FOM: nowhere differentiable/natural
Harvey Friedman
friedman at math.ohio-state.edu
Thu Aug 5 11:44:22 EDT 1999
Here is a particularly natural example of a continuous nowhere
differentiable selfmap of the closed unit interval.
Let (a,b),(c,d) be points in the plane where a < c. It can be verified that
there exists unique points (p,q),(r,s) such that
a) p-a = r-p = c-r;
b) the slope from (a,b) to (p,q) is 9m/4, the slope from (p,q) to (r,s) is
-3m/2, the slope from (r,s) to (c,d) is 9m/4, where m is the slope from
(a,b) to (c,d).
Let us call these unique two points the special intermediate pair for (a,b)
and (c,d).
Now start with (0,0) and (1,1). Throw in the special intermediate pair for
them. Now we have four points. For each of the three resulting consecutive
pairs, throw in their special intermediate pairs. Now we have ten points.
For each of the nine resulting consecutive pairs, throw in their special
intermediate pairs. Continue this forever. The result is a countably
infinite subset of the unit square whose closure is the graph of a
function. That function is a continuous nowhere differentiable function
from the closed unit interval into itself.
**********
Soare 5:35PM 8/3/99 seems to claim to have an example of a finitely
presented group whose word problem is of intermediate degree which is at
least as natural as natural examples of a continuous nowhere differentiable
function. This is going to be very interesting to see.
**********
I have not carefully stated a position on the value of research on the
lattice of r.e. sets and r.e. degrees, etcetera. Obviously, I have said
enough already to indicate that these things are overdone when taking into
account their general intellectual interest.
But I want to clarify a point here.
It is certainly possible for there to be results so striking and/or
clarifying about these structures that it overcomes the kind of
reservations people have. So in that limited sense I am favorable to work
along these lines. But that is a very very high standard.
And one must compare intensive research on these structures with intensive
research on other topics which have the distinct advantage that they are
not only relatively new and unmined, but also have more general
intellectual interest.
There are two reasonable rejoinders to my statements. Firstly, how can
there be such striking and/or clarifying results about these structures if
people are not free to intensively work towards them? Secondly, why is
Friedman singling out recursion theory for such critcism?
As for the first, it is a question of balance. One must consider the
opportunity also to obtain results at least as striking and/or clarifying
about other matters of more general intellectual interest. If the recursion
theory community as a whole tends to reward mainly work on these limited
topics, then people in the field cannot spend much time venturing out into
matters of greater general intellectual interest. This might be especially
true of junior people. If it proves too risky for people - especially
junior people - to become more reflective and get involved in projects that
are not steeped in forty years of intense complications, because the early
results are going to be not as "hard" - well, then, there is a major, major
problem.
Let me rush to the second, because it is very important. Why is Friedman
singling out recursion theory?
It would be very unfair if I were singling out recursion theory, but I am
emphatically not. Very similar remarks apply throughout much of
mathematical logic these days. I have already criticized lots of things
going on in mathematical logic having nothing to do with recursion theory,
and I expect to be doing this sort of constructive criticism in much
greater detail and much more broadly in the future.
For example, I have constructively criticized the following:
1. Negative: Disappearance of f.o.m. as a motivation for work in
mathematical logic. Positive: F.o.m. can be powerfully used as a motivation
for work throughout mathematical logic, with examples.
2. Negative: Overconcentration on bigger and bigger notation systems (proof
theory). Positive: Use of notation systems for independence results.
Structure of actual mathematical proofs.
3. Negative: Overconcentration on contexts too general and far removed from
standard mathematical contexts (set theory). Positive:
Borel/discrete/finite independence results. New axiomatizations of set
theory.
4. Negative: Rejection of foundational issues and removal of mathematical
logic structures in favor of direct applications to specialized topics in
mathematics (model theory). Positive: Foundational investigation of tame
structures. Reconsideration of philosophical model theory.
5. Negative: Overconcentration on r.e. sets and degrees (recursion theory).
Positive: Reverse mathematics, and the use of recursion theory there.
Reconsideration of Church's Thesis.
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