Categorical Foundation of Mathematics? (jodmos.horon)
Timothy Y. Chow
tchow at math.princeton.edu
Wed May 25 23:42:47 EDT 2022
Jodmos Horon wrote:
> Harvey Friedman wrote:
>> No one has challenged me on here to show how to present the usual
>> f.o.m. with philosophical coherence. And no one has tried to present
>> categorical foundations in a philosophically coherent way on here.
> I do not see the need, honestly. For neither.
I think that the above exchange, brief though it is, highlights perhaps
the main point of disconnect in these types of debates/discussions.
Let's consider an analogy from another area of philosophy. Descartes
famously questioned how we can have certain knowledge of anything. What
if there is a malicious demon trying to fool us at every turn? He
concluded that the only secure starting point for building an edifice of
secure knowledge was "cogito ergo sum." Not too many subsequent
philosophers accepted Descartes's entire train of thought---Edmund Husserl
perhaps came closest, but did not accept Descartes's argument for the
existence of God---but Descartes did have a powerful influence on Western
epistemology, which ever since then has been fascinated by what is often
called "foundationalism." Roughly speaking, foundationalism is the idea
that the way to secure a body of knowledge, and justify it, is to analyze
its logical structure, decomposing complex concepts into simpler ones,
until a bedrock of clear and sound concepts and methods is reached. Then
one constructs, or reconstructs, the body of knowledge carefully on the
basis of the solid foundation.
There is a big divide between people who are attracted to foundationalism
(as I've defined it) and people who aren't. The former typically see
foundationalism as the *only* way to arrive at truly secure knowledge, or
at least that if a foundationalist project can be carried out, then it is
indispensable. The latter see no need for foundations at all, or if they
accept that a foundationalist project is valuable, then its value is
similar to that of a bitter pill that cures you of a disease, or the
scaffolding used to help construct a building. Once you are cured, or
once the building project is finished, you can throw away your pills or
scaffolding and never think about them again. You don't need to worry
that you are sawing off the branch that you are sitting on.
Is the body of mathematical knowledge that we have accumulated over the
centuries sound and secure? Most would answer yes (modulo the inevitable
smattering of mistakes and gaps, but these can all be ferreted out and
corrected with enough determination and patience). But foundationalists
maintain that the reason we know the answer is yes is that a successful
program of reducing all mathematics to a comprehensible core (set theory,
or perhaps arithmetic if you care only about "countable" mathematics), and
then analyzing that core thoroughly, has been carried out. Moreover, that
foundational structure remains important to this day if we want to
continue to have confidence in mathematics.
By contrast, there are others who see no need for a foundationalist
project. They argue that we acquire confidence in mathematics via a
sociological process of checks and balances, not through some formal
process of logical dissection and mechanical verification. Or, if they
concede that the foundational work of the 20th century had some value, it
was valuable in the sense that Prozac is useful for treating anxiety.
Once we are cured of our anxiety, we can throw away the Prozac. Now we
can go about our business freely and ignore everything we used to be
worried about, because all those problems were solved once and for all.
Don't like sets? Throw them out. Why worry? Just be happy!
Once again, Maddy's question is crucial: What do we want a foundation to
do? There is little chance of reaching agreement if (for example) one
party is obsessed with Risk Assessment and the other party prefers extreme
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