[FOM] Harvey on invariant maximality

Sam Sanders sasander at cage.ugent.be
Thu Mar 29 18:16:45 EDT 2012

Chow and Friedman are discussing 'naturalness in FOM'.  I weigh in with my opinion.

There are indeed sociological aspects to naturalness in FOM.  However, there are sociological
aspects to almost everything: the very notion of sufficient mathematical proof evolved over time
(towards more rigour) to what we have today.  Does that mean that mathematical proof is
an entirely sociological construct? No!  It has sociological aspects because the practitioners
of math form a sociological group for practical reasons.  Indeed, we believe that mathe-
mathical proof has some deeper meaning independent of these sociological underpinnings.
These are not just fictions we happen to construct!  By the way, what kind of global intercultural mass-delusion
would it be that mathematical proof would just be a sociological construct?  A much simpler 
explanation is that there is some deeper meaning to mathematical proof, which we do not
fully grasp yet (and maybe never will).  

I would like to make a similar claim for 'naturalness': while it indeed has sociological aspects
(e.g. the higher weight of the opinion of Big Names on naturalness) these aspects by themselves
are not all there is to naturalness.  In my opinion, there is a deeper (poorly understood) meaning to naturalness
beyond this superficial level.  

The way I see it, mathematics is a growing body of knowledge, for which
at each moment in time, there is a notion of naturalness, which evolves along.  It makes sense
to talk about "Mathematics at this moment in time (March 29, 2012)", however, this object
came about by an intricate evolutionary process of thousands of years.  To capture the full meaning of such an evolutionary object in static (i.e. non-evolutionary)
terms, seems difficult. The same seems true in general for other evolutionary phenomena,
such as languages, scientific disciplines, life, intelligence, …:..   These phenomena cannot be derived from
basic principles via pure reason in a stint of intellectual labour: they have come about via
countless (but admittedly finitely many) iterations.  This evolutionary origin sets them apart
from "static" things and gives them holistic properties (like naturalness, robustness,…).

A thought experiment: imagine if we could program a computer to generate new facts 
from a certain body of knowledge based on known rules.  I believe that one would
observe that "static" (i.e. non-evolutionary) bodies of knowledge (e.g. the Harry Potter 
novels) would degenerate quickly into nonsense/inconsistency, whereas evolutionary
bodies of knowledge would be stable (for significantly longer time).  

Of course, such a test is impossible at present, and it is still qualitative in nature 
(long vs short term stability).  i believe that is the nature of naturalness: it is 
fundamentally qualitative in nature, but that does not make it a sociological construct.     


ps: I intend to write a more concrete post regarding the discussion at hand
(the naturalness of Harvey Friedman's new discoveries) soon.

pps: This is not an underhanded attempt at re-opening the laid-to-rest thread on fictionalism.        

On Mar 27, 2012, at 6:43 PM, Timothy Y. Chow wrote:

> Harvey Friedman wrote:
>> This has nothing directly to do with raising social status of f.o.m. 
>> That is a by product.
> Similarly, he wrote:
>> It is my view that "naturalness" and "inevitability" are NOT 
>> sociological. In particular, these notions are timeless and independent 
>> of the human condition. The only extent that they may depend on the 
>> human condition is the overall brain capacity of humans, given by 
>> numerical quantities.
> On the other hand, he also wrote:
>> I am NOT doing Concrete Mathematical Incompleteness for the purpose of 
>> showing that large cardinals exist. I am attacking Conventional Wisdom 
>> concerning the profound and intrinsic irrelevance of so called Abstract 
>> Nonsense of which higher set theory is generally included.  
>> Conventional Wisdom supports the total disregard of the Incompleteness 
>> Phenomena as a silly distraction from real mathematics.
>> First this Conventional Wisdom must be profoundly destroyed. One is  
>> then beginning to be armed with new tools needed for dealing with  
>> further issues about which nothing convincing is being currently said.
> Frankly, I think it is disingenuous to claim that all your talk of 
> naturalness has nothing to do with sociology.  Why the obsession with 
> mathematicians who have won prestigious awards?  Why the use of the term 
> "victory"?  Victory in what kind of battle, if not a sociological one?  
> Mathematicians do not usually use the term "victory" to refer to their 
> technical achievements.
> Suppose you devise a theory of "naturalness" and show that according to 
> the notion of naturalness explicated by the theory, a certain statement is 
> both natural and independent of ZFC.  Then you might go around trumpeting 
> the fact that you have solved the longstanding problem of exhibiting a 
> statement that is both natural and independent of ZFC.  This is a free 
> country, after all; we can all say what we want.  However, unless the 
> statement in question is *accepted by the mathematical community* as 
> natural---either because it directly affirms it as such, or because it 
> accepts your theory of "naturalness" and accepts that the statement in 
> question is natural in your sense---such a "victory" will be a hollow one.
> In particular, the Conventional Wisdom will remain the dominant point of 
> view, and sociologically, all you will have accomplished is to convince 
> *yourself* even more strongly that the Conventional Wisdom is wrong.
> Call that a "victory" if you want, but I would reserve that term for a sea 
> change in the way mathematicians in general think about f.o.m.  Using the 
> term "victory" for what is admittedly a very impressive technical 
> achievement, but that does not convince anyone who is not already 
> convinced, is a tactic that in my opinion will ultimately be detrimental 
> to the social status of f.o.m.  And even if you declare that the social 
> status of f.o.m. is only of secondary interest, it is still important 
> enough that it should not be ignored.
> I would go even further and say that the tactic of arguing that the word 
> "natural" is not sociological, *even in the context of the search for 
> "natural" independent statements*, is also detrimental to the social 
> status of f.o.m.  Of course, there's nothing wrong with trying to develop 
> a theory of mathematical naturalness that captures many of the intuitions 
> we have about it.  However, when people ask for a natural statement 
> independent of ZFC, most of them are probably looking for something that 
> has already occurred in the literature of core mathematics, or connects 
> strongly to it.  In particular, they are using the word "natural" in a 
> sociological sense.  If you respond to them that such-and-such a proposed 
> statement is "natural" in a non-sociological sense, and respond to their 
> protests that that's not what they meant by telling them that their notion 
> of "natural" is wrong, it will strike them as a semantic trick.  They will 
> not be persuaded.
> Tim
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