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[Vaughan Pratt]

From: Reuben Hersh <rhersh at math.unm.edu>
>Fine.  Suppose that for the moment I agree, "LT was true whenever there
>were piles of pebbles."  There were not always pebbles.  The history
>of the cosmos as now told by cosmologists has a considerable period (I
>forget if it was hundreds of thousands, millions or billions of years)
>when its physical state did not allow matter in the solid state such as
>rocks  or pebbles to exist.
>
>So the pebbles on a Jurassic beach are a diversion.

Two remarks on this, one on each side of the main issue.

First, replacing pebbles by elementary particles, similarly configured
into squares, moves the time back to whenever particles first came
into existence.  Here cosmology is considerably more speculative:
for all we know particles of some kind or other "have always existed",
whatever that means (e.g. is "before the big bang" meaningful given what
we know about the early universe)?

So while pebbles may be a diversion, particles are less obviously so.

Second, the possibility of configuring things into arrangements of
squares bears on the SETI argument that was raised by a couple of
people in January including me, namely that one of the premises of
the SETI enterprise is that alien civilizations share our theory
of numbers, including our understanding of primes and factoring.
Any civilization that has advanced to the point of being able to perceive
such configurations is likely to have a general theory which accounts
for them, inter alia.

Here however I must side with Hersh in not leaping from those arrangements
to the conclusion that the aliens' theory of those arrangements must
be number theory as we understand it.  Why should it be?  There may
be some quite different theory, say a mathematics founded on size and
shape instead of number and set, for which the aliens perception of
this configuration of squares constitutes experimental confirmation.
Whether we view this mathematics as "incredibly complex" is beside the
point so long as the aliens obstinately persist with it.

Speculating on the possible forms alien mathematics could take may
be enjoyable, but no amount of such speculation can demonstrate that
every alien mathematics will inevitably include number theory in the
long run, because we cannot claim to have anticipated all such possible
forms ourselves.

Whether *our* mathematics is true even on the aliens' planet is meaningful
only to those civilizations that can comprehend our mathematics.  If there
exists an alien mathematics that we can't comprehend, the truth of that
mathematics on Earth is meaningless to everyone on earth: only those
aliens in possession of it can address its truth.

In this I fully concur with Carsten Butz' rhetorical question in his
remarkable 400-line posting

	http://www.math.psu.edu/simpson/fom/postings/9802.45

"(4) How can anybody claim that there is absolute truth out there?"

Granted that *our* truth is out there, but to say that our truth exists
without us has about the same content as saying that the concept of homo
sapiens exists without us.

The separability of our truths from ourselves is an illusion.  A widely
held one, to be sure, but maybe one that will be widely rejected in the
22nd century.  Or maybe not, if abandoning that illusion proves in the
long run to be too unsettling for the majority.

Vaughan Pratt