sees the future.
      Inconsistent with the free will of both humans and God <

      There is no inconsistency here. The following quote probably explains it better than I can. (It might also help if you can distinguish between the "A-series and B-series" of time. I trust that you can because you have already invoked these terms yourself.)

      "God does not properly forsee anything; he simply sees it as it is actually taking place. He sees nothing as past or future to him, though he knows what time things happen for us." (source; pg. 240, "The One and Many" by by W. Norris Clarke, S.J.)

      > God is perfect
      Too vague. I'm not sure that "perfect" really makes sense as a concept on its own <

      The divine attributes are derived form what is called "perfect being" theology. The attributes we have been discussing are some of the attributes a perfect being would have. I don't see how you can say that that's vague.

      > God is a non-physical spiritual being which thinks and plans <

      That shouldn't be a show-stopper for you. After all, you believe that the only things that exist are nonphysical things. Right?

      > How does it work if it's not governed by rules? <

      By logic? However, I would argue that logic is an expression of the mind of God, not something that governs God's mind. IOW, God's intellect, power, and will are one. (The concept of divine reason (a.k.a. the "Logos") goes to the very beginning of philosophy and theology. It probably finds its fullest expression in the Hegelian dialectic.)

      > What is an infinite mind? What does that even mean? How does it work? How did it come to be? How does it exert causal power? How did it create the universe? Why did it create the universe?

      There are no good answers to these questions. That's why it's not parsimonious - it raises more questions and causes more problems than it solves.<

      Well, Tegmark (according to Massimo) argued that it would take an "infinitely intelligent mathematician" (clearly a euphemism for God) to fully understand mathematical monism. So, unless God truly exists, there is no mind that can fully understand mathematical monism.

      "No created intellect can comprehend God wholly." - St. Thomas Aquinas

      > On the MUH, the universe is maths just as the Mandelbrot set is maths. End of story. No mystical causal powers needed. <

      Are you completely dispensing with all causality? Because if you are, then all we are left with are correlations and observations, without any causal explanation whatsoever. (In other words, MUH doesn't appear to explain anything. That's what happens when you completely dispense with "mystical causal powers.")

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    11. @ lalanguedesetoiles

      > Why is a QM 'particle' less abstract than a 'wave'? Both are equally abstract, mathematical concepts with only approximate correspondences in classical mechanics/everyday life, like a water wave, or a ball. For example, is the concept of an infinitely small point mass not pretty abstract? <

      Actually, I have already made a similar argument on another blog post. It appears that contemporary physics reduces everything to abstractions, But I'm not sure exactly what you are getting at? Are you arguing for mathematical monism? Because if you are, then there is one thing you have left out (besides matter and energy) - namely, the act of observation. The standard (or "Copenhagen") interpretation "asserts that an observation produces the property observed." (source: pg. 100, "Quantum Enigma" by Bruce Rosenblum and Fred Kutner)

      Delete
    12. @Alastair

      I'm not ignoring the citations you have given from Wikipedia. I explained how it might make sense to regard even probability waves as physical, and you ignored that argument.

      As for Shermer's definition of pseudoscience, I have already answered that point with reference to the criteria of plausibility and evidence. The MUH does not fit the definition even though it is not testable.

      >It's presented as an interpretation of what the scientific theory means. (It's not presented as a scientific theory, in and of itself.)<

      I think that's splitting hairs. It is one of several interpretations promoted by and taught by scientists in science classrooms. You yourself are trying to use it to make points by appealing to it as a scientific fact about the universe.

      In any case, If it were presented as science by one individual, would that make it pseudoscience? If Tegmark made it clear that the MUH is an interpretation of science rather than science itself would that satisfy you?

      >I said "Platonism" was theistic, not "mathematical Platonism." But mathematical Platonism has basically co-opted Plato's "theory of forms" (hence the name, "mathematical Platonism).<

      Classical Platonism is not relevant to this discussion.

      As for whether God is incoherent, I'm probably not going to get into a detailed theological debate with you right now. I'll concede that there are answers to several of the points that I have made, particularly with regard to omniscience and omnipotence, and that other points I made are straw men with regard to yourself.

      I still think your God is incoherent though, as I think a B-theory of time is itself incompatible with free will, especially if the future is known.

      I also think that there is no coherent picture of how God's mind might work. You say it thinks according to logic, but how does it decide what logical rules to apply and to what?

      In my view, the only coherent model for intelligence is the CTM. If God's mind is an algorithm, I'll accept it as coherent. If God's mind is not an algorithm, then I don't understand how he is supposed to think or make decisions.

      As an explanation for existence, God is not parsimonious because "God simply is" is no more satisfying than "The universe simply is". If that's the kind of answer you're happy with we can dispense with God altogether.

      >Well, Tegmark (according to Massimo) argued that it would take an "infinitely intelligent mathematician" (clearly a euphemism for God) to fully understand mathematical monism.<

      No he didn't. I understand mathematical monism. It's not that hard.

      It would take an "infinitely intelligent mind" to perceive at once all the possible coherent mathematical objects, but nobody is postulating such a mind and indeed the concept of such a mind is brought up only to explain a point and is perhaps not itself coherent at all.

      >Are you completely dispensing with all causality?<

      No, but the existence of the universe is not something that has a cause on the MUH. The universe exists necessarily the same way the Mandelbrot set exists: without a cause.

      To explain causality in the context of the MUH you need to look at mathematical analogies with iterative development of a system over (an analogue of) time.

      Cellular automata are ideal. Each state, combined with the rules for the evolution of the system, is a cause of later states.

      This is how causality works in the context of a mathematical system. I don't see how you have a problem with this. It's really quite simple.

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    13. @ Disagreeable Me

      > I'm not ignoring the citations you have given from Wikipedia. I explained how it might make sense to regard even probability waves as physical, and you ignored that argument. <

      Okay. You stated in your previous response: "In the Copenhagen interpretation, it's a probability function, true. That doesn't mean it's not physical. Massimo can say there is a difference between the probability function of an actual physical quantity and a probability function on paper, and that's that the former is physically manifested."

      The wave function represents an abstract realm of possibilities, not something physical. (A nonphysical entity can theoretically cause a physical effect. But that does not make the said entity physical. In the double-slit experiment, we can observe an interference pattern - a property which is suggestive of a wave. However, we cannot observe any physical wave.)

      Rosenblum and Kuttner (professors of physics at UC/Santa Cruz) write "The mathematical representation of the wave is called the "wave function." In some real sense, the wavefunction of an object is the object." (pg. 72, "Quantum Enigma" by Bruce Rosenblum and Fred Kuttern.)

      > I think that's splitting hairs. <

      I agree. But do you agree that Massimo is splitting hairs when he characterizes Tegmark's MUH, not as a metaphysical interpretation, but as a "metaphysically informed scientific speculation?"

      > If Tegmark made it clear that the MUH is an interpretation of science rather than science itself would that satisfy you? <

      Yes, it would.

      > Classical Platonism is not relevant to this discussion. <

      I think it is. Because if we are obligated to presuppose that some Platonic forms/ideals (such as mathematical abstractions) are objective in order to do science, then we should also be obligated to presuppose other Platonic forms/ideals (such as truth, beauty, and goodness) are objective in order to do philosophy.

      > I still think your God is incoherent though, as I think a B-theory of time is itself incompatible with free will, especially if the future is known. <

      There are two separate issues here - the coherence of my concept of God and the coherence of my concept of free will. So, even if I concede (for the sake of argument) that my concept of free will is incoherent, this really does not have any bearing on the coherence of my concept of God.

      > If God's mind is an algorithm, I'll accept it as coherent. If God's mind is not an algorithm, then I don't understand how he is supposed to think or make decisions. <

      There is a logical structure to God's infinite thought-process. (Whiteheadian metaphysics is called "process theology" because it describes the God-world relationship in terms of a process, not unlike the process that occurs in an information processing system.)

      > It would take an "infinitely intelligent mind" to perceive at once all the possible coherent mathematical objects <

      Yeah, and why exactly is such an "infinitely intelligent mind" (which is clearly a euphemism for God) so incoherent?

      > This is how causality works in the context of a mathematical system. I don't see how you have a problem with this. It's really quite simple <

      I understand how software and hardware interacting together can process information. I even understand how a nonphysical mind can process nonphysical information. But I am having a very difficult time seeing how nonphysical mathematical abstractions can accomplish this feat. (And I am not the only one.)

      It seems to me that you are presupposing two domains: a "temporal" one and a "nontemporal" one. And the temporal domain appears to be causally-dependent on the nontemporal one.

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  32. DM,

    > Precisely. I do actually believe unicorns exist in some sense. <

    To me that’s playing with words. Yes, they do “exist” in *some* sense, but that sense adds nothing to the discussion we are having. I think numbers exist in a much more robust sense than unicorns, and chairs in an even more robust sense. After all, I can’t throw unicorns or numbers at people, but I can hit them very well with a chair…

    > Will you get fatter if you just think about pizza? No, because you need to physically interact with it. You can't physically interact with the number four because it is not an object within your universe. That's the difference. <

    And that’s all the difference I need…

    > Five can "eat" Four and become Nine because they are in the same "universe" of numbers. <

    Forgive me, but this sounds like the definition of nonsense on stilts.

    > to have a less tortured, and misfitting analogy we should be instead be comparing physical pizza to virtual pizza in a simulation, or comparing George Lucas to Luke Skywalker. <

    Yes, and you know what I think about the difference between physical and simulated objects…

    > You've missed the point again, I'm afraid. You deny monism. OK. I'll accept that monism is false for now for the sake of argument, if you accept Platonism and CTM. Now, where does that take us? <

    I have most certainly not missed the point. Look at what you just did: you *had* to introduce the CTM, which assumes (or is at the very least exceedingly, almost incestuously, friendly to) monism in order to get back on the saddle.

    > The mathematical universe contains self-aware substructures <

    Sounds like magic to me.

    > per your own description of Occam's razor, we must not multiply unnecessary entities. Since the mathematical universe must exist, given Platonism, the physical universe is unnecessary. <

    You definitely misapply Occam: the key word there is “necessary.” Like Ladyman and Ross (and unlike Quine), I’m ok with a lush (as opposed to a desert) ontology, if that’s what’s required. By saying that it isn’t you are simply begging the question.

    > I'm not sure your confidence is shared by all physicists (although Tegmark seems to agree with you). Personally I'm agnostic on the question. <

    But the fact that Tegmark does agree ought to bother you.

    > I'm just expressing entirely reasonable uncertainty about how close we have come to the ultimate laws of reality. My point is not specifically that quarks and gluons are made of more basic particles. <

    Sure, but you can’t just make up examples out of possibilities that are currently nowhere to be found in physical theory, just because it is convenient for your position, and then think you’ve made a positive point. “Possibility” (or, even worse, Chalmers’ much dreaded “conceivability”) is just too weak to buttress any serious argument.

    > I am invoking the anthropic principle specifically to counter the argument that the universe is too simple to have been selected randomly from the set of all possible mathematical structures. <

    And what, exactly, makes you think that life wouldn’t be possible in a much more complex universe?

    > my argument is that only simple structures are likely to be conducive to life <

    Why? How do you define simple? Which constraints are imposing, and why?

    > What do you mean when you say every successful physical theory would violate the CUH? <

    *I* didn’t mean anything. I was simply quoting one of Tegmark’s critics. And Max seemed to acknowledge the point in his reply. Perhaps we should both go back to the original exchange.

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    1. Hi Massimo,

      >After all, I can’t throw unicorns or numbers at people, but I can hit them very well with a chair…<

      That example proves my point. I see numbers and unicorns both as abstract conceptual objects, so of course they are more similar to each other than to chairs.

      >And that’s all the difference I need…<

      Pizza is present as an object in your universe. The number four is not. That accounts for the difference while being consistent with the MUH.

      >Yes, and you know what I think about the difference between physical and simulated objects…<

      Again, please suspend your skepticism of the CTM so as to understand the argument for the MUH. We can then fall back to debating the CTM or Platonism on which it depends.

      >you *had* to introduce the CTM, which assumes (or is at the very least exceedingly, almost incestuously, friendly to) monism in order to get back on the saddle.<

      The CTM is greatly helpful in arguing for the MUH . It's not necessarily required - there are competitors to the CTM which would also allow for self-aware substructures to exist in mathematical objects, for example Roger Penrose's ideas.

      It's news to me that the CTM is incestuously close to monism. I think most CTM advocates would disagree with you.

      So, again, my argument is that if the CTM is true and if Platonism is true, there must be purely mathematical analogues of our physical selves who also perceive themselves to be real.

      Would you agree that this follows from the CTM and Platonism?

      Would you not also agree that if this is true then the distinction between mathematical and physical is redundant and meaningless for the reasons I have outlined, and that there is no reason to believe in the physical world?

      I think this argument is pretty solid, but you don't have to accept it because you can reject either the CTM or Platonism.

      At this point, I would hope that you accept that the MUH makes sense given these premises and we can go back to discussing biological naturalism vs the CTM.

      >You definitely misapply Occam: the key word there is “necessary.”<

      I'm not asserting the physical universe is unnecessary, I'm arguing for it. If you followed my argument, then the physical universe is unnecessary given the CTM and Platonism. Given these premises only, Occam's razor suggests the physical universe hypothesis can be discarded. This is not a misapplication of the razor.

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    2. >But the fact that Tegmark does agree ought to bother you.<

      I'm not a disciple of Tegmark's. I came up with the MUH independently and while I'm mostly happy with how he argues for it I'm not bound to consider his word gospel.

      And Tegmark is not agreeing that the universe is simple, he only leans towards optimism that it is.

      Agnosticism is the only appropriate position, and as such the possibility that the universe is very complex is an appropriate response to the assertion that it is too simple for the MUH to be true.

      >Sure, but you can’t just make up examples out of possibilities that are currently nowhere to be found in physical theory<

      Examples are for illustrative purposes only. I'm only making the point that nobody knows how complex the laws of nature are, so any argument from the incredible simplicity of nature is premature. Your confidence that nature really is simple is, I feel, unwarranted.

      >And what, exactly, makes you think that life wouldn’t be possible in a much more complex universe?<

      Life depends on certain regularities in nature so as to evolve to exploit those regularities.

      My hypothesis is that a universe with a huge (or infinite) multitude of significant forces, laws of nature and arbitrarily complex equations of physics will behave chaotically and unpredictably.

      An unpredictable environment leaves no opportunity for life to adapt to it, as what works in one moment will not work in the next.

      I think this is a testable hypothesis. With cellular automata such as Conway's Game of Life and Wolfram's 2D automata we have a model for how simple rules can lead to great complexity and structure.

      If my hypothesis is correct, we could add hundreds or thousands more rules and possible states to the laws governing these toy universes, and what we will see emerging will not be more complex and sophisticated structures but more apparent chaos and randomness.

      >Perhaps we should both go back to the original exchange.<

      I have done so and I don't find therein anything like the statement that the CUH is incompatible with known physics.

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  33. Philip,

    > There is actually no "the number four" just like there is no "the pizza". You can have a bunch of fours just like you can have a punch of pizzas. <

    That’s an old debate in metaphysics about particulars vs universals. You are a bit too confident that it has been settled your way.

    Patrick,

    > I don't understand the argument that mathematics is unreasonably effective. Is it? As a tool to do biology or philosophy? How many equations did Darwin use? <

    Mostly, in physics. The fact is that many times over mathematicians have proceeded their own way, investigating abstruse concepts that had seemingly nothing to do with the physical world, and then it suddenly turn out that those concepts had a hell of a lot to do with that world. Regardless of whether one accepts Platonism or not, it is something that philosophy of mathematics simply has to factor in.

    > One can construct a zillion different mathematical models to describe something. Are they all effective? No, a limited number are and it's not mathematics that decide which models are useful, but experiment. <

    True, but of course a Platonist would respond that that’s because mathematics describes all coherent formal systems, and the universe must be one such system.

    Paul,

    > Tegmark presents his argument. It appears to be something like the following:

    Basic physical objects (particles, say) are fully described by mathematical properties.
    Objects of pure mathematical structures are fully described by mathematical properties.
    Thus, particles do not differ from objects of pure mathematical structures.
    Thus, the universe is a pure mathematical structure. <

    Ah. that reminds me of Harris’ so-called “argument” that, because the brain uses the same structures in response to statements of facts and to statements of value, therefore facts and values are the same. Easily refuted by pointing out that the brain also uses the same structures when we think about having sex and when we actually have sex…

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    1. "the brain also uses the same structures when we think about having sex and when we actually have sex"


      Those who have awoken from a wet dream find out they weren't actually having sex, but during the dream they think they were.

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    2. Massimo,

      > Mostly, in physics. The fact is that many times over mathematicians have proceeded their own way, investigating abstruse concepts that had seemingly nothing to do with the physical world, and then it suddenly turn out that those concepts had a hell of a lot to do with that world. Regardless of whether one accepts Platonism or not, it is something that philosophy of mathematics simply has to factor in. <

      I certainly wish the philosophy of mathematics would factor in all those moments too when physicists did unmathematical things with great success. The development of renormalization theory is a good example. I wonder what the Platonists were thinking when physicists were happily subtracting infinities. Those ideal forms must have looked surprisingly meek, malleable by less than ideal humans.

      > True, but of course a Platonist would respond that that’s because mathematics describes all coherent formal systems, and the universe must be one such system. <

      That Platonist isn’t beating about the bush, is he? I would say something else: aided by experiment and a solid dose of not necessarily mathematical intuition, physicists use mathematics to describe those properties of systems that can be described with coherent formal systems. In some areas of physics – but not all - that is equivalent to a large part of system being studied, as far as we know (*). But is this “unreasonable”? Physics studies objective reality, that part of reality about which we all have to agree. Mathematics too studies properties about which we all have to agree. Is it really surprising that both are meeting on a regular basis? (Although the meeting is far from trivial … as every physicist who has measured a non-trivial property of a physical system will tell you. The Platonist view is only sensible after a heavy dose of idealization.)

      (*) This statement is true or not depending on the definition of the system. Take a collision between a proton and an anti-proton. What’s the system? Those two particles? Or the particles and the collider? The particles, the collider and the detectors? If you take the last definition mathematics only describes a relatively small part of the system – there’s certainly no mathematical description of the individual atoms in the detectors …

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    3. >I wonder what the Platonists were thinking when physicists were happily subtracting infinities. Those ideal forms must have looked surprisingly meek, malleable by less than ideal humans. <

      I'm not too partial myself to the argument for Platonism from the effectiveness of mathematics, but I do want to defend Platonism against your disdain.

      As a Platonist, I'll say I'm happy for you to subtract infinities as much as you like. As long as you are consistent in your application of whatever rules you make up, you're doing mathematics. Mathematics encompasses all formal systems, including systems that don't work quite the way arithmetic normally does.

      >The Platonist view is only sensible after a heavy dose of idealization.<

      Only if the "unreasonable effectiveness of mathematics" were the only argument for Platonism. It isn't.

      >there’s certainly no mathematical description of the individual atoms in the detectors<

      Doesn't matter. According to the Platonist view, there is such a description in principle, even if it is too complex for any human mathematician to ever work with it in practice.

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    4. Patrick -

      Renormalization theory gives physicists the ability to approximate certain situations where our theories encounter infinities. This is precisely why our current theories aren't the final word! The fact that we now use renormalization techniques isn't an argument against platonism. The final description will likely have no need for renormalization.

      In addition, you seem to be able to discard infinities because you don't see their effects on the physical or the areas of mathematics that are finite in nature. In fact, there is evidence that we can see the effects of infinite cardinal numbers. Read this article for a further description (you may be amazed):

      http://www.telegraph.co.uk/science/8118823/Large-cardinals-maths-shaken-by-the-unprovable.html

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    5. Disagreeable,

      > As a Platonist, I'll say I'm happy for you to subtract infinities as much as you like. As long as you are consistent in your application of whatever rules you make up, you're doing mathematics. <

      The problem with the development of renormalisation was that physicists did mathematically inconsistent things, although they were in a certain sense consistent in their inconsistency.

      > Only if the "unreasonable effectiveness of mathematics" were the only argument for Platonism. It isn't. <

      No, but take away that effectiveness and Platonism becomes a lot less convincing and a lot less interesting.

      I still feel that some people here underestimate how complicated the link between all those very messy observable phenomena and mathematics really is.
      Don't you think one should visit a sausage factory before one starts to philosophize about sausages?


      Delete
    6. Hi Patrick,

      I don't know anything about renormalisation, however it sounds like they were being internally consistent, which is all that matters. This from Wikipedia:

      "renormalization eventually was embraced as an important and self-consistent tool in several fields of physics and mathematics."

      Anything that is internally consistent and formal is fine and counts as mathematics. There is no One True Way of doing maths. There are an infinite number of axiomatic formal systems and this just sounds like another one.

      >take away that effectiveness and Platonism becomes a lot less convincing and a lot less interesting. <

      Unless the MUH is true, in which case Platonism forms part of the explanation of all of reality, fine tuning and lets us know the intriguing fact that parallel universes exist, doing away in one stroke with any need to invoke a divine creator of any kind. That's pretty interesting, in my book. I also think it plays a role in philosophy of mind.

      >underestimate how complicated the link between all those very messy observable phenomena and mathematics really is.<

      I don't think so. I have even speculated in comments here that reality may be infinitely, fractally complex. Complexity doesn't matter as long as I don't have to personally do the math.

      Delete
  34. "The best point in its favor is the so-called “no miracles” argument, the idea that mathematics is too unreasonably effective (at predicting things about the world) for it to be just a human invention."

    I think a lot of people look at this and say something like the following:

    Mathematics is a language developed to describe regularities. If mathematics is astoundingly good at describing the universe, it means that the universe is, at a basic level, highly regular. And this statement (the universe is mathematical insofar as it behaves regularly) is something close to trivial.

    I think the "ontological status" thing ends up muddying the waters. Giving something an "ontological status" is often just a conceptual way of trying to think of something cognitive as being more like an "object". But the mathematical-ness of the universe is, if anything, related to processes rather than objects. And we're no more clear on the ontological status of processes than we are on the ontological status of mathematics.

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  35. The Universe is as math is, is equals =.

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  36. Suppose there is a computer (C) running a simulation (S) of some part (P) of the world (W).

    (C, S, P, and W are all made of "elementary particles".)

    But you (observer O) look at S and look at P and see two different things. To O, S is not P. S just "describes" or "models" P.

    But suppose there is also a simulation of you (avatar A) running in the same computer C interacting with S as O interacts with P.

    O <-> P in W.
    A <-> S in C.

    Now A wonders, "Suppose there is a computer ..."

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    Replies
    1. S is not made of particles. S is software, mathematical. Any particles you might consider to be part of S are actually part of C.

      But yeah, other than that you're spot on.

      But that's the simulation hypothesis, which is dodgy because there's no reason to believe it would ever be feasible to run a universe simulation on a computer.

      It is useful as a stepping stone towards the MUH though.

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    2. Just an afterthought ... but I find it amusing that Tegmark invokes Occam's razor.

      "The last refuge of the scoundrel", I sometimes jokingly say. You could also suggest that Occam disfavors mathematical Platonism – the idea that triangles and mathematical groups etc. exist. I don’t think that there’s anything new in what I’m going to write, but I’ll do it anyhow.

      What’s a right triangle? IF you have a set and IF you define subsets (called ‘points’ and ‘straight lines’) and IF certain relationships between these subsets hold etc. etc. then you can find three points A, B and C such that AB is orthogonal to AC. The IFs are very important; if you forget the fifth postulate then forget about right triangles of the usual kind.

      The chain of if-then statements is true. Perhaps I’m naïve, but I find it more economical to admit that truth than to admit that truth and postulate that right triangles exist.

      Sure, I can use these geometrical objects to describe something (lines and points on a piece of paper). Perhaps, after some work with a ruler, I’ll find out that the Pythagorean theorem holds (within the uncertainties induced by my imperfect ruler). But that doesn’t show mathematical right triangles exist. It only shows that I’m describing something that matches (within measurement error) the IF’s with which Euclidean geometry starts. If I take three points on a globe, no Pythagorean theorem. I need another chain of if-then statements to describe what happens on a globe.

      Mathematics is, to me, a giant chain of if-then statements.

      Massimo writes “For instance, the set of real numbers has a number of structures.” I don’t feel this is correct, strictly speaking. There are a number of ways to construct the reals. You can start with a set and impose certain relationships: two operations with a field structure, an order, the Archimedean property etc. Et voilà! You have the reals (up to an isomorphism). But the set doesn’t “have” these properties; you gave them to it. IF you work with them, then … You can also start with the axioms of set theory, construct the natural numbers, define an order on them, construct fractions with equivalence relations and reals with Dedekind cuts etc. I’m simplifying, but et voila! You get the reals. But again, it’s a giant if-then chain of reasoning. I can take a set and impose different properties and end up with a non-Archimedean field, with infinitesimals etc. I can impose additional properties on an Archimedean field (like in Internal Set Theory) and still end up with something that has all "infinitesimal-like" reals I need, if I need infinitesimals.

      I’m willing to admit the truth of it, but Occam’s razor forbids me to believe that the reals are real. I don’t need to believe that; the truth of the if-then’s is enough.






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    3. Patrick -

      You saying you think mathematics to be a giant chain of if-then statements in no way takes away from the fact that it has been extremely useful ("unreasonably effective") in describing the world. The only thing I see from your last comments are that you might be trying to embrace a sort of formalism about mathematics, which doesn't negate Platonism in any way. The if-then statements are describing a structure that happens to be indispensable to our best theories, so yea, Occam would say your'e ok to believe in their existence.

      I think of Einstein's quote when people start throwing around Occam's razor with reckless abandon: "Make things as simple as possible, but not simpler."

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    4. Hi Patrick,

      I agree with everything you said, other than your conclusion that Platonism is false.

      >The chain of if-then statements is true.<

      Sure. And the chain of if-then statements is isomorphic to a triangle. I'm saying that chain of if-statements exists. Denying Platonism is saying that it doesn't. So you're asserting properties (truth) about something you deny exists, which I think is probably dangerously close to nonsense.

      So there is some limited sense in which that chain of if-thens exist. Not some weird ghostly mystical way, but some limited way that lets us discuss it and assert properties about it, such as truth.

      This is precisely the way that mathematical objects exist.

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    5. How could real numbers exist? If you had access to a computer that could execute a transfinite number of simple computational steps in finite time (a physical hypercomputer).

      Delete
    6. Pete, Disagreeable,

      > The if-then statements are describing a structure that happens to be indispensable to our best theories, so yea, Occam would say your'e ok to believe in their existence. <

      Would he? I don't see why. The existence of mathematical right triangles is not indispensable for the usefullness of mathematics, nor for the truth of all those if-then statements.

      > So you're asserting properties (truth) about something you deny exists, which I think is probably dangerously close to nonsense. <

      If Superman is the guy who shares an office with me, then he can fly. That's a true statement. But Superman doesn't exist and as far as I know my colleague can't fly. Although he can make himself invisible when you need him. And there may be alternative universes in which Superman exists and shares an office with me, of course.

      > And the chain of if-then statements is isomorphic to a triangle. <

      I don't know what that is supposed to mean. An isomorphism is a well-defined mathematical property. How would you apply it to if-then statements and right triangles?

      Very loosely speaking there is (within measurement error) some sort of property-preserving relationship between points and lines on a piece of paper and mathematical points and lines in plane Euclidean geometry. I don't mind people using the word isomorphism for that relationship, as long as it's clear we're not talking about a mathematical isomorphism.

      Delete
    7. Hi Patrick,

      Thanks for continuing the debate. I'm enjoying it.

      >If Superman is the guy who shares an office with me, then he can fly. That's a true statement. But Superman doesn't exist and as far as I know my colleague can't fly.<

      Well, Superman does exist, as a concept. Otherwise statements such as those you have outlined would be meaningless. But he is not physically present as an object in this universe, which is what you mean when you say he doesn't exist.

      This is exactly like mathematical objects. They exist as concepts but are not physical objects.

      >An isomorphism is a well-defined mathematical property. How would you apply it to if-then statements and right triangles? <

      I could define a function that would translate any statement about triangles to the language of your if-then statements, and an inverse function that would do the reverse in a one-to-one bijective fashion. As far as I can see, this is damn close to the idea of a mathematical isomorphism (although perhaps not quite there?).

      In this sense, your chain of if-then statements is essentially identical to the concept of a triangle. Your assertion that the chain of if-then statements is true then implies that the chain exists which means that triangles exist.

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    8. To find the truth is as simple as removing the ifs, thens, and uncertainties, the errors of our measurements or judgement s we have been taught to practice and believe. These untruths are what confound, compound, and obscure the mathematical equation that the Universe truly is; not if, just is! Most simply stated: the Universe is =.

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    9. > Would he? I don't see why. The existence of mathematical right triangles is not indispensable for the usefullness of mathematics, nor for the truth of all those if-then statements. <

      ????

      I guess the existence of electrons is not indispensable for the usefulness of science, nor the truth of all those elementary particle physics statements.

      It's ok. These mathematical abstractions can be real and the universe won't explode!

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    10. Pete,

      I thought our discussion had run its course, but apparently there's a misunderstanding. I just wanted to point out that it's strange Tegmark invokes Occam, because one could use Occam just as well to argue that ideal, matematical objects don't exist.

      Delete
    11. The discussion has officially run its course. After 15 rounds of aggressive punching by each fighter, Platonism still stands

      Delete
  37. I have only one rhetorical correction to make: it "bollocks" not "bullocks." A bullock has bollocks.

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    1. @ Erik Weissengruber

      > I have only one rhetorical correction to make: it "bollocks" not "bullocks." A bullock has bollocks. <

      Actually, a bullock is a castrated bull. So, I guess it technically doesn't have any bollocks. At any rate, "bollocks" is British vulgarity, not American. (Americans use "bullshit" as a vulgar expression for "nonsense.")

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    2. Sorry I didn't get the relationship right. The bollocks are derived from the bullock. Never mind those bullocks, let's get back to philosophy.

      Delete
  38. Hi Massimo,

    Before you leave this topic (if you have not already), could you indicate if the MUH now makes any more sense to you (especially on what it means for reality to be made of mathematics)? I hope I have offered an explanation that is more than a category mistake.

    Keeping in mind your skepticism of the Computational Theory of Mind, I don't think that acknowledging this means you ought to find the MUH convincing. I just hope it is no longer as incoherent or incomprehensible as you say you and Julia initially found it.

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    1. DM, MUH has always made sense to me, I just don't buy it. And I still think it is likely predicated on a category mistake, because I think there is a fundamental distinction between the ontology of numbers and the ontology of physical things. Sorry. But it was fun! ;-)

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    2. When I say it doesn't make sense to you, I mean the following problem:

      "The problem is in what sense, if any, can a mathematical structure, so defined, actually be the fundamental constituent of the physical world, i.e. being the substance of which chairs, electrons, and so on, are made."

      This is the problem I had hoped to answer.

      Thanks for the conversation Massimo.

      Delete
    3. About the possible physical existence of infinite mathematical things (infinite Turing machines in black holes), see Joel David Hamkins, CUNY math professor.
      http://www.3ammagazine.com/3am/playing-infinite-chess/

      Delete
  39. @Alastair

    Not sure my reply will end up in the right place. The comment form is acting funny.

    Anyway, some good points there.

    >The wave function represents an abstract realm of possibilities, not something physical.<

    I disagree with you that this means it is not physical. Define physical. I could propose a definition which means something that can be empirically detected and interacted with. I think the wave function meets those criteria.

    If you have a different definition that's fine, but you don't get to insist that yours is right, and Massimo is entitled to his original opinion that your point about duality proves nothing.

    >But do you agree that Massimo is splitting hairs when he characterizes Tegmark's MUH, not as a metaphysical interpretation, but as a "metaphysically informed scientific speculation?"<

    No, because he didn't specifically reject the characterisation of the MUH as a metaphysical interpretation. He would probably be perfectly happy with that.

    >> If Tegmark made it clear that the MUH is an interpretation of science rather than science itself would that satisfy you? <

    Yes, it would. <

    Good. We're agreed on that at least.

    So, what he's doing is pretty shaky as science, but pretty good metaphysics in my book. That doesn't make it pseudoscience.

    I don't think you've really answered my earlier point that there are many traits of pseudoscience not exhibited by Tegmark. He's not fraudulent. He's not denying evidence. Nothing he says is contradicted by the empirical facts. He is not even making baseless claims, as he has a rational argument to back up what he proposes. I think to call him a pseudoscientist is to lump him in unfairly with much more objectionable characters. Tegmark's mistake seems to me to be pretty minor in comparison.

    >if we are obligated to presuppose that some Platonic forms/ideals (such as mathematical abstractions) are objective in order to do science, then we should also be obligated to presuppose other Platonic forms/ideals (such as truth, beauty, and goodness) are objective in order to do philosophy.<

    We are not obligated to presuppose mathematic realism in order to do science. Plenty of great scientists are mathematical anti-realists.

    But even if your premise were true I don't see how the need for classical Platonism in philosophy would follow form the need for mathematical realism in science.

    >There are two separate issues here - the coherence of my concept of God and the coherence of my concept of free will<

    Not unless you don't think God has free will. If you do, then free will is part of your concept of God.

    >There is a logical structure to God's infinite thought-process.<

    Does this mean you accept that God's mind must be equivalent to some algorithm? In that case, your concept of God's mind may not be incoherent, but I think you are in pretty rare company.

    >and why exactly is such an "infinitely intelligent mind" (which is clearly a euphemism for God) so incoherent?<

    Firstly it's not a euphemism for God. It's a hypothetical entity which he postulates to make a point. I'll grant that it does bear a strong similarity to the concept of God, but this is incidental to the point he was making.

    If minds are algorithms, then the concept of an infinite or perfect mind is incoherent.

    Think about how God as algorithm looks in light of Godel.

    In my interpretation, this means there are true statements which God cannot know are true. This seems to fly in the face of omniscience, but of course it is open to you to answer by pointing out that omniscience does not entail knowing that which it is logically impossible to know.

    The problem is that for any such statement, there exists a "greater" algorithm that could know them. The concept of a mind which knows everything it is possible to know is therefore like the concept of a largest integer: incoherent.

    ReplyDelete
  40. (continued)

    >I understand how software and hardware interacting together can process information.<

    Don't think of it as processing. Think of the universe as a complete mathematical structure with some parts entailed by other parts. This entailment is what we see as causation.

    Consider Wolfram's Rule 30.

    http://www.wolframalpha.com/input/?i=rule+30

    Starting from the same initial conditions, that cellular automaton will produce precisely the same pattern every time you run it. That pattern exists Platonically as a mathematical structure. On Platonism, it is not recreated anew every time the cellular automata program computes Rule 30, instead it already exists and has always existed abstractly. Processing only allows us to generate graphics of the structure, but the structure exists independently.

    Yet there is causation inherent within the structure, because each horizontal row of the structure is determined by a rule which generates rows iteratively, each time based on the preceding row.

    Each row is a state of the universe. The rule is the laws of physics. Later states are determined by prior states. This is pure mathematical causation within the context of a mathematical object, and is very close to how I view causation in light of the MUH.

    ReplyDelete
    Replies
    1. @ Disagreeable Me

      > Define physical. <

      The "physical" is objective and amenable to the third-person perspective. It also exhibits physical properties.

      > If you have a different definition that's fine, but you don't get to insist that yours is right, <

      If you redefine the "physical" to be compatible with the "nonphysical," then all you have accomplished is to render both terms meaningless. (Mathematical abstractions are clearly not physical.)

      > No, because he didn't specifically reject the characterisation of the MUH as a metaphysical interpretation. He would probably be perfectly happy with that. <

      Massimo rejected that characterization by re-characterizing the MUH as "metaphysically informed scientific speculation."

      Why do skeptics (like Massimo) characterize Sheldrake's hypothesis of morphic resonance and morphic fields (a Platonic information field or probabilty structure) as pseudoscience? Answer: Because they believe it cannot be reliably tested. (Massimo also believes that Tegmark's MUH cannot be reliably tested. So, he can't have it both ways. If Sheldrake is engaging in pseudoscience, then so is Tegmark.)

      > So, what he's doing is pretty shaky as science, but pretty good metaphysics in my book. That doesn't make it pseudoscience. <

      It is considered pseudoscience according to Shermer's definition of the term.

      > Tegmark's mistake seems to me to be pretty minor in comparison. <

      Sheldrake's 'mistake' is no different than Tegmark's.

      > But even if your premise were true I don't see how the need for classical Platonism in philosophy would follow form the need for mathematical realism in science. <

      You have to presuppose the absolute truth unless you believe all truth is relative. You have to presuppose absolute beauty unless you believe all beauty is relative. You have to presuppose absolute good unless you believe all morality is relative. And, of course, you have to presuppose the laws of logic (which are clearly nonphysical) to do philosophy.

      > Not unless you don't think God has free will. If you do, then free will is part of your concept of God. <

      I was conceding (and only for the sake of argument) that creatures may not have any free will, not that God doesn't.

      > Does this mean you accept that God's mind must be equivalent to some algorithm?

      Do you consider the Darwinian process (random variation and natural selection) to be some kind of algorithm? (The "process" in process thought is an evolutionary one.)

      > The problem is that for any such statement, there exists a "greater" algorithm that could know them. The concept of a mind which knows everything it is possible to know is therefore like the concept of a largest integer: incoherent <

      You're not thinking big enough.We are talking about an infinite mind.

      "God is that being of whom no greater can be conceived." - St. Anselm

      That being said, Roger Penrose's argument holds that human thought is not a computable algorithm. He invokes "Godel's incompleteness theorems" to make his argument.

      > Don't think of it as processing. Think of the universe as a complete mathematical structure with some parts entailed by other parts. This entailment is what we see as causation, <

      Well, if it is not a process, then it is simply an infinite mathematical structure without any causality. (There may be mathematical relationships; but there aren't any causal ones.)

      To reiterate: "It seems to me that you are presupposing two domains: a "temporal" one and a "nontemporal" one. And the temporal domain appears to be causally-dependent on the nontemporal one."

      Delete
    2. Hi Alastair,

      I'm back on the pleasantries for now, because though it may appear disingenuous I do genuinely respect you. Your intelligence, knowledge and thoughtfulness is obvious, and appreciated.

      I know I've been a bit caustic to you in the past. That has more to do with irritation at redundant citations than anything else. Your own arguments are usually pretty interesting.

      Anyway, more good points.

      >The "physical" is objective and amenable to the third-person perspective. It also exhibits physical properties.<

      I don't see why this might not describe the probability function. Again, the difference between the probability function and a mathematical abstraction on paper is that we can interact with and indirectly observe and measure the probability function.

      >Massimo rejected that characterization<

      I disagree. I don't see why something cannot be both a metaphysically informed scientific speculation while also being a metaphysical interpretation of science. The two phrases seem to me to be equivalent.

      >Answer: Because they believe it cannot be reliably tested<

      Everything I know about morphic resonance I gleaned from wikipedia in the past five minutes, however...

      Morphic resonance is an empirical claim about how nature works. It's not a metaphysical interpretation.

      In particular, Wikipedia states "critics [cite] a lack of evidence supporting the concept and its inconsistency with data from genetics and embryology".

      There is no such inconsistency with evidence for the MUH.

      >It is considered pseudoscience according to Shermer's definition of the term.<

      We're going in circles on this. I reject Shermer's definition if it lumps Tegmark in with genuine pseudoscientists for reasons already outlined. I'm not convinced it does meet Shermer's definition anyway because it doesn't lack plausibility or evidence.

      >You have to presuppose the absolute [truth/beauty/good/logic]...<

      Truth and logic, yes. Morality and beauty are relative in my view.

      >I was conceding ... that creatures may not have any free will, not that God doesn't. <

      One of us is confused. If God has free will, and if I have identified an inconsistency between free will and knowledge of the future, that means that God is incoherent.

      >Do you consider the Darwinian process (random variation and natural selection) to be some kind of algorithm?<

      Yes, since there are algorithms that simulate and exploit the Darwinian process to, e.g., design circuits.

      >"God is that being of whom no greater can be conceived." - St. Anselm<

      Such a being is incoherent. I can conceive that a being greater than God could turn up tomorrow, knowing stuff God does not know and being able to overpower God with his mind. And a being greater than that, ad infinitum. There is no greatest conceivable being for the same reason there is no greatest number.

      And, for the same reasons that infinity is not a number, an "infinite being" is not a being, it is a limit or ideal which can never be reached.

      Even infinity is not the greatest possible. There are always greater infinities. Greatest possible just doesn't work as a concept.

      >That being said, Roger Penrose's argument holds that human thought is not a computable algorithm.<

      Please read http://disagreeableme.blogspot.co.uk/2013/02/strong-ai-godel-problem.html and post a comment there if you disagree.

      >it is simply an infinite mathematical structure without any causality.<

      It will look indistinguishable from causality to observers within the structure. To me, that's what causality is. You say that's not true causality. OK. In that case, I disbelieve in your concept of "true causality".

      >"It seems to me that you are presupposing two domains...<

      I don't understand this. Please explain.

      Delete
    3. >"It seems to me that you are presupposing two domains: a "temporal" one and a "nontemporal" one. And the temporal domain appears to be causally-dependent on the nontemporal one."<

      Actually, I've been thinking about this and I think I do understand you.

      You think I'm presupposing a non-temporal domain, which is the mathematical ensemble of the MUH, and a temporal domain, which is our universe, and that this temporal domain is causally dependent on the abstract ensemble in a way which doesn't make sense because there is no time in the ensemble.

      OK, if that accurately reflects what you mean I see what you're getting at.

      My response is that there are not ultimately two different domains but two different perspectives. Fundamentally, there is only the non-temporal domain (although the temporal domain also exists as a subset of this, and it isn't really temporal after all).

      From the non-temporal perspective, there is only entailment, and our universe is a static structure with a time dimension.

      From the perspective of an observer within this universe, we perceive entailment as causation, and we perceive the time dimension of our universe as a temporal flow.

      An analogy might help.

      The universe is a novel. From outside the novel, the novel is a static unchanging structure. This is the nontemporal perspective.

      From the point of view of a character inside the novel, time flows and the world changes. This is the temporal perspective.

      Delete
    4. @ Disagreeable Me

      > I don't see why this might not describe the probability function. <

      Because the wave function doesn't have any physical properties, it is not located in space or time, and it collapses into a particle upon measurement.

      > I don't see why something cannot be both a metaphysically informed scientific speculation while also being a metaphysical interpretation of science. <

      There's a difference (at least in Massimo's mind) between a "metaphysically informed scientific speculation" and a "scientifically informed metaphysical speculation."

      > Morphic resonance is an empirical claim about how nature works. It's not a metaphysical interpretation. <

      The following is the entry for "morphic resonance" from the online "Skeptic's Dictionary" (based on a published book by the same name by Robert Todd Caroll).

      "In short, he [Rupert Sheldrake] prefers metaphysics to science, though he seems to think he can do the former but call it the latter. Perhaps it would be fairer to say that he sees no borderline between science and metaphysics."

      > In particular, Wikipedia states "critics [cite] a lack of evidence supporting the concept and its inconsistency with data from genetics and embryology". <

      A hypothesis only has to make a testable prediction in order to qualify as a scientific hypothesis. So, of course, it lacks evidence. It hasn't been tested and verified. The same holds true for the MUH.

      > Truth and logic, yes. Morality and beauty are relative in my view. <

      Then we agree that we presuppose some form of Platonism (beyond mathematical Platonism) in order to do philosophy. (Virtue ethics presupposes the perfect (ideal) Good, Massimo's objection to the contrary notwithstanding.)

      > One of us is confused. If God has free will, and if I have identified an inconsistency between free will and knowledge of the future, that means that God is incoherent. <

      You're confused. You previously argued that God's foreknowledge of the future precludes (human) free will. For the sake of argument, I granted you that. But I fail to see why you believe that strips God of free will. It doesn't. It simply means that God's "will" determines everything..

      > There is no greatest conceivable being for the same reason there is no greatest number. <

      If you can conceive of a greater being, then your present conception of God is less than adequate.

      > And, for the same reasons that infinity is not a number, an "infinite being" is not a being, it is a limit or ideal which can never be reached. <

      We clearly can intuit the infinite, even though we cannot fully grasp it.

      > Even infinity is not the greatest possible. There are always greater infinities. Greatest possible just doesn't work as a concept. <

      It does work. Cantor called it the "Absolute Infinite" (which he equated with God).

      > It will look indistinguishable from causality to observers within the structure. To me, that's what causality is. You say that's not true causality. OK. In that case, I disbelieve in your concept of "true causality" <

      You're simply redefining "causality" to be compatible with "acausality." By doing so, you render both terms meaningless. A computer program doesn't process any information unless it actually executes. That's how you get "causality."

      > Fundamentally, there is only the non-temporal domain (although the temporal domain also exists as a subset of this, and it isn't really temporal after all). <

      Then there really isn't any causality. And any "self-aware" structures are eternally self-aware.

      Delete
    5. Hi Alastair,

      >Because the wave function doesn't have any physical properties<

      Not according to the definition of physical property you linked to.

      >it is not located in space or time<

      It sort of is, in that the value of these functions have different values at different points in space and time. Alternatively, you could regard the function as a whole as something like a ubiquitous field.

      >it collapses into a particle upon measurement<

      Which seems to me like a physical interaction.

      >There's a difference (at least in Massimo's mind) between a "metaphysically informed scientific speculation" and a "scientifically informed metaphysical speculation."<

      Where do you get this from?

      >It hasn't been tested and verified. The same holds true for the MUH<

      Read what I posted again. There is evidence which contradicts it. It has been tested and found wanting, unlike the MUH.

      >Then we agree that we presuppose some form of Platonism (beyond mathematical Platonism) in order to do philosophy<

      I would say truth and logic are mathematical concepts, or can be construed as such. So no, I'm not sure I would presuppose Platonism beyond the mathematical.

      >You previously argued that God's foreknowledge of the future precludes (human) free will.<

      Actually, I said that it was inconsistent with the free will of either humans or God.

      >I fail to see why you believe that strips God of free will<

      Because if God can see what he will do, then he has no freedom to do otherwise.

      >If you can conceive of a greater being, then your present conception of God is less than adequate. <

      If you cannot, then your imagination is inadequate. Say tomorrow some evil metagod reveals himself to Yahweh and demands (and enforces) supplication. What about that scenario is incoherent?

      >It does work. Cantor called it the "Absolute Infinite" (which he equated with God).<

      Just because Cantor proposed it and equated it with God does not convince me that either concept works.

      >You're simply redefining "causality" to be compatible with "acausality."<

      Or you're just redefining "causality" so as to be incompatible with mathematical entailment. Either way, it's just a semantics question. If so, it would seem that I do deny that causality exists, according to your specific definition. I do believe in causality of a different kind, however. Either way, what I claim is true is consistent with what we observe.

      >And any "self-aware" structures are eternally self-aware.<

      Sure, but "eternally" is misleading when talking about things from an atemporal perspective, where time becomes like another spatial dimension.

      If we're not careful it could become a bit like saying that a sphere is smooth everywhere and inferring from that that the sphere exists at all points in space. Like the sphere, the self-aware structure does not exist at all coordinates in spacetime, but from an atemporal perspective it is eternal and unchanging at those places where it does exist.

      Delete
  41. @ Disagreeable Me

    This is part one of a two-part response.

    > Not according to the definition of physical property you linked to <

    It's not located in space and time. Therefore, it cannot possibly have any physical properties. (And it certainly doesn't have any physical properties that were listed in the Wikipedia article.)

    > It sort of is, in that the value of these functions have different values at different points in space and time. <

    It is not located in space and time.

    "1 According to Bohr, the concept of a physical state independent of the conditions of its experimental observation does not have a well-defined meaning. According to Heisenberg the wavefunction represents a probability, but not an objective reality itself in space and time." (source: Wikipedia: Interpretations of quantum mechanics: footnote 1 to the Copenhagen interpretation.)

    > Which seems to me like a physical interaction. <

    What is observed upon measurement is a physical particle, not a wave. Also, the standard (or "Copenhagen") interpretation "asserts that an observation PRODUCES the property observed." (source: pg. 100, "The Quantum Enigma" by by Bruce Rosenblum and Fred Kuttner) So, no physical properties exist independently of observation.

    > Where do you get this from? <

    I essentially asked Massimo what was the difference and he evaded the question by responding with a snide remark (that's his typical M.O.). You can read it for yourself here.

    ReplyDelete
  42. @ Disagreeable Me

    This is part two.

    > It has been tested and found wanting, unlike the MUH. <

    I stand corrected. It has been tested, but the test results were inconclusive. (Steven Rose (a skeptic) and Rupert Sheldrake disagree on the interpretation of the experimental results.)

    ".An Experimental Test of the Hypothesis of Formative Causation"

    > I would say truth and logic are mathematical concepts, or can be construed as such. <

    "Logic" is more encompassing than "mathematical logic. "

    > Because if God can see what he will do, then he has no freedom to do otherwise <

    Three points:

    I have already explained to you that God's existence is timeless. So, from God's perspective, there is no past or future, only the ever-present "now." That being said, God does know what is past or future relative to our perspective.

    Also, divine free will and determinism are compatible, because there is nothing external to God. IOW, God's "will" determines everything. So, self-determinism is completely compatible with determinism when the self in question is making all determinations.

    Finally, "determinism" does not necessarily imply that the act of creation was a "necessity." IOW, God's self-determinism (will) is compatible with a contingent creation. And if you really think that that's a show-stopper, then all I have to do to make my theology coherent is to make the creation a necessity. (In fact, process theology holds that the creation is a necessity.)

    > If you cannot, then your imagination is inadequate. <

    The problem is with your argument, not my imagination. You cannot conceive of anything greater than the absolute infinite. And if you could, then you were not conceiving of the absolute infinite to begin with. (IOW, your argument is inherently self-refuting.)

    > Just because Cantor proposed it and equated it with God does not convince me that either concept works. <

    I believe Cantor's "Absolute Infinite" is considered to be mathematically proven. (Whether or not one can prove that God and the Absolute Infinite are one and the same is another issue. At any rate, I suspect that Tegmark's "mathematical universe hypothesis" presupposes it.)

    > Either way, it's just a semantics question, <

    Just semantics? Semantics is important, because it is concerned with the meaning of words. And we must agree on the meaning of the words in order to have a fair debate. If you're tack is to redefine "causality" in order to furnish your metaphysical belief with a fatuous "causal" explanation, then there is no point to continue this debate. (It's easy to score points when you have the luxury of "moving the goal posts.")

    > Sure, but "eternally" is misleading when talking about things from an atemporal perspective, where time becomes like another spatial dimension. <

    If time is a subjective illusion, then this necessarily implies that subjectivity itself is eternal - eternal in the sense of being timeless.

    ReplyDelete
  43. Hi Alastair,

    A bit busy with Christmas, so I apologise for having been slow to get back to you.

    It's a bit unfortunate that we're covering so many topics in one chain of correspondence. It's hard to keep within the 4096 char limit. Hopefully there's one or two things we can close off.

    >It's not located in space or time<

    Arguably. It also could be considered to be everywhere, with probability values for different places and times, a bit like a physical field. That's a kind of location in space or time.

    >Therefore, it cannot possibly have any physical properties.<

    "A physical property is any property that is measurable whose value describes a state of a physical system."

    The probability function has such properties, i.e. amplitude at a certain location.

    Your points Heisenberg, Bohr and the Copenhagen Interpretation are interpretative, not authoritative. They don't settle the question.

    Then you assert that a particle is physical, but the wave is not.

    Thanks for that link to your exchange with Massimo. I had missed that. My reading of it would be that Massimo did not engage with the question out of irritation, not that he would have a problem with the characterisation of the MUH as either metaphysical speculation or interpretation.

    >"Logic" is more encompassing than "mathematical logic. "<

    Perhaps. But perhaps I only assume the existence of mathematical logic.

    >God's existence is timeless<

    Then he doesn't have a mind, because minds are processes. Yeah, you gave the example of an infinitely fast computer, but even that has a concept of "internal" pseudo-time. Certain computations happen before others. The concept of a truly timeless mind does not seem coherent to me.

    >And if you could, then you were not conceiving of the absolute infinite to begin with.<

    You conclude from this that my imagination is lacking. I conclude that the absolute infinite is inconceivable (because it is incoherent).

    >Just semantics?<

    Semantics are important. But when an argument boils down to semantics only, there is no metaphysical issue in dispute, only what terms we prefer to use. So while it's good to clarify semantics, if it ever becomes clear that a question has boiled down to semantics only, then the (metaphysical) dispute is over.

    It's not my tack to redefine causality. I'm not moving the goalposts. My goalposts just weren't where you thought they were if you don't accept my definition. I'm fine with you saying that I deny causality as long as you recognise that I have my own account of pseudo-causality which is compatible with what we observe.

    >If time is a subjective illusion, then this necessarily implies that subjectivity itself is eternal - eternal in the sense of being timeless.<

    OK?

    ReplyDelete
  44. @ Disagreeable Me

    > Arguably. It also could be considered to be everywhere, with probability values for different places and times, a bit like a physical field. That's a kind of location in space or time <

    All you're doing here is co-opting the theological concepts of transcendence and immanence in order to argue for the omnipresence of a nonphysical field of mathematical abstractions.

    > "A physical property is any property that is measurable whose value describes a state of a physical system."

    The probability function has such properties, i.e. amplitude at a certain location. <

    The object that has a measurable property here is the particle, not the probability function.

    > Your points Heisenberg, Bohr and the Copenhagen Interpretation are interpretative, not authoritative. They don't settle the question. <

    I never argued that the Copenhagen interpretation was not interpretative (such an argument would clearly be oxymoronic). What I did argue (and supported) was that the Copenhagen interpretation is the most widespread interpretation in the physics community. Bohr and Heisenberg, both of whom formulated the Copenhagen interpretation, clearly stated the wave function was not a physical object.

    Also, your position on this matter is clearly contradictory. It is contradictory to argue that the probability wave function is an actual physical object when you have already already gone on record and explicitly stated there is no physical stuff!

    > Massimo did not engage with the question out of irritate >

    People become annoyed whenever they are put on the spot and forced to defend that which is ultimately indefensible.

    > But perhaps I only assume the existence of mathematical logic. <

    It doesn't make a difference, because your response implies that all logic (so defined as 'mathematical logic') is required to do philosophy.

    > Yeah, you gave the example of an infinitely fast computer, but even that has a concept of "internal" pseudo-time. Certain computations happen before others. <

    Pseudo-time is not real time. So, the idea that a mind can think thoughts without taking any time to do so is coherent.

    > I conclude that the absolute infinite is inconceivable (because it is incoherent). <

    The "Absolute Infinite" is considered proven by Cantor. (Set theory is based on it. Also, Tegmark's MUH, according to Massimo, presupposes a NONPHYSICAL, actual infinity!)

    > Semantics are important. <

    Yes they are. And your counterargument is nothing more than a semantical game (a.k.a. as the fallacy of "moving the goalposts.")

    > I'm fine with you saying that I deny causality as long as you recognise that I have my own account of pseudo-causality which is compatible with what we observe. <

    Pseudo-causality is not real causality. As such, your metaphysics furnishes us with no causal explanation whatsoever.

    > OK? <

    So, if time is an illusion, then our consciousness (or, I should just say consciousness itself) is timeless and not a byproduct of some temporal, 'physical' process.

    Also, I agree with Lucas and Penrose, (and apparently, Godel himself) that "Godel's theorem seems to me to prove that mechanism is false, that minds cannot be explained as machines." - J.R. Lucas, "Philosophy" 36 112

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    Replies
    1. Hi Alastair,

      The amplitude is the property of the probability function, not the particle. Perhaps it cannot be measured directly but it can be inferred and predicted. I think that might be close enough to the definition of physical property for some people.

      > clearly stated the wave function was not a physical object<

      Even if that were so (and I'm not so sure it's as clear as you think), that doesn't settle the question. Another interpretation might consider the wave function to be physical as it is what the physical world is built of.

      >Also, your position on this matter is clearly contradictory<

      I can see why it might seem so.

      My position is that there is no physical stuff, and that the concept of an objectively physical universe is incoherent.

      However, it is also my position that wave-particle duality and other QM stuff is not sufficient to establish this, and indeed that these are different concerns altogether.

      I may agree with the some of the conclusions you seem to be drawing from QM, but I don't agree with the argument you're making.

      >So, the idea that a mind can think thoughts without taking any time to do so is coherent. <

      Agreed, but not that such a mind is omniscient. It cannot know what it will think next, because thoughts occur in a sequence, even if that sequence is infinitely fast.

      >The "Absolute Infinite" is considered proven by Cantor.<

      It's not clear from the Wikipedia article that this is accepted as a valid or useful concept by mathematicians in general.

      >Also, Tegmark's MUH, according to Massimo, presupposes a NONPHYSICAL, actual infinity!<

      An actual infinity is not the same as an absolute infinity.

      >And your counterargument is nothing more than a semantical game<

      It's really not.

      If I was insisting that causality exists, perhaps. But as I'm willing to admit that perhaps I don't believe in causality as you define it (actually I'm not sure about this as I'm unclear on your definition), I'm not moving the goalposts. We just have different interpretations of what "causality" means. I could just as easily accuse you of moving the goalposts, or even of No True Scotsman (since I have explained how what we view as cause and effect arises but you say this is No True Causality). Let's engage with the argument instead of accusing each other of logical fallacies.

      >Pseudo-causality is not real causality. As such, your metaphysics furnishes us with no causal explanation whatsoever.<

      OK, so even if that's the case, what's the problem? What required explanation is missing from the concept of (pseudo-)causality as arising from mathematical entailment? What I have described matches what we observe about the universe, and allows us to continue to understand the world in terms of cause and effect, at least from our perspective as temporal beings within the mathematical universe.

      >So, if time is an illusion, then our consciousness (or, I should just say consciousness itself) is timeless and not a byproduct of some temporal, 'physical' process.<

      OK?

      >Also, I agree with Lucas and Penrose, (and apparently, Godel himself)<

      I don't.

      Read this please: http://disagreeableme.blogspot.uk/2013/02/strong-ai-godel-problem.html

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    2. Hi Alastair,

      Sorry, the link to my argument on Lucas/Penrose/Godel was broken.

      http://disagreeableme.blogspot.co.uk/2013/02/strong-ai-godel-problem.html

      Delete
    3. @ Disagreeable Me

      This is part one of a two-part response.

      > I think that might be close enough to the definition of physical property for some people. <

      The Copenhagen interpretation says that it isn't. And that's the interpretation that is accepted by the majority of physicists.

      > Another interpretation might consider the wave function to be physical as it is what the physical world is built of. <

      But we're debating what the Copenhagen interpretation says about the issue, not what another interpretation might or might not say. (Perhaps I should refresh your memory. I presented Massimo with the argument that nature is fundamentally dualistic - the wave/particle duality. I also argued that this implies that nature has a physical aspect (the particle aspect) and a nonphysical aspect (the wave aspect). I cited numerous sources to support that claim. You took exception to my claim, saying that that was my interpretation of the "wave/particle" duality. I countered that by arguing that that is the Copenhagen interpretation, not my interpretation.)

      > My position is that there is no physical stuff, and that the concept of an objectively physical universe is incoherent. <

      Then it necessarily follows that you neither believe the wave is physical nor the particle. So, by arguing for the physicality of the wave, you are actually arguing against mathematical monism and undermining your own position. (IOW, you're engaging in an inherently self-defeating argument. It's bizarre!)

      > Agreed, but not that such a mind is omniscient. It cannot know what it will think next, because thoughts occur in a sequence, even if that sequence is infinitely fast. <

      I have explained this repeatedly to you. God's existence is timeless. Unfortunately, you keep placing God in time and presupposing that God has a past and a future. But God does not have any past or future. So, God does not have any past thoughts or future thoughts. In fact, properly speaking, God only has one thought, and it is happening in the ever-present "now." The whole of time (past and future) is, as it were, spread out before God. (Tegmark presupposes something analogous to this timeless state - a.k.a. the "block universe or block time" - for his MUH. So, I am failing to see why you can't grasp this.) From God's nontemporal perspective, God sees and responds to all temporal contingencies all-at-once - in one simultaneous act, in one simultaneous thought.

      > It's not clear from the Wikipedia article that this is accepted as a valid or useful concept by mathematicians in general. <

      "Classical set theory accepts the notion of actual, completed infinities." (source: Wikipedia: Actual infinity)

      > An actual infinity is not the same as an absolute infinity. <

      The Absolute Infinity is an actual infinity (or, more specifically, it is the actual "infinity of infinities").

      "The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type." - Georg Cantor (source: Wikipedia: Absolute Infinite)

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    4. @ Disagreeable Me

      This is part two.

      Anyway, all this is a digression. You originally argued (contrary to Anselm) that God cannot be that "being of whom no greater can be conceived," because you believe that you can always conceive of a greater being. And my response to that argument is that if you can conceive of some being greater than the Absolute Infinite, then you weren't conceiving of the Absolute Infinite to begin with. (It's impossible, by definition, to conceive of an infinity greater than the Absolute Infinite. Why? Because the Absolute Infinite is the "Infinity of infinities".)

      Also, you failed to address the point that I made in regards to Tegmark and his position on actual infinities. He considers all actual infinities to be NONPHYSICAL. So, he seems to be presupposing some kind of dualism - the duality between physical mathematical abstractions and nonphysical mathematical abstractions. This appears to be not only unintelligible, but also contradictory - contradictory to the "monism" in mathematical monism.

      > I'm not moving the goalposts. We just have different interpretations of what "causality" means. <

      You originally claimed that your mathematical universe was uncaused. So, I countered by arguing that if you are completely dispensing with causality, then all you have are correlations and observations without any causal explanation. And since the universe is clearly evolving and undergoing change, then it requires a causal explanation. However, your metaphysical theory offers none; therefore, it has no explanatory power whatsoever. After digesting my argument, you panicked and immediately started to engage in spin-doctoring by "moving the goalposts." The rest is on record and I don't think there is any need for me to rehash all the tedious details.

      > OK, so even if that's the case, what's the problem? <

      The problem is that pseudo-causality is not real causality by definition. You're making a spurious argument that explains nothing. Clearly we are experiencing a world of change. And if your metaphysical theory cannot explain that, then it has no explanatory power whatsoever.

      > OK? <

      So, if consciousness is not a byproduct of some temporal 'physical' (actually, in your case,"informational" rather than "physical") process, then the "computational theory of mind" is patently incorrect.

      > I don't. <

      Godel's theorem demonstrates that "there are true statements in any theory which a computer can never prove, but which we can see are true. It appears that conscious minds can learn things about any logical system that a computer, following the rules of the system, can never discover." - Euan Squires, pg 150, "Conscious Mind in the Physical World"

      Delete
    5. Hi Alastair,

      I too think we have lost track of what we're talking about with the different interpretations of QM and wave/particle duality.

      My point was, and continues to be, that this duality does not establish that there is a physical and an abstract component to reality. Those who believe reality is physical can simply define "physical" so as to describe whatever reality is fundamentally made of, whether that be waves or particles or probability functions or whatever.

      This is not a superficial semantic moving of the goalposts, as the physical/abstract distinction is still maintained. There remains a difference between (pseudo-physical?) mathematical functions that describe aspects of the physical universe and those (abstract)functions that do not.

      >Then it necessarily follows that you neither believe the wave is physical nor the particle<

      Kind of, at least in an ultimate, objective sense. There is no such thing as an objectively physical universe. But it might make sense to describe the particle or the wave as physical from the point of view of an observer in the universe. "Physical" in this sense means "present in my universe".

      >I have explained this repeatedly to you.<

      I feel the same way. This is indeed frustrating.

      God can be timeless, and God can have a mind, but God cannot have a mind that perceives itself and the universe timelessly. The idea of a mind that operates entirely outside of time is nonsensical, as the only coherent concept of a mind that has ever been presented is that of mind as a process, and processes need something analogous to time to function.

      >"Classical set theory accepts the notion of actual, completed infinities." <

      I didn't deny actual infinities. I denied the absolute infinity.

      >The Absolute Infinity is an actual infinity (or, more specifically, it is the actual "infinity of infinities").<

      Maybe the hypothetical absolute infinity would be an actual infinity if it existed. My claim is that the absolute infinity (the infinity of infinities), like the set of all sets in Zermelo–Fraenkel set theory, is not a coherent concept.

      (to be continued)

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    6. On whether a greatest being is conceivable, you didn't answer my point. If I can conceive of a being greater than God, that has two interpretations. Yours is that I have not correctly conceived of God. Mine is that a greatest being is actually inconceivable. Why should I prefer your interpretation to mine?

      Suppose I ask you to conceive of the greatest possible number. Now I ask you to raise that number to the power of itself. This number must now be greater than the original number. So, does this mean that I didn't correctly conceive of the greatest number, or does it mean that a greatest number is inconceivable? In this case it's clearly the latter. I see the concept of God the same way.

      >So, he seems to be presupposing some kind of dualism<

      Not really. He's just doubtful that this particular universe contains any actual infinities. He also has doubts about whether there can be uncomputable universes (which infinities might lead to), a view I don't share.

      >And since the universe is clearly evolving and undergoing change, then it requires a causal explanation.<

      Or a pseudo-causal one which is consistent with and explains our observations.

      Which I have given, but you have not understood. On the B theory of time, the universe is not changing, but we perceive it to be changing as we move through time. Mathematical entailment, as we see in cellular automata, is perfectly consistent with what we perceive as cause and effect, even though you reject it as true causation.

      >So, if consciousness is not a byproduct of some temporal 'physical' (actually, in your case,"informational" rather than "physical") process, then the "computational theory of mind" is patently incorrect.<

      I don't see why. If time is an illusion, nothing is ultimately temporal, including brains and computers. I see no conflict with the computational theory of mind.

      >Godel's theorem demonstrates that...<

      Alastair, I've already refuted the Lucas/Penrose argument. Again, please read the blog before mentioning Godel to me again. There's really no point in discussing it until you've read my take on it.

      Here's the link for you again:
      http://disagreeableme.blogspot.co.uk/2013/02/strong-ai-godel-problem.html

      Delete
  45. @ Disagreeable Me

    > I too think we have lost track of what we're talking about with the different interpretations of QM and wave/particle duality. <

    Correction. You lost track; I did not.

    > My point was, and continues to be, that this duality does not establish that there is a physical and an abstract component to reality. <

    My argument is that the Copenhagen interpretation holds that the probability wave function is not an actual physical object. I have furnished you with numerous sources to support that claim. You have furnished me with nothing to counter that claim.

    > This is not a superficial semantic moving of the goalposts <

    That's exactly what it is.

    > Kind of, at least in an ultimate, objective sense <

    Metaphysics is concerned with ultimate reality. And since we are having a metaphysical debate, that's is what we are concerned with. Clearly, mathematical abstractions are nonphysical. Therefore, mathematical monism qualifies as a (twisted) form of "immaterialism."

    > The idea of a mind that operates entirely outside of time is nonsensical, as the only coherent concept of a mind that has ever been presented is that of mind as a process, and processes need something analogous to time to function. <

    You have already conceded in a previous post that information can be timelessly processed if given infinite computing power.

    > Maybe the hypothetical absolute infinity would be an actual infinity if it existed. My claim is that the absolute infinity (the infinity of infinities), like the set of all sets in Zermelo–Fraenkel set theory, is not a coherent concept <

    It's easy to demonstrate that one infinity can contain multiple infinities. For example, the set of real numbers, the set of natural numbers, the set of even numbers, and the set of odd numbers - all four sets - are infinite. Nevertheless, the set of real numbers contains the set of natural numbers, the set of even numbers, and the set of odd numbers. So, with this simple example, I have demonstrated that an "infinity of infinities" is not only a coherent concept, but also a mathematical fact.

    > Yours is that I have not correctly conceived of God. Mine is that a greatest being is actually inconceivable <

    Paradoxically, it's actually both. In one sense, our finite minds cannot conceive of the infinite. Yet, in another sense, they can. That's why the term "infinite" has meaning for most of us who can intuitively embrace this paradox. It's exactly the same thing with "God."

    "God is that being of whom no greater can be conceived." - St. Anselm

    > On the B theory of time, the universe is not changing, but we perceive it to be changing as we move through time. <

    I have already addressed this. The "A-series," "B-series," and "C-series" are terms coined by .J.M.E. McTaggart in "The Unreality of Time" in order to argue for his pantheistic idealism, not mathematical monism. (If time is a subjective illusion, then something still has to be causing the illusion. In idealism, that something is "mind" or "consciousness," not some ridiculous mathematical "pseudo-cause.")

    > I don't see why. If time is an illusion, nothing is ultimately temporal, including brains and computers. I see no conflict with the computational theory of mind. <

    The computational theory of mind requires that information be processed. According to you, time is required for any kind of processing to take place. So, if time is illusory, then there can be no information processing.

    > I've already refuted the Lucas/Penrose argument. <

    Correction. You have deluded yourself into believing that you have.

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    1. @Alastair

      I'm not sure I'm really enjoying this conversation any more.

      I would like to make the general observation that you seem to have very rigid definitions of certain concepts such as "physical" and "causality", and when others take different interpretations you find citations which you think bolsters your claim but really are not that persuasive in light of the differing or more flexible interpretations of your interlocutors.

      For example, you take as definitive that the universe is not entirely physical because modern physics thinks of reality of consisting of functions. I'm just telling you that many people will not find this argument persuasive. They will adjust the definition of physical to accomodate these findings rather than concluding that the universe is not physical. I think this is reasonable as long as we understand that what physical must mean, ultimately, is that which the universe consists of. If you think that this is moving the goalposts, that is your prerogative, but I'm just telling you now that you won't persuade anyone by quoting physicists about wave-particle duality because you are ignoring that difference of interpretation.

      Your answer to my point about infinity was misguided. I never denied that an infinity can contain infinities. I denied that it could contain all infinities, just as a set cannot contain all sets in ZF set theory (that does not mean that a set cannot contain sets).

      I've probably had enough of going in circles explaining why I think God is incoherent. You asked, I answered. I remain unconvinced, although I will admit that I am not *certain* that your God is incoherent. It just seems rather so to me.

      If you still can't see how my claim about causality makes sense, then I'm lost. It's really simple. You're tying me in knots with semantic games, but it's really obvious to see what I mean if you just imagine it for a second. Imagine a simple computer program that simulates a ball bouncing. The entire future of that bouncing ball is defined by the source code. The program is a mathematical object. As it sits on the computer, the program itself is not changing, but change does exist within the scope of the simulation - the ball does move after all. Time exists as a concept only within the simulation - the program itself exists eternally and changelessly as an abstract object. I am suggesting the universe is like this. Honestly, how can you not get this?

      If you think I'm deluded about Lucas/Penrose then kindly tell me where I went wrong, otherwise don't mention Lucas/Penrose or Godel to me again.

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    2. @ Disagreeable Me

      > I'm not sure I'm really enjoying this conversation any more. <

      That probably explains why you have dispensed with the pleasantries.

      > I would like to make the general observation that you seem to have very rigid definitions of certain concepts such as "physical" and "causality", and when others take different interpretations you find citations which you think bolsters your claim but really are not that persuasive in light of the differing or more flexible interpretations of your interlocutors <

      You have already gone on record in this particular thread and stated explicitly that both physicality and causality are illusory. You changed your tune afterwards only because you felt it was necessary to muster some kind of counterargument to mine. (If you completely dispense with causality, then all you have are correlations and observations without any causal explanation. And if your metaphysical theory has no causal explanation, then it explains nothing whatsoever.)

      > Your answer to my point about infinity was misguided. I never denied that an infinity can contain infinities. I denied that it could contain all infinities <

      My point about infinity was spot on. You previously argued that an "infinity of infinities" is an incoherent concept. I clearly demonstrated to you that it is not only a coherent concept, but a mathematical fact.

      > You asked, I answered. I remain unconvinced, although I will admit that I am not *certain* that your God is incoherent. It just seems rather so to me. <

      I asked, you argued, and I promptly responded. You have completely failed to expose any incoherent aspects with my concept of God.

      > You're tying me in knots with semantic games. <

      Puhlease! The only one here playing semantic games is you.

      > The program is a mathematical object. As it sits on the computer, the program itself is not changing, but change does exist within the scope of the simulation - the ball does move after all. <

      Information processing is a dynamic process, not a static structure.

      > Time exists as a concept only within the simulation <

      By your own admission, there can be no process (simulation) without time.

      > Honestly, how can you not get this? <

      The mathematical universe hypothesis is a ridiculous hypothesis for reasons already stated.

      > If you think I'm deluded about Lucas/Penrose then kindly tell me where I went wrong, otherwise don't mention Lucas/Penrose or Godel to me again. <

      The onus is upon you (not me) to refute Squire's argument. I have already stated it in a previous post. It was a very succinct argument. No counterargument was forthcoming. (If you can't furnish me with a succinct counterargument, then you really don't have one.)

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  46. > I clearly demonstrated to you that it is not only a coherent concept, but a mathematical fact.<

    No you didn't. You demonstrated that an infinity can contain infinities. You manifestly did not demonstrate that the absolute infinity is coherent. Your argument is just like an argument for the set of all sets by giving examples of sets which contain sets.

    >Information processing is a dynamic process, not a static structure.<

    Yes, a running computer process which processes information is a dynamic process, e.g. a process which renders the Mandelbrot Set to a computer screen. What it actually ends up computing is a static mathematical structure, e.g. the Mandelbrot Set itself. The program itself (as opposed to the process which runs the program) is also a static mathematical structure.

    >The onus is upon you (not me) to refute Squire's argument. I have already stated it in a previous post. It was a very succinct argument. No counterargument was forthcoming.<

    It is very disingenuous to say that no counterargument was forthcoming. Once again, read this: http://disagreeableme.blogspot.co.uk/2013/02/strong-ai-godel-problem.html

    >(If you can't furnish me with a succinct counterargument, then you really don't have one.)<

    Yeah, but then you're going to respond to that, and my answer to your response is probably already covered on my blog. If you want to discuss it then my blog is a better place to do it because it's of minimal relevance to this post.

    But if you want a succinct argument, here goes:

    Squires:
    >there are true statements in any theory which a computer can never prove, but which we can see are true. It appears that conscious minds can learn things about any logical system that a computer, following the rules of the system, can never discover.<

    This argument is based on a gross oversimplification and overinterpretation of the incompleteness theorems. This concise, succint argument is riddled with problems.

    Firstly, it has not been established that there are true statements which no computer can ever prove. It has been established that for any specific computer, there are true statements that specific computer can never prove. This is actually quite different, because for any statement unprovable by computer program A, there may exist a computer program B which can prove it.

    However, it is actually quite possible that there are statements that no computer can prove, but then it is just as possible that these statemetns cannot be proven by any human mathematician. There are plenty of unproven conjectures in mathematics which seem to be true but have never been proven. There is no reason to suppose they ever will.

    It's also not true to say that there are true but unprovable statements in *any* theory. There may be theories too complex for any human to find those Godel sentences, such as those that correspond to the algorithms of their own brains.

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    1. >It's also not true to say that there are true but unprovable statements in *any* theory. <

      Sorry, what I meant to say here was that it is not true that there are sentences in all systems which humans can see are true but cannot prove within the system. Some systems may be too complex for humans to construct such Godel sentences.

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    2. @ Disagreeable Me

      > You demonstrated that an infinity can contain infinities. You manifestly did not demonstrate that the absolute infinity is coherent <

      The Absolute Infinity is an "infinity of infinities." And you previously argued that an "infinity of infinities" is not a coherent concept. I responded to that argument by clearly demonstrating that it was not only a coherent concept, but also a mathematical fact.

      > Your argument is just like an argument for the set of all sets by giving examples of sets which contain sets. <

      You appear to be mistaken. Russell's paradox (I believe that is what you are referring to) never invalidated Cantor's naive set theory. (It was Frege's naive set theory, not Cantor's.)

      "Cantor was aware of some of the paradoxes and did not believe that they discredited his theory. Gottlob Frege explicitly axiomatized a theory in which the formalized version of naive set theory can be interpreted, and it is this formal theory which Bertrand Russell actually addressed when he presented his paradox." (source: Wikipedia: Naive set theory)

      > Yes, a running computer process which processes information is a dynamic process, e.g. a process which renders the Mandelbrot Set to a computer screen. What it actually ends up computing is a static mathematical structure, e.g. the Mandelbrot Set itself. The program itself (as opposed to the process which runs the program) is also a static mathematical structure. <

      The bottom line is that information processing is a dynamic process, not a static structure. And there is nothing in your above response that would suggest otherwise. (That a computer program and the input information it processes may be static does not change the fact that the information processing itself is not.)

      > It has been established that for any specific computer, there are true statements that specific computer can never prove. This is actually quite different, because for any statement unprovable by computer program A, there may exist a computer program B which can prove it. <

      The bottom line is that human intelligence will always be able to take one step more beyond any computer program that has ever been written.

      > It's also not true to say that there are true but unprovable statements in *any* theory. <

      It is according to Godel.

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    3. Hi Alastair,

      >The Absolute Infinity is an "infinity of infinities."<

      The absolute infinity is an infinity greater than all other infinities. (I think it's clearer to talk in terms of greater than or less than. I'm not really sure it's clear what an infinity of infinities means.)

      You gave examples of an infinity (the cardinality of the set of real numbers) which is greater than certain other infinities (e.g. the cardinality of the set of natural numbers). Yet there are infinities which are greater than the cardinality of the set of real numbers. You have not established that there is a coherent concept of an absolute infinity than which there is no greater infinity.

      > Russell's paradox (I believe that is what you are referring to) never invalidated Cantor's naive set theory.<

      I didn't say it did. I said the set of all sets was invalid in Zermelo-Fraenkel set theory, and that the argument you had presented for the infinity of infinities was analogous to an argument for the set of all sets. Since your argument wasn't specific to any specific set theory, the conclusion you are drawing is questionable at best.

      However, if it's true that Cantor's set theory is not axiomatised then it's impossible to say what does or does not work in it because it's vague. Nothing can be proven either way without axiomatisation.

      >The bottom line is that information processing is a dynamic process, not a static structure.<

      Agreed. So what? I'm arguing that the universe is a mathematical structure not a running computer process. A computer process allows us to explore certain mathematical structures. It is not the structure itself. If you want to explore causality under the MUH, I have already posted a new start to that conversation I hope you will pick up.

      >The bottom line is that human intelligence will always be able to take one step more beyond any computer program that has ever been written.<

      So what? We're agreed that no human-intelligent computer program has ever been written.

      If you mean that humans can in principle take one step further than any conceivable computer program, you're dead wrong. Godel doesn't prove that at all. If there was a straightforward way to do that trick, you could write a computer program to do it. If there isn't a straightforward way to do it (and there isn't), eventually it becomes too difficult as the program becomes too complex until you get to a point where no human can take that step. Godel only proves that it's possible to do it in principle, not that a human can do it in practice.

      >It is according to Godel.<

      No it isn't. Godel's proofs do not apply to all theories, but only to theories of a certain minimum power. But that's nitpicking. Unfortunately you seem to have entirely ignored the correction I posted shortly after this:

      "what I meant to say here was that it is not true that there are sentences in all systems which humans can see are true but cannot prove within the system."

      Delete
  47. Hi Alastair,

    I think the issue of causality in the MUH is something that deserves it's own thread. I want to explain how I account for what appears to be causality and I want to understand your objections to it.

    Firstly, some questions.

    1. Are you familiar with Conway's Game of Life? If not can you read about it a bit?

    2. I want to adopt the universe of the GoL as a toy model for a mathematical universe. Can we for the sake of argument pretend that this universe contains intelligent observers?

    Can you perhaps outline your initial thoughts on whether this universe has causality or not, and compare the causality in this universe with that of ours? How do we know we are not in such a universe?

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  48. I like Tegmark's simple idea that reality is a mathematical structure, but it seems inconceivable how our physical world with material objects might emerge from (or within) this mathematical structure. I wonder if Godel's incompleteness theorem may actually be the bridge between purely mathematical structures and physical structures, or even consciousness.

    I am no expert on Godel's theorem but from what I've seen it says that within a mathematical structure of sufficient complexity (being able to incorporate arithmetic or something like that) necessarily exist entities or truths that cannot be derived from this structure. So these strange entities are actually new axioms that expand what seemed like a complete structure, and it turns out that the structure is bigger than it seemed.

    So how do you derive physics from arithmetic, or consciousness from arithmetic? The answer might be that you can't. And yet arithmetic may give rise to both physical objects and consciousness - in a Godelian fashion.

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    1. Hi Tomas,

      Personally, I don't think there is anything inconceivable about how the physical world or consciousness could emerge from a mathematical structure, even without bearing Godel in mind.

      I would be very wary about invoking Godel as an explanation because there is a general tendency to explain consciousness by a vague appeal to whatever the mystery of the day might be. For example, there are many people who think consciousness must have something to do with quantum mechanics simply because both are weird and defy intuitive explanations.

      Your appeal to Godel seems to be of this nature to me. I honestly don't see how you can get from the idea that there are true unprovable statements to an explanation of physicality and consciousness, unless by some form of very loose analogy.

      I would resist your characterisation of true unprovable statements as axioms. I think an axiom is a fundamental statement of a mathematical system that is chosen somewhat independently of other axioms, whereas the true unprovable Godel statements are actually derived (indirectly) from the other axioms.

      (Although, admittedly I have said in comments elsewhere that we can adopt these Godel statements as axioms for new systems).

      I'm not sure it's fair to say that the original system is bigger than it seemed, as the full scope of the derivable statements of any system is generally unknown. We start with axioms and then the mathematicians see what they can make of them. Godel only shows that what is true is a larger set than what is provable. I'm not sure that is really of any great significance.

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    2. To clarify my last sentence:
      > I'm not sure that is really of any great significance... <
      ... to any but mathematicians and philosophers of mathematics.

      Delete
    3. Hi DM,

      "I honestly don't see how you can get from the idea that there are true unprovable statements to an explanation of physicality and consciousness, unless by some form of very loose analogy."

      Yes, I don't know how to add much clarity to this idea, also due to the fact that I don't really understand the details of Godel's theorem. It just seems to me intuitively that if you can get an underivable object (a true unprovable statement) from a mathematical structure then perhaps this object might have a quality that seems more than just mathematical, simply because it cannot be derived from the mathematics.

      Consciousness seems to be a similar example of this phenomenon: it is believed to be resulting from a pattern of neuronal firings but is notoriously resistant to derivation from this pattern. You arrange neurons in a certain way, let them fire in some sequence or accord, and bang! - red color appears. In this case, the Godelian transition would be from a physical structure to a consciousness structure.

      Delete
    4. Hi Tomas,

      I think it's not really correct to call the Godel statement more than just mathematical. It really is mathematical, in syntax and in semantics and even in derivation. It is derived mathematically, just not within the mathematical operations allowed within a specific system. I really think it's a stretch to think it is metaphysically somehow more.

      Even without Godel sentences, there are plenty of perfectly ordinary mathematical conjectures which may well be true but have resisted attempts to prove them (The Goldbach Conjecture is my go-to example). Some of these likely are both true and impossible to prove.

      That does not, to my intuition, imply that they are in any way mysterious or transcendent. Your mileage may vary!

      Delete
  49. "There are other problems with MUH. For one, several critics of Tegmark’s ideas have pointed out that they run afoul of the seemingly omnipresent (and much misunderstood) Gödel’s incompleteness theorems."

    Are you saying there is conceptual evidence against MUH? Or are you only saying that it is premature to claim that MUH is at all probable rather than theoretically possible?

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  50. I’m not sure why Max Tegmark felt it was necessary to jump from positing a reality either controlled in a computational sense by mathematics, or described by mathematics, to a reality which is mathematics. I’ve read his book now after enjoying the podcast. I’ve found many useful ideas and concepts explained in the book, such as the importance and predictive power of inflation and the implications of non-collapse of the wavefunction, and I really don’t have a problem with most of this material – it’s very good.

    However I am mystified about what evidence is driving the apparent position that reality is mathematics. It’s clear enough that the most basic properties of particles of strings can be described as being equivalent to a mathematical description, but can’t see why this would mean that these mathematical functions would not be being played out, or realised, in the form of operations applied to a substrate.

    For me, the nub of the problem is this: there is no evidence that mathematics can exist in the absence of a substrate.

    I’m a chemist. I know that physicists and mathematicians are used to distinguishing between abstract and applied mathematics, so at first this idea itself may sound surprising. But when you think about it, there can be no such thing as completely pure or abstract mathematics, because no mathematical operations can be enacted outside a physical framework, or at least there is no evidence that this could happen.

    Dubious? How do you calculate 2 x 2?

    It’s not abstract at all: you need a consciousness housed inside a brain to calculate 2 x 2. Energy is expended.

    There’s no evidence that 2 x 2 computations are being carried out in some non-physical non-reality which somehow gives rise to a physical reality, as a consequence of their being carried out nowhere and notime in particular.

    I’m not saying it might not be happening – I’m just saying that there is no evidence that mathematics independent of substrate could happen, and no examples exist that I know of, and I might not understand Max Tegmark’s arguments but I don’t think he provides any. A second related problem from the brain analogy is that realization of mathematical operations should also require energy, or motive power, whatever this means and wherever it comes from...

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  51. ...I’m writing this after the recent groundbreaking discovery of the imprint of gravitational waves in the cosmic background radiation – another reminder that our space-time itself is a substrate. Okay spacetime itself might be made of mathematical operations, but why would those operations not be on some other, more fundamental, substrate?

    The jump from reality acts in a mathematical way to reality is math seems to me to be the same type of leap as moving from mind as embodied and realized in a brain, and a soul or spirit that is somehow independent of the brain. Again, it’s possible, but the evidence so far isn’t very good. Even if that were true, we could posit that the wavefunction of any disincarnate soul would probably need to exist within some other undiscovered substrate, rather than being substrate-free.

    I wonder whether Max Tegmark would mind admitting back a possibility that in our observable Universe, the likely Type I and II Multiverses, the postulated Type III Multiverse, and possible Type IV Multiverses, mathematical operations might always require substrata, and probably motive power, for their realization? Why not entertain the idea of math plus substrata, and see where that leads?

    An implication of this line of thought (positing math + substrate + energy which I think might equal computation) might be that the nature of the substrate determines which mathematical operations would be allowed, and which are forbidden (in a quantum mechanical sense).

    Existence of a single unified substrate could therefore rule out many types of Type III or Type IV Multiverses. Alternatively, substrata might vary between Universes, with the mathematical operations playing out in each being limited by constraints imposed by the substrate or substrata within each. This would also rule out many types of Type III or Type IV Multiverses, but we’d never know which ones.

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    1. I think the conversation here is over, and this website is somewhat retired.

      But I can invite you to read the following for a discussion of the motivation for the MUH.

      http://disagreeableme.blogspot.co.uk/2013/12/the-universe-is-made-of-mathematics.html

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  52. Thanks DM

    Thank you for setting out your arguments there. From your blog post I see that the hypothesis rests on this statement as one of three "crucial premises":

    All mathematical objects exist abstractly and independently of minds

    ...and what I'm saying is that there is no evidence that mathematics, or mathematical systems or patterns, can ever exist in the absence of a physical framework. There is simply no evidence for this, whether the math is being done in a brain, on paper, in a computer, or anywhere else. It would appear that the enactment of any mathematical operation presupposes the existence a medium in which the operation is carried out.

    I think it is the idea that abstract mathematics is inherently so pure and unsullied (so abstract) that it is able to exist independently (of anything) where this theory seems to have gone off the rails.

    To my mind all this does is replace the God concept with Abstract Mathematics. Is this really an improvement?

    - Nick


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    1. Hi Nick,

      Mathematical Platonism is quite a popular view among philosophers and mathematicians, and perhaps physicists too. There are many who doubt it but it is far from a fringe view.

      Granted, I think it would have been good for Tegmark to make the dependency of the MUH on mathematical Platonism more clear. He seems to take Platonism as a given without giving much thought to defending it.

      That said, I do think mathematical Platonism is more defensible than the alternative. I have some thoughts on this on my blog also.

      http://disagreeableme.blogspot.co.uk/2013/10/mathematical-platonism-is-true-because.html

      This doesn't specifically address your argument, but if I could respond very quickly to your point I would say that it is a mistake to think of calculations or computations taking place to breath life into the universe. The mathematics defines (or is) the universe, there is no computational process which sustains it. Processes require time and there is no time outside the universe.

      If you would be interested in a more detailed discussion then you can reach me via my blog.

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  53. I suspect that Mathematical Platonism (MP), as a philosophical position about a “true” aspect of reality, is area that sits outside the empirical reach of science. It's adjacent. However at the same time I think that pairing Max Tegmark’s ideas with MP would be an unjustified conflation.

    I think the MP position, for example that “there exists” an infinite number of prime numbers regardless of whether we know about them or not, or that mathematical entities are not constituents of the spatio-temporal realm, is not the same as a position that abstract mathematical structures are a self-creating engine of creation.

    The first is passive (“prime numbers exist even if we don’t”), whereas the second is active (“the prime numbers got together one day somewhere outside of time and space and decided to create a Universe”). So I don’t see how Max Tegmark’s ideas are the same thing as MP – maybe they do start there but they quickly leave it in the dust.

    “Abstract Math as Creator” implies that mathematical machinery is operating in a manner whereby things including spacetime are created, whether or not this engine of creation is envisaged as occurring inside or outside of time and space. This is equivalent to the concept of God creating everything from outside time and space. A new label for an old idea?

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    1. Hi Nick,

      I would prefer to have this conversation via another forum because I feel that Massimo has better things to do with his time than to have to approve moderation on comments between us that nobody else is likely to be reading, especially now that he has moved on from Rationally Speaking and onto Scientia Salon.

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