foundations of mathematics

[Stephen G Simpson]

Harvey said:
 > I am unboundedly excited about the responses from Lou, Anand, Dave, Steve,
 > and indirectly John Baldwin and John Burgess about foundations of
 > mathematics.

I'd like to echo that.  Finally, a serious dialogue with serious
people about foundations of mathematics!  This is something I have
wanted for a long time.

Like Harvey and everybody else here, I have a lot of other commitments
(in particular I'm struggling to get the finished LaTeX manuscript of
my book on reverse math etc to Springer-Verlag by December 31 as
promised, plus I'm struggling to finish writing some papers, plus a
heavy teaching load, departmental duties, etc etc, plus I got married
a few months ago!).  But I can't resist taking a little time now to
respond however quickly and inadequately to some of Dave's and Lou's
very thoughtful points.  I hope to find time to respond more carefully
before Thanksgiving.

Dave Marker writes:
 > I do not except your conclusion that the Frege-Hilbert-Godel line
 > is the only approach to the foundation of mathematics.

I'm not saying that at all.  In my essay at
www.math.psu.edu/simpson/Hierarchy.html, I defined "foundations of
mathematics" generically, as the study of the most basic mathematical
concepts (actually there is more to it than that, but that's it in a
nutshell).  I emphasized the importance of the F-R-H-G line, but I
gave other instances too, e.g. Cartesian geometry.  I also envision a
revival of foundations of mathematics beyond the F-R-H-G line in other
directions.  For instance, one line I hope to pursue is an
Aristotelean (non-set-theoretic!) analysis of truly basic concepts
such as number and geometrical figure, with emphasis on applications,
to give a rational basis for what Wigner called the "unreasonable
effectiveness" of mathematics.

 > I get the impression that you feel that once Descarte reduced geometry
 > to algebra there was nothing left to do.

Not at all.  I never said that and I certainly don't think that.

 > To my mind there are many results which I consider foundations which
 > are clarifying the interaction:
 > Three examples (all of which I've mentioned before):

All of these examples are important and interesting mathematics, but I
wouldn't call them foundations of mathematics, because they are not
close enough to the most basic mathematical concepts.  Skew fields,
Desarguesian projective planes, complex curves, Zariski topology, etc
are many levels higher in the conceptual hierarchy.

 > Though mathematical all of these results are also foundational. They
 > clarify the connection between the fundamental mathematical concepts of
 > geometry and number.

I would put it this way: they state fascinating relationships between
certain kinds of abstract geometrical objects (projective planes,
etc.) and certain kinds of generalized number systems (skew fields,
etc).  But that doesn't make them foundations.  If you think that any
theorem about numbers or geometrical figures is foundations, then you
would have to classify all of number theory, e.g. Wiles' theorem, as
foundations.  I can't agree with such a classification.  What
distinguishes foundational stuff from non-foundational stuff is: how
close is it to the base of the conceptual hierarchy.

 > I am also sympathetic with Anand's point about cohomology. While
 > you are right that cohomology is a mathematical tool with
 > mathematical applications, one might wonder about the
 > meta-mathematical significance of the pervasive role of cohomology
 > in modern mathematics.

Yes, one can wonder about the metamathematical significance of
cohomology, but such speculations are a far cry from systematic
foundations of mathematics.  The term "foundations of mathematics" has
a definite scope, and cohomology simply isn't included in that scope,
at least not in the present state of knowledge.

 > My guess is that most mathematicians simply view the foundations as
 > "done". ZFC provides a strong enough system (which is probably
 > consistent) to prove any theorem which is currently accepted as
 > proven. If forced any result they want to prove can be traced back
 > to a formal proof in ZFC, so there is no need to worry about
 > foundational questions and they can carry on.

Yes, many mathematicians do hold this extremely unenlightened and
dysfunctional view of foundations.  I blame Bourbaki's influence for
this lack of serious interest in foundations.  There was an article in
the JSL by Bourbaki which promulgated exactly this view, in an
incredibly crude way.  I have never forgiven Bourbaki.

 > What convincing argument do you give them that they should care that
 > their results can be proved in a much weaker system? 

There are plenty of valid arguments, but my experience is that
mathematicians influenced by Bourbaki are typically very closed-minded
when it comes to anything outside pure mathematics (present company
excepted!).  They can't be convinced if they don't want to be
convinced.

 > As for "reverse mathematics".
 > -First from a (unviersity wide) intellectual level:  While the existence
 > of such
 >  a classificaton is interesting:  a)Why is this true?  What does this tell
 > us about mathematics. ( I mean what does it tell us about mathematical
 > knowledge and the activity of doing mathematics, rather than what
 > mathematical knowledge
 > do we gain)

Yes, this is an interesting question on a "university-wide" level.
This question is discussed in my book.  But note: this kind of
question is totally uninteresting to mathematicians of the dogmatic
Bourbaki-ZFC stripe.

Lou van den Dries writes:
 > I don't think Atyiah and Horgan should be mentioned in the same breath.

You are correct and I won't do it again.  Atiyah's anti-foundational
foray was probably an innocent mistake.  By contrast, Horgan is an
evil monster.

 > Cohomology: this has to do with the very broad issue of passing from
 > local information to global, and measuring the obstructions. It's
 > potentially not that far removed from the most basic mathematical
 > concepts. 

Have you got any specific ideas for relating cohomology to the most
basic mathematical concepts?

 > And it is quite imaginable to me that it could have a very wide
 > intellectual impact.

But not as wide as G"odel's 2nd incompleteness theorem.

Lou quotes my essay at www.math.psu.edu/simpson/Hierarchy.html:
 > "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". 

and then retorts
 > This sounds like a very big claim for Foundations.

Well, yes, you are right.  I should probably sharpen this formulation.
It's not really what I wanted to say.  The point I was trying to make
is something like this: By focusing on the most basic concepts of
field X (e.g. X = mathematics), you are viewing field X from an
interesting and important perspective, a broader intellectual
perspective than is usually the case for specialized work within field
X.  And most of the key connections between field X and the rest of
human knowledge (by "key connections" I mean the connections needed
for the unity of human knowledge) rely crucially on the basic concepts
of field X, not the higher-level concepts.  I'll try to say this
better when I have more time.

 > Does, say, set theory help us focus on the "conceptual unity of
 > math"?  Or "provide links etc." ?

I'm not a big fan of set-theoretic foundations, especially dogmatic
set-theoretic foundations as propounded by Bourbaki.  (In fact, I
frequently rant and rave against set-theoretic foundations, and the
set theorists view me as a threat.)  But you have to recognize that
there is a lot more to the F-R-H-G line than set-theoretic
foundations.  Lou seems to be identifying "foundations" with
"dogmatic, stupid, Bourbaki-style set-theoretic foundations."  This
identification is incorrect.  I point to the Friedman volume for a lot
of examples of foundations of mathematics on a broad front.

 > I think Weyl would have resisted the idea that he had *reduced*
 > the notion of Riemann surface to that of the notion of set. 

In fact he did exactly that, whether he would admit it or not.  But
the interesting question to my mind is, are such reductions
illuminating?  Is set theory the appropriate foundational framework?
I think not, but we are mired in the legacy of Bourbaki and have to
work our way through it and past it.  I view my own work in
foundations (reverse mathematics etc) as a serious attempt to get past
this stupid Bourbaki-ZFC dogma.

 > 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. 

Amen, I agree.

 > More emphatically: they are much more *permanent* than 
 > the stuff of Foundations.

YES, if you mean dogmatic Bourbaki-style ZFC foundations.  But
emphatically NO, if you mean the broad subject of foundations of
mathematics, as I defined it in my essay.  Foundations of mathematics
is of permanent value.

Also, I'm glad Lou is not trying to say that Riemann surfaces *are*
foundations of mathematics.  The clarification of the concept of
Riemann surface is a great mathematical achievement, but Riemann
surfaces are a higher level mathematical concept, not a basic
mathematical concept, so it's not foundations of mathematics, at least
not in the sense that I defined "foundations of mathematics."

Maybe there is need for a term (other than "foundational") to describe
far-reaching mathematical results (such as Weyl's work on Riemann
surfaces) that clarify high-level concepts and/or elucidate
connections between different branches of mathematics.  How about
"conceptual mathematics" or "penetrating mathematics" or something
like that?  But I think there is also need for a term like
"foundations of mathematics", referring to analysis of the most basic
concepts, rather than higher-level concepts.

 > 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). 

Yes, of course.  It's important to be precise about these matters.
But there is such a thing as informal discussion in e-mail where we
play with definitions, try to refine and sharpen them, etc.

 > 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. 

Sure, that's fine.  I do agree that, for instance, model-theoretic
algebra is an interesting subject with its own identity, somewhat
distinct from algebra, and the distinction is based on the use of
model-theoretic methods.  Fine.

 > 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.

I never said that all interesting connections between branches of
mathematics have to be stated in terms of reduction of less
fundamental concepts to more fundamental ones.  In short, I never said
that all interesting mathematics is foundational.  If anything, I
think it has been Anand, not I, who tends to apply the word
"foundational" to any and all interesting mathematical developments.

My view is, there is a lot of deep, interesting mathematics that isn't
foundational, but we also have to recognize foundations of mathematics
as a deep, interesting subject in its own right, transcending
mathematics itself in some ways.  To use the term "foundational" to
describe all interesting mathematics would be a very serious mistake,
because it would obliterate foundations of mathematics as a subject
distinct from mathematics itself.

 > 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.

Well, I think this is right.  Everybody with a high-school education
thinks in terms of graphs like the GDP over 50 years, and this is more
or less based on Descartes, no?

 > 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.

Lou, I think you are struggling to deny the obvious: that Cartesian
geometry is foundational and includes a reduction of geometry to
algebra.

 > 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.

Reasonable people can differ in their formulation of why Cartesian
geometry is of such general interest to the educated public.  I don't
think you and I are really very far apart on the substantive point.

-- Steve