FOM: What is mathematics?

[Gordon Fisher]

Vladimir Sazonov wrote:

> Insall wrote:
> >
> > Miguel Lerma wrote:  ``Proofs may be an important part of mathematical
> > activity but not essential in defining mathematics.''
>
> What else, if not proofs, rigor and formal reasoning, distinguishes
> mathematics from other sciences? Intuition, abstractions? They are
> widely used in any science, e.g. in philosophy, (and often in so
> non-mathematical manner!). WHAT ELSE???

Something here hinges on what counts as formal reasoning, or
as rigor.  Example:  Is Newton's _Principia_ a work of
formal reasoning?  To be sure, there are mistakes here and
there in that work, but then any kind of formal reasoning is
susceptible to that, humans being what they are.  Have any
of the scarce mistakes been found by recasting contents of
the _Principia_ into axiomatic form, and then into some
symbolic form in the manner of symbolic logic?  How many
physicists or engineers have made egregious errors in
applying Newtonian mechanics and dynamics, where it is
applicable, because (and _only because_) the reasoning
involved was not _sufficiently_ formal in a good textbook
on Newtonian dynamics?

On a different tack, is there such a thing as a formal system
presented, say, in notations of symbolic logic, or so-called
mathematical logic, which doesn't have a close relationship
with ordinary languages?

>
>
> Formalisms are used also in other sciences, but in that case
> it is usually interpreted as application of mathematics.
> Say, rigor is not so important in physics, unlike facts,
> experiments and general understanding the nature.
>

Here, do you suggest that the intuitions on which mathematicians
base their formal systems (formal in some strict sense) are different
in nature from those on which physicists base their expositions?
So that they don't need rigor?  Or perhaps is it the case that
physicists can't be sure of anything until their mathematical
apparatus is imbedded in formal systems which justify their
use, and provide a kind of certainty physicists can't obtain
without such formal systems having been constructed?

>
> By the way, let me give a small addition to my favorite definition
> of mathematics as science on formal systems, IN DEFENSE OF INTUITION
> and the like:
>
> Only those formal systems are considered mathematical which are
> based on and formalize some intuition or abstraction, etc. And only
> those intuitions, abstractions, informal considerations are considered
> mathematical which are formalizable. We have two inseparable sides
> of the same "medal" - Mathematics. And this is a proper
> FORMALIST VIEW on mathematics as I suggest to understand it
> (unlike it is interpreted by many others).
>

Agreed.

>
> Any separation of a pure intuition or a pure formalism is fruitless
> when discussing the general nature of mathematics.
> Moreover, pure mathematical intuition is just impossible.

I wonder.  Remember Poincare's story about how he came
to discover theta functions?

>
> Pure formalism is possible, but mathematically uninteresting.
> Also, this view needs no Platonism, Intuitionism, etc. as
> philosophies of Mathematics. The relation between intuition and
> formalism is extremely delicate thing and should be discussed
> in each concrete situation separately. In general, we can say
> only about some acceleration of thought and making it and
> intuition more powerful, reliable, solid, etc. by formalisms.
> Also, nothing is asserted on the concrete nature of formalisms
> (based on First Order Logic or quite arbitrary?). They should
> be just meaningful, helpful for thought! That is all,
> in general.

Also, I suggest, the relation between ordinary languages and
formalisms is inseparable without distortion.  But then I believe
that falls in with what you have said.  To this, I add the
nature of human brains, considered as influences of the
sort Kant had in mind, though of course without providing
physiological considerations.

[remainder of message from Prof Sazonov answering Matt Insall
deleted]

>
> --
> Vladimir Sazonov                        V.Sazonov at csc.liv.ac.uk
> Department of Computer Science          tel: (+44) 0151 794-6792
> University of Liverpool                 fax: (+44) 0151 794 3715
> Liverpool L69 7ZF, U.K.       http://www.csc.liv.ac.uk/~sazonov

Gordon Fisher
Prof Emeritus, Mathematics and Computer Science
one-time Senior Lecturer in Mathematics, and History &
             Philosophy of Science, Univ of Otago & Unive
             of Waikato, New Zealand
James Madison Univ
Harrisonburg VA 22801 USA
gfisher at shentel.net