Why do constructive mathematicians claim that mathematical truth is temporal?

Question

It seems to me (and correct me if this is a misconception) that the traditional divide in the interpretation and practice of mathematics is between platonists, who believe that mathematical objects exist eternally, independently of our capacity to proof them and who accept classical logic, and constructivists, who only accept constructive reasoning and hold that mathematical objects are constructable and thus temporal. The latter think that mathematical statements become true by constructing instances or proofs. (This is defended explicitly in Dummett's Elements of Intuitionism)

But why can't I just hold that mathematical objects are constructable and what is constructable is eternally so? Dummett says that this essentially involves a platonist understanding of the existence of constructions, but what does this mean? The only reasonable understanding I can imagine is that Dummett says that when I say "The possibility of proofs exist eternally." I am using a non-constructive notion of "exist". But this seems false because I am precisely saying that all constructions are constructable. The method of construction is just the activity of doing math...

So again, why can't one just be an eternalist constructivist?!

In a nutshell, because there is no "platonic" eternal truth but a mathematical facts is so when we have a proof of it and the "book of proofs" is a human construct that evolves (increases) in time. See e.g. Carl J. Posy, Mathematical Intuitionism (Cambridge UP, 2020) — Mauro ALLEGRANZA, Dec 13 '23 at 12:59
"mathematical objects are constructable and what is constructable is so when we have constructed it, i.e. when we have a proof of it". — Mauro ALLEGRANZA, Dec 13 '23 at 13:16
See Enrico Martino, Intuitionistic Proof Versus Classical Truth: The Role of Brouwer’s Creative Subject in Intuitionistic Mathematics (Springer, 2018) and Carl J. Posy, Mathematical Intuitionism (Cambridge UP, 2020) — Mauro ALLEGRANZA, Dec 13 '23 at 13:18
@Mauro ALLEGRANZA: Your first comment seems to provide an answer to something I've often wondered about but haven't tried very hard to research the literature about (e.g. some of the discussion in this MSE answer), namely whether constructivists allow non-constructive proofs for the existence of a constructive proof. I'm guessing the answer is NO. — Dave L. Renfro, Dec 13 '23 at 13:19
The question is not what constructivists say but what they should say. I.e. where is the problem in assuming that the totality of proofs exists eternally — user9812063, Dec 13 '23 at 13:28

score 0 · Answer 1 · answered Dec 13 '23 at 14:54

I'm no expert on this matter, but I'll try to answer. It seems to me that these two questions are important.

1) Can the eternalist constructivist (EC) position be used as an argument against classical constructivism (CC)?

The argument might go something like this:

Since it is predetermined which constructions we can actually perform, then we can consider all potential constructions instead of all actual constructions. However, potential constructions exist regardless of time or the mental activity of individuals. Thus, there actually are eternal abstract objects.

The crucial point here is probably the considered predetermination. I think CC would either reject it, or reject that it necessarily leads to postulating the actual existence of abstract objects.

If mathematics is invented in some predetermined form, does that mean that it is actually discovered and not invented? It would be important to analyze what we actually mean by such predetermination.

2) Would it be possible to develop the EC position independently?

Probably yes, but there would be several challenges. For example, explaining why EC is concerned with constructions at all if it postulates eternal objects -- why not adopt unrestricted realism instead? Or what would be EC's relationship to such logical principles as the Law of Excluded Middle?

Nevertheless, I find EC's position interesting and worthy of consideration.

Why do constructive mathematicians claim that mathematical truth is temporal?

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