FOM: What is mathematics?

Vladimir Sazonov V.Sazonov at
Mon Feb 18 13:23:58 EST 2002

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??? 

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. 

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). 

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. 
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. 

> I submit that mathematics needs no definition.  Mathematics just is.  

It is not enough for understanding what are we doing in mathematics and 
what are its foundations. (Why then the whole this FOM forum?) By the 
way, the definition which I like so much suggests that no unique formal 
system like ZFC and its intuition can be considered as a unique formal
foundation of mathematics. The distinguished role of ZFC is only the 
fact of contemporary history of mathematics. The future might be quite
different - a lot of formal systems (probably interrelated in some way)
without a unique foundation or with a (small?) number of foundations 
for various parts of mathematics. (Like in a town, each building has 
its own base.)

This somewhat changes the view on and the direction of thought in f.o.m. 
and on the whole mathematics. Is not this interesting and (potentially) 
important? Look for NEW informal concepts and formalisms which,
are not embeddable in ZFC or its natural extensions, in principle 
(as Cantor's set theory was not embeddable in the old mathematics; 
it was radically new theory and needed an additional defense of such 
great peoples as Hilbert). This is a positive impulse from the 
definition of mathematics discussed. 

In any
> given situation, a mathematician, or other professional, may desire, or
> need, to adjudicate in a matter about whether a particular activity is
> mathematical in nature.  In these cases, the judgement need not be held up
> to so much scrutiny that it becomes a ``law''.  It may, however, be
> reasonably cited as a ``precedent''.  In any case, however, I do not mean to
> suggest that the discussion about ``What is mathematics?'' is somehow
> unworthy of pursuit.  It can be quite enlightening and/or entertaining to
> see what is on some particular person's list of ``mathematical'' items, and
> what is on that person's list of items that they deem to be definitely
> ``nonmathematical''.
> Matt Insall

Vladimir Sazonov                        V.Sazonov at 
Department of Computer Science          tel: (+44) 0151 794-6792
University of Liverpool                 fax: (+44) 0151 794 3715
Liverpool L69 7ZF, U.K.

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