[FOM] Mathematics ***is*** formalising of our thought and intuition
Vladimir Sazonov
vladimir.sazonov at yahoo.com
Fri Jun 4 14:59:52 EDT 2010
----- Original Message ----
> From: Shay Logan <logan110 at umn.edu>
http://cs.nyu.edu/pipermail/fom/2010-June/014803.html
> I'm unclear on the meaning of the word formal as it is being used in
> this discussion.
>
>
> If by formal we mean what the word etymologically should mean; that
> is, that truth is a function EXCLUSIVELY of the form of our sentences,
I suggest just this understanding or even a little bit weaker one
with "EXCLUSIVELY" replaced by "mostly" because in this general
context, how to check that "EXCLUSIVELY" does hold? This would rather
applicable for a full formalisation (like FOL and ZFC are formalised
with axioms and formal proof rules). But we want to also have a more
flexible concept of "formal".
Also I suggest not to use the word "truth" here. A mathematical proof
cannot be true but rather formally correct -- satisfying formal rules
(or not).
> it seems odd to me to claim that mathematics is formal. There is far
> more going on in mathematics than merely a verification that the
> shapes (forms) of the sentences of mathematical statements are of the
> right type.
What more? Evidently we assume that we have a formal system
(such as Euclid's geometry) which *deserves* to be considered
*by formalising some kind of our intuition* (on the real world
or on any our fantasies, does not matter). We usually assume
that the *theorem proved is interesting enough*. Then ANY formally
correct proof of this theorem is good independently how we obtained
it or who is its author. We usually prefer an intuitively clear
well-structured proofs, desirably presented in not too formal
way, only to be able to evidently see that it is potentially
formalisable according to standards of the given formal system.
Contemporary standards of (potential) formalisability are very
high.
Additionally, appropriate (subconscious) "maniacality" in getting
more and more formal versions of formal systems and proofs is assumed
in mathematics. For contemporary mathematics this means potential
possibility of absolute formalisation.
This kind of "maniacality" is not a bad thing. It is just a very
strong adherence to the idea of mathematical rigour. It is because
of this adherence not very formal proofs of Euclid were shown
(essentially by Hilbert) fully formalisable with filling several
clearly localised gaps.
If such consistent filling the gaps could ever be possible in other
sciences then they would be parts of mathematics. But they are not.
Even physics!
> However, if by formal we mean something more colloquial, something
> more along the lines of "rigorously thought through and carefully
> argued", then a great deal of political science does seem formal to
> me. So does mathematics. Mathematics may be (somehow) more formal than
> political science when looked at from these lights.
Not just "somehow" more formal, but in SO DIFFERENT WAY!
Mathematical rigour is so strong and so much different from what
any other sciences do that they are almost incomparable.
By mathematical rigour I understand the formality in the above
sense (+ "maniacality" in formalising).
See also my previous posting
http://cs.nyu.edu/pipermail/fom/2010-June/014804.html
Vladimir Sazonov
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