[FOM] Foundational Challenge

[Paul Blain Levy]

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.