FOM: Invitation to Soare

Harvey Friedman friedman at
Mon Jul 19 07:33:44 EDT 1999

Robert Soare has written a response to Simpson and me on the COMP-THY
mailing list, which I saw through


Soare writes:

>WELL, this has been a busy time... This week alone there have been two from
>Simpson and one from Harvey, another from Simpson before that.  They
>have made all those comments about ME with more than the usual
>invective and disparagement.  Gosh, fellas, I didn't know I merited
>all this attention.  I am really flattered!!
>(See <>.)

I call attention to "They have made all those comments about ME with more
than the usual invective and disparagement."

Speaking for myself, Soare is referring to my FOM posting, Open Criticism
10:56PM 7/14/99.

Reading this over, I can't see what in that posting is "about ME (Soare)
with more than the usual invective and disparagement." In fact, when I
composed that posting, I was determined to add only light and no heat to
this debate.


Soare wrote a revised Conclusion of 7/16/99, also available from Apparently this was (in part)
inspired by my Open Criticism posting. So we definitely have the start of a
dialog here. In his revised Conclusion, he writes:

>Suppose [Simpson] had a "thesis" that "priority
>methods are almost completely absent from applied recursion theory."
>... he could have brought it up OPENLY with
>the roughly 60 participants since they included most of leading
>experts in computability in the world.  He could have brought this up
>either in several of the lectures, in small groups, or privately
>experts on this question.

>... this thesis is immediately
>provable or refutable from readily available FACTS [and] could have been
>quickly answered by perhaps half of the participants...  Instead the
>"reviewer" ... advanced it as fact to a relatively CLOSED forum of FOM
>subscribers, a
>list which contains few of the Boulder participants, and relatively
>few who were in a position to supply the information necessary to
>decide the issue.

The FOM list is not closed. There is a well known well maintained archive
of all postings that is accessible worldwide on the web. This is not true
of all academic e-mail lists. Also, the list of subscribers is publicly

There are 385 subscribers to the FOM list all over the world, and many many
more people frequently access the archives to keep up with what's happening
there. The moderation is very light, serving mainly to ensure that there is
serious FOM content to every posting. "Rules" have been spelled out by the
moderator. These rules are minimal.

It is clear that the moderator has never injected opinions on issues into
any decision regarding allowable content on the FOM list. The moderator
routinely puts up postings - without modification - that disagree
vehemently with positions taken by the moderator.

The only objections to the moderator that I am aware of are that he has
opinions that differ with many subscribers, and his postings often tend to
be very critical. This is also felt about other subscribers. I have never
heard any objection to the *moderation* itself.

Come to think of it, I have indeed heard two objections to the moderation.
One is that the moderator allowed a posting of mine which was a superharsh
response to a posting by a nonacademic trashing my work and programs
without discussion. The posting I was responding to should never have been
posted, and was probably a major reason why the moderator later put forth
"rules" about f.o.m. content. The moderator was roundly criticized by me
for allowing the original posting, and by many subscribers for allowing my
response! Poor moderator. He did a great job under the circumstances.

A second objection came from one or two people who decided to leave the
FOM, and complained that their last postings were withheld. These last
postings were complaints about the f.o.m. list, with no f.o.m. content.
They were "parting shots." Such postings are obviously not appropriate.

If the moderation wasn't so minimal and so even handed, then people should
complain because of the strong opinions of the moderator. But in the
present circumstances, we should all be grateful to the moderator for
maintaining the FOM list with the minimal amount of objective moderation
that is needed to avoid complete chaos. The fact that the moderator has
strong opinions is completely irrelevant.

And if anyone thinks that a mailing list on the topic of f.o.m. can be
conducted without this minimal level of moderation, they are in for a rude

The FOM list is a very OPEN forum, ideally suited for a discussion of all
aspects of the foundations of mathematics. And recursion (computability)
theory is relevant to the foundations of mathematics. So is model theory,
proof theory, and set theory, philosophy of mathematics, and aspects of
computer science. Model theorists, proof theorists, set theorists,
philosophers, and computer scientists are not objecting to any discussions
taking place on the FOM list, just because not enough of their colleagues
are on the FOM list.

In fact, I routinely ask people for feedback on postings I make on the FOM
list who I know are not on the FOM list, and I routinely send some of my
postings for feedback beyond the FOM list.

In fact, this is exactly what Simpson did. He posted on the FOM, while
seeking feedback from wider sources. Maybe you can quibble about the
timing. That Simpson should have extensively sought feedback from wider
sources first. But the FOM list is an informal give and take forum, where
people routinely modify their postings according to feedback, make
corrections, etcetera.


Soare discusses a Simpson thesis:

*priority methods are almost completely absent from applied recursion theory*

Soare writes that Simpson's thesis can be immediately provable or
refutable. However, I disagree with this since there is an ambiguity in the
phrase "applied recursion theory" and even the word "uses." Such short
phrases typically mean different things to different people.

Here is one interpretation of "applied recursion theory" that is probably
close to what Simpson has in mind:

**The use of priority methods for proving something whose natural
formulation does not mention recursion theoretic concepts, and which
naturally arises outside the framework of recursion theory**

NOTE: The SIGNIFICANCE of Simpson's claim is another matter that should be
revisited after we clear the air about what it means and the extent to
which it is true.

In order to refute what Simpson intends to claim:

***The use of priority arguments either has to be apparently necessary, or
at least give the simplest proof for the general mathematical logician. The
priority arguments cannot be routinely eliminated***

Soare lists a huge number of references that he has obtained from several
experts at Do
any of these papers contain a refutation of Simpson's claim? It is hard to
tell, there being so many of them. But the overwhelming majority of them
mention recursion (computability) theoretic concepts in the title of the
paper, and so are not good candidates for this.

In particular, I have been doing reverse mathematics off and on since my
basic setups from the 70's. To say the least, I am out of practice with
priority arguements. And if I get stuck with something that I sense could
even remotely involve such arguments, I would certainly ask an expert in
recursion (computability) theory - of which there are many suitable
choices. This hasn't happened to me yet. It might happen later. I
understand that Simpson is looking at a case that was presented to him in
reverse mathematics.

The Nabutovsky/Weinberger material is very intriguing, and Simpson and I
have talked to Weinberger and had e-mail correspondence with him, as have
other people. But our understanding of this material is far too weak to
come to any definite conclusions about genuine applications of priority
arguments, ***even given his talk and Soare's talk at the Boulder
meeting***. And the impression I got at the Boulder meeting was that my
view - that it is too early to tell what is really going on - was the
prevailing view among the experts at that meeting. I had a discussion with
Soare as to the possible importance of this development if things go
certain ways, and we agreed with each other. But our mutual enthusiasm was
provisional on a much better understanding of what is going on.


In this rough vein, I do have an interpretation of recursion
(computability) theory in terms of the dynamics of piecewise linear maps
where both the logic and the mathematics are directly accessible to both
logicians and mathematicians. So both sides can see exactly what is going
on from the outset. Of course, the usual nondegree concepts from recursion
(computability) theory, descriptive set theory, and effective descriptive
set theory can be recouched in this strictly mathematical context. I expect
the degree concepts can also be couched nicely as well. What remains to be
seen is how interesting various facts about degrees look when reinterpreted
in this mathematical context. Perhaps I can get an idea of how they should
or could look by studying Nabutovsky/Weinberger. I'll be posting on this
dynamics stuff on the FOM list fairly soon.


There are a number of timely issues concerning the value and relative value
of directions of research in recursion (computability) theory. For that
matter, concerning the value and relative value of directions of research
in model theory, set theory, proof theory, f.o.m., philosophy, computer
science, pure mathematics, applied mathematics, science, and intellectual
acitivity. All of these have been discussed at least a little bit on the
FOM already, and the intention of many subscribers was to discuss all of
such issues in great depth on the FOM. For a provacative general posting
about such issues, there is my old FOM posting:

F.O.M. & Math Logic  12/14/97 5:47AM

I perceive a great interest on your part in getting involved in such wider
issues, as least with regard to recursion (computability) theory. You
probably already know of my general opinion that there is a great deal of
high level brain power in the recursion (computability) theory community,
and that it could be used far more productively than it is now on new
directions that have much higher general intellectual interest. There is no
question that this has already started to happen, but we have much further
to go. I also claim to have a clear enough and objective enough concept of
"general intellectual interest" to be able to generate such new directions,
and to defend them as more productive. And I think I can say with some
objective force what is lacking in many current directions that are not

The previous paragraph applies also to every standard branch of
mathematical logic. It also applies to Philosophy. Maybe to everything.

Are you willing to get involved on an interactive basis with such a
discussion, at least with me? This kind of thing was the intention of the
founders of the FOM list.

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