On Sun, Dec 16, 2018 at 7:21 PM Bruce Kellett <[email protected]> wrote:
> 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. > Admittedly then your believe that physics is not derivative from anything more fundamental is a quasi religious belief--it's held without any evidence for or against (in your view). On the other hand, there is evidence that physics is derived from more fundamental structures. But you reject them. Why? > > >> >> >>> 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. > A simple set of axioms allows us to define the Integers as well as computation, but those axioms can only scratch the surface regarding all the truth about the integers and their relations. > Are you claiming that there is an objective arithmetical realm that is > independent of any set of axioms? > Yes. This is partly why Gödel's result was so shocking, and so important. > 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? > It will be either incomplete or inconsistent. > > 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. > > The integers can be defined, but no axiomatic system can prove everything that happens to be true about them. This fact is not commonly known and appreciated outside of some esoteric branches of mathematics, but it is the case. For example: https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems "*Gödel's incompleteness theorems* are two theorems <https://en.wikipedia.org/wiki/Theorem> of mathematical logic <https://en.wikipedia.org/wiki/Mathematical_logic> that demonstrate the inherent limitations of every formal axiomatic system <https://en.wikipedia.org/wiki/Axiomatic_system> capable of modelling basic arithmetic <https://en.wikipedia.org/wiki/Arithmetic>. These results, published by Kurt Gödel <https://en.wikipedia.org/wiki/Kurt_G%C3%B6del> in 1931, are important both in mathematical logic and in the philosophy of mathematics <https://en.wikipedia.org/wiki/Philosophy_of_mathematics>. The theorems are widely, but not universally, interpreted as showing that Hilbert's program <https://en.wikipedia.org/wiki/Hilbert%27s_program> to find a complete and consistent set of axioms <https://en.wikipedia.org/wiki/Axiom> for all mathematics <https://en.wikipedia.org/wiki/Mathematics> is impossible." And https://en.wikipedia.org/wiki/Halting_problem#G%C3%B6del's_incompleteness_theorems "Since we know that there cannot be such an algorithm, it follows that the assumption that there is a consistent and complete axiomatization of all true first-order logic statements about natural numbers must be false." Jason -- You received this message because you are subscribed to the Google Groups "Everything List" group. 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