John, As I was typing that post, I thought about the fact that I was leaving out the word "positive" but I left it out anyway. I should have typed "a prime is a positive integer having no factors other than 1 and itself."
While I'm at it, I wanted to correct something else in the same post. Instead of 1) The reductionist definition that something can be predicted by the sum of atomic parts and rules. I think I should have said 1) The reductionist definition that something is determined by the sum of atomic parts and rules. Saying "is determined by" is theoretical and so covers the cases like with a coin when limitations and uncertainties prohibit actual prediction. I guess this is a difference between the primes and tossing a coin. But perhaps it's only a difference of degree, because when you add and multiply there is always the chance that you will make an error. But then you can repeat the computation, whereas you can't repeat the *same* coin toss. How does this relate to free will? I probably don't want to talk about that too much. There's *probably* a *deterministic* reason for that too. ;) Tom -----Original Message----- From: John M <[EMAIL PROTECTED]> To: firstname.lastname@example.org Sent: Tue, 4 Apr 2006 11:57:21 -0700 (PDT) Subject: Re: Do prime numbers have free will? Tom, I did not shoot my mouth about free will, because of my esteem for Bruno. Now, however, your definition of the primes tickled my mathematical ignorance and I ask you: IF - as you wrote, ">a prime is an integer having no factors other than >1 and itself. < (I heard that somewhere already) My question: is a 'number' the same as its negative, eg. is 2 = -2? because if not, then a prime number "p" is both equal to p.1 and 1.p, (so far so good,) but it is also p = -1.-p -- factors different from the prime itself and 1. (And please spare me of the [..] absolut values) What say you? John --- [EMAIL PROTECTED] wrote: > > 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: email@example.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/ > > > > > > --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. 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