[FOM] top/down, bottom/up

[Harvey Friedman]

I just got back from my UT Austin visit where I gave the lecture
Adventures in Incompleteness. The notes for this lecture can be found
at

https://u.osu.edu/friedman.8/foundational-adventures/downloadable-lecture-notes-2/
#68

The advertised audience was Mathematics, Philosophy, Computer Science.

I was told that the audience was predominantly from Computer Science,
which is expected since my host was Scott Aaronson of quantum
computing fame. I had some substantial followup after the presentation
with Scott Aaronson, and an intense interaction with Marijn Huele,
http://www.cs.utexas.edu/~marijn/, a major figure in SAT solvers. See
especially there:

Marijn J.H. Heule, Oliver Kullmann, and Victor W. Marek (2016).
Solving and Verifying the boolean Pythagorean Triples problem via
Cube-and-Conquer.
In Theory and Applications of Satisfiability Testing – SAT 2016, pp.
228-245. Lecture Notes in Computer Science 9710, Springer. [pdf, doi,
slides]
Files and tools of the ``largest math proof ever" are available at
Everything's Bigger in Texas.

I had earlier written about using work in Concrete Mathematical
Incompleteness in order to

*confirm the consistency of mathematics and also extensions of ZFC by
large cardinals.*

I am now rethinking this program using the new Emulation Theory, and
plan to write about it on FOM soon.

Meanwhile, it is now clear that my "confirmations" can be viewed as
obtaining certificates for SAT problems. The bigger the SAT problems
used the more convincing the confirmation is.

Meanwhile, I have become acutely aware of the enormous divides that
exist between researchers in adjacent areas, even if they are close
enough to often find themselves looking at the same papers.

I am referring to huge unexpected divides between what immediately
resonates in one community and what immediately resonates in another.

In particular, I am just beginning to see the enormous divides of this
kind that exist between these communities:

1. Math Logic.
2. Theoretical Computer Science.
3. Mainstream Mathematics.
4. Scientifically Oriented Philosophy.

I have been totally convinced that Emulation Theory and many aspects
of the wider Concrete Mathematical Incompleteness should immediately
resonate with all of the above 4 communities, and to various extents,
to many other communities.

After having giving Adventures in Completeness, somewhat differently,
to Stanford and now to Austin, I am still convinced that it should
immediately resonate with 1-4, BUT ESSENTIALLY THE SAME MATERIAL MUST
BE PRESENTED VERY DIFFERENTLY IN VERY DIFFERENT ORDERS WITH VERY
DIFFERENT EMPHASES AND VERY DIFFERENT BACKGROUND INFORMATION in order
to work with 1-4.

At least this is very clear to me in a lecture setting. In print, one
has SOME more commonality for 1-4, but it is very easy to
underestimate the divide even with print.

The practical upshot of this is as follows.

1. I want to understand these absolutely major differences in what
resonates better. At the heart of it, there are absolutely fundamental
differences in the ways these communities generally process
information and formulate research programs and evaluate findings.

2. I have a deadline of May 1, 2017 for a Hillary Putnam volume, for
my paper CONCRETE MATHEMATICAL INCOMPLETENESS: Emulation Theory. I
certainly cannot address all of the myriad issues created by these
absolutely major differences between these communities. However, one
major difference that arises between 1-4, and even also in individuals
within 1-4, is this.

3. When presented with a new area of research, for some, a TOP DOWN
presentation resonates best, whereas, for others, a BOTTOM UP
presentation resonates best. In fact, for many, only one of these two
kinds of presentation are at all palatable. With the wrong choice of
these two, I have seen, quite often, completely blank stares. This
phenomenon, which I find unexpectedly strong and pervasive, is so
powerful that it often completely destroys a presentation - especially
in the lecture format. There is a little more leeway in written form
as people are somewhat used to having to read and reread and think
some about written materials. But as professionals develop and narrow
their perspective and have this reinforced by close colleagues, the
tolerance for how new things are presented greatly lessens. This is
all especially true when the presentation concerns material and issues
that are not quite in the same field, but rather in adjacent fields.

4. Another aspect is the use of EXAMPLES. I have often felt that
developing examples is not a high priority IF the definitions are
extremely simple and natural. When it is easy enough to give simple
examples, I have put that as a high priority. When the notions are, in
my opinion, utterly straightforward and obviously natural, but working
up examples that are not misleading take some time to develop, I have
usually put it as a lower priority. This turns out to be a DISASTROUS
decision The reason being that for a range of people in 1-4 there is
almost no new notion wroth anything that is sufficiently simple and
natural and compelling.

5. Having said this, I am willing to bet half of my entire net worth
that in the near future essentially everybody in 1-4 will wonder why
they even hesitated to immediately understand Emulation Theory (the
claims, not the proofs). They will in fact blame me - if only I said
this or said that or didn't say this or didn't say that -- or only if
my slides were better, etcetera. No - it's just the nature of the
beast, and I am stuck simply presenting it in several radically
different ways in order to resonate with fundamentally diverse
thinkers.

6. So now what? For the May 1, 2017 paper, I plan to give TWO
PRESENTATIONS, with some cross referencing. One is TOP DOWN, and the
other is BOTTOM UP. Furthermore, I will work up some serious examples.
I personally am a TOP DOWN person. I will not even listen to BOTTOM
UP, and when I am confronted with it when I can't interact with the
presenter, I will look at what I see and invent possible ways where it
could have arisen from TOP DOWN and try to guess or create the TOP
DOWN stuff that might have led to it. Often there is no such thing, or
it is beyond me to create it on the spot, in which case I will be
entirely lost and just want to see the ms. later. So for me, there is
essentially no BOTTOM UP whatsoever. Of course, there has to be BOTTOM
UP, or else there really is nothing, ultimately. But there is, for me,
only BOTTOM UP as an endpoint only.

7. EXAMPLES, yes. Even when non misleading examples take a lot of
effort to develop, this now seems to me to be much more important than
I thought before.

8. But I think the novelty here is the DUAL PRESENTATION idea. TOP
DOWN and BOTTOM UP. Each of these two is just worthless for too many
people, one or the other.

Harvey Friedman