[FOM] Arnon Avron's reply to me

[Patrick Caldon]

On Wed, Oct 24, 2007 at 12:15:26AM -0700, John McCarthy wrote:
> I think I agree with Martin Davis that human practice of mathematics
> is a human social activity and is sensitive to historical events, but
> the basic facts of mathematics are independent of that.  I'd like to
> be sure.
> 
> Consider Hans Freudenthal's 1960 Lincos project which proposed to
> establish a common language with an alien species by sending radio
> signals beginning with arithmetic examples, advancing to higher level
> mathematical examples, and going on to physics and even morality.
> 
> Are the mathematical facts independent of humanity so that
> Freudenthals's project might succeed?  I think yes.

Another question is whether the aliens would find the arithmetical
propositions sufficiently interesting *to them*, as opposed to trivial
artefacts of the number system.

For instance in the "Contact" film, the aliens send us prime numbers,
which we instantly recognize.  One could imagine without much effort
aliens who considered prime numbers a mathematical triviality, but
were instead obsessed with some other collection of numbers (say
Catalan numbers, or the powers of 5) and had prizes for proving
properties we consider obscure but they consider fundamental about
that sequence.  Maybe number theory itself is uninteresting and
unimportant to the aliens, and "real mathematics" is geometry.

We say that some proofs and theories contain insight and others lack
it.  This could be psychological, or it could be a property of the
proofs and formal systems themselves.  I would very much like to know
if (or the degree to which) "fundamental" or "profound" mathematical
observations and proofs are profound on account of some internal
inherent property they have (e.g. proof length, or possibly some way
of counting "distinct" proofs which show the same result, or some way
of counting the different "distinct" methods for axiomatizing the same
underlying mathematical system, or possibly some "visualizabilty"
property where the proof can be mapped to "visualizable" 3-dimensional
objects, or some notion of "generalizability" where a simple proof
about simple things can be mapped into essentially the same proof
about complex objects, or ...)

Patrick.