Categorical Foundation of Mathematics?
Timothy Y. Chow
tchow at math.princeton.edu
Fri May 20 09:25:57 EDT 2022
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
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