> On 17 Dec 2018, at 02:21, Bruce Kellett <[email protected]> wrote: > > On Mon, Dec 17, 2018 at 11:36 AM Jason Resch <[email protected] > <mailto:[email protected]>> wrote: > On Sun, Dec 16, 2018 at 4:14 PM Bruce Kellett <[email protected] > <mailto:[email protected]>> wrote: > On Mon, Dec 17, 2018 at 9:04 AM Jason Resch <[email protected] > <mailto:[email protected]>> wrote: > On Sun, Dec 16, 2018 at 4:01 PM Bruce Kellett <[email protected] > <mailto:[email protected]>> wrote: > On Mon, Dec 17, 2018 at 8:56 AM Jason Resch <[email protected] > <mailto:[email protected]>> wrote: > On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker <[email protected] > <mailto:[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.
Only because they are hidden under the rug. Like the application of logic and mathematics throughout, or simply consciousness and qualia that we all live and experience. A priori physics is neutral on consciousness and even on the ontological nature of matter, but physicalism has the problem to relate the experience of matter with matter, and using Mechanism break the mind-brain identity link, making the necessity to justify the physical laws by a sum on all computation seen from some first person view. Physics fails on consciousness. Which is not its cup of tea, normally. But Physicalism fails too, and that is more problematical, as this tends to eliminate the person and its experience. > 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. Same in mathematics. > > > 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, After Gödel, we know the the arithmetical reality is beyond *all* effective theories. What is remarkable, is that a very tiny part of arithmetic emulate all universal machine, bringing a total mess in the arithmetical reality. But our ignorance is quantifiable, notably through the number of alternating quantifier on a decidable predicates: sigma_1 = computable,(ExP(x,y)), pi_1, AxP(x,y), sigma_2,(ExAyP(x,y,z), pi_2, (AxEyP(x,y,z), sigma_3, pi_3, …are more and more unsolvable, etc. It is not arithmetic which depends on a small set of axioms, it is theology and physics which depends on a very small set of axioms once we assume mechanism. Any first order specification of a Turing universal machine (Turing), or creative set of numbers (Post) of a degree four diophantine polynomial is enough. > 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? Yes, known as the standard model of Arithmetic (Model in the logician’s sense: it is the arithmetical reality thought intuitively in primary school). It is reality that RA, PA, ZF, can only scratch on the surface. > And our axiomatisations are attempts to provide a theory of this realm? For arithmetic, theoretical computer science, … yes. > In which case any particular set of axioms might not be true of "real" > mathematics? There are other branch in mathematics. But with the digital mechanist assumption, we just cannot not use theoretical and mathematical computer science, which is indeed where the universal machine are born (except Babbage, but perhaps not) > > 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. That is not true. Integers and rational are as complex as the natural numbers. Simplicity comes with real … polynomials, but get complex back with trigonometric functions, which reintroduce the natural numbers in disguise. All those theories (N, Z, Q) with addition and multiplication are essentially undecidable. Not only you cannot complete them to get the whole truth, but no consistent extension ever get it. And unlike set theory or analysis, you can define sequence of approximation of truth notion, and PA can already define all sigma_i and p_i truth, but not the general case, and with mechanism, so can't we (without assuming more). For the polynomial, they are Turing universal on N and Z, but it is an open problem with Q. Even the number, to survive, have to believe in much more than the numbers, but at some point they learn to distinguish the ontology from the phenomenology, and get their (G*) theology right, as they can test locally through the “arithmetical-self-referential physics”. Bruno > > 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] > <mailto:[email protected]>. > To post to this group, send email to [email protected] > <mailto:[email protected]>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <https://groups.google.com/d/optout>. -- 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.
