Dear John, If we grant your point that:

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> So while the natural numbers and the integers have a rich internal structure > (rich enough to contain the whole universe and more, according to most > subscribers on this list, I suspect), the reals can be encoded in the single > 'program' of tossing a coin. How do you distinguish the generation of the Reals from the "'program' of tossing a coin"? Are they one and the same? If so, I can go along with that, but what about "complex numbers"? The main problem that I have with your reasoning is that it seems to conflate objective existence (independent of implementation or representation) with representable existence, the latter being those that can be known by finite entities, such as us humans (or Machines pretending to be humans). Your reasoning also neglects the "meaningfulness" of the NP-Complete problem. Kindest regards, Stephen ----- Original Message ----- From: "John Collins" <[EMAIL PROTECTED]> To: "Stephen Paul King" <[EMAIL PROTECTED]>; <[EMAIL PROTECTED]> Sent: Thursday, January 22, 2004 6:02 AM Subject: Re: Is the universe computable > > ----- Original Message ----- > From: "Stephen Paul King" <[EMAIL PROTECTED]> > To: <[EMAIL PROTECTED]> > Cc: <[EMAIL PROTECTED]>; <[EMAIL PROTECTED]> > Sent: Wednesday, January 21, 2004 5:39 PM > Subject: Re: Is the universe computable > > SPK wrote: > > > You are confussing the postential existence > of a computation with its "meaningfulness". But in the last time you are > getting close to my thesis. We should not take the a priori existence of, > for example, answers to NP-Complete problems to have more "ontological > weight" than those that enter into what it takes for "creatures like us" to > "view" the answers. This is more the realm of theology than mathematics. ;-) > > > > ..This is rather like an argument I like to put forward when I'm feeling > outrageous, and one which I've eventually come to believe: That the real > number line 'does not exist.' There are only countably many numbers you > could give a finite description of, even with a universal computer (which > the human mathematical community probably constitutes, assuming we don't die > out), and in the end the rest of the real numbers result from randomly > choosing binary digits to be zero or one (see eg. anything by G. Chaitin). > So while the natural numbers and the integers have a rich internal structure > (rich enough to contain the whole universe and more, according to most > subscribers on this list, I suspect), the reals can be encoded in the single > 'program' of tossing a coin. By this I mean that the only 'use' or 'meaning' > you could extract from some part of the binary representation would be of > the form 'is this list of 0s and 1s the same as some pre-chosen lis of 0s > and 1s?', which just takes you back to the random number choosing program > you used to create the reals in the first place. > -- Chris Collins