On 18 Jul 2011, at 19:14, meekerdb wrote:

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On 7/18/2011 2:48 AM, Bruno Marchal wrote:On 17 Jul 2011, at 20:28, meekerdb wrote:On 7/17/2011 10:11 AM, Bruno Marchal wrote:On 15 Jul 2011, at 18:41, meekerdb wrote:On 7/15/2011 2:15 AM, Bruno Marchal wrote:Numerology is poetry. Can be very cute, but should not be takentoo much seriously. Are you saying that you disagree with thefact that math is about immaterial relation between nonmaterial beings. Could you give me an explanation that 34 isless than 36 by using a physics which does not presupposeimplicitly the numbers.|||||||||||||||||||||||||||||||||||||||| ||||||||||||||||||||||||||||||||||||||||||Nice, indeed. We do agree that 34 is less than 36, and what thatmeans.I am not sure your proof is physical thought. Physics has beenvery useful to convey the idea, and I thank God for not havingmade my computer crashed when reading your post, but I see youonly teleporting information. That fact that you are using thephysical reality to convey an idea does not make that ideaphysical.I was expecting a physical definition of the numbers.Of course there is no physical definition of the numbers becausethe usual definition includes the axiom of infinity.You don't need the axiom of infinity for axiomatizing the numbers.The axiom of infinity is typical for set theories, not naturalnumber theories. You need it to have OMEGA and others infiniteordinals and cardinals.As finite beings we can hypothesize infinities.Yes, but we don't need this for numbers. On the contrary, theinduction axioms are limitation axioms to prevent the rising ofinfinite numbers.By thinking that I can understand your proof, you arepresupposing many things, including the numbers, and the way tocompare them.On the contrary I think you (and Peano) conceived of numbers byconsidering such such examples. The examples presuppose verylittle - probably just the perceptual power the evolution endowedus with.That is provably impossible. No machine can infer numbers fromexamples, without having them preprogrammed at the start. You needthe truth on number to make sense on any inference of any notion.Nothing can be proven that is not implicit in the axioms and rulesof inference.

OK.

So I doubt the significance of this proven impossibility.

?

`It means, contrary of the expectations of the logicist that the`

`natural numbers existence is not implicit in many logical system.`

`We cannot derive them from logic alone, nor from first order theories`

`of the real numbers, nor from most algebra, etc. So, if we want`

`natural numbers in the intended model of the theory, they have to be`

`postulated, implicitly (like in wave theory, set theory) or`

`explicitly, like in RA or PA.`

So it is a funny answer, which did surprise me, but which avoidsthe difficulty of defining what (finite) numbers are.It *is* a theorem in logic, that we can't define them"univocally" in first order logical system. We can define them insecond order logic, but this one use the intuition of number.If you agree that physics is well described by QM, an explanationof 34 < 36 should be a theorem in quantum physics,I'm sure it is. If you add 34 electrons to 36 positrons you gettwo positrons left over.Physics is not an axiomatic system.That is the main defect of physics. But things evolve. Withoutmaking physics into an axiomatic, the whole intepretation problemof the physical laws will remain sunday philosophy handwaving.Physicists are just very naïve on what can be an interpretation.The reason is they "religious" view of the universe. They take itfor granted, which is problematic, because that is not a scientificattitude.Accepting what you can feel and see and test is the antithesis oftaking it for granted and the epitome of the scientific attitude.

`That is Aristotle definition of reality (in modern vocabulary). But`

`the platonist defend the idea that what we feel, see and test, is only`

`number relation, and that the true reality, be it a universe or a god,`

`is what we try to extrapolate.`

`We certainly don't see, feel, or test a *primitive* physical universe.`

`The existence of such a primitive physical reality is a metaphysical`

`proposition. We cannot test that. This follows directly from the dream`

`argument. That is what Plato will try to explain with the cave.`

The trouble with axiomatic methods is that they prove what you putinto them. They make no provision for what may loosely be called"boundary conditions". Physics is successful because it doesn't tryto explain everything. Religions fall into dogma because they do.

`I don't criticize physics, but aristotelian physicalism. which is, for`

`many scientists, a sort of dogma.`

`Religion fall into dogma, because humans have perhaps not yet the`

`maturity to be able to doubt on fundamental question. To admit that we`

`don't know if there is a (primitive) physical universe.`

Physicists use mathematics (in preference to other languages) inorder to be precise and to avoid self-contradiction.That is the main error of the physicists. They confuse mathematicswith a language.And the main error of mathematicians is they confuse proof with truth.

`That is unfair because all what I use here is the (big) discovery of`

`Gödel that arithmetical truth escapes all possible effective or`

`axiomatizable proof systems. So mathematicians are able to distinguish`

`mathematically, in many case, the difference between proof and truth.`

`Only intuitionist confuse proof and truth, (like S4Grz!) but classical`

`mathematicians note that not only proof does not entail truth, but`

`that even in the case where proof entails truth, the contrary remains`

`false: truth does not entail proof.`

`The whole AUDA is based on the fact that arithmetical truth is beyond`

`all correct machines (proofs).`

`Let me comment a little part of your dialog with Jason. I comment also`

`Jason.`

"True" is just a value that is preserved in the logical inferencefrom axioms to theorem. It's not the same as "real".True is more than inference from axioms.

`I think Jason said that. I agree. Truth is preserved in the`

`application of sound inference rules, but truth is far bigger than`

`anything you can access by inference rules and axioms. Arithmetical`

`truth is, compared to any machine, *very* big. The predicate truth`

`cannot even be made arithmetical.`

For example, Godel's theorem is a statement about axiomaticsystems, it is not derived from axioms.

`Well, the beauty is that Gödel's second incompleteness theorem is a`

`theorem of arithmetic. BDt -> Bf (or ~Bf -> ~B~Bf) is a theorem of PA.`

`It is the whole point of interviewing PA about itself. It can prove`

`its own Gödel's theorem. That is missed in Lucas, Penrose, and many`

`use of Gödel's theorem by anti-mechanist. Simple, but not so simple,`

`machine have tremendous power of introspection. Löbian one, have,`

`actually, maximal power of introspection.`

Sure it is. It's a logical inference in a meta-theory.

`Not at all. The second (deeper) theorem of Gödel, like the theorem of`

`Löb, is a theorem of Peano Arithmetic. The tedious part consists in`

`translating the "Bx" in arithmetic, but Gödel's succeeded famously in`

`the task (cf beweisbar ('x')).`

`G axiomatise all such metatheorem that a theory can prove about`

`itself, and G* formalize all the truth that the theory can prove +`

`that the theory cannot prove about itself. In that way, Solovay closed`

`the research in the modal propositional provability/consistency`

`logics, by finding their axiomatization, and this both for the`

`provable part of the machine (which contains BDt -> Bf), and the non`

`provable part (which contains typically Dt, DDt, DDDt, DBf, DDBf, etc.).`

Bruno

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