[FOM] Mathematics and rigour

[Vladimir Sazonov]

Quoting Henrik Nordmark <henriknordmark at mac.com> Mon, 05 Mar 2007:

> It seems as if Mathematics is defined by its methodology and not so
> much or at all by its underlying ontology whatever that may be.

When I was a student of math I had come to similar conclusion (in 
writing a philosophy "referat"). The method mathematics uses - rigour - 
is what is fully defining it. Which "ontological" objects of our 
intuition or application domain it is applicable to does not matter in 
general. It could be ANYTHING.

But ANYTHING is so unclear, superwide to be subject matter! So now I 
think slightly differently: what is usually considered as method (the 
rigour) is, in fact, the subject matter of mathematics.

Indeed, the so called rigorous considerations (or proofs) are 
characterised by the fact that they are, in a definite sense, 
"detached" (of course, temporary!) from the intuition concerning what 
are these proofs about. Only formal structure of the proof does matter 
when we check its correctness. (For example, this was true to a high 
degree to Euclid in his developing the Geometry. Recall also well-known 
joke that doing Geometry is an ability to get to right conclusion on 
the base of wrong geometrical pictures.)

Moreover, mathematics is devoted not only to proving theorem but, may 
be mainly, to creating (and developing by proving theorems, etc.) 
formal systems. (I always mean that these formal systems should have a 
meaning, should help to our intuition and thought in any abstract or 
real domain of discourse. Otherwise this play with symbols has no 
serious relation to mathematics). Examples of such formal systems can 
be various, beginning with the formal rules we use to multiply numbers 
in decimal notation and ending with ZFC. All of them strengthen our 
thought abilities in a sense like engineering (mechanical and other) 
devices make us stronger, faster, etc.


Quoting "Timothy Y. Chow" <tchow at alum.mit.edu> Tue, 06 Mar 2007:

> Actually, I still prefer my original formulation, in terms of "degree of
> precision" rather than "methodology vs. ontology."  To favor methodology
> over ontology is to bias the formalist/platonist debate in favor of the
> former, whereas I would say of both methodology and ontology that they are
> mathematical if they are sufficiently precise.
>

As I wrote above the "ontology" of mathematics is ANYTHING whichever we 
can imagine.

I think it is clear from the above why I replaced "precision" by 
"rigour" in the subject. Precision is too vague and does not 
distinguish mathematics from other sciences and human activities at all 
(virtually any science tries to be precise, and what does it mean 
"sufficiently precise"? what is sufficient and what is not? it is 
highly non-specific!) in contrast to the mathematical rigour which is, 
in fact, nothing else as professional developing formal tools 
strengthening our abstract thought abilities.

Also it is not a precision what distinguishes various views on the 
foundation of mathematics (Intuitionism, Predicativism, etc.) 
Mathematics has one and the same, a very specific kind of precision - 
the rigour (a specific formal view). We may be inclined, say, to 
predicatevism or to anything else in mathematics by a quite different 
reason than a precision - a specific kind of intuition we want to 
explore by appropriate formal methods.


Vladimir Sazonov


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