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PhilosophyGround >> Calculus is stupid!


1/10/05 2:35 PM
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FudoMyoo
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Edited: 10-Jan-05
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"Since most people are nominalists now"

on a sidenote, I have heard different from many of my math-friends; that condidering the different scientific fields you find most of the realists/platonists among matematicians.

But perhaps you can back up your assertion Dogbert?

1/10/05 2:49 PM
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Dogbert
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Edited: 10-Jan-05
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I was speaking about physics. And, yes, platonism is pretty popular among mathematicians. But they don´t believe the world of mathematics is the ühysical world around us as our Socrates does.
1/10/05 5:15 PM
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Socrates
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Edited: 10-Jan-05
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"It does matter, that´s why differentiable functions are all continuos." I don't understand your response. You seem to agree that there is no motion in one single instant, but you imply that if many instants stand together, there is motion. How many instants? Two? Three? You also seem to imply that continuous means something like "composed of many instants", which is the opposite of what it means for something to be continuous. "This means?" It means that you cannot get motion by putting together individual instants, none of which have any motion. You cannot build a line out of points. "No. With different other points, you get different speeds at that instant (or none at all)." But you are still determining the speed at one unique instant, right? It's not the speed before or after that instant. "The speed is a number. The sequence of average speeds around an instant can eventually become constant, hence the speed at an instant gets reached." see above "No quantity you talk about in ordinary calculus, based on limits, is infinite." You clearly talk about an instant in calculus, at least when it is applied to physics. And isn't an instant an INFINITELY small section of time? I believe that this refutes your claim. "Since most people are nominalists now, there should be no confusion." I don't understand. Are you implying that instants don't actually exist in nature? Remember, my claim is that calculus cannot give an intelligible account of nature, but that people foolishly try to use it for that. If you grant that instants don't actually exist, then you are quite close to granting my point. "While there is no need for calculus to speak about infinitely small quantities, they lead to no contradictions whatsoever. This follows from the compactness theorem of first order logic." I disagree. Perhaps we could take an example. Let us grant that there are infinitely small quantities. Since they exist, how many infintiely small quantities is a one inch line composed of? I suspect we will quickly run into contradictions...
1/10/05 5:40 PM
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Socrates
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Edited: 10-Jan-05
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"Zeno demonstrated his mathematical inability and little more." Do you think that Zeno actually believed that the Achilles could never catch a tortose, or that arrows don't fly through the air? Did he simply not notice that motion DID exist, despite his proofs? Your assumption is that Zeno is a complete retard. Perhaps he wasn't a complete retard; perhaps he was using his paradoxes to highlight something about the nature of time, space and infinity. Perhaps they are reductio ad absurdums. If you assume he is an idiot, you will never learn anything from him. By the way, you never responded to my earlier point: Aristotle was able to explain away the paradoxes, and he did not have to use calculus to do so. "Since one cannot measure anything with arbitrary precision, one cannot differ between sufficiently good approximations and "reality". " Good approximations? What makes an approxiamation "good", if not that it is close to reality? Furthermore, I am claiming that calculus is fundamentally incapable of providing an intelligible account of nature; I am not disputing its predictive powers. "I think the nature of time and space is an issue of physics, not armchair philosophy. How come one cannot derive Zenos paradoxes from the models used in physics?" Time and space are issues for philosophy, not bean counting physicists. Moreover, one can indeed derive Zeno's paradoxes from physics that attempts to apply calculus to nature. As a matter of fact, I have begun to show that above. "We can actually measure the difference between the predictions and the observed outcome. This is a different story." It still proves that one can have good predictive power based with an account of nature that is fundamentally flawed.
1/10/05 5:46 PM
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Socrates
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Edited: 10-Jan-05
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"Than I feel sorry for you. You want something you will never have." Do not feel sorry for me. For someone who attempts beautiful things, it is beautiful even to suffer whatever it befalls him to suffer. It would be far worse if I desired and had petty things.
1/11/05 1:42 PM
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Dogbert
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Edited: 11-Jan-05
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"I don't understand your response. You seem to agree that there is no motion in one single instant, but you imply that if many instants stand together, there is motion." "Instant at a motion" is IMO a sensible concept, defined on points of certain curves. Usually these are functions from some euclidean space into another. "How many instants? Two? Three?" Uncountably many. "You also seem to imply that continuous means something like "composed of many instants", which is the opposite of what it means for something to be continuous." No. A functions is continuous, intuitively speaking, if small changes have small effect. A longer answer would practically nvolve the whole history of mathematical analysis and topology.
1/12/05 9:08 AM
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Socrates
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Edited: 12-Jan-05
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""Instant at a motion" is IMO a sensible concept, defined on points of certain curves. Usually these are functions from some euclidean space into another." I assume you meant to say "motion at an instant". Now, I have already said that motion requires the passing time, and that there is no passing of time at an "instant", so there can be no motion at an instant. If you still disagree, please tell me what in my logic is flawed. Otherwise, I think you must concede the point. "Uncountably many." By "uncountably", do you mean a finite number or an infinite number? "No. A functions is continuous, intuitively speaking, if small changes have small effect." Intuitively speaking? Perhaps my intuition is not as refined as yours, because I have no idea what "small changes having small effects" means. In math, doesn't "countinuous" usually mean "not composed of discrete elements", like a line is continuous, in that it is not composed of points. "A longer answer would practically nvolve the whole history of mathematical analysis and topology." Have we reached an impasse? I sense that you would like to end this discussion.
1/12/05 2:38 PM
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Dogbert
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Edited: 12-Jan-05
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"Now, I have already said that motion requires the passing time, and that there is no passing of time at an "instant", so there can be no motion at an instant." There can be passing time "around" an instant and that is all you need for the definition. "Speed at an instant" describes the behavior around that instant. "By "uncountably", do you mean a finite number or an infinite number?" A infinite number. As much as there are points in an interval. "Have we reached an impasse? I sense that you would like to end this discussion." "In math, doesn't "countinuous" usually mean "not composed of discrete elements", like a line is continuous, in that it is not composed of points." No. Even more intuitively (and kinda sloppy) speaking, a continuous function can be drawn without putting the pencil of the paper. There are "no jumps". I´m just not sure how to do a discussion on the foundations of mathematics without some formal stuff.
1/12/05 6:52 PM
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jgibson
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Edited: 12-Jan-05
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Dogbert's about to break out the epsilons and deltas....run for your lives! I can't believe I missed this discussion. Excellent thread. By the way, has anyone read David Foster Wallace's "Everything and More: A COmpact History of Infinity"? It goes well with this conversation.
1/13/05 1:45 AM
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FetFnask
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Edited: 13-Jan-05
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I must say that I agree with Dogbert, but then again I'm a statistician. I like approximations and I think they are the only way to describe the universe. I see what Socrates is getting at though and he is correct in a way.
1/14/05 1:38 PM
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Socrates
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Edited: 14-Jan-05
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"There can be passing time "around" an instant and that is all you need for the definition. "Speed at an instant" describes the behavior around that instant." Let me see if I understand you. You appear to be agreeing with me that motion "at" an instant is nonsense, but arguing that I am wrong to claim that calculus implies that there is motion "at" an instant. Calculus, you claim, implies motion "around" an instant. If calculus did imply motion at an instant, it would be nonsense, but it does not. I have thus created a straw man. Is that your claim? "A infinite number. As much as there are points in an interval." This is a perfect example of my original complaint about calculus. You are treating the infinite as though it is a definite quantity. You can never have an "infinite number" of anything. It's nonsense, and treating motion as though it is composed of an infinite number of instants will lead to many, many logical paradoxes. For instance, let's say you move from A to B. How many instants did that take? According to you, and infinite number. How many did it take to get halfway to B? Also an infinite number. But now the part is equal to the whole. The only way to avoid this is to speak of bigger and smaller infinite numbers, which is meaningless jibberjabber, not intelligible speech. "No. Even more intuitively (and kinda sloppy) speaking, a continuous function can be drawn without putting the pencil of the paper. There are "no jumps"." I like that definition, but I do not see how it differs from what I said. I think this definition means that a continious thing is not composed of discrete elements, which would require "jumps".
1/14/05 1:40 PM
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Socrates
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Edited: 14-Jan-05
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I welcome more thoughts from anyone else who wants to jump into the discussion.
1/15/05 5:47 AM
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Dogbert
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Edited: 15-Jan-05
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"Let me see if I understand you. You appear to be agreeing with me that motion "at" an instant is nonsense," I agree that one cannot move without some time passing. "but arguing that I am wrong to claim that calculus implies that there is motion "at" an instant." Yes. "Calculus, you claim, implies motion "around" an instant. If calculus did imply motion at an instant, it would be nonsense, but it does not. I have thus created a straw man. Is that your claim?" Kinda. You have created a strawman, but I have no problem with nonstandard versions of calculus that allow for this concept to be defined even when no time passes "around". These would be versions that work indeed with infinitesimal numbers. "This is a perfect example of my original complaint about calculus. You are treating the infinite as though it is a definite quantity. You can never have an "infinite number" of anything." In the real world yes. But I've said nowhere that the points in an interval have a quantity attached to them. "It's nonsense, and treating motion as though it is composed of an infinite number of instants will lead to many, many logical paradoxes." No. "For instance, let's say you move from A to B. How many instants did that take? According to you, and infinite number. How many did it take to get halfway to B? Also an infinite number. But now the part is equal to the whole." Yes. A subinterval of an interval contains as much points as that interval (with boring exceptions). The only way to avoid this is to speak of bigger and smaller infinite numbers, which is meaningless jibberjabber, not intelligible speech." "The only way to avoid this is to speak of bigger and smaller infinite numbers, which is meaningless jibberjabber, not intelligible speech." I don't think that's jibberjabber, but that's beside the issue. Say we have the time interval (1,3) and the interval (1,2) on travels along that interval with speed 1. So the time neede to go from 1 to 3 is 2 and the time to go from 1 to 2 is 1. Now you think that these times must be the same if we have a mathematical model in which there are as many points in both intervals, but that's a confusion. in the sense of set theory both interval have the same "size". Tthis is shorthand notation, it doesn't mean that these are quantities that can be added, that every set has a quantity etc. For this stuff we use the algebraic properties of the numbers. and fact like 3-2=1. One can treat the same objects from different points of view, dependend on what the problem is.
1/23/05 10:43 PM
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willconley777
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Edited: 23-Jan-05
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Side note. For years I have been in the (now almost unconscious) habit of applying the concept of limits to philosophy. I love calculus' admittance of proximity and I hate the hubris of the illusion of exactitude.
1/27/05 11:04 AM
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Subadie
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Edited: 27-Jan-05
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Cool discussion. Now, everybody vote: I say : Socrates
1/28/05 3:33 PM
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jgibson
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Edited: 28-Jan-05
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No, Dogbert by logical PWNAGE.
3/25/05 2:40 AM
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WoodenPupa
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Edited: 25-Mar-05
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Dog by the set of all PWNAGES.
5/24/05 1:48 PM
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Jbraswell
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Edited: 24-May-05
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Jesus Christ. I think you guys are going around in circles here. Socrates, the problem is your lack of knowledge of math history, and unregistered use is correct. The vague notion of a limit as originally conceived vexed mathematicians, too, which is why Cauchy and Weierstrass developed a way of looking at things that isn't vague at all. This modern view of a limit, which has indeed changed drastically since Newton's day, clearly defines a limit without involving any such notions as infinitely small or instantaneous speeds. Yes, mathematicians still use the terminology because it's convenient to do so, just as we continue to use terms such as "average family," even though there is no individual entity that answers to the term. Another good example is the reinterpretation of imaginary numbers as ordered pairs of real numbers. "I agree, but I don't think any numbers exist like, say, cats and dogs exist. Numbers are part of our language for talking about the world, they aren't the world." I think this debate is different from the one between nominalism and Platonism. THIS discussion is about what kind of terms we are even going to allow to BE in our mathematics. It's a seperate story as to what ACCOUNT of mathematical terms we are going to put forth. (I myself am a nominalist, but as Fudo mentions, there are plenty of hardline Platonists around still.)
6/9/05 11:51 PM
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Socrates
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Edited: 09-Jun-05
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I've been gone for a while. Sorry. Hopefully, some of the new participants in the discussion can help me understand my error. Now, my original complaint was that calculus treats the infinite as a definite quantity, which it is not; this leads to all sorts of problems when it comes to using calculus as a way of understanding the world (although I never disputed its predictive power). So, Jbraswell said: "The vague notion of a limit as originally conceived vexed mathematicians, too..." I am glad I am not alone. "which is why Cauchy and Weierstrass developed a way of looking at things that isn't vague at all. This modern view of a limit, which has indeed changed drastically since Newton's day, clearly defines a limit without involving any such notions as infinitely small or instantaneous speeds." Well, that's a relief, but I can't just take your word for it. What is this "clear" definition? If you could use a clear example of its application, as well, I would be in your debt. "Yes, mathematicians still use the terminology because it's convenient to do so, just as we continue to use terms such as "average family," even though there is no individual entity that answers to the term." 'To speak not beautifully is not only not beautiful in itself, but makes something not beautiful in our souls' - Socrates (in Plato's 'Phaedo')
6/13/05 4:53 PM
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Dogbert
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Edited: 13-Jun-05
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"To construct a reliable defense against Berkeley's objections, you have to work out a rigorous mathematical theory of approximation processes, which neither Newton nor Leibniz was able to do." Or a rigorous theory of infinitesimals, like nonstandard analysis or smooth infinitesimal analysis.
6/14/05 12:32 PM
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Socrates
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Edited: 14-Jun-05
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Hmmm... In my first post, I said, "At best, [calculus] is a very useful approximation." Now, Mr. Devlin claims that "if you regard h as a variable, and concentrate not on the given function, but rather on the process of approximation that arises when h approaches 0, then Berkeley's argument can no longer be sustained." So, it sounds like we are in agreement. However, this still leaves open the problem of the ontological status of the limit, and the question of the way in which calculus should (or should not) be used to understand nature (as opposed to just manipulating nature), seeing as it is a sort of approximation. "Indeed, it was not until 1821 that the Frenchman Augustin-Louis Cauchy developed the key idea of a limit, and it was a few years later still that the German Karl Weierstrass provided a formal definition of this notion." Is this the elusive 'clear definition' that was alluded to earlier?
6/15/05 8:09 AM
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Jbraswell
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Edited: 15-Jun-05 11:20 AM
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Here's the definition: L is the limit of f(x)as x approaches a if and only if given any e > 0, there exists d > 0 such that 0 < |x - a| < d implies that |f(x) - L| < e Note that there's nothing fishy here. It just says (loosely) that with the function f and point a in question, I can get the function value as close as you want to L by picking a number close to x and feeding it to the function.
6/17/05 10:21 AM
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Socrates
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Edited: 17-Jun-05
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I was hoping for a defintiion with more words :) Would you give a concrete example of how one might use this definition in everyday life? Let's say a planet is rotating around the sun, for instance. Could you walk me through how one would use this idea of a limit in that case? I think if you did this, my objections would become clearer.
6/17/05 2:55 PM
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Dogbert
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Edited: 17-Jun-05
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Planets have nothing to do with mathematics.
6/17/05 3:33 PM
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Socrates
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Edited: 17-Jun-05
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"Planets have nothing to do with mathematics." Don't planets obey mathematical laws? Are you saying that one should not try to use mathematics as a way to explain planetary motion? That would be odd, given the origins of calculus. Look, people use calculus in an attempt to gain an understanding of nature, right? I am just trying to find a simple example of how calculus is used in science, so we can discuss how the limit is being understood- the philosphic implications of the use of calculus, if you will. If you have a better example than planetary motion, please let me know.

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