[FOM] Mathematics and formalizability

Vladimir Sazonov V.Sazonov at csc.liv.ac.uk
Mon Aug 2 16:35:47 EDT 2004

Martin Davis wrote:
> At 12:21 PM 7/30/2004, Vladik Kreinovich wrote:
>> There are two different notions of what mathematics is.
>> We mathematicians usually define mathematics by the level of rigor, while
>> others define mathematics by objects of study: if it is about abstract
>> mathematical objects, it is mathematics, eevn if there is no rigor.

What are "abstract mathematical objects" participating in such a
definition of mathematics is unclear. This looks circular.

>> Mathematicians all (99% probably) agree that mathematics is something 
>> that is
>> formal or at least formalizable.
> This narrow approach flies in the face of history and mathematical 
> practice. 

It depends how to read the above phrase and what Vladik Kreinovich
had in mind (that is, the whole context should be taken into account).

Most mathematicians aim for "rigor" but give little if any
> thought to formal systems. Rigor (and ultimately formalizability) can be 
> regarded as goals, but that's all. 

Is not this "that's all" (which is "regarded as goals"!!), exactly what
distinguishes mathematics from any other human activity - the "rigor"
of such a level (both ideal and practical) which does not exist
anywhere else.

Is not "rigorous" just a slightly weaker version of "formal"? It
is not necessary to understand "formal" in the sense of contemporary
mathematical logic. Mathematics (with some level of rigor/formal
reasoning) existed long time before that. Anyway, no working
mathematicians write absolutely formal (non-trivial) proofs. But we
know some contemporary ideal and try to follow it, in principle (of
course not literally because we are not computers), and that is enough
for doing mathematics in a reliable way and to realize its formal
character as an ideal only (but we need to achieve this ideal if
a computer proof system should be created).

(I think this is a reply to the posting "The Role of Formalization"
by Henrik Nordmark.)

No question, there are preliminary heuristic considerations in
mathematics (does not matter who is doing them, mathematician or
not and how soon these preliminary considerations will become
rigorous). This may be also called "intuition" which is used in
any human activity and is not a prerogative of mathematics only.

What is the subject of mathematics if not rigorous considerations
related to whatever we want? Or is mathematics rather about
quantitative and space forms (and what else?) according to Engels?
What is the other alternative to the rigor as the main point of
mathematics? Is not the quantitative and space view both too narrow
and indefinite?

As to the rigor, we have a good, seemingly unique formal ideal.
I do not mean ZFC or even FOL - any formal system of any kind
could potentially be considered as mathematical and mathematics
can be considered, in a first approximation, as a science  on
(or study of) formal systems having an intuitive content.

By the way, is not mathematical heuristics just a way to create some
kind of a formal (or semi-formal) system? If a heuristic does not give
anything like that (say, allowing to make some calculations) what
is the point of this heuristic and why it is related with mathematics
at all? The Heaviside operator calculus mentioned by Martin Davis is
far from my interests, but is not this calculus itself just a kind of
formalism? Was not ZFC, as well as any calculus, axiomatic system or
a theorem and its proof, also created by some heuristic?

What is the point to stress on these heuristics? Is not this self
evident? Does anybody doubt on the necessity of intuition in
mathematics (and everywhere)? And what is then that "narrow"
formalistic (or formalizability) approach which does not take all
of this into account, and who is the proponent of this grotesque
approach? Was it Hilbert, who suggested that geometrical "plane",
"line" and "point" may be replaced by "table", "chair" and "beer glass"
in the axiomatic geometry? May be he, saying that, did not understand
the role of geometrical intuition and had no geometrical images in
his mind? Or was it Abraham Robinson, the creator of Nonstandard
Analysis (based on a non-trivial interplay of a formalism and
intuition), who explicitly expressed his formalist views?
Why not to take these views in a rational, non-anecdotal way?

Vladimir Sazonov

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