Re: Mathematical Logic, Podnieks'page ...
Dear John, At 11:19 07/07/04 -0400, John M wrote: Dear Bruno, I don't know how tolerable our discussion may be for the list, but for me it is enjoyable. Amazing, in how many things (aspects?) we DO agree, coming fundamentally from quite different worldviews. I'm sure we agree on something and be confident we will be able to agree about what we disagree ... Anyway discussion is only interesting between people who disagrees (as far as they are willing to search the disagreement point). I will be obliged now to write the paper for Amsterdam, which will be an english version of my Paris paper (except I will not talk about Changeux and Connes). I hope George and Kory are not angry because I don't give them enough Holiday work ;) What should I do? Explain my work top down or bottom up. Teach logic? Wait for people reading Smullyan's FU. To be frank I do think Comp and QM are more universal than human, and perhaps what *is* human is to considere comp and QM as human thinking 'YOU do think' (!) In our 'human' restrictions we cannot even think about 'other' ways of thinking. If it went through our mind, it DID become 'human thinking'. If 'it' happened to go into and through it at all. Whatever we imagine as 'non human' IS human with a twist. Including your argument Godel II, a gem of the 'human' thought. Here I don not understand you at all. Why are you so stuck by our human nature. We are mammals too. Why don't you say that whatever we think it did become mammal thinking, that our imagination get a mammal twist. The same with earth-like creature. Actually even without the comp hyp, we are (at least) universal machine (in the church thesis sense, I can prove it to you). So why don't you say that whatever we think it gets the universal machine twist? I really don't understand. Furthermore, why should such a twist, as human (mammal, life being, universal machine, whatever ...) prevent us to bet on some reality beyond us? Whatever we are. And perhaps to bet wrongly and precisely enough to be able to learn ... (Sci fi is the worst human violent emotional stupidity, neither sci nor fi. So (just not so bad) are our dreams about 'nonhuman' thinking). Come on! *Some Sci Fi* are perhaps bad, but some could be master piece of reasoning, but too much in conceptual advance to be appreciate by current academical institution or even by the sc fi author himself. belongs to all possible universes Consider the impossible ones. But I told you that is exactly what Godel II makes it possible to do in a clean sharable (between universal machine) way. And that a way to be more independent of our human nature. But of course whatever I do you will always be able to say: human!. I will try your article with my rusty French, I never read math-based science in French (and that was ~50 years ago) so I have doubts about understanding it. French and english are quite alike for math: look function is fonction relation is relation collection is collection theorem is theoreme lemma is lemme proposition is proposition Ok, some exception (exception) the french for 'set' is 'ensemble'. But you can wait for my amsterdam paper. You can also read my 2000 Computation Consciousness and the Quantum http://iridia.ulb.ac.be/~marchal/publications/CCQ.pdf Or my older 1991 Mechanism and personal identity: http://iridia.ulb.ac.be/~marchal/publications/MPI_15-MAI-91.pdf I am afraid to read 'Smullyan's little bible' - it may be too good. Besides you would not believe how many people suggest such an inescapble 30,000th ONE book to read. It is up to you of course. But please don't think I want to convince you or anyone on any truth ... I certainly would like to share a feeling of supertranscendantal *beauty* in the way it seems the laws of nature should arise from all machine dreams ... What I am reading now is Krakatoa, with a cultural history of the Dutch British takeover of the Portugese South-of-Asia world - part of my cultural debt of information. This is certainly interesting too. Although no clearly in the scope of the list. Then you wrote: ? after my par including the 'model' view I carry. To attempt an inadequate rambling about this point: I consider the universe(s) parts of the wholeness they emerged from, the plenitude (not Plato's), an unimaginable-undescribable everything in infinite invariance of infinite symmetrical changes (no time involved). Such emergences are inevitable by the fact that the 'everything' involves (local - without a space-concept, transitionally occurring) asymmetries ie. universes, You are quick here. I imagine things ... I dont' figure that clearly. I must read it again. I tell you if I understand when I understand. I think, John, that we agree that wholism is important and reductionism should be avoid. But what I would like to explain is that the idea of being a machine is not a reductionism. It is only a reductionism in the eyes of those who have a reductionist conception of machine. Such a conception
Re: Mathematical Logic, Podnieks'page ...
Dear John, At 16:50 05/07/04 -0400, you wrote: Bruno, I really cannot work this way. I still prepare to reply to your earlier post (to me) and here I have the repost on the 1st part with lots to be replied upon. G. Take it easy. I am in debt with ~30,000 books I did not read. Never will. This is 29,999 books too much. Just read the Smullyan little bible: Forever Undecided IF you want ease the understanding of the reversal ... How much time may I have left? 30-40 years? (I am pushing 83). Will my mind give up? My hip did already. (My fingers did not, I still perform classical piano-music for a local-public audience - next: October). Nice. I'm about 50, and serious jaws problems ... You know one of my best Saturday Course student is 87. So I gave up checking on past millennia wisdom and work on the present - absorbed and developed for myself since retirement. The oldies speculated in a cognitive inventory of the mind which was much poorer than the lately absorbed enrichments. I appreciate their wisdom, as 'function of mind', but the conclusions MAY be old. I am not a judge of that, but can stay out of such argumentations. Comp, QM, you ask? Aren't they within the mindset of the minds within THIS universe, which I deemed human ways of thinking? To be frank I do think Comp and QM are more universal than human, and perhaps what *is* human is to considere comp and QM as human thinking. Got argument that those belongs to all possible universes, or better all possible dreams. Same good old math-conceptualization. I am talking about something not-matching. Cohen and Stewart played such tunes in their enjoyable books (Collapse of Chaos and Figments of Reality) - their aliens, the Zarathustrans, with their octimalization (8?). Of course that was still sort of human switch, a bit of Tao etc. I read many of your web text. You did defend sort of naturalism isn't it? I tend to consider that the opposition natural/artificial is ... artificial (and thus natural). truth is an object of study by logicians. My best wishes for them. I went through many 'thruths' - different religious ones, reincarnational, pragmatic natural science, astrology, Indian, We are not talking of the same truth. I talk about many different ways to just talk about the concept. Just tools for being able to go through many truth is a non confusing way. Marxist, Leninist, atheist (who require a god to deny), In the first page of the introduction to Conscience et Mecanisme I explain that atheist are believers indeed. every one had something attractive, but... and settled with my scientific agnosticism: not even the contrary is true of what people believe. (That really came from politics). I am a scientific agnostic too. (Agnostic to nature, universe, matter, ...). But I cannot doubt 1+1 is 2, except for some minutes before the first (or the second I never know) cup of coffee ... If you go into variables: my wholistic views allow no fixed conditions and unlimited variabilities upon which a mathematician friend remarked: well, this is a bit steep. Only models can have boundaries, quantities, fixed qualia etc. That goes also for QM (Comp I don't know, never let it clarify in my mind). ? Even topics are cut out from the extratopical wholeness. Limited Models. A map is a model, a territory a wider one. Most minds (on this and other lists) work within a certain modeling (we cannot do better, that's the way we can manage with the material tool we apply for thinking: the neuronal brain, restricting the mind into human logic (oops!). Is my wholistic thinking inept for achieveing practical conclusions? you bet it is. We just started to tackle with such ideas, have to find suitable concepts and (formulate?) words to express them. This is what I try to propose right now. ...(UDA) *forces* us to do: if comp is true we have to explain the physical appearances by a sort of mean on all consistent belief systems. (if!) Yes. IF! I cannot provide more. - now the 'physical appearances' are the mind's interpretations upon impact inknown, lately observed by instruments WITHIN this system of ours. And I did not ask for CONSISTENT belief systems, before I even know what kinds may exist at all. Arithmetical consistency is very large (by Godel). Indeed even inconsistency is consistent ! (Godel' second theorem). Just let your mind accept classical logic for a while, if only for the sake of the argument. Why not? We know SOME, here and now, pertinent to our cultural basis (human mindset of the present local(?!) societal conditions). I am consistent in my agnosticism. All argumentative support from within is useless for without. OK. Now I can return to thinking about math (for the 2nd part reply), although I don't know much about it. It was my elective in my Ph.D. work (1948), never used it later, beyond arithmetics, mostly by my slide ruler, while inventing and implementing a pioneering-worldlevel industrial branch, 38 patents, consulting (and
Re: Mathematical Logic, Podnieks'page ...
Hi Bruno: At 01:15 PM 7/2/2004, you wrote: Hi Hal, At 12:44 02/07/04 -0400, Hal Ruhl wrote: By the way if some systems are complete and inconsistent will arithmetic be one of them? As I understand it there are no perfect fundamental theories. So if arithmetic ever becomes complete then it will be inconsistent. Yes, if by arithmetic you mean an axiomatic system, or a formal theory, or a machine. No if by arithmetic you mean a set so big that you cannot define it define appears to be a two sided activity. When you define a thing you also define the thing which it is not - a bag of the remainder of all. Most of the time the latter may not be useful. Since all of arithmetic [and mathematics] is in the Everything and the Everything in my system is the definitional pair to the Nothing, defining the Nothing [or the Everything] automatically defines all of arithmetic along with all of mathematics. A Something is less than the Everything and may or may not contain mathematics or a portion thereof. in any formal theory, Well my theory seems concerned with the form of its statements that is the Somethings and how they alter. I think my theory defines mathematics the way that The first number that can not be described in less than fourteen words defines a number that we nevertheless may never actually have in hand. Hal
Re: Mathematical Logic, Podnieks'page ...
At 16:44 04/07/04 -0400, John M wrote:
I think we got into a semantic quagmire. I feel a different meaning in my
(5th language English) TRUTH from what I read as the (4th language French)
'verité'. I use 'truth' as the OPINION one accepts as being not false.
Yes but then you will misunderstand the Theaetetus definition of knowledge
(as true opinion). I think it is just a question of vocabulary. See below...
What
you imply sounds to me as 'constructing a reality. Truth has nothing to do
with decisionmaking. Decision comes into the picture only in the 1st
person thinking to decide whether the item is not false. If I agree, it is
(my) truth as well.
?
JM:
the fact
that anything we may know (believe or find), is interpreted by the ways
how our 'human' mind works -
BM: SURE! (but it is invalid to infer from that that truth itself
depends on
our beliefs and findings).
JM: Sorry, Bruno, you sound in the parethetized remark as a person who
believes
in some eternal 'truth' chisled in the (nonexistent) stone of (nonexistent)
supernatural 'law', - or rather: takes something like 'truth' as the
installations (facts??)of the world. There is no such thing as THE TRUTH -
ITSELF at least not among people who think... Maybe some religious fanatic
fundamentalists know the truth, the only ONE, worthwhile killing (-dying)
for.
Nobody in this list pretend to know the truth. A theory is always a
(hopefully consistent) set of beliefs, mainly.
But we can privately hope our beliefs are true. The point is: do you find
comp inconsistent? Do you find QM inconsistent? Do you find PA inconsistent?
(and then do you find that plausible, etc.)
Also, can you conceive that QM could be true, independently of us knowing
it, except in the thaetetus sense of just believing it, and that by chance
it *is* true.
Also, truth is an object of study by logicians. In classical
propositional logic
truth is just a function from the propositional variable {p, q, r, ...}
into {O, 1}, in
the company of rules to extend those truth values to compound propositions,
like saying that pq is true in case p is true and q is true, which for
the logician
means only the function above send p and q on 1. But logicians considers
many, many, many other sort of truth valuation (abstractly they are
sub-object
classifier, truth being the object itself, so in classical logic truth can
be represented
by a set, in intuitionistic logic truth can be represented by a
topological space,
in quantum logic truth can be represented by a Hilbert space, but that's
for much latter ...).
Even the facts are explanations for observations - and we saw lately
discussions on observers.
The flat Earth: a fact (Ptolemaios), hell: a fact (A. Dante), the atoms in
the molecules I synthesized: facts, then all these things turned into
fiction. Props of some belief system.
I doubt very much Ptolemaios maked flat earth a fact. For Dante I don't know
but I would have believe he wrote a fiction (?)
Anyway, we are interested in ALL belief systems. I should have give you a
better
answer last day when you asked:
With the ideas about 'quite' different universes why are we
closed to the idea of 'quite' different mathematical thinking?
I should have told you that this is exactly what the Universal Dovetailer
Argument (UDA) *forces* us to do: if comp is true we have to explain
the physical appearances by a sort of mean on all consistent belief
systems. And giving the fact that the tool exists to study the basic shape
of that means (the interview of the universal machine), we can do it, and
compare with the empirical physics.
Now let me take a deep breath and if I am still 'on' this list, later I will
come back to 'math'.
(I don't know Wilfried Hodge, will not read him for this purpose.)
You know John to tell you the truth I would like to confess you that I am
a believer indeed in the sense that I really feel bad (like lying to myself)
when I try to put into doubt the laws of the excluded middle concerning
arbitrary
arithmetical sentences. I believe the 667nth fortran program running on
the data
766 will either stop or ... not stop.
This does not prevent me to appreciate many other logics. (and classical logic
is the simplest to talk
It is Wilfrid Hodges, I put a e because it is a common Flemish name here,
I probably mess up with the s as I always do.
Apology to Wilfrid Hodges.
I recommend it much for the non mathematically minded people who
want a first rate introduction to (classical) logic. Hodges defines logic
as the study
of the consistent set of beliefs, and show quickly and simply the relation
with the more common definition of logic as science of the valid argumentation.
He wrote also good (but more technical) books in model theory.
Hodges' Logic book is a nice cheap companion to Smullyan's Forever
Undecided.
Shortcuts to G. (The key mathematical tool to transform the reversal
between the
physical universes/psychological universes forced in the UDA.
Re: Mathematical Logic, Podnieks'page ...
Bruno, I really cannot work this way. I still prepare to reply to your earlier post (to me) and here I have the repost on the 1st part with lots to be replied upon. G. I am in debt with ~30,000 books I did not read. Never will. How much time may I have left? 30-40 years? (I am pushing 83). Will my mind give up? My hip did already. (My fingers did not, I still perform classical piano-music for a local-public audience - next: October). So I gave up checking on past millennia wisdom and work on the present - absorbed and developed for myself since retirement. The oldies speculated in a cognitive inventory of the mind which was much poorer than the lately absorbed enrichments. I appreciate their wisdom, as 'function of mind', but the conclusions MAY be old. I am not a judge of that, but can stay out of such argumentations. Comp, QM, you ask? Aren't they within the mindset of the minds within THIS universe, which I deemed human ways of thinking? Same good old math-conceptualization. I am talking about something not-matching. Cohen and Stewart played such tunes in their enjoyable books (Collapse of Chaos and Figments of Reality) - their aliens, the Zarathustrans, with their octimalization (8?). Of course that was still sort of human switch, a bit of Tao etc. truth is an object of study by logicians. My best wishes for them. I went through many 'thruths' - different religious ones, reincarnational, pragmatic natural science, astrology, Indian, Marxist, Leninist, atheist (who require a god to deny), every one had something attractive, but... and settled with my scientific agnosticism: not even the contrary is true of what people believe. (That really came from politics). If you go into variables: my wholistic views allow no fixed conditions and unlimited variabilities upon which a mathematician friend remarked: well, this is a bit steep. Only models can have boundaries, quantities, fixed qualia etc. That goes also for QM (Comp I don't know, never let it clarify in my mind). Even topics are cut out from the extratopical wholeness. Limited Models. A map is a model, a territory a wider one. Most minds (on this and other lists) work within a certain modeling (we cannot do better, that's the way we can manage with the material tool we apply for thinking: the neuronal brain, restricting the mind into human logic (oops!). Is my wholistic thinking inept for achieveing practical conclusions? you bet it is. We just started to tackle with such ideas, have to find suitable concepts and (formulate?) words to express them. ...(UDA) *forces* us to do: if comp is true we have to explain the physical appearances by a sort of mean on all consistent belief systems. (if!) - now the 'physical appearances' are the mind's interpretations upon impact inknown, lately observed by instruments WITHIN this system of ours. And I did not ask for CONSISTENT belief systems, before I even know what kinds may exist at all. We know SOME, here and now, pertinent to our cultural basis (human mindset of the present local(?!) societal conditions). I am consistent in my agnosticism. All argumentative support from within is useless for without. Now I can return to thinking about math (for the 2nd part reply), although I don't know much about it. It was my elective in my Ph.D. work (1948), never used it later, beyond arithmetics, mostly by my slide ruler, while inventing and implementing a pioneering-worldlevel industrial branch, 38 patents, consulting (and solving technical production-problems) on 3 continents over 4 decades. All in the simplest reductionist technical common sense creativity. I am ready for a coffee, myself. John Mikes - Original Message - From: Bruno Marchal [EMAIL PROTECTED] To: [EMAIL PROTECTED] Sent: Monday, July 05, 2004 10:52 AM Subject: Re: Mathematical Logic, Podnieks'page ... At 16:44 04/07/04 -0400, John M wrote: SNIP I copied out some sentences, the rest is in the archives.
Re: Mathematical Logic, Podnieks'page ...
At 06:57 03/07/04 -0400, John M wrote: (Bruno: am I still in your corner?) OK. Let us see. Dear Kory, an appeal to your open mind: in the question whether we discovered math or invented it..., many state that the first version is 'true'. Beside the fact that anybody's 'truth' is a first person decision, Then I would decide to have food when I am hungry, to have water when I am thirsty. I would decide Riemann hypothesis true and even proved by me, and I would decide to get those million dollars. I would decide you to be a platonist, my friend, ... I would decide peace everywhere, ...if truth was a matter of first person *decision*. Seriously, I am afraid you confuse the luckily adequate first person feeling the first person lives in front of truth and truth itself. the fact that anything we may know (believe or find), is interpreted by the ways how our 'human' mind works - SURE! (but it is invalid to infer from that that truth itself depends on our beliefs, and findings). including comp and all kinds of computers, as we 'imagine' (interpret, even formulate) the thoughts. I find the above distinction illusorical. We may FIND math as existing 'before' we constructed it, or we may FIND math a most ingenious somersault of our thinking. Then you will miss the discovery that a big part of math and actually the whole of physics is a most ingenuous somersault of the universal machine thinking. (and the discovery that comp imply that in a testable manner) To 'believe' that 17 is prime? of course, within the ways as we know (and formulate) the concept 'prime'. Axioms, conventions. Are you not confusing sentences/theories with proposition/truth? Read Wilfrid Hodge Penguin's Logic page 39. (I can quote it if you insist). With the ideas about 'quite' different universes why are we closed to the idea of 'quite' different mathematical thinking? Because for some reason we are (or we will be) studying different sort of mathematical thinking, and so, to avoid confusion, it is better to agree at the start on the elementary principle we share. Logic, is the science of different thinking, actually. Boolean (classical) logic is the simplest to use in math (but not the simplest to describe in boolean logic because the similarity of the object and the subject ...) We don't have to go to another universe: the Romans subtracted in their calendar (counting backwards from the 3 fixed dates in a month) like minus 1 = today, minus 2 = yesterday and so on. I wonder how would've done that Plato (before the invention of 0)? Our list-collegues think about math(s) in quite different concepts from the classic 'constructivist(?)' arithmetical equational thinking. Should I understand you are realist for intuitionistic arithmetic ? It is enough for the reasoning I propose. how far can go a quite differently composed mind - maybe in an organizational thinking/observing system of a universe NOT based on space - time? You underestimate the hardness to understand ourselves despite our probable common space time background. What can be called 'mathematics'? (Theory(s) of Everything?) Here you jump to an infinitely difficult and controversial question. I have criticize Tegmark for relying on that problem. One of the power of comp is that it made possible to give information on fundamental matter without needing to define 'mathematics. Vive le 'scientific agnosticism'! Right! (At the condition that this principle does not discourage us to propose theories ...) (Bruno: am I still in your corner?) If you really believe truth is just a matter of first person decision, you are not. Neither if you belief the primality of 317 is a matter of convention. Only the language is (partly) conventionnal, not the proposition, including their intended meaning. We must agree on a minimal amount of reasoning if only to be able to talk about others ways of reasoning. If not: it will be confusing from the start. Bruno - Original Message - From: Kory Heath [EMAIL PROTECTED] To: [EMAIL PROTECTED] Sent: Friday, July 02, 2004 4:10 PM Subject: Re: Mathematical Logic, Podnieks'page ... At 02:45 PM 7/2/2004, Jesse Mazer wrote: As for the non-constructivism definition, is it possible to be a non-constructivist but not a mathematical realist? If not then these aren't really separate definitions. It may be that all non-constructivists are mathematical realists, but some constructivists are mathematical realists as well (by my definition of mathematical realism). So Platonism == mathematical realism and Platonism == non-constructivism are two different statements. I can imagine a non-constructivist asking Are you a Platonist? (thinking Do you accept the law of excluded middle?), and a constructivist answering Yes. (thinking, yes, valid constructive proofs are valid whether or not any human knows them or believes them.) This miscommunication will lead to confusion later in their conversation. -- Kory http
Re: Mathematical Logic, Podnieks'page ...
At 14:20 03/07/04 -0400, Kory Heath wrote: Yes, but some confusions are so easy to avoid! Confusions will always appear in the middle of conversations, but I want them at least to be unexpected ones...! Anyway, I didn't mean to derail the conversation with my jargoning; I was just pointing out that whenever I see platonism in one of these conversations, I'm never sure what we're really talking about. No problem. Let us use arithmetical realism, (for the belief that any (close) arithmetical formula is either true or false, independently of us). I mean first order logic formula ... for those who know what I mean (cf Podnieks page if some wants to know that urgently). Now I recall the problem: by UDA physics (in world/state /situation A) is given by a measure on all computationnal histories going through A and as seen from A. The strategy I have followed consist to ask a sound universal machine what she thinks about that question. I translate the world/state/situation A by a (finite or infinite) set of provable (DU accessible) arithmetical propositions, and I translate all computationnal histories by the set of all maximal consistent extensions of A. Then I show that the measure one or probability one propositions p must satisfy the following conditions: 1) to be true everywhere (= true in all maximal consistent extensions, = []p) 2) to be true somewhere (= true in some consistent extensions, = p) (by Godel 1) does not imply 2) from the machine in A perspective!) This is enough to prove that the probability 1 is quantum like. The miracle comes from the strange and counter-intuitive behavior of the Godel beweisbar (provability) [] predicate. Bruno http://iridia.ulb.ac.be/~marchal/
Re: Mathematical Logic, Podnieks'page ...
Dear Bruno, let me segment your long reply (thanks) and reflect now in the 1st part to your comments on truth. (I may come to the others later, I just beware of milelong posts). I interleave my response. John Mikes - Original Message - From: Bruno Marchal [EMAIL PROTECTED] To: [EMAIL PROTECTED] Sent: Sunday, July 04, 2004 10:32 AM Subject: Re: Mathematical Logic, Podnieks'page ... (1st part) At 06:57 03/07/04 -0400, John M wrote: (Bruno: am I still in your corner?) OK. Let us see. Dear Kory, an appeal to your open mind: in the question whether we discovered math or invented it..., many state that the first version is 'true'. Beside the fact that anybody's 'truth' is a first person decision, Then I would decide to have food when I am hungry, to have water when I am thirsty. I would decide Riemann hypothesis true and even proved by me, and I would decide to get those million dollars. I would decide you to be a platonist, my friend, ... I would decide peace everywhere, ...if truth was a matter of first person *decision*. Seriously, I am afraid you confuse the luckily adequate first person feeling the first person lives in front of truth and truth itself. JM: I think we got into a semantic quagmire. I feel a different meaning in my (5th language English) TRUTH from what I read as the (4th language French) 'verité'. I use 'truth' as the OPINION one accepts as being not false. What you imply sounds to me as 'constructing a reality. Truth has nothing to do with decisionmaking. Decision comes into the picture only in the 1st person thinking to decide whether the item is not false. If I agree, it is (my) truth as well. the fact that anything we may know (believe or find), is interpreted by the ways how our 'human' mind works - SURE! (but it is invalid to infer from that that truth itself depends on our beliefs and findings). Sorry, Bruno, you sound in the parethetized remark as a person who believes in some eternal 'truth' chisled in the (nonexistent) stone of (nonexistent) supernatural 'law', - or rather: takes something like 'truth' as the installations (facts??)of the world. There is no such thing as THE TRUTH - ITSELF at least not among people who think... Maybe some religious fanatic fundamentalists know the truth, the only ONE, worthwhile killing (-dying) for. Even the facts are explanations for observations - and we saw lately discussions on observers. The flat Earth: a fact (Ptolemaios), hell: a fact (A. Dante), the atoms in the molecules I synthesized: facts, then all these things turned into fiction. Props of some belief system. Now let me take a deep breath and if I am still 'on' this list, later I will come back to 'math'. (I don't know Wilfried Hodge, will not read him for this purpose.) Till then, I celebrate July 4th John Mikes SNIP the rest
Re: Mathematical Logic, Podnieks'page ...
At 16:10 02/07/04 -0400, Kory Heath wrote: At 02:45 PM 7/2/2004, Jesse Mazer wrote: As for the non-constructivism definition, is it possible to be a non-constructivist but not a mathematical realist? If not then these aren't really separate definitions. It may be that all non-constructivists are mathematical realists, but some constructivists are mathematical realists as well (by my definition of mathematical realism). So Platonism == mathematical realism and Platonism == non-constructivism are two different statements. I can imagine a non-constructivist asking Are you a Platonist? (thinking Do you accept the law of excluded middle?), and a constructivist answering Yes. (thinking, yes, valid constructive proofs are valid whether or not any human knows them or believes them.) This miscommunication will lead to confusion later in their conversation. True, but if we want to make sure no confusion will ever appear later in the conversation we will never start. So it is better to tackle confusion when they appear. You will tell me that CMR and me were in such state of confusion. I am not so sure. Well, I don't know, and to be clear, using the less confusing _expression_, I will avoid platonism and use arithmetical realism instead. Please pardon me CMR but I will quote your answer, so as to be able to answer you and illustrate my point to Kory at the same time (without sending different cross-referent posts). CMR wrote: Would it not be more to the point to ask whether I believe in an ideal computer, the affirmation of which might be construed as an essentialist view? If in fact all things are subject to entropy, including quantum objects (http://www.maths.nott.ac.uk/personal/vpb/research/ent_com.html), then would not any hardware eventually degrade to a halt? I suppose if the decrepit computer remained structurally complex enough to be potentially universal (Wolfram has suggested a bucket of rusty nails is, for instance !?!) than it could (would?) eventually re-self-organize and start running a new routine. BM: OK. Here I see you postulate physical realism. But I am more sure of the non existence of a highest prime than of entropy or quanta. I don not postulate physical realism, but I postulate arithmetical realism. To come back on Kory, Kory wrote also: KORY: 1. Platonism == Mathematical Realism. 2. Platonism == The belief in Ideal Horses, which real horses only approximate. 3. Platonism == Non-constructivism. So I propose we choose 1. By Godel's theorem 1 implies 3 (even for the intuitionist (= those who discard the excuded middle principle), but non-constructivism will acquire a different meaning, and I never refer to it so let us forget it. So to be clear and simple I will always use the terme platonism in the sense of Classical Arithmetical Realism (Classical = Boolean = admission of all classical tautologies excluded middle principle included (if I can say). For 2, I would say that comp does not entail it, unless you define the ideal horse by the set of its digital approximation done at some level. Obviously ideal computer exist, any definition of something capable to emulate any turing machine will make the job, from c++ to universal unitary transformation in a Hilbert space. With Church thesis we can say the existence of an ideal computer can be proved in and by Peano Arithmetic, like PA can prove the inexistence of two numbers p and q such that (p/q)^2 is 2. With comp the existence of the ideal computer entails the *appearance* of many relatively concrete computer which seems to obey quantum entropic decay ... ... I could suspect CMR of physical platonism and perhaps physical essentialism ;) I must go now (saturday course!), but I want to say something about physical essentialism, and Aristotle substantialism ... Bruno http://iridia.ulb.ac.be/~marchal/
Re: Mathematical Logic, Podnieks'page ...
At 02:17 PM 7/2/2004, CMR wrote: Would it not be more to the point to ask whether I believe in an ideal computer No! It isn't more to the point. You may believe that all physical things are subject to entropy, and that therefore no physical computer could last forever, but you should still be able to talk about whether or not some program *would halt* if it were allowed to run forever. Look at the following program: 1: X = 1 2: X++ 3: if X 1 then halt 4: goto 2 This program would clearly never halt if it were run on an ideal computer - and we can recognize that fact even while believing in entropy and the physical impossibility of running a program forever, etc. So the question is, for every possible finite program, do you believe there's a fact of the matter about whether or not it would halt if we *were* able to run it forever? -- Kory
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At 10:12 AM 7/3/2004, Bruno Marchal wrote: True, but if we want to make sure no confusion will ever appear later in the conversation we will never start. So it is better to tackle confusion when they appear. Yes, but some confusions are so easy to avoid! Confusions will always appear in the middle of conversations, but I want them at least to be unexpected ones...! Anyway, I didn't mean to derail the conversation with my jargoning; I was just pointing out that whenever I see platonism in one of these conversations, I'm never sure what we're really talking about. -- Kory
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At 10:14 01/07/04 -0400, Hal Ruhl wrote: Re the discussion on mathematical realism etc. I ask for comments on whether or not definition that is the division of ALL in to two parts is a mathematical process. To me definition seems arbitrary but some definitions result in mathematical concepts such as the one I use which results in the concepts of incompleteness and inconsistency From this I can infer you are not following classical or more general standard logic where inconsistent theories are trivially complete in the sense that *all* propositions are provable (all the true one + all the false one!). This explains probably why it is hard to me to follow your post. I suggested to you (some years ago) to follow simpler paths, for pedagogical reasons. I read your posts but I have not yet a clue of what are your more primitive beliefs. You over-use (imo) analogies, which can be inspiring for some constructive path, but you don't seem to be able to realize the lack of clarity of your most interesting posts in that regards. I respect your willingness to try, and I hope my frankness will not discourage you. Bruno http://iridia.ulb.ac.be/~marchal/
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Hi Bruno: The idea of my model is that the foundation system has two components one is inconsistent because it is complete - it contains all - and the other is incomplete - it is empty of all. These two components can not join but the incomplete one must attempt to do so - leading to the creation of metaverses. Hal At 10:36 AM 7/2/2004, you wrote: At 10:14 01/07/04 -0400, Hal Ruhl wrote: Re the discussion on mathematical realism etc. I ask for comments on whether or not definition that is the division of ALL in to two parts is a mathematical process. To me definition seems arbitrary but some definitions result in mathematical concepts such as the one I use which results in the concepts of incompleteness and inconsistency From this I can infer you are not following classical or more general standard logic where inconsistent theories are trivially complete in the sense that *all* propositions are provable (all the true one + all the false one!). This explains probably why it is hard to me to follow your post. I suggested to you (some years ago) to follow simpler paths, for pedagogical reasons. I read your posts but I have not yet a clue of what are your more primitive beliefs. You over-use (imo) analogies, which can be inspiring for some constructive path, but you don't seem to be able to realize the lack of clarity of your most interesting posts in that regards. I respect your willingness to try, and I hope my frankness will not discourage you. Bruno http://iridia.ulb.ac.be/~marchal/
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At 03:21 01/07/04 -0400, Kory Heath wrote: At 03:25 PM 6/30/2004, CMR wrote (quoting www.fact-index.com): Mathematical realism holds that mathematical entities exist independently of the human mind. Thus humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same. The term Platonism is used because such a view is seen to parallel Plato's belief in a heaven of ideas, an unchanging ultimate reality that the everday world can only imperfectly approximate. This is a perfect example of what I'm complaining about. The quote implies that the term Platonism can be used as just another term for mathematical realism, but then it immediately provides a definition that goes beyond simple mathematical realism. The belief that mathematical entities exist independently of the human mind - that humans discover mathematics rather than invent it - does not automatically entail the belief that there's a heaven of ideas containing (say) the Essence of Horseness which everyday horses only imperfectly approximate. These two ideas are logically distinct, and it seems sensible to call them by two different names. I prefer mathematical realism and essentialism, or maybe Platonic essentialism. I'd prefer not to use the term Platonism all by itself, but if I had to use it, I'd use it to refer to Platonic essentialism, not mathematical realism. Perhaps you could say more on Platonic essentialism, but I would have attributed the beginning of Essentialism to the Aristotle reading of Plato. Plato is too vague on these question imo. Aristotle essentialism is much more clear especially through the development of modal logic (Aristotle's invention). But it is a complex problem which I find premature. Quine criticized the use of quantifier in modal logic because, he argues, this would reintroduce essentialism in the scientific field. Comp is vaccinated in that respect because the modal logic G and G* have quantifier entirely defined by their arithmetical interpretations, so that there is a clear non essentialist view of them, and at the same time, it explains why some form of essentialism is just inevitable once we listen to the (sound) machine's point of view. Note that in my these I have not use the Gq and Gq* (G and G* first order extension). Ruth Barcan Marcus wrote a book on that Quantifier-in-modal-logic/essentialism question. See http://www.fordham.edu/gsas/phil/klima/ESSENCE.HTM for a nice link with references. Now I agree with you, let us avoid the use of the term platonism (only mathematicians use it for (mathematical) realism. Note that I avoid it most of the time, but I could defend it's use as well, giving that Pythagore and Plato have appreciate it so much. With comp, note, there is a sense to say that not only the almost-one-horse lives in Platonia, but all possible apparently concrete one too. But that is probably a good reason to avoid the terme platonism before being sure everyone grasp that aspect of comp. Sometimes I define an arithmetical realist as someone who believes in all the the propositions of the form (A or not A) with A an arithmetical proposition. That's enough for my use of the term. G. Boolos make a case that there is no notion of alternative world without the use of the (A or not A) exclude middle propositions. I have order his book logic, logic and logic and don't know yet his argument, which I find a priori astonishing giving that you can do (and people does that) intuitionistic modal logic (that is manage a notion of possible world without the exclude middle principle). To finish, Kory, I would avoid the term essentialist giving that its modern philosophical use is more precise than our admittedly rather imprecise use of it. It is better not to use the word more precisely than the way we are using them This reminds me one of my favorite replies by Bruno in the (not so well known) Sylvie and Bruno by Lewis Carroll. By memory: There was a herd of sheeps near Bruno who was talking with the Professor somewhere in the country, and Bruno said oh, look there is about 1004 sheeps there in the field. The Professor told him that he should not say about 1004 but about 1000 giving that about is in contradiction with the precise use of 4. Bruno replied that he was absolutely sure about the four, seeing them near here, and that he was using the about concerning the use of 1000 giving that he could hardly be sure of that! Since, I am used to call that error (suspected by the Professor in Bruno's exclamation), the 1004 error: It is the error consisting of using words in a way more precise than the way you are using them. Not all jargon are 1004 errors, but 1004 errors lead always in the limit toward jargon. Kory, I am not pretending that your are jargoning but I would like to avoid the risk of pointing to the essentialist debate too early, especially without the modal logical tools. But I will try to avoid platonism, and this should
Re: Mathematical Logic, Podnieks'page ...
Hi Bruno: By the way if some systems are complete and inconsistent will arithmetic be one of them? As I understand it there are no perfect fundamental theories. So if arithmetic ever becomes complete then it will be inconsistent. In the foundation system which I believe contains mathematics from the beginning arithmetic is complete so its inconsistent. Hal
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Hi Hal, At 12:44 02/07/04 -0400, Hal Ruhl wrote: By the way if some systems are complete and inconsistent will arithmetic be one of them? As I understand it there are no perfect fundamental theories. So if arithmetic ever becomes complete then it will be inconsistent. Yes, if by arithmetic you mean an axiomatic system, or a formal theory, or a machine. No if by arithmetic you mean a set so big that you cannot define it in any formal theory, like the set of all true arithmetical sentences. That set cannot be defined in Peano arithmetic for exemple. Some logician use the word theory in that generalized sense, but it is misleading. Now the set of true sentence of arithmetic is that large sense is obviously consistent gieven that it contains only the true proposition! (but you cannot defined it mechanically). In the foundation system which I believe contains mathematics from the beginning arithmetic is complete so its inconsistent. No, because if it is complete, it will not be a mechanical or formal system. Only a theory will be inconsistent if both complete and enough rich. Not a model. To borrow Boolos title, I would like to say I get the feeling this list is missing the key road: Logic, logic and logic BTW an excellent introduction to elementary logic is the penguin book by Wilfried Hodges : http://www.amazon.co.uk/exec/obidos/ASIN/0141003146/qid=1088787942/sr=1-2/ref=sr_1_26_2/026-1716457-4246007 Only the first sentence of the book is false. (will say more on that book later ...) Bruno http://iridia.ulb.ac.be/~marchal/
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To finish, Kory, I would avoid the term essentialist giving that its modern philosophical use is more precise than our admittedly rather imprecise use of it. I see what you mean, but we need *some* way of referring to specific (although perhaps imprecise) ideas or beliefs. I might feel comfortable defining Platonic essentialism as the belief that there exists a world of essences in which (say) the Ideal Horse exists, and all physical horses are imperfect copies of it, because I don't think this group already has multiple conflicting definitions of the term Platonic essentialism. However, this group definitely does have multiple conflicting definitions of the generic term Platonism, and people usually just assume their own definition when they hear the term. So someone asks someone else if they're a Platonist, and that person ends up answering a totally different question. Hi-larity ensues! Kory, I am not pretending that your are jargoning but I would like to avoid the risk of pointing to the essentialist debate too early I agree, and in fact, avoiding the essentialist debate is exactly what I'm trying to do. My point is that every time we use the term Plantonism simply to refer to arithmetical realism, we run the risk of starting an essentialist debate (or a constructivist debate) that we didn't intend, because for many other people Platonism implies essentialism, or non-constructivism. -- Kory
Re: [InfoPhysics] Re: Mathematical Logic, Podnieks'page ...
At 03:09 PM 7/1/2004, Jim Whitescarver wrote: Platonist reasoning is the antithesis of constructionism. Thanks for the clarification. In this short discussion I've seen at least three conflicting ways that people use the term Platonism: 1. Platonism == Mathematical Realism. 2. Platonism == The belief in Ideal Horses, which real horses only approximate. 3. Platonism == Non-constructivism. -- Kory
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Greetings Bruno, This is equivalent to say yes in the test for "platonism" given in the Podnieks page.CMR, do you believe that a running program (on an ideal computer) will stop, or will not stop? Would it not be more to the point to askwhether I believe in an "ideal" computer, the affirmation of which might be construed as an essentialist view? If in factall "things" are subject to entropy, including quantum objects (http://www.maths.nott.ac.uk/personal/vpb/research/ent_com.html), then would not any "hardware" eventually degrade to a "halt"? I suppose if the decrepit computerremained structurally complex enough to be potentially universal (Wolfram hassuggested "a bucket of rustynails" is, for instance!?!) than it could (would?) eventually re-self-organize and start running a new "routine". Cheers CMR- insert gratuitous quotation that implies my profundity here -
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Kory Heath wrote: Thanks for the clarification. In this short discussion I've seen at least three conflicting ways that people use the term Platonism: 1. Platonism == Mathematical Realism. 2. Platonism == The belief in Ideal Horses, which real horses only approximate. 3. Platonism == Non-constructivism. Roger Penrose uses the word mathematical Platonism to describe his philosophy of math, which is clearer in that it obviously does not require believing in such a beast as the Ideal Horse. As for the non-constructivism definition, is it possible to be a non-constructivist but not a mathematical realist? If not then these aren't really separate definitions. Jesse
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Just so my friend Jim's comments to Kory will have some context: From: Jim Whitescarver [EMAIL PROTECTED] Subject: Re: Re: Mathematical Logic, Podnieks'page ... Yes Kory, one needs to be explicit about what they mean by Platonist. I try to be explicit, by Platonic thinking, logic or reasoning I mean: 1. Platonic logic: law of excluded middle, a proposition may be true or false, there is no third alternative. Proof by induction is not questioned. Logical systems are necessarily incomplete. 2. Platonic existence: that which exists need not be constructible, infinities may be invoked at will and are attributed actuality. Platonist reasoning is the antithesis of constructionism. In constructionism you can have a set of points equal distance from one point but the set of all such points is considered imaginary, not real. You may have irrational numbers but only those generated by the countable set of algorithms exist. Others are random and cannot be constructed by any algorithm and therefore cannot exist. Jim Kory Heath wrote: At 09:19 AM 6/30/2004, Bruno Marchal wrote: Also, you said that your are not platonist. Could you tell me how you understand the proposition that the number seventeen is prime. (I want just be sure I understand your own philosophical hypothesis). CMR - insert gratuitous quotation that implies my profundity here -
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At 02:45 PM 7/2/2004, Jesse Mazer wrote: As for the non-constructivism definition, is it possible to be a non-constructivist but not a mathematical realist? If not then these aren't really separate definitions. It may be that all non-constructivists are mathematical realists, but some constructivists are mathematical realists as well (by my definition of mathematical realism). So Platonism == mathematical realism and Platonism == non-constructivism are two different statements. I can imagine a non-constructivist asking Are you a Platonist? (thinking Do you accept the law of excluded middle?), and a constructivist answering Yes. (thinking, yes, valid constructive proofs are valid whether or not any human knows them or believes them.) This miscommunication will lead to confusion later in their conversation. -- Kory
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At 03:25 PM 6/30/2004, CMR wrote (quoting www.fact-index.com): Mathematical realism holds that mathematical entities exist independently of the human mind. Thus humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same. The term Platonism is used because such a view is seen to parallel Plato's belief in a heaven of ideas, an unchanging ultimate reality that the everday world can only imperfectly approximate. This is a perfect example of what I'm complaining about. The quote implies that the term Platonism can be used as just another term for mathematical realism, but then it immediately provides a definition that goes beyond simple mathematical realism. The belief that mathematical entities exist independently of the human mind - that humans discover mathematics rather than invent it - does not automatically entail the belief that there's a heaven of ideas containing (say) the Essence of Horseness which everyday horses only imperfectly approximate. These two ideas are logically distinct, and it seems sensible to call them by two different names. I prefer mathematical realism and essentialism, or maybe Platonic essentialism. I'd prefer not to use the term Platonism all by itself, but if I had to use it, I'd use it to refer to Platonic essentialism, not mathematical realism. -- Kory
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Re the discussion on mathematical realism etc. I ask for comments on whether or not definition that is the division of ALL in to two parts is a mathematical process. To me definition seems arbitrary but some definitions result in mathematical concepts such as the one I use which results in the concepts of incompleteness and inconsistency. Hal
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Hal, I agree (whatever it is worth) with the (empty or not) 'set' being 'something' in the nothing. (I started my 'naive otology (1991) with 'nothingNESS', which is also more than (your) nothing: containing an ontological qualifier, so it became SOMETHINGNESS which was the startup of the world. (Never mind that now, just reminiscence). - ) I had no 'smart'(?) ideas on what moves, but had some headaches with the vacuum energy (I read about it by D. Bohm) and concluded that it may be a physicistical quantizing of the (OOPS!) creation: the vacuumenergy-amount of allegedly 10^124 times that of the total energy content of the material universe - contained in 1 ml of vacuum (I did not make that up) - was assigned in my mind to the 'work' to make 'nothing' into 'something'. (Just for the fun of it.) Silly idea but at that time I had no better one - nor do I have now. ^ To your more recent post upon Kory's rather technical complaints to CMR's quote: (Mathematical realism holds that mathematical entities exist independently of the human mind...) I have a principal complaint: How do we learn about things existing independently of our mind? by some 'unidentifiable' input how it is interpreted by the mind (the essence of 1st person ideas). So it is our mind that 'makes up' the mathematical concepts which may exist in the natural world quite differently. Our response. As I recall I mentioned D. Bohm's remark that numbers do not exist in nature, they are human inventions or something of that kind, and as I remember it was CMR who retorted (correct me if I remember wrong) that your mind is part of the world and the numbers exist in it, how can you maintain Bohm's statement? [quote approximate]. Discounting what came first WE may conclude that if mathematics came from nature - it came from our interpretation. An inventive discovery. Just like the space-time coordination which led to motion. Sorry for the common sense rambling John Mikes - Original Message - From: Hal Ruhl [EMAIL PROTECTED] To: [EMAIL PROTECTED] Sent: Wednesday, June 30, 2004 2:38 PM Subject: Re: Mathematical Logic, Podnieks'page ... Hi Stephen: At 01:14 PM 6/30/2004, you wrote: Dear Hal, Could the Nothing be a generalization of the notion of the Null or Empty set? I think the Null or Empty sets are more particular than my Nothing since they include all the underpinnings supporting the idea of set. One question that I have is what moves? It seems that I am merely re-asking Zeno's question... How is motion, whether it is the UD moving infinitely slowly from string to string or your example of a shackwave, what is the reason MOTION exists? What necessitates motion and change a priori? In our universe we identify something called a vacuum energy. I see the incompleteness of the Nothing as such a prime mover if you will. The initiator is sort of a symmetry breaking when the Nothing must answer a meaningful question. Once this starts it acts rather like a formal system attempting to complete itself - an empty quest. This provides the motivator for the evolution of the particular metaverse associated with this particular symmetry breaking. I do see the evolution process as digital so there is no motion as we usually interpret it. A universe just winks out between successive states. In this case relativity and quantum mechanics seem to me to be simple consequences. Hal
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At 09:02 29/06/04 -0700, CMR wrote: Here's one reasonably functional definition of science: sci·ence( P ) Pronunciation Key (sns) n. 1. The observation, identification, description, experimental investigation, and theoretical explanation of phenomena. 2. Such activities restricted to a class of natural phenomena. 3. Such activities applied to an object of inquiry or study. BM: OK. That is very large. You know there are borderline. Bohr has dismiss the EPR paper as metaphysics, and he did that by imposing its own metaphysics, etc. And of course such a definition is a user-description, it is not an attempt to define it for a deeper epistemological study. What *is* science, is also a object of inquiry. CMR: I find its not uncommon for those who may chafe at the inconvenient constraints of science as defined above to be somewhat dismissive of its special utility in generating knowledge about our world(s). The creationists often leverage this tactic, for instance. Just as often the label science is co-opted by occultists to lend credibility to otherwise incredible claims They'd all like to cast it as just another world view intrinsically no more valuable than any other. But it's not.. It's not because science as a methodology ignores that which is by necessity matters of faith, be it religion, mysticism, metaphysics (or Platonism?). BM: Absolutely. Science as a methodology ignores that which is by necessity matters of faith. But how many scientist are aware that the existence of a *physical* universe is a matter of faith? Many scientist quickly consider (like Bohr) question which they cannot solve or formulate in the language of their field as metaphysical, but in general the frontier between science and metaphysics are either methodological or metaphysical, or historical. Also, why do you put platonism along with faith. The level of clarity and seriousness of a text like the Thaetetus is rarely met these days. And Plato has less ontological commitment than Aristotle, and many scientists today keep some Aristotelian act of faith without ever mentionning it, apparently they are not aware of their act of faith. *This* is unscientific attitude, no? CMR: Is science sometimes (often?) malpracticed by agenda driven egos? Certainly, but that doesn't diminish the utility or validity of science well executed. BM: Surely John and me were a little ambiguous in our discussion on science. I thought we were discussing what science *is*, not the shape of actual human science. That's why I say science is merely the product of inquiry, humility, and curisosity. In front of unsolved hard problem, it is also the ability to recognize our prejudice, and to keep an open mind. CMR: Any and all philosophers, mystics and mathemiticians can and are welcome to minimize, reject and even appropriate science as they will. And so it should be in a free society. But if and when they claim their faith-based musings are scientific or as good as same, then they are charlatans in deed as well as name, IMHO. BM: I agree 100% (if you add physicists, biologists, ... in your list). The pity, today, is that most scientist are specialized, and their keep their scientific attitude only in their discipline, and lack it completely once they talk about anything outside (except perhaps on soccer). In particular a lot of naturalist (materialist, physicalist), like Changeux, are just totally UNscientific when they pretend that all honest scientist should be naturalist, or materialist, ... The same for the platonist. Platonism is unscientifical only when it is presented as being the only way science should be. But naturalism or platonism per se are quite respectable views or departure point. Now the UDA shows that the first is logically incompatible with the computationalist hyp., the other is not. Have you see this? Also, you said that your are not platonist. Could you tell me how you understand the proposition that the number seventeen is prime. (I want just be sure I understand your own philosophical hypothesis). Bruno http://iridia.ulb.ac.be/~marchal/
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At 12:42 29/06/04 -0400, Hal Ruhl wrote: I have enjoyed my first looks at Podnieks' page. Bruno thanks for the URL . My issue is that my model while it has changed many times seems to persistently return me to the idea that while some metaverses may be otherwise Turing computable all metaverses are subject to input from what might be considered an external - to them - random oracle. The system that embeds these metaverses - a dual simultaneous existence of a Nothing and an Everything seems inconsistent and incomplete so its not Turing computable as I understand the term. This seems to put my view in conflict with Comp. If your system is inconsistent then it is obviously Turing computable (just write a generator of ALL arithmetical formula). But I am not sure your system is inconsistent. Well, I am not sure it is a system, or perhaps you just fail to present it as such, probably. Bruno
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Dear Hal, Could the Nothing be a generalization of the notion of the Null or Empty set? One question that I have is what moves? It seems that I am merely re-asking Zeno's question... How is motion, whether it is the UD moving infinitely slowly from string to string or your example of a shackwave, what is the reason MOTION exists? What necessitates motion and change a priori? Stephen - Original Message - From: Hal Ruhl [EMAIL PROTECTED] To: [EMAIL PROTECTED] Sent: Wednesday, June 30, 2004 12:18 PM Subject: Re: Mathematical Logic, Podnieks'page ... Hi Bruno: At 09:34 AM 6/30/2004, you wrote: If your system is inconsistent then it is obviously Turing computable (just write a generator of ALL arithmetical formula). But I am not sure your system is inconsistent. Well, I am not sure it is a system, or perhaps you just fail to present it as such, probably. Bruno As for my model and its system I was referring to my post of June 8 which because I can not get on the escribe site to get the URL right now I have copied below. Ok so if I accept that the Everything half of the system is Truing computable what about the Nothing half which is the incomplete part. In this case there is no output. So if indeed evolving metaverses are the result of an interaction between the two then they can only be incomplete and evolve inconsistently. xx Prior post: 1) Given that the following definitions are sound: The Everything: That which contains all. The Nothing: That which is empty of all. A Something: A division of the Everything into two subparts. 2) These are unavoidable because at least one must exist 3) They are interdependent so that you can not have one without the whole set. 4) Notice that Definition is the same as establishing a boundary between what a thing is and another thing that is all that the first thing is not. 5) The Nothing has a logical problem: It can not answer any meaningful question about itself including the unavoidable one of its own stability. 6) To answer this unavoidable question the Nothing must at some point penetrate the boundary between itself and the Everything in an attempt to complete itself. 7) However, the boundary is permanent as required by the definitions and a Nothing remains. 8) Thus the penetration process repeats in an always was and always will be manner. 8) The boundary penetration produces a shock wave [a boundary] that moves into the Everything as the old example of Nothing tries to complete itself. This divides the Everything into two evolving somethings - evolving multiverses. Notice that half the multiverses are contracting. 9) Notice that the Everything also has a logical problem. Looking at the same meaningful question of its own stability it contains all possible answers because just one would constitute a selection i.e. net internal information which is not an aspect of the all content of the Everything. Thus the Everything is inconsistent. 10) Thus the motion of a shock wave boundary in the Everything must be consistent with this inconsistency - That is the motion is at least partly random. 11) Some of these evolving Somethings will admit being modeled as UD's with true noise.
Re: Mathematical Logic, Podnieks'page ...
Greetings Bruno and Kory, Also, you said that your are not platonist. Could you tell me how you understand the proposition that the number seventeen is prime. (I want just be sure I understand your own philosophical hypothesis). A quick aside: It might be better not to even use the term platonist in these discussions, because it means at least two different things. It can be used to refer to Plato's essentialism - the idea that there's a world of Forms in which exists (for instance) the Ideal Horse, and all physical horses represent imperfect copies of this Horse. This is certainly a more elaborate belief than mathematical realism (or arithmetical realism, or computational realism). One can be a mathematical realist without being an essentialist. I am. So some people would call me a Platonist, and some wouldn't, but that's just a disagreement about a definition. I prefer just to use the term mathematical realism or essentialism, depending on what I'm talking about. There would seem to be some difference of opinion on this view: Mathematical realism holds that mathematical entities exist independently of the human mind. Thus humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same. The term Platonism is used because such a view is seen to parallel Plato's belief in a heaven of ideas, an unchanging ultimate reality that the everday world can only imperfectly approximate. Plato's view probably derives from Pythagoras, and his followers the Pythagoreans, who believed that the world was, quite literally, built up by the numbers. This idea may have even older origins that are unknown to us. http://www.fact-index.com/p/ph/philosophy_of_mathematics.html I'd have to agree that mathematical realism smacks of essentialism to me as well. Thus my reservations regarding it. But my real point here is that, for myself, all isms including Platonism are merely maps (models?) and the world the territory, to paraphrase Korzybski. Mathematics is, I believe, one of those maps. Hard pressed for a label, I'd guess that I probably fit most well as a non-Aristotelian if anything (but I'm not sure they'd have me). But in truth I tend to be like bacterium where my world view is like it's genome: I take a little here, a little there from various compatible isms and assimilate the parts that seem to fit well, averaging across many maps to better grok the territory. Ultimately though, I suppose my main man would be Socrates, if I had to choose one (and apparently I just did). Plato would have done well to assimilate more of his mentor's methodology, IMHO. He might have been more competitive with the Ionians had he done so. On the science topic: Natural History magazine this month has an article on the anthropic universe, the cosmological constant and cosmology. It cites the multi-verse as one theory gaining popularity in explaining the constant's otherwise apparently arbitrary value. The author quotes Tegmark as well as some soft multi-versers and of course the skeptics who tend to see a meta-verse solution as a cop-out and, in at least one view, akin to a religious mythological tale. These last bemoan what they see as a premature abandonment of rigourous physics methodology in pursuit of an instant TOE. To this lot the rumors of the end of science are greatly exaggerated, I would imagine. Look, all I know is that the world(s) apparently proceeds from state to state and exihibits patterns of varying degrees of order and (psuedo/) randomness. I suspect that this is likely the consequence of simple underlying rules or a single rule. Thus the world contains information, IMHO. That these patterns map reasonably (remarkably?) well to a meme that we call mathematics and that first appeared at a recent juncture(s) of that procession seems clear. Am I a mathematical realist? You tell me.. Cheers CMR - insert gratuitous quotation that implies my profundity here -
Re: Spam Alert: Re: Mathematical Logic, Podnieks'page ...
CMR:( I quote your earlier points here about 'science' to explain why I called them reductionistic.) 1. The observation, identification, description, experimental investigation,and theoretical explanation of phenomena.2. Such activities restricted to a class of natural phenomena.3. Such activities applied to an object of inquiry or study. #1 works with (observed/able) phenomena: models according to our epistemic enrichment at the time of study. Identification, description, especially explanation is topical, observing the boundaries within which we stay put - disregarding the rest - (whether known or unknown) within our reduced models. This is what I call 'thereduction of the total'. It is even enhanced in #2, "restricted to the chosen class" - while #3 puts the crown on its reductionistic head: applying activities [only?] to the limited (topical, boundary-enclosed) models. If your vocabulary sais different from reduction of the total into limited models, we have to smoke the calumet for using different vocabularies in peace. You wrote: "A methodwith clearly identifiedacceptable methodology. No more. No less." And I seek understanding. I don't believe to find it in your "scientism"G. I may have used the wrong adverb: not "reductionist science", only "science" as we know it.Conventional. It is a topically reduced segregated-parcelled modeling of nature into topics all considered assubstantial units - while really in an interconnected total where separating barriers exist only in our organizing effort. Complexity is a loaded historical noumenon, almost as unidentifiable as consciousness. I am not talking about the "reductionistic limited models" that are complex, have a theory and work in a formalism of acknowledging the limited model values as 'complete values' in the equational math treatment. If I have to use the word, I mean the complexity of the total in unabridged interinfluencingreciprocity. The word allows flexible semantics. I prefer to say wholeness. Not even 'hole-ism'. It is hard to skip the belief-system we were brainwashed into during our early studies. Maybe you can find more in http://pages.prodigy.net/jamikes/SciRelMay00.html This was written before I foramlized my thinking about reductionism, but applicable. And NO End of Science! The good old reductionist edifice is very good and useful, a NEW WAY of doing scientific activity may be in the works. Give it 2-300 years. "[Conventional]...science issignificantly less lousy than all the alternative approaches ..." which is not good enough for me. Finally as you could see from all the above: Igladlyagree with your final remark: "I like, respect and even largely share your apparent philosophy, John. But it ain't science." I hope so and appreciate the preceding to it. John M - Original Message - From: CMR To: [EMAIL PROTECTED] Cc: John M Sent: Tuesday, June 29, 2004 5:06 PM Subject: Spam Alert: Re: Mathematical Logic, Podnieks'page ...
Re: Mathematical Logic, Podnieks'page ...
At 15:38 28/06/04 -0400, John M wrote: 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. Reply-BM: We surely differ. I am not sure the word science really refers to anything. Scientific attitude exists though. About it the words and expressions like *curiosity*, *modesty*, *clarity*, *willingness to share*, etc.. comes to my mind. I agree there has been, in the human story, attempts to build reductionist theories, but they have all failed, and with comp, by Godel II, it is necessarily so. 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. Reply-BM: I'm afraid I don't understand. 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. Reply-BM: I suspect a terrible confusion due to a probably subtle point which has begin to be clear to me only when I begin to understand the abyssal gap between the notion of total computable function and partial computable function. Or Godel's incompleteness theorems. Cf the diagonalisation posts. COMP is just the (religious? meta-religious?) belief that there exists a level of description of you such that you are not aware of any difference in your life after a digital substitution has been made at that level. (+ Church thesis, + a minimal amount of arithmetical realism). It is the nuance brought by GODEL II which makes COMP not reductionnist. JM: (I evaded: 'quantities'). Which means the omission of aspects OUTSIDE such qualia and rules. Reply-BM: Yes, but apparently just because you evade quantities, it seems to me. JM: 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). Reply-BM: Which limitations ? I am not sure I understand. JM: 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. Reply-BM: ? You loose me. JM: do we have ANY other knowledge-base? Proof (Popper's no-no) is within the belief system. True is a 1st pers. judgement. Even an 'accepted' 3rd p. truth is 1st p. accepted. Reply-BM: I agree. JM: I haven't (yet?) included the universal mchine into my vocabulary. It is not 'simple' (see above). Reply-BM: Thanks for your admission. It is the key notion of comp. JM: One remark to math vs science: I consider math a human language, a mental activity (again this term!) on its own, (uninhibitied by observational models - only by its intrinsic connotations). Reply-BM: I really do not consider math as a language. Math papers are written (mostly) in English (or in German, French, Russian, etc.). Mathematicians uses abreviations, drawings, and are keen to abstract by the very often use of symbolic variables giving the impression it is a language by itself, but it is not. Mathematicians like Pyhtagore Cantor, would never have hide results if that was only languages. Godel's theorem is often use to defend platonism at least in computer science and arithmetic, and I find the argument compelling. But any book on number theory is enough to illustrate this. Even many physicists agree there is a mathematical reality. The irrationality of the square root of 2 is neither a piece of language, nor a convention, but a (startling) observation. A discovery. JM: Science, however, is a reductionist parcelling of observations - according to the epistemic level of the age, the cognitive inventory and its connectional capabilities of the by that time acquireds. Reply-BM: I understand why you say that (given the amount of reductionnist scientist), but such a reductionnism is the product of a betray of science spirit. We should not confuse the often use reductionnist parcelling of observations, which could be a good method of observation, with the attempt to guess the reality beyond. To be short I would say that science for me is just honesty. The confusion between reality and the parcels is produced by sleepy conscience (of course that occurs all the time, and science asks for ever vigilance). JM: Science applies math in its formalizing of deductions, but such math is quantitatively distorted - adjusted to the models and the observations it pertains to. Which is also subject to the actually
Re: Mathematical Logic, Podnieks'page ...
Reply-BM: We surely differ. I am not sure the word science really refers to anything. Scientific attitude exists though. About it the words and expressions like *curiosity*, *modesty*, *clarity*, *willingness to share*, etc.. comes to my mind. I agree there has been, in the human story, attempts to build reductionist theories, but they have all failed, and with comp, by Godel II, it is necessarily so. Here's one reasonably functional definition of science: sci·ence( P ) Pronunciation Key (sns) n. 1. The observation, identification, description, experimental investigation, and theoretical explanation of phenomena. 2. Such activities restricted to a class of natural phenomena. 3. Such activities applied to an object of inquiry or study. I find its not uncommon for those who may chafe at the inconvenient constraints of science as defined above to be somewhat dismissive of its special utility in generating knowledge about our world(s). The creationists often leverage this tactic, for instance. Just as often the label science is co-opted by occultists to lend credibility to otherwise incredible claims They'd all like to cast it as just another world view intrinsically no more valuable than any other. But it's not.. It's not because science as a methodology ignores that which is by necessity matters of faith, be it religion, mysticism, metaphysics (or Platonism?). Is science sometimes (often?) malpracticed by agenda driven egos? Certainly, but that doesn't diminish the utility or validity of science well executed. Any and all philosophers, mystics and mathemiticians can and are welcome to minimize, reject and even appropriate science as they will. And so it should be in a free society. But if and when they claim their faith-based musings are scientific or as good as same, then they are charlatans in deed as well as name, IMHO. Cheers CMR - insert gratuitous quotation that implies my profundity here -
Re: Mathematical Logic, Podnieks'page ...
I have enjoyed my first looks at Podnieks' page. Bruno thanks for the URL . My issue is that my model while it has changed many times seems to persistently return me to the idea that while some metaverses may be otherwise Turing computable all metaverses are subject to input from what might be considered an external - to them - random oracle. The system that embeds these metaverses - a dual simultaneous existence of a Nothing and an Everything seems inconsistent and incomplete so its not Turing computable as I understand the term. This seems to put my view in conflict with Comp. Hal
Re: Mathematical Logic, Podnieks'page ...
Reply to Bruno's Tuesday, June 29, 2004 10:13 AM
post
Subject: Re: Mathematical Logic, Podnieks'page
Dear Bruno, it seems our ways of expressing
thoughts and sights is so different that in spite of many agreeable
pointsa detailed discussionwould grow out of the framework of the
list.
I want to concentrate on a few minor(?) points -
leaving out the rest of the posts.
Science.
I am in your corner, however I spoke about the
"official" terror of science establishment, the editors, tenure-professors,
Nobel people, etc. control freaks. This type of science is perfectly described
in today's post of CMR in his points, identifying "reductionist
science":
1. The observation, identification, description, experimental
investigation,and theoretical explanation of phenomena.2. Such
activities restricted to a class of natural phenomena.3. Such activities
applied to an object of inquiry or study.
all pertinenet to mind-interpreted and
boundary-enclosed models as observations in the topics we study.
I was shocked when you wrote:
"...I am not sure the word "science"
really refers to anything." and after a while I agreed. Chacqu'un a son go^ut.
Today's fashion is emphasizing in the west the applied math -involved
formalistic 'language' (which is a topic I will come back
to).
I would not degrade the reductionist ways:
whatever we achieved in technology is based on them. (Read e-mail, use a car,
eat cooked food, take an aspirin, etc.) they are just not efficient in
"understanding the world" - anymore.
Simplicity.
In my wholistic view everything is within
unlimited interinfluencing in the universe (this one). No random, no
singularity, so everything is infinitely complex - unless we cut it off into
boundaries of our attention and disregard the off-limits. Then things become
simple.
Special thanks to Hal for his today's post, in
which he emphasized a qualifier ('to them'):
"...input from what might be considered an
external - to them - random oracle." I read this as: 'random', irrelevant as in
'having nothing to do with circumstances of a Turing computability - and ONLY in
this respect. We cut our models to be
considered.
I referred th "The Cause" (one) for effects,
that indeed are the synthesis of unlimited occurrences (influences, two-way
functions) whatsoever, except for our limiting (topical?) boundaries which allow
ONE to be overwhelmingly acknowledged. (Reductionistically).
"evade quantities"
The incomplete 'scientific' (reduced) models
omit connotations beyond their boundaries
(topical, qualia, magnitudes, etc.) so a
definite "quantizing" value should become feasible. It is not the value
(quantity) of the named (concept) item, only of the model in attention.
Formalism works with them and practical results
are obtained for technology. Applied math serves for assuring the equational
'truth' in such 'science'. That's what I called an "edifice" of sci.
(Sorry, 'nonreductionistic' Comp by Godel II
is beyond me).
Limitations:
compare the limited model with the unlimited
(natural?) "maximum model", an image of the named item as connected to the total
of the world. A silly example: you expect the Board of Co. 'C' to vote according
to the well established interest of Co. 'C' (= limited model). Yet board
members are also board members of companies X,Y,Z,R,L,M and have vested interest
in legal processes, educational aspects, international affairs, relatives,
lovers, health problems, perversities, hobbies, so all these influence (in the
wider model) the voting outcome. It may not fit the interest of Co. 'C' at all.
The Chairman cuts off all those esoteric side-interests in a reductionist
limitation and will get the limited-model voting FOR Co.'C' only. It is still
not wholism, just an illustration of thewidening of the
boundaries.
Wholistic thinking is in its early embryonic
stage, has no adequate language, just as a
toddler (sorry for writing embryo) does not
(yet) have the words to confer about Godel.
And I did not even mention understanding, just
the words.
Language I mean as much more than syntax and
semantix, I consider it a way to communicate symbols as they occur in the
development. Matematicians try to describe their "math-language" (ideational
symbolics?) in diverse human vocabulary-talks, yet what they 'think' in is still
math. Feelable, as J.v.N. said. In this respect I value it as aprimary
item in the human mind (not the way Platonists say), comparable maybe to the
mother-tongue.
Not so with 'that' reductionistic
establishment-science I talked about above.
I am strongly with you in the (free)
science-concept with the connotations you mentioned.
I think this was more than I wanted to write onlist.
Thanks for your considerations, it helps in clarifying my obscure thinking.
John
----- Original Message -----
From
Re: Mathematical Logic, Podnieks'page ...
Science. I am in your corner, however I spoke about the "official" terror of science establishment, the editors, tenure-professors, Nobel people, etc. control freaks. This type of science is perfectly described in today's post of CMR in his points, identifying "reductionist science": With respect, no it's not "reductionist science". It is in fact and precisely, just science. We don't have to like it or find it particularly useful in determining the true "nature" of nature. But that's what it is.A methodwith clearly identifiedacceptable methodology. No more. No less. Qualifying it with adjectives tend totake it out of the realm of a practice and into the morass of a "school", like structuralism, existentialism, Marxism, holism..(pick your ism). I in fact tend towards a holistic philosophy of lifeand world view. ButI don't confuse my model with the practice of science. Rigorous scientific investigation of complexity theory, for example, lends support to my model. But it is the very "complexity" you allude to that limits the utility of the scientific method in generating accurate knowledge about the world(s). This is the dim boundary of that ever expanding circle of knowledge Einstein alluded to. Just beyond it lies the fuzzy but oh so satisfying realm of conjecture. Beyond that lies matters of faith. The End of Science? Hardly, but it does reflect the difficulty we embedded monads (another model of "reality") have in objectifying our world. Fortunately like Democracy, science issignificantly less lousy than all the alternative approaches (including any "ism"one might favor) to gaining knowledge of equally embedded systems. I like, respect and even largely share your apparent philosophy, John. But it ain't science. Cheers CMR- insert gratuitous quotation that implies my profundity here -
Re: Mathematical Logic, Podnieks'page ...
To try to avoid confusion on what I meant I find my model telling me that all metaverses will experience the injection of new information to some non zero degree. Some metaverses are Turing computable between such events. The new information is as if from a random external oracle. The to them was to modify external. If one steps out to the system that contains all metaverses one finds that it is not Turing computable because it is both incomplete and inconsistent. Any metaverse is already incomplete so it is the full system level inconsistency that leaks in as the metaverse evolves trying to complete itself. Each of these leakage events can be looked at as resetting the computer with new program/data. This has interesting potential application. For example if our universe indeed has a maximum informational density then the added information must cause space to expand. As our universe gathers more information the area of its interface with the information in the full system increases so we get a positive feed back situation - The so called Dark Energy and its acceleration effect on the expansion. Hal
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 pre
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
Re: Mathematical Logic, Podnieks'page ...
Dear Bruno,
I did some browsing in the Podieks website and
found interesting statements.
Without connotation and order:
*
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.
Remark: provided that YOUR mind is "out of this
world" and stays unchanged 'as is' after (the rest of) the universe was
gone.
Another point is "science" but I let it go now.
(cf: Is math 'part of science'?)
*
The JvNeumann quote:
In mathematics you don't understand things. You just get used to
them.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?) - 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.
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.
*
Let us assume that PA is consistent. Then only computable
predicates are expressible in PA.
("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.
No 'discrete objects': everything is
interconnected at some qualia and interactivity level.
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." -
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.
John Mikes
- Original Message -
From:
Bruno Marchal
To: [EMAIL PROTECTED]
Sent: Saturday, June 26, 2004 11:30
AM
Subject: Mathematical Logic,
Podnieks'page ...
Hi George, Stephen, Kory, All.I am thinking hard finding to
find a reasonable way to explain thetechnical part of the thesis, without
being ... too much technical.The field of logic is rather hard to
explain, without beinga little bit long and boring in the beginning
:(At least I found a very good Mathematical Logic Web page:http://www.ltn.lv/~podnieks/index.htmlThe page
contains also a test to see if you are platonist (actually it testsonly if
you are an arithmetical realist!). Try it!From that page I will be
able to mention easily set of axioms, and rules.For example below are
the non logical axioms of Peano Arithmetic.Does it makes intuitive sense
?I suggest you try to find the logical axioms and the inference rules
inPodnieks page.
SKIPAny comments ?BrunoPS I have finished my french
paper, and I will write the paper forAmsterdam. The goal is always the
same: how to be clear, short andunderstandable (given the apparent
"enormity" of the result!)
http://iridia.ulb.ac.be/~marchal/
