[FOM] The unreasonable soundness of mathematics

[Josef Urban]

On Sat, Apr 30, 2016 at 10:17 PM, Timothy Y. Chow <tchow at alum.mit.edu> wrote:

> It seems to me that much of the skepticism about the existence of abstract
> infinitary objects has its roots in physical intuition.  That is, it is
> tacitly assumed that the acquisition of knowledge about physical objects
> through sense perception is a well-understood phenomenon.  After all, I have
> never heard anyone go on and on about the "unreasonable effectiveness of
> vision in providing knowledge about the physical universe"; it is not
> regarded as puzzling, and in need of further explanation, how human beings
> can possibly acquire knowledge of physical objects---or at least, it is
> regarded as much *less* puzzling than how we could possibly acquire
> knowledge of an abstract object such as an elliptic curve.  But if physics
> is regarded as a gold standard of some kind, then we can argue that
> knowledge of abstract objects is in fact entirely analogous to knowledge of
> physical entities that are not directly perceptible.  An "electron" in
> quantum field theory is not something that is directly accessible to our
> senses.  Rather, it is an *abstract* theoretical entity that is posited in
> order to explain the results of experiments, and it cannot even be correctly
> conceptualized in commonsense terms such as "particle" or "wave."
> Similarly, elliptic curves are theoretical objects that are posited to
> explain the results of computer searches, and so we should not balk at
> accepting their reality any more than we should balk at accepting the
> reality of electrons.

Petr Vopenka has spent considerable part of his later life trying to
build alternative foundations of math that would be more "intuitive"
in this sense, allowing to model infinitary objects "more naturally".
Physics and Husserls's phenomenology ("extending the horizon", the way
physicists work with infinitesimals, etc.) were an important
motivation. He wrote and lectured quite a lot about it in a very
interesting and pretty deep way, analyzing the history and
"subconsciousness"  of math and physics. Unfortunately, I am so far
not aware of any English translations of that.

Josef