friedman at math.ohio-state.edu
Thu Jan 26 23:59:57 EST 2006
On 1/26/06 7:08 PM, "Eray Ozkural" <examachine at gmail.com> wrote:
> I am not so much interested in the polemical discussions,
> but I find the concept of a "natural" mathematical statement
> or technique interesting. Has there been any attempt to
> objectively distinguish natural statements from unnatural
> statements, without appeal to authority?
I was intending to post something about this later.
Here is a plan to "objectively distinguish natural statements from unnatural
statements". Of course, the plan does not directly head on analyze the true
notion of naturalness that is used in actual mathematics. But this is to be
expected in this very early century of f.o.m.
PLAN A. This is probably worthless for present purposes. A statement is
natural according to its length in the primitive notation of set theory.
Worthless for the purpose at hand because almost anything one writes down is
quite long, and primitive notation is too crude to reflect naturalness in
normal mathematical contexts.
However, even this ridiculously crude plan is quite interesting.
Let's go to the primitive notation of set theory - epsilon and =, with the
usual five connectives and two quantifiers.
An obvious notion of length is the number of occurrences of variables.
(There are other closely related measures).
What can we say about the shortest sentence in primitive notation which is
not provable or refutable in ZFC?
Probably the shortest known one is in my paper
Three quantifier sentences, Fundamenta Mathematicae, 177 (2003), 213-240.
I did not make an attempt there to optimize length, but instead optimize
"number of quantifiers". A 5 quantifier sentence independent of ZFC is
explicitly given there. (3 quantifiers are not independent, and 4
quantifiers is unknown).
PLAN B. More promising. The independent statements that I come up with seem
to be quite short RELATIVE to some languages for mathematics based on VERY
COMMONLY USED primitives.
One can state the results in the following form:
*In such and such a language, there is a statement of such and such size
which is independent of ZFC*.
In order to reflect other crucial aspects of the statements (e.g., they are
explicitly Pi01 in some appropriate sense),
*In such and such a language, there is a statement of such and such size and
form, which is independent of ZFC*.
Of course, there is the issue of
**What languages are appropriate for this purpose?**
Let L be a language based on some primitive mathematical concepts. In order
to justify its use for this purpose, one would normally claim that these
primitive concepts are in constant and pervasive use in mathematics, or in
certain branches of mathematics.
One can attempt to formally justify the constant and pervasive use by taking
some major Journals and textbooks, and counting up the number of uses, or
counting up the number of implied uses, and comparing these numbers to the
numbers for concepts that seem relatively specialized.
Or at a meeting of mathematicians, or of people in a field of mathematics,
what concepts can be named without any obligation on the part of the speaker
to give their definition?
For instance, in an investigation as to the "naturalness" of a statement in
finite graph theory, the concepts
vertex of a finite (di)graph
edge of a finite (di)graph
path in a finite (di)graph
independent set in a finite (di)graph
complement of a set of vertices in a finite (di)graph
induced sub(di)graph of a finite (di)graph
isomorphism between induced sub(di)graphs of a finite (di)graph
would be principal components of an associated basic language. In a meeting
of graph theorists, it would in fact be *wholly inappropriate* for a speaker
to be given these definitions. (Maybe a comment that all (di)graphs are
simple - for non graph theorists, this means no loops and no multiple
Of course, a graph theorist (and others) would instantly recognize the
appropriateness of such a language, without a formal justification.
Statements in finite graph theory, independent of ZFC, would be measured
according to their length in such a language.
These lengths would then be compared with the lengths associated with:
***existing main Theorems in the subject***
whose naturalness is NOT in question.
I regard this as very promising - even though it is far beyond our (at least
my) capabilities to prove Theorems to the effect that: yet shorter
statements in such languages are all provable or refutable in ZFC. Of
course, baby theorems of this kind would be extremely challenging, but
YES - we know that, given any language, we can always cut down the lengths
of statements considered in that language, so that it becomes trivial to
check that all such statements are provable or refutable in ZFC.
BUT, if we inch the length up ONE BY ONE, we get to a point where, generally
speaking, it becomes challenging and doable to show that all statements of
that length and less are provable or refutable in ZFC.
SO, there should be profound threshold phenomenon FOR ANY REASONABLE
Perhaps we should get to work on this...
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