From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: Computability of the universe
Organization: Computer Science Department,  Stanford University.
Date: Fri, 25 Sep 1992 06:56:28 GMT

In article <1992Sep23.094901.1@sscvx1.ssc.gov> doctorj@sscvx1.ssc.gov writes:
>pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>How about: Physics is science, while math is not.  By science, I mean 
>the sequence:
>  1: Look at the existing data and formulate a theory which explains it.
>  2: Use that theory to predict new results.
>  3: Perform experiments (ie, collect more data) to test the theory.
>  4: Go to 1.
>Math does not follow this sequence.  It does not contain falsifiable theories
>in the way that physics does.
>...
>This use of the word "experiment" is ill conceived.  Computer calculations 
>are not experiments.  Neither are slide rule or hand calculations.

Let me not dispute your sequence here, but rather opine:

(A) Math is perfectly capable of following your sequence.

(B) Hypotheses can be falsified in math.

(C) Falsification isn't all it's cracked up to be in either math or
physics.

Here are my arguments for each of these.

------------------

(A) Consider the number of ways to fold n stamps.  It is known (as a
mathematically proven fact) that this is asymptotically more than c n^k
and less than C n^K for irrelevant constants c,C and certain k,K.  I'm
not sure what k and K are at the moment but k=11 and K=13 are not too
far off the mark.

In 1989 Don Knuth and I experimentally estimated the actual exponent to
be in the neighborhood of 12.2633, based on the exact values for
folding a *loop* of n stamps, n<18 (loops are easier to count and the
exponent is known to be the same).  We later learned that W. Lunnon at
Cardiff had much earlier, in an unpublished 1981 paper still sitting on
his desk, estimated the exponent at 12.2626.  Our estimate was based on
a little more data than his (faster computers by then), so we felt our
estimate had more justification.  But either way you can see that we
were independently homing in on the same answer on the basis of data.
We had more data, hence closer estimates.

Not everyone interested in this problem was impressed by our estimate,
so I made the following offer to one of them.  "Can I interest you or
[xyzzy] in even odds on its lying in (12.262,12.265)?"  He didn't take
me up on it, which I took as a sign that he didn't completely distrust
our methods but didn't want to admit it.  We had no rigorous basis for
our exponent, and indeed the truth could well turn out to be say 12.5
without thereby entailing any inconsistency in mathematics.
Nevertheless I feel extremely confident about (12.26,12.27) (better
than 100:1 odds, so I wasn't really being fair to him about the even
odds for (12.262,12.265)).

Now what would you call our approximate determination of the exponent?
Let's look at Webster's definition of "experiment":

# an operation carried out under controlled conditions in order to
# discover an unknown effect or law, to test or establish a hypothesis,
# or to illustrate a known law

We were trying to determine the general law followed by the sequence.
We constructed several hypotheses about what the coefficients of a
formula might be, and tested them against the data to get some idea of
the robustness of the different formulae being used to model the
sequence.

Does not our activity fall well within Webster's requirements for an
experiment?

------------------

(B,C) Consider the following two remarkably parallel stories, both
beginning with Newton.  The first concerns the mathematical existence of
infinitesimals and infinities, the second the physical existence of
light particles and light waves.

Newton begat the calculus.  Whereas Newton saw from the outset that
limits at infinity offered more sanity than infinitesimals, Leibniz,
who came later but published first, preferred infinitesimals, whose
treacherous logic let him down only occassionally.  But subsequent
generations could not bring themselves to trust infinitesimals as more
than metaphysical motivation for limits.  The most notable exception
was Cauchy, but even he however later joined and contributed greatly to
Weierstrass's epsilon-delta program formalizing the notion of limit.
And when Dedekind defined reals as rational cuts in 1872 the door of
mathematical rigor appeared to have closed forever on infinitesimals.
It was not until nonstandard analysis reconciled infinitesimals and
infinities in 1961 through the remarkable insight of Abraham Robinson
that harmony was achieved.  Calculus texts today based on this approach
use whichever of infinitesimals or epsilon-delta is more appropriate to
the task at hand.

Newton unwittingly authorized (but did not beget) the corpuscular
theory of light.  His mildly expressed preference for particles over
waves was turned by others into a century of dogma that was finally
relegated to metaphysics by Young, Fresnel, and Fizeau, whose
investigations appeared to show conclusively that the observed
phenomena of diffraction and interference were compatible only with
Huygens wave model.  The rout of particles was so complete that, a
century later, even Planck as the father of the quantum could not
accept Einstein's inference from the photoelectric effect that light
must have a corpuscular character after all.  It was not until quantum
mechanics reconciled particles and waves in 1926 through the
extraordinary insights of Heisenberg, Schroedinger, and Born that
harmony was achieved.  Descriptions of quantum phenomena today use
whichever of particles or waves more naturally describe the situation
at hand.

Now what I claimed in (B) was that hypotheses can be falsified in
math.  Here Liebniz' hypothesis of the mathematical existence of
infinitesimals, later endorsed by Cauchy, was eventually *falsified* by
Dedekind.  (You can *prove* via reals as cuts that no real can be an
infinitesimal, the only reals greater than 0 are too large to be
infinitesimals.)  And my evidence for (C), the impermanence of
falsification, is Robinson's restoration of infinitesimals to
mathematically rigorous existence.

The parallels with the corresponding story for particles and waves
involve both (B) and (C) with physics in place of mathematics.  These
intimate parallels makes it *very* hard to tease apart the processes of
mathematics and physics in any useful way.

When particles were outlawed, were they being banned from math or
physics?  We say physics, but it seems to me that this is more a
cultural assertion than any real difference between math and physics.
The list of differences that I gave constitutes all the important
differences I could come up with.  I still haven't seen anyone propose
a convincing fifth important difference.

-- 
=================================================== Trouble is, son
Vaughan Pratt   pratt@cs.Stanford.EDU  415-494-2545 The farther you run
=================================================== The more you feel undefined


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