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Robert I. Soare soare at
Thu Aug 5 02:56:17 EDT 1999

TO:	fom at
FROM:	Robert Soare, University of Chicago
DATE:	Thursday, August 5, 1999 
RE:	Claims, Withdrawals, Proofs.


In mid July in his office, Leo HARRINGTON volunteered to me that in
1979 he "had PROVED a statement "about reverse mathematics
by a priority argument."  [His exact words.]  He wrote the statement
carefully on a PAD so I could get it exactly right.  I later wrote
exactly these words in (FOM, Aug. 3),

BEFORE posting this on his web page, I went to JOHN STEEL (also at
Berkeley and a member of FOM), who wrote an email msg on July 29 (see
the above web page, "steel.text") in which he confirmed the statement
of Steel's 1975 theorem, and stated both precisely.  I then posted the
statement exactly as Harrington had written and Steel had

Simpson pointed out the Harvey Friedman had proved it.  When
reminded of this, Steel agreed and wrote this msg to FOM:


>  Steel (FOM, Wed. Aug. 3, 1999) wrote:
>  The errors in Bob Soare's statements about Sigma_1^1-AC versus Sigma_1^1-DC
>  are due to me (and perhaps also Harrington), not to Soare. Harvey proved 
>  that
>  Sigma_1^1-AC does not imply Sigma_1^1-DC. I confused this with some other
>  work, and this led to Soare's mis-attribution. Bob was careful enough to
>  ask me about it several times, so the fault is all mine.
>  Harrington did prove Delta_1^1-CA does not imply Sigma_1^1-AC. I believe his
>  proof had a priority argument in it, although to my mind its most ingenious
>  feature was its use of Sacks' "shiny little box" technique, which is based
>  on Kleene's Recursion Theorem.
>  John Steel


Nevertheless, regardless of the sources, it was I who posted the
incorrect fact; it is I who now WITHDRAW it; and it is I take FULL
RESPONSIBILITY for it.  I APOLOGIZE to Mr. Simpson and to any FOM
members who may have been inconvenienced by this incorrect
attribution.  When Leo Harrington returns from vacation, perhaps he
can clarify on FOM more of what he had in mind that day in his office.


Now that we are beginning an OPEN PUBLIC DIALOGUE on FOM on the
issues of computability theory as devoutly wished by Mr. Simpson and
Mr. Friedman, let me request that Mr. SIMPSON NOW BE EQUALLY GRACIOUS
and OPEN with the multitude of questions which SOARE has raised in
the long paper (FOM, Soare, Aug. 3).  For EVERY one of these
questions, I expect Mr. Simpson to either:

(1)  Make a withdrawal and issue an apology as SOARE has done above;  or

(2)  Present a full demonstration of his assertion with a complete
	logical argument and supporting evidence and facts
	(not just the casual statement of bias and prejudice with
	no supporting argument).

Failure to do EITHER one of these, even for ONE question, would be
irresponsible, because: it would leave us uninformed about the truth;
it would allow Mr. Simpson to make further unsupported allegations
with no responsibility for either proving them or withdrawing them.
This would be a serious miscarriage of justice and truth.


There has apparently been some misunderstanding of my position on
debate of other issues outside those raised in my long paper
(FOM Aug. 3) by Mr. Friedman, Mr. Shipman, and perhaps Mr. Simpson.
I stated this clearly in my cover letter (FOM, Aug. 3), but let
me state it again in the terms of William Tecumseh Sherman:



1.  On June 21, Soare, an nonmember of FOM, received the FOM July 21
	statement from Simpson, which contained MANY errors in
	Soare's field of expertise.

2.  Soare's aim is to have these errors corrected, and either withdrawn
	or proved.  To achieve this Soare wrote a long paper and posted it 
	(FOM, Aug. 3).

	of the (Soare, FOM, Aug. 3) paper by (1) either WITHDRAWING and 
	APOLOGIZING; or (2)  GIVING a PROOF of his original assertion.   
	To do less is scientifically and personally unacceptable.
	(FOM subscribers: please do NOT send me any NEW questions until 
	then.  There are PLENTY in the 15 pages for us to discuss now.
	AFTER that I will be happy to discuss ANY new question.)


I choose NOT to discuss any OTHER parts of Simpson's memos (of June
21, July 13, July 19, July 22, July 28-1, 28-2 etc.) which may be
objectionable, excessively critical, and so on,
NOT to debate or excessively criticize, but JUST to
correct the MATHEMATICAL MISTAKES.  Namely to paraphrase the slogan
about the economy in President Clinton's 1992 presidential campaign,

               "IT'S THE ERRORS, JUST THE ERRORS!"


-------------	HELPING TO RECALL THE QUESTIONS  ----------------

Since the Soare's (FOM, Aug. 3) paper is so long, pointing out so
many Simpson errors, let us concentrate on just a FEW at the
beginning and ask Mr. Simpson some SPECIFIC questions.



At the AMS Boulder meeting Nabutovsky (Toronto) and Weinberger
(Chicago), announced two very interesting papers, NW-1 and NW-2,
applying computability theory and specifically computably enumerable
(c.e.) sets to problems in topology and differential geometry.
Simpson (FOM, June 22) quoted from NW-1, but got it WRONG.  He then
tried again (FOM, July 16) and STILL got it WRONG.  He protested,
>     (Simpson, FOM, July 16):
> However, I don't think this a very SERIOUS error on my part.
> It is an error of ATTRIBUTION, not SUBSTANCE.

But his ORIGINAL mistake (NOT corrected in his later July 16 posting)
was INDEED an ERROR of SUBSTANCE; in fact it was a MAIN POINT as
the topologist had EARLIER stated.


Mr. Simpson, do you acknowledge that YOUR version of the topologists'
theorem from NW-1 quoted by you in FOM June 21 after you "read the
paper on the plane" does NOT agree with the topologist's view as
stated in the message of the topologist co-author posted by me July 9
on Comp-thy, of which you were aware when you made your July 16
"corrections" to your June 21 FOM posting?




Simpson (FOM June 21) did NOT mention AT ALL one of the most exciting
parts of the Boulder conference and of paper NW-2, an announcement of
a theorem on comparing the depths of local minima on manifolds, and
the distance between them, using the Sacks Density Theorem [1964] for
computably enumerable (c.e.) degrees, proved by a priority argument,
and did even give an ANNOUNCEMENT of ANY of the exciting topologists'
results announced at Boulder relating computably enumerable (c.e.)
sets to scales on the manifold, a topic already presented in numerous
seminars and in TWO lectures at Boulder.  Simpson explains that he had
>                 Simpson (FOM July 16):
Simpson does not say EXACTLY WHAT he mistrusts here.  


WHAT EXACTLY does mistrust Simpson here?  Does Simpson mistrust:

(1)   The THIRTY-FIVE YEAR OLD Sacks Density Theorem (SDT) 
	[Annals of Math. 1964];  or 

(2) The ANNOUNCEMENT by two very EMINENT TOPOLOGISTS, Nabutovsky
	and Weinberger that from SDT (and NO other results in 
	Computability Theory (CT) ) that they could derive a new 
	theorem on relative depths of local minima for manifolds 
	and fractals?

It HAS to be either (1) or (2) because that's all there is.  ALL that
is necessary FROM LOGIC or from ANY LOGICIAN is the Sacks Density
Theorem.  From that the topologists have announced a result on local
minima for the space RIEM.  Simpson could NOT be doubting any OTHER
result in logic or any logician.

QUESTION 2B.  Is is just that Simpson has such a virulent hatred of
c.e. sets and degrees, which he has often expressed, that he cannot
bring himself to admit to FOM members that c.e.sets might be USEFUL
in topology and other areas central to mathematics? 
Did he fail to mention the applications of c.e. sets to
topology so that he could preserve the reverence and awe for the
f.o.m.  enterprise (Simpson, FOM, June 10), and the INTELLECTUAL
STATUS  for found of math and Rev Math from which he can look
down on all other logicians?  
	Simpson admits he understood the result but deliberately
withheld it even though he shared many less important facts, such
as the fact that the group attending his OWN special session
"was a distinguished group even if I [Simpson} do say so myself."
     Why not also share this startling connections of c.e. sets
to differential geometry.?

(Simpson states in FOM Aug. 3) that c.e. sets have no role in
undecidability results, but Boone after getting undecidability for
finitely presented groups he proved that there was one of every
nonzero c.e. degree thereby sharpening his previous undecidability
result.  Undecidability just says that the degree is not 0, while the
second result gives a much broader spectrum of the achieved degrees.)



Mr. Simpson DO YOU admit that the following argument of yours about
priority arguments and PURE and APPLIED computability theory makes
ABSOLUTELY NO SENSE in first order logic or even in a socratic

  Simpson (FOM, July 22), when confronted with the list above of
  nearly 200 applications, replied there were still many MORE applications
  of the priority method in PURE recursion theory than in APPLIED
  recursion theory and THAT is what was his original point in stating
  Simpson's Thesis.
  >     ---[Simpson (FOM, July 22)]:  ---
  >  First of all, I think everyone agrees that priority arguments are
  >  *much less* prevalent in ``applied recursion theory'' than in the
  >  ``pure'' theory of r.e. sets, r.e. degrees, etc.  That contrast was my
  >  original point in stating Simpson's Thesis, and I think it is a valid
  >  and important point for anyone interested in recursion/computability
  >  theory to keep in mind."                    

  This argument CONFOUNDS first order logic and BEGS a Socratic dialogue.
   	"But Alcibiades, if there be 200 applications of A to B, and
  	in ADDITION there be 500 applications of A to C does the
  	multitude of 500 diminish the effect of the 200,
  	or does it merely demonstrate the enhanced intellectual
  	vitality of A, and thereby increase its chance of still MORE
  	applications of A to B in the FUTURE?
  	"And anyway, Alcibiades, if your thesis (even as MODIFIED) 
  	mentions ONLY applications of A to TOPIC *B*
  	what relevance is it if there are any connections of A to C 
  	at ALL or a multitude of them?   The point is CONNECTIONS
  	of TOPIC A TO TOPIC B.  Any other topics are entirely 
  [Adapted from Plato's work, "Symposium," in which a dialogue between
  Socrates and Alcibiades takes place, as Alcibiades takes one
  unreasonable position after another.  Socrates, with gently probing
  questions, gets him to abandon each one.  See also the EPILOGUE here
  for MORE on Socratic dialogues and their relevance to this discussion
  of FOUNDATIONS of mathematics.]


==================ON COMPLAINING ======================

>       H. FRIEDMAN, FOM June 5, 1999 to FOM scolders:
>  Stop complaining about f.o.m. researchers. Either express your own thoughts
>  on the matter or ask appropriate questions or both. There are a lot of real
>  problems for f.o.m. researchers to work on, rather than listen to you
>  inappropriately scold them.

        R. SOARE, FOM August 5, 1999, REPHRASING the Above Friedman argument:

                To Reverse Mathematics People Who Criticize
                Computability Theory (CT) Researchers Without Cause:

Stop complaining about CT researchers. Either express your own
thoughts of the problems of relevance of Reverse Math to DISCOVERY of
NEW results in mathematics (e.g. topology and differential geometry)
or answer the appropriate questions raised by Soare, (FOM, Aug. 3) or
both. There are a lot of real problems for CT researchers to work on
which will give NEW DISCOVERIES in core mathematics like differential
geometry; rather than listen to you inappropriately scold them.



As a new subscriber I want know to more of the opinions of the vast
silent majority of the 380 FOM subscribers, not just the few who send
in their opinions very often.  So far about 8 or 10 have responded
some with positive and some with negative comments about FOM.

In addition to any public comments posted on FOM about this paper,
please SEND ME PRIVATE EMAIL with your comments and suggestions,
because one can often say things in private which one cannot say as
candidly or as completely in public for a variety of reasons.  I
GUARANTEE to keep your response absolutely private, not even quoting
from your msg anonymously.  This is the only way I can get a feeling
for this group, since the remaining 20 are the same respondents all
the time.  If I do not hear from you I will assume that almost no one
of the 385 subscribers is listening, and I will not waste time writing
more submissions.

I would like ESPECIALLY to hear your OPINIONS on the MODERATOR and
whether you can SPEAK FREELY.  Some have said they are afraid of
speaking up because the Moderator is so harsh and they are afraid of
being sharply criticized, especially if their views do not agree with
his.  One person wrote me that one had to be very "thick skinned" to
speak up.  How do you all feel?  I have the skin of a Sherman tank,
and a 75 mm gun turret to match.  I have not had any trouble
speaking out.  Indeed the Moderator has been MOST polite to me thus
far as you can all see, and I have done my best to RECIPROCATE.

Robert I. Soare
Paul Snowden Russell 
Distinguished Service Professor
    of Mathematics and Computer Science			
Department of Mathematics 	     PHONE: (773) 702-6029, Secty: 702-7100 
The University of Chicago            FAX:   (773) 702-9787
5734 University Avenue		     E-MAIL:	soare at
Chicago, IL 60637-1546	USA  	     WEB:

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