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

[Timothy Y. Chow]

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.

Tim