[FOM] Queries for a constructivist

Richard Heck rgheck at brown.edu
Sat Oct 22 11:50:28 EDT 2005

I'm not an expert on these matters, but my understanding is that, for an
intuitionist, the answers are:

>A: It is meaningful to say of a mathematical proposition:
>   "There is a (constructive) proof of this statement but no-one has yet found one."
Meaningful, yes. The question is whether that could be true. And that
will depend upon what one means by "Someone has found a proof of A". We
know that there is either a proof the 10^10^10 + 1 is prime or there is
a proof that it is not. But there is a natural sense in which no-one has
found such a proof. So if that is what one means, then there are true
statements of that form. If, on the other hand, one has a looser
understanding of what it is to have a a proof---if one counts having a
method that would yield such a proof as having such a proof---then we do
have a proof or one disjunct or the other. On that way of understanding
the key phrase, one could never have grounds to assert a sentence of
form (A). That is not to say that (A) is necessarily false. We have only
weak counterexamples to it, I take it.

>B: Every mathematical statement is in exactly one of these 3 states:
>   (c) It has been (constructively) proved;
>   (d) It has been (constructively) refuted;
>   (e) It is in neither of states (c) or (d).
The issues mentioned in connection with (A) will arise here, too, and so
will some other ones. That is true if it is decidable whether a
proposition has been proved. That is typically held to be so. There is
some idealization here. One can imagine that, say, Goldbach's Conjecture
has been constructively proved, but only by a long-dead race of
creatures living in some distant part of the universe. If one means to
be including such possibilities, then one's view about whether we should
(f) Either there is a proof of A or there is not a proof of A
as true will depend upon one's views about "The Reality of the Past", to
borrow a phrase. That is not how people usually think about it, though.

Richard Heck

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