I recently attended a talk by Daniel Garber (Princeton University) on the topic of “God, Laws and the Order of Nature in the Scientific Revolution.” While Garber’s talk was mostly historical in nature, it raised some interesting points about why and how we talk about laws of nature at all. And the connection was reinforced just a couple of days ago when I went to the New York Film Festival and saw a screening of “Particle Fever,” a documentary about the Higgs boson during which the concept of the fundamental (lawful, according to supporters of supersymmetry; random, according to people who favor the multiverse) architecture of the universe was the truly big question lurking in the background.

The point of departure of Garber’s talk was that the very concept of laws of nature is actually pretty recent, and it basically starts with Rene Descartes. Indeed, both Galileo and Hobbes [1] — who were aware of Descartes’ work — were definitely unenthusiastic about the idea.

Descartes introduced the idea of law of nature in his Le Monde and in the Principles of Philosophy, explicitly using that term. The concept did have precursors, of course, for instance in the Stoics, but Descartes gave arguments in its favor (as opposed to simply assume it, as the Stoics did) and explored it in quasi-mathematical terms (which the Stoics certainly didn’t). One of Descartes’ examples was the principle of inertia, which was again taken up and elaborated upon by Galileo, and eventually Newton (becoming incorporated in the latter’s principles of general mechanics).

Descartes, however, was a figure perched in equilibrium between the old Scholastic view of the world and the soon ongoing scientific revolution, so it shouldn’t exactly be surprising that he used the classical theological doctrine that the world is kept going by a continuous action of God as grounding for his concept of laws of nature. According to Descartes, laws of nature are not chosen by God and then imposed on the world (contra Leibniz, for instance). Which, interestingly, means one cannot use the study of nature to derive conclusions about God’s plans (so much for natural theology). God, in a sense, becomes the efficient cause of all things, to use the Aristotelian classification of causes.

For Descartes, God doesn’t think of the regularities of nature in terms of laws, but that’s the way we conceptualize how God sustains the world from moment to moment. Descartes thought that the world was created in chaos, and it developed into its current structure by way of the laws of nature. This, by the way, includes the development of plants and animals, which could be reasonably interpreted as an early picture of evolution. The influence of the Greek atomists (via Lucretius’ poem, De Rerum Naturae, which had been recently rediscovered) on Descartes’ thinking should be obvious.

As I said above, Garber stressed that Galileo wasn’t too happy with the Cartesian picture. Galileo’s main concern was the behavior of bodies in motion (yes, yes, he was distracted for a couple of decades by telescopic observations and the issue of Copernicanism, but motion apparently truly was his main interest), particularly the issue of the best definition of accelerated motion, from which he derived his framework for a mathematical description of naturally falling bodies.

Galileo’s findings, however, cannot reasonably be considered examples of laws, because they were not meant to be sufficiently general by their author. For instance, Galileo was not sure that his description of motion on earth could generalize to motion as it occurs on other celestial bodies (though he did have a theory of how planets behaved with respect to the Sun). Indeed, we had to wait for Newton to get a complete theory of general mechanics. It is this partiality of Galileo’s account that in turn generated Descartes’ skepticism about the former’s theory of motion. Galileo’s work was just not general enough (i.e., universal) for Descartes’ taste.

What about Hobbes? Here too we don’t find either the term or the concept of law of nature. But we know that Hobbes was aware of Descartes’ ideas on the matter, so his avoidance of the term was certainly deliberate. (It is worth remembering that Hobbes very likely was an atheist, and certainly thought that God doesn’t play any role in a natural philosophical account of the world.)

Hobbes didn’t recognize the existence of any conservation principle, unlike Descartes. But again, Descartes' conservation of physical quantities was nothing like what we think of today, as it had a direct link to God, without whom there would be no reason for things not to change (presumably, at random, thus falling apart).

Hobbes was also a great admirer of Galileo, whom he considered the first physicist. In fact, Hobbes thought that everything reduces to motion of bodies, and he did think of motion in universal terms, though there is no reason to believe that he saw his statements about motion any differently from how Galileo saw his — i.e., not as expressing “laws.” Interestingly, Hobbes (unlike Descartes) didn’t see any crucial distinction between real physical bodies and geometrical idealizations: i.e., the regularities of motion for him were analogous to mathematical generalization. It therefore follows that universals in physics are analogous to universals in mathematics, and since mathematical statements for Hobbes are not laws of nature... (mathematical structural realism, anyone?)

Jumping to modern times, there is a serious question about whether physicists actually got rid of some of the unsavory implications of the concept of laws of nature that apparently left both Galileo and Hobbes somewhat cold. While I’m sure that physicists like Steven Weinberg would vehemently reject such teleological (if not downright theological) implications, philosopher Nancy Cartwright provocatively wrote that there is no sensible way to avoid the connection between laws and some kind of law maker, so that talk of law should be rejected in favor of a more Galileian (Garber would say) approach based on generalizable (but not necessarily exceptionless!) regularities. This criticism isn’t limited to philosophers, as it has been explicitly endorsed by physicist Lee Smolin, for instance. Indeed, discussions on the philosophy and conceptual science of natural laws have now generated a sizable technical literature.

Which brings me to the fundamental debate underlying the Particle Fever movie mentioned at the onset. To make a long (and complex) story short, before the Hadron Collider at CERN actually discovered the Higgs there were two rough predictions of how massive the new particle would be: either around 115 GeV (Giga electronVolts) or around 140 GeV. The difference is crucial, because supersymmetric theories predict the lower amount, while versions of the multiverse theory hover around the higher number. Since the Higgs is the particle that “gives” mass to all other particles, its characteristics are crucial to explain why the universe is the way it is.

Well, what do you know? The current best estimate from the Hadron Collider experiments (which will resume in a bit more than a year at higher energy levels) is around 125 GeV, a bit too high for supersymmetry, a bit too low for the multiverse. (And, moreover, right in the instability range for the Higgs itself: and if the glue that holds together the universe is unstable, it may one day go. And if

*it*goes, well...)

How does this fit with the issue of natural laws (or unexceptional regularities, or whatever you want to call them)? A big deal in Particle Fever is made of the fact that if the Higgs is confirmed to be outside of the reasonable range of supersymmetric theories, thus providing “strong circumstantial evidence” (though not proof) for the multiverse, to put it as one of the interviewed physicists did, then this would in some sense be “the end of physics” (as we know it). Why? Because if the multiverse scenario is true then

*there is no answer*to why the “laws” of our universe (and its physical constants) are the way they are: it’s just one of an infinite number of possible random outcomes. Which would mean that, in a deep sense, there are no “laws of nature,” just regularities that happen to hold in our particular universe.

If, however, the Hadron data will eventually bring back the Higgs firmly into supersymmetric territory, then we will be able to pursue further questions about why the universe is structured the way it is and in no other way, thereby reopening the issue of what it means to have a world in which there are “laws” that admit of no exception (i.e., the question of where such laws come from). Nobody’s thinking of a Cartesian lawgiver, of course, but in an interesting sense supersymmetry vs multiverse rehashes the Descartes vs. Galileo debate over whether there are laws of nature or just regularities which admit of exceptions (albeit with the “exceptions” now taking place in other parts of the multiverse).

Any actual physicists out there who would like to chime in on this?

______

[1] Few people nowadays think of Descartes, and even less of Hobbes, as physicists. But they wrote stuff about the topic that was good enough to be taken seriously by Galileo and later by Newton.

ReplyDelete...philosopher Nancy Cartwright provocatively wrote that there is no sensible way to avoid the connection between laws and some kind of law maker...I'm with her on this one. Even if we agree to rule out exceptions to the regularities that scientists observe and report, the choice to name these regular observations "laws" still strikes me as a vestige of our superstitious religious past.

It's like when an atheist says "God forbid!" or "Thank God!" He may not literally believe what those words mean in the minds of theistic listeners, but he's still speaking their language and thereby reinforcing their pet themes.

I think it's perfectly fine to call them laws as long as we understand the context in which we use the term.

DeleteA scientific law in physics is just a statement about the relationships of physical quantities which brooks no exceptions. I don't think there is any good alternative word for this than "law", but that doesn't mean there is a law giver.

I too agree with Cartwright, and with Mufi's reasoning as to why better language would be preferable.

DeleteAs for Galileo, his experiments with cannon balls and such show that he was definitely interested in gravity, and as derivative from that, issues of motion in general. To tie this more to Descartes, Galileo was able to show that some Aristotelian ideas about motion weren't true, which is surely more reason for him to distrust Descartes' laws, since, as you note, Massimo, their idea for Descartes derived from Aristotle (as recycled through the Church).

That all said, given modern ideas about consciousness, subselves, etc., even Descartes' bedrock statement fails to work.

Massimo already introduced an alternative: "regularity." I suppose that another option is "pattern", both of which seem to better convey the human phenomenal/sensorimotor basis of scientific inquiry.

DeleteAside from the theistic cultural origins of "law", I suspect that scientists may enjoy some prestige from the connotation that

theyare in fact the "lawgivers", in the sense of handing down knowledge from a privileged position of epistemic authority.But now I'm starting to sound like a postmodernist, which is definitely not my tribe. :-)

I guess regularity or pattern might be OK, but they don't really seem to connote that so far as we can tell there are no exceptions, which is supposed to be the case for real physical laws.

Delete"Law" just seems like a very suitable word for this idea, but it may be that I've just become used to it.

While the origins of the word might have been theological, meanings and connotations change over time, and I don't think there's any problem with the current usage in science.

Deletemeanings and connotations change over timeAgreed, but I think "law" in this day and age still connotes to most folks something other than what you suggested is its technical meaning (basically, as jargon).

Also, I'm not so sure that these regularities

arewithout exception, unless by that one assumes the provision "as far as our scientists have observed (either directly or indirectly via inference)."But, even if these regularities truly are without exception (which, as Massimo might say, is not an empirical/scientific statement, but rather a philosophical/metaphysical one), "law" doesn't convey that to me, so much as it conveys a general rule or custom, to which there may or may not be exceptions.

I think laws are by definition without exception. If there are exceptions to a proposed law, then it is not actually a law and the hypothesis that it is is falsified. It's not so much a metaphysical position as a definitional one.

DeleteFor example, it is in some ways only a hypothesis that the law of gravitation exists, although this hypothesis is pretty well evidenced at this stage.

If we found exceptions to it, then we would need to refine the law or else accept that it isn't a law at all.

So I think a good way to look at it is that we see patterns and regularities in observations and data. We then make a hypothesis that these patterns and regularities arise because of a law of nature. The law of nature could be falsified, but the originally observed patterns and regularities are still there.

That's why I think "regularity" or "pattern" is not the best word. But then, perhaps "law" isn't either.

DeleteI think laws are by definition without exception.You think that, and perhaps most scientists think that, as well. But I don't get that connotation from the dictionary, let alone from the street.

That said, what's more important than this semantic point is whether or not the premise is justifiable. On that point, I hear what you're saying about the hypothetical basis of these "laws", but then (to paraphrase an apocryphal conversation between Laplace and Napoleon): I see no need for that hypothesis.

I know his theological commitments hark back to earlier times, but Descartes was amazingly modern in many ways, and his skepticism was quite breathtaking. (Like when he looked down from his apartment onto the street below and wondered whether perhaps those hats and cloaks were being moved along by strange machines rather than men. All he could

ReplyDeleteseewere hats and cloaks.)That notion that God would not conceive of what we see as laws in the way we do may sound a bit presumptuous or just silly to our ears, but I think it is part of a very sophisticated approach to reinterpreting the worldview he was born into. Perhaps a way of saying: they are not really laws; we just see them as such.

There are, of course, very different ways of conceptualizing 'the laws of nature'; and it could be seen in fact as a relatively harmless metaphor (so long as it is appreciated as such).

I seem to recall also that Descartes was somewhat skeptical of Natural Law approaches to ethics and politics.

Hi Massimo,

ReplyDeleteUnfortunately I'm no physicist, but I do have a keen interest in physics so I hope you won't mind me opining.

Firstly, I could be wrong here, but I think you're getting two different ideas mixed up. String theory is the idea that all particles etc are vibrations in strings. Supersymmetry is the idea that all particles have analogues we don't know of yet, that every boson (force-carrying particle) has a counterpart fermion (matter particle) and vice versa. The two can be combined into sypersymmetric string theory, (or superstring theory), but the two ideas are independent as far as I know. Either one or both could be correct or false.

Again, as far as I know, the findings at the LHC show that supersymmetry is unlikely to be true. String theory is still fine.

I know of no meaningful connection between supersymmetry and multiverses.

But string theory and multiverses are also orthogonal concerns. In fact the two ideas are very compatible indeed. As far as I understand it, there are many ways to solve the equations of string theory and each leads to different physical laws and constants. It is possible that each solution corresponds to a universe and that each universe exists. Some string theorists, Brian Greene for example, endorse this view as in this entertaining TED talk (in particular about half way through):

http://www.ted.com/talks/brian_greene_why_is_our_universe_fine_tuned_for_life.html

I want to say again that I think that it makes no sense to suppose that a multiverse could ever be ruled out by science. It's important to understand that there are different kinds of multiverse, operating at different levels. As you are aware, there's the many worlds interpretation of quantum mechanics, there's the aforementioned string theory multiverse, there's regions of space so distant from us as to be forever entirely causally disconnected, there's the constant creation of new universes in some inflationary models, but there's also the idea that there are completely separate realities which have no bearing whatsoever on our universe or anything connected to our universe. This latter concept in particular is unfalsifiable (and so the subject of reason and philosophical analysis rather than empirical evidence), and it certainly makes no predictions as to the value of the Higgs boson.

As for the status of natural law, you probably know my position by now. I'd go with mathematical structural realism. I think the universe is fundamentally a mathematical object, and the fundamental laws of physics are essentially the mathematical axioms that define it. There need be no lawgiver, because in mathematical Platonism all mathematical structures exist and need no creators.

The only laws in the Universe are the ones that we create. The Universe is free except for you and me. The proof is in the measure and the solution can be written mathematically. =

ReplyDelete@ Massimo

ReplyDeleteThe way I see it, there are only two possibilities regarding this issue: either there are "laws of nature" (what you are calling

the regularities of nature) or "habits of nature" (what you are callingthe regularities of nature with exceptions). The former presupposes determinism, the latter indeterminism. The mechanistic view presupposes determinism (albeit, determinism and mechanism are not interchangeable terms). The organismic view presupposes indeterminism.As was noted above in your blog post, the laws of nature have teleological and theological implications - namely, it suggests a lawmaker (i.e.God). However, the habits of nature (the regularities of nature with exceptions) also have teleological and theological implications. In fact, it provides the basis for Whitehead's "philosophy of organism" which influenced the thought of both Rupert Sheldrake and David Bohm. Below is a link to a short YouTube video (less than five minutes) in which Rupert Sheldrake discusses the

habits of naturewith Daniel Dennett, Stephen J. Gould, et al. (It would appear that the latest discoveries in particle physics have vindicated Sheldrake's view on this issue.)"Rupert Sheldrake: The Habits of Nature"

Hi Alastair,

DeleteI'm not sure indeterminism is the same idea as there being exceptions to the "habits" of nature. I think instead you're talking about supernaturalism.

Indeterminism isn't really about having exceptions to laws. Quantum Mechanics is not deterministic but it defines laws giving the probability of certain events. It's not that we see exceptions to the laws, it's that the laws are fundamentally statistical.

@ Disagreeable Me

DeleteMerriam-Webster defines "indeterminism" as "

a theory that the will is free and that deliberate choice and actions are not determined by or predictable from antecedent causes," "a theory that holds that not every event has a cause," and "the quality or state of being indeterminate; especially. unpredictability."Does

indeterminismimply something supernaturalistic? Well, it does imply that at least some event isnotphysically determined by an antecendent cause. If we define naturalism as being interchangeable with physicalism, then it does imply something supernaturalistic.Remember that the primary issue here is whether there are

"regularities in nature with exceptions," not whether there areonly"regularitiesanywithoutexceptions." So, if there are onlyregularities with exceptions, then you have to ask yourself why are there exceptions. There are exceptions because there are some events that are not determined by an antecendent cause. IOW, indeterminism holds true.Merriam-Webster defines "habit as "

a usual way of behaving : something that a person does often in a regular and repeated way.What's the difference between a behavior or action that is

law-likeand one that ishabit-like? Thetrulylaw-like behavior or action doesn't have any exceptions. The habit-like behavior or action does.Again, I say "mu" to the whole idea of free will as classically understood, especially free will versus determinism. So 20th century and earlier.

DeleteMark,

ReplyDelete> Perhaps a way of saying: they are not really laws; we just see them as such. <

Indeed, I did not take Descartes’ notion to be silly at all.

> it could be seen in fact as a relatively harmless metaphor (so long as it is appreciated as such) <

Yes, but my experience with scientific metaphors is that they are rarely harmless: http://philpapers.org/rec/PIGWMM

Alastair,

> the habits of nature (the regularities of nature with exceptions) also have teleological and theological implications. <

I don’t see that at all. Indeed, not even “laws” in the sense of exceptionless regularities necessarily have those implications. Just ask (almost) any physicist.

DM,

> A scientific law in physics is just a statement about the relationships of physical quantities which brooks no exceptions. <

Yes, but two questions immediately arise: 1) how do we know that there are no exceptions, since we have direct access only to a small portion of the cosmos? (This is one of the sources of Nassbaum’s skepticism about laws of nature.) 2) If there are no exceptions, why not? I.e., what compels matter to behave in a way that seems more appropriate for logic and math? (Yes, yes, I realize I’m giving you an opening to talk about mathematical realism, see below...)

> The two can be combined into sypersymmetric string theory, (or superstring theory), but the two ideas are independent as far as I know. <

You are correct, it was sloppy writing on my part. But I did mean to highlight the connection, i.e., to talk about sypersymmetric string theory.

> I know of no meaningful connection between supersymmetry and multiverses <

Except that they make different predictions about the mass of the Higgs, apparently.

> there are many ways to solve the equations of string theory and each leads to different physical laws and constants. It is possible that each solution corresponds to a universe and that each universe exists. <

Correct, that’s my understanding too. Again, we should be contrasting supersymmetric theories with the multiverse, not string theory per se.

> I want to say again that I think that it makes no sense to suppose that a multiverse could ever be ruled out by science. <

True, but the question is: can it be ruled *in*? That is, is it possible to find evidence for it? As I said, earlier this year I’ve read reports of very large scale asymmetries in the background radiation that *may* be interpreted that way.

@ Massimo

Delete> I don’t see that at all. Indeed, not even “laws” in the sense of exceptionless regularities necessarily have those implications. Just ask (almost) any physicist. <

What about physicist Lee Smolin? You stated in your post that he is explicitly endorsing Daniel Garber and Nancy Cartwright's criticism concerning the use of the term "law".

DM,

ReplyDelete> there's the many worlds interpretation of quantum mechanics <

My understanding is that that has nothing to do with the idea of the multiverse, though the two are often confused.

> As for the status of natural law, you probably know my position by now. I'd go with mathematical structural realism. <

Yup, and so increasingly do I.

> There need be no lawgiver, because in mathematical Platonism all mathematical structures exist and need no creators. <

Although, to be fair, it still makes sense to ask where *those* come from and why they are the way they are.

> I guess regularity or pattern might be OK, but they don't really seem to connote that so far as we can tell there are no exceptions, which is supposed to be the case for real physical laws. <

Smolin has suggested that there may be intriguing empirical evidence that some of our laws are not actually universal, as in the case of large scale breakdown of general relativity. But at the moment this is highly speculative.

mufi,

> the choice to name these regular observations "laws" still strikes me as a vestige of our superstitious religious past. <

It may be worse than that. A physicist talking about laws implies a strong inference beyond just regularities, an inference that is not, strictly speaking, warranted by the empirical evidence (again, as Cartwright points out).

> I suspect that scientists may enjoy some prestige from the connotation that they are in fact the "lawgivers" <

Yup...

Hi Massimo,

DeleteI had thought I had replied but since it hasn't shown up by now I think maybe something went wrong with the submission. If not, sorry for the double post.

>1) how do we know that there are no exceptions, since we have direct access only to a small portion of the cosmos?<

We don't. But a law is only a law if it has no exceptions. If there are exceptions, the law needs to change to accommodate them or else be thrown out or downgraded to a heuristic.

>If there are no exceptions, why not?<

As you say, mathematical realism.

>> I know of no meaningful connection between supersymmetry and multiverses <

Except that they make different predictions about the mass of the Higgs, apparently.<

...

>we should be contrasting supersymmetric theories with the multiverse, not string theory per se<

I still don't understand what you mean, as supersymmetric string theorists are often (perhaps usually) proponents of the idea of a multiverse, Brian Greene for instance. If that is the case, then I don't understand what you mean when you say supersymmetric string theory predicts X while multiverses predict Y, as the two ideas are not just compatible but positively complementary.

...

OK, I've just done some Googling and I've found the original paper that this came from. It seems not that the multiverse makes a prediction about the Higgs boson, but that a certain value for the Higgs boson would imply that there had been fine tuning, leading us to conclude that there was a multiverse much as we debated on the last article. So, not multiverse predicts Higgs at 140GeV, but Higgs at 140 GeV predicts multiverse.

I haven't seen the film, so it's possible that this was not well communicated or misleading. In any case, the failure of the "multiverse prediction" is not actually indicative that there is no multiverse.

>True, but the question is: can it be ruled *in*?<

It depends on what kind of multiverse. Some kinds of multiverse are not entirely disconnected from this one and so could be detected empirically. Other kinds of multiverse are in principle entirely disconnected and can only be inferred by philosophical reasoning (e.g. the anthropic principle).

I've recommended this to you before, but if you didn't read it then I suggest you do now, as it gives a good explanation of the different kinds of multiverses there may be.

Max Tegmark's paper "The Multiverse Hierarchy"

>> There need be no lawgiver, because in mathematical Platonism all mathematical structures exist and need no creators. <

Although, to be fair, it still makes sense to ask where *those* come from and why they are the way they are.<

In mathematical Platonism, all possible mathematical structures exist necessarily. They don't come from anywhere. They exist because of logical necessity. Most people don't feel the need to ask who created the natural numbers or the rules of logic, because the question makes no sense.

They are the way they are because all possibilities are exhausted, as all possible mathematical structures exist. If you take the ensemble as a whole, therefore, there is no other way it could be.

>Smolin has suggested that there may be intriguing empirical evidence that some of our laws are not actually universal<

Then the laws as currently conceived are not laws. They can be made into actual laws by making their scope of applicability part of the law. Another (complementary) approach might be to identify the underlying laws that govern this variation.

Great post Massimo,

ReplyDeleteLet me first start off by saying that understanding the laws of nature and why they got that way is one of the most interesting things I can think of. I've always thought one of philosophy's biggest questions deals directly with why the world evolves in a manner that is described to extreme precision by mathematical equations. Now as far as mathematics goes, Massimo you know I'm a strong realist in that department. In fact, I honestly believe that its the only sensible position in the face of the facts, and I think too many others immediately think of the connection between Platonic ideals and religion and get scared away towards a nominalist/fictionalist account of mathematics that looks to me to be entirely untenable.

As far as calling them "laws of nature," I like to think of it in terms of just being a description of the regularities/patterns that exist in the world. Personally, I think the term comes closer to describing what is actually going on, because to our knowledge these are hard and fast laws that have no exception (and yes of course that might not be the case, but its hard being skeptical of this in the face of countless experiments over the decades. If you seriously believe in something akin to Bayesian reasoning, your prior probability of any well founded law of nature NOT being exceptionless should be close to zero in the face of the evidence).

In relation to Massimo describing the multiverse idea and the possibility that this excludes there being any foundational law of nature, I don't think that's entirely true. Even in the case of string theory, for instance, there is an overarching edifice, or meta-law if you will, known as M-theory (Cumrun Vafa has made some progress on so called F-theory, but that's a whole other conversation). Indeed, I think this needs to be the case no matter what. Regardless of whether or not our Universe is the only one, its pretty safe to assume that the "space" of all possible universes would indeed have some meta-law in itself that governs the evolution of the entire multiverse. This mathematical law would be the foundation for the dynamics of the multiverse, and this shouldn't seem too incredible. A scientist who has seen the power of mathematical physics and its ability to describe more and more of the world in overarching equations (starting from Newton and continuing to this day), would be hard pressed to then conclude that this doesn't extend to the multiverse itself. In that way, the "laws of nature" are rescued and indeed do exist for the whole of existence. Whether or not we should still be calling them "laws" I will leave to the linguists.

Any thoughts from Massimo and others?

DeleteI think too many others immediately think of the connection between Platonic ideals and religion and get scared away towards a nominalist/fictionalist account of mathematics that looks to me to be entirely untenable.At least in my case, as a lay person who's moved back and forth between religious and skeptical circles, I'd say that's an inaccurate way to characterize the opposition. I simply have no more need for Platonic ideals than I have for the God of the Abrahamic faiths.

Same goes for "laws of nature." Identifying the regularities/patterns in human experience is interesting enough, without introducing these "strong inferences" (to quote Massimo from his comment above), particularly as this minimal information bears strongly on our prospects for survival and flourishing in this world.

IOW, such threats and opportunities are "real" enough for me. The rest seems more dubious and safe to ignore.

I can definitely understand that position, but its not about you "needing" a realistic account of mathematics (again this is very different from Plato's theory of forms, even though we use the term Platonism) any more than you have a need for God. This is about the empirical realization that mathematical equations are deeply ingrained in modern science, and fundamental physics in particular. Once you get to the most foundational theories we have, everything turns into mathematical structure, with physicists literally following the math to whatever it logically entails, then finding out that their results are borne out in experiment.

DeleteThe indispensability of mathematics and its status as our best tool for understanding reality at a fundamental level lend support to the idea of mathematics being independent of us. If that isn't "real" enough for you I'm not sure what else can be.

Hi Pete,

Delete>its pretty safe to assume that the "space" of all possible universes would indeed have some meta-law in itself that governs the evolution of the entire multiverse<

I'd agree with most of what you said, but I think there is a very strong argument to be made for an ultimate mathematical multiverse which consists of every possible mathematically consistent universe, and which does not evolve over time and so has no meta-laws.

This ultimate multiverse itself has no concept of time since time only applies within a particular universe, and this is why it does not evolve. It's just the set of all possible universes, like the set of all mathematical objects in mathematical Platonism.

I still don't see why they have to be 'mathematically consistent'. It's not clear that ours is.

Deleteplato.stanford.edu/entries/mathematics-inconsistent

iep.utm.edu/math-inc

Hi Philip,

DeleteI'm not sure if my meaning is the same as that you quote, but perhaps it is.

I just find it hard to conceive of a universe where the laws contradict each other, and you get different true predictions based on how you calculate the outcomes.

If there are contradictions then they must be reconcilable, perhaps in much the same way that a photon is both a particle and a wave, or that an electron can pass through two slits at the same time.

I don't think I could accept a universe where a photon is both absolutely a particle and absolutely not a particle at the same time, or one where an electron both does and does not pass through a given slit.

Perhaps if I understood inconsistent mathematics a little better I might change my mind.

The notion of mathematical platonism is really about having two worlds: the earthly world of everyday (material) things and the heavenly world of unearthly mathematical objects. (Kurt Gödel, for example, believed in the heavenly world, at least according to Rudy Rucker.) Now if the universe is a quantum computer (that computes itself, see Living in a quantum game, Pablo Arrighi and Jonathan Grattage), there is just this "earthly" world. ("Platonism is unsatisfactory because it violates our instinctive drive to obey Ockham's principle of parsimony", Jan Mycielski.) The collection of working (quantum or otherwise) computers is the "theory of everything". ("Working" would be the key word, not necessarily "consistent" is the traditional sense.)

Deletepete:

DeleteThe indispensability of mathematics and its status as our best tool for understanding reality at a fundamental level lend support to the idea of mathematics being independent of us.No one doubts that math is a useful tool in the sciences and in other human endeavors. But then, for that matter, so is language and color, neither of which, in my estimation, are "independent of us."

Perhaps this is what Richard Rorty had in mind when (in Philip's quote below) he said: "The suggestion that truth is out there is a legacy of an age in which the world was seen as the creation of a being who had a language of his own." But my thinking on this matter is more influenced by the work of philosopher Mark Johnson and cognitive linguist George Lakoff.

I suppose if we are "Living in a quantum game" (Pablo Arrighi and Jonathan Grattage, COSMOS Magazine), one would talk about Rules instead of Laws.

ReplyDeleteOh how universcentric we are... ;-) Seriously though, I'm still not clear on why the multiverse scenario necessarily entails the "laws" of our universe to be inexplicable. Is there a philosophical problem here and what exactly is it? Apologies for being obtuse.

ReplyDeleteI would say that in science the word "law" is used in two senses, both to refer to scientists' models of reality, and to refer to the properties of reality which make those models good ones. (This is not counting the ordinary non-scientific sense, in which a law is a prescription for behaviour.) I think these two senses tend to get conflated, which may lead to confusion. When we ask why the laws of nature are the way they are, we're not asking (using the first sense) why scientists have created the models they have; we're asking (using the second sense) why reality is the way it is.

ReplyDeleteIncidentally, I would say something similar is true of grammatical "rules". The word can refer both to patterns in the way we speak, and to our models of those patterns. However, in the case of grammatical rules there's a third usage: a grammatical rule can be a prescription for how to speak. In the case of grammatical rules people definitely do conflate these senses, leading to a lot of confused talk about linguistic prescriptivism and descriptivism.

The idea that "law" must be universal and never have exceptions is not in accord with how physicists use the term. For example the "second law of thermodynamics" is violated routinely (on a small scale for short periods of time).

ReplyDeleteWhat is a "law"? It's really just a description of a regularity in nature. The description need not be absolute (e.g. "Dollo's Law, which is really just a rule of thumb).

We still call things "laws" even when we know full well that they break down (e.g. Newton's Law of Gravity).

By "law" we just mean a description of nature that can be summed up in one sentence or one equation. Anything that requires a longer exposition tends to get called a "theory" instead, though all of this is mere semantics.

I wrote an account of this at What are laws of physics?

I would say that the second law of thermodynamics, if stated precisely, should allow for these temporary violations. It should not say that entropy always increases, it should say that an increase in entropy is inevitable over non-trivial time periods.

DeleteNewton's law of gravity is perfectly good within its scope of applicability. If it breaks down at all it's only in extreme situations like in singularities. I'm not sure that general relativity reflects a breaking down of Newton but rather a refinement.

Since today would have been Richard Rorty's (October 4, 1931 – June 8, 2007) birthday:

ReplyDelete"To say that truth is not out there is simply to say that where there are no sentences, there is no truth, that sentences are elements of human languages, and that languages are human creations. The suggestion that truth is out there is a legacy of an age in which the world was seen as the creation of a being who had a language of his own."

Contingency, Irony, and Solidarity

I would say that truth simply is. =

ReplyDelete(The humanist tries to puzzle out math and physics. This is gonna be ugly)

ReplyDeleteThe Hobbes idea of the distinction between mathematical regularities and physical laws is interesting. It complicates even further some thoughts I was already having along those lines. I was reading Poincare on how mathematics relates to the laws constructed out of experimental evidence. He suggests a relationship that persists despite divergences and convergences in the work of mathematicians and physicists, or in naïve experience:

“the idea of the mathematical continuum is not simply drawn from experiment. If that be so, the rough data of experiment, which are our sensations, could be measured … a law has even been formulated, known as Fechner's law, according to which sensation is proportional to the logarithm of the stimulus. But if we examine the experiments by which the endeavour has been made to establish this law, we shall be led to a diametrically opposite conclusion. It has, for instance, been observed that a weight A of 10 grammes and a weight B of 11 grammes produced identical sensations, that the weight B could no longer be distinguished from a weight C of 12 grammes, but that the weight A was readily distinguished from the weight C. Thus the rough results of the experiments may be expressed by the following relations: A=B, B=C, A < C, which may be regarded as the formula of the physical continuum. But here is an intolerable disagreement with the law of contradiction, and the necessity of banishing this disagreement has compelled us to invent the mathematical continuum. We are therefore forced to conclude that this notion has been created entirely by the mind, but it is experiment that has provided the opportunity. We cannot believe that two quantities which are equal to a third are not equal to one another, and we are thus led to suppose that A is different from B, and B from C, and that if we have not been aware of this, it is due to the imperfections of our senses. (Science and Hypothesis 22-23)

I like the way that Poincare points out the inescapability of a particular construction of a mathematical idea. It is a construction but not an arbitrary one, one that persists because of a contradiction between the fallible sense impressions of the experiments’ subjects and basic logical principles, not despite them. The mathematical continuum was not deduced from direct sense experience, but the fallibilities of the senses as demonstrated in experiment were subjected to deductive thinking that supports the mathematical construction. What is the relationship between these two realms? I would hesitate to bring in the word dialectical because … well, I am not even sure what that means. And how does a slightly evolved primate brain working in meatspace become informed of these principles and bring its activity in line with them? Is the disjunct between cognition of mathematical regularities and undertaking physical work another hard problem of consciousness as well as an ontological puzzle?

Thought you would like this:

ReplyDeleteB?

And if A = B, and B = C, then A = C

But what about B?

They all look different to me, so what is truth,

What can it B?

Different or equal?

What should it B?

To B or not to B?

That is the question.

The Nature of B,

Aristotle, Shakespeare and Me.

=

MJA

I think that the problem of the laws of nature is not put in its proper context unless it is noted that the general laws of physics are, by far, our most certain form of knowledge. What I am referring to is general laws, like the first law of thermodynamics (the conservation of energy.) The conservation of energy is so certain that is is used, in engineering and scientific practice, to check if some specific idea will work or not. Perpetual motion machines are (correctly) rejected by the patent office without looking at the details, simply because they violate the first law.

ReplyDeleteI am not saying that the first and second laws are absolutely certain, however, they are much more certain than anything else. This even covers specific everyday facts: they also are much less certain than general scientific laws. For instance, I think I know where I my car is parked, but my memory sometimes fails, or my car could have been stolen or towed away. All of these eventualities are much more probable than a violation of the first law. It is an interesting exercise, trying to find something we know that is more certain than the first law. I could never think of anything, except basic math. Advanced math has problems like what to do with the axiom of choice. And, of course, math is not really knowledge about the real world.

Note that the problem of the certainty of the law of conservation of energy cannot be resolved by claiming that it is just a convention. A simple convention would not allow us to predict if a particular machine will work or not. The conservation of energy is routinely used to make such predictions.

[This is the first installation of my contribution. The next one will be more technical.]

mufi,

ReplyDelete> I simply have no more need for Platonic ideals than I have for the God of the Abrahamic faiths. <

No sure what sort of need you are referring to, but there are good arguments for mathematical Platonism. One may or may not buy them, but the discussion is of a very different nature from those about Abrahamic faiths.

Randall,

> I'm still not clear on why the multiverse scenario necessarily entails the "laws" of our universe to be inexplicable <

It’s not that they would be inexplicable, it’s that they would be random, which means they wouldn’t need an analytic explanation like the one provided by supersymmetric theories.

Alastair,

> What about physicist Lee Smolin? <

What about him? He certainly wouldn’t go for a theological interpretation of the laws of nature.

coelsblog,

> For example the "second law of thermodynamics" is violated routinely <

Well, that gets tricky. Since the law is stated in statistical terms, it’s not clear that the occasional deviation actually counts as a violation. But that raises the interesting question of whether it makes sense to state a “law” in statistical terms to begin with.

> The description need not be absolute (e.g. "Dollo's Law, which is really just a rule of thumb). <

Very true, but the way physicists understand the sort of laws they are concerned with is definitely not the way biologists regard their laws (which, as you say, are understood from the beginning as generalizations with exceptions).

pete,

> In relation to Massimo describing the multiverse idea and the possibility that this excludes there being any foundational law of nature, I don't think that's entirely true. Even in the case of string theory, for instance, there is an overarching edifice, or meta-law if you will, known as M-theory <

That is my understanding too. But the point made by the physicists in the documentary was that if the multiverse is true we will simply never get access to the meta-level, which means that we will never get an explanation of why *our* laws are the way they are (beyond a random occurrence in the multiverse lottery).

> The indispensability of mathematics and its status as our best tool for understanding reality at a fundamental level lend support to the idea of mathematics being independent of us. <

I’m confused, isn’t that an argument for mathematical Platonism? I thought you rejected the idea.

Thanks for the response Massimo. Oh no, as I wrote in the post above I'm a firm believer in mathematical Platonism and definitely agree with you in that regard. In fact, I'd probably take that up as my strongest position in the entire field of philosophy.

DeleteDM,

ReplyDelete> a law is only a law if it has no exceptions. If there are exceptions, the law needs to change to accommodate them or else be thrown out or downgraded to a heuristic. <

Agreed, but that of course doesn’t settle the question of whether in fact there are laws (as opposed to only generalizations), and if so why.

> If that is the case, then I don't understand what you mean when you say supersymmetric string theory predicts X while multiverses predict Y, as the two ideas are not just compatible but positively complementary. <

I’m not a physicist, ask the two guys in Particle Fever. They made that point over and over throughout the documentary: supersymmetric theories predict a low value for the mass of the Higgs, while multiverse predicts a high value. And what we got so far is precisely in between and therefore highly inconclusive.

> not multiverse predicts Higgs at 140GeV, but Higgs at 140 GeV predicts multiverse. <

That’s not the way they put it in Particle Fever, but ok. Let’s talk in terms of the multiverse being “compatible” with a certain value of the Higgs mass and incompatible with other values.

> the failure of the "multiverse prediction" is not actually indicative that there is no multiverse. <

Correct, and I don’t think I said that. The way the documentary put it was that a certain value of the Higgs mass would make for strong circumstantial evidence for the multiverse, but neither proof nor disproof.

> In mathematical Platonism, all possible mathematical structures exist necessarily. They don't come from anywhere. They exist because of logical necessity. <

I understand, but that still leaves open the question of whether all those structures are actually manifested in physical universes. I don’t think there is any strong reason to think that.

> Most people don't feel the need to ask who created the natural numbers or the rules of logic, because the question makes no sense. <

Maybe, but I’m not so positive.

> They are the way they are because all possibilities are exhausted, as all possible mathematical structures exist. <

But what makes a mathematical structure impossible? That is, where do the constraints come from? (Yes, we’re getting pretty far from the topic of the post...)

> Newton's law of gravity is perfectly good within its scope of applicability. <

But than it’s not a law, not in Descartes’ sense of something that applies everywhere under all conditions. It becomes something more akin to what Galileo had in mind.

Philip,

thanks for the Rorty quote. It reminded me why I hate the guy with all my guts.

> "Platonism is unsatisfactory because it violates our instinctive drive to obey Ockham's principle of parsimony", Jan Mycielski. <

That seems like an exceedingly bad argument. First, because our instincts can be wrong, sometimes seriously so. Second because Occam’s razor is just a heuristic, not a law of epistemology.

Filippo,

> It is an interesting exercise, trying to find something we know that is more certain than the first law. I could never think of anything, except basic math. <

I hear you, but even your own use of “more certain” brings up the idea that perhaps we should think of our generalizations about how the world works along a continuum, depending on how confident (in, say, a Bayesian sense) we are about them. It still leaves open the questions whether there is anything that actually does reach a prior of 1 (you seem to think so), and if so of what distinguishes those cases from all other generalizations.

Massimo,

DeleteMy point is not that the conservation of energy has probability 1, but that it is a generalization that has an higher probability than any specific fact. Specific "facts" have less-than-1 probabilities because people lie, have hallucinations, delude themselves, etc.

Also, I give a much higher prior to the possibility that an experimental result showing a violation of energy conservation is due to experimental error, experimental fraud or both, rather than being due to energy actually not being conserved. This was the case in the "superluminal neutrinos" experiment a couple of years ago.

Hi Massimo,

Delete>Let’s talk in terms of the multiverse being “compatible” with a certain value of the Higgs mass and incompatible with other values.<

I'm afraid that's still missing the point. There are no values which are incompatible with a multiverse. A few scientists thought at one point that a particular value for the Higgs mass would be indicative of a multiverse. That's not the same at all as other values being incompatible.

>I understand, but that still leaves open the question of whether all those structures are actually manifested in physical universes. I don’t think there is any strong reason to think that.<

Sure, that's fine, there is certainly no obvious reason to think they are physically manifested. However, I think there are non-obvious reasons, but that's another topic. I think there is a very strong argument for it which I would love to get into with you at some point.

>But what makes a mathematical structure impossible? That is, where do the constraints come from?<

I would say incoherency. For example, the idea of a square circle, or a real root of negative one.

>But than it’s not a law, not in Descartes’ sense of something that applies everywhere under all conditions.<

I disagree. If the conditions of applicability are considered to be part of the law, then it is true everywhere under all conditions, even in circumstances where the law doesn't apply, and it is this truth that matters, not applicability.

For example, if there is a law "All red balls have mass 1kg", you can't say it isn't a law just because you've found a blue ball of mass 2kg. Similarly, if we imagine that Newton's laws only apply outside of black holes, then as long as we include that as a caveat in the laws themselves, there are no exceptions.

Massimo:

ReplyDeleteNo sure what sort of need you are referring to, but there are good arguments for mathematical Platonism. One may or may not buy them, but the discussion is of a very different nature from those about Abrahamic faiths.Since I didn't specify a

kindof need, I assumed that it would be clear that I meant: no need ofanykind.Of the arguments that I've seen for mathematical Platonism, I'd agree that they are better than those for theism, but only by degree. In the end, neither metaphysical doctrine warrants taking on board - thus the analogy.

I just thought I would come in here and say how interesting – and frustrating – I am finding some of the issues being discussed here. I read the Tegmark article recommended by Disagreeable and don't like his conclusions at all – I'm with the frog all the way! Tegmark's bird's eye view (God's eye view?) Platonism irks me, but I can't see any easy way to dismiss it. Have much more sympathy with the sort of approach Philip appears to favor and Seth Lloyd's ideas – the universe as a quantum computer computing itself, that sort of thing. What relation does this have I wonder to the computer simulation idea (which was mentioned in passing by Tegmark, by the way)?

ReplyDeleteI was also going to say something not particularly friendly but not quite as negative as Massimo's comment about Rorty, but that can wait.

And Massimo, I hope you continue to do pieces on mathematical realism and related topics. Why is it so difficult (for some of us at any rate) to come to firm conclusions?

Mycielski's theory of quantificational relativity makes it easier to dismiss Platonism. (Just like Darwin's theory of natural selection makes it easier to dismiss Creationism.)

DeleteJan Mycielski's work does look very interesting, but it would take me a bit of time to come to terms with it properly. It certainly does seem to weaken the case for mathematical Platonism.

DeleteMy general view is that, though there are (and will remain) many possible ways of conceptualizing mathematics, it does seem that traditional Platonistic views have been undermined by a number of developments (including digital computers and the theoretical work which accompanied their development; and findings in physics which suggest that nature is fundamentally discontinuous).

Also, I see that Max Tegmark has – due to challenges (some based on Gödel's theorem) to his original hypothesis – backed away from the MUH, and proposed a much more restricted Computable Universe Hypothesis (which however even he sees problems with).

Alexander Vilenkin was one of the critics of the MUH, it seems.

>Tegmark's bird's eye view (God's eye view?) Platonism irks me, but I can't see any easy way to dismiss it.<

DeleteWell, if you can't see any easy way to dismiss it, why not be open-minded? Why let it irk you? If nothing else, it's food for thought.

I read Mycielski's quantificational theory over 10 years ago, first via Lavine's "Understanding the Infinite".

DeleteAs for Tegmark, in his paper he does go from MUH, to CUH, but then to CFUH (where "F" is for "Finite".) That's pretty close to PUH ("P" for "Programmatic".)

@Mark English,

Delete>Also, I see that Max Tegmark has – due to challenges (some based on Gödel's theorem) to his original hypothesis – backed away from the MUH, and proposed a much more restricted Computable Universe Hypothesis (which however even he sees problems with).<

It's not so much that he's backed away from the MUH, it's more that he's entertaining the idea that perhaps the CUH is the better formulation. As far as I'm aware he's still quite open to the MUH being correct. Even if he did move more firmly to the CUH I'm not sure that would be much of a concession, as it still has pretty much the same explanatory power and implications as the MUH.

In any case, I really don't agree that Gödel is any kind of problem to the MUH. I've read the paper this came from and the criticism seems incoherent to me. I'm surprised Tegmark felt the need to propose the CUH at all, unless it was something he was considering for other reasons.

I disagree that traditional Platonistic views have been undermined and would love to know why you think it is that they have.

>I disagree that traditional Platonistic views have been undermined and would love to know why you think it is that they have.<

DeleteI agree. Gödel himself never considered his work as opposing traditional Platonism. Computational schemes, on the other hand, have severe problems with Gödel, Turing, and even more with computational complexity issues.

P ?= NP anybody?

Disagreeable

DeleteMy claims about the status of mathematical Platonism were fairly modest and qualified, or were meant to be. (I used the word 'seem' for example.) I do think that

traditional formsof mathematical Platonism are challenged by the developments I mentioned, but my position is simply that of a curious onlooker. I have a strong leaning towards non-Platonistic approaches – something along the lines of neo-Meinongianism might suit me, and I like Timothy Gowers's ideas – but I have come to no firm conclusions. In the end, I distrust my own instincts and intuitions quite as much as I distrust everyone else's. :-)I haven't researched the Gödel-related criticisms of the MUH so I can't comment on that.

Filippo

My (passing) reference to Gödel was specifically related to Tegmark's ideas, not to mathematical Platonism in general.

Also, I think we should be wary of privileging Gödel's interpretations of the significance of his own work.

I can see, however, how what you call 'computational schemes' (by which I suppose you mean non-Platonistic philosophies of mathematics which see mathematics in algorithmic terms) are at risk of falling foul of Gödel's ideas (the limits of a fixed axiomatic system, and so on).

But I also think that ideas from algorithmic information theory and related areas may not only provide support for or arguments against mathematical Platonism as traditionally defined and understood, but will also inevitably change the way we conceive the various options and the whole framework of the debate. And, given the lack of progress over the centuries on this one, coupled with the dizzying proliferation of increasingly refined options, maybe a new framework is required.

Hi Mark,

DeleteOn my part, I realise my "I would love to know" may sound a bit passive aggressive and sarcastic, but I mean it genuinely. I would be very interested to discuss with you why you think that Platonism (traditional or otherwise) has been undermined.

I agree with you that we ought not to privilege Gödel's own interpretations, but the fact that he remained a Platonist ought at least to give you pause before assuming that his theorems pose grave problems for Platonism.

Similarly, I don't see any reason why discrete mathematics or the development of digital computers pose any problems for Platonism. I don't even understand why you would think that they do. I'd like to learn about your views because I honestly haven't a notion of where you are coming from, and it's always great to learn about new viewpoints.

DM

Delete> >Tegmark's bird's eye view (God's eye view?) Platonism irks me, but I can't see any easy way to dismiss it.<

Well, if you can't see any easy way to dismiss it, why not be open-minded? Why let it irk you?<

If I admit that I can't find a way to dismiss a view which I dislike, I would have thought I am displaying an open mind.

>I agree with you that we ought not to privilege Gödel's own interpretations, but the fact that he remained a Platonist ought at least to give you pause before assuming that his theorems pose grave problems for Platonism.<

I did not claim – and do not assume – that Gödel's theorems pose grave problems for Platonism.

>Similarly, I don't see any reason why discrete mathematics or the development of digital computers pose any problems for Platonism. I don't even understand why you would think that they do...<

Again, you are misrepresenting (but not quite so grievously this time!) what I actually said. But rather than trying to elaborate here and now on my approach, I will just note that commenters here are obviously using the term 'mathematical Platonism' to mean very different things, and progress will only be made when we tease apart the various claims etc.

There is the issue of finitism, the issue of what we mean by 'exist', and crucially the issue of the relationship between mathematics and the world. Cosmological (and, for some, religious) questions are at the heart of this.

Hi Mark,

DeleteSorry if you think I've misrepresented you. It wasn't intentional.

I accept that you're open minded. But why are you irked?

I didn't mean to imply that you claimed that Godel's theorems posed grave problems for platonism. I was just responding what you said about being wary of privileging Godel's interpretation. I agreed with you, but wanted to make the point that Godel's intepretation still probably has some bearing on the issue, because we can at least be confident that he had a good understanding of his theorems, whereas some others who have used them in argument may not.

I'm disappointed you don't want to discuss the implications of discrete mathematics and digital computers, as I would have been intrigued to learn about your thinking. I agree of course that it's always best to clarify terms and as you say "tease apart the various claims".

I think the issue of what we mean by 'exist'" is at the very heart of the problem. Indeed, I suspect that's the entirety of it. Non-Platonists have a perfectly valid and consistent definition of existence, whereas that of Platonists is also valid and consistent though broader. I don't think there's any objective way of saying which is correct, since it's just a semantic issue, although I do find Platonism to be useful in thinking about and discussing mathematics and certain related philosophical issues such as the computational theory of mind.

I don't think we have actually discovered any fundamental law yet. Most physical laws are low-energy approximations, like Newtonian physics is the small-velocity and low-gravity limit of general relativity, or electro-magnetism the low-energy limit of electro-weak theory.

ReplyDeleteGeneral relativity is not the final answer, because it also breaks down at very high energies, when quantum effects start to show up. Even theories that are extremely well confirmed experimentally, like QED, are believed to be not fundamental by most physicists. Because of symmetry reasons, and because it has happened before.

Also the thermodynamical laws are problematic. The first law does not hold in quantum mechanics, where due to the energy-time uncertainty energy virtual particles appear (and disappear shortly after) using energy borrowed from nowhere. Energy is also not conserved in general relativity and an expanding space.

Candidates for fundamental laws such as GUT or string theory are out there, but no experimental evidence for any of this has been found. Even if that were the case, how could we know whether such a theory is not yet another brick in a hierarchy of laws?

A multiverse is also not of any help, since it necessarily defines a structure of some kind, which again has to be explained. Notions of all mathematically possible and/or consistent structures to exist as universes quickly opens a can of worms. For example, such a multiverse has to contain universes that contain other universes. Thus there are those universes that contain themselves, and those that don't and, even although everything so far is consistent, we run into Russell's paradox, making the whole multiverse inconsistent.

FYI:

Delete1 The first and second laws of thermodynamics apply to quantum field theories (QFTs.)

2 Energy is conserved in General Relativity (GR) and in an expanding universe.

3 Because quantum theories of gravity have been shown to be equivalent to some QFTs (the AdS/CFT correspondence,) the first and second laws will survive in any future quantum theory of gravity. (This is not as rigorous a result as the first two points, of course.)

Looking for a foundation to stand on, a theory that unites everything, well look no further than the Universe, the proof is right here. =

DeleteMassimo, mufi, Philip,

ReplyDeleteI have noticed the use of loaded words like “Creationism” and “theism,” by the opponents of mathematical Platonism. Using such words amounts to “arguments from invective” that have no logical validity.

Given that infinite sets can be eliminated from mathematics, what then is the argument in defense of Platonism? (I just don't see what it could be.)

Delete@Philip

DeleteLike Massimo, I don't see why the possibility of doing mathematics without infinite sets is any problem for Platonism.

My own argument in defense of Platonism is here:

Mathematical Platonism is True Because it is Useful

Filippo: I suspect that theists would find your comment to be at least as insulting as the mathematical platonists found mine.

DeleteBut I'll concede this much: There isn't time or space for a lay person like me to convey a full counter-argument here to mathematical platonism (although I have alluded to experts who have done so elsewhere). So, it's just quicker & easier for me to say how I categorize it: as a conjectural metaphysical doctrine, for which I have no need, but which I also cannot disprove once and for all.

In that sense, it is quite like theism - at least for me.

I am not aware of a Platonist mathematician who restricts their "ontology" to finite sets.

DeleteSay

P = Platonist

N = Non-Platonist

I = Infinitist

F = Non-Infinitist

Seem to go together: P-I N-F

Could go together?: P-F? N-I?

If you say there are P-Fs, then that would be interesting.

@Philip

DeletePlatonists believe that all mathematical objects exist, and infinity is a mathematical object. It's not surprising that Platonists accept the idea of infinity.

But as I said to you elsewhere, just because you can do mathematics without infinite sets does not mean that infinite sets don't exist. And Platonism is not just about defending infinite sets, as you'd see if you read my article. Indeed, I didn't once mention infinite sets as a motivation for Platonism.

I don't see why you think the scarcity of finitist Platonists means that Platonism is wrong.

mufi,

ReplyDelete> Of the arguments that I've seen for mathematical Platonism, I'd agree that they are better than those for theism, but only by degree. In the end, neither metaphysical doctrine warrants taking on board - thus the analogy. <

I guess my objection is that you can reject mathematical Platonism without having to insult its proponents by bringing in the comparison with Abrahamism... ;-)

Mark,

> I just thought I would come in here and say how interesting – and frustrating – I am finding some of the issues being discussed here. <

What can I say, welcome to philosophy? ;-)

lala,

> General relativity is not the final answer, because it also breaks down at very high energies, when quantum effects start to show up <

Well, that’s one possibility. Or one could go with Smolin and begin to think of natural laws as being domain specific, so that the apparent contradiction between relativity and quantum mechanics is simply generated by our insistence that there “ought” to be a fundamental unifying law.

> The first law does not hold in quantum mechanics, where due to the energy-time uncertainty energy virtual particles appear (and disappear shortly after) using energy borrowed from nowhere <

Same as I just wrote above...

> such a multiverse has to contain universes that contain other universes. Thus there are those universes that contain themselves, and those that don't and, even although everything so far is consistent, we run into Russell's paradox, making the whole multiverse inconsistent. <

Interesting thought, but I think the response there is that the multiverse only contains universes that are the physical instantiation of mathematically coherent structures, thus avoiding paradoxes.

Filippo,

> My point is not that the conservation of energy has probability 1, but that it is a generalization that has an higher probability than any specific fact. <

Yes, I understood that. I was just asking whether we have anything we’d attach priors of 1, and whether that would therefore qualify as a true law.

> I have noticed the use of loaded words like “Creationism” and “theism,” by the opponents of mathematical Platonism. Using such words amounts to “arguments from invective” that have no logical validity. <

Yup.

Philip,

> Mycielski's theory of quantificational relativity makes it easier to dismiss Platonism. (Just like Darwin's theory of natural selection makes it easier to dismiss Creationism.) <

I have no idea of that means, but I doubt it. There are large numbers of mathematicians and very capable philosophers of mathematics that apparently missed that “easy” dismissal. That doesn’t mean Mycielski is wrong, but I don’t think this is a no-brainer either.

> Given that infinite sets can be eliminated from mathematics, what then is the argument in defense of Platonism? <

I don’t see the connection, please explain.

Delete"infinite sets ... leads naturally to Platonism"

The Meaning of Pure Mathematics [ jstor.org/stable/30227216 ]

So without the need to defend the existence of infinite sets, what is the defense of Platonism?

In finite mathematics (e.g. of relative quantifiers, above), math consists only of finite things in the brain, marks on paper, in computer programs, etc. I would say this is not Platonist mathematics. I suppose there are Platonist mathematicians who are finitists in the above sense, but I am only aware of Platonists who accept the existence of infinite sets.

Do you know of a Platonist mathematician who is a finitist?

Sure, Platonists believe in infinite sets. At least I do. But just because you can do mathematics without infinite sets doesn't mean that infinite sets don't exist. And there are many more reasons to believe in Platonism than because of a need to defend the existence of infinite sets. See the post I linked to you above for more if you're interested.

DeleteGive me an example of an infinite set that you believe exists.

DeleteI believe {1, 2, 3, 4} exists. The code in my brain made me type those keystrokes, and there it was! (Well, I guess it will really exist when this comment has appeared.)

There is no infinite set (that exists) that I am aware of!

An infinite set of "natural numbers"? That doesn't exist. Where is it?

>An infinite set of "natural numbers"? That doesn't exist. Where is it?<

DeleteIt's not spatio-temporal. It is in no spatial location, because the way in which abstract objects exist is not like the way in which physical objects exists. It's a different, though related, sense of the term "exist".

Even if I accepted that only physical objects existed (which would rather contradict my Platonism, don't you think?) it is not certain that there are no physical infinite sets.

It is possible that the universe is infinite in spatial extent, so the set of all particles could be infinite. Indeed, this is possibly the most mainstream view. It is also possible that there are an infinite number of events stretching back to the past (it is not a settled question whether the big bang is the start of time) or forward to the future.

You have identified why I reject mathematical Platonism:

DeleteIt's a different, though related, sense of the term "exist".(I called it "earthly" and "heavenly" existence. Sounds like the writing of New Testament Paul!)But the point about the possibility that "the set of all particles could be infinite" in our universe is indeed worth considering, and it would would argue for an infinity in mathematics. But I think that most big-bang/inflationary cosmologists think there are (in reality) only a finite number of particles, or say they don't know.

But I have only one sense of "exists", and if there are infinite entities in the cosmos, I might become an N-I: non-Platonist infinitist!

>Yes, I understood that. I was just asking whether we have anything we’d attach priors of 1, and whether that would therefore qualify as a true law.

DeleteI've been following the conversation between Massimo and Filippo, which I thought was interesting. But I have trouble with the above quote. Where did a probability of 1 come from? Whether there are fundamental laws or not, and whether the laws we imagine to be fundamental laws are actually fundamental laws or not are two separate problems, aren't they?

@Philip

Delete>You have identified why I reject mathematical Platonism<

But there's no need to reject it so categorically. You just need to understand that we're using words with subtly different meanings than you do. You have your own interpretation of existence, and I don't think you're wrong. But similarly, I don't think you should regard Platonism as wrong, you should just think of it as a way of thinking about mathematics which you don't personally find useful or intuitive.

But surely you must agree there is certainly some sense in which mathematical objects pseudo-exist, even if fictively. The sentence "There exists no solution to this problem" makes sense, as does "There exists no generalised Pythagorean triples for exponents greater than 2", as does "There exists no greatest prime number".

If you don't like the concept of existence as applied in these sentences, then please provide an alternative. If you do accept it, but regard it as metaphorical, then feel free to take all of mathematical Platonism as talking about metaphorical existence and we'll get along famously, as long as you allow me to use that language.

"There exists no greatest prime number"?

DeleteAre you sure? I'm not totally sure. Or, I'm at least a bit unsure. Maybe more than a bit.

en.wikipedia.org/wiki/Ultrafinitism

"Are you sure? I'm not totally sure. Or, I'm at least a bit unsure. Maybe more than a bit."

DeleteAnd that's a perfect example of why denying Platonism makes communication more difficult.

Because even if there is a largest prime number, on the grounds of ultrafinitism, that is entirely different from what I mean when I offer the conjecture that there is a largest even number which is expressible as the sum of two primes (thus falsifying the Goldbach conjecture).

Even if you are an ultrafinitist in practice, Euclid's prime number theorem ought to be meaningful to you. Rejecting it is throwing out some genuinely useful and important mathematics.

Entering the conversation a bit late, but I must say I can't really understand how Mycielski's theorem truly comes to undermine the mathematical realists position. In addition, I find it odd that the many books/articles I have read on the philosophy of mathematics don't posit this "Mycielski theorem" as being a serious threat to the Platonist position. In fact, the large portion of google results from "Mycielski theorem platonism" are either commentary from Phil Thrift or a paper by Mycielski himself.

DeleteThe first thing I'd like to add is that this debate between finitary vs infinitary mathematics misses the point a bit. Mathematical Platonism is a realist position that suggests the structures and relations in mathematics would exist independently of humans in some form. Some of you finitists/ultrafinitists may know of Doron Zeilberger, a brilliant mathematician who advocates for ultrafinitism. He also happens to be a card carrying mathematical Platonist, who believes he is discovering mathematics truth and relationships. Simply avoiding talk of infinity in no way undermines a realist position.

In addition, Godel's theorems don't have a bearing on whether or not Platonism is true, which is something that is constantly mentioned in discussions on this. The incompleteness theorem simply states that any sufficiently powerful mathematical system will either be inconsistent or incomplete. If it is incomplete, we cannot know every theorem and every truth, but we can continue to prove results and add axioms that make intuitive sense (as we have for ZFC set theory). We are thereby uncovering more and more mathematical truths as we go, if one is a Platonist about math.

One last word on what has been mentioned by Phil earlier in these comments, in regard to paraconsistent logic and how that might undermine the Platonist position. The fact of the matter is it really doesn't accomplish that, at all. If one is a dialetheist, as Graham Priest is, then one could subscribe to all of the truths of mathematics while also believing that contradictions exist in some strong sense (while not collapsing into trivialism, which no philosopher/mathematician would ever want to endorse). Even before entertaining that notion, many individuals think that paraconsistent logic simply isn't a real logic because the negation operation is not really negation. In addition, some philosophers, David Lewis among them, simply find it absurd to think that something can be both true and false at the same time.

Some food for thought. I'm opened minded about everything, but the idea that we created mathematics in our heads and it just happened to "map" to the patterns of the real world (when no other human invention ever has) is to me far more unwarranted than believing that mathematical structures exist in some way.

DM, pete,

DeleteGood job defending mathematical Platonism (MP.) In particular, I like @DM's observation that it is nearly impossible to actually talk about math without using the language of MP. @pete, you saved me the effort of reporting my own research about Mycielski's theorem. I also liked your example of an ultrafinitist Platonist.

I would like to conclude with a quotation from David Deutsch, the quantum computing pioneer:

“Thus, abstract mathematical entities we think we are familiar with can nevertheless surprise or disappoint us. They can pop up unexpectedly in new guises or new disguises. They can be inexplicable and then later conform to a new explanation. So they are complex and autonomous, and therefore by Dr Johnson's criterion we must conclude that they are real. Since we cannot understand them either as being part of ourselves or being part of something else that we already understand, but we can understand them as independent entities, we must conclude that they are real, independent entities.”

Note that Deutsch is also an ultrafinitist, because he recognizes that mathematical proof is limited by the laws of physics. (A position I agree with.)

Great stuff, Pete, and I'd agree with all that.

DeleteI maintain there's no real fact of the matter though, it just depends on how broad a definition of existence you find intuitive. Platonism is to be preferred because it's useful in thinking about and discussing mathematical objects.

Any thoughts on that?

DM to Philip:

DeleteIf you don't like the concept of existence as applied in these sentences, then please provide an alternative. If you do accept it, but regard it as metaphorical, then feel free to take all of mathematical Platonism as talking about metaphorical existence and we'll get along famously, as long as you allow me to use that language.Do phenomena, such as colors and shapes, also exist? How about emotions? or language?

I don't mind saying that these "exist" in some meaningful sense (e.g. as human sensory-motor processes, neural instances, and psycho-social artifacts). What's problematic, to my mind, is saying that they exist

independentlyof us (or perhaps of other creatures in this universe who are physiologically and behaviorally similar to ourselves).Same goes for math, which, as far as I can tell, is no more "indispensable" to our survival and flourishing as these other human capacities, notwithstanding its importance to our culture.

Hi mufi,

Delete"Do phenomena, such as colors and shapes, also exist? How about emotions? or language?"

I'm going to answer your questions according to my honest intuitions, but I'm not going to make any great effort to defend them or give an argument for them.

Insofar as they can be given precise definitions, they exist. Colors and shapes can even be described mathematically. If you're talking about the qualia of colour, then I'm less inclined to think they exist, certainly not outside our own minds.

Languages, yes.

Emotions - seem kind of like qualia to me. There is no way to know if your emotion of anger is the same as mine, for instance.

But I subscribe to the computational theory of mind in any case, so for me the mind itself is a type of mathematical object. Emotions could be considered to be a property of that object.

So, I guess it's not a requirement that all abstract objects be independent of us, if we consider "us" to be our minds, because I think that our minds are actually abstract objects.

Another way of looking at it: if all life became extinct, then the concept of life and of various species would still abstractly exist. Anger would still exist as a property of some of these abstract objects.

"mathematical proof is limited by the laws of physics"

DeleteThat's what I think, naturally, but one would call that "mathematical Platonism"?

I just go with what Mycielski said Platonism is in his "The Meaning of Pure Mathematics" paper.

On a related note, I found this claim interesting:

Delete[Derek] Abbott [Professor of Electrical and Electronics Engineering at The University of Adelaide] estimates (through his own experiences, in an admittedly non-scientific survey) that while 80% of mathematicians lean toward a Platonist view, engineers by and large are non-Platonist. Physicists tend to be "closeted non-Platonists," he says, meaning they often appear Platonist in public. But when pressed in private, he says he can "often extract a non-Platonist confession."source

Anyway, as you can see from the rest of the article, the guy has a piece on this topic published in

Proceedings of the IEEE.>Or one could go with Smolin and begin to think of natural laws as being domain specific, so that the apparent contradiction between relativity and quantum mechanics is simply generated by our insistence that there “ought” to be a fundamental unifying law.<

ReplyDeleteThe fundamental equation for unity is =,

Nature's single absolute.

Truth.

=

You're wrong. The fundamental equation of unity is =, Nature's duplicate absolute. Truth 2.0

Delete=

You're right, = is

Delete==

Delete===, === = =. ====!

+?

=

Not even Einstein could find the equation that unites everything, because his belief in the constant of light stood in his way. Beyond Nature's immeasurability is Nature's truth, a truth much more simple than thought. Don't let the uncertainty of measure, the probability of measure that science has proven itself to be stand in your Way! Physics 101 =

ReplyDeleteDM,

ReplyDelete> A few scientists thought at one point that a particular value for the Higgs mass would be indicative of a multiverse. That's not the same at all as other values being incompatible. <

No, but it does mean that the actual value of the Higgs makes the multiverse more or less likely, in the words of the two physicists featured in Particle Fever.

> I would say incoherency. For example, the idea of a square circle, or a real root of negative one. <

Yes, indeed. But one could always push the question one further (just for the sake argument): why are certain things incoherent? Or do we stop at incoherence (however defined) as the ultimate constraint on reality?

> If the conditions of applicability are considered to be part of the law, then it is true everywhere under all conditions <

Sorry, that’s just cheating. It erases the difference btw Galileo and Descartes, which is important.

> if we imagine that Newton's laws only apply outside of black holes, then as long as we include that as a caveat in the laws themselves, there are no exceptions. <

Down that path lies pseudoscientific unfalsifiability. How many “exceptions” are acceptable before we throw the “law” away?

brainoil,

> Where did a probability of 1 come from? Whether there are fundamental laws or not, and whether the laws we imagine to be fundamental laws are actually fundamental laws or not are two separate problems, aren't they? <

Yes, I was simply suggesting that a law could be a pattern to which we are willing to apply a prior of 1 (i.e., no exceptions, pace DM).

pete,

> this debate between finitary vs infinitary mathematics misses the point a bit. Mathematical Platonism is a realist position that suggests the structures and relations in mathematics would exist independently of humans in some form. <

Yup, that was my thought too. I see no reason whatsoever why one has to be an infinitist in order to be a Platonist.

> Godel's theorems don't have a bearing on whether or not Platonism is true, which is something that is constantly mentioned in discussions on this. <

Again, yup, thanks for the clarification.

> on what has been mentioned by Phil earlier in these comments, in regard to paraconsistent logic and how that might undermine the Platonist position. The fact of the matter is it really doesn't accomplish that, at all. <

I’m getting repetitive: yup.

> the idea that we created mathematics in our heads and it just happened to "map" to the patterns of the real world (when no other human invention ever has) is to me far more unwarranted than believing that mathematical structures exist in some way. <

Which is otherwise known as the “no miracle argument” for mathematical realism (if people don’t like the loaded word, “Platonism”). It is analogous to the same argument deployed in favor of scientific realism, and I find both of them compelling, though neither one is a knock out.

>No, but it does mean that the actual value of the Higgs makes the multiverse more or less likely, in the words of the two physicists featured in Particle Fever.<

DeleteI would suspect that this is a minority view within physics, but I could be wrong. It's a pity that there is no more authoritative physics expert in this conversation to weigh in.

I think that the nature of the argument is that these guys thought that a value of 140 GeV, which there was some experimental evidence for at the time the paper was originally written, was supposed to in some way be evidence of fine tuning, and so indicative of a multiverse. The fact that it is not 140 GeV therefore only weakens the possibility of a multiverse in the trivial sense that this argument doesn't apply. There's still many other reasons to believe in a multiverse, e.g. all the other kinds of fine tuning.

> Or do we stop at incoherence (however defined) as the ultimate constraint on reality?<

I would. I think we need to take logic on faith before we can partake in any reasoning exercise. Questioning logic is "going nuclear" in the way of presuppositionalist apologetics (and to a lesser extent, Plantinga's Evolutionary Argument against Naturalism). If logic can be used to prove that something is nonsense, then it's nonsense, and doesn't refer to any realisable concept.

>Down that path lies pseudoscientific unfalsifiability. How many “exceptions” are acceptable before we throw the “law” away?<

I can see what you're saying, but I don't think that's the same thing. A law is an accurate model of reality, which is no more complex than it needs to be to explain the observed phenomena. Refining a law to more accurately reflect the situations where it applies is not the same as coming up with post hoc special pleading to explain why it is that the spirits were not in the mood to talk to the psychic today.

We throw the law away when we have found a more elegant way to model the same phenomena, with greater scope and greater predictive power. Epicycles were a valuable and pretty accurate refinement to geocentrism. We threw away geocentrism altogether when we found heliocentrism to be more elegant and powerful in the ways described.

Just to be clear: I think there can be more than one model to describe a given phenomenon (I suppose this is a bit like the interpretations of Quantum Mechanics). If those models are both perfectly accurate and always agree, then I think both have equal status as laws, even if they are very different conceptually. If, hypothetically, geocentrism with epicycles and similar refinements were just as accurate as heliocentric models, I would consider it to be just as true a model of reality as heliocentrism, suffering only from unnecessary complexity.

"I see no reason whatsoever why one has to be an infinitist in order to be a Platonist."

DeleteSo assume infinite objects are out.

Finite objects can be implemented in computer programs that run on a computer. (And I mean a real, physical computer, not an "abstract" computer.)

I don't see one single bit of Platonism in that.

Phil,

DeleteUnfortunately I don't think you're understanding what the Platonist is saying, nor do you exactly understand what your "non-Platonist" position entails. For one, talking about a "real, physical computer" doesn't really get you anywhere as far as arguments against Platonism goes. You seem to have some problem with the idea of mathematical structures existing, and I would go so far as to say its ideological rather than borne out by serious consideration of the arguments/evidence (maybe your inflating it with mysticism, as I think many nominalists do). Not only that, but your "real, physical computer" seems to be mathematical in some sense by itself. After all, our best description of reality would seems to suggest that its composed of some 10^27 atoms that are really all quantum wavefunctions evolving in an infinite dimensional Hilbert space (according to our most accurate scientific theory to date). So your comfort with that "real" computer should really get crushed once you actually sit down and think about (i.e. do actual philosophy) whats going on based on our current scientific theories. Not only that, but like I've mentioned above and in a previous post, there are ultrafinitists like Zeilberger who are also hardcore Platonists.

You have to actually take these things into account when you argue against Platonism. Simply saying "I don't see one single bit of Platonism in that" holds no weight whatsoever. Implementation by computer programs also does nothing as an argument against it. Especially when we consider that the computer seems to break down into very high level mathematics when we look at it at its most fundamental representation.

I'll leave you with a direct quote from Zeilberger: "I am a platonist, and I believe that finite integers, finite sets of finite integers, and all finite combinatorial structures have an existence of

their own, regardless of humans (or computers)."

"The application of Platonic reality to physics is fraught with problems."

Deletehuffingtonpost.com/victor-stenger/materialism-deconstructed_b_2228362.html

The same it true of mathematics. The elimination of Platonism from physics and mathematics is a noble task.

Speaking of Doron Zeilberger, I'm more in tune with this:

Mathematics is so useful because physical scientists and engineers have the good sense to largely ignore the "religious" fanaticism of professional mathematicians, and their insistence on so-called rigor, that in many cases is misplaced and hypocritical, since it is based on "axioms" that are completely fictional, i.e. those that involve the so-called infinity.math.rutgers.edu/~zeilberg/Opinion126.html

Sadly I read that article on Huff Post when it came out last year, and even though I respect Victor Stenger as a physicist and proponent of the scientific worldview, the article is seriously lacking in any philosophical justification of anti-Platonism. Aside from that one sentence where he even mentions Platonism, he then goes on to discuss wave-particle duality in Quantum Field Theory, which is a very noble endeavor in its own right, but somewhat unrelated to our discussion about the reality of mathematical structures.

DeleteIn addition, his arguments against his Platonist counterparts in theoretical physics and mathematics amounts to:

1) Our physical ideas are temporary

2) Quantum theory, as anything else, is a model - an invention of the human mind

These two arguments are, unfortunately for Victor, garbage from the start when it comes to trying to defend a particular view. Any skeptic can say these things when we discover new regularities and equations that describe reality, and it really carries no weight. Structural realism destroys these arguments pretty handily, showing one how mathematical structure is actually constant even as our scientific theories get better and more advanced, while at the same time advocating for the ontological commitment to the entities (whether physical/mathematical) that are a part of the theory.

I really don't know why you think of the anti-Platonism cause as "noble" unless year fear that its too religious or anti-materialistic, in which case it is again ideology. Ironically enough, Stenger is a vehement opponent of religion, and this clearly might influence his hatred for all things Platonic (even though mathematical Platonism has nothing to do with Gods, angels, and other ghouls). The fact is, I consider myself a materialist, believing everything to be matter. Then I sat down and wondered, what the hell is matter? Our most advanced theory suggests its vibrating strands of energy, and others posit equally spectacular fundamental entities? Well what on Earth are those in fact? Can you really call that matter?

One might be able to pragmatically call oneself a materialist (which I often do), while understanding that in reality we embrace a completely naturalistic worldview while removing the concept of concrete matter, which looks more and more ephemeral the more our scientific theories improve.

Note on your Zeilberger quote: Its anti infinite sets, not anti-Platonist. And we clearly know where he leans on that issue from the quote I took from a paper of his.

Philip,

ReplyDeleteComputers are not an argument against Platonism. In fact, when computers will develop enough intelligence, some of them could be Platonists. After all, truly intelligent computers should be able to surprise and disappoint you.

If you believe in the computational theory of mind, you already know Platonist computers: the brains of Plato, Gödel, Penrose, Deutsch, etc.

This is what scares me: Some computers might be Republicans.

DeleteOh, no!

Philip,

ReplyDelete> "mathematical proof is limited by the laws of physics" That's what I think, naturally, but one would call that "mathematical Platonism"? <

Funny, I would say precisely the opposite: it is logic and mathematics that impose constrains on physics, not the other way around.

DM,

> I would suspect that this is a minority view within physics, but I could be wrong. <

Didn’t look like from the documentary. These are both mainstream theoretical physicists at Princeton.

> I would. I think we need to take logic on faith before we can partake in any reasoning exercise. <

Ugh, while I agree with your argument, please don’t use the word “faith.” It may cause irreparable damage to my good humor... ;-)

> Refining a law to more accurately reflect the situations where it applies is not the same as coming up with post hoc special pleading to explain why it is that the spirits were not in the mood to talk to the psychic today. <

Agreed, but my point is that your approach erases the difference between a law (sensu Descartes, without the theology) and an empirical generalization (sensu Galileo). A distinction, again, that I think makes sense to retain.

> We throw the law away when we have found a more elegant way to model the same phenomena, with greater scope and greater predictive power. Epicycles were a valuable and pretty accurate refinement to geocentrism. <

But then it wasn’t a law, it was an empirical generalization! By the way, there may be some confusion here between the concept of law, which simply says that the universe behaves in a certain way under all circumstances, and a theory to explain a particular phenomenon. Epicycles were never laws, only theoretical entities (which turned out to be superfluous).

> If those models are both perfectly accurate and always agree, then I think both have equal status as laws <

Again, see comment just above about not confusing laws with theories (or models).

> If, hypothetically, geocentrism with epicycles and similar refinements were just as accurate as heliocentric models, I would consider it to be just as true a model of reality as heliocentrism, suffering only from unnecessary complexity. <

No, no, no my friend. All that would mean is that we would have a case of underdetermination of theory by the data. Only one model can be correct (either the sun or the earth is at the approximate center of the solar system), but we may not a way to tell based on the available empirical data. And at any rate this has nothing to do with laws.

>Only one model can be correct (either the sun or the earth is at the approximate center of the solar system),<

DeleteI don't agree. If a geocentric model were refined to the point where it worked just as well as heliocentrism, then whichever point is the center would be a relative question, as long as we define center to have something to do with the axis of rotation. (If on the other hand we define center to mean something like center of gravity, you'd certainly find that it was inside the sun somewhere due to the overwhelming mass of the sun).

If we take the analogous view that the center of the universe is the point from which it is expanding, then every point in space has equal claim to that title. It might be a bit like that.

From my point of view, spinning on the spot is pretty much the same as the universe revolving around me. I could be wrong here, but I seem to remember reading somewhere that if the universe actually did revolve around me in this way, I would even experience the same centrifugal forces as if I had just spun on the spot.

If all predictions work out the same, I still think both explanations are equally true. It's just that some are much more elegant and useful than others.

Massimo,

ReplyDelete>Didn’t look like from the documentary. These are both mainstream theoretical physicists at Princeton.<

Please don't take the multiverse argument in “Particle Fever” seriously. The “paper” about the 140-GeV-Higgs-multiverse connection is not even a paper. That is, it was never published in a refereed journal. It never passed peer review. It is not really part of the physics literature.

I checked on the arXiv and the “paper” only exists as an arXiv preprint. Since many people (me included) can “publish” on the arXiv without any peer review, you should not consider an arXiv “paper” much more authoritative than a contribution to this blog, unless it has been subsequently published in a refereed journal. When this happens, it is noted on the arXiv page for the paper.

The SUSY prediction is more mainstream but, since SUSY makes several other predictions that have been refuted already by LHC, it can be considered experimentally disproved.

Physics has many problems: “mainstream Princeton theoretical physicists” and the arXiv are some of them.

For more information, please read the report on “Particle Fever” on Peter Woit's blog.

Massimo, Philip,

ReplyDelete>> "mathematical proof is limited by the laws of physics" That's what I think, naturally, but one would call that "mathematical Platonism"? <

Funny, I would say precisely the opposite: it is logic and mathematics that impose constrains on physics, not the other way around.<

Mathematical theorems are established by mathematical proofs. Such proofs are processes that happen in brains, on paper, on white-boards and (increasingly) in computers. All of these are physical entities: mathematical proofs are physical processes. It would be a ridiculous form of mysticism to assume that mathematical facts are obtained by mystical visions of a (non-existent) celestial world.

Given the fact that mathematical proofs are physical processes, the laws of physics (mostly QM and the first and second laws) limit what can be proven. Such limits are much stricter than the limitations following from Gödel's theorem. The restrictions from Gödel's theorem (and/or algorithmic information theory) state that some proofs are impossible even given an infinite amount of energy and an infinite amount of time. In reality, neither the time nor the energy available for proving mathematical theorems is infinite. So the effectively provable mathematical theorems are extremely limited. Still, the results of mathematical proofs are the most certain statements we can produce. They are the only statements I would give probability 1. (Note, however, that this is still a subjective probability.)

Mathematical realism or Platonism is the statement that the results of (correct) mathematical proofs are not (totally) arbitrary human creations, that they restrict physical laws and theories and that they have a reality of their own, regardless of humans (or computers.) It is not affected by the fact that mathematical proofs are physical processes. Besides the no-miracles argument, mathematical Platonism is supported (psychologically) by the fact that mathematical results (even the ones obtained by computational methods,) can surprise, disappoint and delight us.

As pointed out by @DM, it is almost impossible to talk (and even think) about mathematical objects without using a realist language. For a modest applied mathematician (like me,) giving up realism takes considerable effort, an effort that is only justified by a rabid, irrational and mystical ideology. It is the same kind of rabid mysticism needed to believe in the nonexistence of your own consciousness, for instance.

Hi Filippo,

DeleteI'm very much on board with what you've said here, however I would have one minor semantic quibble.

"mathematical proofs are physical processes"

I would disagree with this as I would see mathematical proofs as abstract mathematical objects, existing independently of the mathematicians that find or analyse the proofs. I would distinguish the proofs themselves from the act of proving or from a physical representation of the proof.

Lets say that theories of reality can be reduced to a quantum lambda calculus.

ReplyDeleteThat leaves the programming of the 'the good society'.

That's what philosophy is for.

Cartwright is correct that anyone can misunderstand the phrase "laws (of nature") to imply a law-giver. And it is equally correct that if one correctly conceives natural law to be different from human or divine law, that there is then a strong inference to metaphysical naturalism. It seems to me that most scientists intend the whole phrase, laws of nature, but some scientists and nonscientists do smuggle in the lawgiver. It also seems to me that most objections are not really to any left over religious notions confusing things, but to the materialism.

ReplyDeleteFurther, the favorite alternative on offer, regularities or some such phrasing, seems to me to be most popular because it implicitly repudiates naturalism. It's the kind of thing popular with those scientists who claim science doesn't prove anything, or even tell us about reality, it just makes predictions about the outcome of experiments. The real question is not why anyone should pay these people for their hobby, but, whether this position really is justified by scientific experiment? Or, is it in fact, a religious and philosophical prejudice re-expressed as their science?

Platonic metaphysics was never tenable, as Aristotle demonstrated. Inasmuch as the foundations of mathematics do not, to my knowledge, sustain the a priori logical necessity demanded by philosophers, I'm still not sure why mathematical Platonism is imagined to be any more tenable.

There is no cure for the confusion to be found in terminology. Those who are offended by materialism will misunderstand, misinterpret and misrepresent no matter what.