Re: pragmatic foundation

[Zinovy Diskin <zdiskin@gsd.uwaterloo.ca>]

>>
>>> I invite everyone to read the interesting interview of Yuri Manin
>>> published in the November issue of the Notices of the AMS:
>>
>> Manin is always entertaining but not very careful about what he says.
>>

Hm, Manin is never just entertaining: he wrote several papers
concerning physics, linguistics, psychology, and his  thinking is an
example of how a true mathematical mind works in complex areas like
the humanities, generates unexpected views, reveals deep connections
etc. If the results are readable and enjoyable, it just shows the
literary talent of the author... :)

I also wouldn't say that Manin is not very careful about what he says.
The parts of the interview about foundations and physics say,
basically, this. After Bourbaki, a correct mathematical text should
consist of two parts:
(a) definition of the structure  in question (structure in the sense
of Bourbaki),
(b) deductions about this structure in some logic (perhaps, non-classical).
Manin says that texts generated by physicists do have (b) but not (a).
These are deductions about something that has not been defined and
hence, for a mathematician, that does not exist at all  (the Eiffel
Tower is in the air). This situation is not unique, of course: Manin
mentions Cantor's set theory at the time of invention, and it was and
is so for engineering theories. Software engineering should be of
special interest for this list because modern software executes
deductions about categorical structures.

It is not in the interview explicitly, but the following model of a
mathematical text would be probably close in spirit to what Manin
says. Mathematical texts form a span:
PM <--- MM --->FM
with
PM -- the universe of "physical" mathematical texts (physics, computer
science, engineering etc),
MM -- the mathematician's universe of mathematical texts; they are
written in a special subset of the natural language (nowadays, in
accordance with Bourbaki or category theory),
FM -- the universe of formal (machine-readable) mathematical texts.

A physicist is interested in the left foot, a logicist  -- in the
right one, but mathematics is about the entire span (well, for a true
mathematician, P stands for Platonic rather than Physics). If you
want: the logicist view is more normative because it insists on the
right right leg, but Bourbaki concerned about the entire span and did
not want to fix neither right nor the left legs (unless P is for
Platonic). So, they proposed a reasonable structure for MM for which
the left and right sides of the whole could be added (if needed). It's
indeed more about practical foundations...

After all, Eduardo said it best:

> Well, the fact that he is not very careful is precisely what makes his
> saying
> meaningful, interesting, fresh and enjoyable. He does not place himself
> within
> any philosophical or political frame. He feels free to say what it crosses
> his
> mind just as it comes. beautiful !
>


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]