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

[Marc Alcobé]

> That is why formalising is the goal of mathematics and why
> mathematical rigour is in the blood of all mathematicians
> even when it is not directly seen that it will lead
> to so strong formal tools for our thought.

I would regard that more as a technical means (hence the word "tools"
you use) by which mathematical goals (the solution of certain
problems, and maybe even only those amenable to formalization) are
pursued, rather than as a goal in itself.

> All you mentioned above (idealization, abstraction, etc.) is
> covered by the phrase
>
>
>         formalising our ***thought and intuition***
>
>
> but not vice versa! How abstraction, etc. without assuming
> formalisation would lead to, e.g. the Calculus?

The way I see it, without those skills, formalization would be
something impossible for us to achieve.

> If formal character of mathematics and mathematical rigour are
> considered only as some additional feature of mathematical thought
> then it becomes unclear why it should be ever compulsory and why all
> mathematical proofs are actually formalisable, and it is also unclear
> what is mathematics about since then it has no clear "pivot".

We humans share the right devices (I certainly do not pretend to know
which exactly they are) to perform such tasks as testing the truth of
certain assertions against what we could call our "knowledge
database", checking the performance of (sufficienlty simple)
algorithms, etc. Couldn't these be the "pivot" you are looking for?

> Formalist view on mathematics covers everything (essentially
> by one simple phrase) and does not reject (but rather assumes
> implicitly) anything whatever we could want to reasonably
> include in mathematics. Abstractions, idealisations, etc.
> do not explain the nature of mathematics because they belong
> to ANY kind of human thought about any subject matter and so
> cannot clearly distinguish mathematics from anything else.

Also, many human activities involve a certain care for the form:
poetry, music, painting, ... The purpose to solve some specific kind
of problems must not be lost of sight.