[FOM] 486:Naturalness Issues

[Timothy Y. Chow]

Harvey Friedman asks for feedback on his latest examples of statements 
that require large cardinals and that he hopes are "natural."

It may help to step back a bit and ask why one is interested in "natural" 
statements in this context.  As I understand it, one of the main 
motivations is the feeling that finding such statements will raise the 
social status of f.o.m.  Friedman's preoccupation with the social standing 
of the mathematicians whose opinions he solicits seems to confirm this.

The word "natural" can mean many things.  In my opinion, what would most 
raise the social status of f.o.m. would be a "natural" independent 
statement in the following sense of the word "natural":

1. The statement in question originally arose in the course of "ordinary 
   mathematical research"; that is, mathematicians pursuing research on 
   questions with no known or even suspected connections with f.o.m. were 
   led to formulate the statement as a question or conjecture of great 
   interest, or at least moderate interest.

In this sense of the word "natural," any statement that is devised with 
the express purpose of demonstrating some point about f.o.m. is doomed to 
be criticized as "unnatural."  In particular, Friedman's examples cannot 
be natural in this sense unless he manages to demonstrate the independence 
of some statement that is already floating around in the literature.

Supposing, though, that we set our sights a little lower.  That is, we 
resign ourselves to considering statements that are constructed with the 
express purpose of demonstrating some point about f.o.m., and so are 
"unnatural" in the above sense.  However, within these constraints, we 
still wish to maximize the gain in the social status of f.o.m. by finding 
statements that are as "natural" as possible.  What sense of the word 
"natural" could we hope for that would boost the social status of f.o.m.?  
Here's one possibility:

2. The statement, considered in isolation, enjoys some kind of aesthetic 
   properties that lead considerable numbers of mathematicians with high 
   social status to endorse it with the word "natural."

This isn't bad, since a good way to increase the social status of 
something is to get people with high social status to endorse it.  Still, 
as I've stated it, #2 is somewhat unsatisfactory, since it does not give 
us a good sense of *what sorts of aesthetic properties* will elicit the 
desired endorsement from mathematicians with high social status.

In fact, I would go further and say that what is unsatisfactory about #2 
is its implicit assumption that "naturality" is some kind of intrinsic 
property about a statement that can be judged by examining it in 
isolation.  On the contrary, a statement is natural insofar as it connects 
with ideas and theories that are already recognized to be part of ordinary 
mathematical research.  For example, Friedman's definition of "order 
invariance" is judged by most mathematicians to be natural.  Why?  I would 
argue that this is because similar definitions, or perhaps even exactly 
the same definition, have appeared repeatedly in the course of ordinary 
mathematical research.

The function Z+up (by the way, Friedman clarified for me, in response to a 
private email, that "Z+" means "the positive integers"; I wasn't sure 
about this from his original post) is bound to raise some eyebrows because 
it is not a familiar concept, nor is it highly similar to concepts that 
almost every mathematician encounters on a routine basis.  I strongly 
suspect that even with the endorsement of several mathematicians of high 
social status, it will still not be widely regarded as "natural" *until* 
a strong connection with existing mathematical research is demonstrated.  
It's not necessary, of course, that one discover the exact definition of 
Z+up in an existing mathematical paper of importance, but at least the 
theory surrounding the concept needs to be developed far enough that the 
connection with existing mathematical research is clear.  For example, if 
the theory could be applied to prove some existing important mathematical 
conjecture, then its "naturality" would be established, despite its 
"artificial" origins and perhaps eccentric ("unnatural") appearance upon 
first inspection.

Claiming victory at this early stage is therefore, in my opinion, highly 
premature, even though I'm a fan of Friedman's work and would like to see 
the social status of f.o.m. raised.

Tim