On Thu, Jan 22, 2015 at 6:21 AM, Bruce Kellett <[email protected]> wrote:
> Jason Resch wrote: > >> >> >> On Thu, Jan 22, 2015 at 12:48 AM, Bruce Kellett < >> [email protected] <mailto:[email protected]>> wrote: >> >> John Clark wrote: >> >> On 18 January 2015 at 18:27, Jason Resch <[email protected] >> <mailto:[email protected]> <mailto:[email protected] >> >> <mailto:[email protected]>>> wrote: >> >> > Do you believe that *one and only one* of the >> following >> statements is true? >> 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 >> Either you answer yes, or no to that question. If you >> answer >> yes, I don't see how you can escape mathematical realism. >> >> >> Seth Lloyd has estimated that the maximum number of >> computations that could be performed in the visible universe is >> about 10^121 operations on 10^90 bits, if this is insufficient >> to find your number is it meaningful to say pi has a >> 10^(10^(10^100))th decimal digit? I don't know, it depend on if >> mathematics gave rise to physics or physics gave rise to >> mathematics. >> >> >> Realist and constructivist approaches to mathematics do not cover >> all the possibilities. You can believe that one of the above >> statements is true without knowing which is true. It is logically >> necessary that one of the statements is true, given the meanings of >> the terms involved. This does not entail mathematical realism. >> >> >> 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. 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. >> It follows then that if these properties don't depend on physical processes >> of this universe, that even if this universe did not exist at all, those >> properties would not be affected. And from that it follows that >> mathematical properties and truth statements concerning them have an >> existence independent of physics, hence mathematical realism. >> >> Jason >> > > No, your conclusion does not follow -- unless you have some non-standard > meaning of mathematical realism. It is all quite simple, really. We have a > certain set of axioms. Those axioms encompass the definition of pi which > can be shown to be a transcendental number. These are simple consequences > of the axioms. My example above shows that there are statements that are true under some set of axioms that are physically unprovable. However, Godel went further and showed that what can be proven about numbers trancends any system of axioms. For any system of axioms there are statements that are true but not provable. So if mathematical truth is independent of physics (as my example shows) due to true but unprovable statements, then with Godel's result we must also accept that mathematical truth is independent of any set of axioms, for the same reason. Nothing at all need exist, or in any sense be /real/, for all of this to be > true. This is exactly my point. Truth does not depend on the existence of anything that's physically real. 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.
