FOM: Re: pointless numbered postings?

[friedman@math.ohio-state.edu]

Reply to Silver Monday, July 1 2002 05:30 pm

> Friedman:
> > 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".
>
Silver:

>     I'm a bit puzzled what is meant by "normality".  Does it mean something
> like "present day consensus," and if so, consensus of what group?   One
> should, I think, take into consideration that "normal mathematicians"
> typically neglect foundational concerns entirely.

As I indicated, a lot can be said about "normality". There is a keen sense of 
"normal mathematical theorem" in the culture, with a lot of agreement. The 
homogeneity over this is quite striking, especially when one restricts to 
reasonably high status full time professors in mathematics departments, and the 
homogeneity is very substantial even if one widens the group considerably. 
Metamathematics is definitely not "normal mathematics". It is abnormal 
mathematics done for a foundational or philosophical purpose.  

The key sufficient ingredient is that there should be a compelling kind of 
simple, memorable, formulation involving familiar everyday objects and everyday 
notions. Beautiful implies normal, but not vice versa. Here I mean "beautiful" 
as asserted by professional mathematicians - not by others. Beautiful is used 
only in a certain semiprecise way. 

I can go on and on about this, but choose not to at this point. One can immerse 
oneself in this professional mathematical culture and see what I mean. There is 
nothing particularly controversial about it in the mathematics community.

> Friedman:
> > The choice of systems, and the philosophy behind the whole enterprise, [the enterprise of reverse mathematics] is obviously not purely technical. Some aspects are largely nontechnical.
>
>     Sure.  I'm just asking for this to be spelled out.   

Why don't you look at Simpson's book and ask about specific points to be 
clarified? Will that work?

>Incidentally,
> though I'm unconvinced that ZF set theory is **the** one and only foundation
> for mathematics, I've never doubted for an instant Harvey's sincere
> philosophical concerns in foundations.   I've just wanted to read more of
> the views that underly his results.   (For example, off-line several times,
> I've found myself in the unfortunate position of *defending* some extremely
> critical comments Harvey's made that several ex-subscribers found offensive.
> My defense, such as it was, relied on my--admittedly shallow--understanding
> of some of Harvey's underlying philosophical views on certain foundational
> issues.)

Now that much time has passed, you should consider revisiting these issues on 
the FOM. E.g., I said things like "categorical foundations does not serve as a 
foundation for mathematics as normally presented because of crucial missing 
elements", and "to the extent that categorical foundations serves as a 
foundation for mathematics, it is through its equivalence, in many senses, with 
set theory", and "Quine's New Foundations does not, at this point, serve as a 
foundation for mathematics. Some foundational value may emerge after we figure 
out how to interpret it in set theory, possibly with large cardinals", and "the 
prevailing view is that there is no natural r.e. set of intermediate Turing 
degree, and that makes it nontrivial - but not impossible - to clearly state 
the clear value of work done on the structure of the intermediate degrees". Are 
these the kind of things that ex-subscribers found offensive? Are there more? 

> Friedman:
> > Do the postings of Martin Davis and Richard Heck form an adequate
> explanation
> > for you?
>
>     Both of them have been helpful, but you are the best expositor of your
> own underlying philosophical motivations.   I do understand that you would
> prefer to concentrate on achieving results of foundational interest, rather
> than spelling out the views motivating those results.   Nonetheless, I'd
> appreciate an occasional tip of the hat toward philosophical concerns.
> Actually, a book, or something like one, on the philosophy of mathematics by
> you would be especially welcome.    And, it seems to me that in the back of
> your mind you have already "written" such a book.
>
For me, there has always been a conflict between the writing of such material - 
in articles, on the FOM, and/or in a treatise - versus the unabashed 
unrestricted full time development of ideas. I have never been able to resolve 
this conflict satisfactorily. Some promising ideas hold out the promise of 
permanently changing the whole complexion of f.o.m. So one idea is that IF the 
whole complexion of f.o.m. is about to change permanently, THEN various 
articles and extended postings, and especially any treatise, will be obsolete 
even before it is assimilated. FOM postings would make the most sense under 
these circumstances, as they are not "engraved in stone" and are a low overhead 
way to publish ideas with the possibility of immediate feedback. But of course 
even extensive, polished, FOM postings are a major undertaking. Responding to 
focused inquiries seems like a good compromise.