On Sun, Dec 16, 2018 at 6:02 PM Brent Meeker <[email protected]> wrote:
> > > On 12/16/2018 2:04 PM, Jason Resch wrote: > > > > On Sun, Dec 16, 2018 at 4:01 PM Bruce Kellett <[email protected]> > wrote: > >> On Mon, Dec 17, 2018 at 8:56 AM Jason Resch <[email protected]> wrote: >> >>> On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker <[email protected]> >>> 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. 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.
