Did Gödel think certain math could only be understood if platonism is correct? (and correspondence and nominalism)

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.