[FOM] Harvey on invariant maximality

[Harvey Friedman]

> On Mar 27, 2012, at 7:43 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.

Naturalness, fundamentalness, and inevitability are not sociological,  
but fundamental.

I use interaction with core mathematicians to provide evidence that I  
am not making errors in judgment concerning naturalness,  
fundamentalness, and inevitability.

Core mathematicians, particularly luminaries, accepting naturalness,  
etc., under the present circumstances, strongly suggests that the  
statements are in fact natural, etc.

However, under the present circumstances, if core mathematicians do  
not accept statements as natural, then that is rather poor evidence  
that they are not natural.

Recall that I asked the FOM subscriber list what form of "Victory"  
this constitutes, and indicated that I have not settled on the terms  
in which this "Victory" should be cast.

> 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.

You are talking about sociology, and I am talking about naturalness,  
fundamentalness, and inevitability. So if I or anyone else "devises" a  
really good theory, then that is a great achievement of fundamental  
importance.
>
> 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.

Well, I asked for advice about how to state "victory" here, and you  
have given it. Thank you.

> 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.

Just the opposite is true. The notion of "natural", which is  
fundamental, and crucial to the development of mathematics, in the  
past, now, and in the future, begs to have some nontrivial analysis. I  
haven't seen any interesting analysis of this, even in limited  
contexts. In the present environment, we cannot look to the  
mathematics community to do this sort of work.

I have no idea how you view what you call the "social status of  
f.o.m." It appears that very very few mathematicians are aware that  
there is an area of mathematically rich research called f.o.m. They  
are somewhat aware that there is what they call a "branch of  
mathematics" called mathematical logic. Very very few mathematicians  
have any idea what is going on in mathematical logic.

When you say "detrimental to the social status of f.o.m.", I think you  
are talking about the social status of f.o.m. in the mathematics  
community, in which case we are starting with essentially a total lack  
of awareness. It doesn't make any sense to me that this can become  
worse by talking about truths like " "natural" is fundamental and not  
sociological".

> 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.

Invariant Maximality connects "strongly" to existing mathematics, in  
some very clear senses of "strongly connects". This kind of "strong  
connection" can probably be subject to illuminating theory - or at  
least this is promising.

> In particular, they are using the word "natural" in a
> sociological sense.

There are of course some sociological meanings of "natural" which  
Invariant Maximality falls under. E.g., the sociological notion of  
"combining two fundamental notions, both in common use, in simple ways".

> 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.

I don't know whether you are talking about core mathematicians,  
logicians, or observers of f.o.m. On the contrary, I expect many  
people to be rather convinced by Invariant Maximality and the  
fascinating prospect that there might well be a theory behind  
fundamental notions of naturalness in mathematics and other facets of  
human thought.

EVERY INVARIANT SUBSET OF Q[0,1]^2k HAS AN INVARIANT' MAXIMAL SQUARE

for which basic notions of invariance is this statement true?

is convincing enough to meet many sociological and non sociological  
notions of natural, fundamental, and inevitable.

The examples given, where this is independent, and Pi01, are enough to  
launch the full investigation. The full investigation of course is at  
a very early stage.

Harvey Friedman