FOM: Formalists (reply to V. Sazonov)

[Matthew Frank]

On Monday (June 5), Vladimir Sazonov asked if formalists discussed
questions like:  "why such and such formalisms are considered?", "could
they be related one with another and with the real world?", "how it is
possible that some of formalisms are so useful, say, in physics,
engineering?"

I was never satisfied with what I saw from the historical formalists on
these issues.  However, as a contemporary formalist, I have some
responses:

Often we consider a formalism because we have a good set of intuitions
somehow associated to it:  we have a picture of the mathematical objects
to which it is supposed to apply, we have an idea of the sorts of
arguments it is supposed to encode, etc.  (One can also start with the
formalism and try to develop the intuition from it, but that generally
seems to be more difficult and less rewarding.)  Then we can create new
formalisms by generalizing or reformulating old ones; and we can decide
not to work on certain formalisms because we find them inelegant or
isolated from other formalisms of interest, etc.

Formalisms can certainly be related to each other; indeed, this is a major
subject of proof theory, and is important in model theory in the context
of interpretability.  As for whether formalisms can be related to the real
world, I prefer to consider the next question....

The usefulness of various mathematical formalisms in physics or
engineering is largely an evolutionary phenomenon:  people create many
formalisms, and the ones which do not seem useful are not studied, and
disappear from view.  It is also unsurprising given the strong historical
connections between math and physics or engineering:  through the 18th
century, these subjects were not distinguished in the way that we
distinguish them now; in the 19th century, most work on analysis or
differential geometry was connected with these areas.  Even in the 20th
century, the idea of an abstract Hilbert space was created by von Neumann
in a 1927 paper on the mathematical foundations of quantum mechanics, and
affine connections were introduced into differential geometry by Weyl (and
others) in order to elucidate Einstein's general theory of relativity.  I
find that this retelling of the history (and corresponding devaluation of
the "unreasonable effectiveness of mathematics in natural science") lends
considerable support to formalism.

--Matt