On Sat, Dec 15, 2018 at 7:39 PM Brent Meeker <[email protected]> wrote:
> > > On 12/15/2018 2:58 PM, Jason Resch wrote: > > > > On Saturday, December 15, 2018, Brent Meeker <[email protected]> wrote: > >> >> >> On 12/15/2018 7:43 AM, Jason Resch wrote: >> >> >> >> On Sat, Dec 15, 2018 at 1:09 AM Brent Meeker <[email protected]> >> wrote: >> >>> >>> >>> On 12/14/2018 7:31 PM, Jason Resch wrote: >>> >>> On Fri, Dec 14, 2018 at 8:43 PM Brent Meeker <[email protected]> >>> wrote: >>> >>>> Yes, you create a whole theology around not all truths are provable. >>>> But you ignore that what is false is also provable. Provable is only >>>> relative to axioms. >>>> >>>> >>> 1. Do you agree a Turing machine will either halt or not? >>> >>> 2. Do you agree that no finite set of axioms has the power to prove >>> whether or not any given Turing machine will halt or not? >>> >>> >>> 3. What does this tell us about the relationship between truth, proofs, >>> and axioms? >>> >>> >>> What do you think it tells us. Does it tell us that a false axiom will >>> not allow proof of a false proposition? >>> >> >> It tells us mathematical truth is objective and doesn't come from axioms. >> Axioms are like physical theories, we can test them and refute them if they >> lead to predictions that are demonstrably false. E.g., if they predict a >> Turing machine will not halt, but it does, then we can reject that axiom as >> an incorrect theory of mathematical truth. Similarly, we might find axioms >> that allow us to prove more things than some weaker set of axioms, thereby >> building a better theory, but we have no mechanical way of doing this. In >> that way it is like doing science, and requires trial and error, comparing >> our theories with our observations, etc. >> >> >> Fine, except you've had to quailfy it as "mathematical truth", meaning >> that it is relative to the axioms defining the Turning machine. Remember a >> Turing machine isn't a real device. >> >> Brent >> > > > Ahh, but diophantine equations only need integers, addition, and > multiplication, and can define any computable function. Therefore the > question of whether or not some diophantine equation has a solution can be > made equivalent to the question of whether some Turing machine halts. So > you face this problem of getting at all the truth once you can define > integers, addition and multiplication. > > > There's no surprise that you can't get at all true statements about a > system that is defined to be infinite. > But you can always prove more true statements with a better system of axioms. So clearly the axioms are not the driving force behind truth. 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 https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
