FOM: ReplyToBarwise

[Harvey Friedman]

>My earlier
>message was already beyond the bounds of what most people will read in
>email.  And then email does not inspire the same kind of careful
>formulation one expects in, say, a paper.  And, too, if I explain what I
>really mean, no doubt many of you will think I have really gone off the
>deep end.  But so be it.

I have thought twice about writing more than what most people will read in
e-mail, but I thought that people can store it in a file, and come back to
it. I was hoping that I could get enough of a reputation for putting a lot
of useful ideas out, so that people will come back to what I say when they
have more time and want some stimulation.

Jon, you haven't gone off the deep end. I do think that you are, however,
on the edge of the deep end. And I am replying to you to rescue you from
going over.

>When I wrote of first-order, I assumed that people would read this as
>meaning "expressible in standard first-order logic" in some sense of
>expressible and I do indeed believe that mathematical reasoning is not
>expressible in standard first-order logic.

If by standard first-order logic, you mean something like: the usual axioms
of ZFC plus standard first order predicate calculus, then obviously
mathematical reasoning is not directly expressible, since there aren't even
any provisions for abbreviations!!!

The challenge for you is to fix this statement up so that it has some
punch. Randy Dougherty and I have the real goal of creating a suitable form
of user friendly ZFC plus first order predicate calculus that is clearly
sufficient for mathematical reasoning in various strong senses. Rather than
go into what the criteria are for success in this regard - which is not
easy (but possible) to formulate - I would like to know if you are
dismissing out of hand that this can be done in an appropriate sense.

>The basic point here is that standard first-order logic takes only part of
>the mathematician's vocabulary seriously, what we have come to call the
>logical operators.  The rest it takes as uninterpreted and forces one to
>try to axiomatize in terms of the operators it takes having a fixed
>meaning.  That just does not seem to me faithful to the way mathematicians
>work.  Why?  Well, I think that mathematicians mean what they say when they
>speak of natural numbers, for example.  They are not talking about finite
>ordinals and they certainly are not talking about elements of non-standard
>models of set theory or of number theory.

Of course "they are not talking about finite ordinals and they certainly
are not talking about elements of non-standard
models of set theory or of number theory." This is a straw man. The serious
one of these is "they are not talking about finite ordinals."

Now I think you make a serious well known point here. There does seem to be
some sort of construction principles that are not intellectually faithfully
captured by throwing everything into set theory. And these construction
principles have not really been attractively, smoothly, and neatly
formulated in a suitably interesting way. Agreed. And this could be very
interesting. Agreed. And the category theorists have tried to do this. And
it really hasn't been all that attractive or convincing.

But let me say this. The real issue for me is: is this a minor point or a
major point? I think it is a minor point. Let me make a connection with
engineering. People have all sorts of ideas about devices. People build
devices differently. Engineering only gets going as an effective practical
force when standards are created. These standards always have a certain
kind of artificiality to them. Why should a byte be 8 bits? Why should
electrical outlets have certain dimensions? Why should serial ports obey
certain protocols? Why should text files be based on ASCI?

Without rigid standards, engineering collapses from incompatibility. That
is also the ultimate secret of mathematics. Your way of constructing things
and imagining things may be different than mine. But if we have a simple
common language in which we all can go back to in order to resolve any
failure of communication, just as in engineering, then we can progress.
This is what set theory provides in a way that nothing else provides right
now.

The engineers don't endlessly fret over the fact that one can build
electrical outlets with different dimensions, and the vast difficulties in
accomodating them. The engineers pick one way to incorporate everything
that is relevant, so as to facilitate compatibility. The engineers live and
die by interpretability! So does, in a sense, foundations of mathematics.

>I have not thought a lot about the question of whether mathematics is
>expressible in an INTERPRETED first-order language.  My guess is that it
>would be and that is part of what Harvey's intuition rests on.

I'm not sure what you mean by an INTERPRETED first-order language, but if
it is what Randy and I have in mind, then I of course concur.

>However, Harvey later goes on to write:
>
>>	So the present way of setting this up, which goes to the other tidy
>>extreme, is to just have sets and membership (and equality). Despite the
>>undeniable fact that this has its unsatisfactory aspects, it is quite
>>satisfactory, simple, and neat, and quite adequate for a great many
>>purposes. Furthermore, nothing comparable has been proposed and developed
>>that does the same things so well.

>To forestall this was the whole point of my remarks about modeling.  Set
>theory gives us a way to MODEL many of the objects of mathematics, but it
>is only that: modeling.  Modeling has its uses but one should never forget
>the difference between the model and the things being modeled.

The engineers realize that they are not going to get anywhere talking like
this, and so they settle on particular modelings - i.e., standards. Saying
that the real thing is not the same as a MODEL is, to me, from a productive
point of view, the same as saying that we should keep our mind open and be
flexible in order to perhaps create better models.

> I spend a
>lot of my life building mathematical models, myself, and don't mean to
>belittle them.  I think the offer a lot of insights.  But in my experience,
>confusing models with reality only breeds confusion.  Modeling mathematics
>in set theory has its uses, and its insights.  The existence of
>non-standard models of various sorts shows us that it also has it
>shortcomings.  But even without these non-standard models, if we were to
>work in some higher-order set theory, say, we would be well advised not to
>confuse the two. I think Harvey and I will just have to disagree about this.

I think you might be using the term "model" in two different ways in your
e-mail. I think of set theory as, among other things, modeling number
systems. I don't see what this has to do with models of set theory. From
this point of view, the adequacy or inadequacy of modeling number systems
in set theory doesn't seem to me to have anything to do with the
mathematical fact that there are nonstandard models of set theory.

>Harvey:
>>The real dichotomy to me is: is there a
>>precise syntax and a precise finite set of axioms and rules of inference
>>that are used to model mathematical reasoning?
>
>While I don't think it is THE question, I do think that is a very
>interesting one.  My bet is that the answer is "NO".  If the rules of
>inference are finitary, and so subject to the compactness phenomenon, I
>don't see how there can be.  More importantly, though, I think there will
>always be room for creative insights that lead to new constructions [give a
>nod in Vaughan's direction here] and with these constructions new methods
>of proof.  In other words, I suspect that mathematics is open-ended in a
>way that no fixed finite set of axioms and rules of inference is.  (Isn't
>that what the search for higher infinities is about?)

This is a point of view that I suspect many mathematicians have in order to
justify their rejection of the current foundation for mathematics. This
point of view is, in all due respects, out of touch with the observed
development of mathematics over two thousand years. How do you account for
the fact that every paper in the Annals of Mathematics for the last fifty
years in group theory, representation theory, algebraic geometry, analytic
number theory, algebraic number theory, differential topology, differential
geometry, Lie groups, harmonic analysis, complex variables, several complex
variables, etcetera, is trivially and routinely translatable into a tiny
fragment of ZFC?

And do you think that they may have used a Measurable Cardinal in one of
these papers without the referree or author noticing it becuase it was
simply the result of "a creative insight" and a "new method of proof?"
People dining at Chinese Restaurants generally notice adult elephants
running around the premises.

>So in summary, my position is a consequence of my views about interpreted
>versus uninterpreted languages, and my convictions about mathematical
>modeling.  I understand that these are not universally shared, but I am not
>sure just how heretical they will seem to the readers of this distribution
>list.

Jon, I think many of us will want to hear more details and examples about
the distinction you draw between "interpreted" versus "uninterpreted"
languages, in order to judge how heretical your views really are.

Thank you for your contribution to fom. Hope you find the time to give us more.

Your conservative friend, who defends the status quo --- HMF.

PS: Maybe I'm getting old.