FOM: pointless numbered postings?
friedman at math.ohio-state.edu
Sat Jun 29 19:42:32 EDT 2002
Some time ago, I made a flurry of numbered postings about incompleteness, and
things have calmed down considerably since then. Questions were raised about
the point of them, their significance for f.o.m., etcetera. See
charles silver" <silver_1 at mindspring.com>
Date: Wed, 27 Mar 2002 06:40:58 -0600
wiman lucas raymond <lrwiman at ilstu.edu>
Date: Wed, 27 Mar 2002 11:28:38 -0600 (CST)*
*see wiman lucas raymond <lrwiman at ilstu.edu>
Date: Wed, 27 Mar 2002 20:04:32 -0600 (CST)
Martin Davis <martin at eipye.com>
Date: Wed, 27 Mar 2002 11:23:53 -0800
Richard Heck <heck at fas.harvard.edu>
Date: Wed, 27 Mar 2002 15:26:42 -0500
sandylemberg at juno.com
Date: Wed, 27 Mar 2002 17:16:57 -0700
"Insall" <montez at rollanet.org>
Date: Thu, 28 Mar 2002 02:55:22 -0600
> I have a question: Whatever is "foundational" about Harvey Friedman's
copious, exceedingly technical postings?
They are meant to address the question "Does normal mathematics need new
axioms?" positively by means of examples. There are criteria for "normality"
according to current mathematical culture. These criteria have evolved into
their present state over hundreds of years, in fits and starts. There is much
than can be said about this notion of "normality".
Given recent unexpected successes (see my postings #126, #150, #151). and the
reactions, I can advantangeously rephrase this question as "Does normal
beautiful mathematics need new axioms?"
>Perhaps that's the
inevitable evolution of mathematicians with philosophical interests: towards
increasingly more technical work, I don't know.
Not for me. If one is going to show the neccessity of new axioms for normal
beautiful mathematics, then one is going to have to give examples, and show the
requisite neccessity/sufficiency, and that is going to be technically
challenging, both in terms of coming up with examples, and in terms of showing
For example, isn't Goedel's most famous paper on incompleteness exceedingly
technical? And what about his monograph on the consistency of the continuum
hypothesis? These are very old, and exceedingly technical, aren't they?
>At any rate, it seemed to
me, anyway, that work in so-called "foundations" deteriorated into the
purely technical, where "results" were arrived at that fell into one of the
areas of mathematical logic.
I am not going to personally admit to any "deterioration" in what I do.
>Or perhaps belonging to a new area. For
example, Harvey's and Steve's and others' work in reverse mathematics seems
to me to belong to a somewhat new area (though perhaps some persons would
>assimilate it under proof theory).
Again, the overarching project of classifying mathematical theorems according
to an appropriate foundational scheme, is of obvious importance for f.o.m.,
assuming that the framework is informative. The RM framework is, indeed,
informative. But carrying out the actual classification is going to be
There are overarching issues about the RM enterprise - e.g., improving it and/
or modifying it in various ways. Some of these involve introduction of new
lines of research, and the presentation of such things can have a very large
nontechnical component. But still there has to be a technical component in
completely precise and workable formulations, and also in carrying out the
>As far as I can see, besides a nod in
the direction of first-order logic and some brief motivational talk of
>various kinds of induction, work in r.m. is almost exclusively technical.
The choice of systems, and the philosophy behind the whole enterprise, is
obviously not purely technical. Some aspects are largely nontechnical.
>I come now to Harvey's postings. I am not able to evaluate the importance
of Harvey's postings, because I do not have the requisite technical savvy.
I further doubt that the majority of people in FOM can understand his
My postings #126, #150, #151 are readily understandable by major mathematicians
outside logic, and also by a large majority of active mathematical logicians
working in mathematics departments. Perhaps many don't quite remember what the
Mahlo cardinal hierarchy is, but they know that such cardinals are some sort of
exotic inaccessible type cardinals outside of ZFC.
>They seem to me to epitomize technical results for the
sake of technical results.
This is completely wrong.
>Harvey has described his recent results as
As was made clear earlier, I was giving the explicit reaction of important
luminaries. And as I explained earlier, "beautiful" has come to mean something
quite specific among mathematicians, even if it is difficult to explicitly
>Perhaps they are, I'm in no position to judge. All that
seems clear to me is that there is nothing or at least relatively little in
>his posts to explain their "foundational" content.
As I said earlier on the FOM, I hoped to come back to explain this.
Do the postings of Martin Davis and Richard Heck form an adequate explanation
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