[FOM] Wittgenstein Inspired Skepticism

[Timothy Y. Chow]

On Sun, 5 Mar 2017, Thomas Klimpel wrote:
> If the skeptic just tries to be difficult, then I see no reason to
> enter into "retreat mode" at all. If he is serious, then the challenge
> is to answer his specific concern.
[...]
> But that would give up the consensus to use ZFC as the undisputed 
> foundations of mathematics.

My perception of the current landscape of "working mathematicians" is a 
bit different from yours.

I do not believe that ZFC is the "undisputed foundations of mathematics." 
We have skeptics right here on FOM.  Nik Weaver doesn't find the usual 
justification for the cumulative hierarchy convincing, for example.  The 
fact that there is so much discussion and debate about the consistency of 
PA among working mathematicians also shows that just because something 
("PA is consistent") is an established theorem of ZFC, it won't 
necessarily be considered as settling the matter.

I agree that the challenge is to answer the specific concern, but since 
there are so many varieties of doubts out there, I think that those of us 
who are conversant with f.o.m. do well to have a "triage" plan, to quickly 
home in on where the sticking point is.  For this purpose, I believe that 
starting with Turing machines is, nowadays, a better opening gambit than 
starting with ZFC.  This way, you'll immediately establish a foundation on 
which you can agree (unless you really do meet a Wittgenstein-inspired 
skeptic) and can build up from there.

You could, perhaps, instead start with ZFC and work your way down, but I 
don't think this works too well.  Most mathematicians have only a very 
hazy understanding of what ZFC is and couldn't tell you the axioms, so you 
kind of have to explain formal logic and Turing machines anyway.

Tim