Categorical Foundation of Mathematics?

[Timothy Y. Chow]

Mikhail Katz wrote:
> To make the idea of philosophical coherence clear and precise (and 
> therefore workable), it may be helpful for starters to distinguish it 
> from "truly existing".  Otherwise one will have to fall back on the 
> working hypothesis above, no matter how implausible it may seem.

Harvey Friedman is the only one who can say exactly what he means by the 
term "philosophically coherent," but since I have interrogated him on this 
point in the past, I can give my approximate understanding as a starting 
point that may be useful for others.

The idea here is that to build a foundation, one wishes to start with very 
basic concepts that, to the best of our ability to determine, are 
unproblematic, and that cannot be analyzed into even simpler and more 
basic concepts.  From these basic concepts, one builds up more complicated 
concepts in a careful, step-by-step fashion, being alert to any additional 
assumptions that are introduced.

This might just sound like the axiomatic method, but it's not quite the 
same.  Formalists are happy with a kind of axiomatic method, in which one 
writes down any axioms one pleases, and explores their consequences.  But 
one of the fundamental features of mathematics is that we have a strong 
pre-theoretic understanding of what constitutes "logically correct 
reasoning," whereas a formalist is happy playing around with any syntactic 
rules whatsoever, regardless of whether they bear any relationship to 
"logically correct reasoning."  Furthermore, as soon as they are presented 
with some formal axiomatic system, mathematicians immediately want to ask 
about the *meaning* or *interpretation* of the system.  They also want to 
know if the interpretation has anything to do with the mathematics that 
they already know and love.  Formalists have no answers to such questions; 
indeed, many formalists insist that such questions are pointless or 
senseless.

Even if one is not a formalist, there is still a difference between the 
"axiomatic method" as broadly construed, and the step-by-step construction 
of mathematics from maximally simple concepts.  The distinction can be 
illustrated by considering axioms for category theory, or for model 
theory.  Someone who proposes such axioms is motivated by actual 
mathematical practice and has not just plucked an arbitrary axiomatic 
system out of thin air.  So far so good.  But the point is that the 
concept of a "category" or a "model" is complicated and sophisticated. 
We do not expect elementary school students to understand what a category 
is or what a model is.  If an undergraduate student asks what a category 
is or what a model is, we do not respond by saying that the concept is so 
simple that we can barely think of a way to describe it in terms of 
anything else.  For categories, we launch into a short lecture giving 
examples of categories, with the examples themselves requiring rather 
sophisticated concepts to grasp.  For models, we launch into a short 
lecture about formal languages and interpretations.  The fact that we have 
written down some formal axioms for categories or models is not adequate 
from a foundational point of view, unless we are willing to fall back on 
the formalist dodge of saying that we don't care what the axioms "mean," 
and insisting that our foundations are adequate as long as we know how to 
play the formal game.  If our explanation of what the axioms are supposed 
to mean relies on concepts that need to be explained in terms of more 
basic concepts, then we're sort of cheating, at least as far as the 
project of building the foundations of mathematics is concerned; we're 
"taking as primitive" something that every sophisticate knows (wink, wink, 
nod, nod) to be far from being primitive in reality.

In short, a "philosophically coherent" development of f.o.m. (a la 
Friedman) is supposed to be a development that builds up the entire 
conceptual structure of mathematics from the most basic concepts possible. 
It is not enough to formally mimic existing mathematical practice while 
punting on the hard work of analyzing compound concepts in terms of atomic 
ones.

As an aside, when Friedman complains of "incoherence" he just means that 
something fails to be coherent in the above sense, and he doesn't 
necessarily mean that it is nonsensical or contradictory.

Tim