Tom, - ha ha, I would have asked the same stupid question, because a "poisitve" integer is just so the product of -1 and the NEGATIVE of the integer plus the (positive) integer itself and 1. You did not want that either. I think a better restriction is in order, but let me stop here. I don't want to start math 101. Sorry for having been facetious.

John --- [EMAIL PROTECTED] wrote: > > 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: everything-list@googlegroups.com > 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: 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/ > === message truncated === --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. 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