FOM: What is the motivation behind the Kreiselian school?
Soren Moller Riis
smriis at brics.dk
Mon Jan 5 12:04:29 EST 1998
I am new to this fom list. I received my Ph.D. on Bounded Arithmetic
in 1994 from the University of Oxford. I am about to go on leave from
a research fellowship at BRICS (Aarhus, Denmark) and begin a research
fellowship at the Fields Institute in Toronto. With my limited academic
status, I will mostly be listening rather than posting. I expect to
ask questions from time to time. Here is my first question:
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What is the motivation behind the Kreiselian school?
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Introduction:
As a student at Oxford I was intensively exposed to Kreisel and his
followers (the most prominent perhaps being Angus Macintyre).
The line of thought (when it is neutral) has perhaps most clearly been
put forward on FOM by Lou van den Dries. On Thu, 16 Oct 1997 Lou van den
Dries wrote:
>5. KEYPOINT (overlooked by many logicians and its popularizers):
>For this to be an effective strategy in shedding light on the original
>structure the language better be not too rich or expressive, on
>penalty of producing too complicated a theory to be of any use.
>
> And that seems to me one of the lessons of Goedel's Incompleteness
> Theorem: if in some way your object contains a certain minimal
> amount of discrete arithmetic structure AND your language allows to express
> that fact, then its theory in that language reflects too much the
> object itself and is---as all logicians know in exhaustive
> detail---extremely complicated; in fact, this theory is then beyond
> any kind of effective description and unlikely to play a role in
> creating "effective, positive understanding" of the object it was
> supposed to describe. (Of course, knowing that the Goedel phenomena
> apply to a certain mathematical object as described in a certain
> language, is itself valuable knowledge, but is here interpreted as
> a negative: the language, acting here as a kind of binoculars,
> was too strong and prevents one from seeing the forest beyond the trees.)
I hope we on FOM can agree that Lou have a very important uncontroversal
point. Furthermore the view have proved itself by leading to some beautiful
mathematics.
>From the discussions I had with Kreisel I think one can summerize
his position as being one that puts an extreme emphasis on the "effective,
positive understanding" Lou refers to.
But unlike Lou's fair and balanced position, Kreisel have a much stronger
bias towards the "effective, positive understanding" at the expense of less
applied logic.
Let me illustrate this: at one occasion Kreisel told me (and some fellow
students) that a proof of "P \neq NP" would very likely be a completely
useless with very little impact on mathematics (as well as on our
understanding of mathematics). On the other hand a proof of "P=NP" would
very likely represent very new deep understanding with a major impact on
mathematics as a whole.
My question "what is the motivation behind the Kreiselian school" concerns
this strong bias against f.o.m. I think the heart of the matter is that
in Kreisel's understanding, f.o.m. should rather be called "foundation of
logic", while foundation of mathematics (if there is such a thing) is
something rather different. In Kreisel's understanding (as far as I can see)
f.o.m (which he presumably he denies exists as a topic) is something much
closer to a unifying understanding (covering many different fields in
mathematics) obtained by the experts in mathematics.
On this account if a logician wrote a book on the foundations of number
theory, it would teach an expert in number theory (like Wiles) virtually
nothing of interest about "foundations of number theory". It would do
the same for foundations of number theory as the "introduction of the
category of wheels would do for foundation of engineering" (an expression
of Kreisel taken from a somewhat different context).
The correct approach is that of humility. Find out what insights the
leading mathematicians have achieved and try to give these insights a
logical foundation.
Kreisel and his followers undoubtedly have some important noncontroversial(?)
points. I expect that many subscribers on FOM are very happy some of the
above enterprises being pursued.
What I find problematic is the underlying bias (which places the logicians
in a rather observing and passive position). A bias which very uncritically
admires the leading pure mathematicians' perception of f.o.m. (the heart of
the method?). A bias which admires any "insights of great unity" proclaimed
from the leading pure mathematicians, while apparently disregarding the
tremendous unity and insights which demonstrately have been achieved by the
logicians. It is this bias which makes me ask:
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What is the motivation behind the Kreiselian school?
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By all appearances, after discussions with Angus Macintyre, the "Kreisel
school" (Kreisel is very opposed to schools or isms; this is my expression)
seems to be based on an admiration of certain mathematicians including
about a hundred number theorists, and especially Deligne and Grothendieck.
The same admiration for leading logicians was clearly missing (what did
they do with "impact" on mathematics?). I discussed this bias with Macintyre
openly on many occasions. Many of these discussions (with Angus) took place
in a pub after two pints of beer, but there were no doubt that this reflected
an attitude which I think is important to understand. I recall one
discussion in which I was outraged that Cohen was the only logician
who got the Fields medal. I remember I told him that I thought there
were logicians who deserved two Fields medals.
Also I felt that Robertson and Seymour who solved the graph minor problem
from graph theory should have gotton it (although they may have been too old).
Angus immediately suggested a list of people in number theory who in his
opinion was more qualified to get the prestigious Field medal (more
qualified in the sense of having done something of large "impact" and
providing a positive unified understanding in mathematics (=number theory?))
I came to the conclusion that Kreisel and Macintyre were expressing
extreme bias against logicians (perhaps even logic), and I was speculating
that there were some psychological factors behind this bias.
In general it is not a good idea to inject psychological considerations
into such matters, but I don't see how else to account for the extremeness
of these views.
It was not just Angus Macintyre who expressed such views. Also Kreisel,
(proof theorist by education) has taken extreme views against logicians.
While in Oxford we students often had discussions with Kreisel. Previous
to one of these discussions I had read an interview of Kreisel [I could
not find it in the library, but I know it has been published; do some of
you have the reference?]. In this interview Kreisel expressed some quite
negative opinions on Cohen (who proved the independence of CH in 1963).
In the discussion I asked Kreisel (in order to help him into the right
mode) if it was correct that the first versions of Cohen's paper had
been flawed.
He told me that Cohen had been "completely ignorant" of Kreisel's result
(which dated 10 years back earlier). Apparently at some stage Cohen had
assumed that there is a minimal model (with no reference to transitivity).
I tried to defend Cohen against the many insults from Kreisel. At some
stage Kreisel referred to Cohen as a Charlatan. When I asked him to
explain he answered in a parable about a witch-doctor!
Kreisels parable:
There was a witch-doctor. He did not know any medicine. His colleagues
knew a lot. The witch doctor became very successful by using a very
clever trick. He followed the big animals into the forest to see what
plants they were eating.
If he was confronted with a patient who looked like a pig, he would
treat him with the plants that the pigs usually were eating....If the
patient instead looked like a cow he would treat him with the plants that the
cows usually were eating.
This made the witch doctor very successful (more than his colleagues).
But what did he know about medicine?
If we think of Cohen as the witch-doctor (keep in mind that Cohen
was not trained in logic) it is not hard to see what the parable is
hinting at. Why this makes Cohen a charlatan remains however completely
unclear to me. I suppose the idea is that had he been serious
(not a Charlatan) he would have studied medicine the proper way
to gain real insight.
I suggested to Kreisel that one could get the impression that his
response about Cohen was guided by an inferiority complex. He got visibly
upset at which point our conversation stopped.
The next day we meet. He told me he was very pleased to see me.
He had forgiven me! He told me that I was young so I had an excuse.
He knew a full professor, who was forty and who shared "my views".
In Kreisels opinion this professor had no excuse. Well I am not sure
he realised I had been away from academia and was already 35 at the time!
While in Oxford I organized a series of junior seminars where we went
through Angus Macintyres' paper "The strength of weak systems"
[In: Schriftenreihe der Wittgenstein-Gesellschaft, 13 logic, Philosophy
of Science and Epistemology, Wien 1987 pp 43-59].
Towards the end of this paper Angus Macintyre discuss the insights of
R. Thom and compared them with those (achieved before 1986) from reverse
mathematics. R. Thom who won the Fields Medal in (1958) is perhaps most
known (among general educated intellectuals) for his Catastrophe theory.
I am sure most researchers in f.o.m find R. Thoms' "insights" that there
are four principles in Mathematical analysis and that the implicit
function theorem is one of them, interesting. Also I hope you appreciate
that researchers (Wilkie, A. Macintyre, Van den Dries) many years AFTER
Angus paper were able to "reconstruct" parts of R. Thoms insight in more
exact mathematical terms:
The implicit function theorem follows from o-minimality, and conversely
there is an axiomatisation of the real closed fields with exponentiation
(which Wilkie proved to be an o-minimal structure) in which the
implicit function theorem appear as an axiom scheme (in a non-dummy
fashion).
Macintyre finished his paper (the strengths of weak systems) by writing
insulting things about Reverse mathematics being pale stuff compared to
Thom's insights.
In my modest opinion, this "conclusion" is unwarranted intellectually,
in the same way that Kreisel's critique of Cohen is unwarranted.
But I am at a loss to explain these extreme views without resorting to
psychological explanations.
I was struck by how people in this camp continue to appeal to authority
(e.g., R. Thom) while making no arguments themselves. What is at stake
here? Their approach seems entirely outside the world of rational debate.
Why does this "school" have such a radically negative view of logic?
These people seem to be overly anxious to please the leading pure
mathematicians, and at the same time express unrestrained contempt for
logic. Does the explanation for this reside in the realm of psychology?
It would be interesting to get some outstanding mathematicians to
participate on this list. Well I know they usually do not "waste" time on
philosophy (R. Thom is an outstanding exception).
Usually they prefer to just work on pure mathematics and talk to other
pure mathematicians. But perhaps we can invite them to share their insights
with us on the FOM list. Thom's insights seem stimulating and helpful, and
have sparked some beautiful mathematics.
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Here are some manuscripts (actually most on weak systems) you should feel
free to download at http://www.brics.dk/~smriis/
Regards,
Søren Riis
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