FOM: Mathematics, the world of experience, and the concept of proof

[Jon Barwise]

I have been tempted to jump into this interesting discussion, but nothing
gave me the quite opening for what I wanted to say.  But yesterday Harvey
wrote, among other things:
>
>The informal concept of integer, or at least of natural number,
>speaks to everyone, and is fundamental to everybody's understanding of the
>world. Integers occur in bank accounts, baseball statistics, salaries, date
>of birth, etcetera; and this is even with the ordering. E.g., your salary
>is higher than mine. Also the idea of zero - e.g., zero salary - is
>universally familiar.
>

This begins to get at what to me is a fascinating question in the FOM:
	How does mathematics connect to the everyday world
	of direct experience?
I take it that there are things like numbers, functions, sets, and the
like.  I further take it that these objects are somehow grounded in our
direct experience of the world, and it is this grounding which accounts for
the applicability of mathematics to the world.  I would like to better
understand this grounding process, what it says about mathematical activity
and about the nature of applied mathematics.

How is this related to what others have been calling FOM in this
discussion?  I suggest that
	the more basic concepts of mathematics are those that have the
	more direct connection with the world of ordinary experience.
That is why I mention numbers, sets, and functions, but I should also
mention more geometric notions related to shape, and things connected with
differential equations, e.g., time and rates of change.

A reason for mentioning numbers, functions, and sets is that they are
implicated in the semantics of ordinary language.  All human languages have
ways of talking, for example, about functions; e.g.,.  ``There is no way to
assign the guests to the rooms that does not have at least three guests
sharing one room'' is, in logical terms, expressing a universal
quantification over functions.  (Also relevant here is Boolos' discussion
of plurals and second order logic.) (It was the realization that ordinary
language allows us to talk about basic mathematical objects that first
seduced me into thinking about the semantics of natural language some years
ago.)  Let me make a second proposal:
	the more basic concepts of mathematics are those that
	are more directly expressed in human languages.

This proposal is not intended to be a definition of the basic concepts of
mathematics, but rather an empirical hypothesis about them.  What makes it
plausible is the hunch that the concepts of mathematics that are most
directly grounded in our experience of the world would, for language
evolution reasons, be most likely have made it into our ordinary ways of
talking about the world.

(Application to Harvey's question about the number 3: Human languages allow
us to talk both about 3 people but also about the third person in a row.
There is no obvious reason to suppose that either the cardinal or the
ordinal notion is more basic than the other.  They are both very directly
grounded in our experience of the world.  They are related in various ways.
But neither needs to be lower than the other in the "more-basic-than"
ordering.)

Discussions of foundations often get sidetracked by reductionism: the
doctrine that everything must be reduced to something else that is
ultimately reduced to some primitives, perhaps sets.  One used, for
example, to see a lot of comments to the effect that set theory (maybe ZFC)
provides "the foundations" for mathematics.  This doctrine seems often to
be based on an elementary fallacy, namely, confusing a model of something
with the thing models.  (I am not suggesting that anyone part of this
discussion has made this fallacy, of course!) Only a child would confuse a
model of the solar system with the solar system, but even adults sometimes
seem to confuse models of the natural numbers with the natural numbers.
The number 3 is, among other things, the number of fingers I am holding up
now (trust me).  In ZFC it is usually modeled by a certain set, the set
denoted by, well, spare me writing out the correct sequence of {'s and }'s.
So too, 2/3 is a certain ratio, or fraction, say the fraction of my three
fingers I have just put down.  It is usually modeled in ZFC by a certain
equivalence class of ordered pairs of integers.  These may be good or poor
models but they certainly are not identities.  (See the first half of
Tait's "Truth and Proof: The Platonism of Mathematics," Synthese 69 (1986):
341-370.) The reason that this is relevant to the current discussion is
that the ability to model one thing by another in no way shows that the
latter if more basic than the former.  For example, we all know how to
model the solar system by balls and wires and electric motors.  This does
not make the latter more basic than the former.  We also know how to model
functions by sets (of ordered pairs) and sets by (characteristic)
functions.  This does not make each more basic then the other.

This leads me to Hilary's suggestion to think about proofs.  This seems to
me one of the most interesting foundational issues, at least for logic, if
not for mathematics.  What are proofs, anyway?

Proofs are surely part of the world of (mathematical) experience,
mathematicians proving things to convince themselves and others of the
truth of various mathematical propositions.  How are these proofs related
to the various models of proof developed over the years in logic, say some
notion of first-order proof?  Just how good are these models?

Martin Davis once used "Hilbert's Thesis" to name the claim that the
informal concept of proof is adequately modeled by the first-order model of
proof (though he did not put it quite this way).  This thesis is analogous
in status to Church's Thesis.  It is not a mathematical conjecture, subject
to proof, but it is surely a foundational question, one subject to rational
inquiry.  I don't recall whether Davis subscribed to Hilbert's thesis or
not.  My own belief is that the thesis is false.  Mathematical reasoning is
often just not first-order.  (And no, Neil, its is not rules of inference
that determine what counts as a proof in ordinary mathematics.  All it
takes is a clear preservation of truth.  Any argument that does that is
fine.) Another reason for thinking Hilbert's Thesis is false is that there
are many aspects of real mathematical proofs that are ignored in the
first-order model as witnessed by the fact that it is so blasted difficult
to formalize much of mathematics in first-order logic.  (In the most recent
issue of Linguistics and Philosophy, Joseph Almog has an interesting
article which can be read as an attack on Hilbert's Thesis.)

It sometimes get in a mood where I find it a bit shocking that we have as
little to say about mathematical proofs as we do.  This is not to belittle
the work in proof theory, which has flourished as a part of mathematics in
recent years.  Rather, what I can find shocking in this mood is that we
have not done more with the task of trying to understand the relation of
the informal concept of proof to various formal models of proof, and with
it the task of trying to improve the our formal accounts to better model
the informal concept used in mathematics.  Perhaps the reason we have not
done more is that it is not clear to what extent the task is a logical one,
rather than philosophical, cognitive, sociological, or heaven knows what
else.  But still I hope to see a day when there is what I would consider a
rich "theory of proofs".  The theory may not be part of mathematics, or
even of the FOM, but it should greatly enrich the FOM and our discussions
of mathematical activity.

Best, Jon




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Jon Barwise
Professor of Mathematics, Philosophy, and Computer Science

Dept of Mathematics   	URL: http://www.phil.indiana.edu/~barwise/barwise.html
Indiana University 	Phones: (812) 855-2054, 323-8552, 323-7633 (evening)
Bloomington, IN 	Fax: (812) 855-0046
47405
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