RE: Platonism vs Realism WAS: ROADMAP (well, not yet really...
Peter Jones writes: > > Is it possible that we are currently actors in a single, deterministic, > > non-branching > > computer program, with the illusion of free will and if-then contingency in > > general > > being due to the fact that we don't know the details of how the program > > will play > > out? > > Lots of things are possible. The question is what to believe. True, but I thought you were saying that such a thing was incompatible with consciousness, and I see no reason to believe that. You could replace "computer program" with "machine" and have a description of the universe. Actually, you could leave out "non-branching" as well: the MWI is branching but deterministic, and still leaves room for first person indeterminacy. Stathis Papaioannou _ Be one of the first to try Windows Live Mail. http://ideas.live.com/programpage.aspx?versionId=5d21c51a-b161-4314-9b0e-4911fb2b2e6d --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
RE: Platonism vs Realism WAS: ROADMAP (well, not yet really...
Peter Jones writes (quoting SP): > > What about an inputless computer program, running deterministically like a > > recording. > > Would that count as a program at all, > > It would be a trivial case. Trivial does not mean false. > > and could it be a conscious program, given that > > computationalism is true? > > Obviously not, since people have inputs. There are several possible responses to that. Firstly, it isn't necessarily true that only people can be conscious, even if people (and maybe some animals) are the only entities we have good reason to believe are conscious. Secondly, there is always the possibility that an inputless, deterministic machine might be able to receive input. We might open the case, connect a lead with alligator clips, and start communicating with it in morse code. To the program, it would be like a message from God. You would then have to decide at which point the program became conscious: when the case was opened, when the message was sent, or was it conscious all along? If you say it was not conscious until the message was actually sent you would have to say that an entity was only conscious while receiving input, and stopped being conscious while it was analysing the input already received. Finally, it is possible to have the same kind of interaction that a machine with environmental inputs has by connecting two previously inputless machines to each other. They can then have a two way conversation, each surprised by the other's responses, but the system as a whole remains inputless and deterministic. The ultimate example of this is, as Brent Meeker suggested, a self-contained virtual reality. The universe as a whole could be seen as just this, unless you believe that God speaks to us from outside. Stathis Papaioannou _ Be one of the first to try Windows Live Mail. http://ideas.live.com/programpage.aspx?versionId=5d21c51a-b161-4314-9b0e-4911fb2b2e6d --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: ROADMAP (well, not yet really...
Hi, Bruno - Original Message - From: "Bruno Marchal" <[EMAIL PROTECTED]> To: Sent: Friday, August 18, 2006 11:23 AM Subject: Re: ROADMAP (well, not yet really... Bruno wrote: Hi John, Le 18-août-06, à 03:03, <[EMAIL PROTECTED]> a écrit : > Why has 6 'divisors'? because my math teacher said so? then ... >Now I know you are joking. I know that you know that six has divisors. It follows from the elementary definitions...< [JM]: Yes, I was. Not now: WHO supplied those "elementary definitions"? Surely a math teacher or the ancestor of such. Not even Gauss was born with the knowledge that 6/2=3. * Just as those "axioms" Stathis finds 'explanatory' stem from necessity to hold a "theory" valid. Another view of the world would require different "axioms". * Brent wrote about the crows who "counted" to 5, with some uncertainty even higher and concluded that 'numbers exist in nature' - I did not examine what the crows think, whether it was just a waiting period until their memory expired, and the wise experimentors assigned it to 'counting' - all in their human logic? * BTW I have a problem with the "perfect" 6: ITS DIVISORS are 1,2,3,6, the sum of which is 12, not 6 and it looks that there is NO other perfect number in this sense either. (If 1 is a divisor - meaning 1x6 = 6 then 6 is also one: 6x1 = 6). An exclusion of '1' would give a sum of 5 - Unless you want to exclude the 'number itself' from the "sum" - in which case the sum of "1" would be zero (excluded the 1). NOW I was joking. John I say to my students that in case they are saying a falsity (in math), they will get a bad or a good note, depending on the way they will defend the proposition. If they defend it by saying "because you say so during the course", then they will get a *very* bad note, indeed! Even, and I would say *especially* if it is true, that I have said that falsity. Actually I teach like that, I make error all the time (mostly intentionally but of course not always). It works. Students eventually understand that they must understand math by themselves. Each year I have student (about 20 years old) just realizing what math is all about. Now I know you are joking. I know that you know that six has divisors. It follows from the elementary definitions. And I will not repeat them, because that would be sort of an insult (of course a number is "perfect" if it is equal to the sum of its proper divisors ... by definition. Why using the word "perfect"? Pythagorean superstition or folklore, but mathematicians are not sanguine about words and representations. In the lobian interview all natural numbers are represented by strings like 0, s(0), s(s(0)), s(s(s(0))), etc. :-) Best regards, bon week-end, Bruno http://iridia.ulb.ac.be/~marchal/ -- No virus found in this incoming message. Checked by AVG Free Edition. Version: 7.1.405 / Virus Database: 268.11.3/423 - Release Date: 08/18/06 --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Are First Person prime?
Stathis Papaioannou wrote: > Peter Jones writes: > > > > > > But the important point is that the temporal sequence does not itself > > > > > make a difference > > > > > to subjective experience. > > > > > > > > We don't actually know that it is possible that > > > > there might be some flicker effect. > > > > > > Not necessarily. I'm suggesting that the actual physical events are > > > *exactly* the same, > > > just their order is different. If the world were created 5 minutes ago, > > > complete with > > > fossils, ruins, false memories etc., you could not be aware of this on > > > the basis of any > > > observation - by definition, otherwise the illusion would not be perfect. > > > This is of course > > > no reason to believe that the world was created 5 minutes ago; but it > > > does mean that > > > the absence of a sensation of having just flickered into existence is no > > > evidence *against* > > > this theory. > > > > My original point stands. There is no evidence *for* the theory. If > > the present > > state is determined by more than a 0-width time slice preceding it, > > then > > a physical process cannot be arbitrarily sliced up. > > Your original point was that the continuous flow of consciousness is evidence > against a block > universe. It is not, whether the time slices are of finite or infinitesimal > duration. It is, because however you slice a dynamic sequence, you don't remove the dynamism. You just get lots of little dynamic slices. >I'm not sure > what you mean by the last sentence either: are you suggesting that time is > quantised rather > than continuous, and if so how is that evidence against a block universe? No, I am suggesting that 0-width slices don't contain enough information to predict future states in physics. > > Computationalism does not help, because computationalism requries > > counterfactuals. > > I don't see why it does, or why it makes any difference to the present > question if it does. Computer programmes contain conditional (if-then) statements. A given run of the programme will in genreal not explore every branch. yet the unexplored branches are part of the programme. A branch of an if-then statement that is not executed on a particular run of a programme will constitute a counterfactual, a situation that could have happened but didn't. Without counterfactuals you cannot tell which programme (algorithm) a process is implementing because two algorithms could be have the same execution path but different unexecuted branches. > > > > > Would you say that it is in theory possible for the subjective > > > > > passage of time to be as we know it if the blocks were not > > > > > infinitesimal, but lasted for > > > > > a second, so that the whole ensemble of blocks lasted for a second? > > > > > > > > There is still duration within blocks > > > > > > Yes, and... > > > > > > > > Then what if you > > > > > make the blocks shorter in duration and larger in number, > > > > > progressively down to > > > > > infinitely many blocks of infinitesimal duration: is there room for > > > > > dynamism in an > > > > > infenitesimal interval? > > > > > > > > There are such things as infintiessimal velocities... > > > > > > So if there is room for movement in infinitesimal intervals (or through > > > combination of > > > infinitesimal intervals) in a linear theory of time, why not with a block > > > universe? > > > > A block universe with movement is just as dynamic universe > > (specifically, > > a growing universe). > > The effect of movement would be the same in a block universe as in a linear > universe. If time > is discrete then in a linear universe movement is the result of a series of > static frames of finite > duration, like the frames in a film. Finitism doesn't imply stasis. New frames could be popping into existence dynamically. > If time is continuous then in a linear universe movement is the > result of a series of static frames of infinitesimal duration. Likewise. > There is no room for movement within > a frame in either case - There is room within an infinitessimal frame. dx/dt is not necessarily zero. > that is what defines it as a frame - but the series of frames creates the > effect of movement. > > Stathis Papaioannou > > _ > Be one of the first to try Windows Live Mail. > ht --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Platonism vs Realism WAS: ROADMAP (well, not yet really...
Le 18-août-06, à 22:59, complexitystudies a écrit : >> > > As 1Z has so nicely put, existence implies causal interaction. > Numbers cannot causally interact, therefore they do not exist, > save as thoughts in our brains. Don(t say this to a logician. there are as many notion of "causality" than there are modal logics. (But I guess you assume a physical world and define "causal" by some "interaction"). I don't need that. I need only math "causality", like if p divide 8 then p divides 24. > > Of course I do not believe that 37 could be a non prime number, > simply because what it means to be "prime" has been exactly > defined in arithmetic. You reassure me. > I just say that these are thought constructs > with no independent existence (independent of human brains, not of a > concrete human brain). > > You might say, that 37 was prime even in the Jurassic, but I say: > nobody had invented arithmetic yet, so it's about as true as the > fact that James Bond was played by Sean Connery was in the Jurassic. > > I define a system: > > 1 + 1 = 2 > 2 + 1 = 1 > 1 + 2 = 1 > > That's all. Okay, it doesn't describe much and probably isn't very > useful, but other than that it is not inferior to peano arithmetic. > Does my system now exist mind-independtly for all eternity? Yes, and even Robinson Arithmetic can prove that. It exists because among all fortran programs there exist an infinity of programs which compute your table. (it is a Wi or a Fi) Of course "existing" does not mean interesting. Some numbers could be (relatively) boring, but that is hardly provably so in general. > > >> I have not yet seen a book on human brain which does not presuppose >> the >> understanding of the natural numbers. > > Of course, because it is a useful way to describe reality. But in our > brains, not numbers operate, but chemicals. I agree. The question here is what are eventually the chemicals. Comp answers: it is necessarily an average of point of views on an infinity of possible computational continuations. Put in that way, it even looks like some formulation of QM. The details are tricky of course (both with comp and with the QM only). > > > I let you >> discover that, and feel free to ask questions if you have a problem >> with UDA. > > Thank you, I will. > > >> >> It is really the point of the UDA. It shows that computationalism (the >> idea that I am a digitalizable machine) is incompatible with "weak >> materialism" (the idea that there is a primary stuff or matter or >> aristotelian substances). > Ok, I'll watch for this in UDA. >> > >> But until now, comp leads only to weirdness, not contradiction. And >> then that weirdness seems to explain the quantum weirdness ... >> Intuitively and qualitatively (already by UDA), and then technically >> through the interview of some universal turing machine. > > But wouldn't this universal turing machine need to be composed of > matter, and then the whole caboodle starts from the beginning? No. The additive+multiplicative structure of the non negative integers (what logician called the natural numbers) is already a video-game rich enough to generate coherent and sharable set of computational first person plural histories. That *is* the point. Don't take this for granted at all. UDA explain why. The lobian interview explain how. The question is not does Humty-Dumpty "exist"? The question is "does not comp entails too much probable Humpty-Dumpty in my neighborhood" and "how to test that". The question is "does comp entails *more*, or *less* white rabbits than QM already predicts apparently with some accuracy". Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Rép : ROADMAP (well, not yet really...
Le 18-août-06, à 19:01, 1Z a écrit : > That is quite different from conjuring up existential conclusions > from non-existential premises. I believe there exist numbers, prime numbers, even numbers, etc. and in relative universal numbers, ... (with Church Thesis "universal" need not to be accompanied by "turing"). Relative universal numbers can be defined through addition and multiplication only. They associate to each number relative ways to generate the diophantine sets, which are the Wi (and thus coding of the Fi) which I explained to Tom and George. So I believe in the collection of all (mathematical) relative computations. A well defined highly non trivial structure, especially "n-viewed from inside". So I have ontological assumptions: the non negative integers together with their additive and multiplicative structure. Ontologically: nothing more, nothing less. Epistemologically: the same + induction axioms. (on the natural numbers, or not). From that I explain how discourse about quanta and qualia emerge in a sufficiently precise way so that we can compare the comp physics (physics from the number) and empirical physics. (A test has already be done and confirmed comp, the rest are mathematical conjectures). The UDA shows it has to be like that. Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Platonism vs Realism WAS: ROADMAP (well, not yet really...
Stathis Papaioannou wrote: > Günther writes: > > > Well, let's see: in Alice in Wonderland, Humpty Dumpty fell off a > > wall. This is true, isn't it? It is certainly true independent > > of our minds. Indeed, it is true in such a way that even when > > all humans have died, this universe will have a contained a life-form > > which produced an author who wrote a book in which Humpty Dumpty > > fell off a wall. But neither Humpty Dumpty nor the fact that he > > fell off a wall were ever true in this universe - only that this > > story was written, and that many people read about it and could > > converse about it. > > > > So if you believe that numbers have an independent existence, then you > > would definitely also have to believe that Humpty Dumpty exists. > > Both are products of the mind. Either both exist, or both don't > > (other than as brain patterns). > > > > As much as I would like Humpty Dumpty to exist, I'm afraid that > > it is not so. > > The existence of numbers is not like the existence of objects, and I don't > think that most mathematical Platonists would say that it is. Most Platonists ar not Mathematical Monists. M-Monists *do* have to think it is the same. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Platonism vs Realism WAS: ROADMAP (well, not yet really...
Stathis Papaioannou wrote: > Is it possible that we are currently actors in a single, deterministic, > non-branching > computer program, with the illusion of free will and if-then contingency in > general > being due to the fact that we don't know the details of how the program will > play > out? Lots of things are possible. The question is what to believe. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Platonism vs Realism WAS: ROADMAP (well, not yet really...
Stathis Papaioannou wrote: > Peter Jones writes (quoting SP): > > > > I've never really understood why computationalists insist that a system > > > must be able to handle counterfactuals in order for consciousness to > > > occur, > > > > I've explained that several times: computer programmes contain > > if-then statements. > > > > > other than that otherwise any physical system could be seen as > > > implementing > > > any computation, which does not seem to me a good reason. In any case, > > > Maudlin shows that the requirement for handling counterfactuals leads to > > > a situation where of two systems with identical physical activity, one is > > > conscious and the other not. > > > > If two systems differ counterfactually, they are not physically > > identical. > > What about an inputless computer program, running deterministically like a > recording. > Would that count as a program at all, It would be a trivial case. > and could it be a conscious program, given that > computationalism is true? Obviously not, since people have inputs. > Stathis Papaioannou > _ > Be one of the first to try Windows Live Mail. > http://ideas.live.com/programpage.aspx?versionId=5d21c51a-b161-4314-9b0e-4911fb2b2e6d --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Platonism vs Realism WAS: ROADMAP (well, not yet really...
Bruno Marchal wrote: > Le 18-août-06, à 17:02, 1Z a écrit : > > > It is for Pythagorenas and Platonists to explain what they mean by > > "exist". > > > > However, if you are going to claim that we are actually *in* Platonia, > > (mathematical monism) there must be some equivalence between the > > existence we > > have and the existence numbers have. > > > > Of course! But that is what I am currently explaining. If you have > follow the UDA, then, even if you could not yet be completely convinced > by each steps, you should at least be able to figure out in which sense > "you", here and now, in the "shape" of an OM, to borrow the list > vocabulary, exist as a relative number. I cna't be persuaded of that without first being persuaded that numbers exist. > Third personally you exist in > aleph-zero relative number form (like snapshots of a running program). > First personally you exist in the same shape (although you cannot know > that) but you are "multiplied" by a continuum: the computational > histories generated by the UD, in the mathematical sense, going through > those third person states. > > Bruno > > > > http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Platonism vs Realism WAS: ROADMAP (well, not yet really...
Bruno Marchal wrote: > Le 18-août-06, à 17:38, 1Z a écrit : > > > That is an explanation of mind-independence, not of existence. > > The anti-Platonist (e.g. the formalist) can claim that > > the truth of mathematical statments is mind-independent, > > but their existence isn't. > > > "Their" existence ? Mathematical statements needs "chatty" machines. Mathematics proceded for centuries without any machines at all. > Mathematical truth, a priori doesn't (although "free structures" can > feature some chatty aspects many times). > > Let us make the following convention. When I will say "I believe there > exist a perfect number", it is a shorter expression for, I believe the > proposition "There is a perfect number" is true (satisfied in (N,+,*)) > independently of me, or of any theory of cognition. (A good thing, > giving that a theory of cognition is build *from* digital machine, that > is from number theoretical relations). If AR makes no existential commitments, it cannot lead to the existential conclusion that there is no such thing as matter. > Bruno > > > http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Platonism vs Realism WAS: ROADMAP (well, not yet really...
Le 18-août-06, à 17:38, 1Z a écrit : > That is an explanation of mind-independence, not of existence. > The anti-Platonist (e.g. the formalist) can claim that > the truth of mathematical statments is mind-independent, > but their existence isn't. "Their" existence ? Mathematical statements needs "chatty" machines. Mathematical truth, a priori doesn't (although "free structures" can feature some chatty aspects many times). Let us make the following convention. When I will say "I believe there exist a perfect number", it is a shorter expression for, I believe the proposition "There is a perfect number" is true (satisfied in (N,+,*)) independently of me, or of any theory of cognition. (A good thing, giving that a theory of cognition is build *from* digital machine, that is from number theoretical relations). Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Platonism vs Realism WAS: ROADMAP (well, not yet really...
Le 18-août-06, à 17:02, 1Z a écrit : > It is for Pythagorenas and Platonists to explain what they mean by > "exist". > > However, if you are going to claim that we are actually *in* Platonia, > (mathematical monism) there must be some equivalence between the > existence we > have and the existence numbers have. Of course! But that is what I am currently explaining. If you have follow the UDA, then, even if you could not yet be completely convinced by each steps, you should at least be able to figure out in which sense "you", here and now, in the "shape" of an OM, to borrow the list vocabulary, exist as a relative number. Third personally you exist in aleph-zero relative number form (like snapshots of a running program). First personally you exist in the same shape (although you cannot know that) but you are "multiplied" by a continuum: the computational histories generated by the UD, in the mathematical sense, going through those third person states. Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Platonism vs Realism WAS: ROADMAP (well, not yet really...
Le 19-août-06, à 08:48, Brent Meeker wrote quoting Stathis Papaioannou >> >> What more could we possibly ask of a theorem other than that it be >> true relative to some >> axioms? That a theorem should describe some aspect of the real world, >> or that it should >> be discovered by some mathematician, is contingent on the nature of >> the real world, but that >> it is true is not. > > That it is a true description of the real world, or that it is a true > theorem > relative to the axioms. It is a mistake to conflate the two, which I > suspect is > done by people claiming mathematical theorems are true. No. It is done by people claiming true mathematical propositions are theorem. Robinson Arithmetic (Q or RA) and Peano Arithmetic (PA), which in our context are better seen as a (mathematical) *machines*, are SOUND with respect to the so-called (by logicians) standard model of arithmetic, which is the mathematical structure (N,+,*) given by the non negative integers N together with addition and multiplication (learned in high school). Now RA and all its consistent extensions (and thus PA, "ZF", ...) are INCOMPLETE with respect to that mathematical structure (N,+,*), in the sense that for any of those theories there exist always infinitely many true propositions, "true" meaning really: satisfied by (N,+,*) which are unprovable by those theories. There is no complete TOE for the "standard" additive and multiplicative behavior of the natural numbers. But there is nothing wrong asserting that a theorem of PA is true (always with that meaning of being statisfied in (N,+,*)), because nobody (serious) doubt the axioms of PA, or doubt truth couldn't be preserved by the modus ponens inference rule or by the quantifier rules (and thus nobody doubts in the theorems proved by PA). Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
RE: Platonism vs Realism WAS: ROADMAP (well, not yet really...
Brent Meeker writes (quoting SP): > >>But the fact that a theorem is true relative to some axioms doesn't make it > >>true > >>or existent. Some mathematicians I know regard it as a game. Is true that > >>a > >>bishop can only move diagonally? It is relative to chess. Does chess > >>exist? > >>It does in our heads. But without us it wouldn't. > > > > > > What more could we possibly ask of a theorem other than that it be true > > relative to some > > axioms? That a theorem should describe some aspect of the real world, or > > that it should > > be discovered by some mathematician, is contingent on the nature of the > > real world, but that > > it is true is not. > > That it is a true description of the real world, or that it is a true theorem > relative to the axioms. It is a mistake to conflate the two, which I suspect > is > done by people claiming mathematical theorems are true. OK then, I agree. The two should not be conflated. Stathis Papaioannou _ Be one of the first to try Windows Live Mail. http://ideas.live.com/programpage.aspx?versionId=5d21c51a-b161-4314-9b0e-4911fb2b2e6d --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
