Bruno,

To help us understand this:  How is this different from saying the toss 
of a coin is both unpredictable and yet determined by laws?

Another thought is that there are the two extremes of the meaning of 
"law":

1) The reductionist definition that something can be predicted by the 
sum of atomic parts and rules.
With the primes it is the integers and addition and multiplication.  
With a coin supposedly it is "atoms" and the laws of physics.
2) The statistical definition that something follows a certain 
distribution over many trials.
With the primes it would be the prime number theorem or more precise 
bounds on the distribution of the primes.  With a coin it would be the 
binomial distribution.

This brought up another thought.  The definition of the primes is a 
negative definition, an integer having no factors other than 1 and 
itself.  Of course this is what makes it difficult to determine if a 
large number is prime.  But is there something about a negative 
definition that sets us up for... what... not being able to understand 
something?  This also reminds me of the diagonalization process, 
defining something by saying it is not something else, like Chaitin 
does with his Omega, and of course Cantor with the reals (resulting in 
the mystery of the continuum hypothesis).  Another famous negative 
definition is that of infinity, which causes so many weirdnesses in 
divergent series, and talking about the multiverse, etc.

Perhaps free will is such a mytery because it can be defined only 
negatively.  Free from what?

Tom

-----Original Message-----
From: Bruno Marchal <[EMAIL PROTECTED]>
To: FoR <[EMAIL PROTECTED]>
Cc: everything-list@googlegroups.com
Sent: Tue, 4 Apr 2006 17:42:03 +0200
Subject: Do prime numbers have free will?

Hi,

I love so much this citation (often quoted) of D. Zagier, which seems
to me to describe so well what is peculiar with ... humans, which
behaviors are simultaneously completely determinated by numbers/math or
waves/physics and at the same time are so much rich and unpredictible.
I find instructive to see that primes already behaves like that ....


"There are two facts about the distribution of prime numbers of which I
hope to convince you so overwhelmingly that they will be permanently
engraved in your hearts. The first is that, despite their simple
definition and role as the building blocks of the natural numbers, the
prime numbers...grow like weeds among the natural numbers, seeming to
obey no other law than that of chance, and nobody can predict where the
next one will sprout. The second fact is even more astonishing, for it
states just the opposite: that the prime numbers exhibit stunning
regularity, that there are laws governing their behaviour, and that
they obey these laws with almost military precision."




Bruno


http://iridia.ulb.ac.be/~marchal/



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