[FOM] non-Euclidean geometry and FOM programs

S. S. Kutateladze sskut at math.nsc.ru
Mon Oct 29 03:51:04 EDT 2007

Antonino Drago> 23/10/2007  wrote:
Antonio Drago>... Hilbert's solution of the historical problem
Antonio Drago>of non-Euclidean geometries was  wrong.
Antonio Drago>In historical terms, it suggested a mere change in an axiom
Antonio Drago>(this is the version of vulgar historians).

Whether these  statements are of a vulgar provenance is irrelevant.
What is  relevant is that these statements are  incorrect.

The flow of history is turbulent but  happens in time.
When we average the events and ideas over time, we see
less but understand more. Mathematics is the art and science 
of  calculating  truth by proof. Philosophy  explains the origin 
and meaning of truth. Mathematicians avoid metaphysical discussions
of truth. They admitted readily that truth is what is proved in the
pre-Frege time and what is provable since then for the time beeing.
What is provable and improvable in principle was out of mathematics
prior to the uprise of mathematical logic.

The present-day understanding of noneuclidean geometry
is based on the modern content of mathematical logic
which incorporates the concept of model.
This concept  was absent in the standards of rigor of
the pre-Frege--Hilbert times, but nevertheless indispensable for
establishing noneuclidean geometries in mathematics.
Kastner, Gauss, Lobachevsky, and and Bolyai used the logic
of Aristotle.   The problem was not with the type of logic, but with the
inadequacy of Euclidean axiomatics and the lack of
tools for discerning improvability in principle.
The geometrical contributions of Hilbert and Poincare are 
relevant, beautiful, and immortal. By the way, Hilbert
was awarded the first Lobachevsky prize on the overview 
of his research into axiomatics which was written by Poincare.

Antonio Drago>Is the above reconstruction of the history of FOM research a possible one?
You exposition of this reconstruction proves that it is possible.
In my opinion, you overestimate the interest  in the fifth postulate
as a major quest for the proper FOM. Moreover, you underestimate
the general trends of mathematics. The call for proper foundations
was universal.
Antonio Drago>No hint came from Cantor's set theory or more advanced mathematical
Antonio Drago>theories.
This is definitely not so in whatever context.

                                           S. Kutateladze
Sobolev Institute of Mathematics
Novosibirsk State University
            mailto: sskut at math.nsc.ru
            copyto: sskut at academ.org       

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