> On 16 Dec 2018, at 02:42, Jason Resch <jasonre...@gmail.com> wrote: > > > > On Sat, Dec 15, 2018 at 7:39 PM Brent Meeker <meeke...@verizon.net > <mailto:meeke...@verizon.net>> wrote: > > > On 12/15/2018 2:58 PM, Jason Resch wrote: >> >> >> On Saturday, December 15, 2018, Brent Meeker <meeke...@verizon.net >> <mailto:meeke...@verizon.net>> wrote: >> >> >> On 12/15/2018 7:43 AM, Jason Resch wrote: >>> >>> >>> On Sat, Dec 15, 2018 at 1:09 AM Brent Meeker <meeke...@verizon.net >>> <mailto:meeke...@verizon.net>> wrote: >>> >>> >>> On 12/14/2018 7:31 PM, Jason Resch wrote: >>>> On Fri, Dec 14, 2018 at 8:43 PM Brent Meeker <meeke...@verizon.net >>>> <mailto:meeke...@verizon.net>> 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.
Indeed. None of my book on number theory provides any axioms for it. Only logicians introduce axioms, which provides just machine, because they are interested in the relation between those machine/theory and truth, not in arithmetic. People confuse easily a theory (seen here as a machine) and the reality partially explored by such machine. Like you said (roughly) that is confusing Hubble telescope and the galaxies. Bruno > > 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 everything-list+unsubscr...@googlegroups.com > <mailto:everything-list+unsubscr...@googlegroups.com>. > To post to this group, send email to everything-list@googlegroups.com > <mailto:everything-list@googlegroups.com>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <https://groups.google.com/d/optout>. -- 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 everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
