FOM: What is mathematics?

[Vladimir Sazonov]

Gordon Fisher in his posting from 13 Feb 2002 15:20:21 -0500 
suggests that there exists a special way of geometrical 
thinking in mathematics which is not a formal by its nature 
and which arose in pre historical and even pre human times. 

May be Gordon Fisher also considers embryo (say, of a human) 
and the human (for example, Albert Einstein with his scientific 
achievements) which will arise from it as the same thing (entity)? 

Of course, it is interesting how mathematics arose. But 
I would not identify "pre mathematics" with mathematics. 
The main point is that proper mathematics arose with 
(sufficiently) rigorous proofs. 

As to geometry, say, in comparison with algebra (which looks 
as more formal) I would like to tell about my personal feeling 
after Soviet (10 years) school. (As I know, school education in 
various countries are quite different. Sometimes some school 
educated peoples think that proof is something from jurisprudence, 
not from mathematics. In Soviet and in the contemporary Russia 
school even for children week in mathematics it was/is known very 
well that mathematical, especially geometrical theorems should 
be proved.) Geometry for me was the only part of school mathematics 
which contained some serious (for the school level) proofs. 
Algebraic or trigonometric proofs were more like (non deterministic)
calculations. When I became a student in a math. department of 
a university, I understood that algebra also has (logically based) 
proofs. 

Let me also add that (school and actually, Euclidean) 
geometrical proofs were based not only on a naive, but 
clearly recognizable logic (say, using reductio ad absurdum 
- a very formal way of reasoning). They used geometrical 
figures - forms - a kind of formulas. Anyway, they were quite 
formal (rigorous) even for the school level. 

I finish that, for all defendants of intuitions and abstractions 
in mathematics, I myself could defend them, if necessary. But 
whichever important they are, the main thing which distinguishes 
mathematics from other sciences is rigorous, formal proofs and, 
more generally, using formal systems as accelerators of thought 
making our intuitions stronger and abstractions more solid and 
reliable, not just words. Mathematical (in difference with pre 
mathematical) intuitions and abstractions exist only together 
with and on the base of formalisms. Pre mathematical intuitions 
are aimed by mathematicians to be (after some work) mathematical, 
i.e., formalized. Before that, even if it arose in the head of 
a greatest mathematician, it is only a candidate to be called 
mathematical. It could be just a wrong way of thought, some 
raw working material, if you want. 


Vladimir Sazonov