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

[Joe Shipman]

Although what you said is correct, there is enough consensus about arithmetic that no one (who is speaking as a mathematician) is going to declare Euclid’s assertion that there are arbitrarily large prime numbers to be a matter of opinion.

Even for more complicated statements like the Dirichlet theorem (all variables are natural numbers)

For all m, n > 0 there exists a prime p such that  either p divides m and n or n divides p-m

there is no one who seriously claims this is a matter of opinion (by which I mean a statement on which people may disagree about its truth value, not a statement on which people may disagree whether it has been established). There is no conceivable state of affairs where professional mathematicians will insist that that statement is false (a computable bound exists, the proof is constructive).

As for tautologies, non-complete systems can have them. One can make a good case that all theorems of Peano Arithmetic (and the Euclid and Dirichlet theorems can be proven in PA) are (or can be seen to be equivalent to) logical truths, the difficulties for logicism arise when one needs an axiom of infinity.

The interesting question to me is why we don’t expect there to be permanent disagreements about statements of arithmetic, as we do about statements like AC or CH.  One can be of the opinion that the Riemann Hypothesis (in its arithmetical form that a certain computation never halts) is true, or that it is false, but we certainly don’t think it will ever occur that one school of mathematicians will come to believe that it has been proven true and another that has been proven false. Unlike with CH, nobody expects that among the (consistent) axioms that will ever be seriously proposed, there are some that prove RH and some that prove not-RH.

— JS

Sent from my iPhone

> On Dec 28, 2019, at 10:50 AM, Michael Lee Finney <michael.finney at metachaos.net> wrote:
> 
> 
>> I distinguish between a “fact” and a “tautology”, although if you make a
>> good enough argument for logicism I would be willing to reclassify
>> mathematical theorems as tautologies rather than facts.
> 
> Since tautologies must be valid in all interpretations, they can be identified
> with theorems only where the axiom system is complete with respect to the
> semantics. Since (at least in my opinion) non-complete systems are far more
> interesting and potentially more useful, I would argue that facts are limited
> to theorems -- and even then, only with respect to a specific axiom system.
> In a complete system, it could be argued that there are only facts and no
> opinions, since every fact has a theorem.
> 
> For example, we know that the Axiom of Choice is independent of ZF, so its
> truth is merely an opinion relative to ZF and not a fact. Yet, in ZFC -- which
> contains some equivalent axiom -- its truth is a fact relative to ZFC. And,
> likewise, the truth of CH is an opinion relative to ZFC.
> 
> I doubt the existence of universal facts in mathematics. Virtually every axiom
> in formal logic has been challenged -- or if one has not, it almost certainly
> will be. Set theory itself has been challenged as the best basis for general
> mathematical research.
> 
> I think that all mathematical facts are relative rather than absolute. And
> physical facts are limited to observation and thus are also not absolute.
> Political facts are pretty much entirely in the eye of the beholder, they
> certainly cannot be absolute.
> 
> For me, an axiom system includes the proof system, the deductive rules and, of
> course, the set of axioms (I actually go further, but that is not relevant
> here). All of those can be varied from one axiom system to another, so just
> the list of axioms is insufficient.
> 
> I would even argue that if a person does not have an axiom system (or at least
> a model inside an axiom system), that they are not yet working in mathematics,
> because there is no ground to stand on. Without an axiom system (such as ZFC),
> a person is not even in a position to reason about anything, because the
> axiom system defines valid reasoning. At most they will be assuming an
> implicit axiom system that supports whatever reasoning they favor.
> 
> Michael Lee Finney
> 
> 
>