From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: Report on Philosophies of Physicists
Organization: Computer Science Department,  Stanford University.
Date: Tue, 15 Sep 1992 06:02:13 GMT

In article <TORKEL.92Sep14100615@isis.sics.se> torkel@sics.se (Torkel Franzen) writes:
>If you (consistently) add to ZFC an axiom saying that a
>certain Diophantine equation has a solution (and thus, perhaps,
>implying that lots of interesting Diophantine equations have solutions), why
>should I, as a mathematician, be interested in the resulting theory?

Well, maybe you'd like a Nobel prize in physics.  Here's how to get one
using foundations.

Trying to capture the discreteness of quantum mechanics in a natural
way, Ticklehammer has spent most of the 1980's and early 1990's trying
to come up with an integer-coefficient polynomial in eight variables
having formal properties that make it satisfy a certain integer version
of Dirac's equation involving partial difference operators in place of
partial derivatives.  In this model the roots of the polynomial turn
out to be states (or some such concrete thing that any model needs to
have to make sense--you needn't take this too literally, I'm only
trying to make a foundational point here).  The integers are of course
critical, since this is intended to be a discrete theory with
everything having only discrete solutions.

At JIPC'94 Ticklehammer announces a polynomial with these properties.
Great excitement, both the professional and popular press are full of
reports on Ticklehammer's polynomial, affectionately nicknamed Tick,
and in the ensuing months many amazing properties of Tick are found and
reported.  Integer quantum mechanics based on Tick seems to be taking
shape.

Then at JIPC'95 the other shoe falls.  Squozz proves an amazing
property of Tick: for any countable cardinal b it is undecidable
whether Tick has b roots.  Uproar: everyone realizes at once that this
means Ticklehammer's theory might have no model, not in the
mathematical sense that it is inconsistent but in the physical sense
that it may have no states.  (How could this discovery have taken so
long, the newspapers wanted to know?  People replied that it just never
occurred to them to actually *look* for an example of a state, everyone
had been spending all their time investigating formal properties of
Tick, using it to make predictions of measurements, and comparing the
model's behavior with that of Hilbert space and path integral models.
One group said they had recently written a code to plot phase spaces
but it was unexpectedly having trouble plotting even the first point,
which Squozz's result now seemed to explain.)

Now this is where you come in.  Since the question at hand is
undecidable you take the bold step of simply deciding it on your own
initiative.  You decide you want aleph-null states, so you simply
postulate that Tick has that many solutions, call this postulate T.
You proceed to work in ZFC+T.

Your resulting theory is consistent.  Yet you have not harmed a single
fact among the hundreds that people have discovered about
Ticklehammer's polynomial in the past year.  And all the consequences
of the discreteness of Ticklehammer's charming setup are preserved.  It
remains the discrete theory that Ticklehammer envisaged from the
beginning.  Moreover this is also true for future researchers: the fact
that things are now happening in ZFC+T doesn't prevent you from
ignoring T and just working in good old ZFC, except whenever you
actually need T at some point.

That was the easy part, no prize yet.  Now you're going to have to sell
this to physicists who aren't going to like your playing mathematical
God in this way one little bit.  "Cheap logician's shot," they'll say.

This is about the point where Abraham Robinson was, halfway through his
development of nonstandard analysis.  Cauchy had claimed that his
infinitesimal-based treatment of calculus was consistent, but did not
have the logical tools to either state or prove what was intuitively
obvious to him.  Robinson observed formally that simply postulating
infinitesimals was consistent with analysis, proving that Cauchy was
right.  But how to sell it to mathematicians, and ultimately to
freshman calculus classes, without making first order logic a
prerequisite and forcing everybody to write all their proofs in a
formal system.

Answer: find and describe a natural model of analysis, not the standard
one, that could nevertheless be described in standard terminology.
Various models are possible, e.g. countable models might be thought of
as small and therefore attractive.  However the conceptually most
natural and simplest approach seemed to be to add "nonstandard reals."
Though "nonstandard," these still behave in all respects exactly like
ordinary reals as far as analysis is concerned.  This expanded model of
analysis is called nonstandard analysis, not a very user-friendly name,
they ought to call it something like Cauchy analysis or take back the
name infinitesimal calculus.

Now this is where you get your Nobel prize.  Like Robinson you work out
a natural kind of nonstandard integer to add to the standard model,
call these "Ticklehammer integers."  These will have infinitely many
predecessors in your model, as viewed in ZFC.  (This is always how a
nonstandard model of the ZFC integers looks when viewed in the standard
model.)  Ticklehammer's setup won't be at all perturbed about this
because the meaning of "infinity" within ZFC+T will have changed
relative to its meaning in the standard model of the integers so as to
allow the set of integers below each nonstandard integer to appear to
be finite and hence indistinguishable in all *visible* respects from
any ordinary integer.

Now you have a more intricate notion of "Ticklehammer integer" to
explain to people.  But it may well be no worse than Hilbert space,
which isn't as simple as the integers either or they'd teach it in high
school.  Since your nonstandard integers aren't that complicated
there's a fair chance they'll teach yours in high school in the not too
distant future.

Two years later you and Ticklehammer triumph at Stockholm.  You are
cited for having developed the mathematical foundations that made
Ticklehammer's theory physically meaningful.

Now to argue that ZFC+T is not a foundations for physics you will have
to first argue that Robinson's nonstandard analysis is not a
foundations for elementary calculus.  If you want to argue that ZFC+T
is not a foundations for mathematics, go argue that in 1995 with the
burgeoning cottage industry of applied mathematicians out there
exploring the amazing new world of ZFC+T.  But not now, please, I have
a paper to write.

-- 
======================================================| God found the positive
Vaughan Pratt   pratt@cs.Stanford.EDU   415-494-2545  | integers, zero was
======================================================| there when He arrived.