Categorical Foundation of Mathematics? (jodmos.horon)

[Timothy Y. Chow]

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 
sports.

Tim