Re: mathematicians utter contempt for common sense
From: KRamsay (kramsay_at_aol.com)
Date: 12/22/04
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Date: 22 Dec 2004 21:10:21 GMT
In article <32s1luF3qmpbbU1@individual.net>, "John Morrison"
<john.morrison@tesco.net> writes:
| Many years ago, at school, my senior (head of department) mathematics
|master used to quote (and, no, I haven't look up the source) the idea that,
|in mathematics, the key is not truth but _consistency_.
I disagree with that. We want to know whether there are infinitely many
twin primes, which is a matter of truth.
It can be thought of as a matter of consistency, if we take consistency
in a relatively broad sense. In particular, it's not a matter of *formal*
consistency. One can give a list of properties that characterize the
natural numbers, including in particular the inductive property. Whether
there are infinitely many twin primes can then be thought of as being
a matter of these properties being consistent with the existence of
infinitely many twin primes in that structure. But this is a somewhat
eccentric way of describing the situation, since those axioms
characterize the natural numbers up to isomorphism. Saying that some
structural property is consistent with them is equivalent to saying that
that property holds *true* in the natural numbers. Saying that some
structural property is inconsistent with them is equivalent to saying
that that property is *false* in the natural numbers.
|Consistency is the
|key to so many proofs (or "proofs", or ""proofs"", and so on ...), provided
|that one does not utterly reject /tertium non datur/, and even if one does,
|can still be adjusted in subtle ways to complete proofs.
I don't know what this remark is supposed to mean.
In classical logic, where "tertium non datur" is assumed, a proof, of B
from premises A1,...,An say, can always be thought of as a proof of a
contradiction from A1,...,An and not-B. So searching for a proof can
be thought of as searching for a contradiction, there. One might say
for this reason that looking for contradictions holds a prominent
position in the doing of classical mathematics. I still would be
reluctant to call it "key".
In intuitionist logic, where "tertium non datur" is not assumed, proving
A->B and proving that (A & not-B) leads to a contradiction are not the
same thing. I don't know what kind of "adjustment" to the notion of
consistency would make it more relevant.
Keith Ramsay
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