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

[Michael Lee Finney]

> 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