[FOM] Epistemology for new axioms

[Joe Shipman]

Of course, the same thing could be said of ordinary mathematicians who assume unproved hypotheses all the time in their research; but there are two important differences.

1) The set theorists who believe V=L, in my example, would be more sure that their axioms wouldn’t be disproved than number theorists who assume the Riemann Hypothesis, and less willing to speak of their results as conditional.

2) Both sides of the open question can be fruitful in set theory; but I’m not aware of a comparable situation in ordinary mathematics (there aren’t a lot of papers on the consequences of ~RH). The closest I can think of is the P-NP question but we don’t have any community of researchers who are working on the consequences of P=NP because they expect that to be true.

— JS

Sent from my iPhone

> On Sep 7, 2019, at 9:48 PM, Joe Shipman <joeshipman at aol.com> wrote:
> 
> Tim, the difference is that finite group theorists’ results and infinite group theorists’ results are not logically incompatible, they can use each others’ work. Those who favor V=L and those who favor large cardinals (large enough to contradict V=L) can build up independent and incompatible developments of mathematics, and both will not like being forced to always reframe their results as implications of unproven axioms when in their own world and that of all their colleagues those axioms are always assumed.
> 
> Sent from my iPhone
> 
>> On Sep 7, 2019, at 10:50 AM, Timothy Y. Chow <tchow at math.princeton.edu> wrote:
>> 
>> Joe Shipman wrote:
>> 
>>> I can foresee the "pluralism" possibly leading in an unhealthy direction, with the kind of fruitlessness that is characteristic of the endless debates about "interpretations" of quantum mechanics. If some mathematical communities concentrate on developing set theory in incompatible directions than others do (V=L vs Large Cardinals, for example), they might not have much to say to each other, especially if the methods and techniques are different enough that it will be uncommon to be expert in both "schools".
>> 
>> I like your analogy with quantum mechanics, but I don't see anything "unhealthy" about your scenario, or at least nothing more unhealthy than the usual perils of specialization.
>> 
>> Suppose set theorists split into two camps as you suggest.  I'm not sure that it will be any "worse" than, say, the split between finite group theorists and infinite group theorists.  The axioms "the ground set is finite" and "the ground set is infinite" are incompatible, and so finite group theorists and infinite group theorists "might not have much to say to each other" and there might not be many who are experts in both subjects.  But so what?
>> 
>> It's true that set theory is closer to philosophy than most other areas of mathematics are, and debates about the foundations of mathematics may be similar in flavor to the debates about the interpretations of quantum mechanics.  But I don't see what is so harmful about such debates.  They don't prevent anyone from getting on with their technical work, since there is still a consensus about what constitutes a technically correct argument or calculation.  Once in a long while, the philosophical debates may even inspire some fresh technical idea.  So what's the problem?  If people want to engage in philosophical debates, why not let them?  If you don't like the lack of "progress" (whatever that means) then just ignore the debates.
>> 
>> Tim
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