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Two ways of saying this:

  1. Each philosophy of math has a direct counterpart mathematics: for intuitionism, it's a mathematics of intuitions (not as "hunches" but in the Kantian sense); for predicativism, it's predicativist concepts; for ante rem realists, it's abstract objects, etc.
  2. Not setting aside the complexities of relating different mathematical subdisciplines, it might be said that e.g. intuitionism goes with geometry (at least, some geometry) or topology or (esp. nonstandard) analysis, ante rem realism with set theory, formalism with algebra, etc. This doesn't seem altogether plausible, since it could be thought that the correlations were more varied (so predicativism might go with both number theory and algebra, maybe; or algebra might go to multiple philosophies-of-math).

Category-theoretically: can each philosophy of math be reformulated as a category theory over the objects promoted by each?

EDIT: or might there be a mathematics "proper to" the philosophy of mathematics "as a whole"/in general? One is tempted to say, "That's what logic is," although this answer is less resolute than desired (for logic is varied, too). Alternatively, perhaps some philosophies-of-math are philosophies-of-philosophies-of-math, e.g. fictionalism might occur on this level of representation.

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    – Philip Klöcking
    Commented Feb 18, 2023 at 21:15
  • @PhilipKlöcking, is there a way to unfreeze a chat? I think there's more discussion to be had here, but the linked chat has been disabled, so to comply with the general principles, it'd be great to get this reactivated.
    – Paul Ross
    Commented Mar 12, 2023 at 18:28
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    I'm going to give answering this a go, though the scope of my answer is going to be fairly restricted - Intuitively, I think you're looking for some kind of distinction between platonism and intuitionism that says intuitionists admit a more constrained body of theory than the platonists do, but the real challenge is going to be drawing a distinction within category theory between the Platonist and the Formalist. Surely one ought to think that formalism is more permissive than platonism, and yet if one accepts Category theory as a foundation, no such distinction seems possible.
    – Paul Ross
    Commented Mar 12, 2023 at 18:44
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    @PaulRoss You can't have discussions here, this is the Philosophy Forum!
    – user4894
    Commented Mar 12, 2023 at 20:00
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    @PaulRoss Done. One needs moderator privileges to do so.
    – Philip Klöcking
    Commented Mar 12, 2023 at 20:16

2 Answers 2

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(apologies for the wild ramble. I'll probably regret this in the future)

I like the core concept at work here, with this idea that there is a manifest difference between various philosophies of mathematics that sees itself play out in the realm of mathematical practice. Certainly, the identification of Platonism with Ante Rem realism over the domain of (higher) set theory seems to be a very common understanding (consider the foundational role of Representation Theorems in many subdomains), and the view of Formalism as a more algebraic concept of mathematical practice (building upon finite methods) echoes Hilbert’s concept of the multiple realizability of ideal mathematical objects.

For Hilbert’s view (neatly introduced in https://plato.stanford.edu/entries/frege-hilbert/#HilbFounGeom ), the grounding foundation of mathematics is in the structures formed by the concepts outlined in axioms. One comes across new kinds of mathematical structure by showing that consistent sets of axioms can be worked provably independently of one another, relative to some familiar background theory.

Far from “axioms” representing an ideal Truth of the matter, as is taken as read by the realists, axiom systems are general technologies. Such systems have at least to be able to demonstrate that they do not collapse into classical trivialism, but that consistency relative to other areas of practice should be enough to prove their viability and potential usefulness, even if we don’t have a specific interpretation for them in mind.

For the realist, like the intuitionist, a demonstration of consistency shouldn’t itself give rise to a substantial existence claim (just like the intuitionist’s concerns about the indeterminate existence proofs by contradiction). For the formalist, the nature of such existence claims does not manifest in something like abstract objects, but is instead understood simply with reference to the axiomatic theory.

If phrased in terms of a simple first order theory, there is nothing more mysterious to mathematical objects than any other kind of object, in that those things that realize the axioms can be things in the positions indicated by the theory – that “Love, Law and Chimney-sweep” might be the placeholders of the existence claims of geometry, in the sense the Platonist thinks of an abstract object domain of points and lines.

So, in some general way, the formalist is not talking about the same domain of objects as the realist – if for a formalist it suffices to talk about the relative consistency of geometric systems interpreted in the algebra of the real number field, then there is no question of a distinct abstract sphere of geometry in the sense in which the real numbers are set-theoretic objects to the realist. For the Ante rem set theoretic realist, this can only pass muster if the objects of geometry are also shown to be definable as sets, where the formalist is content merely that the theory is represented using an algebraic proof/model structure, whatever one might take as the ontic structure of the models about which theorems are proven.

For working mathematicians, this distinction rarely comes into focus. While Godel’s point about the limitations of consistency as an idiom is far-reaching, in actual fact most abstract algebra are well understood as having the legitimacy of concrete set theory with reference to the proof of Representation theorems.

To make some headway into your question, if we want to try to come up with an understanding of what it means to understand relative interpretation of axiomatic theories, some metamathematical structure is probably useful to introduce. (This is an issue posed by Frege, as mentioned in Alan Weir’s SEP article around a challenge for the Game formalist: https://plato.stanford.edu/entries/formalism-mathematics/#GamTerFor. For the sake of your question, let’s take this metatheory to be category theory)

So let’s propose that we might want to try to come up with some sort of Category-theoretic distinction between these two ways of looking at the construction of models over an algebra. Perhaps the morphisms of our category might represent consistent model-theoretic interpretation, with the objects of our category being the “domains” of our proposed theories. Our realist thinks that every category must terminate in Set, while our formalist is happy to disregard the actual objects of the model theory as long as the morphisms are structure-preserving in the key model theoretic sense. (This should indicate to us that we are in some way dealing with a functor category)

Any attempt to demonstrate that there is something specifically distinct about the naïve formalist view, particularly of Hilbert’s initial finitist aspirations, is dashed against the rocks of the Yoneda Lemma (https://en.wikipedia.org/wiki/Yoneda_lemma). By accepting the category-theoretic interpretation of a grounding metatheory to describe the structure of locally small categories, we fall prey to the inevitable realization that we could, without any loss of generality, do this all in the category of sets anyway.

It’s only in the domain of Large Categories, where we start to look at Proper Classes that outstrip the resources of set theory, that this distinction starts to play a factor. For obvious reasons, a proper class cannot be a set. But, we might ask, does this mean that the Class of all Sets/All Ordinals cannot exist? Is there a collection of all algebraic structures of a given type? The Set-theoretic realist seems compelled to say no, even if they can force their way to a set-theoretic model in which one can say that there is.

If we take this seriously, then the exact way Formalism can pull itself apart in a principled way from the Ante-rem set theoretic realist is by countenancing the existence of large categories. To me, this is a fantastic way of understanding just how weird it is to accept the commitment to the complete openness of ontology of mathematics: to the formalist, given a commitment to say that the things of maths are just its terms or the principles of symbol manipulation operating over its theories, there is absolutely nothing wrong with the existence of proper classes.

And why not? Class-theoretic Comprehension can be written down in a sensible way without collapsing into inconsistency (the separation of Set theory has helped with that in the NBG sense), so shouldn’t this be totally reasonable? And in a world where the alternative seems to be a multiverse of set theories to countenance the unanswered questions of the consistency of large cardinal principles, ironically reversing the very worry about the explosive ontologies of consistency in the first place, recognizing that one’s commitment to the stuff of maths is only as much as it represents the logical rules of the game seems like a much needed dose of reality.

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  • Although I disagree with the assertion that proper classes cannot be replaced by "cosmic" sets (the deeper reality is that this replacement requires adjusting for some extremely fine details so as to avoid the paradoxes of unrestricted comprehension over the generic intensional elementhood relation; NF/U works out an eminent historical example of how such a workaround can be pulled off fairly well), your post reminded me that it is a metamathematical question, "Is/can set theory be a 'foundation' of mathematics?" The ability to formulate this question is both philosophical and mathematical ... Commented Mar 12, 2023 at 23:10
  • ... and so represents a zone in which we might look for a subdiscipline, or substyle, of mathematics, specifically appropriate to the philosophy of mathematics. And since category theory/theorizing is partly caught up (though not quite inextricably) in the question of foundations (whether that question be entirely legitimate or not, in the end), then category theory "by nature" will be part of this "appropriate substyle." I wouldn't openly tend to countenance an inference to formalism, except that I think there's a fused realist-formalist theory that's possible and compelling. Commented Mar 12, 2023 at 23:14
  • I don't want to ride off into some Biblical Lovecraftianist sunset, but so to say, if a strong concept of free will is implemented as a claim that we have "semiotic autonomy," or then that whether something is a symbol depends purely on our will, and if objective states/procedures of willing can subsist as abstract possibilities (I mean, we're talking free will here anyway, that's hardly more than possibility as it is, albeit weirdly functioning), then the abstract forms of possible semiotic decisions just are the simultaneously ante rem, in re, and formalist objects at hand. Commented Mar 12, 2023 at 23:19
  • @KristianBerry, I for sure think at some level any sufficiently expressive metatheory (in the Tarskian sense of modelling the truth of the object theory) is going to have to be understood on formalist terms, where the axioms of the theory are chosen for the sake of pragmatic communication rather than as prima-facie laws of reality. But I take the point - if one's choice of grounding theory is a matter of Will then there is a temptation to get really properly old school metaphysical. My gut instinct is to not go there, but I have nothing but impulse directing me in that regard!
    – Paul Ross
    Commented Mar 12, 2023 at 23:21
  • As alluded in my extension to the comment chat, I think the puzzle about the legitimacy of this is where we start to get in to the nitty-gritty about what category theory really is, and the reach of its spindly-arrow-arms. Fortunately, of all parties, the Formalist has the easiest time with this question, as long as it doesn't fall into inconsistency!
    – Paul Ross
    Commented Mar 12, 2023 at 23:23
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I suggest that is rather like asking whether different philosophies 'correspond' to different sub-disciplines in any other field. Setting aside the vagueness of what you mean by 'correspond' (perhaps affinity or compatibility might be better terms here), it seems to me that the whole idea is artificial. The sub-disciplines in mathematics are categorised by one set of fuzzy criteria, and philosophical points of view categorised by other sets of (possibly fuzzier) criteria, so why would you expect a one to one correspondence between them? Why would you not consider some sub-set of mathematics that spanned multiple sub-disciplines to be in greater conceptual harmony with some philosophical viewpoint?

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  • The idea of a direct, exact correspondence does seem misplaced, per the remark at the end of (2). Alternatively, then, different phil.-o'-math would be correlated with different orderings on subdisciplines, e.g. predicativists rank number theory highest, algebra perhaps next, etc. whereas intuitionists rank geometry and typology higher, maybe.; but all philosophies eventually intersect all subdisciplines. Commented Mar 7, 2023 at 17:19

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