[FOM] Formalization Thesis vs Formal nature of mathematics

S. S. Kutateladze sskut at math.nsc.ru
Mon Dec 31 00:02:02 EST 2007

Vladimir Sazonov    wrote :
I do not know what is mathematical 'truth', except the truth in the
real world...
Of course, you know a la Tarski; but I appreciate that
you acknowledge something mathematical in the real
informal world.

Sazonov wrote:
But please note "by way of proof". No other way to 'truth' is allowed.
Proof may be informal like you statement above: you mentioned
something informal and I understand you somehow.

Sazonov wrote:
I would say "fragmentary formal". Epsilon-delta formalization put 
things in sufficiently self-contained way and excluded infinitesimals 
which were also "fragmentary formal" and much more difficult to 
formalize non-fragmentary. Then non-standard Analysis formalized (in a 
way) infinitesimals too.
My point is that calculus was mathematics at the time of invention,
whereas there was no formalization in the modern sense at all.

Sazonov wrote:
I know no particular exclusion from the formalist paradigm...
 Science is not a theology.
Invention, fantasy, semantics, etc. reside partly beyond
formalization. Science does not reduce to theology nor to any other

Sazonov wrote:
It seems you use the term actual infinity in an ambiguous way...
I think
you mean the inverse 1/x of an "actual" infinitesimal x which is 
somewhat different idea from set theoretical infinite sets.
No I do not. The monad of Euclid is an actual infinity, which is
reflected in your understanding. Recall that by Definition I of Book
VII a monad is ``that by virtue of which each
of the things that exist is called one.''

Sazonov wrote:
Because he [Euclid]was sufficiently formal, except some minor points.
So you consider  the absence of any definition of a notion used in
in  a proof a minor point for formalization. I presume always that
the formal paradigm is much stricter.

Sazonov wrote:
Do you actually mean that continuum has a bigger cardinality than 
natural numbers? It is not called continuum hypothesis.
Of course, the cardinality of the continuum is greater than that of the naturals
and this is not the continuum hypothesis. My point is that the problem of intermediate
cardinalities reflects the ancient problem of agreeing  monads and points in
counting and measuring.

Sazonov wrote:
I do not know what is greater. Assume so. And so what?

So the modern standards of rigor are as incomplete and powerful as

Sazonov wrote:
> The independence of the firth postulate  would be  a trifle without
> the treasure trove of the modern knowledge about various spaces of geometry.
And so what? Is this knowledge outside of the formal paradigm (or the
paradigm of rigour) of mathematics?
The quest for mathematical knowledge is mathematical but may be

Sazonov wrote:
Yes, human enterprise of creating and studying formal side and formal
tools of our thought. (Not a meaningless formal game!)
What is misleading?
The nature of math is human whereas the objects of math are formal.

Sazonov wrote:
> Meaning is that which belongs to man.   No man, no meaning.
My question was related with missing your point in your previous post.
I still do not know what you wanted to say there.
Anyway, another man can use computer program in a meaningful way ...

My point is reflected in your reference to "another man."

 Sazonov wrote;
... the goal is formal tools, not truths. Getting truth (in the real
world) is only a side effect of mathematical activity. 'Truth' in 
mathematics is a kind of chess piece, a very important piece, sometimes 
having relation to the truth in the real world.

This reminds me of the mad tailor of Stanislaw Lem's  "Summa
Technologiae." In fact, I am in favor of Mac Lane's approach.

S. Kutateladze
Sobolev Institute of Mathematics
Novosibirsk State University
            mailto: sskut at math.nsc.ru
            copyto: sskut at academ.org       

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