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

[Kapantais Doukas]

As for my suggestion, it consisted in a scenario, where CH (or –CH) become matters of fact, not because either one of them produce new mathematical consequences, when added to ZFC, but because they are theorems of some totally different foundations that the community will be willing to endorse in its own right (i.e. not on the basis of their yielding either one of these).



DK

________________________________
Από: fom-bounces at cs.nyu.edu <fom-bounces at cs.nyu.edu> εκ μέρους του Joe Shipman <joeshipman at aol.com>
Στάλθηκε: Παρασκευή, 27 Δεκεμβρίου 2019 8:55 μμ
Προς: Foundations of Mathematics <fom at cs.nyu.edu>
Κοιν.: tchow at alum.mit.edu <tchow at alum.mit.edu>
Θέμα: Re: [FOM] Απ: Fact and opinion in F.O.M.

I thought I had left open that possibility.

Because both CH and not-CH fail to give any new arithmetical consequences when added to ZFC, my claim was that I did not see how unbelievers could be persuaded of either, because of the consistency of weak systems denying infinite or uncountably infinite sets and thus rendering both CH and not-CH meaningless enough to only be matters of opinion and not matters of fact.

However, I did mention that if fundamental physics were successfully mathematized to a much more rigorous extent than it has been, a Putnam-type argument could be ontologically persuasive and entail the facticity of some infinitary statements. I just don’t see that that is likely anytime soon.

I’ve offered an alternative fictitious history of mathematical physics, where the atomic hypothesis was unproven for long enough that the belief in “continuous matter” led to the adoption of a real-valued measure axiom, and the discovery of general relativity prior to the discovery of atoms then made the appropriate response to the Banach-Taraki anomaly be a rejection of isotropy rather than a rejection of infinite divisibility. In this scenario, by the time the atomic theory and quantum mechanics were discovered, the RVM axiom would still be regarded as intuitively true, and would be preferred to weaker systems like ZFC because it could prove their consistency (Solovay’s result). In that alternative history, not-CH would be a theorem, but the development of modern physics might lead to its losing status as a matter of fact.

— JS

Sent from my iPhone

On Dec 27, 2019, at 1:13 PM, Kapantais Doukas <dkapa at academyofathens.gr> wrote:



On the discussion between Shipman of Chow



I am under the impression that both sides take something for granted, when possibly they ought not to. Perhaps, this is due to the fact that ZFC has acquired a canonical normative status. What I mean is that in the case that ZFC is our final theory, then, I can make a very good sense of Shimpan’s claim that CH might definitely belong to the matter of opinion realm; and I also understand what he means by “matter of opinion” and “matter of fact”, or so it seems to me. However, mathematicians, as any other scientists, do not despair in such kind of situations and try to abductively work their way out. So, suppose that someone is a fervent champion of CH and that CH is a fact. It is not to be excluded that this same person will come up with another kind of foundations, which (i) will yield CH and (ii) will persuade the disbeliever that they are in themselves (i.e. not because they yield CH) a better candidate for the job than ZFC ever was. By the way, this was what Gödel was hoping for, though he was trying to do it without abandoning ZFC. In that sense, I do partake Shipman’s idea that such issues are matters of opinion forever, but only if phrased in some conditional form. Namely, that if ZFC is the final theory, then such and such. NB. Shipman’s point of view is, I take it, in principle, and so “not seeing this or that happening in the foreseeable future” rebuke does not challenge his actual point. The same with not seeing ZFC being abandoned in the foreseeable future with respect to my own.

Doukas Kapantais
________________________________
Από: fom-bounces at cs.nyu.edu <fom-bounces at cs.nyu.edu> εκ μέρους του Timothy Y. Chow <tchow at math.princeton.edu>
Στάλθηκε: Πέμπτη, 26 Δεκεμβρίου 2019 11:58 μμ
Προς: fom at cs.nyu.edu <fom at cs.nyu.edu>
Θέμα: Re: [FOM] Fact and opinion in F.O.M.

Joe Shipman wrote:

> (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.

Okay, how about the following suggested definition of "fact"?

   (*) A mathematical statement X is a fact if no axiom that has ever
       been seriously proposed (or ever will be seriously proposed)
       implies not-X.

This stays closer to your statement (1) above and steers clear of my
objections to your "permanent disagreement" formulation.

If we accept, as I do, that V = L has been "seriously proposed" as an
axiom, then ~CH is not a fact.  Off the top of my head, I can't think of a
seriously proposed axiom that implies ~CH, but maybe someone else can; if
there is one, then that would mean that CH is not a fact either.
(Freiling's axiom of symmetry, maybe?)  If neither X nor not-X is a fact
then we could say that X is not a "matter of fact" (and similarly not-X is
not a matter of fact).

Then your question becomes whether there exist any non-arithmetical facts.

By the way, there seems to be some similarity between your concept of
"fact" and Feferman's concept of a "definite mathematical problem," as in
his paper, "Is the continuum hypothesis a definite mathematical problem?"
(Though Feferman seems to take a different direction from what you're
proposing.)

Tim
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