mm-1819 As do spherical surfaces. On a spherical survace straight lines are great circles. There are no parallel great circles. Any pair will intersect. Bob Kolker robert j. kolker said: That's non-Euclidean, no? Euclidean geometry refers to flat surface 2D geometry doesn't it? No curved surfaces? -- Smiles, Tony So what? You seem to be merely contentious today. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t In the real world. Experimental verification of General Theory of Relativity indicate the spacetime is curved. The geometry of the spacetime manifold is semi-Riemannian geometry which is non-Euclidean in its general formulation. Bob Kolker So Euclidean geometry is false? A theory is still only a theory, i.e. falsifiable. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t [...] When used about the real world, Euclidean geometry runs into trouble very quickly. One of the theorems in E-geometry states that the internal angle sum of a triangle is equal to pi. I'm sure you had to prove this in your high school math class. There are some truly beautiful proofs of this theorem. The one I like best is the one that uses parallel lines. EWhich of course dpends on the paralle line postulate, which doesn't need to be assumed, and ... but let's not go there right now. This theorem is false about triangles on the surface of the Earth. True, the triangle has to be fairly large for the difference bewween its internal angle sum and pi to be larger than the measurement error, but as any engineering student who had to suffer through surveying knows, it's not _that_ large. Doesn't take much to run into the difference, IOW. Surveying a golf course is enough. (How do I know? I found out by doing it. what else?) Does this mean you believe that there are truths that are above and beyond theory? That's how I parse it, but you;ve told me several times now that I don't parse you correctly, so I'd like to know what you intend by this comment. Yes. But I wasn't referring to that in my statement above. Nothing more than what is known throughout science, that all theories are falsifiable. You stated that: Experimental verification of General Theory of Relativity indicate the spacetime is curved. Yet, despite experimental verification, it remains a theory and therefore falsifiable. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t Lines drawn on the surface of a sphere ***are not*** straight lines, in Euclidean geometry. Mati Meron | When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same Indeed. So? They're geodesics defined by the properties of the spece *plus* the constraint defining the surface *within* this space. That's ******not****** straight lines. Again, true, but irrelevant. Lines drawn on the surface of a sphere are not straigh lines and the shape enclosed by three such lines is not a triangle. And in the story about Gauss measuring the sum of the angles of a triangles on the Earth surace, he was measuring the triangle defined by the *lines of sight* between three points, not lines drawn on the surface of the Earth. And his result was that, within the limits of his experimental accuracy the sum of the angles *was* 180 degrees which, as he recognized, didn't mean that space may not be curved, only set a limit on such curvature. Mati Meron | When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same They are geodesics. If you draw circular lines on a cylinder co-axial with the axis of a right cylinder the lines are curved but the intrinsic curvature is 0. Bob Kolker [...] Is there any , um, explanation of what goes in the world that is not a theory, then? If so, what would you offer as an example? Wolf Kirchmeir said: I would say the God did it hypothesis is not a theory in the scientific sense, since it is not testable. -- Smiles, Tony [...] And I would say that's not an explanation. In fact it is an anti-explanation. It is a conceptual obfuscation since nothing quantitative whatsoever can be said about the gods (however many of them there might be). Nor is there a scintilla of evidence that the gods exist. If the gods-hypothesis yields quantitative predictions there might be some excuse for using it. But the gods seem to have whatever properties are needed to cook up an explanation. That makes the gods, however many of them there might be, a major excercise in Ad Hocary. Bob Kolker So if transfinite math yields no quantitative predictions, there's no excuse for using it? As much excuse as for playing chess. Since mathematics is not a science it need not produce quantitative predictions. Mathematics in the abstract, predicts nothing and explains nothing. Bob Kolker Well, Bob, you certainly explain nothing. Isn't it exciting, Bob, to know that in just a few years now, all will be made clear to you and there will no longer be a need to speculate? -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t Speculation is one of the modes of operations of human brains. Mathematicians and other artists will alway speculate and imagine. Bob Kolker So now mathematikers doing not only mathematics and science but art too? Will wonders never cease. THe theory may be false, but the experimental data proving the curvature of spacetime have been verified again and again. Spacetime is curved. Euclidean geometry is an abstraction which does not apply literally to the universe we live in. And you don't need facy relativistic experiments to prove curvature. Just survey a large enough area and you find the angles of a triangle add up to more thant 180 degrees. Euclidean geometry taken as a heuristic applies to very small regions of space. Bob Kolker I never said it was false; I said falsifiable. There is a profound difference. Experimental data /proves/ nothing. There is also a profound difference between verification and proof. It certainly seems so, at least according to current evidence. Yes. So is non-Euclidean geometry. The earth isn't a perfect sphere and the surface is uneven. In some huge depressions I expect you will find some triangles with *less* than 180 degrees. OK. But that doesn't make it false; Just applicable to a different domain. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t This is one way of putting it. But the accurate way of putting it is that it is fairly clear that photons do not move in Euclidean straight lines. Since many calculations in physics are easier if photons move in straight lines, a non-Euclidean geometry makes calculation more convenient. This does not mean that you couldn't use Euclidean geometry to do the same things; it just would be far more inconvenient. Thus, your claim that Euclidean geometry does not apply would appear to be false. At least, Euclidean geometry applies as much as any other geometry, apart from pragmatic concerns. -- Aaron Boyden The main division between the so-called Continental and Analytic traditions has been disputes over whether the task of being unclear should be carried out in natural language or in a formal system. Very well put. If one wants a flat space, then he has to invent forces to account for divergent motion. Which is easier. A non-flat manifold or a flat manifold with inertial forces? Bob Kolker I completely fail to see in what sense the invention of new forces is required. The actions of the forces must be expressed by more complicated mathematical formulae, but where in those complications are the new forces? -- Aaron Boyden The main division between the so-called Continental and Analytic traditions has been disputes over whether the task of being unclear should be carried out in natural language or in a formal system. I doubt Nature is responsive to human questions like Which is easier. I thought we had this all cleared up yesterday. Now you seem to be taking the opposite side. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t Kind of like fiction and non-fiction categories in literature. Or realistic and surreal/abstract in art. Fiction is internally consistent but 'any resemblance to persons living or dead is purely coincidental'. Consistent is one of the meanings of truth. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t Consistent is more related to possibility than truth. As you point out one can construct a cluster of propositions that are internally consistent but who conjunction is false to fact. I have seen consistent systems characterized as systems true in some possible world. Bob Kolker Yes. What I intended, but better stated. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t Set theory was not taught in my '50s math classes. So the 'cardinality' of a set with an infinite number of members would have been viewed with incredulity. what has been said to you on this thread passed by unparsed and therefore not understood. I never found the foundations (algebra, geometry, trig, etc.) particularly difficult. I simply found the lunatic fringe of mathematicians to be illogical and hence boring. Math just never set me on fire. Well, I can agree that literary criticism is crap, but not for the reasons you cite and misunderstand. Now who is getting silly. Take your magic mathematical code ring and wand over to comp.ai.nat-lang and discover the function of context in natural language. You have the perfect example already. Where else, but in mathematics, could one talk of different sized infinities, and a counting of infinities that was a definition rather than a process. No. I would say that 'size' is inappropriately applied to any infinity. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t Why? Cardinalities can be ordered and compared. Bob Kolker It seems to stupid old me that cardinalities could be ordered and compared without recourse to logical absurdities. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t ^^^^^^^^ Thinking a little too much of the epistemology of Empidocles? I could have sworn it was a different volcano. ;-) But then, you never know. No, thinking too little. Oops, quite right ;-) -- Richard Herring I think I said something similar to you, Wolf. Tell me, what about your different sized infinities? Are they just talk or do they describe actually different sized infinities in the world? Or for that matter, are there *any* infinities in the world at all? -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t It was aimed at Wolf in your defense. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t Albert said: -- Smiles, Tony Google for Gregory Bateson. I think you will like him. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t Albert said: -- Smiles, Tony I personally have no problem with saying that there is a mathematics that is a priori and prima causa in Nature. My problem is with those who claim that *all* of modern mathematics is so, when so much of what is claimed by some mathematicians is patently an invention and is totally unrelated to the world. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t Nope. Then you're pointless? Nah, you just don't know what whiffle trees ansd hames are, is all. If you did, you wouldn't have made that pointless reference to Scotland. Well, I'd rather be pointless than witless. [...] There. Wolf Kirchmeir said: I see no contradiction. Perhaps you could be a little more polysyllabic and explain which two statements you see as conflicting? -- Smiles, Tony Wolf was trained in spaces. He has trouble with time. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t ... and never has any application... How can you tell? Wolf Kirchmeir said: You can't necessarily while exploring. Mathematical ruminations are a wonderful pastime in their own right, but are like a pretty tree with no fruit until they find application in, and confirmation from, the real world. It's when they result in a new discovery in the real world that they become truly beautiful, in my eyes. But we all know what they say about beauty. -- Smiles, Tony You should zoom in on Bob Kolker with this issue. I would love to see him try to do with his peers, what he tried to do with me. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t Which part? Why don't you look at what mathematics is actually about. I tried asking you and got nothing but . Well, there seems to be confusion on both sides. You seem to believe that all calculation belongs in applied math and that *authentic* math is only possible for certain rare and highly gifted persons, such as yourself, who have no problem accepting logical absurdities in the name of logic. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t a) the bit about the invention of logic. A philosopher invented formal logic, but that doesn't mean logic is a product of philosophy, if onlyu becasue philosophy on Aristotle's time isn't at all what philosophy means nowadays. It would be far more accurate, both historically and ontologically, to say that logic is the product of Greek science (which they termed philosophy). Or so it seems to me. It's been a long time simce I read Aristotle and Plato, or commentaries on them. b) The bit about math. Math is only partly about measurements. In fact, there is a slew of math which deals with nothing but measurements. Since measurement in actuality entails errors, this area of math has perforce expanded into error-correction theory, and is extremely useful, as well as theoretically intersting. I had to learn a fair chink of measurement theory when I did engineering (which I didn't finish, since pool and poker interested me more than practicing problem-solving. I was very young, you see. :-)) -- FWIW, I suspect that the use of metric in ref to certain spaces has misled you, but without a clearer definition of what you mean by measurements, I can't really tell. [...] Well, it seems there is. I think that arithmetic isn't math, but figuring why arithmetic works is math. (That's all in NL terms, as far as I can tell. :-)) The Sumerians and Egyptians knew a lot of integer solutions to a^2 + b^2 = c^2: they clearly did a lot of arithmetic. We don't know whether they did any math, ie ask a the general questions about integer triplets that satisfy that equation, but on the evdince, they never did. Pythagoras did. He did math. Ditto Euclid, Archimedes, and many other Greeks. And of course doing math requires calculation, sometimes. But calculate is a vague term here. I myself have used it to mean determine the truth values of a complex truth function, which isn't exactly the same kind of process as summing an infinite series. Classical logic (I assume you mean Aristotle's rules) has been subsumed by Boolean algebra, which has been subsumed by set theory. But that's OK, it's all math. Or all logic, if you like. Sometimes I prefer one view, and sometimes the other. Depends on the weather, perhaps - I haven't kept careful records. :-) As to gifts: I have only average gifts in math, and make many errors, some which others have been kind enough to point out. (See stephen's comment on my muddled characterisation of countable/uncountable sets.) I'm a spectator at a sport that I don't do very well, but know enough about to admire the skill of the pros. Like hockey. Where I differ from you, it sometimes seems, and Zick for sure, is that I'm willing to take correction and coaching when I utter opinions. I don't insist that my understanding of what a mathematical term really means must be right, nor do I insist that my insight must be true because everybody else thinks it's daft; etc. Most mathematicians have rather specialised gifts - it's for example rare to find that a whiz at number theory is equally adept at topology. In any case, I enjoy math, I enjoy reading about it, I even from time to time work through a proof just get a better glimpse of how it really works. Mathematics is beautiful. I could now maunder on about the beauty of other arts and sciences, but that would be getting really off topic for this forum. HTH I think some more about the notion that a function among concrete physical objects represents a concrete physical object. Some theories do have a set of all sets that is indeed a set, and the powerset result doesn't apply to that and the powerset result is contradictory in application to that. If ZF is consistent, then certain results are claimed. I think the avenue towards showing ZF inconsistent may lie in regularity. About the concrete physical objects, given three point masses A, B, C why do each of the discrete gravitational forces AB, AC, and BC each exist, as does ABC and perhaps an empty set of the discrete forces upon that system? If you aggregate A, B, C, and AB, AC, BC, and ABC it easy to see why given a set of atomic point masses the discrete gravitational forces among them would represent the powerset of those things. Then we get into the notion of why the powerset of that set, in the physical sense, ie A, B, C, AB, AC, BC, ABC and correspondingly A+B, A+C, A+AB, A+AC, A+BC, A+ABC, ..., would each represent a concrete physical thing by itself or themself. Where that is so it might be possible to consider why that combinatorial amount of those things might not always be the powerset but would diminish in that A+A+ABC would only equal A+ABC. A question would then arise about whether there are any indivisible, i.e., atomic in the sense of being indivisible, concrete physical objects in the universe. If there are, then a question is if they combine or aggregate to make distinct objects, that is, the sum being more than the parts. One way to consider that is that the combinations of the objects are different from their parts in various configurations, where there is infinite divisibility of any physical object or thing. For example, isomers of molecules have varying properties. The orbital electron, besides basically being the orbit and having to do with quantum angular locations. The electron in the hydrogen atom at the tip of your nose might just have been in the rings of Saturn, and probably just recently was. The universe is infinite. Infinite sets are equivalent. Some aspects of technical philosophy are applicable as signposts of deeply explored epistemological phenomenology, or the study of the meaning of knowing things about stuff. Infinity by itself is rife with incipient complication. There's always one more. Classes, model extensions, Burali-Forti, and potential vs. actual are all the same thing. The point to consider is that so is Russell, and the Liar, and they're right that way. eschew dilletantism, although that's educative and shows your rapid comprehension in these discussion, and demand your own, then you can more readily identify those mathematical philosophies that agree with you. Here's what I say: one theory, first-order predicate calculus as a set theory with the ur-element: no axioms, no paradoxes, consistent, complete, and concrete. Ross Finlayson -- Keep on Truckin' .::::. .::::::::. ::::::::::: ..:::::::::::' '::::::::::::::: .:::::::::::' '::::::::::::::. ..::::::::::. ``:::::::::::::: ::::``::::::::' .:::. ::::' ':::::' .::::::::. .::::' ::::: .:::::::'::::. .:::' ::::: .:::::::::' ':::::. .::' :::::.:::::::::' ':::::. .::' ::::::::::::::' ``::::. ...::: ::::::::::::' ``::. ```` ':. ':::::::::' ::::.. '.:::::' ':'````.. grounds TF?) The first statement is pure assertion. Where is the proof? The second statement is just plain wrong. The set of integers and the set of real numbers are not cardinal-equivalent. See Cantor's famous diagonal proof. Bob Kolker Hi Bob, I've heard of that before, the antidiagonal argument. Because I think that infinite sets are equivalent, to some extent due to the well-ordering and transfer principles, I'm motivated to consider the antidiagonal argument and why it wouldn't apply. One notion of that is the binary antidiagonal and dual representation, another how as the base of the representation of the number diverges to infinity that correspondingly a single digit represents the non-integer portion of the number, another leading zeros, convergence, the inadequacy of expansion as representation, etcetera. Perhaps you've heard of Cantor's first, Cantor's first proof of the uncountability of the reals, and that the reals have a well-ordering, and why that well ordering would necessarily satisfy the same conditions as a bijection from the natural integers to reals, and of iota as the least positive real, and a structuring of the reals as contiguous, and continuous, points on a line as iota-values, and how iota is very similar to the differential or dx, or of non-standard measure theory, infinitesimal measure, or Borel vs. Combinatorics. equivalency function. You probably accept that there is no set of all sets in Zermelo-Fraenkel set theory. The ordinal type of the set of all ordinals would have been an ordinal in said theory, or Burali-Forti, where Ernst Zermelo and A.A. Fraenkel are separate people and Cesare Burali-Forti is one person, and Burali-Forti is that the order type of all ordinals would itself be an ordinal, and vacuously less than ... nothing. Some people do have a set of all sets, a set, in their set theories. In terms of numbers (ordinals) and sets and the powerset result, there is the notion of ubiquitous ordinals, and the dual representation of the ur-element as minimal and maximal element, with only one variable, and as ordinal powerset as successor as order type. Is the ur-element empty or the universal set? It's whatever you want it to be, of those two, they're mechanically interchangeable. I'm somewhat more interested in this notion that there would be infinitely many concrete physical objects in the universe and that each function between them is as well a concrete physical object. To whit: there exists us. That's the primary notion of this post to which you replied that you did not address. You mention Leibniz, his monadology might have each of the concrete physical objects being a monad, or every function comprises the monad. I agree with Leibniz, and much of Stephenson's fictionalized Leibniz, for example As a matter of fact I should like to think that these two sets of rules -- the one governing monads, the other governing the mechanical mind -- will turn out to be one and the same. In more about the universe being infinite, and quantum mechanics and relativity representing ultimately micro- and macroscale essentially combinatoric and continua-based ramifications of very large or small numbers or infinity, my naive perception is that that is what those things are. About the consideration of the continuum in nature, if there are straight lines then there are circles and if there are circles then there are irrational quantities. A raindrop is a droplet. Can you consider a _function_ from a set of physical objects to other physical objects to itself be a physical object? If so, then the universe has infinitely many physical objects. Ross grounds TF?) There does not exist a one to one from the integers onto the reals. Period. Bob Kolker their decades Of course it is moot, becaue as you said they are abstract. Nothing abstract truly exists. Only when you have a physical model as above, it will, and then it will need a physical unit, because an integer is just a property, it is just a form (to use Plato's term). So, we must understand that abstract concepts always exist in and of our intellect and nothing more. However, this is just the bare-bones metaphysics, it doesn't explain much. It doesn't explain how we discovered the concept of an integer to be a useful one. It doesn't explain how you can use integers in finance, in engineering, in measuring time, etc. The most interesting discussion comes from whether the world is lawful or not. Obviously it is, or abstract concepts would not work, etc. You can bring on the rest, it's easy. Leibniz was fascinated by the regularities in the world. Then, we have to take note that the use of an abstract concept is related to the abundance of its physical models. If we could find NO physical model for an integer, then it would be likely that there would be no such thing as an integer in the first place. Maybe there would be no such thing as a number at all, if things did not exist in quantities in this world. (Of course our world is not like that, but we're talking metaphysics) There are other considerations. Quine would suggest that mathematics is continuous with our common sense. I would, too. The only problem is that, common sense is made up of entirely abstract concepts, so the above applies, but there is nothing we can do about it. We can simulate only so much of the world. -- Eray Ozkural Granted, but paradoxically abstractions are very powerful tools, which sort of raises the question as to what we mean by exist. After all, couldn't it be said that selves are abstractions? Nonexistence isn't possible in general terms, only in particular contexts relative to all other contexts. In saying abstractions don't exist, it is only meant that they don't exist like cars or trucks but not that they don't exist at all, claims to the contrary not withstanding. A phrase like don't truly exist is either false or ambiguous. which sort couldn't it Surely selves are abstractions, we can be sure of that because our self-models are almost always grotesquely incomplete. There are so many things that one does not / can not know of himself. But I don't think that this makes the entire system non-existent, which is the other meaning of self ;) -- Eray Ozkural their decades Of course it is moot, becaue as you said they are abstract. Nothing abstract truly exists. Only when you have a physical model as above, it will, and then it will need a physical unit, because an integer is just a property, it is just a form (to use Plato's term). So, we must understand that abstract concepts always exist in and of our intellect and nothing more. However, this is just the bare-bones metaphysics, it doesn't explain much. It doesn't explain how we discovered the concept of an integer to be a useful one. It doesn't explain how you can use integers in finance, in engineering, in measuring time, etc. The most interesting discussion comes from whether the world is lawful or not. Obviously it is, or abstract concepts would not work, etc. You can bring on the rest, it's easy. Leibniz was fascinated by the regularities in the world. Then, we have to take note that the use of an abstract concept is related to the abundance of its physical models. If we could find NO physical model for an integer, then it would be likely that there would be no such thing as an integer in the first place. Maybe there would be no such thing as a number at all, if things did not exist in quantities in this world. (Of course our world is not like that, but we're talking metaphysics) There are other considerations. Quine would suggest that mathematics is continuous with our common sense. I would, too. The only problem is that, common sense is made up of entirely abstract concepts, so the above applies, but there is nothing we can do about it. We can simulate only so much of the world. -- Eray Ozkural for mathematicians, But don't mathematical is exist. As pretence, forgivable, but as a definite metaphysical position as Ketland asserts it is far from that. First of all, it is being naive. Secondly, it seeks to give mathematicians a special status. According to this view, when I write down a mathematical theory, I do not only create a consistent system of rules, etc. etc., but I also arrive at a pre-existing truth. That is childish. Everybody knows that consistency is not identical to reality. What should we next believe in? That when we think of any new idea, it had existed all along? These Platonist hallucinations do not make any sense. If all novelists thought their characters actually existed, we would have to put them in sanitariums. However, of course, it may be easier for a novelist to assume that his novel-world existed, and think more thoroughly, live the emotions directly. That is forgivable pretence as you say. But we call seriously Platonist novelists madmen. This is the confusion between knowledge and existence. Some people think that nominalism is a cure for such fundamental confusions. You can know something about a mathematical theorem, that does not mean that its referents exist outside of our physical universe, or anything absurd like that. First and foremost it's just a couple of squiggles on a piece of paper, it's just an idea. There is no such thing as a number existing outside of theory before we can find a meaningful physical example to that. (For number 1, we have the number of electrons in the hydrogen atom, it's a _physical_ property) (Unless that is done) It's merely a theoretical concept, an abstract concept. It is a refined and useful concept, and indeed there are such useful concepts in our common sense, for instance past. The past does not exist, but it's a really useful abstraction. Why do you say I drank some tea yesterday? Does yesterday exist? The epistemological status of yesterday is a little like the concept of 2. Think about this a little if you would like to. There is more. I can say There is Mr. Spock in Enterprise spaceship. This is like me saying There are uncomputable reals in the (0,1) interval. Surely both sentences are true. But their referents are complex things we call fiction or theory. Unless we find a physical example for this theory, there is no physical sense in which these are true. Surely, mathematics is a lot like metaphysics, it talks about possible worlds, systems, but it's all _imagination_ you see. Sometimes the imagination is so powerful that it can address even engineering, the hardest kind of physical achievement for man, but this only shows that mathematics is just another intellectual tool, it's thinking hard and well, that's all. I show the difference between abstraction, theory and existence in the above examples, those who can understand it, will. Mr. Ketland, on the other hand, thinks that whatever relevant or irrelevant mathematical theory you conjure, somehow its objects become automatically real, or _was_ real before you thought it up. That is mathematical realism. And on the face of it, it is _absurd_, because we have something called metamathematics whose object is mathematics. Then, all formal mathematics, including _all_ formal axiomatic systems, not just real numbers or hyperreals or large cardinals, existed before we thought them up! What a consistent schizophrenia. Now, I suppose if we had a mathematical theory of evolution, this would mean that human beings existed since the dawn of time:) Not satisfied? You can say that every computation exists, when you make a new computation you don't 'make' it, you just arrive at something that already existed. Oh! There are an endless number of absurdities you can draw from this philosophical position. -- Eray Ozkural Sure. Physical existence, like your brain. Mathematical existence is different, it means something like, being Alice character in Alice in Wonderland. The theory is useful, though, it is not fiction like Alice in Wonderland, because although it arises from certain definitions, the definitions are sometimes derived from the world. Etc. etc. that is the sense many mathematicians use the word exist in their writings, even if they have no metaphysical concerns about these. You can also call this textual truth if you are linguistically motivated. The word exist is used in so many other senses you would be surprised. have First of all, Quine does NOT contradict with what I say, he is merely pulling off another linguistic word play. I doubt this has any relevance to our subject, but since you thought it had (It does NOT. I am a physicalist like Quine), I will spare some comments. .... It should also be clear that Quine presupposes the physical existence of space-time, which is not granted. I think he is doing a good job of displaying his confusion about knowledge vs. existence there. Note how he avoids explaining what 7 is, another sort of thing, while the issue is precisely what sort of a thing it is. After all, if it is not anything in physical space-time, then why should Quine, as a physicalist, think it exists? He talks as if we don't know that there is and there exists means the same thing! That is, I doubt Quine manages to say anything above. And furthermore, this quote is entirely irrelevant to our discussion, which is about the nature of mathematical objects, *whether* they exist, not about whether exist means is which is the case according to English, which we are using. Maybe you think Quine chose the example of 7 because he indeed thought 7 existed. Hard to say. It's very hard to say what Quine Boston is. and 7 is. don't make equivalent senses. 7 is, I would say, is quite nonsense. Why should anybody say such a thing? It would look absurd even in a mathematics textbook. There is 1 There is 1, repeat that for a while and you'll find yourself saying There is God. we what of Putnam surely doesn't manage to qualify his phrases: t r u t h s about WHAT? What is the truth of mathematics? What is the real thing about mathematics that we're trying to match the theories to? This is just tacit assumption of the truth of mathematical realism, and somehow dressing it up in intellectualism. That is, more of the same scenario as Quine. Written above is nothing more than word play. This same lack of methodological rigor also shows up in his confused commentaries on psychology. He first ruined functionalism by introducing Platonism into it, and then suggested that we should seek a theory of direct perception. Direct what? AFAICT, one doesn't have to be intellectual to be a Platonist. That is, being an excellent professor of mathematics and a wordsmith does not automatically give you an advantage in metaphysics. -- Eray Ozkural Physical objects and mathematical objects are different kinds of objects, but their qualitative difference doesnÇt consist \ in different meanings of exist. The concept of being part of a theory is vague. Theories are sets of coherent sentences, which have only linguistic, or more generally, semiotic items as their parts. So would you accept, e.g., the following definition: The number X exists =def The numeral 'X' is part of a mathematical theory Please surprise me, for in my opinion the concept of existence is totally unambiguous. There are numerous kinds of objects, but there is only one meaning of exist. ? - QuineÇs statement flatly contradicts what youÇve said because he decidedly rejects the more-than-one-sense view of exist. IÇm sorry, but you seem to have missed \ QuineÇs point. Yes, a exists and a is are indeed synonymous. Quine raises no objection to that. Quine intends to points out that the ontic difference between concrete and abstract objects consists in their being different kinds of objects and not in their falling under different concepts of existence. ? (Ever heard of direct realism in the philosophy of perception?) PH On 10 Feb 2005 13:30:31 -0800, Paul Holbach It most certainly does. If a qualitative difference between the existence of A and the existence of C doesn't refer to different meanings of the term exist, what does it refer to? If A and B are two (qualitatively) different objects, then there are some properties possessed by A and not possessed by B. One may say that AÇs existence is different from BÇs existence, with A/B and AÇs/BÇs existence being identical \ (eg, I a m my existence); but that has nothing to do with the unambiguous meaning of the concept of existence. PH On 11 Feb 2005 15:25:18 -0800, Paul Holbach So if the existence of A and C are different, the meaning of existence in A and C is the same? The existence of A/B refers to nothing else but A/B. If A and B are different existents, then that is not due to an equivocality in the concept /existence/. PH On 12 Feb 2005 12:21:54 -0800, Paul Holbach Then how is the existence of different things the same? What do you mean? All existents possess the selfsame property of existence. PH grounds TF?) Exisence is not a property or predicate. Suppose it were. Let e(x) means x exists. Then are reasonable postulate would be -Ex-e(x). Read that as follows: There exists no x which that x does not have the property of Is that what you want? Bob Kolker with concept I am my existence. Do you think that sentence makes much sense? What is your existence? What is a number's existence? -- Eray Ozkural I do think it makes sense. Everything I am. This question is equivalent to: What is a number? PH grounds TF?) Apply the Leibniz Rule. An entity is all the properties it possesses. Bob Kolker So? different They are vastly different uses, so assuming a use theory of meaning I think we'd be forced to that distinction. -- Eray or I did say something in that vein but an even better point would be try to avoid formalizing the distinction of the two senses. Obviously the existence of _a_ set is nothing like the existence of _an_ apple. If I can make no such distinction, believing in Quine's passage, then I would have to presuppose the actual existence of _unique_ abstract objects instead of admitting their fictional character. Even the phrase abstract object carries a Platonist presupposition in its uniquenes, definitive-ness, and actuality. This whole way of talking is vastly impaired in its ability to distinguish reality from *theory*, and we seem to be discussing trivial points that really shouldn't be made. Personally, I cannot see how Quine maintains his monism and at the same time telling us this about actuality of abstract objects. The whole passage seems like an exercise in triviality. Let's think of the word 'exist' as 'is'. This sounds a lot like Let's think of 'put' as 'place'. There are not two senses of existence, but there is difference in the kinds of objects. Why is this difference in kind significant at all? What does it mean to say that a mathematical object like '7' exists? I don't know if I've made my point. Quine there says nothing about what it means to say that a mathematical object exists, which is the very thing we are trying to explain, e.g. your definition above. Surely, in your definition a mathematical theory has to physically exist, which is easy to see. But it should be obvious that the realists whom I am criticizing suggest that numbers actually exist like a rock or a chair, which is an altogether different, absurd, Pythagorean conception of the universe. I cannot believe that Quine would believe in that. It is far more sensible to suggest the theoretical sense of existence, the _hypothethical_ sense. Simply because that is much better than what Quine says, as it applies to more than _mathematical objects_, which is just one class of objects. -- Eray Ozkural Your statement above means nothing else than: Obviously, a set is something which is quite unlike an apple. I certainly agree with that. The question whether the concept of existence is unambiguous or not is not trivial! PH different That's a remarkable underachievement in your use of language. Do you mean to say Alice in Alice in Wonderland exists in the same sense that you exist? -- Eray Is that so? I mean to say that Alice in Wonderland does not exist at all. PH The sense of exists in a novel is fictional. If that were not so, then the novel would be nonsense. (That is a sufficient demonstration) Likewise for a mathematics textbook, taken on its own, say especially on a field where there is no application to think of (large cardinals, etc.). The trouble here may be running deeper than it seems. Perhaps Wittgenstein fans could enlighten us more on this subject. My philosophy of language is a computational version of some of W.'s theories, when it makes sense. In particular, I am mostly content with delineating the use of language, if I am required to show the meaning of a particular expression. In this case, if you want me to show you a logical semantics e.g. some kind of a referential semantics, it is obvious that the referents in Alice in Wonderland are textual . You can think of it like the Bible if that helps it. When I say Alice ate a magic mushroom, I refer to text, a thought, a figment of our mutual imagination. I don't refer to an actual Alice. That would be an obsolete Meinongian theory. And frankly, talking about definite expressions is just so trivial, anybody who's read enough books and engaged in many conversations would know what a definite expresion is, although he may not be able to write a confusing semantics paper about it. What is certain is that, in the real world, words take their senses not only from the definitions of a logicist, but from context which is harder to formalize. If it weren't for the context of practising mathematics (in our daily lives as well as in institutionalized form), there would be very little meaning to such a notion as the existence of a number. But there is! On these forums, this particular point of view was best expressed, in my humble opinion, by Neil W. Rickert's mathematical instrumentalism. When somebody asks a question like What does 'being' mean? he ceases to make sense. When somebody unhealthily mixes logic, language, ontology, epistemology, philosophy of science, etc. as Joshua Stern warns us of, then it is likely that we will end up with absurd notions as the actual existence of mathematical entities. We were talking about logical positivism. Positivism itself would not be such a bad idea, to explain some of this interesting stuff that is popular nowadays, like computers. However, I don't believe that dressing it up in an air of religion serves any purpose. As philosophers, we must be wary of these old religious ways of thinking, otherwise we would have to purchase hoods and start rituals in the fashion of the Pythagoras school. If you approach the world like a logical positivist, very few things make sense, and perhaps as you implied (perhaps) you would have to presuppose the physical existence of mathematical fiction just to be able to be consistent in your belief that science has to make sense. Which was perhaps the point of some the quotes which were shown as evidence. That I deem a very unsuccessful mix of logicism and realism, and I don't think it's a relevant philosophical position. Godel made a good attempt, but that's all, it's fun reading and all that, but he doesn't make any sense in his conclusions. I am trying to think of the first human who found the idea of God. He must be both very fortunate and unfortunate at the same time. For it is only with remarkable intellectual achievement that he must have come to realize the unifying principles of the world and the motivation to find an explanation for the regularity and complexity in our world is well placed, yet in this endeavor the concept of God failed him as it does not exist. For he started science, and countered science at once. The idea of God thoroughly deceived him for all we know. After all, he was just a curious and imaginative ape. That was the reality about him. And God didn't exist except as a fiction. Our skill to invent fiction is both a blessing and a curse at the same time. This relation has been personal for me, too. At a time when I was in dire need of truth and company, I too imagined these fictional realms. I know what it feels like. But on the other hand I'm a scientist. I know that I'm just a dreaming ape. That's all there is to my reality as a human thinking about the actuality of mathematical objects like the empty set! It's just a substitute for God, and a person in that situation is no different than the first ape that thought of God. I cannot bring into existence those things that I dream of. So, if you ask me what really exists in the world, I could tell you that empty containers, empty lists (sheets of paper or computer programs) do exist. I've seen empty buckets. But I've never seen an empty set. It's just an abstract concept. It's theory. There is always a great difference between theory and object. The subjective experience of a theoretical concept should not confuse us. Every idea is just as real inside our heads, after all they are themselves textual entities that extend in space. However, we must stop imagining these silly metaphysical arrows that somehow launch from our head into the heaven. It's the same thing as believing in God and eternal souls. You see that very well in Godel's theology/philosophy. -- Eray Ozkural There is no such phenomenon as fictional existence, for to say that Alice is fictional is just another way of saying that Alice doesnÇt exist. What Quine says about mythical existence goes for fictional existence as well: There is, e.g., the relativistic doctrine according to which Cerberus exists in the world of Greek mythology and not in the world of modern science. This is a perverse way of saying merely that Greeks believed Cerberus to exist and that (if we may trust modern science thus far) they were wrong. Myths which affirm the existence of Cerberus have esthetic value and anthropological significance; moreover they have internal structures upon which our regular logical techniques can be brought to bear; but it does happen that the myths are literally false, and it is sheer obscurantism to phrase the matter otherwise. [Quine, W. V. (1982). /Methods of logic/ (4th ed.). Cambridge, MA: Harvard University Press. (p. 265f)] That many or all singular terms occurring in it refer to nothing doesnÇt render a novel nonsensical. One can certainly use \ the verb exist or corresponding there-is/are constructions meaningfully without pragmatically conferring existential force upon it. But as soon as you claim that a statement is (literally) true, the existential force is also there. Once again, Quine has something to say: The mistaken view that the word 'Cerberus' must name something in order to mean anything turns on confusion of naming with meaning. But the view is encouraged also by another factor, viz., our habit of thinking in terms of the misleading word 'about'. If there is no such thing as Cerberus, then, it is asked, what are you talking a b o u t when you use the word 'Cerberus' (even to say that there is no such thing)? Actually this protest could be made with the same cogency (viz., none) in countless cases where no would-be name such as 'Cerberus' occurs at all: What are you talking about when you say that there are no Bolivian battleships? The remedy here is simply to give up the unwarranted notion that talking sense always necessitates there being things talked about. The notion springs, no doubt, from essentially the same confusion which was just previously railed against; then it was confusion between meanings and objects named, and now it is confusion between meanings and things talked about. [Quine, W. V. (1982). /Methods of logic/ (4th ed.). Cambridge, MA: Harvard University Press. (p. 265)] There is an approach in the contemporary philosophy of mathematics called fictionalism (Field et al.). Is that your favourite? (By the way, Field holds that there are no mathematical entities.) I too reject MeinongÇs impenetrable ontological jungle, but \ when I talk about Alice in W. I donÇt refer to anything at all, neiter to anything inside the text nor to anything outside. The remedy here is simply to give up the unwarranted notion that talking sense always necessitates there being things talked about. [Quine - see above] Of course, contexts and situations do matter very often when it comes to semantic decoding. But I hold that the meaning of exist doesnÇt alter with the contexts. Why should this notion be absurd? If exists actually basically means exists now, then I fail to see why eg The empty set exists actually is absurd. (By the way, I happen to be an actualist.) Have you ever seen a quark or dark matter? PH grounds TF?) But he does say mathematics is useful fiction. There is not doubt that the uses of mathematics are as real as rain. Bob Kolker That all mathematical entities exist actually also means that there are no merely possible mathematical objects. PH are What does this mean? You just made them necessary beings, are you putting forth an Anselmian ontological argument? -- Eray Ozkural grounds TF?) Fictional things are just neurons popping in one's skull. So are real things. The difference is that the cause of the popping in the second case exist external to the skill but not in the first case. Think about out. Our most profound thoughts are made of the same objects and processes as our farts are. Bob Kolker what grounds TF?) Humans beings are just that walks and talks. Democratus was right. Bob Kolker I'm not going to pay much attention to any non-mechanical theory of mind. -- Eray What do you mean? PH grounds TF?) I assume you mean by non-mechanical non-material or non-physical. If so, I concur. I deny the existence of res cogitens. Bob Kolker reply-type=response Hey! --r.e.s. Goodness, I'm happy to claim the position he says is impossible. It's Vol 1, btw, and the next sentence is, In a sense, this means that our intuitions are inconsistent. Well, I'd say his intuitions are correct. Reading down the page, he argues something like that if physical theories are real, then you must give a realist interpretation to mathematics, therefore you can't be a nominalist. But that is nonsense. If one is a nominalist, one does so exactly because one finds the nominalist *usage* to be convenient and pragmatic; among the many varieties of nominalism, what they have in common is that they do not claim any essential or transcendental truth. If all language is a matter of nominalisms, and one is willing to grant that there may be such a thing as a correct physical theory, there is still endless room to negotiate the relationships between the two sides. J. what grounds TF?) Discussion, linux) Big deal. It's Hilary Putnam. He probably claimed the position he said was impossible in the very next paper. I really wish I could have his career. Step one: write a paper. Step n+1: write a paper refuting the arguments of the nth paper. This strategy is optimal, in the sense that requires considerably less reading than any other strategy. You only have to read your own work, However, the plan is not as easy as it looks. I think it takes a Putnam to execute this plan and still get published every time. No one seems to be interested in my attempt at this strategy. Dammit. -- I have to break the code of how [mere humans] work, and I have made a lot of progress over years of effort, and I feel like I am close to figuring out all the inner details of human wiring. -- James S. Harris on the extra problems of conveying his research I am not having any problems finding out the work required to pump out fluid from a container. However I am having trouble finding out the work required to pump out fluid into a container. I guess that I am not clear on how to calculate the weight that I am really lifting. The problem is as follows: You have an inverted cone with height of 10 ft and radius of 5 ft. The density is 62.4 lbs./ft^3. You are pumping in fluid from the base of the cone. How much work is required to fill the cone? Mkajuma It equals the work available by letting the water drain out. fluid required to ft fluid Ok Virgil, what you say makes sense. I'll look at it tomorrow. THANKS! Area at depth d= d*d*sinA*sinA*pi Work done = (d*SinA)**2 *dgp where p is the density =Integaring q*q*q*q*sinA*sinA*pi*p*g/4 Let depth = d Pressure =gpd where p is the density Area of cone = d*d*pi*Sin A where A is the angle of the cone. Hence Work (both to drain & fill) = Int(d*d*pi*sinA*g*p)dd = q*q*q*pi*g*p*sinA/3 Where q is the depth we have filled to Suppose one has n coins, among which there may or may not be ONE counterfeit \ coin. If there is a counterfeit coin, it may be either heavier or lighter than the \ other coins. The coins are to be weighed by a balance. What is the coin weighing strategy for k = 3 weighings and 12 coins? I am trying to figure this weighing scheme out by using optimal Huffman coding scheme to achieve the minimal code length... How to do that? See D.J.C. Mackay, Information Theory, Inference, and Learning chapter on Source Coding, Exercise 4.1, p. 66, andwered on p. 69. Marvellous book. Mackay has a machine readable version of it on his website. Jon C. Closer inspection indicates that Mackay's exercise has a slight difference to the one posed by Lucy, namely that in Mackay's, that one item is different is given. However, as Mike Guy points out, the number of cases is now 25, < 3^3, and some of the 'impossible' leaves in Mackay's tree are possible in this problem, so that (I think) the book strategy is optimal for this problem too. Not so clear however. The more obvious first experiment: weigh 1..6 versus 7..12, in the present problem does have three possible outcomes (information 1.6 bits), whereas in the book problem this experiment yields only 1.0 bits of information. Jon C. Dunno what relevence Huffman has, but the straightforward approach is to use the fact that k weighings can distinguish at most 3^k cases. Initially, we have 25 cases (each coin heavy/light, or all coins OK) which is less that 27, so we are OK. After the first weighing, these cases must be divided into 3 groups of <= 9 each. If we initially weigh n coins against n coins, what value must n have? Mike Guy In [1], a decoding algorithm for errors and erasures for Reed-Solomon codes is briefly stated. The algorithm uses the Massey-Berlekamp algorithm to find the error-locators, and Forney's algorithm to find the values of the errors and erasures. However, the decoding algorithm is for the Reed-Solomon codes generated by g(x) = (x - a^1)(x - a^2)...(x - a^[n-k]). How should I adapt Forney's algorithm when I want to use the generator polynomial (x - a^0)(x - a^1)...(x - a^[n-k-1])? Your time, effort and suggestions will be highly appreciated Jaco [1] S.B. Wicker and V.K. Bhargava, Reed-Solomon codes and their applications. New York: Institute of Electrical and Electronic Engineers, Inc., 1994. I had a similar problem while writing a RS decoder a while back. There are two things that you need to take caree of: 1. Syndrome computation should be modified to compute at roots alpha^i, i=0..2*t-1. 2. The forney algorithm should be computed using the relation e(i,j)=[(Xi)^(2-m)]*Omega(1/Xi)/Lambda'(1/Xi) where m is the lowest root of the RS generator polynomial. I need to look back at my code to verify that this expression is absolutely correct [I was succesfully able to get my code to work for arbitrary start powers of the roots of the gen poly]. Finally, the book The Art of Error correction coding by Robert Morelos Zaragoza has some very relavent information in this very topic. Hope that helps, Vikram In pursuing Goldbach Conjecture in PA, I've come across something that looks like a new axiom for ZFC; but I don't know for sure so I post it here for discussion. Before formulating that [axiom] candidate, let me first present PN, a 1st order system of general prime numbers. The non-logical symbols of PN are 0, 1, *, f; where 0, 1 are 2 numbers, * is a binary operation, and f is a function. The axioms of PN are: a1) Ax,y (x*y = y*x) a2) Ax (x * 0 = 0) a3) Ax (x * 1 = x) The p in a4) is called a prime number. Let P the set of those p's then: a5) P is Dedekind-infinite under f: there exists a proper subset P' of P that the function f maps P' 1-1 and onto P. Certainly, a model of PA in ZFC would be a model of PN, w.r.t. the binary operations *'s of both systems. Let's card(s) means the cardinality of s [a ZFC set], then consider the following formula in L(ZF[C]): (*) If there exists a model M1 of PN, then there exists a different model M2 of PN, in which card(P1) < card(P2), and where P1, P2 in turn are the sets of primes [see a4)] for M1, M2 respectively. ---Nam Nguyen There's no such thing as a new axiom for ZFC. ZFC is a fixed set of axioms. A new axiom for _set_theory_, that is, an axiom whose objects of discourse are sets, well, that's another matter. Were you? I sort of doubt it. It sounds like you were trying to use methods involving sets, which don't directly formalize in PA (though PA is strong enough to prove certain things about finite sets of naturals, properly coded). But I have this atrocious suspicion that my response above may be to a mis-parsing of your phrase, that you may have intended Goldbach's conjecture in PA as the thing you were pursuing. If so, then I have to point out that Goldbach's conjecture is not in any sense in PA (unless of course it's provable in PA, which is perhaps likely, but kind of beside the point). Goldbach's conjecture is in _arithmetic_; that is, its objects of discourse are natural numbers with + and *. Those are also the objects of discourse of PA, but they are not defined by PA, and it isn't necessary (and is a bit misleading) to mention PA every time you talk about them. Order matters, here. If you are at first-order then the natural numbers ARE NOT the objects of discourse of PA; First-order PA has plenty of models that include infinitely many things that are bigger than any natural number. Or by any other classical first-order theory either. Well, if you are going to insist on staying down at first order, it arguably IS necessary; and if you are going to go up to 2nd then it is entirely PROPER, which tends to make it more nearly necessary. Conceded, but the fact that this objection is true doesn't stop it from being bull. FOL is in some sense ALL YOU HAVE. Everything MORE powerful is MORE incomplete and MORE ineffective, in terms of humans' ability to use it. If you are encoding your knowledge about N in terms of what you can prove from a recursive set of first-order axioms, then the sum total of your knowledge about N is NOT sufficient to enable you to even distinguish the identity of things in N from things outside it. In that sense, your objects of discourse are NEVER the natural numbers. You HAVE to go to 2nd-order get categoricality. And once you go there, the 2nd-order language of PA *is* the *right* language. So encouraging people not to talk about PA is ass-backwards. It is legitimate to encourage people not to talk about PA when talking about the natural numbers, but the more IMPORTANT point there is NOT about the legitimacy but about the IMPOSSIBLITY: at first-order, you simply CANNOT talk intentionally, specifically, about the natural numbers. So if you are going to begin by crippling yourself with what few things first-order languages CAN say about the natural (and supernatural) numbers, you might as well be clear WHICH first-order language you're using, and you might as well be using PA, since it IS the right language at second-order. I simply mean a [possible] new axiom that could be added to ZFC, the way CH/GCH are. The reason I tend to look at ZFC's models of many theories in the quest for GC (Goldbach's Conjecture) is because I sort of suspect that there is a competition between 2 major mathematical principles, Induction and Choice, and that the difficulty about GC seems to rest with this competition. . [Of course Choice is known as a set axiom; but there is way to look at it as a principle, in the same Principle-footing with the Induction.] I'm not sure about the misleading but perhaps it's not necessary. The only reason I mention PA a lot of times in conjunction with GC is Torkel Franzen's asking what formal system I was referring to, one time in the past. Coming from him, I could only suspect that there are systems in which there are solutions to the question of GC's [un]decidability. So I've just been careful. But it's true that PA is the system I'm looking at, going after GC. My guess is that this was one of Torkel's little rhetorical-ironical thingies. The first thing to check, whenever you're not quite sure what Torkel's talking about, is whether he might be making fun of you. That's the general you, not you, Nam. However in this case I think the possibility I've suggested is especially likely due to the fact that you, Nam, tend rather systematically to mix up syntax and semantics, which is exactly what I called you on this time. But I did it directly rather than ironically. More like sadistically. You did it when you said, which is just ridiculous. It's in PA BECAUSE it is easily phrased While this may not be enough to make something in PA, it certainly IS enough to make it WRONG to claim that something is not in any sense in PA. THIS IS *one* sense in which it IS in PA. You continued, and this is just idiotic -- the whole FOL paradigm is completely mute on the objects of discourse; it never even MATTERS what the objects of discourse are. In this context it is always SUFFICIENT for the objects of discourse to be the terms of the theory, unless you have some existential axiom requiring anonymous objects-of-discourse. In that case the proper thing to do is rewrite the axiom There are a whole lot of theories that can add GC AS AN AXIOM TO PA; whether their resulting objects of discourse will still include the natural numbers is not currently known, but it does not really MATTER, EITHER. Supposing for the sake of argument that that _is_ ridiculous, nobody but you sees how it's _sadistic_. If you say so. What can and what cannot be expressed in the language of PA is a little too slippery for me, but let's assume this is so. Regardless, GC is not equivalent to PA |- GC. So GC is not in PA. No, it's not just idiotic, in fact it's true. Yes, fol is mute on the objects of discourse. But GC is not a question about fol, it's a question about arithmetic. Or maybe it is a question about fol. Can you give a set A of axioms such that (i) you can prove all the elements of A are true in the natiural numbers (ii) you can give a procedure for _recognizing_ whether a given wff is or is not an element of A? (iii) you can prove that GC is true if and only if A |- GC ? If you can I'd be curious to see you do it. If you can't then I don't see how you can say that GC is a question about fol. (The point to (ii) is to rule out silly examples; silly because although they do have the property that GC is true if and only if A |- GC, that fact cannot possibly be of any use in determining whether GC is true. For example we could take A to be the set of all theorems of arithmetic. In that case GC _is_ equivalent to A |- GC, but that's a little useless, because if GC is in fact true then A |- GC holds simply because GC is an element of A, and unfortunately there is no procedure to determine whether or not a given wff is an element of A. So this is silly, for _this_ set A saying that GC is equivalent to A |- GC says nothing more than just that GC holds if and only if GC holds.) Or maybe you have some totally different explanation for why GC is a question about fol. If so what? Does not matter? It matters a great deal. To heck with existing theories: let A = {GC}. Then GC is an axiom of A. So what? This tells us absolutely nothing about whether GC is actually _true_. ************************ David C. Ullrich this should do it: (x is a Prime number) with m the bounded minimization operation and Div the divisibility relation. D. to me was how we express things like addition _formally_; of course one has seen informal definitions of addition in PA. Not that that's hard, I just never thought about it: If p is a statement involving addition, then the what you get by replacing x+y=z in p with Add(x,y,z) and defAdd is something like ************************ David C. Ullrich I think you'll find it's actually a theorem of ZFC, a consequence of the Loewenheim-Skolem theorem. in the relationship between it and (*). Hi all, I am facing with an equation: s=exp(lamda((1-p)^2+2*p*(1-p)*s+p^2*s^2-1))) I want to know if it is possible to write out the express for s in closed form(the solution to the above equation) without numerical computing(e.g. Matlab, Mathematica, etc.) Your parentheses are unbalanced. Assuming you mean (using t for lambda): s=exp(t*((1-p)^2+2*p*(1-p)*s+p^2*s^2-1)); Maple gives: (1/2) * {-2*t +- 2*sqrt ( t^2 + t * ln(s) )} / (t * (s - 1)) I reckon you could easily check it by hand by taking ln of both sides and solving the quadratic. --Lynn Eh? What's that meant to be? Lucy asked for an expression *for s*, but that involves s. Mike Guy s=1 is a solution. If 2*lambda*p=1, I expect it is the only solution. Probably not. The solution to s = exp(2p(1-p)s) is (through mathematica) s = W(2(p^2-p))/2(p^2-p) where W(x) is the Lambert W function (or in mathematica the product log) If both s -and- s^2 are in the exponent then even then it is not simplifiable (at least mathematica tells me) to something with W(x) (of course it may still be possible depending on the structure, but just unlikely) -- Mitch Harris (remove q to reply) I happen to find this question in mathforum.org and it is quite an interesting one, atleast according to me. Question: Alice is at the front of a line of 100 people ready to board a 100-seat airplane. Bob is at the back of the line. Alice has lost her boarding pass, so she simply takes a random seat. Everyone else takes their proper seat if it is available, or takes a random empty seat if not. What is the probability that Bob gets to sit in his properly assigned seat? Direction, I took: The '+' stands for the required arrangement. If there were only 1 person and 1 seat then the probability that the last person will sit on the correct seat is 100%. 1+ If there were only 2 persons and 2 seats then the probability that the last person will sit on the correct seat is 50% 12+ 21 If there were only 3 persons and 3 seats then the probability that the last person will sit on the correct seat is 50% 123+ 213+ 231 321 If there were only 4 persons and 4 seats then the probability that the last person will sit on the correct seat is 50% 1234+ 2134+ 2314+ 2341 2431 3214+ 3241 4231 I do not know whether my above argument is correct this far. So, someone please comment on it. Also, I have no idea how I might generalise and find the solution for the question. Any direction on the question will shed light for other would be See the thread, entitled Airplane puzzle: Slick solution?, which I elementary explanation. -- Stephen J. Herschkorn sjherschko@netscape.net And after Cantor has changed the diagonal digit d_nn then there remains the diagonal digit d_n+1,n+1 to be exchanged. As Cantor treats only lines with finite line number n,there always remains anotehr line untreated. WHAT IS THE DIFFERFENCE ? But by another proof which has exactly the same relevance, we arrive at the contrary conclusion. This shows that set theory is inconsistent. What are you talking about? What you call a proof is not a proof. What you call a proof has no relevance. Speak for yourself. No, it shows that you are thick. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Elegance is an algorithm Iain M. Banks, _The Algebraist_ Of course, this cannot be proved for fixed x,y because there is no largest n. But why should the x not depend on n? There is no reason at all why it must be fixed. The point is: It must exist! And it does exist. It is one of the numbers Dn + Pi, whereas the y is, say, sqrt(2)/2. By the way: Do you know the difference between pointlike convergence and strong convergence for sequences of functions? (I am not sure whether pointlike and strong are the correct English translations of the German words. I f - f_n I < epsilon. In the second case epsilon depends of n_0 alone, whereas in the first case epsilon depends of n_0 and the argument x. This distinction was discoverd at Cantor's times, by the way. Here we have a similar case: x is not fixed, nevertheless, it exists. Discussion, linux) You can only prove a universal if the domain is finite? Learn some logic. Preferably before the next time you enter a classroom. In any case, if it cannot be proved, then you have not proved a contradiction. You said you proved a contradiction, that you proved P and NOT P. You said that P was the theorem: For all irrational x,y, there is an n and a z such that |z| = n and B(x,z,y). You now have to prove NOT P. You haven't. You're babbling. The issue is nothing more complicated than quantifiers here and your utter inability to keep them straight. -- Jesse F. Hughes Usenet is demonstrably dangerous. It needs to be regulated. --James S. Harris, voice of reason and moderation The question is about volumes in space represented by functions F(r) Most of the questions discussed in: http://www.aizu.com/mirror/F-rep/ For example, 1D region with center at C and halfsize A could be represented by F(x) = -((x-C)^2-A^2). Intersection of two figures could be expressed in the following manner: H(F(r),G(r)) = F(r) + G(r) - sqrt{F^2(r) + G^2(r)}. So the intersection of two 1D regions is: H(Fc,Fa,Gc,Ga,x) = (x-Fc+Fa)*(Fc+Fa-x)+(x-Gc+Ga)*(Gc+Ga-x)-sqrt{((x-Fc+Fa)*(Fc+Fa-x))^2+((x-G c+Ga)*(Gc+Ga-x))^2}. (Fc, Fa, Gc, Ga - centers and halfsizes of regions.) My question is the following: is it possible to calculate the approximation of volume for such a figure? The best thing is to express it in closed form but any ideas (except numeric solution) are welcome. Numeric solution is prohibited because I need a closed form. ;) Or series expression. I have to calculate partial derivatives by Fc and Gc later. That's why I need a closed form solution and hope for a best variant. I tried take an volume integral (1D for time being) of 1/(1+e^(-kH(x))), but failed miserable and now is out of ideas. (1/(1+e^(-kx)) is an approximation of theta (?? always forget it's name) function: by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: \ 1.9 primary) id j1BCcNJ18014; If the series sum_n a_n does not converge, then there exists such that b_n := ( sum k=1 to n (p_k a_k) )/p_n does not converge Does someone know a simple construction for p_n? [the converse is also true and is not difficult to prove: if the series sum_n a_n is convergent then for any increasing the result seems to belong to Kronecker]. V. Anisiu Yes, the p_n can be taken to be integers. I'm not sure whether the following proof is simple, but it can't be too non-simple, because I didn't know this result and came up with the proof in a few minutes (in much less time than I suspect it's about to take me to write it down...): We can suppose the a_n are real. The fact that the sum diverges says that the partial sums are not a Cauchy sequence; hence, multiplying the a_n by a suitable constant to simplify the typing, there exist integers n_1 < m_1 < n_2 < m_2 ... such that for all j. We will show that there exists a strictly increasing sequence of integers N_j such that if we define p_k = N_j for n_j <= k < n_{j+1} then b_k does not tend to 0. (you can define p_k any way you like for k <= n_1.) Suppose that N_1, ... N_{j-1} have already been chosen, and hence p_k has been determined for k < n_j. Now (sum_{k < n_j} p_k a_k + sum_{n_j <= k <= m_j N a_k}) / N if N_j is large enough. QED. (I know that what I have in mind is correct; it's too simple to be wrong. If you don't believe something above it means I bollixed the notation; ask and it will be fixed.) ************************ David C. Ullrich As nobody has answered, let me try - it will not be a *solution* to your problem, though. It seems to me rather difficult to explain general simplexes and thus even more simplicial complexes to a 'general audience' ... This is because general simplexes are n-dimensional objects with n not restricted to 0 until 3 ... And some 'infinite' simplicial complexes need an infinity of coordinates dispite they are made of finite-dim. 'components' ... So you will have to say that simplexes are generalizations of the following geometrical objects one can 'see': points, segments, (full) triangles ans tetraeders ... where this continues in 4 and more dimensions - for those who can imagine what that might mean, and for the others: suffices they know that with these 4 examples from dim. <= 3 one omits some things, but it is possible to show many examples of what comes next - the simplicial complexes - using only these limited 4 types ... Next difficulty: even with only 2 dim. at most there are 'finite' simplicial complexes definig a (topol.) space that cannot be properly imbedded in ordinary 3-space ... I think it will not be possible to mention the concept of a topological or even metrical space for a general audience, so it depends on what your exact theme is, whether examples taken in 3-space will suffice. There a simplicial complex would be a union of simplexes (or of 'distorted' simplexes) where the intersection of two of these has to be a face of both ... so one has to define what these faces are - which will have to be explained separately for each couple (m,n) of numbers such that 0 <= m <= n <= 3 where m,n are the dim's of a face and the simplex of which one considers faces ... Hi all, Sir Edward Wright (*the* Wright from Hardy & Wright) died on February 2, aged 98. You can read an obituary at: http://www.telegraph.co.uk/core/Content/displayPrintable.jhtml?xml=/news/200\ 5/02/10/db1001.xml&site=5 Jose Carlos Santos Strange that Wright does not get a mention here: http://www-groups.dcs.st-and.ac.uk/~history/Indexes/W.html but that eminent mathematician Ludwig Wittgenstein does. Wright is best known as the junior coauthor with Hardy of the classic textbook: An introduction to the theory of numbers. As the obituary indicates, after WWII he became increasingly involved in university administration which, no doubt, left him precious little time for serious mathematical research. have number theory as a primary or secondary classification. Browsing the titles it seems his primary number theoretic interests were in combinatorial and analytic number theory, especially the theory of partitions and Waring problems. 11 of these 43 papers are cited by other papers/reviews in the Math Reviews database (they are marked with a '*'). --Bill Dubuque ----------------------------------------------------------------------------\ -- Matches for: Author/Related=Wright, E. M. AND MSC Primary/Secondary=(10 or 11) Number of Matches: 43 ----------------------------------------------------------------------------\ -- Number theory and other reminiscences of Viscount Cherwell. Notes and Records Roy. Soc. London 42 (1988), no. 2, 197-204. (Reviewer: Edward L. Cohen) 01A70 (11-03 11K99) 90b:01096 Corrigenda: A quadratic recurrence of Faltung type [Math. Proc. Cambridge Philos. Soc. 88 (1980), no. 2, 193-197; MR 81i:10016]. Math. Proc. Cambridge Philos. Soc. 92 (1982), no. 2, 379. 10A35 83k:10024 A quadratic recurrence of Faltung type. Math. Proc. Cambridge Philos. Soc. 88 (1980), no. 2, 193-197. (Reviewer: Ian Anderson) 10A35 81i:10016 *The Tarry-Escott and the easier Waring problems. J. Reine Angew. Math. 311/312 (1979), 170-173. (Reviewer: O. H. Koerner) 10J06 (10A45) 82f:10059 An introduction to the theory of numbers. Fifth edition. The Clarendon Press, Oxford University Press, New York, 1979. xvi+426 pp. ISBN: 0-19-853170-2; 0-19-853171-0 (Reviewer: T. M. Apostol) 10-01 81i:10002 Stacks. III. Quart. J. Math. Oxford Ser. (2) 23 (1972), 153-158. (Reviewer: A. L. Whiteman) 10J20 45#8623 Arithmetical properties of Euler's rencontre number. J. London Math. Soc. (2) 4 (1971/72), 437-442. (Reviewer: J. Riordan) 10A10 45#3304 Stacks. II. Quart. J. Math. Oxford Ser. (2) 22 1971 107-116. (Reviewer: A. L. Whiteman) 10.55 (05.00) 44#174 *Stacks. Quart. J. Math. Oxford Ser. (2) 19 1968 313-320. (Reviewer: A. L. Whiteman) 10.48 37#5178 Rotatable partitions. J. London Math. Soc. 43 1967/1968 501-505. (Reviewer: M. S. Cheema) 10.55 37#4043 Coefficients of a reciprocal generating function. Quart. J. Math. Oxford Ser. (2) 17 1966 39-43. (Reviewer: M. S. Cheema) 10.48 33#308 An extension of a theorem of Gordon. Proc. Glasgow Math. Assoc. 7 1965 39-41 (1965). (Reviewer: B. Gordon) 10.48 31#2231 Equal sums of like powers. Canad. Math. Bull. 8 1965 193-202. (Reviewer: L. Carlitz) 10.46 (10.13) 31#117 *An enumerative proof of an identity of Jacobi. J. London Math. Soc. 40 1965 55-57. (Reviewer: L. Carlitz) 10.48 30#69 Approximation of irrationals by rationals. Math. Gaz. 48 1964 288-289. (Reviewer: T. M. Apostol) 10.32 29#4737 Partition of multipartite numbers into k parts. J. Reine Angew. Math. 216 1964 101-112. (Reviewer: B. Gordon) 10.48 29#3453 *A closer estimate for a restricted partition function. Quart. J. Math. Oxford Ser. (2) 15 1964 283-287. (Reviewer: N. J. Fine) 10.48 29#1191 *Proof of a conjecture of Sudler's. Quart. J. Math. Oxford Ser. (2) 15 1964 11-15. (Reviewer: N. J. Fine) 10.48 29#1190 Direct proof of the basic theorem on multipartite partitions. Proc. Amer. Math. Soc. 15 1964 469-472. (Reviewer: H. Gupta) 10.48 28#5049 *Partition of multipartite numbers into a fixed number of parts. Proc. London Math. Soc. (3) 11 1961 499-510. (Reviewer: H. Gupta) 10.48 24#A2573 A simple proof of a known result in partitions. Amer. Math. Monthly 68 1961 144-145. (Reviewer: G. J. Rieger) 10.48 23#A3114 The frequency of prime-patterns. Quart. J. Math. Oxford Ser. (2) 11 1960 60-63. (Reviewer: S. Chowla) 10.42 24#A98 *A functional equation in the heuristic theory of primes. Math. Gaz. 44 1960 15-16. (Reviewer: D. H. Lehmer) 10.42 23#A873 *Partitions into k parts. Math. Ann. 142 1960/1961 311-316. (Reviewer: G. J. Rieger) 10.00 22#12088 The asymptotic behaviour of the generating functions of partitions of multi-partites. Quart. J. Math. Oxford Ser. (2) 10 1959 60-69. (Reviewer: S. Chowla) 10.00 21#3386 Prouhet's 1851 solution of the Tarry-Escott problem of 1910. Amer. Math. Monthly 66 1959 199-201. (Reviewer: D. H. Lehmer) 10.00 21#3375 *A definite integral in the asymptotic theory of partitions. Proc. London Math. Soc. (3) 8 1958 312-320. (Reviewer: N. J. Fine) 10.00 21#1286 *A generalization of a result of Mordell's. J. London Math. Soc. 33 1958 476-478. (Reviewer: T. M. Apostol) 10.00 21#18 Partitions of large bipartites. Amer. J. Math. 80 1958 643-658. (Reviewer: R. A. Rankin) 10.00 20#3111 Einfuhrung in die Zahlentheorie. (German) R. Oldenbourg, Munich 1958 xvi+480 pp. 10.00 20#828 The number of partitions of a large bipartite number. Proc. London Math. Soc. (3) 7 (1957), 150-160. (Reviewer: R. A. Rankin) 10.1X 19,16k Partitions of multi-partite numbers. Proc. Amer. Math. Soc. 7 (1956), 880-890. (Reviewer: N. J. Fine) 10.1X 18,793c An introduction to the theory of numbers. 3rd ed. Oxford, at the Clarendon Press, 1954. xvi+419 pp. (Reviewer: N. G. de Bruijn) 10.0X 16,673c A simple proof of a theorem of Landau. Proc. Edinburgh Math. Soc. (2) 9, (1954). 87-90. (Reviewer: L. Carlitz) 10.0X 16,448e A class of representing functions. J. London Math. Soc. 29, (1954). 63-71. (Reviewer: I. Niven) 10.0X 15,288d The calculation of large primes. Math. Gaz. 37, (1953). 104-106. (Reviewer: D. H. Lehmer) 10.0X 14,1063c Functional inequalities in the elementary theory of primes. Duke Math. J. 19, (1952). 695-704. (Reviewer: H. N. Shapiro) 10.0X 14,355d The elementary proof of the prime number theorem. Proc. Roy. Soc. Edinburgh. Sect. A. 63, (1952). 257-267. (Reviewer: H. N. Shapiro) 10.0X 14,137d A prime-representing function. Amer. Math. Monthly 58, (1951). 616-618. (Reviewer: W. H. Mills) 10.0X 13,321e Equal sums of like powers. Proc. Edinburgh Math. Soc. (2) 8, (1949). 138-142. (Reviewer: D. H. Lehmer) 10.0X 12,11e The Prouhet-Lehmer problem. J. London Math. Soc. 23, (1948). 279-285. (Reviewer: D. H. Lehmer) 10.0X 10,510b Equal sums of like powers. Bull. Amer. Math. Soc. 54, (1948). 755-757. (Reviewer: N. G. W. H. Beeger) 10.0X 10,101c *The `easier' Waring problem. Quart. J. Math., Oxford Ser. 10, (1939). 190-209. (Reviewer: G. Pall) 10.0X 1,69a http://www.telegraph.co.uk/core/Content/displayPrintable.jhtml?xml=/news/200\ 5/02/10/db1001.xml&site=5 Hmmm. Did Ludo contribute anything of value to mathematics? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Elegance is an algorithm Iain M. Banks, _The Algebraist_ I'd say he did. He was also a mathematician, studied with Bertrand Russell, and made aportations to foundations of mathematics, logic, philosophy of mathematics, etc. He even gave, I believe, lectures on phil. of mathematics at Cambridge. Being one of the 20th century's greatest intellects it must be at least interesting to check his work. Tonio Wittgenstein would have rejected any suggestion that he was a mathematician. What mathematical contribution of Wittgenstein's do you have in mind. Emphasizing that nothing he said was mathematics. Suppose that I can colour the n(n-1) edges of a complete digraph on n vertices using k colours, in such a way that no two successive edges have the same colour i.e. no vertex has an ingoing and an outgoing edge with the same colour. Is there an upper bound for n in terms of k? (I have not thought about this much yet - I am just being lazy, and hoping somebody will know the answer!) Derek Holt. # Suppose that I can colour the n(n-1) edges of a complete digraph on # n vertices using k colours, in such a way that no two successive edges # have the same colour i.e. no vertex has an ingoing and an outgoing edge # with the same colour. Is there an upper bound for n in terms of k? # # (I have not thought about this much yet - I am just being lazy, and hoping # somebody will know the answer!) # # Derek Holt. For every vertex v, let S(v) denote the set of colors of the in-going edges. Then any two such sets S(u) and S(v) must be in-comparable: Hence, these sets form a Sperner-family, and we must have n <= (k choose k/2). This bound is tight. --Gerhard ___________________________________________________________ Gerhard J. Woeginger http://www.win.tue.nl/~gwoegi/ but I see now that there is not! Derek Holt. by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: \ 1.9 primary) id j1BFWOO02980; Can someone please help me rearrange this equation as i havent done it sinc \ school either!! X = e (1-b) / p + b (X + 0.01) I know the values of all except X so i need the X's on one side. it sinc school either!! Parentheses would help. I don't know if b(X + 0.01) is in the denominator of the fraction or if it is a separate term added to a fraction e(1-b)/p. If what you have is this: X = e(1-b)/[p + b(X+0.01)] then you need to multiply both sides by the denominator: X[p + b(X+0.01)] = e(1-b) pX + bX^2 + 0.01 b X = e(1-b) bX^2 + (p + 0.01 b)X - e(1-b) = 0 and you have a quadratic equation, which you can solve with the standard formula. - Randy by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: \ 1.9 primary) id j1BFWOs02986; When demanding x != +/-y (mod N) this chance becomes even 100%. Johann Not if x-y=1 and x+y=N Jack I did not so. Just today the 7 Tesla tomograph was used for the first time. Can anyone just envision a similar progress concerning cardinality? I guess Mueckenheim can: He argues for declaring it nonsense. The brain DOES things in our world, understanding the brain does NOTHING. Psychology is like commisioning DaVinci to paint the Saturn V. The government uses LIDAR microwave laser imaging of the SOUNDS that perpetually eminate from your voice box. They can HEAR all your thoughts, even the sped up conglomerate of phrases that is your subconsious you cannot hear. You make a constant monology of ideas as you think, the show THE WONDER YEARS is what happens in real life, people can listen to your contant thoughts as you walk around. They can read out your thoughts to you a second before you think them yourself. What is a tomograph going to do? Herc by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: \ 1.9 primary) id j1BG8QH06446; ^^^^^^^^^^^^^^^ Radix? hp: Let Asubseteq R^k and f_n(x,y) be a sequence of L2(AxA) functions converging almost everywhere to f(x,y). th: Then for a.e. x* in A f_n(x*,y) converges to f(x*,y) for almost every y in A. Is this true? Yes, and it really has nothing to do with L2; it's just Fubini's theorem. by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: \ 1.9 primary) id j1BHIwM12826; with the first question. But for question two, I have a theorem in my books which says that the if we have some irreudicble poly with splitting field E over F, then |Gal(E:F)| has degree less then or equal to n with proof? I think it's true, but you are saying otherwise I am a little confused about this. Your theorem does not specify what n is. If it is the degree of the polynomial the result is false. If it is the degree of the extension then it is true. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland in In trying the proof (backwards) starting from statements about base 3 Is it not possible then for k to actually be smaller than (2j + 2 - 2*(3^c))/3(c+1) ? Any help appreciated. k is an integer, so a fortiori is an integer multiple of 1/(3^(c+1)). There are no integer multiples of 1/(3^(c+1)) in the open interval ( (2j + 1)/3^(c+1) - 2/3, (2j + 2)/3^(c+1) - 2/3 ). Mike Guy Hi! Do you know any list of books from the same author that teach math from algebra and to the top? Everything! Would prefer a pedagogic language if possible :0) Howard Anton of Drexel U has written textbooks from Calculus to Functional Analysis. I am uncertain if he has written anything at the High School level. I have 2 exercices about differential equations but i can't do anything I don't know how to start ... If anybody can help me, it would be wonderful Exercice 1 Let x be wealth and let u(x) be an individual's utility function, depending on wealth. Define the function [Micro](x) = - u(x)/u'(x). Then [Micro](x) is called the Arrow-Pratt (AP) measure of absolute risk aversion. Consider a function that has constant AP absolute risk aversion given by K a)Obtain a second order differential equation for utility, u(x). b) Solve the equation to obtain a functionnal form for utility functions that have constant AP absolute risk aversion c) Show that this soltion satisfies the second order differential equation Exercice 2 The market for a product is given by the following demand and supply equations. qd(t) = 500- p(t) -p'(t) qs(t)= 3p(t) + 1/2 p'(t) -16 Assume that the market always clears. a)Obtain a second order differential equation for prices b)Solve the differential equation assuming initial values p(0)=20, p'(0)=10 c)Determine whether or not process converges to equilrium d) What is the frequency of oscillations about equilibrium? Merci beaucoup ...c'est tr.8fs important ! If mu(x) = K, then -u''(x)/u'(x) = K. Then you have that -u''(x) = Ku'(x) or \ u'' + Ku' = 0. This is a particular instance of equations of the form Au'' + \ Bu' + Cu + D = 0, which you might know a method to solve. It involves using \ the quadratic formula, if that rings a bell. I've already mentioned one path to a solution, but it might be helpful in this case to consider f = u', so that f' = -Kf. You reduce a second order equation to a first order separable differential equation. This form comes up quite often, so you ought to be able to solve for f, then for u, since you know that one is the derivative of the other. This is just a matter of plugging the solution you derived in (b) back into \ the equation. Are you sure you typed the problem correctly? If you know that the market clears, then you have that qd = qs for all t. This gives you an equation in \ terms of p and its derivative alone. There's no second derivative that I can \ see. Anyway, once you set qd = qs, after some algebra you'll get to an equation of the form Ap' + Bp + C = 0. You can solve this by using an associated equation, solving for roots, etc. Perhaps the OP has covered solving equations of the form you've given by finding an integrating factor. Given Ap' + Bp + C = 0, (*) p' + (B/A)p = -C/A The integrating factor I is found from dI -- = (B/A) dt I Having found I, multiply each term of (*) by ( I dt ), and the solution follows immediately. by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: \ 1.9 primary) id j1BI9m817516; ... I was hoping for some compact formula for p_k but the construction is simple indeed! Here is your proof again with 2 tipos removed. We can suppose that a_n are real. The fact that the sum diverges says that the partial sums are not a Cauchy sequence; hence, multiplying the a_n by a suitable constant to simplify the typing, there exist integers n_1 < m_1 < n_2 < m_2 ... such that for all j. We will show that there exists a strictly increasing sequence of integers N_j such that if we define p_k = N_j for n_j <= k < n_{j+1} then b_k does not tend to 0. (you can define p_k any way you like for k <= n_1.) Suppose that N_1, ... N_{j-1} have already been chosen, and hence p_k has been determined for k < n_j. Now ( sum_{k < n_j} p_k a_k + sum_{n_j <= k <= m_j} N a_k ) / N if N_j is large enough. QED. V. Anisiu It is well known that the sum of the inverse of all primes is divergent, but starting with what smallest prime will this sum be convergent? A question that probably can not be answered by singling out a specific starting prime. Dan If a series diverges at all, then any tail , the portion from some term onward, will still diverge. So there is no such prime. Go back to the definition of convergence of a series: it implies that if you drop, insert, or alter a finite number of terms, the fact of its convergence or divergence will remain the same. So, to help the series to converge, you need to drop infinitely many terms. In case of primes, it is known that if the set of prime twins (pairs that differ by 2) is infinite then the sum of the (infinite series of their) reciprocals is finite. Number theorists may know other interesting examples of pruned prime sequences with finite sums of reciprocals. What is not interesting? Anything that is already easy to show to be dominated by a known (elementary) convergent series, such as reciprocals of squares of positive integers. Originator: grubb@lola If the set is finite, that sum is still finite ;) --Dan Grubb The sum would still diverge if you threw out a few terms, so how can it be convergent? This doesn't answer your question, but it led me in this direction: Let P be the set of all polynomials over Q with positive coefficients and 0 constant coefficient. Then, if a is any number in A, then surely for all p in P, p(a) is also in A (this is just the closure of a and Q under addition and multiplication). I'll write P(a) for {x : exists p in P s.t. x = p(a)}. Then b must also be in A if there exists p, q in P such that p(a) = q(b). So there's a sort of minimal ideal which we could define around any element a: I(a) = {b : exists p,q in P s.t. p(a) = q(b)} P(1) = Q, and I(1) contains all positive, real roots of polynomials over Q with positive coefficients except for the constant coefficient, which would be (strictly) negative. That's a lot of the algebraics, but not all (not to imply that your conjecture is false...) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: \ 1.9 primary) id j1BIxo522079; I'd like to know how you tackle such an equation. Alain. I think an equation with x+1 is easier to visualize than one with x^2. So let x=2^2^v, then x^2 corresponds to v+1. We should solve g(v+1)=5*2^(3*2^v)*g(v). For integer v, the solution is g(v) = c*5^v*8^(2^v), so our original has solution f(x) = c*5^(log[2](log[2](x)))*x^3 with constant c. Of course for more general solution we can allow c non-constant, but of the form c(log[2](log[2](x))) where c is any function of period 1. First I'd note that it looks like it has a lot of freedom. We should be able to choose arbitrary behaviour on [2,4) and generate values for the successive square roots and squares of this interval by using the formula. g(u + ln 2) = (some mess) g(u), for any u in R. In this form, it is more obvious that we can divide the domain up into intervals and generate the whole function from any one of these intervals. - Tim by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: \ 1.9 primary) id j1BJ8dV23358; Yes. It is sufficient to assume that f_n are measurable. Let N be the negligible set where f_n does not converge to f. Then a.e. x in A, the section N_x is negligible (Fubini). It results that a.e. x in A, for each y in A N_x, (x,y) is not in N i.e. f_n(x,y) converges. V. Anisiu Actually you don't need to know the f_n are measurable. Let me summarize: If N is negligible in AxA, then Nx .OR. Ny is negligible; moreover Nx is negligible for a.e. x in A .AND. Ny is negligible for a.e. y in A It's ok? Sorry, here with Nx i mean the projection of N on the first component (i.e A). next, I mean what you mean. moreover Josh. There may be some specific geometries that would allow you to avoid the horizon problem. As a general issue, though, if one takes two widely separated points, their light cones do not intersect at the time when the CMB formed. Actually, at the time of the formation of the CMB, the temperature of the Universe was only about 3000 K, IIRC. The temperature really doesn't matter, though, from the standpoint of the horizon problem. In general, two points will have the same temperature only if light can travel between them. Only if the entire Universe were only 300,000 light years across at the time of the CMB formation should the temperature be the same. -- Lt. Lazio, HTML police | e-mail: jlazio@patriot.net No means no, stop rape. | http://patriot.net/%7Ejlazio/ sci.astro FAQ at http://sciastro.astronomy.net/sci.astro.html What? I can only warm my arse against a fire if my arse is as hot as the fire? Pass the jalapeno beans! [Eyes water] Errr, so by night my skin is the 6kK approx of the Sun's surface, and by night I freeze to Olber's paradox (or glow a medium yellow, depends on your understanding of Olber's P.) -- Aidan Karley, Aberdeen, Scotland, Location: 57Á10'11 N, 02Á08'43 W (sub-tropical Aberdeen), 0.021233 Come off it! If an event has no duration, then it doesn't/never did exist. It would be unobservable and unmeasurable. Is this quasi QM thinking? If one event (in this case a circuit), is two (in your arguement), how many events are in two circuits? I realise that Relativity had to invent math constructs to avoid implacable contradictions; now I'm seeing the extent to which it also has to change definitions and meanings within language. This IS a sporting-type event; the two clocks leave the blocks together, and race around the sun together, returning to their starting point TOGETHER! But the clocks read a different time!!! off, unbrainwashed contradiction The insult was not directed at you- I apologise if you took that to be the case. Jim G c'=c+v No, it is normal English language. For example if I said the key event that started the First World War was the assassination of Archduke Ferdinand, you wouldn't be surprised. The event was the transition from when he was alive to when he was dead. That change happened at a particular instant. The moment of his death has no duration, but the effects were observable. This is meaning 1. on the page below and item 3 really just repeats the same. If the orbits are contiguous, there would three events: Event A: satellite passes overhead to start the first orbit. Event B: satellite passes overhead to end the first orbit and start the second. Event C: satellite passes overhead to end the second orbit. Note event B is the transition from being on the first orbit to being on the second. That change occurs at a particular instant but the change doesn't have a duration, the orbits have durations. There's nothing complex about this Jim, I don't know why you think what I'm saying is unusual. Science often makes strict terms out of words which are vague or ambiguous in conversation, but is there any real difference in meaning between items 1 and 3 on the Princeton page? Two clocks orbiting together round the Sun would be an entirely different situation (they would measure the same time of course) but we were talking of one clock on the ground and the other being onboard a GPS satellite. Check what you said: Well I have to admit I did assume George ... you better smarten up! was directed at me so I had better apologise for my unwarranted arrogance ;-) Anyway, I would far rather take your argument on its merits. If you think you can show some sort of contradiction in this feel free to try. The language needn't get in the way even if it does produce some temporary misunderstandings. George by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: \ 1.9 primary) id j1BKiP132142; If x+y=N, then x = -y (mod N). Hence, this case is included in my condition. Johann Jack Matt271829,. (18, 27, 29 - goood numbers ;) I've been simulating that kind of Lotto system's expectations and guaranties long time now, using 3 ordinary loops in a plain basic code. The script is coupled with a side sub-program which monitors/processes eventual updates from a number of external files (on the online basis). I'm very rational person, preferring maximum simplicity. (my first replay on this ng was a little bit more aggressive ;) as I was pretty sure that with only mentioning the word Lotto in the subject line will provoke enough, mostly negative reactions ;) Well, your status in this group does not allow you too much of out of the pattern activity; some rules must be obeyed, even recreationally ;) Nothing new in the west ;( Like many other groups, people (5-10) sheltered in a kind of, almost private-like correspondence, keep proving that 0.9999=1. ;(( (read, inventing the famous wheel) Following that theorem, the point isn't the point, but a circle in 2-dimensional system, or even sphere in 3-dim. system ;) Sorry, I've just got a plankt, nothing serious, an apple only ;) 0.9999 will never be equalized with 1.0000, as long as we have faces like Einstein, Tesla, Newton.... PS. Wolfram, a new Messiah??!! (Bullsh@@) --------------------------------------------------------------------- All personal names are randomly chosen in this joke and do not refer to this ng. --------------------------------------------------------------------- A crusty old man walks into a bank and says to the woman at the window, I want to open a damn checking account. The astonished woman replies, I beg your pardon, sir. I must have misunderstood you. What did you say? Listen up, damn it. I said, I want to open a damn checking account now! I'm very sorry sir, but that kind of language is not tolerated in this bank. The teller leaves the window and goes over to the bank manager to inform him of her situation. The manager agrees that the teller does not have to listen to that foul language. They both return to the window and the manager asks the old geezer, Sir, what seems to be the problem here? There is no damn problem, the man says. I just won $200 million bucks in the damn lottery and I want to put my damn money in this damn bank. I see, says the manager, and is this bitch giving you a hard time? --- well, Matt, no hard feelings ;) Oh yes, I've almost forgot ,) Apart from English, I also speak Italian, French, German and 2 other languages... -- Gramophone free design - no mechanics, pure silicon! djura Save your dough. These gaps have no meaning. If the balls were not labeled with numbers, but with Chinese characters or with the names of the Caesars, the game would be identical. LH yes, generally you're right ;) but, first of all, I should stop smoking, then ... -- Gramophone free design - no mechanics, pure silicon! It is for simulation purposes, from a hat ;). I could choose any other. But, what is the essence: - I wouldn't do that if I knew (by a formulae, pattern) how to calculate exact number of Lotto combinations with (6 only one), 5, 4, 3 numbers caught for a given gaps A, B, C, D, E... 1. When I statistically choose 5 gaps (A, B, C, D, E) and create a system (in the way you understood), OK., let's say - simulate or whatever you like and that's the point ): a) There is 1 Lotto comb. with 6 balls caught. (this is for shure ;) b) How many Lotto combinations will have 5 numbers caught? c) How many Lotto combinations will have 4 numbers caught? d) How many Lotto combinations will have 3 numbers caught? I know these numbers vary (odd or even gaps numbers) as you noticed, choosing the one with odd numbers ;) - but it's not relevant at this moment. (a little question. What about 3 even - 2 odd, or vice verse?) Anyway, the guarantee is 'rock solid'; there must be min. 1 Lotto comb. with 6 balls caught, 1 with 5 balls caught, 1 with 4 balls caught and 1 with 3 balls caught. OK? see above. What's nice with this system: - I do not like to think about the numbers in the play. All 36 play. - As about the guarantee, it's rock solid but VARIES. (nobody knows the final epilogue :) - To open another thread (lucky me) take care there's one, very nice trap: The guarantee changes unexpectedly if sum(A+B)