[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
Of course, you know a la Tarski; but I appreciate that
you acknowledge something mathematical in the real
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.
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.
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
It seems you use the term actual infinity in an ambiguous way...
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.''
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.
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.
I do not know what is greater. Assume so. And so what?
So the modern standards of rigor are as incomplete and powerful as
> 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
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.
> 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."
... 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.
Sobolev Institute of Mathematics
Novosibirsk State University
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