[FOM] Fact and opinion in F.O.M.

[Joe Shipman]

I think you are taking overly seriously the social convention to state one’s results in the form of propositions deducible in ZFC. There are plenty of self-proclaimed Platonists out there among set theorists, who have strong beliefs in the definiteness and truth value of non-absolute propositions such as CH, and they don’t all agree with each other. My point is not that they have different concepts of what “proof” means, but that they have different axioms they believe to be true, and it seems likely they will have no way to ultimately persuade each other.

This is based on empirical observation of their professional behavior. Quite simply:

(1) all the arithmetical consequences of all the axioms that have ever been proposed appear to be compatible with each other. 
(2) this is completely untrue for statements of higher type.

I see no reason why different “schools” of set-theoretical thought could not arise with different and incompatible axioms. One could argue that they already have arisen, but for reasons of professional courtesy do not publish results following from their axioms without noting the dependence on statements independent of ZFC.

In fact, one could go further and argue that some of the researchers who reject AC are already examples of “permanent disagreement” because it seems impossible to conceive of an argument that would dissuade them from their (provably consistent) systematizations.

— JS

Sent from my iPhone

> On Dec 24, 2019, at 8:28 PM, Timothy Y. Chow <tchow at math.princeton.edu> wrote:
> 
> On Tue, 24 Dec 2019, Joe Shipman wrote:
>> I am making a strong claim here, which I have made here before and which no one took up my challenge to rebut: that although there may be permanent disagreement about whether some statement S of arithmetic has been proven or not, there will never be permanent disagreement of the type “mathematical school X believes that S has been proven and mathematical school Y believes that S has been disproven”.
> 
> I won't claim that my response to you---
> 
> https://cs.nyu.edu/pipermail/fom/2019-September/021652.html
> 
> ---was a "rebuttal", but I did point out that your "permanent disagreement" statement above is very weak.  In particular, I believe that it holds with S = CH.  With S = CH, I think that the most likely situation is that neither X nor Y will come to exist, but I don't believe that *both* X and Y will come to exist.  That would require mathematicians to come around to a totally different concept of what "proven" (with no further qualification) means, and relinquish their cherished notions about the objectivity of mathematics.  I don't see that happening in the foreseeable future.
> 
> Returning to your question:
> 
>> But can you refine this? Is there a statement of mathematics, not provably equivalent to an arithmetical statement, which is still a matter of fact?
> 
> Whether there is such a statement appears to me to be a matter of opinion.
> 
> In particular, set-theoretic platonists regard CH as a matter of fact. Now maybe you will object that the sense in which they regard CH to be a matter of fact differs from the sense in which you intend the phrase "matter of fact," but I maintain that you haven't clarified your sense of the term.  As I argued above, your "permanent disagreement" criterion, as stated, fails to exclude CH.
> 
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