Re: Fw: Something for Platonists
But in fact, the only thing that privileges the set of all computational operations that we see in nature, is that they are instantiated by the laws of physics. I would dispute this. The set of computable operations may also be privileged in that only a universe with laws of physics that instantiate all of these operations and none others can evolve intellegence (or alternatively these universes have the greatest chance of evolving intelligence). It is only through our knowledge of the physical world that we know of the di.erence between computable and not computable. So it's only through our laws of physics that the nature of computation can be understood. It can never be vice versa. So Deutsch is basically saying that we should not rule out the possibility that we may discover a new law of physics that will allow us to solve the halting problem, for example. I agree with this, given that we don't know that what I wrote above is actually true (instead of just a possibility).
Re: Fw: Something for Platonists
Wei Dai wrote: But in fact, the only thing that privileges the set of all computational operations that we see in nature, is that they are instantiated by the laws of physics. I would dispute this. The set of computable operations may also be privileged in that only a universe with laws of physics that instantiate all of these operations and none others can evolve intellegence (or alternatively these universes have the greatest chance of evolving intelligence). It is only through our knowledge of the physical world that we know of the di.erence between computable and not computable. So it's only through our laws of physics that the nature of computation can be understood. It can never be vice versa. So Deutsch is basically saying that we should not rule out the possibility that we may discover a new law of physics that will allow us to solve the halting problem, for example. I agree with this, given that we don't know that what I wrote above is actually true (instead of just a possibility). Deutsch has mantained consistently that the Church-Turing Hypothesis (essentially Computable = Turing Computable) is undercut by the Bennett- -Church-Turing Hypothesis (essentially that Physically Computable = Turing Computable). Bennett bever agreed to this but that may be beside the point, these days. A few people have been involved in what is called SuperTuring computing spiked by the whole Quantum Computing revolution but not limited to it... The following paper deals with these issues specifically with some of what Jesse Mazer brought up in this discussion: http://arXiv.org/abs/math.GM/0305055 or http://alixcomsi.com/The_formal_roots_of_Platonism.htm Check it out... -Joao Leao -- Joao Pedro Leao ::: [EMAIL PROTECTED] Harvard-Smithsonian Center for Astrophysics 1815 Massachussetts Av. , Cambridge MA 02140 Work Phone: (617)-496-7990 extension 124 VoIP Phone: (617)=384-6679 Cell-Phone: (617)-817-1800 -- All generalizations are abusive (specially this one!) ---
RE: Fw: Something for Platonists
Please take [EMAIL PROTECTED] off this mailing list.Michael Annucci [EMAIL PROTECTED] wrote: Please take [EMAIL PROTECTED] off of this mailing list.-Original Message-From: CMR [mailto:[EMAIL PROTECTED] Sent: Monday, June 16, 2003 12:50 PMTo: Joao LeaoCc: [EMAIL PROTECTED]Subject: Re: Fw: Something for Platonistsshameless indeedCheersCMR--enter gratuitous quotation that implies my profundity here--- Original Message -From: "Joao Leao" <[EMAIL PROTECTED]>To: "Stephen Paul King" <[EMAIL PROTECTED]>Cc: "" <[EMAIL PROTECTED]>Sent: Monday, June 16, 2003 9:19 AMSubject: Re: Fw: Something for Platonists Dear Stephen, Given that, were it not for Plato the question you ask me would not make sense and could not probably be formulated, I should not have to answer it.! t; If that is what you driving at: Mathematical Realism or Platonism is not a religion, but a conviction which most working mathematician have been reasonably led to in their practice. As for physicists it is a prejudice that most share but few find a need to confess. My only distinction is that I am quite shameless about it... -Joao Stephen Paul King wrote: Dear Joao, Is this the statement of a person that bases their belief in faithor reason? Sincerly, Stephen - Original Message - From: "Joao Leao" <[EMAIL PROTECTED]> To: "Lennart Nilsson" <[EMAIL PROTECTED]> Cc: "Everything List" <[EMAIL PROTECTED]> Sent: Monday, June 16, 2003 11:18 AM Subject: Re: F! w: Something for PlatonistsSpeaking as a devout Platonist I see nothing much to contemplatein Deutsch's statement! Whether the Universe is computable, as hestates without argument, or the computable subrealm of themathematical world coincides with the physical, which he believesfor unstated reasons, is of no concern to me or anyself-respecting Platonist. The Realm of Forms is entirely separate from the physical universe which is nothing but an inept andcorrupt model of it. Our physical theories, and Deutsh'sspeculations are even crappier versions of that model whichcapture nothing but mere glimpses of the Platonic World and thusare destined to be surpassed. Computation may be indeed a fairly acceptable measure! of our ineptitude to see into Platonia: that is a plausiblehypothesis. But the fact that we know of the realm of theuncomputable and that we can access its truths irrespective of our finite computational capabilities is an entirely more profoundstatement than any of Deutsch dubious speculations... -Joao Leao -- Joao Pedro Leao ::: [EMAIL PROTECTED] Harvard-Smithsonian Center for Astrophysics 1815 Massachussetts Av. , Cambridge MA 02140 Work Phone: (617)-496-7990 extension 124 VoIP Phone: (617)=384-6679 Cell-Phone: (617)-817-1800 -- "All generalizations are abusive (specially this one!)" --- Do you Yahoo!? SBC Yahoo! DSL - Now only $29.95 per month!
Re: Fw: Something for Platonists
Lennart Nilsson wrote: But in fact, the only thing that privileges the set of all computational operations that we see in nature, is that they are instantiated by the laws of physics. It is only through our knowledge of the physical world that we know of the di.erence between computable and not computable. So it's only through our laws of physics that the nature of computation can be understood. It can never be vice versa. I don't agree. I think computability is a pure abstract property describing the reachability of some states (or state descriptions) from others via a set of incrementally different states (or state descriptions). I think computability is tied to notions of locality. But computability may define locality and not the other way around. Eric -- We are all in the gutter, but some of us are looking at the stars. - Oscar Wilde
Re: Fw: Something for Platonists]
At 10:46 16/06/03 -0700, Hal Finney wrote: Jesse Mazer writes: Yes, a Platonist can feel as certain of the statement the axioms of Peano arithmetic will never lead to a contradiction as he is of 1+1=2, based on the model he has of what the axioms mean in terms of arithmetic. It's hard to see how non-Platonist could justify the same conviction, though, given Godel's results. Since many mathematicians probably would be willing to bet anything that the statement was true, this suggests a lot of them are at least closet Platonists. What is the status of the possibility that a given formal system such as the one for arithmetic is inconsistent? Godel's theorem only shows that if consistent, it is incomplete, right? Are there any proofs that formal systems specifying arithmetic are consistent (and hence incomplete)? As Jesse Mazer said we all have an intuitive model of Peano Arithmetic (PA), and this should convince us of PA consistency. (We learned that model in secondary school). We can formalize such an argument in a set theory like ZF, that is, a model of PA can be constructed in ZF, as a first order citizen. Now this should not really convince us that PA is consistent because the ZF axioms are more demanding, and we would be entitled to ask for a proof of the consistency of ZF. By Godel second incompleteness theorem PA cannot prove the consistency of PA, ZF cannot prove the consistency of ZF. But ZF can prove the consistency of PA. Note that this latter fact *can* be proved in PA, that is: PA can prove that ZF can prove the consistency of PA, of course PA cannot prove the consistency of ZF, so this is not very useful here. A perhaps more relevant question is: does it exist a *finitary* proof of PA consistency? It all depends of course of what is meant by finitary. If by finitary you mean arithmetically representable, then by Godel, the answer is no. But many logicians consider that transfinite induction toward some reasonable ordinal can be considered as finitary. Actually Gentzen succeeds in presenting a proof of the consistency of PA through a transfinite induction up to \epsilon_0 (which is omega up to omega up to omega up to omega ...). This shows (by Godel again) that transfinite induction up to \epsilon_0 cannot be done in PA, although it can be shown that all transfinite induction up to any \alpha little than \epsilon_0 can be done in PA. This has lend to ordinal analysis of formal theories where the strongness of provability of a theory is measured in term of ordinal. Remember that computability is an absolute notion (Church thesis), and formal provability is a necessary relative notion. I conjecture that the consistency of COMP should need at least a transfinite induction up to the Church-Kleene least non constructive ordinal (omega_1^CK). This should reflect the fact that the consistency of COMP is not provable by any consistent machines ... (although machines could bet on it, at their own risk and peril). Bruno
Fw: Something for Platonists
- Original Message - From: Lennart Nilsson [EMAIL PROTECTED] To: [EMAIL PROTECTED] Sent: Sunday, June 15, 2003 9:14 AM Subject: Something for Platonists Here is something from David Deutsch for Platonists to contemplate...I think LN We see around us a computable universe; that is to say, of all possible mathematical objects and relationships, only an in.nitesimal proportion are ever instantiated in the relationships of physical objects and physical processes. (These are essentially the computable functions.) Now it might seem that one approach to explaining that amazing fact, is to say the reason why physical processes conform to this very small part of mathematics, 'computable mathematics,' is that physical processes really are computations running on a computer external to what we think of as physical reality. But that relies on the assumption that the set of computable functions - the Turing computable functions, or the set of quantum computable operations - is somehow inherently privileged within mathematics. So that even a computer implemented in unknown physics (the supposed computer that we're all simulations on) would be expected to conform to those same notions of computability, to use those same functions that mathematics designates as computable. But in fact, the only thing that privileges the set of all computational operations that we see in nature, is that they are instantiated by the laws of physics. It is only through our knowledge of the physical world that we know of the di.erence between computable and not computable. So it's only through our laws of physics that the nature of computation can be understood. It can never be vice versa.
Re: Fw: Something for Platonists
Speaking as a devout Platonist I see nothing much to contemplate in Deutsch's statement! Whether the Universe is computable, as he states without argument, or the computable subrealm of the mathematical world coincides with the physical, which he believes for unstated reasons, is of no concern to me or any self-respecting Platonist. The Realm of Forms is entirely separate from the physical universe which is nothing but an inept and corrupt model of it. Our physical theories, and Deutsh's speculations are even crappier versions of that model which capture nothing but mere glimpses of the Platonic World and thus are destined to be surpassed. Computation may be indeed a fairly acceptable measure of our ineptitude to see into Platonia: that is a plausible hypothesis. But the fact that we know of the realm of the uncomputable and that we can access its truths irrespective of our finite computational capabilities is an entirely more profound statement than any of Deutsch dubious speculations... -Joao Leao Lennart Nilsson wrote: - Original Message - From: Lennart Nilsson [EMAIL PROTECTED] To: [EMAIL PROTECTED] Sent: Sunday, June 15, 2003 9:14 AM Subject: Something for Platonists Here is something from David Deutsch for Platonists to contemplate...I think LN We see around us a computable universe; that is to say, of all possible mathematical objects and relationships, only an in.nitesimal proportion are ever instantiated in the relationships of physical objects and physical processes. (These are essentially the computable functions.) Now it might seem that one approach to explaining that amazing fact, is to say the reason why physical processes conform to this very small part of mathematics, 'computable mathematics,' is that physical processes really are computations running on a computer external to what we think of as physical reality. But that relies on the assumption that the set of computable functions - the Turing computable functions, or the set of quantum computable operations - is somehow inherently privileged within mathematics. So that even a computer implemented in unknown physics (the supposed computer that we're all simulations on) would be expected to conform to those same notions of computability, to use those same functions that mathematics designates as computable. But in fact, the only thing that privileges the set of all computational operations that we see in nature, is that they are instantiated by the laws of physics. It is only through our knowledge of the physical world that we know of the difference between computable and not computable. So it's only through our laws of physics that the nature of computation can be understood. It can never be vice versa. -- Joao Pedro Leao ::: [EMAIL PROTECTED] Harvard-Smithsonian Center for Astrophysics 1815 Massachussetts Av. , Cambridge MA 02140 Work Phone: (617)-496-7990 extension 124 VoIP Phone: (617)=384-6679 Cell-Phone: (617)-817-1800 -- All generalizations are abusive (specially this one!) ---
Re: Fw: Something for Platonists
Joao wrote: Speaking as a devout Platonist ... About 7 years ago I realized there was a severe contradiction resident in modern concepts of Being. Godel's Incompleteness Theorems have established a condition-of-knowledge which seem to challenge if not negate Platonic thought. I'd like to get your ideas on the following: Consider the Platonic Ideal of 'apple'. I can almost guarantee that your mind immediately came up with an image of 'apple' including stem, colorful skin, other qualities, etc. As Godel designated -system internally consistent-, we might at first presume the two depictions to be isomorphic. But I submit that per Godel, 'apple' includes only those characteristics or qualia evident up to but not external to the bounds of the system, whatever they may be. That being the case, 'color' of any existential ideal-apple exists only in the out-space where the platonic apple per se -does not-. Therefore 'color' and 'apple' - in any platonic sense - must be mutually exclusive. Which seems to press the 2500 year old standing impression of 'ideal apple'. Another discontinuity. If you climb Mount Everest and sit down on it, does the mountain now satisfy the platonic ideal of chair? Thanks in advance for your thoughts, James Rose
Re: Fw: Something for Platonists
Dear Joao, Is this the statement of a person that bases their belief in faith or reason? Sincerly, Stephen - Original Message - From: Joao Leao [EMAIL PROTECTED] To: Lennart Nilsson [EMAIL PROTECTED] Cc: Everything List [EMAIL PROTECTED] Sent: Monday, June 16, 2003 11:18 AM Subject: Re: Fw: Something for Platonists Speaking as a devout Platonist I see nothing much to contemplate in Deutsch's statement! Whether the Universe is computable, as he states without argument, or the computable subrealm of the mathematical world coincides with the physical, which he believes for unstated reasons, is of no concern to me or any self-respecting Platonist. The Realm of Forms is entirely separate from the physical universe which is nothing but an inept and corrupt model of it. Our physical theories, and Deutsh's speculations are even crappier versions of that model which capture nothing but mere glimpses of the Platonic World and thus are destined to be surpassed. Computation may be indeed a fairly acceptable measure of our ineptitude to see into Platonia: that is a plausible hypothesis. But the fact that we know of the realm of the uncomputable and that we can access its truths irrespective of our finite computational capabilities is an entirely more profound statement than any of Deutsch dubious speculations... -Joao Leao
Re: Fw: Something for Platonists
Dear Stephen, Given that, were it not for Plato the question you ask me would not make sense and could not probably be formulated, I should not have to answer it. If that is what you driving at: Mathematical Realism or Platonism is not a religion, but a conviction which most working mathematician have been reasonably led to in their practice. As for physicists it is a prejudice that most share but few find a need to confess. My only distinction is that I am quite shameless about it... -Joao Stephen Paul King wrote: Dear Joao, Is this the statement of a person that bases their belief in faith or reason? Sincerly, Stephen - Original Message - From: Joao Leao [EMAIL PROTECTED] To: Lennart Nilsson [EMAIL PROTECTED] Cc: Everything List [EMAIL PROTECTED] Sent: Monday, June 16, 2003 11:18 AM Subject: Re: Fw: Something for Platonists Speaking as a devout Platonist I see nothing much to contemplate in Deutsch's statement! Whether the Universe is computable, as he states without argument, or the computable subrealm of the mathematical world coincides with the physical, which he believes for unstated reasons, is of no concern to me or any self-respecting Platonist. The Realm of Forms is entirely separate from the physical universe which is nothing but an inept and corrupt model of it. Our physical theories, and Deutsh's speculations are even crappier versions of that model which capture nothing but mere glimpses of the Platonic World and thus are destined to be surpassed. Computation may be indeed a fairly acceptable measure of our ineptitude to see into Platonia: that is a plausible hypothesis. But the fact that we know of the realm of the uncomputable and that we can access its truths irrespective of our finite computational capabilities is an entirely more profound statement than any of Deutsch dubious speculations... -Joao Leao -- Joao Pedro Leao ::: [EMAIL PROTECTED] Harvard-Smithsonian Center for Astrophysics 1815 Massachussetts Av. , Cambridge MA 02140 Work Phone: (617)-496-7990 extension 124 VoIP Phone: (617)=384-6679 Cell-Phone: (617)-817-1800 -- All generalizations are abusive (specially this one!) ---
Re: Fw: Something for Platonists
Joao Leao wrote: James N Rose wrote: Joao wrote: Speaking as a devout Platonist ... About 7 years ago I realized there was a severe contradiction resident in modern concepts of Being. Godel's Incompleteness Theorems have established a condition-of-knowledge which seem to challenge if not negate Platonic thought. That just happens to be totally orthogonal to what Godel himself expressed as his own opinion on the consequence of his theorem... Godel is possibly the most consequent of all XXcent. self professed Platonists. I'd like to get your ideas on the following: Consider the Platonic Ideal of 'apple'. I can almost guarantee that your mind immediately came up with an image of 'apple' including stem, colorful skin, other qualities, etc. As Godel designated -system internally consistent-, we might at first presume the two depictions to be isomorphic. Why? Is there any reason why my apple need to fit a consistent system of appleness? I don't think so... But I submit that per Godel, 'apple' includes only those characteristics or qualia evident up to but not external to the bounds of the system, whatever they may be. That being the case, 'color' of any existential ideal-apple exists only in the out-space where the platonic apple per se -does not-. Therefore 'color' and 'apple' - in any platonic sense - must be mutually exclusive. Which seems to press the 2500 year old standing impression of 'ideal apple'. Not at all. You are confusing images with things and forgetting a good deal of what platonism is about. An apple, this apple, the apple I am thinking of, all partake the form of appleness whatever that is. The color of this apple, the color of that bird, this red, the red you are thinking of right now, all partake of the form of redness in the Patonic world. There is no contradition here. There are no forms here! You have glossed over the issue I was establishing. Godel pretty well specified a disconnect between certain ceptualizations - uniform agreements even with varieties involved - in that specificities are subject to alteration upon inclusion of external (not currently available) information. Platonic thought - to satisfy the extensive nature and the inclusive scope you indicate in your remarks - requires that all possibilities, all variants, all potentia, be taken into consideration, in order to (asymptotically) such ideal of whatever designated. Or, to restrict it according to the regulations you jibe about in remarks further along in your reply (a table can be sat upon, but it is not a 'chair'). The ideality of 'apple' includes the former condition of being 'ideal' only when the totality of environments are included - the exterior realm which Godel says can -never- be holistically involved in any ultimate _experiential_ sense. So if Godel counted himself a Platonist, he necessarily had to conclude that no platomic ideal (conditions-of-knowledge) could have any relevance with the material (conditions-of-being) since there would be no way to secure - permanently and reliantly - what 'ideal' would be expansive enough, and, because any window to 'ideal' cannot help but be rooted in (conditions-of-being) .. the expeiential. I.e., there would be no way of knowing if any 'knowing' a mind held had any real mappings with a purported 'ideal'. My personal arguement with Platonism is that Plato never took into consideration the requisite conditions relative to information conveyance and the issues established by Heisenberg and quantum mechanics. Not only will information influence and alter other information, but there is unavoidable connectivity in order for there to be information conveyance (knowability) in the first place. There is mechanism and process involved (one of Plato's prime beliefs). In fact, all-is-process. There is no one thing, no some thing, nor such a thing whatsoever. But it is from motion or being carried along, from change and from admixture with each other that everything comes to be that which we declare to be (speaking incorrectly), for nothing ever is, but always becomes. (Plato, Theaitetos 152d) In a sense, in fact, to be true to such an extreme idealism - unless one were willing to compromise - if there 'no such a thing whatsoever', then there would be no corresponding 'ideal' ... whatsoever. But, to keep to the argument, even in the Cave, intervening air and lightwaves are conveyors of ideal to real .. which must perforce have relation with both the ideal realm and the real realm .. or whatever conveyor you might agree correlates with the physical indicia of waves. Another discontinuity. If you climb Mount Everest and sit down on it, does the mountain now satisfy the platonic ideal of chair? No, why should it? The form of a chair is not the form of anything I sit on! You can sit on a table or on your head for
Re: Fw: Something for Platonists]
Gödel's incompleteness theorems have and justly should be judged/interpreted purely on the merits of the arguments themselves, not the author's subjective(prejudiced?) interpretation, no? He was as much a victim(beneficiary?) of his discoveries as was anyone... CMR --enter gratuitous quotation that implies my profundity here-- - Original Message - From: Joao Leao [EMAIL PROTECTED] To: Sent: Monday, June 16, 2003 9:51 AM Subject: [Fwd: Fw: Something for Platonists] Joao Leao wrote: James N Rose wrote: Joao wrote: Speaking as a devout Platonist ... About 7 years ago I realized there was a severe contradiction resident in modern concepts of Being. Godel's Incompleteness Theorems have established a condition-of-knowledge which seem to challenge if not negate Platonic thought. That just happens to be totally orthogonal to what Godel himself expressed as his own opinion on the consequence of his theorem... Godel is possibly the most consequent of all XXcent. self professed Platonists. I'd like to get your ideas on the following: Consider the Platonic Ideal of 'apple'. I can almost guarantee that your mind immediately came up with an image of 'apple' including stem, colorful skin, other qualities, etc. As Godel designated -system internally consistent-, we might at first presume the two depictions to be isomorphic. Why? Is there any reason why my apple need to fit a consistent system of appleness? I don't think so... But I submit that per Godel, 'apple' includes only those characteristics or qualia evident up to but not external to the bounds of the system, whatever they may be. That being the case, 'color' of any existential ideal-apple exists only in the out-space where the platonic apple per se -does not-. Therefore 'color' and 'apple' - in any platonic sense - must be mutually exclusive. Which seems to press the 2500 year old standing impression of 'ideal apple'. Not at all. You are confusing images with things and forgetting a good deal of what platonism is about. An apple, this apple, the apple I am thinking of, all partake the form of appleness whatever that is. The color of this apple, the color of that bird, this red, the red you are thinking of right now, all partake of the form of redness in the Patonic world. There is no contradition here. There are no forms here! Another discontinuity. If you climb Mount Everest and sit down on it, does the mountain now satisfy the platonic ideal of chair? No, why should it? The form of a chair is not the form of anything I sit on! You can sit on a table or on your head for all I care... This is a different in extension which is much easier to grasp than one of intention, but it is the same think. Thanks in advance for your thoughts, James Rose I am afraid you are obviously confused about the basis of platonism and the dispute with kantianism, if you will. I suggest you read Stanley Rosen's Antiplatonism in his collection The Ancients and the Moderns for a recent and detailed review of the issue you raise, namely conditions-of-knowledge as conditions-of-being, a sibject prone to post-kantian confusions Regards, -Joao Leao -- Joao Pedro Leao ::: [EMAIL PROTECTED] Harvard-Smithsonian Center for Astrophysics 1815 Massachussetts Av. , Cambridge MA 02140 Work Phone: (617)-496-7990 extension 124 VoIP Phone: (617)=384-6679 Cell-Phone: (617)-817-1800 -- All generalizations are abusive (specially this one!) --- -- Joao Pedro Leao ::: [EMAIL PROTECTED] Harvard-Smithsonian Center for Astrophysics 1815 Massachussetts Av. , Cambridge MA 02140 Work Phone: (617)-496-7990 extension 124 VoIP Phone: (617)=384-6679 Cell-Phone: (617)-817-1800 -- All generalizations are abusive (specially this one!) ---
Re: Fw: Something for Platonists]
CMR wrote: Gödel's incompleteness theorems have and justly should be judged/interpreted purely on the merits of the arguments themselves, not the author's subjective(prejudiced?) interpretation, no? He was as much a victim(beneficiary?) of his discoveries as was anyone... Precisely! The implication I was drawing is that, as he stated quite well, his mathematical results reinforced his Platonist conviction. Unless you are implying that mathematical reality favours the ones who submit to it (an enticing possibility, for sure), I don't see how it could have been otherwise... CMR --enter gratuitous quotation that implies my profundity here-- - Original Message - From: Joao Leao [EMAIL PROTECTED] To: Sent: Monday, June 16, 2003 9:51 AM Subject: [Fwd: Fw: Something for Platonists] Joao Leao wrote: James N Rose wrote: Joao wrote: Speaking as a devout Platonist ... About 7 years ago I realized there was a severe contradiction resident in modern concepts of Being. Godel's Incompleteness Theorems have established a condition-of-knowledge which seem to challenge if not negate Platonic thought. That just happens to be totally orthogonal to what Godel himself expressed as his own opinion on the consequence of his theorem... Godel is possibly the most consequent of all XXcent. self professed Platonists. I'd like to get your ideas on the following: Consider the Platonic Ideal of 'apple'. I can almost guarantee that your mind immediately came up with an image of 'apple' including stem, colorful skin, other qualities, etc. As Godel designated -system internally consistent-, we might at first presume the two depictions to be isomorphic. Why? Is there any reason why my apple need to fit a consistent system of appleness? I don't think so... But I submit that per Godel, 'apple' includes only those characteristics or qualia evident up to but not external to the bounds of the system, whatever they may be. That being the case, 'color' of any existential ideal-apple exists only in the out-space where the platonic apple per se -does not-. Therefore 'color' and 'apple' - in any platonic sense - must be mutually exclusive. Which seems to press the 2500 year old standing impression of 'ideal apple'. Not at all. You are confusing images with things and forgetting a good deal of what platonism is about. An apple, this apple, the apple I am thinking of, all partake the form of appleness whatever that is. The color of this apple, the color of that bird, this red, the red you are thinking of right now, all partake of the form of redness in the Patonic world. There is no contradition here. There are no forms here! Another discontinuity. If you climb Mount Everest and sit down on it, does the mountain now satisfy the platonic ideal of chair? No, why should it? The form of a chair is not the form of anything I sit on! You can sit on a table or on your head for all I care... This is a different in extension which is much easier to grasp than one of intention, but it is the same think. Thanks in advance for your thoughts, James Rose I am afraid you are obviously confused about the basis of platonism and the dispute with kantianism, if you will. I suggest you read Stanley Rosen's Antiplatonism in his collection The Ancients and the Moderns for a recent and detailed review of the issue you raise, namely conditions-of-knowledge as conditions-of-being, a sibject prone to post-kantian confusions Regards, -Joao Leao -- Joao Pedro Leao ::: [EMAIL PROTECTED] Harvard-Smithsonian Center for Astrophysics 1815 Massachussetts Av. , Cambridge MA 02140 Work Phone: (617)-496-7990 extension 124 VoIP Phone: (617)=384-6679 Cell-Phone: (617)-817-1800 -- All generalizations are abusive (specially this one!) --- -- Joao Pedro Leao ::: [EMAIL PROTECTED] Harvard-Smithsonian Center for Astrophysics 1815 Massachussetts Av. , Cambridge MA 02140 Work Phone: (617)-496-7990 extension 124 VoIP Phone: (617)=384-6679 Cell-Phone: (617)-817-1800 -- All generalizations are abusive (specially this one!) --- -- Joao Pedro Leao ::: [EMAIL PROTECTED] Harvard-Smithsonian Center for Astrophysics 1815 Massachussetts Av. , Cambridge MA 02140 Work Phone: (617)-496-7990 extension 124 VoIP Phone: (617)=384-6679 Cell-Phone: (617)-817-1800 -- All generalizations are abusive (specially this one!) ---
Re: Fw: Something for Platonists]
Joao Leao wrote: CMR wrote: Gödel's incompleteness theorems have and justly should be judged/interpreted purely on the merits of the arguments themselves, not the author's subjective(prejudiced?) interpretation, no? He was as much a victim(beneficiary?) of his discoveries as was anyone... Precisely! The implication I was drawing is that, as he stated quite well, his mathematical results reinforced his Platonist conviction. Unless you are implying that mathematical reality favours the ones who submit to it (an enticing possibility, for sure), I don't see how it could have been otherwise... Yes, a Platonist can feel as certain of the statement the axioms of Peano arithmetic will never lead to a contradiction as he is of 1+1=2, based on the model he has of what the axioms mean in terms of arithmetic. It's hard to see how non-Platonist could justify the same conviction, though, given Godel's results. Since many mathematicians probably would be willing to bet anything that the statement was true, this suggests a lot of them are at least closet Platonists. Of course, Platonism in the mathematical realm is a little different than Platonism in the realm of ordinary language. I don't believe there is such a thing as an ideal apple, for example. Jesse _ STOP MORE SPAM with the new MSN 8 and get 2 months FREE* http://join.msn.com/?page=features/junkmail
Re: Fw: Something for Platonists]
Jesse Mazer writes: Yes, a Platonist can feel as certain of the statement the axioms of Peano arithmetic will never lead to a contradiction as he is of 1+1=2, based on the model he has of what the axioms mean in terms of arithmetic. It's hard to see how non-Platonist could justify the same conviction, though, given Godel's results. Since many mathematicians probably would be willing to bet anything that the statement was true, this suggests a lot of them are at least closet Platonists. What is the status of the possibility that a given formal system such as the one for arithmetic is inconsistent? Godel's theorem only shows that if consistent, it is incomplete, right? Are there any proofs that formal systems specifying arithmetic are consistent (and hence incomplete)? Hal Finney
Re: Fw: Something for Platonists
James N Rose wrote: You have glossed over the issue I was establishing. I am sorry if I did. That was not my intention. I still think you are mixing platonic apples with not so platonic oranges, but let us see if I can make out what you are saying. Godel pretty well specified a disconnect between certain ceptualizations - uniform agreements even with varieties involved - in that specificities are subject to alteration upon inclusion of external (not currently available) information. I can't really follow your language here. What Godel's theorems collectively showed is that any well formed axiomatic system will contain statements which, though true, cannot be deductively proved as theorems in that system. There are many ways of expressing this in terms of consistency, undecidability, incompletness what have you. But the bottom lie is the same. I don't see what you may mean by specificities subject to alteration!... Platonic thought - to satisfy the extensive nature and the inclusive scope you indicate in your remarks - requires that all possibilities, all variants, all potentia, be taken into consideration, in order to (asymptotically) such ideal of whatever designated. Not really! The Platonic World only contains true mathematical statements, not all the variety that you seem to believe it requires. In other words it contains presummably less information than most textbooks of mathematics which include unproved conjectures etc... As far as non mathematical ideals the same is the case: there are many different chairs each one a different corruption of the same ideal form of chairness, get it? The Platonic world is very sparsely populated, unlike your brain and mine... Or, to restrict it according to the regulations you jibe about in remarks further along in your reply (a table can be sat upon, but it is not a 'chair'). The ideality of 'apple' includes the former condition of being 'ideal' only when the totality of environments are included - the exterior realm which Godel says can -never- be holistically involved in any ultimate _experiential_ sense. Just the opposite: the idea of the apple is exactly what makes it, the idea, independent of all or any environment unlike this specific apple or that other one ! That is the reason it need not be involved in any ultimate sense, as you put it! So if Godel counted himself a Platonist, he necessarily had to conclude that no platomic ideal (conditions-of-knowledge) could have any relevance with the material (conditions-of-being) since there would be no way to secure - permanently and reliantly - what 'ideal' would be expansive enough, and, because any window to 'ideal' cannot help but be rooted in (conditions-of-being) .. the expeiential. Again you are mistaken. The conditions of knowledge only condition us knowers, they do not condition the ideas which exist independently of whonever knows them. The Platonic World is the unconditioned as the philosophers of the XIX centuries refered to it to distinguish it from the realm of experience. I.e., there would be no way of knowing if any 'knowing' a mind held had any real mappings with a purported 'ideal'. But there isn't! Mathematics is the only form of conviction that we can appeal to in that respect and the fact that math seems to have some bearing in the systematization of our experience in the computational realm, as Deutsch puts it, is the only thing that informs our knowledge, unless you believe in platonic anamnesis... My personal arguement with Platonism is that Plato never took into consideration the requisite conditions relative to information conveyance and the issues established by Heisenberg and quantum mechanics. Not only will information influence and alter other information, but there is unavoidable connectivity in order for there to be information conveyance (knowability) in the first place. Of course Plato did not take Quantum Mechanics into account! But I think he had a good excuse: he lived 2300 YEARS before Quantum Mechanics! On the other hand QM only indictes one form of realism called Local Realism to which Platonic Ideas do not subscribe. This is another subject altogether... There is mechanism and process involved (one of Plato's prime beliefs). In fact, all-is-process. There is no one thing, no some thing, nor such a thing whatsoever. But it is from motion or being carried along, from change and from admixture with each other that everything comes to be that which we declare to ?be? (speaking incorrectly), for nothing ever ?is?, but always becomes. (Plato, Theaitetos 152d) This is the Heraclitean side of Plato and does not have exactly the status you want to give it as it precedes the aristotelian distinction between Beung and Becoming. Still this would describe the world of appearances not the world of Forms. You should read Gadamer's The Beginning of Knowledge which analysis this issue
Re: Fw: Something for Platonists]
The answer is that an incomplete arithmetic axiom system could presumably by consistent, but who cares? If it is incomplete there will be true statements that it cannot prove and we are back to the platonist position! The alternative of an inconsistent system that is complete may actually be more interesting and has been explored in recent mathematics. A great reference is Inconsistent Mathematics by Chris Mortensen (Kluwer 1995). -Joao Hal Finney wrote: Jesse Mazer writes: Yes, a Platonist can feel as certain of the statement the axioms of Peano arithmetic will never lead to a contradiction as he is of 1+1=2, based on the model he has of what the axioms mean in terms of arithmetic. It's hard to see how non-Platonist could justify the same conviction, though, given Godel's results. Since many mathematicians probably would be willing to bet anything that the statement was true, this suggests a lot of them are at least closet Platonists. What is the status of the possibility that a given formal system such as the one for arithmetic is inconsistent? Godel's theorem only shows that if consistent, it is incomplete, right? Are there any proofs that formal systems specifying arithmetic are consistent (and hence incomplete)? Hal Finney -- Joao Pedro Leao ::: [EMAIL PROTECTED] Harvard-Smithsonian Center for Astrophysics 1815 Massachussetts Av. , Cambridge MA 02140 Work Phone: (617)-496-7990 extension 124 VoIP Phone: (617)=384-6679 Cell-Phone: (617)-817-1800 -- All generalizations are abusive (specially this one!) ---
Re: Fw: Something for Platonists]
From: Hal Finney [EMAIL PROTECTED] To: [EMAIL PROTECTED], [EMAIL PROTECTED] Subject: Re: Fw: Something for Platonists] Date: Mon, 16 Jun 2003 10:46:56 -0700 Jesse Mazer writes: Yes, a Platonist can feel as certain of the statement the axioms of Peano arithmetic will never lead to a contradiction as he is of 1+1=2, based on the model he has of what the axioms mean in terms of arithmetic. It's hard to see how non-Platonist could justify the same conviction, though, given Godel's results. Since many mathematicians probably would be willing to bet anything that the statement was true, this suggests a lot of them are at least closet Platonists. What is the status of the possibility that a given formal system such as the one for arithmetic is inconsistent? Godel's theorem only shows that if consistent, it is incomplete, right? Are there any proofs that formal systems specifying arithmetic are consistent (and hence incomplete)? Hal Finney Godel showed that if it's complete, a theorem about its consistency is not provably true or false within the formal system itself. We can feel certain that it *is* consistent nevertheless, by using a model that assigns meaning to the axioms in terms of our mental picture of arithmetic. For example, with the symbols for multiplication and equals interpreted the way we normally do in arithmetic, you can see that x*y=y*x must always be true by thinking in terms of a matrix with x columns and y rows and another with y columns and x rows, and seeing that one can be rotated to become the other. In the book Godel's Proof, Douglas Hofstadter gives a simple example of using a model to prove a formal system's consistency: Suppose the following set of postulates concerning two classes K and L, whose special nature is left undetermined except as implicitly defined by the postulates: 1. Any two members of K are contained in just one member of L. 2. No member of K is contained in more than two members of L. 3. The members of K are not all contained in a single member of L. 4. Any two members of L contain just one member of K. 5. No member of L contains more than two members of K. From this small set we can derive, by using customary rules of inference, a number of theorems. For example, it can be shown that K contains just three members. But is the set consistent, so that mutually contradictory theorems can never be derived from it? The question can be answered readily with the help of the following model: Let K be the class of points consisting of the vertices of a triangle, and L the class of lines made up of its sides; and let us understand a member of K is contained in a member of L to mean that a point which is a vertex lies on a line which is a side. Each of the five abstract postulates is then converted into a true statement. For instance, the first postulate asserts that any two points which are vertices of the triangle lie on just one line which is a side. In this way the set of postulates is proved to be consistent. As I think Bruno Marchal mentioned in a recent post, mathematicians use the word model differently than physicists or other scientists. But again, I'm not sure if model theory even makes sense if you drop all Platonic assumptions about math. Jesse Mazer _ MSN 8 helps eliminate e-mail viruses. Get 2 months FREE*. http://join.msn.com/?page=features/virus
Re: Fw: Something for Platonists
Joao, :-) of course Plato wasn't aware of QM, but, he was also unaware of the importance that -mechanism- -real communication involvements- are resident in any information relation situation, as would be that which connects the Ideal and Real and gives validation/meaning to any correspondences cited or citable. The 'ideal' as posited - and presumptively relied upon by many post facto - is so separated from 'being' and the encounters through which both being and knowing are instantiated, that it would not be unreasonable to populate 'ideal' with all sorts of non-possible existentials. You can't tie 'ideal' to the spectrum of alternative but satisfactory exemplars, and also say there are no requisite relational aspects of the properties or qualia resident in the different domains. Otherwise, you state: The Platonic World only contains true mathematical statements, not all the variety that you seem to believe it requires. In other words it contains presummably less information than most textbooks of mathematics which include unproved conjectures etc... So the platonic world cannot/doesnot contain the ideal called 'unproved/unprovable conjectures? The Platonic World contains -less- information than the instantiated world? Exactly how far can you extend that argument?..to the point that it contains -no- information of relevance? It seems that the Platonic World, as intriguing and frame-of-reference shifting as it may be -- getting people to perceive beyond the immediacy of encounters and the presumptions of observation -- is as flighty and weak as the 'real world' it decries. You hold to it because it infers an eternality that is very appealing, an opiate to the fear of oblivion and total absolute negation of meaning concurrent that comes with complete non-existence (even as potentia). I place it on no such special pedestal. It is not a holy ineffible. If it can't be correlated with being, then there is empty value, use or meaning in presumptively claiming there is - and yet - denying processive ways of having such 'correlations'. I deduce that platonic notions are nice sophomoric ramblings, some interesting relations are enunciated, but in the long run there are more important realite's. James
Re: Fw: Something for Platonists
James N Rose wrote: Joao, :-) of course Plato wasn't aware of QM, but, he was also unaware of the importance that -mechanism- -real communication involvements- are resident in any information relation situation, as would be that which connects the Ideal and Real and gives validation/meaning to any correspondences cited or citable. I still have no idea of what you are talking about! Real communication involvements may be very important, but we are not having one here... The 'ideal' as posited - and presumptively relied upon by many post facto - is so separated from 'being' and the encounters through which both being and knowing are instantiated, that it would not be unreasonable to populate 'ideal' with all sorts of non-possible existentials. Again, I don't know what you mean by encounters through which both being and knowing are instantiated. You can populate all you want but don't blame it on Plato! He was rather economical on his encounters... You can't tie 'ideal' to the spectrum of alternative but satisfactory exemplars, and also say there are no requisite relational aspects of the properties or qualia resident in the different domains. Sorry. You again seem to be confuse the domain of your thoughs with the Platonic Realm. There are no qualia in Platonia so they need not share relational aspects with any other domains, as you insist... Forms are Universals not properties. Otherwise, you state: The Platonic World only contains true mathematical statements, not all the variety that you seem to believe it requires. In other words it contains presummably less information than most textbooks of mathematics which include unproved conjectures etc... So the platonic world cannot/doesnot contain the ideal called 'unproved/unprovable conjectures? I am sure you will agree that those cannot be ideal in the platonic world or in any other, if you reflect for a second. In an ideal world we prove or refute our conjectures. The fact that we can't do that in our world should show you how corrupt it is... The Platonic World contains -less- information than the instantiated world? Exactly how far can you extend that argument?..to the point that it contains -no- information of relevance? I don't think that is the case but it could be! Have you read Tegmark's paper on the Theory of Everything as and Ensemble Theory? It seems that the Platonic World, as intriguing and frame-of-reference shifting as it may be -- getting people to perceive beyond the immediacy of encounters and the presumptions of observation -- is as flighty and weak as the 'real world' it decries. Not quite! The flightiness is yours and mine. The Platonic World is One and the Same for Eternity! You hold to it because it infers an eternality that is very appealing, an opiate to the fear of oblivion and total absolute negation of meaning concurrent that comes with complete non-existence (even as potentia). Or with complete Existence and absolute Potentia and the only certainty of meaning. You keep trying to escape into the Hegelian World instead. I place it on no such special pedestal. It is not a holy ineffible. If it can't be correlated with being, then there is empty value, use or meaning in presumptively claiming there is - and yet - denying processive ways of having such 'correlations'. Wow! You blew me here... I deduce that platonic notions are nice sophomoric ramblings, some interesting relations are enunciated, but in the long run there are more important realite's. I am sorry, I have to laugh (:-). I am talking about the first conception of an integrated system of philosophy of which we know of and you call it nice sophomoric ramblings. I am sure Plato would be delighted... James -Joao -- Joao Pedro Leao ::: [EMAIL PROTECTED] Harvard-Smithsonian Center for Astrophysics 1815 Massachussetts Av. , Cambridge MA 02140 Work Phone: (617)-496-7990 extension 124 VoIP Phone: (617)=384-6679 Cell-Phone: (617)-817-1800 -- All generalizations are abusive (specially this one!) ---
Re: Fw: Something for Platonists]
Jesse Mazer wrote: From: Hal Finney [EMAIL PROTECTED] To: [EMAIL PROTECTED], [EMAIL PROTECTED] Subject: Re: Fw: Something for Platonists] Date: Mon, 16 Jun 2003 10:46:56 -0700 Jesse Mazer writes: Yes, a Platonist can feel as certain of the statement the axioms of Peano arithmetic will never lead to a contradiction as he is of 1+1=2, based on the model he has of what the axioms mean in terms of arithmetic. It's hard to see how non-Platonist could justify the same conviction, though, given Godel's results. Since many mathematicians probably would be willing to bet anything that the statement was true, this suggests a lot of them are at least closet Platonists. What is the status of the possibility that a given formal system such as the one for arithmetic is inconsistent? Godel's theorem only shows that if consistent, it is incomplete, right? Are there any proofs that formal systems specifying arithmetic are consistent (and hence incomplete)? Hal Finney Godel showed that if it's complete, a theorem about its consistency is not provably true or false within the formal system itself. We can feel certain that it *is* consistent nevertheless, by using a model that assigns meaning to the axioms in terms of our mental picture of arithmetic. For example, with the symbols for multiplication and equals interpreted the way we normally do in arithmetic, you can see that x*y=y*x must always be true by thinking in terms of a matrix with x columns and y rows and another with y columns and x rows, and seeing that one can be rotated to become the other. In the book Godel's Proof, Douglas Hofstadter gives a simple example of using a model to prove a formal system's consistency: Suppose the following set of postulates concerning two classes K and L, whose special nature is left undetermined except as implicitly defined by the postulates: 1. Any two members of K are contained in just one member of L. 2. No member of K is contained in more than two members of L. 3. The members of K are not all contained in a single member of L. 4. Any two members of L contain just one member of K. 5. No member of L contains more than two members of K. From this small set we can derive, by using customary rules of inference, a number of theorems. For example, it can be shown that K contains just three members. But is the set consistent, so that mutually contradictory theorems can never be derived from it? The question can be answered readily with the help of the following model: Let K be the class of points consisting of the vertices of a triangle, and L the class of lines made up of its sides; and let us understand ?a member of K is contained in a member of L? to mean that a point which is a vertex lies on a line which is a side. Each of the five abstract postulates is then converted into a true statement. For instance, the first postulate asserts that any two points which are vertices of the triangle lie on just one line which is a side. In this way the set of postulates is proved to be consistent. As I think Bruno Marchal mentioned in a recent post, mathematicians use the word model differently than physicists or other scientists. But again, I'm not sure if model theory even makes sense if you drop all Platonic assumptions about math. You are quite right! The answer is: it doesn't. Model Theory, in which Tarsky built a workable notion of truth is as subject to Godel Incompleteness as any other system of of axioms beyond a certain size. Basically the only mathematical models that do not suffer from this problem are isomorphic to binary boolean algebra of classes (though Set Theory suffers from its own problems). If you want to have an idea of what kind of back-flips people have to do to avoid Platonism in the foundations of math and logic check this paper, (silly as it is): http://philsci-archive.pitt.edu/archive/1166/ -Joao Leao Jesse Mazer _ MSN 8 helps eliminate e-mail viruses. Get 2 months FREE*. http://join.msn.com/?page=features/virus -- Joao Pedro Leao ::: [EMAIL PROTECTED] Harvard-Smithsonian Center for Astrophysics 1815 Massachussetts Av. , Cambridge MA 02140 Work Phone: (617)-496-7990 extension 124 VoIP Phone: (617)=384-6679 Cell-Phone: (617)-817-1800 -- All generalizations are abusive (specially this one!) ---
Re: Fw: Something for Platonists
Joao Leao wrote: James N Rose wrote: Joao, :-) of course Plato wasn't aware of QM, but, he was also unaware of the importance that -mechanism- -real communication involvements- are resident in any information relation situation, as would be that which connects the Ideal and Real and gives validation/meaning to any correspondences cited or citable. I still have no idea of what you are talking about! Real communication involvements may be very important, but we are not having one here... Because there is no way you can leave your mindset, see beyond it. You think it is the ultimate. Se la vie. The 'ideal' as posited - and presumptively relied upon by many post facto - is so separated from 'being' and the encounters through which both being and knowing are instantiated, that it would not be unreasonable to populate 'ideal' with all sorts of non-possible existentials. Again, I don't know what you mean by encounters through which both being and knowing are instantiated. You can populate all you want but don't blame it on Plato! He was rather economical on his encounters... h. Well, -you- and other platonist are quote happy populating it with what -you- are comfortable with. Then shut the door and consider no more. (Especially anomalies or discontiuities left unresolved) You can't tie 'ideal' to the spectrum of alternative but satisfactory exemplars, and also say there are no requisite relational aspects of the properties or qualia resident in the different domains. Sorry. You again seem to be confuse the domain of your thoughs with the Platonic Realm. There are no qualia in Platonia so they need not share relational aspects with any other domains, as you insist... Forms are Universals not properties. If there are no qualia but there are universals -- which cannot be identified except via qualia -- something is awry. If the Ideal need not share relational aspects with any other domains then that right off the bat kills any statements attempted between Ideal and Real. Nice trick, Joao. Otherwise, you state: The Platonic World only contains true mathematical statements, not all the variety that you seem to believe it requires. In other words it contains presummably less information than most textbooks of mathematics which include unproved conjectures etc... So the platonic world cannot/doesnot contain the ideal called 'unproved/unprovable conjectures? I am sure you will agree that those cannot be ideal in the platonic world or in any other, if you reflect for a second. In an ideal world we prove or refute our conjectures. The fact that we can't do that in our world should show you how corrupt it is... strike 'conjectures'; substitute 'presumptions' (weakly founded) A such purety.; No sir, this world is not a -corruption-, it is an exploration of possibilities. Your own words betone a straightjacket spiritualism that comes straight out of western bibilical theology, not Greek adventures in thought. The Platonic World contains -less- information than the instantiated world? Exactly how far can you extend that argument?..to the point that it contains -no- information of relevance? I don't think that is the case but it could be! Have you read Tegmark's paper on the Theory of Everything as and Ensemble Theory? I debate from -my- years of logos. I am courteous to allow all possibilities -- until they are carried to a limit and proved, or, shown problematic. It seems that the Platonic World, as intriguing and frame-of-reference shifting as it may be -- getting people to perceive beyond the immediacy of encounters and the presumptions of observation -- is as flighty and weak as the 'real world' it decries. Not quite! The flightiness is yours and mine. The Platonic World is One and the Same for Eternity! Maybe so. I probably confuse your depictions as accurate on Plato. One point being .. there may be no 'eternity'. Oooops, sorry, that's one of your hallowed anchor principles. Not to be challenged. Damn, I slipped again, phooey. You hold to it because it infers an eternality that is very appealing, an opiate to the fear of oblivion and total absolute negation of meaning concurrent that comes with complete non-existence (even as potentia). Or with complete Existence and absolute Potentia and the only certainty of meaning. You keep trying to escape into the Hegelian World instead. h, there you go again, meaning. You keep talking about connected values and also insist there are no connections to enact 'meaning' concurrently. I'm not escaping anywhere in your A=A=notA = nothing universe. It doesn't ... 'exist' (reality, not pun, intended). I place it on no such special pedestal. It is not a holy ineffible. If it can't be correlated with being, then there is empty value, use or meaning in presumptively claiming there is - and yet
Re: Fw: Something for Platonists
James N Rose wrote: Joao Leao wrote: James N Rose wrote: Joao, :-) of course Plato wasn't aware of QM, but, he was also unaware of the importance that -mechanism- -real communication involvements- are resident in any information relation situation, as would be that which connects the Ideal and Real and gives validation/meaning to any correspondences cited or citable. I still have no idea of what you are talking about! Real communication involvements may be very important, but we are not having one here... Because there is no way you can leave your mindset, see beyond it. You think it is the ultimate. Se la vie. It is written C'est la Vie! -- but let us leave it at this, before we are back in kindergaten... -Joao -- Joao Pedro Leao ::: [EMAIL PROTECTED] Harvard-Smithsonian Center for Astrophysics 1815 Massachussetts Av. , Cambridge MA 02140 Work Phone: (617)-496-7990 extension 124 VoIP Phone: (617)=384-6679 Cell-Phone: (617)-817-1800 -- All generalizations are abusive (specially this one!) ---
Re: Fw: Something for Platonists
Joao Leao wrote: James N Rose wrote: Joao Leao wrote: James N Rose wrote: Joao, :-) of course Plato wasn't aware of QM, but, he was also unaware of the importance that -mechanism- -real communication involvements- are resident in any information relation situation, as would be that which connects the Ideal and Real and gives validation/meaning to any correspondences cited or citable. I still have no idea of what you are talking about! Real communication involvements may be very important, but we are not having one here... Because there is no way you can leave your mindset, see beyond it. You think it is the ultimate. Se la vie. It is written C'est la Vie! -- but let us leave it at this, before we are back in kindergaten... -Joao You're welcome .. for not pointing out your typos and minor spelling faux pas's (did I get that one right?). Let's see, one version of derision - cloaked in academic references - is valid, but, direct enunciation that the debating opponent refuses to consider alternative frames of reference and association -- is not. Another aspect of the unrelated-but-relevant Ideal world, I take it. BTW, I was quite happy platforming at Second Year and moving onward; thought you were up to it as well. Sorry for having presented as so abrasive right off the bat, but I there is no other way to at least get the attention for the entrenched non-apologist for one system of thought or another and really place a dent in the somnambulent inertia one typically mires down into. Refreshed language gets things moving where established rhetoric tends to reinforce home-field advantage; more difficult for the challenger, don't you know! :-) Anyway, to remove the garbage and re-post the nitty gritty: If there are no qualia but there are universals -- which cannot be identified except via qualia -- something is awry. If the Ideal need not share relational aspects with any other domains then that right off the bat kills any statements attempted between Ideal and Real. These are not superfluous issues. They challenge the consistency and fundamentals of Platonism. (They challenge the paradigm, not you its champion.) James
Re: Fw: Something for Platonists]
Joao Leao wrote: Jesse Mazer wrote: As I think Bruno Marchal mentioned in a recent post, mathematicians use the word model differently than physicists or other scientists. But again, I'm not sure if model theory even makes sense if you drop all Platonic assumptions about math. You are quite right! The answer is: it doesn't. Model Theory, in which Tarsky built a workable notion of truth is as subject to Godel Incompleteness as any other system of of axioms beyond a certain size. Basically the only mathematical models that do not suffer from this problem are isomorphic to binary boolean algebra of classes (though Set Theory suffers from its own problems). Actually, I probably shouldn't have used the term model theory since that's a technical field that I don't know much about and that may not correspond to the more general notion of using models in proofs that I was talking about. My use of the term model just refers to the idea of taking the undefined terms in a formal axiomatic system and assigning them meaning in terms of some mental picture we have, then using that picture to prove something about the system such as its consistency. For example, the original proof that non-Euclidean geometry was consistent involved interpreting parallel lines as great circles on a sphere, and showing that all the axioms correctly described this situation. Likewise, Hofstadter's simple example of an axiomatic system that could be interpreted in terms of edges and vertices of a triangle proved that that axiomatic system was consistent, assuming there is no hidden inconsistency in our notion of triangles (an assumption a Platonist should be willing to make). On the other hand, here's a webpage that gives a capsule definition of model theory: http://linas.org/mirrors/www.ltn.lv/2001.03.27/~podnieks/gta.html All these results have been obtained by means of the so-called model theory. This is a very specific approach to investigation of formal theories as mathematical objects. Model theory is using the full power of set theory. Its results and proofs can be formalized in ZFC. Model theory is investigation of formal theories in the metatheory ZFC. I would guess that this means that to prove arithmetic's consistency in model theory, you identify terms in arithmetic with terms in ZFC set theory, like identifying the finite ordinals with the integers in arithmetic, and then you use this to prove arithmetic is consistent within ZFC. However, Godel's theorem applies to ZFC itself, so the most we can really prove with this method is something like if ZFC is consistent, then so is arithmetic. Is this correct, and if not, could you clarify? There would be no conditions on the proof of arithmetic's consistency using my more platonic notion of a model--since we are certain there are no inconsistencies in our mental model of numbers, addition, etc., we can feel confident that Peano arithmetic is consistent, period. This may not be model theory but it does involve a model of the kind in Hofstadter's example. Jesse Mazer _ Add photos to your e-mail with MSN 8. Get 2 months FREE*. http://join.msn.com/?page=features/featuredemail
Re: Fw: Something for Platonists]
Jesse Mazer wrote: Joao Leao wrote: Jesse Mazer wrote: As I think Bruno Marchal mentioned in a recent post, mathematicians use the word model differently than physicists or other scientists. But again, I'm not sure if model theory even makes sense if you drop all Platonic assumptions about math. You are quite right! The answer is: it doesn't. Model Theory, in which Tarsky built a workable notion of truth is as subject to Godel Incompleteness as any other system of of axioms beyond a certain size. Basically the only mathematical models that do not suffer from this problem are isomorphic to binary boolean algebra of classes (though Set Theory suffers from its own problems). Actually, I probably shouldn't have used the term model theory since that's a technical field that I don't know much about and that may not correspond to the more general notion of using models in proofs that I was talking about. My use of the term model just refers to the idea of taking the undefined terms in a formal axiomatic system and assigning them meaning in terms of some mental picture we have, then using that picture to prove something about the system such as its consistency. For example, the original proof that non-Euclidean geometry was consistent involved interpreting parallel lines as great circles on a sphere, and showing that all the axioms correctly described this situation. Likewise, Hofstadter's simple example of an axiomatic system that could be interpreted in terms of edges and vertices of a triangle proved that that axiomatic system was consistent, assuming there is no hidden inconsistency in our notion of triangles (an assumption a Platonist should be willing to make). On the other hand, here's a webpage that gives a capsule definition of model theory: http://linas.org/mirrors/www.ltn.lv/2001.03.27/~podnieks/gta.html All these results have been obtained by means of the so-called model theory. This is a very specific approach to investigation of formal theories as mathematical objects. Model theory is using the full power of set theory. Its results and proofs can be formalized in ZFC. Model theory is investigation of formal theories in the metatheory ZFC. I would guess that this means that to prove arithmetic's consistency in model theory, you identify terms in arithmetic with terms in ZFC set theory, like identifying the finite ordinals with the integers in arithmetic, and then you use this to prove arithmetic is consistent within ZFC. However, Godel's theorem applies to ZFC itself, so the most we can really prove with this method is something like if ZFC is consistent, then so is arithmetic. Is this correct, and if not, could you clarify? You are quite correct, and I appreciate your scrupulous use of model in this context. ZFC is no better than any other system in the sense that it does not escape the scope of Godel's but people are somewhat more confortable with because it covers the other holes in the Cantorian version of set theory (Russell, Buralli-Forti that someone talked about in the list not so long ago)... There would be no conditions on the proof of arithmetic's consistency using my more platonic notion of a model--since we are certain there are no inconsistencies in our mental model of numbers, addition, etc., we can feel confident that Peano arithmetic is consistent, period. This may not be model theory but it does involve a model of the kind in Hofstadter's example. I cannot vouch for whatever mental model one chooses to use to assert the consistency of arithmetic but, platonism insists that I can have access to one. The Kantian alternative is that, as the philosopher Alain, once put it: I have to make the numbers each time I need to think about them --- a much harder undertaking, in my view, at least. -Joao Leao Jesse Mazer _ Add photos to your e-mail with MSN 8. Get 2 months FREE*. http://join.msn.com/?page=features/featuredemail -- Joao Pedro Leao ::: [EMAIL PROTECTED] Harvard-Smithsonian Center for Astrophysics 1815 Massachussetts Av. , Cambridge MA 02140 Work Phone: (617)-496-7990 extension 124 VoIP Phone: (617)=384-6679 Cell-Phone: (617)-817-1800 -- All generalizations are abusive (specially this one!) ---
Re: Fw: Something for Platonists
Joao Leao wrote: James N Rose wrote: If there are no qualia but there are universals -- which cannot be identified except via qualia -- something is awry. Why so? Why can universals only be identified via qualia if they are, by definition, what is not reducible to qualia !!! If the Ideal need not share relational aspects with any other domains then that right off the bat kills any statements attempted between Ideal and Real. What do you mean by statements attempted between Ideal and Real? Give me one such statement and I will let you know... These are not superfluous issues. They challenge the consistency and fundamentals of Platonism. James -Joao -- An etheric uncorruptable realm is a excellent mythos, IMHO. That we act upon and relate to the notion of it speaks to the fact that it is possible to establish an authentic relationship with presumed or virtual extants versus empirical/encounterable extants. Modern Platonists allow that mathematical entities carry this quality and allow exploration of relations that may not have real physical correlates but that eventually, somewhere somehow, expose relations which do. The square root of a negative number has no physical reality (or so it is presumed, because no abject examples have yet been shown/proven) but it has a most definite platonic ideal existence. Plato identifies ideals such as Beauty, Justice, not just the essences of chair and other 'things'. And these -seem- to be requisitely a priori to instantiation, and so, eternal if also intangible. In support of platonism, one correlate would be like trying to educe 'wet' from the equations of QM and atomic interactions. First, most would say it cannot be done (albeit that no one has taken the time to define or make argument doing so). Second, the language of QM doesn't transduce to 'wet' or similar qualia. Yet such qualia would not occur if the primitives (QM) didn't have the relational properties that included eventual conditions and relations which could be labeled as and qualify as this or that 'emerged' qualia. Is 'wet' a platonic realm in the QM tier of existence? Is QM a shadow of instantiated 'wet' which is in turn an instance of the true extant/ideal 'Wet'? So is 'wet' an invisible inherent aspect of QM interactions? There is currently no way to transduce and correlate meaningful information between tiers of systems. But that does not mean it will never be accomplished, or as correlate, that Universals will always stand as some separate perfection. The universe is an holistic operant. Any aspect, meaning or pertinance must have an information relationship with other aspects of existence. Between cannot instantiate except in conjunction with reals. But instantiate it does. It is an intangible, a relation, even as it can be subject and measurement. The Platonic 'ideals' - all of them (however anyone perceives them) are -relations-, and are perforce transcribable information and identifable coordinations, in spite of whether anyone has made effort to clarify the associations and relations which coordinate to the identifiable attractors that qualify as 'ideal'. There is and will be shown to be a way to de-mystify Ideal. James
