FOM: 'constructivism' as 'minimalistic platonism'
Fred Richman
richman at fau.edu
Thu Jun 8 18:01:46 EDT 2000
Jeffrey Ketland wrote:
> (i) The realist thinks the statement "Either A is true or ~A is true,
> but I/we/humanity/etc., might never find out which!" is a coherent and
> sensible thing to say (and perhaps even a priori true),
> whereas
> (ii) The constructivist/intuitionist thinks the statement "Either A is
> true or ~A is true, but I/we/humanity/etc., might never find out
> which" is somehow (conceptually) incoherent (for suitable choice of
> A).
> Is that the right way to state the difference between the realist use
> of "or" and the constructivist use of "or"?
As I might have said already, I don't think there is a difference
between the realist use of "or" and the constructivist use of "or".
I don't find the statement in (i) and (ii) incoherent. A simpler
statement would be "A is true but we might never prove it." Would a
constructivist find that incoherent? "The Riemann hypothesis is true
but we might never prove it." The claim that the Riemann hypothesis is
true is surely intelligible. The claim that we might never prove it
also seems intelligible, if not mathematical. What would make the two
statements incoherent? Because a constructivist cannot justifiably
claim that the Riemann hypothesis is true without possessing a proof?
A classical mathematician couldn't either.
>>How do mathematicians use such statements? At some point in a proof
>>they might have an integer n > 1. This will not in general be a
>>specific integer, like 88001, about which they might conceivably say,
>>"88001 is either prime or composite, but I don't know which". They
>>might say, "either n is a prime, or n = ab for integers a,b > 1", then
>>proceed to argue in each of those two cases. The idea that they do not
>>know which alternative holds would make no sense to them in this
>>situation. It's not like the case of 88001. Here they don't even know
>>what n is, so how would they know if it were prime or composite?
> The real question seems to be: do you need to know what this integer n is
> *before* you're entitled to assert that either it is A or it is ~A?
The question I had in mind was whether a mathematician would pause to
say, of an integer that is completely unspecified, that he did not
know whether it is prime or composite.
> The whole issue turns on having a computational interpretation, which is
> related to some notion of epistemic accessibility---non-constructive
> theorems lie at a more remote "epistemic" distance from human computable
> accessibility. But what's *wrong* with a theorem that doesn't have a
> computational interpretation?
What's usually wrong is that it is misstated. The theorems have
computational interpretations, it's just that the computational
interpretation is not justified by a nonconstructive proof. For
example, you prove "not ( not A and not B ) and say you have proved "A
or B". The latter has a much nicer computational interpretation than
the former. I dare say that is why you phrase it that way. Nobody
likes clumsy theorems like the former.
> Is there a motivating philosophical assumption within constructivism that
> there cannot be mathematical facts which resist constructive proof?
I don't see why one would need that assumption. Of course if you
identify truth with provability, as Brouwer appeared to do, then I
guess that would be a consequence.
> But a realist can still say: why should all mathematical facts be
> epistemically accessible via computation? A liberal constructivist
> (Hellman's terminology) can presumably say "Maybe there are such facts, but
> I'm just *more interested* in the
> epistemically-accessible-via-computation-ones".
I suppose that's more or less what I say, although I hate to be
classified as liberal (and would never use the phrase
"epistemically-accessible"). Perhaps "aphilosophical" instead of
"liberal". I've explained why I like constructive proofs: they justify
the computational interpretation of the theorems. I don't have to have
a position on the nature of mathematical reality. Nevertheless, if you
operate in a constructive framework, your view of mathematical reality
will no doubt be colored by that experience.
--Fred
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