Subject: Re: [HM] What is good math history?
From: Don Cook (tdctdc@surfsouth.com)
Date: Mon Jan 10 2000 - 16:41:35 EST
Dear List,
In an excellent letter, Jan Mycieski made the following comment:
"3. Intuitionism is a rather unsuccessful attempt to develop
mathematics by means of a truncated logic. This attempt was
never clearly justified on philosophical (or any other) grounds,
and it did not generate any superior kind of mathematics. In
fact there is a defect in the practice of intuitionists. Their
proofs yield more information than they state in their theorems.
Thus they hide their results in their proofs unlike classical
mathematicians who try to express as much as possible or
reasonable in their theorems."
I sorta disagree. If I were starting over again, I may study Constructive
Mathematics which uses "truncated logic" (no law of the excluded middle)
I believe that consistency of mathematics is a strong philosophical basis
for any branch of mathematics.
From "A Course in Constructive Algebra", Mines, Richman and Ruitenburg,
Springer-Verlag, 1988, pg vii-viii.
"The constructive approach to mathematics has enjoyed a renaissance
caused in large part by the appearance of Errte Bishop's book "Foundations
of constructive analysis" in 1967, and by the subtle influences of the
proliferation of powerful computers. Bishop demonstrated that pure
mathematics can be developed from a constructive point of view while
maintaining a continuity with classical terminology and spirit; much more
of classical mathematics was preserved than had been thought possible,
and no classically false theorems resulted, as had been the case in other
constructive schools such as intuitionism and Russian constructivism...
It is important to keep in mind that constructive algebra is algebra,
in fact it is a generalization of algebra in that we do not include the
law of the excluded middle, just as group theory is a generalization of
abelian groups theory in that the commutative law is not included."
Peace,
Don Cook
This archive was generated by hypermail 2b28 : Mon Jan 10 2000 - 17:37:33 EST