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

[Marc Alcobé]

I would like to add something to Andrej’s comments, even if it is
quite off-topic.

Yu. I. Manin, in his A course in Mathematical Logic for Mathematicians
(Second Edition), draws a kind of semiotic triangle:

              formal  text
              /             \
written text  -------  interpretation of text.

And he asks his readers to “become used to this trinity, which
replaces the unconscious identification of a statement with its form
and its sense”, as one of the first priorities in their study of
logic.

He also refers to the Saussurian dichotomy “langue-parole” as being
“as relevant to formal languages as to natural languages”.
Yuri also says what he considers to be “reality” for the languages of
mathematics. He says it “consists of certain classes of (mathematical)
arguments or certain computational processes using (abstract)
automata”.

As Ebbinghaus, Flum and Thomas put it in their Mathematical Logic
(Second edition), formal languages and formal systems of logical
deduction have been framed that imitate mathematical proofs in such a
way that logical inferences are represented as formal operations on
strings of symbols. And “the possibilities for justifying methods of
mathematical reasoning (and specifically for justifying a proof
calculus) depend essentially on epistemological assumptions”.

In the real world mathematical objects are free creations of the human
mind. This is what abstraction is about. When an artist paints an
abstract figure he or she wants us not to see that object as something
that represents another thing, but as an object that represents
itself. In the case of mathematics, there is no physical instance of
our abstract creations, at least outside our brains. But it doesn’t
seem to matter much the kind of existence we attribute to the abstract
objects we create as long as we accept certain rules of the game to
deal with them. Actually, it is as a consequence of accepting those
rules that we find our abstractions meaningful. Those rules are the
same that we use when we deal with the objects we find in the real
world. As Kunen puts this in his The Foundations of Mathematics, we
must believe these are the right rules, because to say the rules could
be wrong would amount to saying that we might be crazy. So, ultimately
we would have to ask how we know we are not crazy at all, an
epistemological issue.

Best,

Marc.