[FOM] Intuitionists and excluded-middle

praatika@mappi.helsinki.fi praatika at mappi.helsinki.fi
Wed Oct 12 02:40:00 EDT 2005

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.

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. 
Moreover, it is not clear exactly how the possession of a particular 
solution is so much more useful... 
Further, from a theoretical point of view, such a non-constructive proof 
may be very useful in refuting an universal hypothesis, e.g. "For all x, 
if x is irrational, then x^sqrt(2) is irrational." Finally, I think that 
such proofs are quite useful in suggesting that it may be fruitful, and 
not vain, to search for a particular solution, and a constructive proof.
(In the standard systems, if the theorem is Pi-0-2, there is one.) 

Best, Panu

Panu Raatikainen

Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy
Department of Philosophy
P.O. Box 9
FIN-00014 University of Helsinki
E-mail: panu.raatikainen at helsinki.fi

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