[FOM] Epistemology for new axioms

[Joe Shipman]

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”.

Fortunately, most mathematicians wouldn’t be vulnerable to this, but those working in set theory and other very abstract areas might.

— JS

Sent from my iPhone

> On Sep 5, 2019, at 8:06 PM, Timothy Y. Chow <tchow at math.princeton.edu> wrote:
> 
> Joe Shipman wrote:
> 
>> In fact, for arithmetical statements, I claim the opposite: although there may always be arithmetical statements which some mathematicians will claim to have proven while others will not accept the proofs, there will never be an arithmetical statement such that two groups of mathematicians will permanently disagree about it, with one group claiming it has been proven and the other claiming it has been disproven.
> 
> This last statement about arithmetical statements seems very weak.  Do you think that there will ever arise a "permanent disagreement" about *any* mathematical statement in the sense of one group claiming that it *has been proven* and the other group claiming that it *has been disproven*? We don't have one group of mathematicians going around saying that V=L "has been proven" and another group of mathematicians going around saying that V=L "has been disproven."  We don't even have one group of mathematicians going around saying that Mochizuki has proven abc and another group of mathematicians going around saying that abc has been disproven.  A "permanent disagreement" of this type would seem to represent a kind of major sea change in the way mathematicians think about what the word "proven" means, going well beyond technical distinctions between arithmetical statements and non-arithmetical statements.
> 
>> ***There is no proposition of any mathematical interest that is known (by ZFC proof) to be relatively consistent with and independent of ZFC, such that mathematicians will eventually have a consensus view that it should be considered as a fundamental axiom with which ZFC should be augmented.***
>> 
>> Here "a consensus view" means comparable to the view that Zermelo set theory should be augmented with the Replacement Axiom to give ZF. In my opinion, statements like CH or V=L or their negations will never be settled because there is no way for a believer to persuade a denier, they will never lead to consequences for which any kind of empirical support can be given such as numerical evidence or necessity for science.
> 
> I'm inclined to think that a consensus view, as you've defined it above, isn't going to emerge in the forseeable future for *any* statement.  The reason is that we now live in an age of mathematical pluralism, where some people don't even like infinite sets, let alone ZFC, and some people want to use type theory as a foundation, etc.  There's no way to stuff all these diverse viewpoints back into Pandora's box and return to the good old days when people went along with an axiom set because they could be sweet-talked into thinking that they believed those axioms all along anyway.
> 
> Even something as harmless as the existence inaccessible cardinals seems to be stuck with the status of "something we freely assume when we need it, but which is flagged explicitly when it is invoked."  If even inaccessible cardinals haven't achieved "fundamental axiom" status, then what hope do other candidate axioms have?
> 
> Tim
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