On Sun, Dec 16, 2018 at 10:27 PM Brent Meeker <meeke...@verizon.net> wrote:
> > > On 12/16/2018 4:43 PM, Jason Resch wrote: > > > > On Sun, Dec 16, 2018 at 6:02 PM Brent Meeker <meeke...@verizon.net> wrote: > >> >> >> On 12/16/2018 2:04 PM, Jason Resch wrote: >> >> >> >> On Sun, Dec 16, 2018 at 4:01 PM Bruce Kellett <bhkellet...@gmail.com> >> wrote: >> >>> On Mon, Dec 17, 2018 at 8:56 AM Jason Resch <jasonre...@gmail.com> >>> wrote: >>> >>>> On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker <meeke...@verizon.net> >>>> wrote: >>>> >>>>> >>>>> But a system that is consistent can also prove a statement that is >>>>> false: >>>>> >>>>> axiom 1: Trump is a genius. >>>>> axiom 2: Trump is stable. >>>>> >>>>> theorem: Trump is a stable genius. >>>>> >>>> >>>> So how is this different from flawed physical theories? >>>> >>> >>> Physical theories do not claim to prove theorems - they are not systems >>> of axioms and theorems. Attempts to recast physics in this form have always >>> failed. >>> >>> >> Physical theories claim to describe models of reality. You can have a >> fully consistent physical theory that nevertheless fails to accurately >> describe the physical world, or is an incomplete description of the >> physical world. Likewise, you can have an axiomatic system that is >> consistent, but fails to accurately describe the integers, or is less >> complete than we would like. >> >> >> But it still has theorems. And no matter what the theory is, even if it >> describes the integers (another mathematical abstraction), it will fail to >> describe other things. >> >> ISTM that the usefulness of mathematics is that it's identical with its >> theories...it's not intended to describe something else. >> > > A useful set of axioms (a mathematical theory, if you will) will > accurately describe arithmetical truth. E.g., it will provide us the means > to determine the behavior of a large number of Turing machines, or whether > or not a given equation has a solution. The world of mathematical truth is > what we are trying to describe. We want to know whether there is a biggest > twin prime or not, for example. There either is or isn't a biggest twin > prime. Our theories will either succeed or fail to include such truths as > theorems. > > > This is begging the question. You taking one piece of mathematics, > arithmetic, and using it as a theory describing another piece of > mathematics, Turing machines. And then you're calling a successful > description "true". But all you're showing is that one contains the > other. > I'm not following here. > Theorems are not "truths" except in the conditional sense that it is true > that they follow from the axioms and the rules of inference. > I agree a theorem is not the same as a truth. Truth is independent of some statement being provable in some system. Truth is objective. If a system of axioms is sound and consistent, then a theorem in that system is a truth. But we can never be sure that system is sound and consistent (just like we can never know if our physical theories reflect the physical reality they attempt to capture). 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. 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.
