[FOM] Mathematics ***is*** formalising of our thought and intuition

[Marc Alcobé]

As I told in some post before, I guess one should think about how
mathematics come to existence. Take the problem of Königsberg bridges
as an example. If you pay particular attention to the means by which
it is solved you'll realize that one departs from a more or less
idealized situation with riversides, islands and bridges, and ends up
dealing with entities that not only do have nothing to do with all
that, but also serve equally well to deal with other kind of different
situations.

Mathematics exist because problems exist, and because some of these
problems can be solved by certain means: mainly idealization,
abstraction, generalization, particularization, etc... the kind of
heuristic means Polya used to stress (and others possibly to be
discovered), we call mathematical (proof is only a small, though
important, part of the whole process, I believe).

Physics deals with idealizations of physical situations (with things
that stand for other things, like the painter who paints a landscape,
or a portrait). Similarly for any other sciences: they model realities
different from the objects they use to model them. Mathematics deals
with abstractions, i. e. with objects that are seen as standing for
themselves, and nothing else, no matter where they have come from
(maybe this is not the usual sense of the term abstraction, it could
be thought of as a kind of extreme idealization).

So, the task of the physicist is to test his models against the
reality he pretends to model. But so is the task of the mathematician,
unless he or she is an extreme formalist, because his/her results must
conform to the "self-reality" he pretends to model, be it either
numbers, sets, cats or mathematical statements.