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

Friday, September 14, 2012

On mathematical Platonism


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by Massimo Pigliucci

Recently I have been intrigued by James Ladyman and Don Ross’s ideas about naturalistic metaphysics and in the course of my discussion of their book, Every Thing Must Go, I pointed out that those ideas (as the authors themselves recognize) are compatible with one form or another of mathematical Platonism (hear also Ladyman on the RS podcast). I have also for a while been somewhat sympathetic to the latter notion, which has surprised some of my readers on the ground that it is (allegedly) incompatible with naturalism. It isn’t, but it seems to me time to explore a bit more in detail what one might mean by mathematical Platonism, and what reasons, if any, there are to entertain the notion seriously.

In this post, I will follow closely the excellent summary of the nature of the debate on mathematical Platonism offered by Øystein Linnebo, though the book by James Brown on the philosophy of mathematics is also an excellent, if opinionated, source.

To begin with, just to clear the air of a possible misunderstanding, mathematical Platonism — despite the name — bears little correspondence to Plato’s theory of forms. The latter, it may be recalled, was based on the notion that the world as we perceive it is but a pale reflection, a shadow (as in the famous metaphor of the cave) of the real world of pure concepts, to which, however, it is related. So for Plato there are the chairs of our everyday experience and then there is the ideal of a Chair, there are good things in the world and the ideal of Good itself, and so on.

Mathematical Platonism, instead, is a much more metaphysically circumscribed notion about the ontology of a particular category of abstract objects, those of concern to mathematicians (like numbers, sets, and so on). To be precise, Linnebo defines mathematical Platonism as the conjunction of these three theses:

Existence: There are mathematical objects.

Abstractness: Mathematical objects are abstract.

Independence: Mathematical objects are independent of intelligent agents and their language, thought, and practices.

As we shall see, only the last thesis, Independence, is controversial, and whether one accepts all or only a subset of the above theses defines what sort of ontology one is willing to attribute to mathematical objects.

Let us then start with what is likely the least controversial thesis, that of Existence. Linnebo mentions that the famous logician Frege proposed the following argument in defense of the existence of mathematical objects:

Premise 1 (Truth): Most sentences accepted as mathematical theorems are true.

Premise 2: Let S be one such sentence.

Premise 3 (Classical Semantics): The singular terms of the language of mathematics — such as S — purport to refer to mathematical objects, and mathematical language’s first-order quantifiers purport to range over such objects.

Premise 4: By Classical Semantics, the Truth of S requires that its singular terms succeed in referring to mathematical objects.

Conclusion: Hence there must be mathematical objects, as asserted by Existence.

The second premise is a simple stipulation, so it cannot be challenged. Premise 1 could be challenged, but only at the cost of doing away with much mathematics and its well established applications to science, clearly not a viable route. (Linnebo presents a number of ways to defend the Truth premise, the most convincing of which is the so-called indispensability (of mathematics to science) argument proposed by Quine and Putnam.) Premise 4 is a straightforward derivation of Premises 1-3, so the only thing that could possibly be argued is the truth of Premise 3, but very few philosophers and logicians have seriously questioned classical semantics (see Linnebo’s article for a discussion of this point), therefore we have to agree that mathematical objects exist.

Now that we have Existence, what about the second thesis, Abstractness? Interestingly, this is by far the least controversial piece of the puzzle, as most philosophers think Abstractness so likely that there are few explicit defenses (or criticisms) of it. Linnebo summarizes the situation by simply stating that if mathematical objects were not abstract then mathematicians — like scientists — should be concerned about their location and other physical attributes. Since mathematicians, and — more importantly — the practice of mathematics itself, don’t concern themselves with such things, this is a good prima facie argument for the abstractness of mathematical objects.

Even if we accept both Existence and Abstractness we have not arrived at mathematical Platonism just yet. Rather, we can think of ourselves at this point as anti-nominalists, since nominalism is the philosophical position that there are no abstract objects. Anti-nominalism, it should be obvious, is logically weaker than full fledged mathematical Platonism (which, recall, requires all three theses: Existence, Abstractness and Independence).

Here Linnebo makes an intriguing observation: few philosophers deny the independence of mathematical objects from the existence of minds capable of thinking about them not much because there are a lot of arguments in favor of the thesis, but rather because it would not be at all clear what it would mean for the thesis to fail. Be that as it may, there are some arguments in favor of Independence, most famously those formulated by Kurt Gödel (he of the incompleteness theorems that famously undermined Russell and Whitehead’s quest for self-sufficient logical foundations of mathematics). Gödel proposed two arguments to establish Independence. Due to my limited understanding of mathematical theory, I will simply let Linnebo summarize them both:

First, “the legitimacy of impredicative definitions is best explained by the truth of some form of Platonism [according to Gödel], including something like [the] claim [of] Independence.” Second, “Much of the search for new axioms in set theory is today based on so-called ‘extrinsic justifications,’ where candidate axioms are assessed not just for their intrinsic plausibility but also for their capacity to explain and systematize more basic mathematical facts. Perhaps this methodology can somehow be used to motivate Independence.” Again, however, recall that the stronger argument in favor of Independence appears to be simply how hard it is to fathom the meaning of its failure.

There are, naturally, plenty of objections (and counter-objections) to the notion of mathematical Platonism. Arguably the most obvious one is the issue of epistemic access, which asks how exactly we can gain reliable mathematical knowledge (which we apparently do) if mathematical objects really are abstract and mind-independent. We know how we get epistemic access to mind-independent physical objects (planets, say), but what human sense could possibly be involved in mathematical knowledge? The epistemic access objection is based on a crucial demand for a causal explanation of the reliability of mathematical knowledge, but as it turns out some philosophers have proposed more minimal accounts of reliability that do not involve causality (not even Linnebo gets into this, but he does provide references to the relevant primary literature).

An interesting objection to mathematical Platonism is of a metaphysical nature, and it basically states that there is nothing to mathematical objects (say, numbers) outside relations to other such objects. In other words, there really aren’t “objects” at all, just relations. There are, naturally, counters to this argument too. The idea that natural numbers have only structural properties is apparently rejected by logicist and neo-logicist philosophers on the grounds that numbers are tied to the cardinals of the collections they number. Moreover, structuralist philosophers reject the notion that there cannot be objects characterized only by structural properties. Which, as the attentive reader might have surmised, brings us right back to Ladyman and Ross, since their contention is that even what we think of as physical objects are nothing, at the bottom, but loci of relational properties (hence the title of their book, every thing must go). If that’s not a problem for physical objects it is hard to see why it would be for abstract ones.

To recap, there are strong positive arguments in favor of the Existence and Abstract theses, the acceptance of which at the very least commits one to anti-nominalism in the philosophy of mathematics. In order to be a full fledged mathematical Platonist one also has to accept (or at the least not reject) the further Independence thesis. We have seen that direct arguments in favor of this thesis do exist, but that the most convincing one for philosophers is actually the difficulty of making sense of the failure of the thesis. Of course all of the above can and have been debated, but at the very least implies that mathematical Platonism is not at all a fanciful or irrational position (it is apparently accepted by most mathematicians, not just by philosophers of mathematics and logicians). Notice also that nothing that we have discussed is in any way incompatible with naturalism (as, again, Ladyman and Ross also stress). So until further notice consider me a naturalistic anti-nominalist with strong tendencies toward Platonism in mathematics.

84 comments:

  1. Are you familiar with the book "Where Mathematics Comes From" by Lakof and Nunez?

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    1. Mike:

      Regardless of whether or not Massimo has read the Lakoff & Nunez book, I've recommended it to him before - originally as a counter-argument to mathematical platonism, based in their cognitive-scientific theory of embodied mathematics.

      However, I'm no longer sure that embodied mathematics is a counter-argument to mathematical platonism - partly because of the poor reception to the former by philosophers and (some, though not all) mathematicians (in other words, by non-cognitive-scientists), but also partly because I'm not even sure that the latter is defeasible, as implied by Massimo's statement above that:

      ...few philosophers deny the independence of mathematical objects from the existence of minds capable of thinking about them not much because there are a lot of arguments in favor of the thesis, but rather because it would not be at all clear what it would mean for the thesis to fail.

      Plus, we can't logically rule out the possibility that, as one of Lakoff & Nunez's critics put it:

      ...the math that brains invent takes the form it does because math had a hand in forming the brains in the first place (through the operation of natural laws in constraining the evolution of life). Furthermore, it's one thing to fit equations to aspects of reality that are already known.

      As usual in metaphysical discussions like these, I plead agnostic.

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    2. PS: Sorry, that second quote is awkward. Please ignore from "Furthermore, it's one thing..." onwards. For the the full quote, see here.

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    3. Interesting. To me, the whole idea of mathematical "objects" seems nonsensical. Mathematics are abstract representations of the features and behavior of the physical universe. I don't know that it's coherent to say they "exist" in any real sense.

      But, this kind of stuff reminds me of why I stay away from metaphysics. It seems more like an exercise in semantics than anything else.

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    4. The problem with the claim that "mathematics are abstract representations of the features and behavior of the universe" is that the foundations of mathematics have nothing to do with observations of nature. Mathematical truth is not determined by a relation to observable reality, it's determined by reason and logical rules. These rules are not entirely tautologous (self-evident/true by definition) either, as Godel demonstrated in his Incompleteness Theorem that no system of formal logic can prove every true mathematical statement. In other words, mathematical truth is greater than logic itself.

      It should also be noted that modern physics requires a rigourous form of calculus to work at all. Calculus isn't dependent on its relation to physical facts, but rather physical facts are determined by using calculus. See the Quine-Putnam indispensability thesis for more on that.

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  2. I'm not so familiar with these terms, but on first reading they seem far too loaded and awkward, and they hijack some regular meanings in unusual ways. Just quickly, the simple way to see math is as a language like any other, referring to physical things to describe them by the numbers & logic of math. Physical things have properties understandable by math and by words generally, and math & words are as real as the physicals, but no more so. They are entirely tied to them, and we derive math from physicals, just as we do in languages generally.

    Plato's general approach only has two factors, the universal forms and their particular applications, with the latter being subservient to the former. If numbers & their logic (math) are universals, it's a reversal of my analysis above, where math is subservient to physicals, as accurate descriptions of their properties. To say "math objects" are prima facie "abstract" because physcials are not the concern of math per se, which is concerned with numbers & logic, obviously offends my first para above.

    It offends it by tying math to physicals as an applicable language, and then removing the physicals from consideration on the basis that the language per se is interesting in itself, and the concern of mathematicians. Thus we are left with something " abstract", but not existing except in the mind of the abstractor without physical application. It is a return to the standard differentiation by Plato between ideal forms as number & logic and the real things to which they apply, and simply removes the real things in one fell swoop.

    It would be better, instead, to allow both the math and the physical things as realities, not abstracts, because the math provides accurate descriptions of things (as any language should do). Our ideas are our own constructs, but we can have confidence that those physicals & their math actually exist "out there". Any consideration by mathematicians of number & logic per se would be an abstract pursuit in the absence of real things, just as any language would be in a non-applicable usage. We should always patrol the line between abstract & real, as there is always a flux between them in one's mind. They have made Plato's universals real by a their argument, but its not a real argument, as explained.

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  3. I think the question of 'existence' needs to be scrutinized further. We use phrases like 'there is' and 'there are' in different ways and contexts. Rudolf Carnap talked about 'internal' and 'external' questions, only the latter involving ontological commitments. The position in the philosophy of mathematics which I find most appealing is that articulated by the mathematician Timothy Gowers (who draws on Carnap's idea). According to Gowers, if I say there are infinitely many primes I just mean that the normal rules for proving mathematical statements license me to use appropriate quantifiers.

    Phrases like 'there is' or 'there are' can also be used in a stronger sense - implying ontological commitment - but Gowers's point is that ordinary mathematical discourse does not require that stronger sense.

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  4. @ Massimo

    > I have also for a while been somewhat sympathetic to the latter notion, which has surprised some of my readers on the ground that it [mathematical Platonism] is (allegedly) incompatible with naturalism. It isn’t. <

    Your mathematical Platonism transcends the material world of space and time. Isn't that right? As such, it is compatible with supernaturalism.

    "supernaturalism: the belief of a realm of existence over and above the material realm of existence."

    (source: pg. 304, "The Harper Collins Dictionary of Philosophy")

    Of course, if you truly believe that these mathematical objects are natural objects, then we should be able to measure them employing "methodological naturalism." Right?

    > There are, naturally, plenty of objections (and counter-objections) to the notion of mathematical Platonism. Arguably the most obvious one is the issue of epistemic access, which asks how exactly we can gain reliable mathematical knowledge (which we apparently do) if mathematical objects really are abstract and mind-independent. We know how we get epistemic access to mind-independent physical objects (planets, say), but what human sense could possibly be involved in mathematical knowledge? The epistemic access objection is based on a crucial demand for a causal explanation of the reliability of mathematical knowledge, but as it turns out some philosophers have proposed more minimal accounts of reliability that do not involve causality (not even Linnebo gets into this, but he does provide references to the relevant primary literature). <

    Translation: "I really don't have an explanation how the human mind accesses these immaterial objects."




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  5. A consequence of my view, and illustrative of the importance of math in describing properties of physicals, is Leibniz' principle of sufficient reason for existence. I might boil down to a kind of perfectionism for him in saying that if things exist, they might as well be perfect. This might show as math perfection in balanced equations, for example the measured equality in capacities for the universe to expand & contract after the Big Bang at the broadest level. Math can configure physicals for us in perfect ways, using its logic of equality = , but all other logical symbols for relations between physicals as well. Thus Plato & Leibniz might have confidence in math as fundamental, but not preferential, to physicals.

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  6. What is incorrect with the following argument?

    On Premise 1: The truth of a mathematical theorem is predicated on a set of definitions and axioms. Mathematics contains more than one set of definitions and axioms. For example in hyperbolic, elliptic and Euclidean geometry, each has a theorem that proves the number of degrees in a triangle. The proofs are equally true and the geometries are equally valid but the number of degrees in each is mutually exclusive. Truth, it would seem, is not absolute, but relative to invented definitions and axioms.

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    1. The issue is not whether they are self consistently true as a mathematical abstract, but whether they apply to reality. Perhaps one or other applies to universal expansion, for example (whether under Einstein or some other theory) or does not. The problem with Ontology and issues of truth (including the use of the word in Epistemology as justified true belief) is that it is meaningless. Self-consistency, abstract modelling and so on can all be "true" in their logical structure, but inaplicable to reality and thus meaningless abstractions nonetheless (except to the abstracting mind that enjoys them subjectively).

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  7. Mark,

    > Gowers's point is that ordinary mathematical discourse does not require that stronger sense <

    Well, perhaps, but the same could be said of, say, unobservables in physics (like electrons), and in fact people like Carnap did say it. There is still a question of whether a stronger ontological commitment makes sense even if it is not strictly "required."

    Alastair,

    > Your mathematical Platonism transcends the material world of space and time. Isn't that right? As such, it is compatible with supernaturalism. <

    We've gone down this route before. I see no logical connection whatsoever between the type of ontological commitment implied by mathematical Platonism and supernaturalism.

    > if you truly believe that these mathematical objects are natural objects, then we should be able to measure them employing "methodological naturalism." Right? <

    Nope, did you miss the part about mathematical objects not having spatial extension?

    Marcus,

    > I'm not so familiar with these terms, but on first reading they seem far too loaded and awkward, and they hijack some regular meanings in unusual ways. <

    Yes, it's called technical terminology, the sort of thing that is done in every field of knowledge.

    > Plato's general approach only has two factors, the universal forms and their particular applications, with the latter being subservient to the former. <

    As explained in the post, mathematical Platonism, despite the name, has little to do with Plato.

    > It would be better, instead, to allow both the math and the physical things as realities, not abstracts, because the math provides accurate descriptions of things <

    What about the very large number of mathematical objects that do not describe anything physical?

    > Leibniz' principle of sufficient reason for existence. <

    Which few philosophers take seriously these days.

    Claude,

    > Truth, it would seem, is not absolute, but relative to invented definitions and axioms. <

    That is correct, but even that truth is not arbitrary (i.e., it is objective). Moreover, many choices of axioms immediately yield contradictory results, so they are abandoned.

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    1. Massimo: Bearing in mind my agnostic stance, as stated above...it seems to me like a lot rides on the fact that mathematical objects do sometimes describe things physical.

      By analogy, superstition (as in: supernatural belief/religion) is also non-arbitrary, in the sense that it's a human universal (albeit, a culturally variant one) that social scientists observe among human societies (though not necessarily among all individuals living within them - e.g. skeptics). Yet I suspect that most of us here would agree that superstitions do not describe how the world works in any reliable (let alone economical) way, relative to modern scientific theories, to which math is indispensable.

      In other words, the indispensability (to science) argument seems a lot more forceful to me than the non-arbitrary (or "objective") argument alone. It suggests to me that mathematical objects are not merely "in our heads" (e.g. patterns of imagination that naturally tend to emerge cross-culturally in human minds/bodies - say, as a cognitive bias or HADD) - they're also somehow useful in constructing a map of our environments - and indeed the universe.

      Again, why that is I don't claim to know (although I do think biology should play some role in any plausible explanation), but that physical connection helps me to sympathize more with mathematical platonism (to which I admittedly still harbor some aversion, as well).

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  8. Math is an exapted property of minds (i.e., state machines, but not simply so) and evolution conceptualizing the world for the purpose of survival. Math is just another language (i.e., a means of transferring the state of a state machine to another state machine), that has naturally arisen because of constraints.

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    1. I love the language. Are there any good books you'd recommend that expand on the 'state machine' concept?

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  9. @ Massimo

    > We've gone down this route before. I see no logical connection whatsoever between the type of ontological commitment implied by mathematical Platonism and supernaturalism. <

    The logical connection is this: You're proposing an immaterial realm (or, more specifically, an ontological structure of immaterial mathematical abstractions) that transcends the NATURAL realm of SPACE and TIME. As such, you are proposing a realm that qualifies as a SUPERNATURAL one. (The term "supernatural" literally means that which is "beyond the natural.") I have furnished you a technical definition of the term which supports my argument. I'm afraid you simply don't have the luxury to ignore this fact.

    > Nope, did you miss the part about mathematical objects not having spatial extension? <

    I have explicitly stated, not only above, but also in previous posts that mathematical objects transcend both space and time (that necessarily implies that they are neither spatially nor temporally extended). Obviously, you are not really bothering to read my posts.

    It is noted that you have failed to explain to us why we cannot employ "methodological naturalism" to measure these immaterial abstractions, which you allege to be natural objects.

    It is also noted that you have failed to explain to us how exactly the human mind gains access to these immaterial abstractions. In fact, you actually had the audacity to argue that no explanation is required.

    And, just as an aside, what you are presenting here is textbook "scientism" - the belief that our metaphysics (our worldview) should only be informed by methodological naturalism.

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    1. Alastair,

      Even if you can persuade Massimo that mathematical platonism meets certain definitions of "supernatural", the argument that those definitions are overly broad and/or imprecise is always available to him.

      After all, I can cherry-pick an equally broad and imprecise pair of definitions for "supernatural", which I found at the Merriam-Webster's site:

      of or relating to God or a god, demigod, spirit, or devil

      attributed to an invisible agent (as a ghost or spirit)

      Neither of these definitions seem to fit mathematical objects (or their "structural properties"), which are totally impersonal, whereas I expect that many naturalists have exactly such personal entities (or "invisible agents") in mind when they announce that they don't believe in the supernatural (or at least see no reliable way to test for them).

      None of this is to suggest that I'm convinced of mathematical platonism. But if you want to label that view "supernatural", then you should at least qualify it as the kind of "supernaturalism" which few (if any) religious believers would find comforting and no peddlers of "woo" (e.g. alternative medicine or New Age beliefs) stand likely to cash in on.

      Otherwise, it's like announcing that "Most philosophers believe in God!", only for the audience to learn later on that the referent of "God" in that statement actually bears little resemblance to the personal god that they learned about in church (i.e. someone who hears prayers and takes an interest in how people live, as opposed to an impersonal ensemble of natural laws and/or mathematical relations).

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    2. @ mufi

      > Otherwise, it's like announcing that "Most philosophers believe in God!", only for the audience to learn later on that the referent of "God" in that statement actually bears little resemblance to the personal god that they learned about in church. <

      It's like saying: "I'm an atheist and a naturalist...and...oh...by the way...I believe in buddhahood."

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    3. I would agree that a belief in a literal "state of perfect enlightenment" has supernatural overtones (e.g. given modern scientific knowledge re: cognitive limits), and is therefore unwarranted.

      Of course, language is not always so literal - thus the popularity of "God" among certain scientists (e.g. Einstein, Sagan, and Hawking) who write for public (non-academic) consumption. These folks were (or are) by no means theists in the classic/Abrahamic sense and remain members-in-good-standing among naturalists - presumably because they adhered (or adhere) to the "no miracles" policy of methodological naturalism that Massimo referred to below.

      Language also evolves (if one doubts this, then see the origins of the word "humor"), which is why it's important to be clear about which meaning one uses in any particular situation.

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  10. In what form do 'mathematical objects' exist except as a reference to some 'instance'? As abstractions, they are merely references, correct? They exist as neural connections or patterns of graphite or differences in voltage. The illusion that they are existential is that we cannot think of them otherwise, as the operating system of our consciousness is our only method of understanding them.


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  11. It seems to me that the distinction between abstract and concrete is easily understandable in terms of set theory: a set is an abstraction of its elements or subsets (which are concrete instances of the abstraction).

    Then regarding the Independence thesis of Platonism, an abstraction existing independently of abstractly thinking beings means that there exists, independently of abstractly thinking beings, a set of elements that is the abstraction.

    For example, let's take the abstraction Tree. It is a set of all possible trees. Does such an abstraction (that is, such a set) exist independently of abstractly thinking beings (that is, not just as an idea in their minds)? It seems that it would require the existence of all possible trees, an infinitely large set. Such a set obviously hasn't existed on Earth. But maybe it could exist in a multiverse with unlimited time and space. Such a multiverse might generate all possible trees and these together would constitute the complete abstraction we call Tree, independently of abstractly thinking beings. If abstractly thinking beings emerge they may form a representation of this infinite set in their minds, apparently not by sensory perception but by inductive reasoning (generalization) from the concrete instances of Tree that they can perceive.

    Another interesting fact is that before things come into existence via quantum-mechanical selection (a process known as decoherence) they are in a quantum superposition of all possibilities. So for example all the elementary particles that make up all possible trees are initially in quantum superpositions of possibilities. Does quantum superposition count as existence too? If yes, then we might say that the abstraction Tree (as a set of all possible trees) also exists in this sense. We might call it an "unrealized" abstraction, which might be reflected as a "realized" abstraction in a multiverse where all possibilities are realized by quantum-mechanical selection (and subsequent processes).

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    1. Abstractions can of course exist without abstractly thinking beings. Our knowledge may well outlive the human race, and I don't think any sufficiently intelligent alien discoverers of our knowledge would have an issue translating our abstractions to their own method(s) of abstraction. In other words, the abstractions can exist, but they would be useless without a translator.

      Including more trees, or infinite trees, does not push the abstraction into another category. The abstract set would merely have more instances as referents. Such categorization of "trees" also seems to be cutting nature at the knees. I'm sure there could exist a potential gradient that flows seemlessly from small plants to sequoia's, without any break in the gradient for one of our categories to grab on to as a point of demarcation.

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  12. Bravo - Mark English, and bravo others who also have a problem with elevating things we see to the domain of things that are.

    Craig's response to the Ladyman podcast puts it well:
    I like the way that the question of entity-ness is dissolved by making the notion of individuality scale-relative, but doesn't that also dissolve all of ontology, since all descriptions are now dependent on the scales chosen by (or physically available to) the observer, making them subjective and epistemic?

    Massimo - you pointed me to Giere's scientific perspectivism not long ago. I checked out some of his work. It seems for him, every glimmer of a certifiably objective real object is colored by one's perspective. Yes, he believes in a world out there, but rightly says there is no one view of any aspect of it, nor would the aspect be intrinsic and workable without introducing the observations of others. Not sure if he would extend this view to mathematical objects, but given our trouble visualizing 6-dimensional objects, and a Flatlander's similar problem with visualizing a circle, who knows?

    But it seems clear that Giere rejects objective truth, Ladyman rejects objective stuff, and neither hold knowledge and opinion to be the same thing. They might say gods and politics are in the province of opinion, the results of scientific experiments are not. But doesn't this raise the specter of absolute truth again, with falsifiability the determinant of scientific knowledge? Well, take astrology. I can get up every day and read my horoscope during breakfast, and during dinner tell myself that the prediction came true. Then one day a fortune-teller whose services employ astrological information tells me that I will come into US 10 million dollars by 6 PM that evening. Doesn't happen. Does this tell us anything about whether astrology is BS? No, not for me, because up until yesterday I believed in it, it worked. So it was true for me. until it wasn't. One might say that truth is simply a utilitarian aspect of information.

    And that is a common theme sounded by Ladyman, Giere, and many others. Forget the angels on the head of a pin question and other undecipherable problems. Add value to philosophical inquiry by providing usability. Viewing everything as information without regard to its rightness and wrongness may at first seem to be a waste of time, but there is value here. The value is to always factor in the observer or consumer when talking about anything. The value of equating classic science with stuff you read in the National Enquirer or peddled to you by a Jehovah's witness is to see the perspective of others. The value is in the much wider landscape one is afforded when 'anything goes'. This is the laudable thing that postmodernists have espoused, to the derision of the pre-modernists

    One's perception need not be the only reason to deny objective things. Suppose the information transmitted from Giere's map or color source changed based on who was receiving it. Whether this was done unintentionally by the environment, or gamed intentionally by a god or daemon - we would call this extra information noise. Suppose such altered information was transmitted to entities 1-100 but not 101-999. Then the groups wind up with different common realities, even while taking into account their personal perceptions.

    It comes down to information processing, and truth only comes into the picture when one's perception of the information is judged to be true or false. Either way, the information stands and has value. It is not so much whether there or aren't things like coffee cups, Mack Trucks, circles, gods, and rationality, it is that they are being increasingly seen by professional philosophers as having the same level of physicality. No, not you, Massimo, not yet.

    Anyway it is heartening to read how others came to similar conclusions, and I'm sure we eagerly await publication of "Ontology, Too, Must Go"

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    1. 'Ontology' (what really exists) reduces to the psychology of the mind that both abstracts and identifies realities (and the conncetion between abstract & real). It just bounces back to subjective opinions until the mental creation of the distinction is better understood (read the middle chapters on neurology in my free book to see how this is done www.thehumandesign.net). Until then, terms like "objects" in this article can always be interposed in error to blur the line. An object is either physical and real, or it is not and therefore it is abstract.

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  13. [from Russ Abbott, deleted my mistake, apologies!]

    I originally posted these comments in Massimo's Google+ stream.

    ----

    You quote Frege's argument for the existence of mathematical objects. The third premise is the following.

    Premise 3 (Classical Semantics): The singular terms of the language of mathematics — such as S — purport to refer to mathematical objects, and mathematical language’s first-order quantifiers purport to range over such objects.

    You then write, very few philosophers and logicians have seriously questioned classical semantics ... therefore we have to agree that mathematical objects exist.

    That seems circular to me. Classical semantics consists of rules for relating statements in a language to objects in an assumed domain. Classical semantics doesn't require that the domain consist of objects that exist in any other sense than that we assume a domain that contains them. Why would accepting classical semantics require that objects in every hypothesized domain to which it may be applied force us to accept the existence of objects in those domains?

    You also write that few philosophers deny the independence of mathematical objects from the existence of minds capable of thinking about them [primarily] because it would not be at all clear what it would mean for the thesis to fail.

    From my perspective it is not all clear what it would mean for the thesis to be true. How can a concept be independent of the mind(s) that think it? This seems to be saying that just because we can think about Santa Clause that Santa Clause must necessarily exist.

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  14. Eid,

    > Math is an exapted property of minds <

    And we know this how?

    Alastair,

    > You're proposing an immaterial realm (or, more specifically, an ontological structure of immaterial mathematical abstractions) that transcends the NATURAL realm of SPACE and TIME. As such, you are proposing a realm that qualifies as a SUPERNATURAL one. <

    That's because you have a limited set of possibilities: if something isn't physical than it's supernatural. No serious philosopher would buy that.

    Also, what mufi said.

    > I have explicitly stated, not only above, but also in previous posts that mathematical objects transcend both space and time (that necessarily implies that they are neither spatially nor temporally extended). Obviously, you are not really bothering to read my posts. <

    I read them, believe me. But you said:

    > then we should be able to measure them <

    No, because they don't have spatio-temporal extension. Do *you* read my posts at all?

    > you have failed to explain to us how exactly the human mind gains access to these immaterial abstractions. In fact, you actually had the audacity to argue that no explanation is required. <

    Yeah, I'm an audacious kind of guy. I was simply reporting what a number of philosophers of math think. As for gaining access, I don't know, but apparently we do. Why, exactly is this a problem? Plenty of times in the past science found itself in a position of not knowing how X while at the same time not doubting the existence of X (genes at the time of Mendel for instance).

    > what you are presenting here is textbook "scientism" - the belief that our metaphysics (our worldview) should only be informed by methodological naturalism. <

    You clearly have no idea of what you are talking about, that is not at all what philosophers mean by scientism.

    DaveS,

    > It is not so much whether there or aren't things like coffee cups, Mack Trucks, circles, gods, and rationality, it is that they are being increasingly seen by professional philosophers as having the same level of physicality. No, not you, Massimo, not yet. <

    Nor by any sensible philosopher I know, so I don't know who your "professional philosophers" are. Care to name names?

    mufi,

    > it seems to me like a lot rides on the fact that mathematical objects do sometimes describe things physical. <

    Indeed, I find the connection between science and math one of the best arguments in favor of mathematical Platonism.

    Russ,

    > Why would accepting classical semantics require that objects in every hypothesized domain to which it may be applied force us to accept the existence of objects in those domains? <

    That is not correct, as I explain in the post, it is the truth of S *plus* classical semantics that requires the acceptance of successful reference to mathematical objects.

    > How can a concept be independent of the mind(s) that think it? <

    Not a concept, an abstract entity, which is not the same. As for the difficulty to see what it would mean for independence to fail, I suppose philosophers of math would say that if independence is not true than it is mysterious how math has the consequences for science that it does, and why mathematicians think of what they do as discovering, rather than inventing things.

    Joe,

    > They exist as neural connections or patterns of graphite or differences in voltage. <

    You are looking for physical existence, which is not required for abstract entities.

    > the abstractions can exist, but they would be useless without a translator. <

    That may very well be, I am making no claims about the usefulness of math in the absence of mathematicians. Largely because "usefulness" is a concept that implies the presence of agents.

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    1. @ Massimo

      > That's because you have a limited set of possibilities: if something isn't physical than it's supernatural. No serious philosopher would buy that. <

      Let me try a different approach (since you keep dodging my questions).

      Please explain to us why Sheldrake's hypothesis of "morphic fields and morphic resonance" does not qualify as a scientific one.

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    2. @ Massimo

      > No, because they don't have spatio-temporal extension. <

      Well, if we can't measure these mathematical objects, then they are not amenable to "methodological naturalism" and therefore cannot be classified as natural objects.

      > As for gaining access, I don't know, but apparently we do. Why, exactly is this a problem? <

      In a previous thread, you stated that you weren't proposing anything "mystical." However, it appears that how we gain access to these mathematical objects is completely mysterious. Also, your only evidence for these mathematical objects is subjective, not objective. (I would think that would be problematic for someone who espouses that the only valid form of evidence is objective evidence. Certainly, you would not be in a position to demand that a theist furnish you with objective evidence for God's existence.)

      > Plenty of times in the past science found itself in a position of not knowing how X while at the same time not doubting the existence of X (genes at the time of Mendel for instance). <

      We have already established that we cannot objectively measure these mathematical objects. Therefore, no scientific evidence will be forthcoming.

      > You clearly have no idea of what you are talking about, that is not at all what philosophers mean by scientism. <

      "Scientism is a term used, usually pejoratively,[1][2][3] to refer to belief in the universal applicability of the scientific method and approach, and the view that empirical science constitutes the most authoritative worldview or most valuable part of human learning to the exclusion of other viewpoints.[4]"

      (source: Wikipedia: Scientism)

      I should also remind you that Ladyman and Ross begin their book ("Everything Must Go") by defending scientism.

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  15. Massimo, I am not sure you have responded usefully to my post. The first point would be that it is inappropriate terminology if it obfucates reality, which is what I have proposed, not that jargon or its proliferation are necessarily 'bad' (although my view is that much of science & philosophy proliferates to its own and our confusion). The second point that mathematical platonism has nothing to do with Plato would be an excellent example of hijacking common usage (Plato's ideas) and applying it inappropriately, leading to confusion. However, I say the connection is as explained by me, so I have saved them there nonetheless.

    The third point needs an example from you. Mathematical "objects" are physical, or they would not be objects in common usage of the word. Introducing "object" obfuscates meaning, as explained. If a mathematical symbol does not refer to anything physical (including a ratio between physical units, cancelling them to reveal a number - as in the dimensionless constants derived only from physical measurement) it is an abstract exercise by a mathematician, as explained.

    The fourth misses the point(again). Sufficient reason is probably an idealistic premise rather than scientific, but math would support any underlying motivation to perfection, expressed in math. The example of equations relating the entire expansion & contraction of everything (the universe itself) is a fine example of how a balanced equation supports the notion that, in fact, the universe might be "perfect" in that way. Why should anything exists at all? is answered by that perfection. It doesn't legitimize his principle in itself, but it supports him.

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  16. I guess Rucker and Fredkin are professional philosophers. I assume at a minimum, you will call Rucker a nut (thus making him a nonsensical philosopher??). Mixing the living and not-so-living, Leibniz could be considered some sort of digital physics pioneer when he wasn't doing math. I will guess Wheeler took money for his philosophical ideas when he wasn't pioneering what you've called the best science around, so add him too. And to some extent, the panpsychic stances of Schopenhauer, Russell, Spinoza, and Whitehead if you want to stretch it will get you uncomfortably close to ideas and stuff being the same damn thing. Normally would not include Bohm in this group as he was quite the realist in his own way, but he too said there is no difference between what we imagine and what is external to us, so yeah why not.

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    1. It gets back to the construction of existence in one's mind, by the neuro-physical process explained in my free book. Until that is better understood to sort ideas as abstract and real, anyone might feel as justified as anyone else in proclaiming what exists. They might resist attempts at confinement to physical reality (in its myriad real mathematical forms, including the complexities of neural networks) on the basis that they 'just don't see it that way'.

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    2. Marcus - you are correct - the word real is meaningless - it should be changed to 'real for'

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  17. The main argument for mathematical Platonism (and against all other philosophies of Mathematics) is the “no miracle” argument. Platonism is the only interpretation that makes the applicability of Mathematics beyond really basic problems (like counting up to 10 or so) less than a total miracle. If Mathematics is a property of the mind (presumably produced by evolution in the African savanna,) why is Mathematics still useful? Why is all modern technology based on it? It is sort of ironic that all imaging systems used by neuroscientists, like PET imagers and fMRI machines are based on rather advanced math: any “cognitive” interpretation of Mathematics is refuted by the very tools that cognitive scientist use.

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  18. It might sound petty, but I think the first premise should be: There EXIST mathematical objects AND THEIR RELATIONS.

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  19. Dave,

    > Leibniz could be considered some sort of digital physics pioneer <

    That sounds pretty desperate to me, and at any rate it isn't what you were saying.

    > the panpsychic stances of Schopenhauer, Russell, Spinoza, and Whitehead <

    More desperation, not to mention that none of them falls under the headings of "current philosophers." You might as well bring up Newton's interest in astrology.

    Filippo,

    > The main argument for mathematical Platonism (and against all other philosophies of Mathematics) is the “no miracle” argument. <

    I agree. Interestingly, an analogous "no miracles" argument is the strongest point in favor of realism in science...

    Alastair,

    > Please explain to us why Sheldrake's hypothesis of "morphic fields and morphic resonance" does not qualify as a scientific one. <

    What would that accomplish? How is my opinion about one or another speculative or pseudoscientific notion relevant to the discussion at hand? At any rate, I don't take that particular hypothesis seriously for the simple reason that the relevant community of experts doesn't. I'm not a physicist, as you know.

    > if we can't measure these mathematical objects, then they are not amenable to "methodological naturalism" and therefore cannot be classified as natural objects. <

    Says who? Methodological naturalism is simply the stance that reality doesn't involve miracles, and there ain't no miracle in mathematics.

    > In a previous thread, you stated that you weren't proposing anything "mystical." However, it appears that how we gain access to these mathematical objects is completely mysterious. <

    We do it with our minds. Chop off your cortex and you won't be able to do math.

    > We have already established that we cannot objectively measure these mathematical objects. Therefore, no scientific evidence will be forthcoming. <

    Not surprisingly, you missed the point of the analogy.

    > Scientism is a term used, usually pejoratively,[1][2][3] to refer to belief in the universal applicability of the scientific method and approach, and the view that empirical science constitutes the most authoritative worldview or most valuable part of human learning to the exclusion of other viewpoints. <

    I told you, you really ought to get off Wiki and get down and dirty with the primary literature. Scientism is the view that only scientifically/empirically answerable questions are meaningful or worth asking. It is prima facie defeated by the very existence of math and logic.

    Marcus,

    > Massimo, I am not sure you have responded usefully to my post. <

    I keep failing, as usual.

    > Mathematical "objects" are physical, or they would not be objects in common usage of the word. <

    Once again you ignore what people say, then propose your misunderstanding as a problem in need of a solution, which you are happy to provide, preferably with a reference to your book. It's getting tiresome.

    > 'Ontology' (what really exists) reduces to the psychology of the mind that both abstracts and identifies realities <

    That would be news to any epistemologist.

    > It gets back to the construction of existence in one's mind, by the neuro-physical process explained in my free book. <

    I thought so.

    > An object is either physical and real, or it is not and therefore it is abstract. <

    Ahem, yeah...

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    1. The misunderstanding would be in the use of terms including "object" and "plato". You haven't explained my error at all. Basically, whether its palatable or not, consider revising the appraoch entirely, removing your refernce to "objects" and "plato" and seeing it for what it is, an unnecessary abstraction. Believe it or not, those mathematical expression are as real as the things they describe, and no more. And the mind is behind it all, whether ontologists like it or not.

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    2. I love what Sheldrake did with 'sense of being stared at'. Had not thought about it much and then tried a whole bunch of experiments on my own and felt he nailed this to a T. When I wasn't doing the staring and just watched for others who made sudden moves to see if they were being stared at, I remember that women did this a lot more than men. I don't know much about his other work, but this one? Pseudoscience? Give me a break.

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  20. I once met a man who pretended to be from my home town. He tried to convince other people in the bar that he was from my home town. They liked his attitude, and they didn't want to think critically about him. They wanted this man to be right, because he flattered and reassured them. So when I said, "I know that town quite well, and there is no intersection at Maple and Grand, because those two streets run parallel." People were a little annoyed with me. He laughed and said I must be mistaken. I showed them a map on my phone of the town. He said that Maple and Grand used to intersect, but Grand was renamed. I called my childhood friend and had him tell everyone on speaker phone that he, too, is sure that the streets have never been renamed. The man questioned who my friend was. He questioned our motives. The people wanted to believe him because he said he found the following things on the corner of Maple and Grand: that we are all experts and all equally smart, we all have books to write and interesting things to say, all of our thoughts are not only important, but actually capable of changing the world, we won't every really die. So for most of them, they were satisfied that whether or not the streets ever intersected, the fact that the man had named them was enough.

    Dave and Alastair, you are not from Massimo's home town.

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    1. They make sense to me. I guess I wouldn't be from Massimo's County, unless it were a REAL County.

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    2. While not from Massimo's home town, unlike the guy in the bar, I am not trying to deceive. Unless the guy believed he was from your town for some non-nefarious reason, and simply screwed up the street names... His comment about the streets changing names though sounds like he was happy to say anything and may have had deception in his heart. The other stuff he said, I do agree with, and the dying is just another definitional thing. Yes we die, in the classic and not well thought out sense. But pieces of us are everywhere and away from our bodies, so you want a really good definition of life before you talk about death, and I think all you have to show for it is a body that doesn't really do much post-death. If you have a very serious problem with your arm you are happy to swap it for some fake arm if its not too burdensome. I would swap my brain for a healthy one in a heartbeat. But even that line 'we won't ever really die'.... for all that we think we are, more mind than bodies, no I don't think we die because I do not think it is physically possible. I think we change.

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    3. Marcus, it could be that the entire academic world is full of nothing but liars and you alone see the truth. But just because something cannot be conclusively refuted does not make it true. So the next question becomes, How likely is it, that your free online book is an indispensable addition to the canon? What are our priors on that? What sort of work have you done before? Who have you so far convinced?

      Dave,

      Yes, sorry. I know you hold those beliefs, but I do not suspect you of willful dishonesty. I do think, however, that you are willing to overlook evidence of streets not intersecting that you would like to intersect.

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    4. Onedaymore, it could be, or it might not be, I will leave that question to you as it has nothing to do with anything I have said. The best start to answering your specific questions about my book is by reading it, or don't read it and don't start answering those questions, as the case may be.

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  21. OneDayMore - cool and agreed

    Massimo - not sure if you are painting a fine line between 'all is information' and 'there is no difference between real and abstract'. If so then we have to flip the idea of realism too, because of Bohm's work here. He was very much a realist, and he was very much a proponent of things emanating from one's imagination being as real as coffeepots. But the list of living philosophers who go along with all three ideas is long, I've pointed you to some. Many of them cut their teeth on philosophy because of their prior work with quantum theory. I will concede the desperation of presenting panpsychist theories as proof of this.

    Marcus - you hand your fellow realist Massimo have much to offer. I read your abstract and scanned the book. You talk about design without requiring a designer, could not glean more specifics on why the theory is so different than others, but enjoyed your likening a human body to a universe. But since you have invoked Wheeler on other blogs, I assume you have read his ideas about information, and do not follow why you distinguish between abstract and real, or why can't existence become a very personal thing, such that the reality of a coffee cup becomes a social contract among those who believe in the cup.

    I appreciate that those in the Western non-continental philosophy business have an obligation to break as little china as possible as they advance to newer terrain, so maybe to them ontology is the shop's plate-glass window.

    Whether or not one buys into what Bohm is saying the reason his ideas did not change the world is that their utility has not been found (or created, whatever). I remember in the early seventies crowing about how there was soon going to be a kind of personal computer at our beck and call and getting reaction among the non-technologically inclined that even if true, what would one do with it besides balance a checkbook? Value judgements change dramatically when utility changes, and information theory seems no exception. Sometimes I think it is as useless as string theory, but other times I think - well if decision-makers in academia thought it was a good idea, they could start looking for things from a pure informational point of view (a little like consilience) and abandon the idea that one would find more meaningful information in archaeology than say mythology or astrology. Something like that, but with a smatter of Princetonian Nash divining Cold War secrets using random bits of newspaper in "A Beautiful Mind", but using more analytical methods and relaxed causal, spatial and temporal constraints.

    To be clear, I am saying that 'There is nothing besides information leads to a breakdown of any ontology leads to 'anything is possible'. One could adopt the guideline that higher levels of meaning can be found by things held to be true by larger groups of people. This is why the coffee or tea cup reigns supreme in every walk of human life. It resonates with more people than does any specific science or religion.

    The arguments against 'information above all' hit hard and have been expressed multiple times on this blog. Foremost is the complaint that the mystical spiritual stuff remains hidden from common view, and why should that be so? Why no hard-hitting revelations in Times Square? I would say that mystical and spiritual stuff is a receding target as we learn more. Another complaint is that if anything is possible, then anything goes. My answer there is that anything is Darwinian, all ideas can be thought of having equal value, some will succeed and others will not. And then of course we have the old chestnut "Could Jesus microwave a burrito so hot that he himself could not eat it". I think he could yes

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  22. Dave S, I wont clog up Massimo's Blog with specifics about my book off topic (I'm confining myself to suggestions to have a look where appropriate, which you seem to have enjoyed to the extent you have gone into it). It's quite different from other theories. Wait till you get to the middle chapters on neurology. Wheeler... I might have mentioned him somewhere, perhaps in reference to space-time telling mass-energy how to move and mass-energy telling space-time how to curve?

    I assume he has broad theories relevant to the topic here, as you say. Reality as information might get closer to what Massimo presents as Mathematical Platonism than I would go. I attack it reductively (as I do everything) and look at the components of information (even physical things in digital sequences as computers or DNA). I don't see any mystery in information, and its utiliy would speak for its reality if its processes seem obscure (likewise indeterminacy in QM as a most extreme example under the same principle of reality evident from the processes underlying it).

    Existence can be whatever any individual or group believes it to be, if that belief is realizable in the world we observe, measure and put to rational enquiry. If it is only realizable by a person or group to the extent they buy into it (like hype), they could possibly exist entirely in that state as long as they can manage to feed themselves by mixing in some reality. Your group coffee cup recognition could apply in that context if it was a group hemlock cup recognition, but the principle applies in either case. Our abstracts are continually out there for us to live up to in reality. As for Massimo being a fellow realist, I'm not sure I would agree with that from reading our exchanges and his exchanges with Alastair.

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    1. Hi Marcus - was hoping you could spare a line or two on how and why your theories are different. But will check out the neurology part.

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    2. Sure Dave S, basically I go for predetermination all the way, which is mega-controversial in the current climate. I lay out an environment for evolution that predetermines structures all the way to humans, and is the product of cosmology all the way from the Big Bang, a big task to make the continuing causal connection all the way. In the process I lay out a fully anatomical basis for the structure of human awareness rather than simply relying on neurology (which the full anatomy uses).

      In awareness using neurology, I propose humans have the equal capacity for motor anticipation & sensory recognition (immediate future & past), whereas other species favor the sensory in adaptation to niches in which they are embedded. With anticipation, humans can control their actions and not merely react to the environment in a niche - we can plan, test, & stay a ahead of the game in a universal niche.

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  23. @ Massimo

    > What would that accomplish? How is my opinion about one or another speculative or pseudoscientific notion relevant to the discussion at hand? <

    The relevance is that both Sheldrake and Ladyman/Ross are proposing an immaterial realm of forms - a nonlocal realm that transcends both space and time. So, the question I posed to you earlier is very much relevant to the discussion at hand. Why is Sheldrake's "morphic field" considered to be pseudoscientific while Laydman's and Ross' "mathematical platonism" is not?

    > At any rate, I don't take that particular hypothesis seriously for the simple reason that the relevant community of experts doesn't. I'm not a physicist, as you know. <

    Sheldrake's concept of a "morphic field" is derived from the concept of a "morphogenetic field," originating in developmental biology. You are a biologist, a philosopher science, and a skeptic actively engaged in the business of debunking pseudoscience. Right? So, quit dodging the issue and answer the question.

    By the way, David Bohm (world-renowned quantum physicist) supported Sheldrake's theory, because he recognized the parallels between Sheldrake's theory of morphic fields and his theory of "wholeness and the implicate order."


    > Says who? Methodological naturalism is simply the stance that reality doesn't involve miracles, and there ain't no miracle in mathematics. <

    This is patently false. Methodological naturalism makes no such claim. You're conflating methodological naturalism with metaphysical naturalism - a conflation that qualifies as textbook "scientism."

    Also, you are proposing that the physical world somehow "magically" (that's the only appropriate word) emerged from an immaterial substrate of mathematical abstractions (ontic structural realism) - abstractions that exert no causal efficacy whatsoever. That sounds like a miracle to me!

    It is also noted that you weren't able to proffer any counter point to my argument: "If we can't measure these mathematical objects, then they are not amenable to "methodological naturalism" and therefore cannot be classified as natural objects"

    > We do it with our minds. Chop off your cortex and you won't be able to do math. <

    Your lame attempt at humor simply serves to highlight the fact that you can't provide us with any explanation how the human mind (presumbably physical on your view) accesses nonphysical mathematical abstractions.

    > Not surprisingly, you missed the point of the analogy. <

    It was a false analogy. (You're conflating physics with metaphysics) Based on your own argument, mathematical abstractions cannot be measureed objectively. Therefore, it logically follows that no scientific evidence will ever be forthcoming to establish their objective existence.

    > I told you, you really ought to get off Wiki and get down and dirty with the primary literature. Scientism is the view that only scientifically/empirically answerable questions are meaningful or worth asking. <

    Attacking Wikipedia is a losing strategy.

    "Philosophy in keeping with the new scientism only recognizes the existence of objects that science is already committed to."

    (source: pg. x, "Scientism: Philosophy and the Infatuation with Science" by Tom Sorell)

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  24. @ OneDayMore

    > Dave and Alastair, you are not from Massimo's home town. <

    You score points by attacking an individual's argument, not his "hometown."

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  25. "> They exist as neural connections or patterns of graphite or differences in voltage. <

    You are looking for physical existence, which is not required for abstract entities."

    I was thinking more along the lines of the anchor that abstractions have to something real. If someone is to argue that they exist without such an anchor, I would ask how such a thing could be shown. No abstract entity exists without some anchor to an objective medium, a canvas where it's imprinted.

    The vast unexplored territory of mathematics are potential abstractions, not yet existing as an abstraction, not yet imprinted anywhere.

    Speaking of them as existential entities seems to be an artifact of the human mind, as powerful an illusion as that of free will. Simply because the mind is the canvas where we hold these abstractions, we have no other means to consider them.

    I welcome further reading on this if you have a recommendation.

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  26. Dave,

    > not sure if you are painting a fine line between 'all is information' and 'there is no difference between real and abstract'. If so then we have to flip the idea of realism too, because of Bohm's work here. <

    Nothing of the sort. I was simply objecting to your inclusion of things like gods. I object to any metaphysics that blurs the lines so far that gods get included into the category of existing things, unless by existence one quite strictly means "in one's mind and imagination," which I know is not what you meant.

    > To be clear, I am saying that 'There is nothing besides information leads to a breakdown of any ontology leads to 'anything is possible' <

    Right, and I'm saying your are clearly wrong here, and your conclusion rests on equivocating on the word "real." And remember, this comes from someone whose ontology is broader than strict physicalism, as ought to be clear from this and other posts.

    > Why is Sheldrake's "morphic field" considered to be pseudoscientific while Laydman's and Ross' "mathematical platonism" is not? <

    Because morphic fields are entirely made up, there is no empirical evidence for them and they do no useful work in any area of science (not to mention the dubious association of the concept to Jung's pseudoscientific concept of collective unconscious).

    Mathematical Platonism (MP) is not science, and you shouldn't attribute the idea to Ladyman and Ross, who simply stated that it is *compatible* with their metaphysics. MP is simply a reasonable metaphysical stance to help us understand a very real phenomenon, the effectiveness of mathematics. You can't do science without math, all of science is done without morphic fields.

    > Sheldrake's concept of a "morphic field" is derived from the concept of a "morphogenetic field," originating in developmental biology. <

    I am a biologist, am familiar with the idea, and can tell you that there is no commonality between the two whatsoever. Morphogenetic fields (a term in disuse in this era of molecular biology, btw) simply refers to measurable gradients of chemicals that form during the development of an organism. There is no measurement of any morphic field (likely, because they don't exist).

    > By the way, David Bohm (world-renowned quantum physicist) supported Sheldrake's theory <

    Well, Newton though astrology was a good thing...

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    1. Massimo - ahhh, gotcha. Well however distasteful the notion of gods is to you (or programmer of the simulation we are in or whatever else you think is going on) that is probably the level of distaste I have for the god you call reality that is outside your and everything else's 'mind', and is an idea that is convoluted as kingdom come, and put bluntly, makes no freaking sense to me. It is 'nonsense in kilts', but your emperor is wearing no clothes.

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  27. Alastair,

    > This is patently false. Methodological naturalism makes no such claim. You're conflating methodological naturalism with metaphysical naturalism - a conflation that qualifies as textbook "scientism." <

    I know perfectly well the distinction between the two, and for the thousandth time, I am not inclined toward scientism! I thought it was clear from the context of our discussion and the reference to methodology, but let's spell it out clearly: methodological naturalism is neutral toward the existence of the supernatural (a metaphysical stance), and simply says that one does not need (indeed, should not need) to invoke miracles in understanding the natural world. Happy now? No, probably not.

    > you are proposing that the physical world somehow "magically" (that's the only appropriate word) emerged from an immaterial substrate of mathematical abstractions (ontic structural realism) <

    I am not proposing any such thing. This is the best understand we have from quantum mechanics (if you don't like Ladyman and Ross, read Krauss, an actual physicist).

    > Based on your own argument, mathematical abstractions cannot be measureed objectively. <

    You keep refusing to understand this: the very concept of measuring an abstraction is nonsense, drop it.

    > Attacking Wikipedia is a losing strategy. <

    I am not "attacking Wikipedia," I am simply point out that Wiki in a non specialist source. Whenever it conflicts with a specialist source you need to take the latter more seriously than the former.

    Joe,

    > I was thinking more along the lines of the anchor that abstractions have to something real. <

    Then I would agree, except that I wouldn't contrast "abstraction" with "real," but with physical.

    > No abstract entity exists without some anchor to an objective medium <

    Well, that's precisely what's under discussion. For a mathematical Platonist that's not the case, although it is the case that such anchor has to exist in order for abstractions to be perceived by our minds.

    > Speaking of them as existential entities seems to be an artifact of the human mind, as powerful an illusion as that of free will. <

    I am not at all convinced that free will is an illusion, but we've already had that discussion and I have nothing new to add to it at the moment.

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  28. I hope you don't mind. I was going to let this drop, but the discussion has been going on for so long that I'd like to respond to your answer to my comment.

    My basic concern is with the notion of existence. I don't see a definition in the post of the term. Perhaps it would be useful for you to offer one.

    In the mean time, you said "it is the truth of S [some mathematical statement] *plus* classical semantics that requires the acceptance of successful reference to mathematical objects."

    I'm afraid I still don't get it. What do you mean by "successful reference to mathematical objects"? If a statement is true of a model, but the model is completely hypothetical, why does that establish that the "successful references" from the statement to elements in the model require that those elements exist independently of the hypothesized model?

    I also asked how a concept can be independent of the mind(s) that think it?

    Your answer was that we aren't talking about concepts; we are talking about abstract entities. I'm not sure what you mean by an abstract entity unless you are presupposing that there are abstract entities -- which is the issue under discussion: do abstract entities exist outside of the mind.

    You went on to say, "As for the difficulty to see what it would mean for independence to fail, I suppose philosophers of math would say that if independence is not true than it is mysterious how math has the consequences for science that it does, and why mathematicians think of what they do as discovering, rather than inventing things."

    Those seem like two things. Mathematics has the consequences for science that it does because we have the ability to think. Other than that, I'm not sure what you mean by consequences.

    What mathematicians think (imagine?) they are doing seems to be a completely separate issue. Can an argument for the existence of mathematics objects be based on that?

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  29. This is a delightfully slippery concept. Thanks for the clarification.



    Does such a difficulty in understanding mean the contrary argument is stronger? Doesn't this reflect a possible failure of understanding instead?

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  30. Massimo,

    What mathematicians are doing is, strictly speaking, manipulating symbols in a formal system (axioms, undefined primitive terms, inference rules). Or, let's say, can be reduced to such manipulations. Look as much as you wish in mathematical journals, you will never find a result that could not be so formalized.

    As such, a mathematical object is simply a construct built form these rules and to say that a proposition is true is to say only that it can be deduced from the axioms of a given formal system using its inference rules. Thus, in a real sense, mathematical statements are not about anything at all. If they can be said to be objective, it's only relative to a given formal system.

    Given this, it's not clear at all what are these mathematical objects you're talking about. We certainly use intuitive mental representations when we talk about mathematical abstractions but they are not the “real thing”.

    Of course, it is a remarkable fact that mathematical reasoning can be applied to real objects and allows us to make reliable predictions. This needs to be understood – but, from a consideration of what mathematicians are actually working with, it seems to me this should be done in the context of formal systems.

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    1. I gave a reply to Claude Pellegrino above along the lines that truth can be a misleading concept, as any self-consistent axiom would be true in itself but not necessarily real unless it applies to physical things in their units and with their logical order. In another post on Smolin I began with a basic analysis of number & logical order as a universe of individuals numbered and ordered as we wish mathematically (applicably), and as individuals with relations likewise (applicably).

      So there's a real world where there are numbers and ordering of things individually and relationally, and beyond that there are other self-consistent numbers & ordering which would be abstract to the extent they are not applicable to real units ordering real events. Kant's analytic-synthetic distinction might be relevant, but he seems to overly value 'truth' to remain in an ontological cloud.

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    2. Your description of formalism represents a view of Mathematics that was fashionable one hundred yeas ago. It has now been largely abandoned. There are two different, but related, reasons why this is so. The first reason is the “no miracles” objection. If doing mathematics consists of “manipulating symbols in a formal system (axioms, undefined primitive terms, inference rules)”, then why is the result of such a purely formal process any more useful in the real world than, say, playing chess, which also involves working within well defined, arbitrary rules. If you say that the rules are not arbitrary, after all, then the problem arises of where the rules (axioms and inference rules) of mathematics come from. Formalism cannot provide the answer.

      The second problem is the discovery of Gödel's incompleteness theorem. Formalism has been devastated by Gödel, Turing and company: it is not a viable option anymore. The simplest explanation of this fact was given by G. J. Chaitin: algorithmic information theory shows that any set of axioms contains way too little information to constrain even a simple(?) field of math like arithmetic. You cannot extract more information out of the axioms than the axioms contain to start with. (Information is conserved.) Because of this, progress in mathematics nowadays involves adding to the rules of mathematics.

      New axioms or inference methods need to be added in order to avoid getting stuck. Quoting Gödel: '...Perhaps also the apparently unsurmountable difficulties which some other mathematical problems have been presenting for many years, are due to the fact that the necessary axioms have not yet been found. Of course, under these circumstances mathematics may lose a good deal of its “absolute certainty;” but, under the influence of the modern criticism of the foundations, this has already happened to a large extent...'

      Note that, in the previous quotation, Gödel sounds like a post-modern critic of science (mathematics may lose a good deal of its “absolute certainty”...) In fact, Gödel, arguably the most important mathematician of the 20th century, was an unrepentant Platonist (who are we to disagree...) This is shown in the quote by the use of the word “found” with respect to new axioms. New axioms of mathematics are found out there, like the laws of physics, not arbitrarily defined or fished out of the depths of the human mind.

      I apologize for adding arguments from real mathematics to this content-free discussion of mathematics...

      References

      Gregory J. Chaitin, The Limits of Mathematics, Springer, reprinted in 1998. This book is a model of a modern math book: it is very informal and it is mostly LISP and Mathematica code. You don't have to type the code in yourself, fortunately: it can be downloaded from Wolfram Research (or, at least, it could be last time I checked.)

      Thomas Tymoczco (Ed.), New Directions in the Philosophy of Mathematics, Princeton University Press, reprinted in 1998.

      The quote by Gödel is from his contributions to Philosophy of Mathematics, P. Bencerraf and H. Putnam (eds.), Prentice-Hall (1964)

      Roland Omnes, Converging Realities, Princeton University Press (2005). “...one gains a deeper understanding of the foundations of quantum mechanics when one can rely on the axiom of choice. Said otherwise, this axiom definitely belongs to the language of physical reality, at least as we understand it at present.”

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    3. Hi Filippo,

      Thanks for your thoughts.

      I think Gödel's theorem is a problem for the following view: (1) there exists some kind of an external realm of mathematical objects/truths; and (2) we can get at it through a formal system. This is not what I am saying at all, as I find (1) very problematic.

      And, yes, Gödel was a Platonist. He believed for instance that the continuum hypothesis (although shown to be undecidable using common axioms) is nevertheless true of false in some absolute sense. With all due respect to old Kurt, I must disagree on this one.

      But, whatever one may think of this elusive realm, fact remains that mathematical theories can be formalized. All the theorems of mathematics ever used in physics are deducible from the axioms, and from a relatively small set at that.

      And here's the thing: all these theorems found so useful and applicable to the real world can be deduced in a completely formal manner from the axioms – with no reference whatsoever to spooky mathematical objects and all. Take this as an alternate formulation of the so-called “no-miracle” question: how can purely formal deductions, these (mechanical?) symbol manipulations within a formal system, produce these useful results? I don't see how mathematical Platonism helps to solve this.

      You are right to point out that axioms must be selected and that formalism is no help here. But the problem at hand is the question of predictions and it is the same whatever method we use to select axioms. I also agree all mathematical results are implicit in the axioms (you formulate this in terms of information), the same way all chess games, however brilliant, are implicit in the rules.

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    4. Unfortunately I must avoid much of the detailed referencing as its mostly unfamiliar to me, but I think the problem boils down to the human capacity to abstract mathematics and also apply it to the world. The fact that the world can be understood mathematically, by definition means that the world is mathematical in its properties so described. The fact that chess players can use mathematics in a less useful way means that they can abstract a use for it for their amusement.

      The relation between something abstract and its real application is the purview of rationality (continually matching the abstracts that proceed realization at every moment of thinking). The realtion is not concrete, and the issue is whether there are pure abstracts at all, given there will some elements of an abstract that is of use in some way to model reality, even if put togther in a novel way.

      I would go with an innate capacity in humans to observe, abstract from observation, apply the abstracts to model reality, and to try to model it also for its own sake at times (as games for example). As I suggest, the line of abstract is not concrete and it is an ongoing process of human thought to sort what is extrapolated from observation and fashioned into novelty with greater or less modelling potential in reality. QED

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  31. @ Massimo

    > Because morphic fields are entirely made up, there is no empirical evidence for them and they do no useful work in any area of science (not to mention the dubious association of the concept to Jung's pseudoscientific concept of collective unconscious). <

    Sheldrake has presented "morphic fields and morphic resonance" as a scientific hypothesis, not as an established scientific theory. There's a difference between the two. A "hypothesis" only has to be testable. A theory has to be empirically validated through repeated experiments. So, the question I am actually asking here is: "Why do you believe Sheldrake's hypothesis is not testable?" Because, if it is testable, then it qualifies as a scientific hypothesis.

    (Whether psychoanalysis (Freud and Jung) is scientific may be controversial, but the bottom line is that psychotherapy works.)

    > You can't do science without math, all of science is done without morphic fields. <

    The point is that your immaterial realm of mathematical abstractions qualifies as s kind of morphic field. Moreover, you have provided us with no explanation how the human mind accesses these mathematical abstractions; Sheldrake has. So, you're hardly in a position to be criticizing Sheldrake's morphic fields - especially when you're presupposing what essentially amounts to a morphic field!

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  32. @ Massimo

    > I am not proposing any such thing. This is the best understand we have from quantum mechanics (if you don't like Ladyman and Ross, read Krauss, an actual physicist). <

    Krauss is invoking "magic" too. Of course he doesn't acknowledge it as such. But anyone who asserts that "something can arise from nothing" (and presents it as a naturalistic explanation) is invoking magic. Either that, or he is redefining "something" as "nothing." Either way, we are dealing with someone who is given to intellectual dishonesty.

    > I am not "attacking Wikipedia," I am simply point out that Wiki in a non specialist source. Whenever it conflicts with a specialist source you need to take the latter more seriously than the former. <

    I have already provided you with a direct quote from the "specialist source." Perhaps, I should provide you with the quote once more...

    "Philosophy in keeping with the new scientism only recognizes the existence of objects that science is already committed to."

    (source: pg. x, "Scientism: Philosophy and the Infatuation with Science" by Tom Sorell)

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  34. Massimo,

    Truth, as commonly conceived, is a relationship [it need not be correspondence] between language and an extra-linguistic reality. Thus, 'Belfast is in Northern Ireland' is true because of certain objective sociological and geographical arrangements that obtain in the British Isles. However, what is the extra-linguistic reality that corresponds to the truth of mathematical statements? I have never found much purchase in postulating objectively existing mathematical objects.

    Re: Frege's argument.

    Premise 1 (Truth): Most sentences accepted as mathematical theorems are true.

    Premise 2: Let S be one such sentence.

    Premise 3 (Classical Semantics): The singular terms of the language of mathematics — such as S — purport to refer to mathematical objects, and mathematical language’s first-order quantifiers purport to range over such objects.

    Premise 4: By Classical Semantics, the Truth of S requires that its singular terms succeed in referring to mathematical objects.


    First, if we understand truth in the way I noted in my first paragraph, (P1) is question begging: If I do not accept that there are such things as mathematical objects, I will not accept (P1).

    Second, let us admit as a sociological fact that most sentences expressing mathematical theorems are accepted as true. We need only construe the truth predicate as we apply it to mathematical statements [e.g., '13 + 10 = 23' is true] in a deflated sense – i.e. in much the same sense we apply the predicate to statements of fictional objects [e.g. 'Sherlock Holmes lives at 221B Baker Street' is true].

    This second point leads us also to rejecting (P3): It makes perfect sense to quantify over objects that we believe not to exist. Take again the Sherlock Holmes example. I can express 'Sherlock Holmes lives at 221B Baker Street' unproblematically in the following way: (∃x) (Sx & Lx) [translated: there is some x such that x is Sherlock Holmes and x lives at 221B Baker Street]. This statement is at once meaningful & true. Insofar, then, as I can remain a fictionalist about fictional characters and objects, I can plausibly remain a fictionalist about mathematical objects.

    In short, (P3) attempts to impose upon us a much too simplistic criterion of ontological commitment. There are ways of making ontological commitments to be sure, but we are not committed to the principle that to be is to be the value of a bound variable!

    More generally, Frege (and other purveyors of unseemly metaphysics) essentially argue that certain linguistic practices come along with certain ontological commitments (in some sense they're correct: I think linguistic practice commits one to the existence of interlocutors, e.g.) but my response is simply that for every purported realist-committing sentence, there are straightforward nominalistically acceptable paraphrases.

    P.S. All apologies for commenting late in the game.

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  35. It seems to me that you haven't really come to grips with what it means for a mathematical object -- let's say, the number "31", to "exist". Commenters have asked for clarification, but you haven't provided one.

    I am a professional mathematician. I used to be a mathematical Platonist, but I abandoned it because I could not find any convincing argument that mathematical objects have some existence independent of human minds. The "unreasonable effectiveness" arguments seem to be missing the mark. We use mathematics to describe physical reality because it is useful and effective; if dancing or architecture were more useful, we might use them instead.

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  36. All,

    it has been a fun discussion, but I think I am reaching the point of diminishing returns, so I will move over to the new post on Krauss...

    Russ,

    > My basic concern is with the notion of existence. I don't see a definition in the post of the term. <

    It's implied by the argument for Existence: it deals with successful reference.

    > If a statement is true of a model, but the model is completely hypothetical, why does that establish that the "successful references" from the statement to elements in the model require that those elements exist independently of the hypothesized model? <

    It doesn't, remember that that argument only establishes existence, not mind independence, it is crucial not to confuse the two.

    > I'm not sure what you mean by an abstract entity unless you are presupposing that there are abstract entities <

    The argument in favor of abstraction is different from those in favor of existence and mind independence, as explained in the main body of the post. You may want to check out the original SEP article that I link to, and perhaps some of the references therein.

    > Mathematics has the consequences for science that it does because we have the ability to think. <

    That is not an explanation.

    > What mathematicians think (imagine?) they are doing seems to be a completely separate issue. <

    Correct, but it is a relevant issue.

    JP,

    > it is a remarkable fact that mathematical reasoning can be applied to real objects and allows us to make reliable predictions. <

    Indeed, it is that remarkable fact that mathematical Platonism is a potential explanation for.

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  37. Alastair,

    thanks for the lesson in scientific method 101, though I think I knew the difference between theories and hypotheses already.

    > the question I am actually asking here is: "Why do you believe Sheldrake's hypothesis is not testable?" <

    A better question is: where *is* it testable? Answer: nowhere.

    > The point is that your immaterial realm of mathematical abstractions qualifies as s kind of morphic field. <

    No, it doesn't, partly for the simple reason that the concept of morphic field is empty.

    > Krauss is invoking "magic" too. <

    Add one more professional academic to be replaced by Wikipedia wisdom.

    > "Philosophy in keeping with the new scientism only recognizes the existence of objects that science is already committed to." <

    Congrats for having advanced to at least consider what professionals have to say. But it ought to be clear that Sorell's definition doesn't even apply to people like Ladyman and Ross - who are among the best candidates - so it is hard to imagine how it applies to "philosophy" in general.

    Mike,

    > Mathematics are abstract representations of the features and behavior of the physical universe. <

    Most mathematics has nothing whatsoever to do with representing the universe.

    > this kind of stuff reminds me of why I stay away from metaphysics. <

    Sorry to hear that. Perhaps you will consider reading Ladyman and Ross as an antitdote to your instinctive antipathy for metaphysics.

    Eamon,

    > what is the extra-linguistic reality that corresponds to the truth of mathematical statements? I have never found much purchase in postulating objectively existing mathematical objects. <

    It gives an account of the otherwise miraculous effectiveness of mathematics.

    > If I do not accept that there are such things as mathematical objects, I will not accept (P1). <

    Yes, but that would be rather silly, at least if you know anything about math (and I know you do).

    > It makes perfect sense to quantify over objects that we believe not to exist. Take again the Sherlock Holmes example <

    You may be confusing existence with mind independence. As you probably know, some philosophers would actually say that Holmes does exist, in the very narrow sense that you define. But I don't buy that, I think that the difference is that in the case of Holmes we have to *stipulate* whether statements about him are true or not, in the case of mathematical truths anyone with sufficient understanding of math can arrive at true statements regardless of any such stipulation.

    Jeffrey,

    > Commenters have asked for clarification, but you haven't provided one. <

    I have. As I said above, it is embedded in the argument for existence given at the beginning of the post. You will find more in Linnebo's original article.

    > The "unreasonable effectiveness" arguments seem to be missing the mark. We use mathematics to describe physical reality because it is useful and effective; if dancing or architecture were more useful, we might use them instead. <

    But we don't. You are reversing causal roles here. As a mathematician you should appreciate that most math has nothing to do with the physical world and is not developed because of it. And yet, frequently it happens that very abstract mathematical concepts just happen to match surprisingly nicely with the physical world. Hence the no-miracles argument (which, as I pointed out, works by the same logic as the very convincing one deployed by scientific realists).

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  38. As I said above, it is embedded in the argument for existence given at the beginning of the post.

    OK, then you've made a remarkably weak and unconvincing argument. Fine with me!

    As a mathematician you should appreciate that most math has nothing to do with the physical world and is not developed because of it.

    This is simply historically wrong. From Archimedes to Heaviside to Dirac, much mathematics was developed for its use in physics.

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  39. Massimo,

    I understand you wish to bring this thread to a close, so I won't expect a response.

    I am not claiming (nor do I think it wise to claim) that mathematical propositions are not true. Rather, I am claiming we can say that mathematical propositions are true while, strictly speaking, holding to the non-existence of so-called mathematical objects. [Though, I think framing the truth of mathematical propositions in terms of truth conditions is misguided -- we should instead talk about proof conditions.]

    The bit about fictional objects goes only to show that (P3) is strictly false: We can quantify over objects which we hold not to exist. Of course I agree that there is something disanalogous between the two cases, but I don't need them to be analogous in order to reject (P3).

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  40. @ Massimo

    > A better question is: where *is* it testable? Answer: nowhere. <

    "A experimental test for the hypothesis of formative causation."

    > No, it doesn't, partly for the simple reason that the concept of morphic field is empty. <

    I stand by my previous statement. Your "mathematical platonism" presupposes something very much akin to a morphic field - an immaterial realm of abstract forms. (There's no point to debate this issue any further because you will simply continue to deny the parallel - a denial that doesn't even begin to address the argument I just put forth.)

    > Add one more professional academic to be replaced by Wikipedia wisdom. <

    You're attempting to defend what is ultimately indefensible. Why there is something rather than nothing is a question that science cannot address.

    > Congrats for having advanced to at least consider what professionals have to say. But it ought to be clear that Sorell's definition doesn't even apply to people like Ladyman and Ross - who are among the best candidates - so it is hard to imagine how it applies to "philosophy" in general. <

    To reiterate: Ladyman and Ross begin their book ("Everything Must Go") by defending "scientism." (The first chapter of their book is entitled "In Defense of Scientism.")

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  41. I've only just seen this and haven't had a chance to read all the comments, but I would urge Massimo to be more sceptical of the idea that mathematical platonism (or even anti-nominalism) is compatible with naturalism, not least given the seemingly insuperable epistemological difficulties it gives rise to.

    In a nutshell, the platonist needs to provide a naturalistically acceptable explanation for the consistent reliability of the correlation between mathematical beliefs and the mathematical facts, but since (ex hypothesi) the latter casually inert, exist outside spacetime, and are wholly mind- and language-independent, it's hard to see how any naturalistic explanation is possible even in principle.

    Indeed, if one’s philosophy of mathematics is platonist it seems difficult to avoid Mark Steiner’s conclusion that there must be some kind of pre-established harmony between the rational structure of the universe and the a priori cognitive faculties of the human mind. I take it that if mathematical platonism does lead to such a conclusion, this must be taken by the naturalist as a reductio of mathematical platonism.

    At the very least, a world in which there exists eternal, intelligible mathematical entities and structures knowable purely conceptually suggests a rationalist metaphysics and epistemology which the naturalist ought not to countenance merely on the grounds that the prima facie meaning of mathematical truth statements suggests that there ought to be something to which such statements refer.

    Stewart Shapiro, the leading proponent of mathematical structuralism (of the so-called "ante-rem" or platonist variety), at least recognises where the burden of proof lies here:

    "Any faculty that the knower has and can invoke in pursuit of scientific knowledge must involve only natural processes amenable to ordinary scientific scrutiny. The realist thus owes some account of how a physical being located in a physical universe can come to know about abstracta like mathematical objects. […] The burden is on the realist to show how realism in ontology is compatible with naturalized epistemology".

    Unfortunately, however, Shapiro’s own attempts to sketch such an epistemology by recourse to a just-so story about pattern recognition and abstraction at best provides a hypothetical account of elementary arithmetical *concept-acquisition*. Apart from the fact that cognitive science does not bear out these hypotheses, showing how one might acquire concepts of mathematical objects is obviously not sufficient to establish that the objects to which these concepts purport to refer do in fact *exist*. In response to this objection Shapiro states that when it comes to mathematics "an ability to coherently discuss a structure is evidence that the structure exists". Why this should be the case for mathematics when it is patently not the case for any other area of human discourse Shapiro does not tell.

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  42. There's also Pat Churchland's suggestion that mathematical platonism is *biologically implausible*, given what we know about the kinds of organisms we are and how we obtain information about the world. Not only must the naturalist who is also a platonist explain how mathematicians are supposed to gain access to eternal, causally inert entities or structures, and a nontrivial account of how they are referred to ('nontrivial', that is, in the sense that it cannot amount simply to stipulation, as Shapiro's does), but he or she is also obliged to provide a plausible account of the selective advantage that being able to do so would confer upon our ancestors. To paraphrase Churchland, what would have been the evolutionary pressure for the emergence of a special faculty for discerning immutable rational structures that do not exist in the physical universe? What could have been the nature of such a pressure? Is there a plausible account consistent with natural selection that can explain how humans could come to have such a capacity?

    Finally, far from *explaining* it, it seems to me that platonism makes a *mystery of the applicability of mathematics in science. As Charles Chihara asks, why would empirical scientists need to discover complex and complicated relationships between entities that do not belong to the physical universe in order to develop their empirical theories about what *does* belong to it? Why is knowledge of these undetectable, nonphysical, causally inert entities needed in order to discover facts about the empirical world?

    Unless a feasible epistemological account is given of how such ontologically mysterious objects may be accessed and interacted with such that they *constrain* mathematical theorising, along with an account of how the postulation of such objects explains the applicability of mathematics to the physical world, it seems to me that their postulation is explanatorily redundant and thus ought to be eschewed on grounds of ontological parsimony.

    There are many other objections one could raise to the idea that mathematical platonism is compatible with naturalism, but I’ll leave it at that for now.

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    1. Damian,

      re: Finally, far from *explaining* it, it seems to me that platonism makes a *mystery of the applicability of mathematics in science.

      Well said. What these so-called mathematical objects are and how we can come to know them is under mathematical platonism a far greater mystery than how a non-mathematical realist might go about explaining the utility of mathematics to the sciences.

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  44. @ Filippo

    Hi,

    I would agree with much of what you say. Mathematicians are certainly not mainly concerned with formalizing their work. I should say perhaps that I, too, have done my share of math. That was some time ago but I did study math for the best part of 10 years, with some undergrad teaching thrown in, stopping just short of a thesis (to work in software). My interests were more into analysis and pure mathematics than yours (from what you mention in your comment). But I can fairly say that I know the trade.

    This question we're discussing fascinates me and there are obviously many facets to it. The reason I brought formalism in is not because I take formalism as the most important or even significant part of mathematics. No, of course not (although it is always present in the background).

    I brought it in because I believe formalism provides a very useful point of view in analyzing the predictive power of mathematics. This “power”, I think, expresses itself at least in the following way: we start with some premises (axioms or whatever), apply mathematical reasoning and derive some theorem. Next, we “map” parts of reality (some objects) to the mathematical entities under consideration and expect that the theorem we proved mathematically will also apply to the given objects.

    Now, here's what formalism does for us. The proof of the theorem in question can be written formally in terms of what are essentially mechanical steps. Then, what seems to happen, is that these formal steps can be transposed and apply to real objects (if they fit the premises of our system well enough). I think this process of transposition of formal manipulations to real objects (back and forth) is the key here.

    A simple example. The number 101 is prime – this is our purely mathematical theorem. Now, we (think) we know that, were we to select, anywhere in the universe, 101 small objects (roughly the same size), it would be impossible to distribute them on a plane surface in a (non trivial) rectangular pattern. Why? Because, if we could, this physical manipulation could be transposed back into the formalism of arithmetic and provide a factorization for 101. (Not sure what this says about our universe.)

    So, you see, we have this equivalence between formal and physical manipulations. My feeling is that, perhaps, more complex situations could be analyzed the same way. This is not to say there is nothing to explain - but, at least, this is a situation we can analyze. What do you think?

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  45. So many ism's for a basic issue of abstracting by mind and applying it (or not) to reality. Mathematics is real to the extent it applies to describe physical properties (including spatial & temporal properties of physical individuals and their relations) and otherwise not.

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    1. Having mentioned space & time (which are fundamentals) I realize that physicality is unclear under Relativity Theory, which attemps to reduce the gravitational (physical) "force" to mathematical geometry. The question arises whether mathematical geometry therefore suffices as an "object" without any physical substantiation, if it is strictly geometrical.

      Here we are at the edge of practical examples & knowledge. To avoid matematical platonism of some kind here, it requires the physical substantiation of the force of gravitational attraction (by gravitons, for example, rather than as pure space-time geomtery). It is physical, and the geometry would be the most fundamental property of the physical field gravitons and their interactions with massive particles. Realtivity Theory without gravitons is the best hope for mathematical platonism.

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    2. @Marcus

      We seem to be confusing math and physics here: general relativity (GR) is a physical theory and so it is supposed to be more “substantial” than pure mathematics.

      Also, gravitons are Bosons and so, like all elementary particles, they are indistinguishable. That is, any graviton is indistinguishable from any other. As Ladyman and Ross brilliantly argue in ETMG, indistinguishable particles don't have many of the properties of physical objects envisioned by common-sense realism. A number of gravitons is not so much a plurality of objects as a label denoting a particular quantum state. (The same thing is, of course, true of a certain number of photons.) As Dirac observed, indistinguishable particles don't have definite positions in space at all times. If they did, their positions could be used to distinguish one from the other. Don't be so sure that adding gravitons will make GR any less “Platonic.”

      Also, indistinguishable particles can't have local hidden variables that determine their otherwise random interactions. (Note that the term “hidden” is a misnomer: the “hidden” variables are supposed to determine the results of experiments and so they are not permanently hidden.) The local not-so-hidden variables could be used to distinguish the particles. This is one of the many reasons why all professional physicists were sure that the Bell inequalities would be violated, even before the Aspect experiment was performed: it was not a question of metaphysics. Of, course the consensus of the professional physicists was totally right. This is one of the facts ETMG is wrong about (they don't affect the general argument, however.)

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    3. Filippo, physics is applied math, and pure math can be abstract, that's the only difference. I'm returning to the reality of physics, which is not affected at all by L & R or mathematical patonism. Space & time under Relativity theory are non-physical (just pure geometry). It's an obvious and necessary point to make that if physical gravitons exist, then Relativity is not pure geometry.

      You misunderstand the reference, and the relevance of L & R and mathematical platonism. Returning again to reality, it doesn't matter whether gravitons are considered as a cumulative field and photons likewise. Indeterminacy is a consequence of failure in measurement, and its not necessary to consider photons in cumulative groups. Individual photons are emitted and absorbed with specific energies, so your point is an obfuscation. Gravitons? we will wait and see.

      You, L & R, and Massimo are just obfuscation using defintions and ontology to raise abstracts to reality. Real application is to physicals (physics), and we need accept limitations in measuring physicals. They have common properties measurable in conjunction, including spatial, temporal, interactive, mass-energy, electro., grav., and so on. No need to invent another level of reality for their properties: they are physical properties accurately described by math. Back to reality.

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  46. @ JP
    Hi JP,
    I agree with you that math is mostly formalism, meaning that it mostly involves deriving results from definite axioms using definite derivation rules. After all, I am an algebraist. I cannot think of math except in the form of strings of symbols - my geometric intuition is close to zero, unfortunately. However, formalism is not what math is about. It is about solving problems in the real world and formalism as a philosophy of math is a total failure in explaining why math is successful in such applications. A materialist form of Platonism seems to be the only explanation of this success: mathematical structures are meta-structures of science - the laws of the laws of nature, so to speak. (Note that the former statement is not a mathematical or scientific statement – it is a metaphysical statement. People that say that they don't believe or ignore metaphysics usually are just professing a naive form of metaphysics: metaphysics is unavoidable for any thinking person.) The relationship of this form of Platonism to structural realism (SR) - not necessarily “ontic” - is obvious.

    SR and Mathematical Platonism (MP) are not strange and exotic metaphysical systems. They are the implicit “metaphysic” of the “shut up and calculate” physicists and applied mathematicians. Applied mathematicians, in particular, expect that their math will work for every structurally similar problem, independently from the objects being treated. Is this “object independence” that makes math so powerful. Linking the reality of math to particular physical objects would make its power even more of a miracle than it already is. Also, applied mathematicians are expected to produce results in the real world, not in some sort of formal game. I know: math has paid my mortgage, etc. - until I retired from the Los Alamos National Laboratory. Formalism as a metaphysics of math is a nonstarter.

    Formalism does not explain the progress in mathematics. How does mathematics progresses? The greatest successes of mathematics are not obtained by working carefully within existent systems of axioms. They are not obtained by “playing by the rules.” They are obtained by changing the rules of the game. Think of non-Euclidean geometry: you drop Euclid's fifth axiom, derive new geometries and in a century or so you have general relativity. So, where is the next opportunity for progress in mathematics? My guess is that the next big thing in mathematics will involving changing not only the axioms, but the rules of derivation. In particular, the use of probability methods in the derivation of mathematical “theorems” only probably, but not certainly, valid should significantly extend the reach of useful mathematics.

    [I have made some corrections and removed the last section that was mostly cut-and-paste from the introduction to one of my crazy papers.]

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  48. I'm interested to know if the 3rd statement, of whether mathematics is independent of us, may have any relation to quantum theory of un-limitless possibilities and how our observation (as in the slit experiment) changes things?

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