[FOM] Foundational Challenge

[Joe Shipman]

As you describe it, this is a phony fight. Nobody is asking category theorists to redo their work in ZFC. 

1) There have been once-and-for-all metatheorems proven which have shown that any math done in category theory can be formalized in ZFC plus very mild large cardinal axioms.
2) It was correct for ZFC proponents to ask for that, and not difficult for category theorists to provide it.
3) The reverse process has not been done satisfactorily, therefore category-theoretic foundations proponents cannot offer a general foundation as ZFC proponents can.
4) No one is saying Category Theory is illegitimate or that no one should use it or “challenging the conclusion” that it is the appropriate way to approach certain subfields of math; rather, Category Theorists were asked to engage in a form of mathematical hygiene by showing that their work was compatible with Set Theory AND THEREFORE compatible with everything else. It is precisely the general foundational properties of set theory that allow assurance that work in a specialized area can be connected to and does not threaten and is not threatened by the rest of Mathematics.
5) if you think a practitioner should not be asked to show that his work can in principle be formalized in Set Theory, then he should explain how mathematicians working in other areas can regard his work as relevant to theirs. If his alternative foundations allow their work to be connected to his, then Set Theory can indeed be bypassed! But I’m still waiting for evidence that any alternative foundation can do this for math in general except by a detour through set theory.

— JS

Sent from my iPhone

> On Jan 16, 2020, at 10:04 AM, katzmik at macs.biu.ac.il wrote:
> 
> The "foundational challenge" thread is actually a pair of threads woven into
> one: (1) is ZCF the best framework for implementing proof checker packages;
> and (2) apart from computer verification, is ZFC the best "universal"
> framework for mathematics.  Most of the recent postings have been in thread
> (1) but this one by Levy is in thread (2).  I would like similarly to add a
> comment to thread (2).
> 
> It has been assumed for about 70 years that in homological algebra and
> certain subfields of algebraic geometry, category theory is a more
> appropriate framework than set theory.  This is the position of
> leading practitioners in these fields, such as MacLane, based on their
> own work.  HIstorians like Corry have studied the question in detail,
> and showed in particular that the Bourbaki's slowness in accepting
> category theory as foundation resulted in serious shortcomings in some
> of their volumes.  I think it would be futile for people outside the
> field to challenge such a conclusion.  Moreover, attempts by set
> theorists to challenge the conclusion that category theory is a more
> appropriate foundation in these fields can only lead to such attempts
> being discredited.
> 
> The developments with HTT are more recent and the conclusions seem
> less clear.
> 
> To retain credibility, arguments in favor of ZFC or such as foundation
> should be grounded in a choice of field where such an argument is
> proposed, and preferably made by experts in such fields together with
> set theorists.
> 
> As is clear from previous discussions, many participants here are
> aware of the fact that the discussion is not purely academic, since
> whatever conclusions are reached may affect grant support.  However, I
> think that ostrich insistence on ZFC as universal foundation for all
> of mathematics is no longer credible, and may have the opposite effect
> as far as grants are concerned.
> 
> M. Katz
> 
>> On Thu, January 16, 2020 04:28, Paul Blain Levy wrote:
>> Dear Joe,
>> 
>> I would like to question some of the premises of your challenge.
>> 
>> Firstly, the idea that there is such a thing as "ordinary" or
>> "mainstream" mathematics that is somehow more important than other kinds
>> of mathematics.
>> 
>> I would say that everybody in the world uses mathematics and is
>> therefore a "mathematician".  90% of them (let's say) use only simple
>> arithmetic that children learn.  90% of the rest use only simple algebra
>> that teenagers learn.  90% of the rest use only Peano arithmetic.  90%
>> of the rest use only second order arithmetic.  So that leaves very few
>> who know or care about ordinals.  Even among category theorists,
>> ordinals are a niche concern.  (This is not criticizing anyone, it's
>> just saying that different people have different concerns and
>> interests.)  Does this mean that the theory of ordinals is not "ordinary
>> mathematics"?  Does it make that theory less important than the rest? 
>> If you are saying that, then where are you drawing the line in the above
>> spectrum?  And why?  It seems arbitrary to me.
>> 
>> Secondly, you say there is no evidence that an "alternative" foundation
>> would help these people whom you designate "ordinary mathematicians"
>> achieve what they want to achieve.  Perhaps you are right.  But isn't
>> that equally true for ZFC?  I think you will find that most of them are
>> ignorant of *both* ZFC and alternative foundations, and this doesn't
>> prevent them from functioning effectively in their field.
>> 
>> Thirdly, I think it is necessary to step back and ask what these people
>> are *really* trying to achieve.  As you say, they want to "find and
>> prove theorems".   Surely this requires that their mathematical language
>> is meaningful and their forms of inference genuinely establish truth.  
>> Otherwise they are not "proving" anything.  So foundations (even for
>> these people) have to be guided by the questions of meaning and truth,
>> above all else.  And in this respect, ZFC is wanting, as I have argued.
>> 
>> Paul
>> 
>>> Joe Shipman <joeshipman at aol.com> writes:
>>> 
>>>> ?None of this comes close to giving a positive answer to my challenge,
>>>> and I?m not interested in changing the subject, because my challenge
>>>> is intended to address the PRACTICAL question ?would an ordinary
>>>> mathematician whose primary interest is in finding and proving
>>>> theorems become any more effective at doing so by using an alternative
>>>> foundation than ZFC (plus large cardinals if necessary)??
>>>> 
>>>> If no positive answer to my challenge is forthcoming, I conclude that
>>>> the only mathematicians who might need to care about alternative
>>>> foundations are those working in areas of math that relate directly to
>>>> the alternative foundation.
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