On Monday, December 17, 2018 at 1:21:15 AM UTC, Bruce wrote: > > On Mon, Dec 17, 2018 at 11:36 AM Jason Resch <[email protected] > <javascript:>> wrote: > >> On Sun, Dec 16, 2018 at 4:14 PM Bruce Kellett <[email protected] >> <javascript:>> wrote: >> >>> On Mon, Dec 17, 2018 at 9:04 AM Jason Resch <[email protected] >>> <javascript:>> wrote: >>> >>>> On Sun, Dec 16, 2018 at 4:01 PM Bruce Kellett <[email protected] >>>> <javascript:>> wrote: >>>> >>>>> On Mon, Dec 17, 2018 at 8:56 AM Jason Resch <[email protected] >>>>> <javascript:>> wrote: >>>>> >>>>>> On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker <[email protected] >>>>>> <javascript:>> 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. >
*A deeper explanation is certainly possible. I don't see why you reject it out of hand. OTOH, the issue at hand is whether arithmetic is that deeper explanation. Doubtful IMO. AG* 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.
