[FOM] 486:Naturalness Issues
Timothy Y. Chow
tchow at alum.mit.edu
Thu Mar 15 17:55:36 EDT 2012
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
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
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.
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