[FOM] Harvey on invariant maximality

Timothy Y. Chow tchow at alum.mit.edu
Tue Mar 27 19:43:09 EDT 2012

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.


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