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(crossposted here, wasn't sure where it belongs...)

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?!

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    "mathematical objects are constructable and what is constructable is so when we have constructed it, i.e. when we have a proof of it". In a nutshell, 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. Dec 13, 2023 at 13:20
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    What is a mathematical "object"? Dec 13, 2023 at 14:09
  • Is mathematical truth exclusive?
    – 8Mad0Manc8
    Dec 13, 2023 at 15:44
  • For what it's worth, constructivism is about whether a proof exists, not about the time taken to write down or validate a proof. It turns out that not all mathematical objects can be realized, particularly anything requiring excluded middle. Since choice requires excluded middle, this includes e.g. Banach-Tarski cloning.
    – Corbin
    Dec 13, 2023 at 23:38
  • To intersect this with @MauroALLEGRANZA's note, consider Turing machines (TMs.) Every TM either halts or doesn't halt (on a fixed input), but there's not necessarily a proof for each halting TM which shows that it halts. We say that the set of halting TMs is not decidable/computable. A precise teardown of this example can be found here on Computer Science SE.
    – Corbin
    Dec 13, 2023 at 23:46

2 Answers 2

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See in Enrico Martino, Intuitionistic Proof Versus Classical Truth: The Role of Brouwer’s Creative Subject in Intuitionistic Mathematics (Springer, 2018), Chapter 11 Temporal and Atemporal Truth in Intuitionistic Mathematics, page 97-on:

Nowadays, the most widespread view about constructive truth is that truth should be conceived as a tenseless notion. Prawitz (1987, 153–154) writes:

"[A] mathematical sentence is true if there exists a proof of it, in a tenseless or abstract sense of exists […]. Or we may express the same idea by saying that a sentence A is true if ‘we can prove A’ […]. That we can prove A is not to be understood as meaning that it is within our practical reach to prove A, but only that it is possible in principle to prove A [...]. Similarly, that there exists a proof of A does not mean that a proof of A will be constructed but only that the possibility is there for constructing a proof of A. […] I see no objection to conceiving the possibility that there is a specific method for curing cancer, which we may discover one day, but which may also remain undiscovered."

Martin-Löf (1991) distinguishes between actual and potential truth of a proposition. These notions would be explained intuitionistically by the notions of actual and potential existence of a proof. A proof of a proposition A exists actually if, as a matter of fact, A has been proved; it exists potentially if A can be proved. Here possibility is not understood in the traditional intuitionistic sense as knowledge of a method to prove A, but as “knowledge-independent and tenseless” possibility. Accordingly, a proposition that has been proved becomes actually true, but it was potentially true even before having been proved, and it would be true even if, in fact, it had never been proved. In this way, according to Martin-Löf, the intuitionist can overcome the well-known objection that saying that a proposition becomes true just when it is proved is counterintuitive and in conflict with the standard use of the truth predicate: potential truth is not open to that objection.

Dummett’s position on this point seems to be rather oscillating. On the one hand, he argued for the need of a notion of truth—of some notion of truth—within the constructivist conceptual framework and for some “necessary concession to realism”, and he was probably the first who suggested conceiving intuitionistic truth as tenseless. On the other hand, especially in recent years, he has manifested some perplexities about the compatibility of a tenseless notion of truth with the antirealism of the intuitionists.

Compare with Brouwer's original point of view:

Based on his philosophy of mind, on which Kant and Schopenhauer were the main influences, Brouwer characterised mathematics primarily as the free activity of exact thinking, an activity which is founded on the pure intuition of (inner) time. No independent realm of objects and no language play a fundamental role. He thus strived to avoid the Scylla of platonism (with its epistemological problems) and the Charybdis of formalism (with its poverty of content). As, on Brouwer’s view, there is no determinant of mathematical truth outside the activity of thinking, a proposition only becomes true when the subject has experienced its truth (by having carried out an appropriate mental construction); similarly, a proposition only becomes false when the subject has experienced its falsehood (by realizing that an appropriate mental construction is not possible). Hence Brouwer can claim that “there are no non-experienced truths” (Brouwer, Collected Works 1. Philosophy and Foundations of Mathematics, 1975, p.488).

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The visions constructivists like Zeilberger have of “winning the day” is science is increasingly done with computers, and in hundreds to thousands of years that will be the paradigm. Classical math done by humans will no longer be indispensable to science. Computers being finite are much better at constructive math, and computers are getting more powerful.

What computers can actually compute will be the indispensable math of the future. The computations will terminate and we will use that to form another halting constructive mathematical object.

It’s confusing now because classical math is still indispensable and outcompetes and has outcompeted non-classical math for a while. So the eternal component a la Platonism is there. But according to these futuristic visions, math will no longer have these eternal versions, except by possibly some reclusive classical mathematicians who want to think of the eternal for reasons unrelated to scientific progress. For a lot of mathematical objects of the future, there won’t even be a corresponding eternal version. And where there is, as a whole classical math will be minor, so the temporal aspect will be the conceptual stamp.

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