[FOM] query on the history of the philosophy of mathematics: mathematics as formal

[Colin McLarty]

catarina dutilh <cdutilhnovaes at yahoo.com>
Sun, 2 Mar 2008 10:20:08 -0800 (PST)

asked

> since when is it widely (even if not unanimously) held
> that what is distinctive about mathematics is its *formal*
> character? 

That will depend on what is meant by "formal".  Poincaré was probably
just a little ahead of his time when he held it distinctive of
mathematics that its proofs can be formalized in machine recognizable
form: He found Hilbert had perfected Euclidean geometry by axiomatizing
it so that so that  "One could confide [Hilbert's] axioms to a reasoning
machine, such as the logical piano of Jevons, and one would see all of
geometry come out.'' (p. 95, 1902) 

Of course he was wrong about the power of that piano, which only
evaluated syllogisms; and he held that proofs do not exhaust the
*understanding* of the theorems.

AUTHOR =       {Poincar{\'e}, Henri},
TITLE =        {Les fondements de la g{\'e}om{\'e}trie},
JOURNAL =      {Bulletin des Sciences Math{\'e}matiques 2nd series},
YEAR =         {1902},
volume =       {26},
pages =        {249--72},
  
Translated in P. Ehrlich ed. \emph{Real numbers, generalizations of the
reals, and                         theories of continua}. Dordrecht:
Kluwer Academic Publishers, 1994, 147-68}