On Thu, Jan 22, 2015 at 3:28 PM, John Clark <[email protected]> wrote:
> > > On Thu, Jan 22, 2015 at 6:51 AM, Jason Resch <[email protected]> wrote: > > >> >> >> So one of them is true, but can you (or anyone in this universe) prove: >> >> the 10^(10^(10^100))th decimal digit of pi is 0 ? >> the 10^(10^(10^100))th decimal digit of pi is 1 ? >> the 10^(10^(10^100))th decimal digit of pi is 2 ? >> the 10^(10^(10^100))th decimal digit of pi is 3 ? >> the 10^(10^(10^100))th decimal digit of pi is 4 ? >> the 10^(10^(10^100))th decimal digit of pi is 5 ? >> the 10^(10^(10^100))th decimal digit of pi is 6 ? >> the 10^(10^(10^100))th decimal digit of pi is 7 ? >> the 10^(10^(10^100))th decimal digit of pi is 8 ? >> the 10^(10^(10^100))th decimal digit of pi is 9 ? >> >> >If you answer no to all 10 of those questions, then none of those >> statements is provable by any entity operating within this universe, yet we >> know one of the statements is true. >> > > Do we know that? Do we know that such a digit exists? > It follows from the axioms that there is a certain definite digit. Or do you propose there is some last digit of Pi which varies from place to place according to the available local computing resources of one's local environment? (Whether anyone ever bothers to compute it or not?) > If mathematics is more fundamental then it does, if physics is more > fundamental then it does not. > Neither has to be more fundamental than the other. Mathematics only needs to have an independent existence. > > > So Pi is a mathematical object with properties that don't depend on the >> physical existence of conceptions/proofs realized by entities or processes >> operating physically. >> > > Existence is a property and the existence of that digit may depend on > physical processes, or it might not, we don't know. > If it does. Otherwise this leads to contradictions and violations of common every-day axioms. (Which would also follow from the more general rejection of the idea that one can always add one to any integer to get a bigger integer.) So either one must say mathematics is independent of physics, or accept some ultrafinitism philosophy of mathematics which is incompatible with existing axiomatic systems. > > > It follows then that if these properties don't depend on physical >> processes of this universe >> > > All you're doing is asserting what you're trying to prove. > > I'm merely showing the consequences that result from the either of the two choices one is faced with when they accept or reject the idea that such a physically incomputable digit has a definite value. Here is another example to ponder: I find two prime numbers A and B, each about a million digits long, multiply them together to get a composite number C, write down C, then throw the computer used to generate those A and B into a black whole which won't evaporate until long after all protons in the universe have decayed. The number C is so large it can't be factored in the life of the universe. Do you believe A and B have definite values despite our inability to compute them? (For the purposes of this thought experiment let's assume no significant speed ups in factoring numbers are possible) Jason -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
