[FOM] Harvey on invariant maximality

[Harvey Friedman]

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.

I conjecture that "aesthetics" of mathematical ideas will be found to  
also have nothing to do with sociology, and also be timeless and  
independent of the human condition.

The closest approximation that we have for such notions is  
"simplicity". And the only handle we presently have on "simplicity" is  
"shortness of presentation in a fundamental language".

However, the only handle that we currently have on "fundamental  
languages" for this purpose appears to be

i. identify a group of mathematical notions as primitive.
ii. test their "fundamentalness" by how many hits there are on a  
Google search under quote signs.

Let phi be a mathematical statement that I am proposing as "natural",  
"inevitable", "simple", or the like.

A *presentation* of A consists of the following.

a) An identification of primitive mathematical notions.
b) A series of definitions of new notions starting from a).
c) phi, where phi is in purely logical notation over a),b).

The idea is to strive for

a) should be small in number, each should have a high Google hit number.
b) should be small.
c) should be small.

It is not clear just how to measure the overall "simplicity" of a  
presentation in this sense, but it should become rather clear that the  
statements in question, using Z+up, should be very good under these  
measures compared to typical theorems in Journals.

We can also use the above criteria to judge "simplicity" of the  
Templates. These should also do very well under these criteria.

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.

The definite fine tuning of the results in the way you indicated in  
the last paragraph is certainly true - but for this line of results,  
particularly difficult to pull off. It is certainly a longer range  
goal. Some limited results along these lines is most hopeful for k = 2.

Harvey Friedman

On Mar 24, 2012, at 3:51 PM, Andrew Arana wrote:

These results of Harvey are fantastic & show the stunning  
interpenetration of higher set theory into elementary mathematics. I  
look forward to seeing the magical work behind them, for not only are  
the results breathtaking but so, I am sure, are the methods.

I wanted to remark on the "naturalness" issue. Harvey talks about the  
"naturalness" of his theorems, or even their "inevitability", but I  
worry that this way of putting it suggests that the issue is  
sociological or even aesthetic, a matter of taste. Framing it in terms  
of elementarity seems more promising to me, where its elementarity is  
a matter of the concepts involved in the relevant theorems, i.e. Q, <,  
invariance, maximality, square, etc., & of the way those concepts are  
combined in the theorem's formulation (one might say its "form"). It  
would be good to get sharper in particular about the latter notion:  
perhaps it's just an issue of its syntax.

The purpose of "naturalness" observations is a regressive argument for  
the acceptance of large cardinals: the theorem is natural but its  
proof requires large cardinals, so you should accept large cardinals.  
In addition to the usual worries about how "requires" here should be  
understood, I wonder about the role of "naturalness" in the argument.  
I take it that the normative force of the argument depends on it, but  
how exactly? Does the argument work for weaker normative properties of  
theorems than "natural"? Does it work for "elementary" as I have  
sketched it above? (And what measure of weakness ought be used here?)  
Can we determine which normative properties of theorems enable the  
argument, & which do not? Resolving these would help me get more  
clarity about what exactly is at stake in debates concerning  
naturalness, & what has been accomplished.

Finally, a question for Harvey in particular: fixing the relational  
structure (Q,<), are there values of k,n,m for your templates for  
which the statements are provable in ZFC but not PA? In PA but not  
EFA? That is, if I tweak k, say, can I reduce the provability strength  
of the statements in this systematic way? If not, is such a  
development reasonable to expect, if not for (Q,<), then for some  
other relational structure?