[FOM] Intuitionists and excluded-middle
Ritwik Bhattacharya
ritwik at cs.utah.edu
Fri Oct 14 17:34:28 EDT 2005
praatika at mappi.helsinki.fi wrote:
> So, the question was: what is the argument against the reliability of LEM,
> that is, why should we not be convinced by a proof using LEM ? Lew Gordeew
> suggested that such proofs may be "useless". I then argued that this is
> not sufficiently clear.
no! you tried to argue that mathematics doesn't need to have a use. see
below.
> But now we are told that it means that they are
> not convincing. Surely we haven't made much progress here.
hehe, you have snipped out your original objection to lew gordeew's
argument!!! here's the full transcript:
Lew Gordeew <legor at gmx.de> wrote:
A conventional convincing argument: mathematical proofs using the law of
excluded middle might be "useless". Here is a familiar trivial example
(quoted by A. S. Troelstra, et al).
THEOREM. There exists an irrational real number x such that x^sqrt(2) is
rational.
praatika at mappi.helsinki.fi wrote:
I must say that I find such talk of uselessness quite ... well ...
useless. To begin with, why should one require that pure mathematics,
which is theory building, has to have some use. The general requirement of
usefulness of all scientific theories would certainly paralyse science.
And certainly uselessness of some piece of knowledge does not make it
unjustified or not true.
ritwik at cs.utah.edu wrote:
Ah, but the argument being made was that the *proof* might be useless,
not that the theorem itself was useless. So nobody is suggesting that
mathematics/mathematical "facts" need to have use, but a proof, by
definition, must have the "use" of convincing the reader of the proof.
ritwik
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