FOM: Jurassic pebbles (more on Davis/Hersh)

[Lincoln Wallen]

Martin Davis writes: [extracted]

    Lagrange proved that every positive integer is the sume of <=4 perfect
    squares. This theorem has consequences in the material world where facts
    about positive integers can yield information about counting collections of
    material objects. In fact you note that historically, that's undoubtedly how
    the very notion of natural number arose. 

    ...

    In particular, Lagrange's theorem implies a fact about collection of pebbles
    on a beach, namely that any such pile of pebbles can be rearranged into <=4
    square arrays.

    ...

    You claim that the proposition that Lagrange's theorem was true before there
    were people is untenable.

    ...

    I claim that the fact about pebbles mentioned above was just as true of
    pebbles on a jurassic beach as on a contemporary beach.

I realise this is extracted from an exchange with Hersh, but may I
please abuse it to point out where I think some of the difficulties in
this discussion arise from, and where I think the challenges lie.

The notion that Lagrange's theorem "implies" a fact about a collection
of pebbles on a beach is supported by (at least) the activity of
"rearrangement" and "counting" referred to, both of which are (in the
words of the anthropologists) highly skilled, delicately achieved
activities, decidedly situated within particular, recognisable,
domains of activity.  That is to say what we call counting and
rearrangement depends properly on our purposes.

What stands as an acceptable "arrangement" or "counting" for the
purposes of demonstrating adherence or conseqeunce from Lagrange's
theorem must be explicitly recognised as part of what it takes to be
in a position to assert what Martin Davis asserts above.

Note: I am not calling into question the ability of any
competent/professional member of this applied mathematical-physical
realm to conduct such a demonstration, nor of our ability as members
of that realm to recognise the fact.  Hence presumably Davis' near
certainty that Hersh, if he be such a member, could hardly deny the
early statements.  [I will point out that there are other theorems of
mathematics which any one of us might be severly hard pressed to
demonstrate adherence to in the above way, even though members of
other groups would be able, after considerable instruction, to show us
exactly how "rearrange", "count", "measure", etc, to demonstrate such
adherence.]

Now, coming to the temporal arguments concerning the Jurassic beach.
Isn't it clear that in drawing the conclusion that the rearrangement
and counting can be done *in principle* on a Jurassic beach, Davis is
making a claim about the relationship between the practices of
formulating and proving mathematical theorems and some very particular
human practices of rearrangement and counting, and their relationship
with the situation as we understand it pertaining in Jurassic times.

If the whole frame of reference is conveyed by our hypothesis: that
should there be competent members present on a Jurassic beach, that
they should acceptably carry out such activites of formulating and
demonstrating adherence of the pebbles to mathematical statements,
etc., then, indeed, such a demonstration would be recognisably taking
place on a Jurassic beach.

But now we see that, rather than demonstrating the timeless nature of
mathematics, the argument rests on the presumption that a Jurassic
beach can be considered to function *for the purposes of DAvis'
argument* as a modern beach.  The assertion that the relationship
Lagrange's theorem and its pebble demonstration hold to each other
holds on a Jurassic beach equipped with sufficient machinery of the
modern beach (and its intellectual context of course) for it to be
treated as a modern beach, is surely unsurprising.  Paraphrased: if a
Jurassic beach were *to all relevant intents and purposes* the same as
a beach I can take you to now, the same things would hold.  Of course.
That is the point of stating explicitly, or implictly "all relevant
intents and purposes".

Now, I do believe that Davis is pointing to a, if not the, central
characteristic of the relationship between our mathematical and
social/physical practice.  Interesting questions are how such
intellectual and social/physical practices *are* so aligned to allow
(some would say construct) the physical world to serve as evidence for
our theorising.  How is it done?  (The conceptual order here is
deliberate.)

I do not believe that there is anything at all *necessary* about this.
I believe it is a wonderful achievement borne of experimentation in
both the intellectual and the social/physical world.

F.o.m. would seem to me to have to begin to explain *how we bring this
about*.

In this sense I can agree with Hersh's comments "that mathematics is
social"; nevertheless that recognition is far from making mathematics
entirely subjective; it encourages us to seek its nature in
unravelling exactly how it is possible for Davis to argue as he does
above about Lagrange and pebbles.  As opposed to simply *accepting*
such arguments as self-evident which is the more normal attitude.

[Discalimer: many on FOM will recognise the ideas of a mixed bag of
philosophers, anthropologists and sociologists thinking in the above,
no doubt poorly combined.  I am not suggesting any originality here.]

Lincoln Wallen
Reader in Computing Science
Oxford