[FOM] Mathematical Knowledge

[Studtmann, Paul]

I am curious whether any interesting epistemological conclusions can be
drawn from Godel¹s Second Incompleteness Theorem.  Now, I accept
Benacerraf¹s No Philosophy In/No Philosophy Out doctrine, so I accept that
by itself Godel¹s Incompleteness Theorem does not entail any epistemological
conclusion.  But I am wondering whether some reasonable sounding
philosophical principles might be supported by the opinions of mainstream
mathematicians that would entail interesting epistemological conclusions.
The question is this: what is the strongest mathematical system that
mathematicians regularly employ that we know to be consistent.
 
I should say that I am interested not only in answers to the question but
also in the reasons mathematicians have for thinking their answers are
correct.  (I am also interested in the opinions of those who think that the
question is somehow misguided, perhaps because the terms employed are
insufficiently precise, or some other such reason.)  I also am not concerned
that the question be answered entirely in the terms that I have posed it.
So, if someone thinks that we should not talk about knowing whether some
theory is consistent but rather in terms of degrees of confidence that some
theory is consistent, I am happy to hear what that person has to say.
Finally, I would be interested to know what mathematicians think about the
trajectory of our knowledge relative to strength of mathematical theory.
So, for instance, I would be interested to know that someone thinks
something like -- we can know with absolute certainty that systems strictly
weaker than Robinson¹s Arithmetic are consistent, but we cannot know with
absolute certainty that a system as strong as Robinson¹s Arithmetic is
consistent. Or, just to give another example, I would be interested to know
whether someone thinks something like -- we can know that ZFC minus the
axiom of infinity is consistent but we cannot know that ZFC is.
 
Finally, I would be interested in knowing what people think about the
following argument that makes explicit the relation between Godel¹s Second
Incompleteness Theorem and claims about our knowledge of consistency.
Argument: What we know that we can know in mathematics is whether some set
of sentences logically entails another sentence.  How can we know such
things?  Well, we can write out a proof and check to make sure that the
proof proceeds entirely by valid inferences from sentences in the set in
question.  But such knowledge leaves our knowledge of the consistency of
mathematical theories in a troubling light.  Why?  Well, to prove the
consistency of some sufficiently strong mathematical theory requires
employing a theory that is stronger, and so more likely to be inconsistent,
than the original theory.  But, this simply raises the question as to why we
should believe that the stronger theory is consistent ­ for if it is not, we
should not believe in the truth of the sentences that the theory contains.
So we would need a proof of the stronger theory, but that would require yet
a stronger theory, and so on.
 
I am not saying that the argument I have sketched is a good one.  But it
does show that Godel¹s Second Incompleteness Theorem may have
epistemological consequences.


Paul Studtmann