[FOM] Foundational Challenge

[Joe Shipman]

1) I made no value judgment about “ordinary mathematics” and you should not attribute one to me. The term is intended to denote, to a first approximation, “researchers who publish mathematical papers with proofs in them in subjects taught in mathematics departments of universities, excluding areas related to to foundations”.
2) They are aware of Set Theory because most of the fundamental textbooks employ it (sometimes in introductory chapters or appendices) for clarity and rigor. Even if ZFC is not introduced as such, it is not only implicit, but easy to teach explicitly to those interested in formalization because of this background.
3) I disagree. Before axiomatic set theory comes Logic and practically all forms of foundations can be built on top of the Predicate Calculus. Those who care about meaningfulness in the sense you seem to may simply work in something like ZC rather than ZFC and all their proofs will live inside some finitely iterated powerset of omega with no quantification over arbitrary sets.
— JS

Sent from my iPhone

> On Jan 15, 2020, at 10:12 PM, Paul Blain Levy <P.B.Levy at cs.bham.ac.uk> 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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