[FOM] The irrelevance of Friedman's polemics and results

[Timothy Y. Chow]

Neil Tennant wrote:
> So, that was "thumbing of the nose" #1 on the part of CMs to
> metamathematicians and foundationalists (MFs).
> "Thumbing of the nose" #2 concerned set theory.

Joe Shipman wrote:
> What Friedman is criticizing is the determination of most
> mathematicians to regard Godel's Incompleteness phenomenon as a
> curiosity that is not relevant to mathematics as a whole, rather than
> as a challenge to get involved in the METAmathematical pursuit if
> identifying new axioms to be accepted as true.

While I certainly find Friedman's theorems in this area extremely 
interesting, I am also somewhat troubled by the extent to which some of 
these efforts (I do not speak specifically of Friedman here) seem to be 
motivated by annoyance at the fact that "f.o.m. don't get no respect."

Honestly, what do I care if someone thumbs his nose at me?  There seems to 
be something psychologically unhealthy about channeling enormous amounts 
of effort into winning someone else's acceptance or changing someone 
else's behavior.

For example, it seems to me that one likely outcome of Friedman's results 
is that large cardinal axioms will join the axiom of choice in the 
category of axioms that one learns may be relevant sometimes and which one 
goes ahead and assumes if necessary.  However, "core mathematicians" will 
*still* exhibit no interest in engaging directly in f.o.m. and the "search 
for new axioms."  If new axioms turn up in the course of studying "core 
mathematics," then one will pay due attention to them, but studying f.o.m. 
for its own sake will still be regarded as deviant behavior.  In other 
words, large cardinal axioms will be welcomed into the mainstream, but 
f.o.m. won't.

I see little point in the Sisyphean task of trying to get f.o.m. respected 
in the same way that the more glamorous areas of mathematics are. 
Indeed, there is some danger that this "evangelical" motivation will draw 
effort away from the more meaningful task of formulating and pursuing 
productive agendas within f.o.m. itself.  (Though I don't think this 
potential problem has actually materialized yet in practice.)

Tim