[FOM] Mathematics and precision

[Vladimir Sazonov]

Quoting "Timothy Y. Chow" <tchow at alum.mit.edu> Sat, 03 Mar 2007:

> Some years ago it occurred to me that a possible definition of mathematics
> is that anything that is *sufficiently precise* is mathematics.
>
> The term "sufficiently precise" it itself not sufficiently precise to
> count as mathematical, but perhaps it is sufficiently precise to be a
> useful idea.

<...>

Thus
> mathematics, unlike most other fields of study, is characterized not so
> much by its *subject matter* as by a certain *threshold of precision*.


In principle, I would agree if I read you correctly. Mathematics is not 
about "numbers and geometrical figures" and also not about any kind of 
eternal Platonist truth concerning such abstract objects which are 
considered traditionally as its subject matter. Indeed, its subject 
matter is different.


It is different from "structuralist" views,
> which emphasize the *relations* between mathematical objects rather than
> their intrinsic ontology, because my focus is on precision rather than
> structure.


Looks plausible.


It is different from logicism and formalism, because I am not
> claiming that formal logic has a monopoly on precision.


Why not? And what else besides formalisms and formalisability?

Whatever means "precise", it was best embodied in the idea of 
mathematical rigour which, in turn, was explicated during the previous 
century as formalisability. A formal system can be understood rather 
wide - not only as formal (first order) logic. For example, these may 
be formal rules used in Analysis (called also Calculus), or this may be 
any programming language, it may be any reliable mechanical system. The 
main characteristic of a formal system (based on some formal rules) is 
that once a formal construction (say a proof of a theorem or a program) 
is written down its syntactical "correctness" is only the question of 
(deterministic) computation - that is, something highly reliable (and 
also physically presentable as contemporary computers) and in THIS 
sense precise.

Then mathematics can be simply characterized as science on formal 
systems as TOOLS governing and strengthening our intuition and thought 
abilities.

Pure intuition which is not supported by formal rules (and is not even 
intended to be supported) is anything, but not mathematics.

Pure formal rules whose goal is not to support our intuition and 
thought are anything (say, a chess game) but not a mathematics.

To say about mathematics as of something precise without mentioning the 
real mechanism making it precise (implicitly or explicitly used by 
mathematicians already thousands years) is, I believe, insufficient.

Our understanding on formal systems (and computers) can be changed 
along with the general progress, but I cannot imagine anything really 
precise which cannot be still called a formal system. At least, for the 
contemporary state of affairs nothing better exists. If only some 
ersatz?


Vladimir Sazonov


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