Re: Is reality unknowable?
What are the philosophical implications of unsolvable mathematical problems? Does this mean that mathematical reality, hence physical reality, is ultimately unknowable? It's not clear to me that the root know is terribly useful here; IMHO there is regularity and there is the random (whether it be absolute or effectively so - both are equivalent from the receiving end); the mere fact that we are having this discussion indicates some level of regularity in the interaction; but there is randomness as well; As Gellmann noted, the perceived proportion of each is always a function of a judge (sentient or otherwise) and that implies an inherent subjectivity. when and where there is agreement among judges upon the intersection of recognized patterns, it is labeled shared reality. Where there is not intersection, I call it reality and you call me delusional... Cheers CMR -- insert gratuitous quotation that implies my profundity here --
Re: Is reality unknowable?
CMR wrote: there is regularity and there is the random (whether it be absolute or effectively so - both are equivalent from the receiving end); the mere fact that we are having this discussion indicates some level of regularity in the interaction; but there is randomness as well; I do not know if this note helps. I know that a hamiltonian system is regular, not chaotic, not random, if it has as many constant of motion (integral) as degrees of freedom. The presence of each constant of motion splits the phase space in two parts, that can evolve independently: that is, the two parts do not interact. So I think that when we hear a discussion, we know that it is due to some level of chaos, and not to some level of regularity! Doriano With all due respect, I'll be the judge of that. ;)
Re: Is reality unknowable?
This infamous "definition" is circunscribed to a theory, as in "we say that a physical theory has an EPR if,..." Mathematical reality is not the output of (mathematical) theories but usually its input. But I think mathematical reality does not necessary equate to mathematical truth, nor does the later equate to proof, as everyone knows by now... The mathematical reality that Hardy refers to is the reality of mathematical objects (numbers, geometric figures, functions, equations, algebras...) of which we happen to have knowledge, or rational acquaintance, if you prefer, but not through our senses! It is undeniable that what we can agree about concerning these objects is a lot more certain than what we can agree about our sensorial (physical)experiences. We can surely called them Elements of Mathematical Reality for purposes of comparison (and aknowledge, for example, that Quantum Mechanics contains both EPRs and EMRs and that not all of the later map to the former...) Than my paraphrase would be something along the following lines:"If, without in any way, disturbing the physical support of our mental capabilities, we can ascertain with certainty (not necessarily prove) the attributes of a mathematical object, than there is an EMRcorresponding to it." This is tentative, of course... -Joao -Joao Leao scerir wrote: "If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, there exists an element of reality corresponding to this physical quantity", wrote once EPR. (Of course the strong term here is *predict*, because prediction is based on something, a theory, a logic, a model, ... which may be wrong!) Is there a similar definition, in math? s. -- 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: Is reality unknowable?
Hal Finney If, from a set of axioms and rules of inference, we can produce a valid proof of a theorem, then the theorem is true, within that axiomatic system. I'd suggest that this notion of provability is analogous to the reality of physics. Provable theorems are what we know, within a mathematical system. [...and much much more...] I thank you for that very nice response! (I'm inclined to suppose that math and physics are complementary, and one day we'll see physical solutions of unsolved math problem, a sort of math thermodynamics (Chaitin?), but it is just a lucid dream!). s.
Re: Is reality unknowable?
Hey all, Nice to see some activity on this list again. I think the filament's blown, but then again I'm a physicist :-) Matt. Norman Samish wrote: Perhaps you've heard of Thompson's Lamp. This is an ideal lamp, capable of infinite switching speed and using electricity that travels at infinite speed. At time zero it is on. After one minute it is turned off. After 1/2 minute it is turned back on. After 1/4 minute it is turned off. And so on, with each interval one-half the preceding interval. Question: What is the status of the lamp at two minutes, on or off? (I know the answer can't be calculated by conventional arithmetic. Yet the clock runs, so there must be an answer.Is there any way of calculating the answer?) I've been greatly intrigued by your responses - thank you. Marcelo Rinesi, after analysis, thinks that the problem has no solution. Bruno Marchal thinks that the Church thesis . . . makes consistent the 'large Pythagorean view, according to which everything emerges from the integers and their relations.' George Levy, after reading Marchal, thinks there may be a solution if there is a new state for the lamp besides ON and OFF, namely ONF. Stathis Papaioannou thinks the lamp is simultaneously on and off at 2 minutes. He thinks the problem is equivalent to asking whether infinity is an odd or an even integer. He shows that there are two sequences at work, one of which culminates in the lamp being on, while the other culminates in the lamp being off. Both sequences can be rigorously shown to be valid. Now Joao Leao paraphrases Hardy to say that 'mathematical reality' is something entirely more precisely known and accessed than 'physical reality' So I'm to understand that mathematical reality is paramount, and physical reality is subservient to it. Yet mathematics is unable to determine the on-or-off state of Thompson's Lamp after 2 minutes. What are the philosophical implications of unsolvable mathematical problems? Does this mean that mathematical reality, hence physical reality, is ultimately unknowable? When God plays dice with the Universe, He throws every number at once...
Re: Is reality unknowable?
The lamp is a translated version of the Achilles - Turtle race. My (non-physicist) tupence to the topic: Reality is a tricky concept. WE know the part of it that is interpreted by the mind for our limited appreciation. Tis is OUR reality and we know 'that' - only that. It constitutes the (common sense) world. This pertinent to 'physical reality'. Mathematical 'reality' is IMO an oxymoron, since mathematics is a virtual domain with (Hilbert's) reality of a piece of paper and a pencil (now: computer). My added question: since math is the domain of our internal mental functions - generated within the mind and our knowledge of reality is restricted to the domain of mind-representation upon not mind generated impacts we receive, is mathematical knowledge also an oxymoron? (This is not a phyicistic question) John Mikes - Original Message - From: Matt King [EMAIL PROTECTED] To: [EMAIL PROTECTED] Sent: Saturday, October 25, 2003 8:22 AM Subject: Re: Is reality unknowable? Hey all, Nice to see some activity on this list again. I think the filament's blown, but then again I'm a physicist :-) Matt. Norman Samish wrote: Perhaps you've heard of Thompson's Lamp. This is an ideal lamp, capable of infinite switching speed and using electricity that travels at infinite speed. At time zero it is on. After one minute it is turned off. After 1/2 minute it is turned back on. After 1/4 minute it is turned off. And so on, with each interval one-half the preceding interval. Question: What is the status of the lamp at two minutes, on or off? (I know the answer can't be calculated by conventional arithmetic. Yet the clock runs, so there must be an answer.Is there any way of calculating the answer?) I've been greatly intrigued by your responses - thank you. Marcelo Rinesi, after analysis, thinks that the problem has no solution. Bruno Marchal thinks that the Church thesis . . . makes consistent the 'large Pythagorean view, according to which everything emerges from the integers and their relations.' George Levy, after reading Marchal, thinks there may be a solution if there is a new state for the lamp besides ON and OFF, namely ONF. Stathis Papaioannou thinks the lamp is simultaneously on and off at 2 minutes. He thinks the problem is equivalent to asking whether infinity is an odd or an even integer. He shows that there are two sequences at work, one of which culminates in the lamp being on, while the other culminates in the lamp being off. Both sequences can be rigorously shown to be valid. Now Joao Leao paraphrases Hardy to say that 'mathematical reality' is something entirely more precisely known and accessed than 'physical reality' So I'm to understand that mathematical reality is paramount, and physical reality is subservient to it. Yet mathematics is unable to determine the on-or-off state of Thompson's Lamp after 2 minutes. What are the philosophical implications of unsolvable mathematical problems? Does this mean that mathematical reality, hence physical reality, is ultimately unknowable? When God plays dice with the Universe, He throws every number at once...
Re: Is reality unknowable?
It's also possible that the question, although seemingly made up of ordinary English language words used in a logical way, is actually incoherent. If I say, proposition P is both true and false, that is a sentence made up of English words, but it does not really make sense. I could then demand to know whether P is true or false, and whatever answer you give, I say that it is the opposite. If you say P is true, I point out that we just agreed that P was false, and vice versa. This is a trivial example because the paradox is so shallow, but the same thing is true for deeper paradoxes. The problem is not a failure of our reasoning tools, but rather that the question has no meaning. So you can't always take a sequence of words and expect to get an unambiguous and valid answer. You must always consider the possibility that your question is meaningless. The fact that people can't necessarily answer it does not imply that mathematics is unknowable or that there is no such thing as mathematical knowledge. There may be other reasons to think so, but it does not follow merely because a given sequence of words has no consistent analysis. Hal Finney
Re: Is reality unknowable?
If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, there exists an element of reality corresponding to this physical quantity, wrote once EPR. (Of course the strong term here is *predict*, because prediction is based on something, a theory, a logic, a model, ... which may be wrong!) Is there a similar definition, in math? s.
Re: Is reality unknowable?
Dear Hal, No, it is not the case that such questions have no meaning. The Liar paradox, in its many forms and instantiations, convey a meaning. The problem, IMHO, is in the assumption that the negation is instantaneous. For example, when we read the sentence This sentence is false, we take it in as a whole, it is meaningful as a whole, but we must realize that the reading of the string of letters is not a process that is instantaneous or takes no time to perform. Every physical process requires some non-zero duration. This is at the heart of my argument against proposals such as those of Bruno Marchal. The duration required to instantiate a relation, even one between a priori existing numbers can not be assumed to be zero and still be a meaningful one. You are correct in saying that the question has no meaning, but only in the Ideal sense of ignoring the reality of duration, even within Logic. Kindest regards, Stephen - Original Message - From: Hal Finney [EMAIL PROTECTED] To: [EMAIL PROTECTED]; [EMAIL PROTECTED] Sent: Saturday, October 25, 2003 11:51 AM Subject: Re: Is reality unknowable? It's also possible that the question, although seemingly made up of ordinary English language words used in a logical way, is actually incoherent. If I say, proposition P is both true and false, that is a sentence made up of English words, but it does not really make sense. I could then demand to know whether P is true or false, and whatever answer you give, I say that it is the opposite. If you say P is true, I point out that we just agreed that P was false, and vice versa. This is a trivial example because the paradox is so shallow, but the same thing is true for deeper paradoxes. The problem is not a failure of our reasoning tools, but rather that the question has no meaning. So you can't always take a sequence of words and expect to get an unambiguous and valid answer. You must always consider the possibility that your question is meaningless. The fact that people can't necessarily answer it does not imply that mathematics is unknowable or that there is no such thing as mathematical knowledge. There may be other reasons to think so, but it does not follow merely because a given sequence of words has no consistent analysis. Hal Finney
Re: Is reality unknowable?
Scerir writes: If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, there exists an element of reality corresponding to this physical quantity, wrote once EPR. [...] Is there a similar definition, in math? If, from a set of axioms and rules of inference, we can produce a valid proof of a theorem, then the theorem is true, within that axiomatic system. I'd suggest that this notion of provability is analogous to the reality of physics. Provable theorems are what we know, within a mathematical system. Now, one problem with this approach is that it focuses on the theorems, which are generally about some mathematical concepts or objects, but not on the objects themselves. For example, we have a theory of the integers, and we can make proofs about them, such as that there are an infinite number of primes. These proofs are what we know about the integers, the mathematical reality of this subject. But what about the integers themselves? They are distinct from the theorems about them. Maybe we would want to say that it is the integers which are mathematically real, rather than proofs about them. The question in my mind is how to understand Tegmark's theory that the world is a mathematical structure, something analogous to the integers but more complex. We actually live within a mathematical object, according to this view. What does physical reality mean in such a framework? Would it correspond to mathematical reality, within that one mathematical structure that we live in? Or turning to Schmidhuber's model, where the world is a computer program, what does physical reality correspond to? We have a distinction there between the program and its output, similar to the distinction in math between proofs and the objects about which theorems are proven. If we focus on the output, then that would be the fundamental physical reality of the universe. We could then chunk that output or identify patterns in it, and those would be real as well. In general, any computable function which took as input a region of the universe and produced as output a true/false result would define an element of physical reality. Some functions would be much more useful than others, producing elements of reality which are more stable or more predictable. Conserved quantities, for example, would be elements of reality which were useful for predictions in a variety of situations. But in principle, I think all computable predicate functions would have equal philosophical status. Hal Finney
Re: Is reality unknowable?
Too many messages. I cannot read them all. Is there a user group where these things are more organized? Hope so, else I'll have to block these messages. Stephen Paul King wrote: Dear Hal, No, it is not the case that such questions have no meaning. The Liar paradox, in its many forms and instantiations, convey a meaning. The problem, IMHO, is in the assumption that the negation is instantaneous. For example, when we read the sentence This sentence is false, we take it in as a whole, it is meaningful as a whole, but we must realize that the reading of the string of letters is not a process that is instantaneous or takes no time to perform. Every physical process requires some non-zero duration. This is at the heart of my argument against proposals such as those of Bruno Marchal. The duration required to instantiate a relation, even one between a priori existing numbers can not be assumed to be zero and still be a meaningful one. You are correct in saying that the question has no meaning, but only in the Ideal sense of ignoring the reality of duration, even within Logic. Kindest regards, Stephen - Original Message - From: Hal Finney [EMAIL PROTECTED] To: [EMAIL PROTECTED]; [EMAIL PROTECTED] Sent: Saturday, October 25, 2003 11:51 AM Subject: Re: Is reality unknowable? It's also possible that the question, although seemingly made up of ordinary English language words used in a logical way, is actually incoherent. If I say, proposition P is both true and false, that is a sentence made up of English words, but it does not really make sense. I could then demand to know whether P is true or false, and whatever answer you give, I say that it is the opposite. If you say P is true, I point out that we just agreed that P was false, and vice versa. This is a trivial example because the paradox is so shallow, but the same thing is true for deeper paradoxes. The problem is not a failure of our reasoning tools, but rather that the question has no meaning. So you can't always take a sequence of words and expect to get an unambiguous and valid answer. You must always consider the possibility that your question is meaningless. The fact that people can't necessarily answer it does not imply that mathematics is unknowable or that there is no such thing as mathematical knowledge. There may be other reasons to think so, but it does not follow merely because a given sequence of words has no consistent analysis. Hal Finney
Re: Is reality unknowable?
Too many messages. I cannot read them all. Is there a user group where these things are more organized? Hope so, else I'll have to block these messages. This mailing list is archived at http://www.escribe.com/science/theory/, as well as http://www.mail-archive.com/everything-list%40eskimo.com/. You could read the messages there and not have your mailbox flooded. One problem though is that this list tends to be bursty in its traffic. Your mailbox had to deal with 11 messages so far today, 5 yesterday, 7 over the previous week, and then none back to July. So if you resort to periodically checking the web page, you may find no changes for weeks at a time, lose interest, and then miss out on a new conversation. Personally, I get well over a hundred messages a day, not counting several hundred additional spams. So a dozen from this list is barely a drop in the bucket. It might be worth your while to invest time in learning to manage your mail tools, so that in future years you will be able to handle an increasing flow of information. Hal Finney
