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

Vladimir Sazonov vladimir.sazonov at yahoo.com
Thu Jun 10 17:46:48 EDT 2010

----- Original Message ----
> From: Marc Alcobé <malcobe at gmail.com>
> To: Foundations of Mathematics <fom at cs.nyu.edu>
> Sent: Tue, June 8, 2010 2:16:34 PM

Dear Marc, 

I think, our views actually correlate, although we make different stresses. 

> > 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.

The ***global*** goal of mathematics is to formalise our thought 
to make it stronger by using the resulting formal tools. 
Stronger for what? For any kind of thought activity where 
such tools could be applicable, in particular for resolving 
"certain problems". This completely covers your goals. 
It covers more because the goals may be not just resolving 
tasks but (formal) description of a certain subject matter, 
or whatever other goal we could imagine where formal tools 
of thought could be used. 

> > 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.

Yes, I fully agree, but I stress: "to achieve" formalisation. 
We have this goal (consciously or not). We value formalisation 
because it makes our thought stronger for any imaginable purpose. 

> > 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?

Such kind of processes going in our brains are a "pivot" for 
our thought in general, not only in mathematics. I consider 
formalisation as the center and the global goal for mathematical 

> > 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.

This (possible) purpose is not lost at all as I stressed above. 
(To have a strong thought armed with formal tools ***for what***?) 
As to other activities dealing with forms, only mathematics 
deals with or (potentially) pursues the limit pure forms of 
thought (i.e. contemporary formalisms) usable by anybody else 
as instruments. 

Thanks for the discussion to everybody. Actually, for some 
time I need to stop writing actively to the FOM list or to 
do this with some delays. 




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