[FOM] What is the current state of the research about proving FLT?

[Timothy Y. Chow]

Foundations of Mathematics <fom at cs.nyu.edu>
On Mon, 8 Jan 2018, Harvey Friedman wrote:
> I am not convinced by this line of reasoning. The key possibility is 
> that after seeing such (extremely unlikely, by the way) fundamental 
> inconsistencies, people will also try to prove 1 = 0 using the same kind 
> of arguments that were used to prove the irrationality of sqrt(2). Or at 
> least related totally accepted mathematical theorems.

There's a subtle difference between your objection here and what David 
Fernandez Breton wrote, which was:

> For one, there's always the possibility that there is an inconsistency 
> somewhere in the relevant axiom system (whichever it is that is 
> necessary to carry out the corresponding proof).

I would phrase your objection as, "There is always the possibility that 
the mathematical reasoning used in the proof is incoherent."  This is not 
the same as saying that there is an inconsistency in some particular axiom 
system.  For a start, there is no such thing as "the" relevant axiom 
system that is necessary to formalize a particular argument such as the 
proof of the irrationality of sqrt(2).  More importantly, David Fernandez 
Breton's objection implicitly suggests that the *source* of our belief in 
the correctness of our mathematical reasoning is the consistency of some 
particular axiomatic system that we have concocted.  If we buy that, then 
we are quickly led down the garden path: We then start to worry about the 
consistency of the axiomatic system, and seek a *proof* of that 
consistency to allay our doubts, and then are dismayed that any such proof 
must be formalized in some other axiomatic system whose consistency is 
open to doubt, and so on in an infinite regress.

This is all backwards.  It is only because we have prior confidence in the 
correctness of certain types of reasoning that we are able to define 
axiomatic systems and declare that they are even relevant to the project 
of investigating mathematical reasoning.

In fact I would go further and say that if there is something incoherent 
about the proof of the irrationality of sqrt(2)---and remember that I 
chose this specifically because the proof is so easy and simple---it is 
not going to be exposed by concocting elaborate axiomatic systems and 
discovering inconsistencies in them.  It is going to come from reflecting 
on the proof itself.  That proof is at least as simple as any axiomatic 
systems that we devise.  If we have doubts about the irrationality of 
sqrt(2) then we are going to have doubts about strict reverse mathematics.

Tim