FOM: Open Criticism

[Harvey Friedman]

Well, apparently there's quite a lot of heat generated in the recursion
(computability) theory community by Simpson's review (6/21/99 on the FOM
e-mail list) of the recent Boulder meeting, "Computability theory and its
applications." There is a lengthy and detailed response by Soare at
http://www.math.psu.edu/simpson/cta/soare.html.

This heated exchange does in fact raise some fundamental issues of a
general academic nature, which transcend any special disagreements about
recursion (computability) theory. They have been tangentially addressed on
the FOM before, but this seems to be a good time to focus on them.

It think it useful to put the Conclusion of Soare's response here for
discussion:

"CONCLUSION:

Is it not time that we recognize the mathematical merit, beauty, and
importance of ALL the components of computability theory as listed in
the three areas above, and including more topics as well?  Can we not
celebrate together the inherent beauty of the results and rejoice in
their application and interaction with other areas of logic and
mathematics?

Can we not agree that foundations of mathematics and reverse
mathematics are beautiful and important parts of the subject without
also denigrating and dismissing other parts?  Can we not develop and
explain our own results without deprecating the importance and
interest of results by others?  Can we not recognize that the ideas
and methods of one part of computability stimulate other parts and
enrich the whole?

The subject of computability stands much stronger and more interesting
than even a decade ago, both internally and in its interactions and
applications to other areas.  The scientific community and
mathematical community in general and the logic community in
particular are becoming more aware of the role that the concept of
computability in the sense of Godel and Turing can play.  Let us work
together to ensure that these advances and applications continue and
expand."

Soare's objections to Simpson's review are numerous, but the ones that I
think Soare feels most strongly about are

1. It contains comments that are negative about some research in recursion
(computability) theory, or at least have obvious negative interpretations.
2. These negative comments of Simpson are made without thorough
argumentation and elaboration.
3. Open criticism of areas of logic and the work of mathematical logicians
is to be avoided, and is detrimental to the development of mathematical
logic.

I don't agree with the philosophy behind Soare's objections, and this
transcends any special features of recursion (computability) theory or
mathematical logic.

To begin with, serious and substantial criticism of areas of logic and the
work of mathematical logicians goes on routinely on a daily basis behind
closed doors. Where does this go on? Here are some contexts.

a. In the choice of invited speakers to meetings.
b. In the choice of invited speakers to Department seminars and colloquia.
c. In the choice of hiring priorities.
d. In the choice of job candidates.
e. In the acceptance of papers for publication.
f. In the awarding of prizes.
g. In the awarding of grants.
h. IN THE CHOICE OF RESEARCH PROBLEMS AND PROJECTS.

I emphasize the last item because, ultimately, it is the most critical. It
is normally done behind the most closed of all doors - in one's own head.

This is the normal pattern throughout the mathematics community - that
these judgments are made behind closed doors, and open discussion is rare.
The open heat is pretty much reserved for applied mathematics.

The situation is somewhat different in some critical respects in some other
communities. E.g., in the Philosophy community, criticism of each other's
work is an integral part of the enterprise. This is done very openly - in
publications. In computer science, almost all areas are in such a fast
state of evolution that there are not many fixed targets to criticize, and
most people are constantly improving and overhauling basic approaches on a
continuing basis.

I personally have a vested interest in opening up the evaluation processes
- I will come back to that, perhaps in a later posting. But independently
of that, I am convinced that keeping the evaluation processes behind closed
doors works to the great detriment of mathematical logic and, for that
matter, mathematics and all of academia. In fact, on the FOM e-mail list, I
have previously railed against the closed process that awards the Fields
medal, suggesting that it be opened up completely.

What is to be gained by open criticism of areas of mathematical logic, or,
for that matter, areas of academic research generally? IT MAKES THESE
FIELDS BETTER!!!

And how can it make them better? After all, it may very well demoralize
people who have made enormous investments along certain lines of research.

It makes the fields better because the criticism usually leads to an
invigorating new and related line of research. THIS IS TRUE EVEN IF THE
CRITICISM IS ILL INFORMED AND UNFRIENDLY. Of course, the process tends to
work better when the criticism is well informed and friendly. But that is a
high expectation that is not always met.

Speaking personally, a considerable portion of my own research is inspired
by severe criticisms inherent in the attitudes of mathematicians outside
logic, and of many logicians against f.o.m. I have been operating like this
for so long that I no longer need to actually receive the criticism. I can
anticipate it without it being said, and then base research programs on
such "virtual" criticism. In a sense, it takes on the guise of continuous

**self criticism**.

Here are some personal examples.

CRITICISM. Formal systems are a mere artifact of logicians who need
something to do that they can understand, since they don't know any math.
They have nothing to do with the structure of mathematics.

RESULT. Setup of reverse mathematics. Lots of important Theorems are
provably equivalent to formal systems over a base theory.

CRITICISM. Nice, but the base theory infects everything with logician's biases.

RESULT. Formulation of tamism, and its refutations via severe reverse
arithmetic.

CRITICISM. But the codings needed are not faithful to the mathematics.

RESULT. Theory of coding. Generalized reverse math. (Boulder talk).

CRITICISM. Set theory, beyond trivialities, has nothing to do with normal
mathematics which lives in standard concrete settings.

RESULT. Borel diagonalization, and Borel independence results.

CRITICISM. But Borel functions, better than arbitrary functions, are still
way beyond anything anybody cares about anymore.

RESULT. Discrete independence results from set theory.

CRITICISM. But the particular Boolean inequalities you use are not well
motivated; it's too ad hoc.

RESULT. Program A, with initial result and Grand Conjecture.

CRITICISM. We don't need new axioms for anything concrete and interesting.

RESULT. Program A, with initial result and Grand Conjecture.

CRITICISM. Monsters like the Ackerman function and big numbers don't come
up in real mathematics.

RESULT. n(3). Kruskal's theorem. Extended Kruskal's theorem.

CRITICISM. Extended Kruskal's theorem, with its gap condition, is not all
that natural, and is cooked up.

RESULT. It's use for showing that the Robertson/Seymour graph minor theorem
is independent of pi-1-1-CA_0.

CRITICISM. But that's just combinatorics, and that's not real mathematics.

RESULT. Ackerman's function in chains of algebraic sets. Big numbers in
elementary number theory and elementary topology (not posted yet).

CRITICISM. Relative consistency is a technical remnant of Hilbert's failed
program and Godel's theorem.

RESULT. Relative consistency is equivalent to interpretability.

CRITICISM. Recursive sets, r.e. sets, degrees, arithmetical sets,
hyperarithmetical sets, etcetera - all of these things are not
mathematical, and have no clear mathematical counterparts.

RESULT. Reinterpretation of recursion (computability) theory, including
some effective descriptive set theory, in terms of the dynamics of
piecewise linear maps (not posted yet).

CRITICISM. The largest of the large cardinals are based on elementary
embeddings, which is logical, as opposed to combinatorial. So
mathematicians can't relate to them.

RESULT. Simple combinatorial reaxiomatizations of the largest of the large
cardinals.

CRITICISM. The higher proof theoretic ordinals are too mysterious to be
interesting for consistency proofs, and so are not interesting.

RESULT. Independence results using higher proof theoretic ordinals -
Kruskal's theorem, extended Kruskal's theorem, Robertson/Seymour theorem.

CRITICISM. Your finite forms of Kruskal's theorem are not natural, since
they involve growth rates on the trees.

RESULT. The new "perfect" forms of late 1988 that involve only a single
finite tree.

CRITICISM. Embeddings of trees as posets is simpler than inf preserving
embeddings.

RESULT. Label preserving embeddings (without inf preserving) is already
stronger than predicativity. (Not posted yet).

CRITICISM. There is no unifying nontechnical idea behind large cardinals.

RESULT. Reaxiomatizations of ZF and large cardinals in terms of multiple
universes. Also, transfer principles from omega to V.

**********

I'll stop here. I have to get back to my self criticism. In the words of
Bob Soare, I have just got to get back to "deprecating the importance and
interest of results" of mine. I call on all logicians and academics to
deprecate the importance and interest of my results - but do it openly and
not behind closed doors. That way, I will get the benefit of the criticism
- regardless of its quality - and my work will improve from the new ideas
generated. If you merely do it behind closed doors, then nobody benefits.