FOM: Response to Harvey
barwise at phil.indiana.edu
Mon Oct 27 10:43:34 EST 1997
Harvey asks me to explain what I meant when I wrote:
>>Mathematical reasoning is often just not first-order.
Well, I confess to having been sloppy in this formulation. 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.
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.
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.
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.
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. 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.
>In summary, I think that the first order/not first order dichotomy is a
>false and misleading dichotomy.
You are right, it can be a misleading dichotomy. I think the real
dichotomy is between interpreted and uninterpreted (or partially
>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?)
Harvey then goes on:
>Certainly ZFC meets these criteria.
I certainly don't agree -- for the various reasons mentioned above.
(Harvey, have you forgotten what you wrote long ago in your underground
polemic about justifications of the axiom of replacement and actions of the
digestive tract? It is a different issue, but it makes me wonder if you
really mean what you write here, or if you too are being just a bit sloppy.
If so, you are forgiven. But if you have changed your mind about
replacement, I would love to hear about it.)
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
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