# FOM: Jurassic pebbles (more on Davis/Hersh)

Martin Davis martind at cs.berkeley.edu
Sun Mar 15 18:44:52 EST 1998

```Reuben Hersh recently wrote (in part):

>Martin, I thank you for complimenting my civility and friendliness,
>but I must admit you disappointed me.
>I know we disagree, but from you I expected a more serious response.
>Let me reply to four of your points, if I may.

>1)  The timeless truth of Lagrange's theorem is demonstrated by the
>possibility of arranging piles of pebbles into 1, 2, 3, or 4 squares,
>even before there were people.

>1)  Another member of the list has already commented on this.
>As I remember, Lagrange's theorem is a statement about an arbitrary
>positive integer n, or if you prefer, about all positive integers n.  I don't
>think you will find an arbitrary positive integer n or all positive
>integers n in any pile of pebbles.  That is why you can't prove
>Lagrange's theorem with pebbles.  Where do you find an arbitrary
>positive integer?  It's an idea, a concept, that is understood by many
>people.  Proving Lagrange's theorem, or even stating it or understanding
>it, is an intersubjective or social activity of such people.  Yes,
>it has applications and consequences for piles of pebbles.  No,
>it is not contained in piles of pebbles.  Calling this simple statement
>obvious, or postmodern, or relativist,  suggests that the person
>doing such name-calling doesn't wish to respond to what I am saying.
>You don't stoop to such childishness, but some others do.

I'm sorry that you found my very serious message "disappointing". Reading
the above, I can see that I evidently did not make myself clear. Let me try
to be very explicit. En route, some specific questions for you that I'll
mark with an asterisk.

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.

*Did I read you correctly that you think this last undoubted fact was not
realized before your work?

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.

*Do you agree that Lagrange's theorem has this consequence.

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

*Am I reading you correctly?

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

*Do you agree?

Since I can hardly imagine that you would dispute this, I assume your answer
is "yes". Then I ask, "How do you know?" For me, the answer is clear: it is
a consequence of Lagrange's theorem which was as true then as it is now.

The argument applies to more serious matters than pebbles on a beach.
Consider a calculation telling about an eclipse of the sun that occurred in
jurassic times.

*Do you agree that such a calculation is possible and meaningful?

As fellow materialists I assume we must agree.

Such a calculation involves the differential equations of motion of the
planets and theorems about their solution.

*Were those theorems valid in jurassic times? If not, on what basis are you
led to believe in the calculations? If yes, ...

Going further, cosmologists assume that inferences from the equations of
general relativity can be used to obtain inforamtion about the universe in
its early stages.

*Do you believe that the theorems that justify these inferences were valid
when the universe was very young? If not, how would you justify those
inferences?

Best, Martin

```