FOM: pointless numbered postings?

[friedman@math.ohio-state.edu]

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 

>From Silver:

>    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 
the neccessity/sufficiency. 

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 
technical. 

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 
resulting projects. 

>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
postings either.   

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
"beautiful".   

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 
define.

>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 
for you?