[FOM] Mathematics and precision

[Timothy Y. Chow]

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.  A trained mathematician instinctively knows when some 
problem or concept has been posed precisely enough to allow mathematical 
investigation.  Moreover, once something is sufficiently precise, it can 
be brought into the purview of mathematical knowledge and techniques and 
connected to other mathematical concepts, regardless of its origin.  Thus 
mathematics, unlike most other fields of study, is characterized not so 
much by its *subject matter* as by a certain *threshold of precision*.

I believe that Hartry Field once undertook a project to develop 
"mathematics without numbers," replacing number with something like 
regions of space as the foundational concept.  I did not study this 
project in detail, but my initial reaction was that for such a project to 
succeed, all that would be needed would be to make "regions of space" 
*sufficiently precise*, and then certainly much of mathematics could be 
recovered, and moreover a classical mathematician (who was unconcerned 
with philosophical purity and hygiene in the realm of ontology) would 
probably see different foundations as nearly interchangeable and 
mutually intepretable.

I wonder if this point of view has been developed in more detail by any 
philosophers of mathematics?  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.  It is different from logicism and formalism, because I am not 
claiming that formal logic has a monopoly on precision.

Tim