[FOM] 486:Naturalness Issues

Sam Sanders sasander at cage.ugent.be
Mon Mar 19 18:38:38 EDT 2012

Harvey Friedman and Tim Chow discuss naturalness in FOM.  I would like to weigh in with the following analogy regarding naturalness from Biology.

Note that, in Biology, I understand "natural" to mean "occurs in the biological world"; In FOM, I understand "natural" to mean "occurs in the mathematical world".  

1) Arbitrary genetic modification (performed on plants or animals by man) is not natural in the above sense.

2) However, there are some remarkable natural genetic mutations around:


3) Any similar man-made genetic modification could still be called "natural", in the sense that it could have occurred spontaneously, just like the one for the Belgian Blue,
given enough time (but still within reasonable bounds).  A lot rides on "similar" and "reasonable" in the previous sentence, obviously.

I understand Harvey Friedman's notion of naturalness (or at least aspects thereof) in an analogous way:

1) Arbitrary logical manipulations of mathematical structures or theorems are not natural (See Reverse Mathematics).

2) There are natural mathematical notions around, which are remarkable/interesting from the point of view of logic.
The relations and graph properties considered by Harvey Friedman (order invariance, maximal square,...) are of this sort
(as they can be combined to an unprovable statement, see 3) below). 
These notions are considered in isolation and in different combinations in Mathematics, i.e. they occur already.  

3) Harvey Friedman combines these notions into a new statement which is independent of ZFC.
This new statement is natural in that it is a combination of existing properties already occurring in mathematical practice.  
The particular combination of these properties would have occurred at some point anyway, as they are studied 

It is in this way I understand the naturalness and inevitability of the independent statements: they are 
a "logician-made" combination of existing mathematical properties of which similar variations are in existence.
This particular variation was "waiting to be discovered", and hence inevitable.
Hence, Harvey Friedman's independent statements are not natural in the strict sense of "mathematician-made", but are a close derivative of the latter.  
Although "close" is a very vague term, in the above sense, Friedman's independent statements still deserve the predicate "natural", in my opinion.  

Sam Sanders

ps: In the last paragraph, "made" is used in analogy to "man-made".  

On Mar 17, 2012, at 6:11 AM, Harvey Friedman wrote:

> Thanks for your thoughtful reply!
> This has nothing directly to do with raising social status of f.o.m. That is a by product. 
> I believe that there are reasonably clear notions of 
> 1. Mathematical inevitability.
> 2. Mathematical naturalness.
> They are closely related, and I don't understand these matters quite well enough to make a distinction between 1 and 2. 
> All but a trivial amount of mathematics answers questions not yet asked, and a lot of this mathematics is inevitable and natural.
> Proof: The future is incredibly vaster than the present plus the past. QED
> I strongly believe that in this incredibly short initial segment of mathematical activity that has transpired, nothing stated by core mathematicians is concrete and independent of ZFC. 
> However, there are vast arenas of great mathematics which are both concrete and independent of ZFC - far more of this than there is existing mathematics! Proof: This is because the future is incredibly vaster than the present plus the past.
> What I am doing is making completely inevitable connections between principal ideas in existing mathematics. 
> Invariant Maximality is obviously one such completely inevitable connection, and is totally natural - even compelling. Of course, it is not at all clear just how far Invariant Maximality will go, at this point. 
> Yes, if I weren't trying to jump the gun, it might well take a long amount of time - compared to how old mathematics is today - for Invariant Maximality to be discovered. But it is completely inevitable.
> Likewise, all kinds of great mathematical events in the future - far more than we have seen now - are going to take a long amount of time to discover. 
> Also Boolean Relation Theory is completely inevitable - and entirely mathematically natural. But here, I never found a single compelling instance independent of ZFC. I embedded in a reasonably natural collection of 6561 statements. But the enterprise of BRT - completely natural and completely inevitable. 
> The case (Invariant Maximality) is strengthened, of course, by giving a complete analysis of all of the order theoretic invariance notions that can be used for the maximal square.
> Then one does not have to react to any particular invariance notion such as given by Z+up:Q* into Q*. 
> What I have is close to this, but will be improved.
> Incidentally, Z+up invariance is exactly what arises in set theory with indiscernibles. I don't know how this obvious remark should fit into the discussion.
> Harvey Friedman
> On Mar 15, 2012, at 5:55 PM, Timothy Y. Chow wrote:
>> 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
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