FOM: intuitionistic and classical truth
cxm7 at po.cwru.edu
Fri Sep 4 12:15:27 EDT 1998
Niel Tennant wrote:
>Bill Tait says that for a mathematical proposition A, A is equivalent
>to the proposition that A is true. Moreover, the only warrant for
>asserting A would be a proof of A.
>The classicist and intuitionist would both agree with these points.
>Where they part company is in saying *what truth consists in*. For the
>intuitionist, "A is true" is (analytically, conceptually) equivalent
>to "There exists a proof of A".
But Tait's point was entirely correct. The "intuitionist" you
describe is the Michael Dummett intuitionist, certainly not Brouwer. For
Brouwer, "A is true" is equivalent to "A is founded on intuition".
Dummett might claim that:
0 is not equal to 1
counts as a one-line proof of itself. For Brouwer this is absurd reliance on
language. We know 0 is not equal to 1 simply because we have an intuition of
twoity. And Brouwer makes no division into axioms and theorems--all of math
is based on intuition for him.
According to Brouwer, human mathematicians are always tempted to
make the sinful confusion of means and ends. So we confuse language with
intuition. We are even unable to function without language, and in
particular without proofs. But that is weakness. We have no mathmematical
knowledge so long as we only know that some stretch of wording is a
"proof"--whether that means "proof in a given formal system" or "proof that
satisfied prof. Brouwer" or whatever. Any given person knows mathematical
claim A, according to Brouwer, when that person has created an intuition of A.
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