Re: Mathematical Logic, Podnieks'page ...
Dear John, Thanks for your quotations from (or through) Podnieks. Here are some comments. To the question What is mathematics - Podiek's (after Dave Rusin) answer: Mathematics is the part of science you could continue to do if you woke up tomorrow and discovered the universe was gone. What a pretty quote! It's a good description of what happened to me a long time ago. I woke up, and realized the universe was gone. Only taxes remained ;) Remark: provided that YOUR mind is out of this world and stays unchanged 'as is' after (the rest of) the universe was gone. Sure. Another point is science but I let it go now. (cf: Is math 'part of science'?) I really hope you don't doubt that. math is certainly part of science. With comp and even with weakening of comp the reverse is true: science is part of math. The JvNeumann quote: In mathematics you don't understand things. You just get used to them. I agree. But I think it is the same with loves, cuisine and certainly physics. Children climb in trees before learning the gravitation law ; and even that does not explain things. True. Once you want to understand them you have to couple it with some sort of substrate, ie. apply it to things when the fix on quantities turns the math idea into a (physical?) limited model preventing a total understanding (some Godel?) It is your talk here. I am not sure I understand. Of course we have a sort of build-in theory of our neighborhood, as does cats and birds. But substrate and concreteness are illusion of simplicity. Only many neurons and a long biological history make us forgetting that nothing sensible can be obvious. And then with comp you can have clues why it is so - Isn't this the way with Einstein's form: you first get used to it (in general)(?) then apply it to substrates (shown later in the URL). (My: Aspects of 'model' formation from different directions). * Podnieks: For me, Goedel's results are the crucial evidence that stable self-contained systems of reasoning cannot be perfect (just because they are stable and self-contained). Such systems are either very restricted in power (i.e. they cannot express the notion of natural numbers with induction principle), or they are powerful enough, yet then they lead inevitably either to contradictions, or to undecidable propositions. I agree with Podnieks, as you can guess. Translated into my vocabulary it sais the same as the 1st sentence, (called) 'well defined', topical and boundary enclosed and limited models, never leading to a total (wholistic) result. I generalized it away from the math thinking - eo ipso it became more vague. But that's my problem. I am not sure I understand what you ere saying here. It is too much ambiguous. Remember that comp entails the falsity of almost all reductionist view of numbers, machines, etc. * Let us assume that PA is consistent. Then only computable predicates are expressible in PA. This is ambiguous as it stands. All partial computable predicates, including the total computable predicates are expressible in PA. Incompleteness is linked to the fact that there is no mechanical test to distinguish the total and partial predicates. See my diagonalization posts to get the basic idea. (3.2: In the first order arithmetic (PA) the simplest way of mathematical reasoning is formalized, where only natural numbers (i.e. discrete objects) are used... In (my) wholistic views an (unlimited, ie. non-model) complexity is non computable (Turing that is) and impredicative (R.Rosen). In our (scientific!) parlance: vague. I share with you that idea that the big whole is vague and uncomputable, and that impredicativity is inescapable. Please note that it is indeed provably the case concerning the experience of the universal machine once you accept to define knowledge by true belief (proof) or other theetetic definition of knowledge. No 'discrete objects': everything is interconnected at some qualia and interactivity level. OK (except that interactivity like causality) has no clear meaning (for me). The end of the chapter: We do not know exactly, is PA consistent or not. Later in this section we will prove (without any consistency conjectures!) that each computable predicate can be expressed in PA. - Like Smullyan I believe we know that PA is consistent. With comp that means (by Godel second theorem) that we are superior than PA with respect to our ability to prove theorems in arithmetic. What no machine can ever prove is its own consistency. But machines can bet on it and change themselves. (The logic G and G* will still apply at each step of such transformation, unless the machine becomes inconsistent). underlines my caution to combine wholistic thinking with mathematical (even first order arithmetic only) language. I did not intend to raise havoc, not even start a discussion, just sweeping throught the URL brought up some ideas. Only FYI, if you find it interesting. It is, thanks, Bruno http://iridia.ulb.ac.be/~marchal/
Re: Mathematical Logic, Podnieks'page ...
To the question "What is mathematics" - Podiek's (after Dave Rusin) answer: Mathematics is the part of science you could continue to do if you woke up tomorrow and discovered the universe was gone. Podiek shouldn't have skipped Leibniz in his reading list on philosophy (and should've taken his Newton with a grain of salt?). Monads not only don't "wake up" "outside" the(this) unverse, they have no meaning in isolation from it(them), IMHO. (Guess I'm indeed nota Platonist) Cheers CMR- insert gratuitous quote that implies my profundity here -
Re: Mathematical Logic, Podnieks'page ...
CMR wrote: To the question "What is mathematics" - Podiek's (after Dave Rusin) answer: Mathematics is the part of science you could continue to do if you woke up tomorrow and discovered the universe was gone. Let me make an analogy by paraphrasing: Empty space is the part of the universe that would bew left if you woke up tomorrow and discovered that all stars, planets and galaxies were gone. My paraphrase is only true in the context of classical physics. I don't think Podiek's statement should be so easily accepted and in fact whether it is true at all. As a model of what I am trying to express, think of a creature being simulated together with its own environment inside a computer. The creature wakes up one day to find out that the simulation has been terminated. Obviously such a scenario is impossible. If there is no simulation there is no creature. And there is no math that the creature could do. George
Re: Mathematical Logic, Podnieks'page ...
Dear Bruno, thanks for your detailed reflections(BM). There are some minor points I want to re-address.(R-JM) interleaving intothe orig. post (My text: JM:) John Mikes - Original Message - From: Bruno Marchal To: [EMAIL PROTECTED] Sent: Monday, June 28, 2004 6:27 AM Subject: Re: Mathematical Logic, Podnieks'page ... Dear John,Thanks for your quotations from (or through) Podnieks. Here are some comments. "To the question "What is mathematics" - Podiek's (after Dave Rusin) answer: Mathematics is the part of science you could continue to do if you woke up tomorrow and discovered the universe was gone." BM: What a pretty quote! It's a good description of what happened to me a long time ago. I woke up, and realized the universe was gone. Only taxes remained ;) JM: Remark: provided that YOUR mind is "out of this world" and stays unchanged 'as is' after (the rest of) the universe was gone. BM:Sure.JM:Another point is "science" but I let it go now. (cf: Is math 'part of science'?)BM:I really hope you don't doubt that. math is certainly part of science. With comp and even with weakening of comp the reverse is true: science is part of math. (R-JM): Science in my terms is the edifice of reductionist imaging (observations) of topically selected models, as it developed over the past millennia: subject to the continually (gradually) evolving (applied) math formalism. Will be back to that.The JvNeumann quote:In mathematics you don't understand things. You just get used to them. BM: I agree. But I think it is the same with loves, cuisine and certainly physics. Children climb in trees before learning the gravitation law ; and even that does not explain things. JM: True. Once you want to understand them you have to couple it with some sort of substrate, ie. apply it to "things" when the fix on quantities turns the math idea into a (physical?) limited model preventing a total understanding (some Godel?) BM:It is your talk here. I am not sure I understand. Of course we have a sort of build-in theory of our neighborhood, as does cats and birds. But substrate and concreteness are illusion of simplicity. Only many neurons and a long "biological" history make us forgetting that nothing sensible can be obvious. And then with comp you can have clues why it is so (R-JM): (MY!) Simplicity is the 'cut-off' from the wholeness in our models. Later you mention the causality: it is similarly a cut-off of all possible (eo ipso 'active') influencings, pointing to the ONE which is the most obvious within our topical cut. We make 'cause' SIMPLE. JM:- Isn't this the way with Einstein's "form": you first get used to it (in general)(?) then apply it to substrates (shown later in the URL). (My [_expression_]: Aspects of 'model' formation from different directions).*Podnieks:For me, Goedel's results are the crucial evidence that stable self-contained systems of reasoning cannot be perfect (just because they are stable and self-contained). Such systems are either very restricted in power (i.e. they cannot express the notion of natural numbers with induction principle), or they are powerful enough, yet then they lead inevitably either to contradictions, or to undecidable propositions.BM: I agree with Podnieks, as you can guess. JM: Translated into my vocabulary it sais the same as the 1st sentence, (called) 'well defined', topical and boundary enclosed and limited "models", - never leading to a total (wholistic) result. I generalized it away from the math thinking - eo ipso it became more vague. But that's my problem. BM: I am not sure I understand what you ere saying here. It is too much ambiguous.Remember that comp entails the falsity of almost all reductionist view of numbers, machines, etc. (R-JM): Exactly. Comp (? I am not sure if I know what it is indeed) has IMO brisk rules and definite qualia to handle by those rules. (I evaded: 'quantities'). Which means the omission of aspects OUTSIDE such qualia and rules. The cut-off, ie. limitations, enable comp to become brisk, unequivocal, well defined. Including unidentified and infinite variables, qualia, all sort of influence (quality and strength) - meaning the wholeness-interconnection - makes it more vague than any fuzziness could do (which still stays topical). I don't expect this emryonic branch of thinking (30-50years max?) even using the language of the millennia of reductionist development, to compete in briskness with the conventional - what you and others may call: - science. An embryo would recite Godel in a very vague way. * JM: Let us assume that PA is consistent. Then only computable predicates are expressible in PA. BM: This is ambiguous as it stands. All partial
Re: Mathematical Logic, Podnieks'page ...
Thanks, George (I findyour argumentclose to a wholistic (complexity) vision, only stronger than the assumptionhow I tried to argue it. ) John M - Original Message - From: George Levy To: Everything List Sent: Monday, June 28, 2004 1:28 PM Subject: Re: Mathematical Logic, Podnieks'page ... CMR wrote: To the question "What is mathematics" - Podiek's (after Dave Rusin) answer: Mathematics is the part of science you could continue to do if you woke up tomorrow and discovered the universe was gone.Let me make an analogy by paraphrasing: Empty space is the part of the universe that would bew left if you woke up tomorrow and discovered that all stars, planets and galaxies were gone.My paraphrase is only true in the context of classical physics. I don't think Podiek's statement should be so easily accepted and in fact whether it is true at all. As a model of what I am trying to express, think of a creature being simulated together with its own environment inside a computer. The creature wakes up one day to find out that the simulation has been terminated. Obviously such a scenario is impossible. If there is no simulation there is no creature. And there is no math that the creature could do.George
