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I’m reading Shapiro’s Thinking About Mathematics, and there’s a quote by Godel which I would like to fully understand, both his intended meaning and how it’s viewed in the wider context of mathematics, truth, and philosophy.

First the quote from page 10:

"In traditional philosophical terms, Poincare rejected the actual infinite, insisting that the only sensible alternative is the potentially infinite. There is no static set of, say, all real numbers, determined prior to the mathematical activity. From this perspective, impredicative definitions are viciously circular. One cannot construct an object by using a collection that already contains it. (Shap)

Enter the opposition. Gödel (1944) made an explicit defense of impredicative definition, based on his philosophical views concerning the existence of mathematical objects: (Shap)

...the vicious circle...applies only if the entities are constructed by ourselves. In this case, there must clearly exist a definition...which does not refer to a totality to which the object defined belongs, because the construction of a thing can certainly not be based on a totality of things to which the thing to be constructed belongs. If, however, it is a question of objects that exist independently of our constructions, there is nothing in the least absurd in the existence of totalities containing members, which can be described (i.e. uniquely characterized) only be reference to this totality...Classes and concepts may...be conceived...as real objects...existing independently of us and our definitions and constructions. It seems to be that the assumption of such objects is quite legitimate as the assumption of physical bodies and there is quite as much reason to believe there existence. " (Gödel)

Here's what makes sense to me:

  1. Nominalists do define impredicative mathematical objects without the assumption of their realness in Godel's sense.

  2. These nominalists can do so because their impredicative objects are not the same mathematical objects or semantics Godel is talking about.

  3. Gödel has some idea of a correspondence theory of truth or meaning. Nominalism is closer to pragmatism, which denies correspondence of truth and meaning.

  4. Both the nominalist and the Gödelian realist share the same mathematical language (the same set theory), but the have entirely different meanings, metaphysics, and notions of truth about that language.

  5. We can say in the specific sense of truth and meaning Gödel has in mind, realism is necessary for impredicative definitions to not be absurd or viciously circular.

Are these 5 points correct or fair statements? Am I missing any that should be added that seem relevant? Is this a good exercise within philosophy of mathematics-can I hang my hat on these statements?

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    Re 1. Nominalism in the Phil of Math is basically about rejecting the existence of abstract objects. IMO Poincarè, that rejected impredicative def, was not a nominalist. Commented May 5, 2022 at 14:58
  • I don’t care about Poincare specifically, just wanted the surrounding context for the quote in question. This is about wondering how nominalists and realists can talk about the same objects (eg set theory). Thanks for the link
    – J Kusin
    Commented May 5, 2022 at 15:00
  • According to Poincaré (and Russell of PM) impredicative def generate problems regarding mathematics and infinite collection (for finite collections of "concrete" objects, no issue: the tallest man in the room is "formally" impredicative but define a real person). Commented May 5, 2022 at 15:00
  • And yes, for Godel impredicative def a fine exactly because infinite collections of "abstract" objects are real. Commented May 5, 2022 at 15:01
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    In a sense you're right, a strict pure constructivist/intuitionist won't give up or get defeated here, one can always have some hindsight to conclude they missed some boundary or axioms in the previous construction as they're not the omni-one, so on and so forth, even in an infinitism epistemic fashion, same goes for nominalist and finctionalist here as they essentially share same construction ideas at least for math objects... And in most other cases like classic logic and metaphysics, there's no incompleteness after all, Platonism may be only needed in maths containing certain arithmetics... Commented Jun 5, 2022 at 23:06

3 Answers 3

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The correspondence theory of truth can be framed as the equation of the following two questions:

  1. Does sentence/belief/proposition A correspond to the facts?
  2. Is A true?

However, so microscoped, these questions reveal by their display that we can ask these questions separately anyway. In other words, we could ask whether A corresponds to the facts, or whether it is true, and it could be possible for the first question to have meaningful answers that diverged from the answers to the second question. There could be more to truth than factive correspondence, but there could be more to factive correspondence than 'mere' truth, too.

Now, I think the linguistic overlap between ante rem realists and other schools of mathematical metaphysics is amenable to a fictionalist assessment. Ironically, one barrier to immediately spelling this out would be that it is (epistemically) possible that fictional objects might turn out to be entities in an ante rem world, if also possible that they might turn out to be more semiotic/game-theoretic formalist beings, or something else. At any rate, even if we debated countenancing abstract forms of the stories we tell, we would have to admit that there was a mental icon of these forms, and moreover it seems as if it is the power of the mind that accesses fictional objects by quasi-stipulative force, so by default fictional objects are 'partly' mental, even if this means that our minds are 'partly' ante rem abstractions (composed in a dead Agent Intellect, say).

Correct me if I'm wrong, but I interpret one of your questions as, "If realists and nominalists have separate theories of truth, then how can semantic overlap be so strong between them?" For example, vs. truth-conditional semantics, it would seem that realists and nominalists actually do mean importantly different things by what they say. The plain nominalist has to reject, "2 + 2 = 4," on some level; the only quick maneuver the fictionalist can make is to say, "According to the story of the ante rem realm, 2 + 2 = 4." We might say that when a story's audience can adequately map their list of, "According to the story..." sentences onto the parallel, original list in the author's mind, then their, "According to the story..." sentences are true, and the way they are true is by corresponding to a fact about what the author's list contains. At any rate, alternatively, the nominalist could countenance straight Platonic-sounding claims as corresponding to facts whose proper descriptive form does not involve reference to abstract numbers.

Beyond that, I don't know.

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    Thanks for taking the time to go through my question. You’ve answered a lot of my concerns. As silly as this all sounds it is helpful to hear fictionalist and nominalist accounts. They seem underrepresented
    – J Kusin
    Commented May 5, 2022 at 22:39
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We do not have a definition for existence.

Take a naïve (read best possible) notion of existence:
x is perceivable -> x exists (fails because of hallucinations)
x exists -> x is perceivable (backwards, not a test for existence)

In short, we can't tell whether something exists or not (with certainty). The basic stuff of our world - stones, you, me (perceivable) - exist only if perception is sufficient to prove existence and it is not.

So before Gödel and Shapiro, I suggest we cross the bridge in front of us - do stones exist? May be you should bash me over the head with one and prove that to me.

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  • Rocks aren’t absurd because one can bash you on the head without you thinking about it…anyway, alas I think my question falls under the age old potential vs actual infinity unfortunately, and thus barely gets at the math bit
    – J Kusin
    Commented Sep 28, 2023 at 4:11
  • Gödel was a platonist per some reports, but it is possible that he drew the line at infinity. A mathematical notion of existence does exist though and ... need I say more?
    – Hudjefa
    Commented Sep 28, 2023 at 4:14
  • "A mathematical notion of existence does exist" a or many?
    – J Kusin
    Commented Sep 28, 2023 at 4:44
  • Alas, Gödel is no more (🥀💀) He would've been the go-to-guy for this question. 2x + 3 = 17.
    – Hudjefa
    Commented Sep 28, 2023 at 4:52
  • I actually like this response - this is arguably Hilbert’s answer, that the “stuff” of mathematics is things in the world, and our predicative definitions of subclasses of the math-ish stuff are realised in the world. The finite and countable realm accounts for most of the practically valuable stuff, and so maths is a participant in the formulation of the ontologies of the sciences, not their controlling author.
    – Paul Ross
    Commented Sep 28, 2023 at 6:43
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The passage from Shapiro does not mention nominalism at all, whereas four out of your five conclusions deal with nominalism. One wonders how they are derived from Shapiro's passage.

As far as Gödel is concerned, to understand what is going on, one has to keep in mind the particular kind of Platonism he explicitly declared himself to be an adherent of. For example, mathematicians who wish to stay neutral with regard to philosophical assumptions and background metaphysics, would formulate his incompleteness theorem rougly as follows: if an axiom system (AS) for the natural numbers (or more generally) is strong enough to incorporate Peano Arithmetic, then there will be assertions that be neither proved nor disproved from AS. However, Gödel's preferred formulation was different. He assumed that there are absolute Platonic natural numbers out there, and that, consequently, every assertion is either true or untrue about those numbers.

His formulation of the incompleteness theorem was to say that there are true assertions about the natural numbers whose truth value cannot be determined from AS. To answer to OP's question, it is clearly impossible to understand such a formulation without adopting its Platonist background.

Nominalism being quite at odds with Platonism, it would be hard to derive any views about nominalism from Gödel's writings.

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  • Godel says his platonism is not the least bit absurd, possibly hinting that the non platonist formulation/your first formulation is somewhat absurd. Where he goes to full criticism imo—he doesn’t just say, hey this platonism isn’t absurd guys, he compares it to physical objects. Only crazy people would deny physical objects ergo, he does critique those outside his position to some degree, even neutral positions which only talk about finite collections of the natural numbers I take it. We can’t really understand the world without physical objects, so how can we understand his theorems,
    – J Kusin
    Commented Sep 28, 2023 at 14:49
  • You just chose neutral whereas I chose nominalism as outsiders to his view.
    – J Kusin
    Commented Sep 28, 2023 at 14:51
  • @JKusin, right, to Goedel mathematical entities are comparable to physical objects as far as their mode of existence is concerned. This is of course recognizably a Platonist position. It has its risks, as I elaborated here for example. But I am still not sure what you are asking. Commented Sep 28, 2023 at 15:31
  • With this all said, I’m asking how can we make sense of mathematics, which by and large uses one shared language of existence (there exists infinite primes which..) , when there are lots of nominalists, lots of platonists, lots of neutrals, etc. Outside of math, it would be chaos (how could we understand tables and chairs if many didn’t believe in physical objects, or rather simply said, almost everyone believes in physical objects). We have some kind of existence grounding for the external world (stones hitting heads) but there’s no shared “stones hitting heads” existence grounding in math.
    – J Kusin
    Commented Sep 28, 2023 at 16:33
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    @JKusin, No, I don't think so. But he didn't have your problem because he was a Platonist, so he did not need to make such distinctions. Incidentally, he was very excited about Abraham Robinson's Nonstandard Analysis, and thought that it was "the analysis of the future". Goedel thought of the hyperreals as a discovery of the next significant (and againt Platonic) number system. Robinson explicitly disagreed, and they had a tense exchange about this :-) Commented Oct 2, 2023 at 11:33

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