FOM: Response to Tennant, Simpson, Friedman

[Torkel Franzen]

1.

  Neil Tennant remarks that

   >His [Friedman's] result *puts on an epistemological par* the two kinds
   >of principle---combinatorial and large-cardinal. *That* is why it's so
   >impressive. If I am told that someone has shown "A iff B", when that
   >sort of equivalence would have struck me, prima facie, as highly
   >unlikely, I don't question the significance of the proof of that
   >equivalence by saying "Well, we'll have to wait and see whether A has
   >any application; and we'll have to wait until we've settled the
   >epistemological status of B."

  You are surely aware that no arithmetical sentence (consistent with
ZFC) can imply (in ZFC) any large cardinal axiom.  Friedman's
principle implies the *consistency* of an extension ZFC' of ZFC with
certain large cardinal axioms. So, trivially, does the aritmetical
statement "ZFC' is consistent" itself. Nothing in your comments
depends on any difference between these two cases, so it's unclear why
you are more impressed by Friedman's results than you are by the
results you knew before, on the basis of Godel's theorem.


2.

  Steve Simpson says:

   >Franzen is saying that, in order to establish the general intellectual
   >interest of f.o.m., there is a need to "relate" or "link" f.o.m. to
   >unspecified "intellectual concerns of other people".  But which
   >intellectual concerns?

   That, naturally, depends on your idea of "general intellectual
interest".  The first step is to realize that there are other people,
who do have intellectual concerns. "Directing public relations
campaigns at ignorant louts" may of course be an apt expression
of your personal feelings about the prospect of taking the
intellectual concerns of people not in f.o.m. into account.

   Sifting through your comments for some explanation of the "general
intellectual interest" line, I find

   >First, f.o.m. *already is* related to the concerns of the man in the
   >street.  This is because f.o.m. deals with the logical structure of
   >mathematics, and mathematics is applied to develop technology which
   >benefits the man in the street, whether he understands it or not.

  This argument applies equally to such things as the foundations of
materials science, which are not usually held to be of "general
intellectual interest", and I doubt that you would put f.o.m.s. on a
par with f.o.m. in this regard. I seek further:

   >What then *is* the general intellectual interest of advances in
   >f.o.m.?  Ultimately this is a philosophical matter, concerned with the
   >place of mathematics in the structure of human knowledge as a whole.

  A philosophical matter indeed, but what emerges from this comment is
only that you think f.o.m. is philosophically important.

   You propose a "g.i.i. formulation" of Friedman's results:

   >A coherent body of results in finite mathematics, related to data
   >structures which are familiar in computer applications, have been
   >shown to be provable only by use of speculative mathematical axioms.
   >These speculative axioms are strong axioms of infinity which go far
   >beyond the standard mathematical axioms which have hitherto sufficed
   >for the bulk of mathematical practice.  Such uses of speculative
   >axioms is unprecedented.

  This description may well arouse the interest of e.g. computer
scientists.  But they are not impressed by mere bombast, however
philosophically impressive. If it turns out that "related to data
structures which are familiar in computer applications" means only and
exactly that, and that no interesting applications (comparable e.g. to
the uses of Kruskal's theorem) are forthcoming, their interest will
quickly wane, however strenuously you point out to them the enormous and
objective general intellectual interest of the results.

  I think that what your comments most strongly suggest is that we shouldn't
take this talk about "general intellectual interest" too seriously. What
you mean is that f.o.m. is very interesting stuff, and I quite agree. To
enter into any serious consideration of "general intellectual interest"
we would have to consider such questions as why e.g. Wittgenstein is
regarded by people in the most varying fields as being very interesting,
and more generally try to take into serious account just what is meant
by "intellectual concerns" and how these are manifested. I'm not claiming
to have anything illuminating or well thought out to say on the subject
at this point.


3.

  Harvey Friedman says, in response to my question why non-specialists
should care about his results:

   >Because, e.g., there is plenty of documented evidence that many
   >non-specialists relate very strongly to the incompleteness theorems and the
   >whole idea of unprovability. The above is a very strong and modern form of
   >earlier results which have stood the test of time in this regard. Also, the
   >general intellectual interest in findings relating to the "special
   >objective nature" of mathematics - construed as postitive or negative
   >findings - is completely self evident and well tested.

  This reply is a certainly a great improvement on Simpson's response,
since you actually refer to the intellectual interests of other people.
The fact that many people take an interest in incompleteness and
unprovability does not imply, however, that they either will or should
take an interest in technical refinements (strong and modern versions
of earlier results). Your remark about the "completely self evident
general intellectual interest" of findings in f.o.m. adds nothing to this
(I suspect fruitless) line of discussion.

  In spite of my botching the procedure for making claims in a
serious, formal way you have noticed my claims about the foundational
importance of your results, and state in response:

   >Large cardinal principles, following Godel, are so ingrained in the
   >development of f.o.m., and form such a compelling coherent picture
   >(independently of wide open questions of ultimate justifications, etcetera)
   >that various results concerning them can be obviously crucial steps in the
   >development of f.o.m. totally independently of any future understanding of
   >their "epistemological status."

  I don't think anybody would dispute this. For example, a proof that
the existence of subtle cardinals is inconsistent with ZFC would be an
obviously crucial step. But the question at issue was not about "can
be", but whether the particular results that you announced can be seen
to be an "epochal advance in f.o.m." independently of applications and
epistemological status.

  Of course much here hangs on "epochal advance", as you reasonably
imply in your question what I mean by "epochal". Since Neil Tennant
inexplicably read my earlier comments as a dismissal of your results
as uninteresting, I should perhaps repeat that they are "surely
remarkable" results of obvious foundational interest. When Simpson
states that your results are an epochal advance, he may of course be
indulging in playful hyperbole, but taken literally, his statement
conveys that your results are as important, in terms of the future
course of the subject, as the paradigmatically epoch-making work of
(say) Frege, Zermelo, Godel, Turing. This is what I regard as
premature.

  If you accept the description of your results as an epochal advance,
wherein would you say that this epochal advance consists? Not in the
discovery that combinatorial principles are logically related (in the
sense of these results) to large cardinal axioms - this epochal
discovery has already been made. The epochal advance must be connected
with these specific combinatorial principles, and these specific large
cardinal axioms. And indeed you have strongly emphasized that these
combinatorial principles are simple and natural (to a number of
combinatorists). But even so, if they have no striking mathematical
applications, or if we have no grounds for accepting them, how do
they advance our understanding of mathematics?

  In your response to Feferman, you referred to other work and
projected work of yours. Now, quite a while before this mailing list
got going, I came across some papers that you had (commendably) made
available on the web. These included the precursor to the work now at
issue, a pretty technical paper on "Finite functions and the necessary
use of large cardinals" (containing among other things, as you may
recall, a definition of the k-subtle cardinals). This work looked to
me impressive, but not foundationally epochal. Then there was also
your manuscript on "Transfer principles in set theory". This contained
a series of conjectures that did seem to me to be foundationally very
important if true. But let's not get things mixed up. What has been at
issue in the recent exchanges is only the question of the importance
for f.o.m. of the results you actually announced. What prompted my
questioning that importance was not your announcement - which left
people free to judge the matter for themselves - but Simpson's
ham-fisted follow-up.

  You bring up two examples of other work in foundations. First,
Godel's second incompleteness theorem. In arguing that this is an epochal
advance, it is sufficient in this context to point to only one aspect,
namely that this work establishes the very existence of a logical
connection between combinatorial principles and large cardinal axioms,
and is used and presupposed in practically all subsequent
work. Secondly, you invoke the Godel/Cohen work on the continuum
hypothesis. This work again has several very important aspects, such
as the introduction of the constructible sets and the forcing method,
and indeed the use of these results to establish the eliminability
of many uses of the continuum hypothesis and the axiom of choice.

  About these examples, you ask whether I would make a similar comment
regarding applicability and epistemological status. Of course not.  I
haven't made any comments about what is required for work in
foundations in general to constitute an epochal advance. We can say
many things about how and in what ways the work by Godel and Cohen
constitutes an important advance. My comment about applications and
epistmological status concerned specifically the results you announced.

---
Torkel Franzen
Computer science, Lulea technical university