[FOM] Foundational Challenge

[katzmik at macs.biu.ac.il]

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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