foundations of mathematics

[Lou van den Dries]

As always I found what Harvey had to say very interesting, but I am
also in broad sympathy with the views expressed by Anand and Dave.
Steve Simpson would like Anand and Dave to discuss foundations on his
(Steve's) terms. Perhaps I can give it a try.

1. To avoid confusion, I shall write "Foundations" (F of Frege or
Friedman) if I mean "foundations of mathematics in the sense of 
Frege-R-H-G-Friedman", i.e. foundations of math as defined by Simpson
in his essay "What is Foundations of Mathematics" on his homepage.
 
I agree Foundations represents a coherent and precise style of 
thinking and I don't mind to be informed now and then of what's 
going on there, but I find it of less interest than other broad
issues that arise in mathematics. Simpson says in his essay:
 
"foundations help us to focus on the conceptual unity of the field,
and provide the links which are essential for applications and for 
integration into the context of the rest of human knowledge". 

This sounds like a very big claim for Foundations. But it depends on
what Steve means by "conceptual unity" and "providing links", etc. 
Terms like "conceptual unity" have a somewhat grandiose connotation 
(perhaps not intended by Steve). Does, say, set theory help us focus
on the "conceptual unity of math"?  Or "provide links etc." ?
When such claims are made it seems reasonable to make comparisons
with other conceptual unifications and links. Let me make an attempt
to do that below.
 
A hundred years ago (elementary) set theory began to provide a common
framework for defining mathematical objects in terms of a few simple 
set operations, thus removing uncertainties as to what counted as
correct arguments and allowable constructions. In that role it was
important at that stage in the development of mathematics (and remains
so). For example, the notion of a Riemann surface was only defined rigorously
in this sense by H. Weyl early in this century. Thus a Riemann surface
can be construed as a set in some artificial coding within set theory. Most
mathematicians have picked up this language of elementary set theory,
and haven't looked back since, and to me that is a justifiable
attitude. (For some subjects, like pure measure theory and abstract
model theory one needs a little bit more, not to mention set theory itself.)
I think Weyl would have resisted the idea that he had *reduced*
the notion of Riemann surface to that of the notion of set. 

Riemann surfaces will undoubtedly survive any modifications 
that the present framework for the reduction of mathematics to sets 
might undergo, or even the disappearance of sets if that ever
happens. More emphatically: they are much more *permanent* than 
the stuff of Foundations. One reason: there are such unifying and remarkable
and deep theorems about them as the Uniformization Theorem (see Weyl's
book), another is that they play a key role in another kind of
unification, which also strikes me as more *compelling* than the so-called
"conceptual unification" provided by set theory for all of mathematics.
I have in mind the equivalence between
-  compact Riemann surfaces (topology and complex analysis),
-  function fields of one-variable over C  (algebra) and 
-  (smooth projective irreducible) algebraic curves over C (geometry) 
and all the various other links that come with it: between 
Galois groups (algebra) and fundamental groups (topology), and so on.

NB: This is not a *unification* in terms of a *reduction* to supposedly
more basic concepts. But it does provide unexpected and mutually
enriching links on a deep level between relatively broad areas, very
much more so than the fact that the basic concepts of these areas can
all be defined in terms of sets, and all the proofs formalized in ZFC.  

Doesn't this kind of pervasive phenomenon undercut somewhat Steve's
claims about Foundations? (I am just asking. Steve might deny some
claims I made, or not draw the conclusions from them that I
do. Certainly they are not "conclusions" in the logical sense.)  

This is just one example of how large parts of mathematics strike me as more 
compelling and of more permanent value than the Foundations that one can build
underneath. In addition, much of it is "beautifully" precise compared
to the "ugly" precision of formal systems. (Formal systems have their
place in mathematics, I hasten to say, just as do sheaves and
dynamical systems and ...., and some of them may even be beautiful.)
   I see nothing wrong with building your favorite Foundations and
then go on studying them; some temperaments may find it just what they
want to do. And perhaps the general mathematical community will be
grateful in ten or a hundred years that the study of Foundations has
been pursued so systematically as exemplified in  "Reverse Math". 
And I am certainly grateful for having learnt a lot from the
Frege-Hilbert-Godel line of work, drawing different practical
conclusions from it than Steve though.   


2. Simpson attaches great importance to the proper use of terms like 
"foundations" and "logic", and to who should be properly considered as
a "logician": someone who studies logic (by all available means). 

Many of us model-theorists have at some point in their education or
career spent a fair amount of time thinking about logic and issues in the 
Foundational tradition, and have since gone on to do other things.
We still think about logic, while perhaps not taking it as the
central concern of our mathematical activity, at least not logic as
defined by Simpson. Thus I may not be a logician in his sense. Fine.

Perhaps the answer to Steve's question as to what makes us then
different from geometers when we consider geometric objects like, say 
Zariski geometries, is the model-theoretic perspective that
some of us (Zil'ber and Hrushovski in this case) brought to it. 
There is Zil'ber's remarkable "trichotomy principle" which acted as a
driving force in this case. Again, the kind of unification aimed at by
this principle strikes me as more compelling than the Foundational
unification achieved by, say, reverse mathematics, which I *do*
recognize and respect. Of course, in bringing our own perspective 
and background to subjects we work in, we are no different from 
other mathematicians, but why should we? I happen to find the 
model-theoretic perspective enriching *when suitably integrated and 
combined with other outlooks*, and do not regret my background.  

The point is that such `cross-cultural' connections and links can be
very non-obvious, and mutually enriching. The fact that ideas
originating from mathematical logic can actually make a contribution 
to "ordinary" mathematics is fascinating to me. And ideas
from ordinary mathematics are also changing model theory. 
I reject the idea that these connections have to be in the form of 
some kind of reduction of concepts to supposedly more fundamental ones.
Or in the form of independence results, or other items of the
Foundational kitchen (which have their place of course). The aims of 
Foundations may be explicit and precise, but they say nothing about 
the *conceptual links* within mathematics that many mathematicians
actually care about. In a later email I might try to articulate a
view of "logic & math" that I do find more agreeable and productive.



3.  In his essay Steve refers to Cartesian geometry as showing a 
*reduction* of (much of) geometry to algebra, which fits his idea of 
organizing concepts in a *hierarchy* of more and less fundamental
concepts. But the true mathematical significance of Cartesian geometry
seems to me to lie in the fact that the traditions and intuitions from
one area became available to enrich the other, and that from then on 
they developed together in some synergetic unity, not a unity where
one subject is reduced to another. (Perhaps this is consistent with 
Steve's view.) The key point is that Descartes changed the *dynamics* of the
interaction between the two subjects.   

In one of his responses Simpson even says:

"This foundational aspect makes Cartesian analytic geometry of general
intellectual interest, i.e. of interest to all educated human beings."
                                                                               If you believe that, then your notion of "educated human being"
is probably narrowly taylored to make it true. This Foundational
(reductive) aspect is not what struck me about analytic geometry
when I first learned it in school. Descartes presents it as a
*method* to solve geometric problems by translating them into
algebraic problems. The hundred years or so before him had seen a lot
of progress in algebra, compared to a relative standstill in
geometry. It was Descartes' boast to have found a systematic way to use the
algebra of his time to solve geometric problems which the ancients
couldn't do. He refers to the relation between the two subjects, but
not in terms of a Foundational reduction of one to the other. (Not in
what I read.) Anyway, it is of course not essential what Descartes 
had to say about it, what matters is that the subjects were closely 
intertwined from then on. 

What is of interest to the general educated public in this connection
might perhaps be described as the fact that a quantitative
relationship between numerical variables can be represented and 
thought of as a geometric shape: very suggestive, important in the 
later creation of calculus, of great scientific value, and so on, 
but hardly suggestive of a Foundational reduction of geometry to algebra.    
  

Well, this is getting too long already and I apologize for going on
about this.   
               -Lou van den Dries-