[FOM] Monsieur, x^p = x (mod p), donc Dieu existe. Repondez !

[Andrej Bauer]

My first reaction is that in the real world there are no mathematical
objects, such as numbers and sequences. It is a category mistake to
say that "such-and-such number does not exist in the real world".

It is meaningful to talk about _representations_ or _physical
manifestations_ or _physical instances_ of mathematical concepts. The
number 3 is an abstract concept. It is _not_ identical with any
particular collection of three objects in the real world. It is _not_
identical with the collection of all collections of three objects in
the real world (because the world might be too small to contain three
things).

However, I do not consider myself to be Platonist. An abstract
concept, such as a number, can be meaningful without existing on its
own. It is possible that the only way to make sense of abstract
concepts is in their relationship to the real world, be it in some
structural way, in terms of communications, or in terms of cognitive
explanations (although I'd be weary of simplistic explanations about
areas of brain firing up whenever I hear the word "three" etc.).

Even if we subscribe to an entirely formalist explanation of abstract
mathematical concepts, we will still be forced to conclude that
numbers do not actually exist in the real world. (And we might
conclude that numbers do not exist at all.)

Thus I find it entirely sensible that certain mathematical objects do
not have representations of a particular kind in the real world. The
number 2^2^10 cannot be represented by a very long sequences of
dominos, for example. This tells us something about the number 2^2^10,
but it does rob 2^2^10 of its existence.

With kind regards,

Andrej Bauer