[FOM] Deep Pathology/Typos

[Harvey Friedman]

Some replies to replies to my Deep Pathology postings.

But first I correct some typos in 604:
http://www.cs.nyu.edu/pipermail/fom/2015-August/018920.html on Finite
Emulation Theory:

I wrote there

First, a typo in

should be

First, a typo in http://www.cs.nyu.edu/pipermail/fom/2015-August/018917.html

I wrote there

DEFINITION 2. Let x,y in Z+^k and Z,Y containedin Z+^k.  blah blah blah.

should be

DEFINITION 2. Let x,y in Z+^k and X,Y containedin Z+^k.  blah blah blah.

I wrote there

DEFINITION 3. Let Y be an emulation of X. Y covers A by B if and only
if for all x in A^k\Y there exists y < x from B^k such that {x,y} does
not emulate X.

I didn't explain y < x. I should have wrote

DEFINITION 3. Let Y be an emulation of X. Y covers A by B if and only
if for all x in A^k\Y there exists y from B^k such that {x,y} does
not emulate X and max(y) < max(x).

TOM DUNION WROTE

Sure, it's good to know what contexts give lots of such regularity
properties -- Determinacy, supercompacts, L(R) and so forth -- but who can
tell at this point if future progress in quantum physics for example will
incorporate an expanded concept of probability beyond today's tidy minded
axiomatization in the tradition of Borel and Kolmogorov, and under what
set-theoretic assumptions.

ANSWER: I am nearly certain that the future of both mathematics and
physics is going in the direction of the finite. For physics this
would mean the emergence of serious finite models for reality. For
mathematics, this would mean the emergence of finite forms for as much
of the existing interesting mathematics as possible. The problem with
both is that of uncontrolled complications and ad hoc choices for the
details. But I believe that this can be tamed systematically with new
ideas. Of course, it will likely be the case that a lot of proofs will
be likewise finitary. But it will also likely be the case that
infinitary methods will greatly facilitate a lot of the proofs, and it
will be a challenge to give finitary proofs for the finite statements.
And in extreme cases, even ZFC will not suffice to prove the finnitary
theorems. In other words,
http://www.cs.nyu.edu/pipermail/fom/2015-August/018920.html is the
very start of the tip of the iceberg. Initial quick reactions from a
famous cosmologist are that emulations or continuations resonate in
GR. I have long advocated a systematic reworking of fundamental
mathematics in finite terms. I hope to start a thread on this, but too
much is going on as it is. E.g., I am hoping to give a somewhat
reasonable not too far fetched physical interpretation of
http://www.cs.nyu.edu/pipermail/fom/2015-August/018920.html with the
help of physicists.

JOE SHIPMAN WROTE

Here's a good "pathology" topic for you--the lack of well-defined
"functional square roots" of exponentials (functions defined on a real
interval such that f(f(x))=e^x of f(f(x))=2^x) or the very closely
related lack of a well-defined and non-arbitrary extension of the
tower function to non-integral arguments.

ANSWER: I know about this one, and it is not a good topic, but a great
fundamental topic. I do think it is of a very different character once
we go beyond the obviously similarity. And it is part of a family of
fundamental issues, some of which sharply distinguish the continuous
from the discrete. There needs to be a great new theory merging the
discrete and the continuous properly. Conway Surreals is in this
direction, but it doesn't seem to be enough to address many of the
critical issues.

COLIN MCLARTY WROTE

It gets worse as we go farther from foundational theories,  Finite group
theorists certainly believe the introduction of apt technical ideas like
"thin group" shortens proofs. In some sense it obviously does, but I doubt
this is well understood at all on a formal level.  I would be happy to
learn of references showing it is well understood formally, of course, but
I doubt it is.

ANSWER: The right place to get a good theory of lengths of proofs that
is truly responsive to actual mathematics is in formal systems
associated with real closed fields and algebraically closed fields.
Maybe some theories of the integers and rationals.

Harvey Friedman