[FOM] Mathematics ***is*** formalising of our thought and intuition
Vladimir Sazonov
vladimir.sazonov at yahoo.com
Sat Jun 5 02:00:18 EDT 2010
Hi, Vladik!
Of course I agree with your notes
http://cs.nyu.edu/pipermail/fom/2010-June/014808.html
that other sciences can contain
mathematical formal fragments. This confirms that formalisation
makes our thought powerful and that is why mathematics is so widely
applicable. Also other scientists can contribute to mathematics
by creating some formalisms even not necessary satisfying strong
mathematical criteria. The greatest example is Newton who for
the needs of physics (if I correctly describe the history of
sciences) created the beginnings of Analysis/Calculus.
It took hundreds years to make Newton-Leibniz Analysis to be
formalised according to mathematical standards. But even
in not so perfect form, the Calculus (yet without proper
mathematical foundation) was from the early times at large
degree quite a formal system as I tried to argue in a
recent posting:
http://cs.nyu.edu/pipermail/fom/2010-June/014804.html
I only want to emphasise that sciences other than mathematics
either use existing mathematical formalisms or create their own
only sporadically and ***without*** such maniacality as mathematics
does. In general, they have gaps in their quasi-formal considerations
which cannot be eliminated (as Hilbert eliminated some rare gaps
in the geometry of Euclid). In general, other sciences do not
pursue mathematical rigour with such a passion as mathematics
does it. This would be actually impossible because in general
other sciences have different goals than mathematics.
Studying the Nature cannot be done in fully mathematical way
but can benefit from using mathematics in some particular cases.
Even when they use mathematics, they usually do this in a different
style because their goal is truths (on the nature) rather than proofs.
So they can often neglect the full rigour and it is quite usual that
the full rigour is impossible because they rely on experiments
and on other style of argumentation concerning the real world.
In mathematics it is just vice versa - the goal is proofs rather
than truths.
I am even very suspicious concerning using the word truth in
mathematics and in general non-technical discussions try
to replace it by something scientifically more adequate like
formalising our intuition, reflecting some kind of reality, etc.
Mathematics cannot pretend that proved theorems are true
in the ordinary sense of the real world because it rather
formalises our fantasies. It is better to avoid the unnecessary
temptations on relations with the real world. This relation
is in general not compulsory and only occasional (and in this
case highly valuable). This is another serious difference with
other sciences. They can have fantasies as well, but their goal
is still approaching to the real world. Mathematics is free
in this respect. Its general goal is creating formal tools
for our free and pure thought. Quite happily, these tools
are sometimes applicable in sciences. But forcing mathematics
by our society to create ***only*** applicable formalisms
would kill it.
Best wishes,
Vladimir Sazonov
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