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

[Joe Shipman]

I was careful with my words. Apparently not careful enough.

I distinguish between a “fact” and a “matter of fact”, and between an “opinion” and a “matter of opinion”.

Whether the twin prime conjecture is true or false is, I believe, a matter of fact, but the twin prime conjecture is not a “fact” because we don’t have good enough reason to accept it as a fact yet.

The Continuum Hypothesis is, a believe, a matter of opinion, because there can be people of good will permanently on opposite sides of the question of its truth value.

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”. 

This contrasts with situation for AC or CH. I believe there may come a time when some mathematicians start assuming V=L as an axiom extending ZFC and others start assuming RVM. They will be able to talk to each other and understand each others’ proofs but reject their assumptions. At the moment they are polite enough, or the main journal editors are insistent enough, that they still repor their results from axioms A extending ZFC in the form “A—>B“ rather than B (with an exception for results requiring Grothendieck’s Universe axiom, where omitting mention of it is mostly tolerated).

— JS

Sent from my iPhone

> On Dec 24, 2019, at 12:53 PM, Timothy Y. Chow <tchow at math.princeton.edu> wrote:
> 
> Joe Shipman wrote:
> 
>> Recently I concluded that the biggest problem in internet discussion of public policy was that most people seemed not to competently distinguish between fact and opinion, and that one easy distinguishing feature is that people of good will cannot be in permanent disagreement about a matter of fact, so that people who mistook their opinion for a fact might unfairly regard their opponents as perverse or dishonest.
>> 
>> I now see that this distinction sheds light on some puzzles regarding choice of axioms.
>> 
>> To a first approximation, statements of arithmetic are matters of fact, while provably independent statements like CH are matters of opinion.
> 
> I don't fully understand what you're saying.  You seem to be conflating metaphysics and epistemology.
> 
> Is a provably independent statement of arithmetic a "fact" or an "opinion"?
> 
> If a "fact" is something that people of good will cannot be in permanent disagreement about, then it would seem that the analogue in mathematics would be a theorem, rather than a statement of arithmetic.
> 
> Tim
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