[FOM] Harvey on invariant maximality
friedman at math.ohio-state.edu
Sat Mar 24 20:32:38 EDT 2012
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.
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?
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