On Mon, Dec 17, 2018 at 11:36 AM Jason Resch <[email protected]> wrote:
> On Sun, Dec 16, 2018 at 4:14 PM Bruce Kellett <[email protected]> > wrote: > >> On Mon, Dec 17, 2018 at 9:04 AM Jason Resch <[email protected]> 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. >>> >> >> Physical theories are models of reality -- using the word "model" in the >> physicists sense. >> >> >>> You can have a fully consistent physical theory that nevertheless fails >>> to accurately describe the physical world, >>> >> >> Like Brent's example of an axiomatic description of Trump...... >> >> >>> 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. >>> >> >> Axiomatic system are always going to fail to capture everything we would >> like to capture about any domain. That is why attempted axiomatisation of >> physics have been rather unsuccessful. >> >> >>> It is a completely analogous situation. If you hold the physical reality >>> is real because we can study it objectively and refine our understanding of >>> it through observations, >>> >> >> That is not "why" I hold the physical world to be real. I take the >> physical world to be real because that is the definition of reality. >> > > There is no evidence that physics reality marks the end of our ability to > explain anything deeper. > And there is no evidence that any deeper explanation is possible. Let's face it, you could make such a claim about any theory -- there is no evidence that there is not some deeper explanation -- unless, that is, your theory does not account for all the facts. Physics itself is not a theory. We have theories about physical phenomena that are more or less successful, but the theories are not the physical reality. > > >> then the same would hold for the mathematical reality. >>> >> >> No, mathematical "reality" (note the scare quotes) is a derived realm, >> entirely dependent on the set of axioms chosen in any instance. So it is >> not in any way analogous to physics. >> >> > Did you miss my earlier posts to Brent on this? The integers and their > relations are not modeled by any axiomatic system, they transcend the > axioms and therefore we must conclude have a reality independent from our > attempts to model them. > It is interesting, then, that Bruno is very proud of the fact that arithmetic depends only on a small set of axioms, or even just on the properties of a pair of combinators. Are you claiming that there is an objective arithmetical realm that is independent of any set of axioms? And our axiomatisations are attempts to provide a theory of this realm? In which case any particular set of axioms might not be true of "real" mathematics? Sorry, but that is silly. The realm of integers is completely defined by a set of simple axioms -- there is no arithmetic "reality" beyond this. Bruce -- 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.
