[FOM] Mathematical Experiments.

[Bill Taylor]

This essay is adapted from one I posted to sci.math some time ago.

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This essay, Mathematical Experiments, will overlap considerably with
a more general essay that I've been encouraged to write, on the essential
differences between science and mathematics.  Why some people feel the need
for me to write the more general paper, I am not sure - it seems to me
that the differences have been stated many times before, and are obvious
anyway, despite the claims of many like Lakatos (in his amusing but
not terribly convincing "Proofs and Refutations"). 
But no matter, it will come sometime, and here is a "first installment".

There is undoubtedly an experimental aspect to math; there always has
been, and probably always will be, humans being what they are. At times,
the experimental aspect has dominated, indeed has often been the *only*
aspect of note; but the abstract element which first appeared in
classical Greek times, has been an increasingly dominant aspect as
the millenia have passed, though in this cyber age we are witnessing
a slight turning of the tables.  And it is amusing to note that this
computing aspect had its origins in one of the more abstract of all
areas - the certification of *proofs*, as opposed to discoveries,
associated with the names of Frege, Russell, Hilbert, Godel, and
(especially) Turing.


However.  Experiments in math, as in science and (formerly) geography,
come in two types - exploratory and confirmatory.  Both types of experiment
seek *evidence*, but for different reasons and from differing motivations.

For the exploratory form of experiment, there is little difference in
methodology between science and math.  But for the confirmatory kind,
there is; both philosophically and practically.  The key to all such
discussion is this:-      In science, confirmatory experiments are
the ULTIMATE FORM OF VALIDATION.  There may be others - simplicity,
elegance, Occam, intelligibility, agreement with established results,
even a remnant of agreement with authority.  But ultimately all these
must give way to agreement with confirmatory experiments.  This is
the final court of appeal, and is THE distinguishing feature of science
as a whole.  The scientist, above and after all else, must go to
the physical world.

Contrariwise, the mathematician does NOT go to the physical world.
Not for validation.   He often *starts* there, and it often directs
and informs his efforts, but the physical world is in almost no way,
certainly no direct way, involved in VALIDATION of mathematical results.

For validation, the mathematician goes to the abstract world; he connects
to it not with ears and eyes and noses and hands, as to the physical world,
but (essentially) by pure thought; BUT - thought constrained and directed  
by this ultimate form of validation - the PROOF.  Not experiment, based on 
the physical world; but proof, centred in the abstract world.

There *are* confirmatory experiments in math, but these have a different
status from in science.

"Experiments" can sometimes confirm or disconfirm conjectures, but only
if those are of a pure existential or universal type.  And even then these
"experiments" are in fact little theorems on their own. To confirm that 
27 + 36 = 63 , using (say) long addition in decimal, or to confirm that
there is exactly one prime in the 90's decade, conceals a lot of little
theorems about decimal notation, divisibility and so forth, that we learned
to use long ago without learning how to prove, or even that they
needed proof.  So such theorems often now seem to be experiments
to us; indeed given human limitations in both short and long-term
memory and reasoning power, they *are* experiments.  But not at all
like science experiments, just proofs executed in experimental form.

Equipment may be used too, primarily pen and paper.  Indeed, this *must*
be used, at least to communicate results to others.  More recently
equipment in the form of published tables, and finally calculators
and computers, is much used, and it will probably not end there either!
But note, crucially, these "experimental aids" are not equipment in
the scientific sense; they do not become an integral part of the result
being tested, objects in their own right.  They remain aids, having no
essential role other than to ease our limited brains into the abstract
realm.  Even cases where computer proofs have (so far) turned out to be
indispensable, such as the proof of the four-color theorem, are of this type.
They could easily be humanly checked, but no-one wants to waste the effort.
The equipment is not part of the experiment (except in the most peripheral
way); when we use computers we are not investigating the properties of
semiconductors, or q-bits, or whatever, as a scientist would, we are
just *using* them, and indirectly.


So we have exploratory and confirmatory experiments.  The exploratory
type typically involve number crunching for number theory purpose, prime
number counts, factor sizes etc; they may also be involved with graph
theory or finite group theory, or many other topics.  And the confirmatory
type are not as in science (as explained above), but merely the collecting
of satisying evidence.    Thay have no real mathematical significance.
Such things as searches for counter-examples to the twin prime conjecture
and Goldbach's conjecture are of this type, as are the "almost-proofs" of
large-primeness by using the converse of Fermat's little theorem's.  This
latter type is as close to a confirmatory science experiment as one might get
in math, but they are still not a *validation* - that still requires a proof.

Then finally there are the so-called experiments which are not really such
at all - just the "proof aids" referred to above.  Such as the 4CT proof;
or rounding-out proofs, for theorems proved for numbers greater than some
given integer, by checking all the lower integers one by one.    It is
also possible that computer proofs may one day be used in the rounding
out the (in)famous mega-theorem on exceptional simple groups.

So, in every case there are, as I hope I've manifested, significant
differences between science experiments and mathematical ones.  And there
is yet another significant observable difference.  For the most part,
math experiments are *exact* and thus absolute.  They can be verified
in the minutest detail by other researchers.  But in science, experiments
are usually *approximations*, expected to be overtaken by later and better
approximations.  There is no "last word" on any result, confirmation by
others is at best partial or inexact, and all are liable to be superseded
in future.      Mathematical experiments are *exactly* repeatable,
science experiments are not - just repeatable in a vaguer sense of
essential similarity, rather than exactitude.  Indeed, if the set of
results of one experimenter were uttery identical to another, it would
be taken as evidence not of perfect technique, but of fraud!

And to counter an obvious complaint here, let me agree that yes, there
*are* areas in math where the experimental results are only approximate,
such as solution simulation for differential equations, investigations
into chaos and fractals, and the like.  But again note, these are essentially
"evidence-collecting" confirmatory experiments, rather similar to mere prime
number counts, and of no ultimate validating significance.  Ultimately,
to have full mathematical significance such efforts must be accompanied 
by provable bounds and the like.  So that the apparent imprecision is
covered in an *absolute* way.   Indeed, pure mathematicians may
occasionally be heard to say, 
 "Please don't use the f-word or the c-word in my hearing: fractals & chaos!"

Nor am I saying, (by my concentration of the absoluteness of mathematical
results), that mathematicians are incapable of making mistakes, even
quite profound ones!  Of course they are; they are only human after all. 
There are many cases of "theorems" apparantly proved and agreed to by
others (often for up to a decade or so!), and later found to have
subtle mistakes. But mistakes, even as yet undetected ones, are just
that - *mistakes* - not approximations or re-interpretations or cultural
dissonance that often plague some sciences.  The philosophical distinction
between a mistake, (mathematical), and a "wrong turn", (scientific),
is huge.  Mathematical statements are either RIGHT or WRONG, even if
we don't yet know which.  Such a comment in science would be seen as
unutterably naive and irrelevant.  Even such matters as the continuum
hypothesis, while viewable as neither right nor wrong in itself, are
*finalizable*, in the sense that we may determine (as we have done for CH),
whether or not they follow from various other assumptions.  Again, such
ultimate finality in science has long been abandoned as being naive,
an over-optimistic and unreasonable fancy.


So in all these ways, math experiments differ crucially and irrevocably
from experiments in science or elsewhere.     And these differences, as
alluded to above, are a reflection of the profound differences that lie
at the root of the chasm between math and science - a chasm very narrow,
but extraordinarily deep.

And these differences can be summarized by:-

abstract     v  physical;
exact        v  approximate;
completable  v  supersedable;
proof        v  experiment.

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    Bill Taylor                           W.Taylor at math.canterbury.ac.nz
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    In math we decide on the rules - then try to deduce the consequences.
    In science we observe the consequences, then try to deduce the rules.
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