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12/5/03 1:29 PM
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Socrates
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Edited: 05-Dec-03
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I do not think I can accept that. It strikes me as unintelligible. I don't think a human mind can comprehend these "actual infinities" of any kind. This is one reason why I think modern math is unitelligible. It will lead one to claim that one infinite series is larger than another; you might as well say A is not-A. It is just non-sensical. I strikes me that one should say that these numbers exist as potentialities (although I will be the first one to admit my understanding of potential vs actual is hazy at best... a careful study of the Metaphysics is sorely needed). Potentiality, however, is not sufficient to do modern math. I think we have clarified one of my problems with modern math. Just so you know, I appreciate the pragmatic uses of modern math. I do not mean to say that it is useless or unimpressive. I just mean that it cannot be used to say anything meaningful about the world. Modern math can be used to interact with the world, but it cannot be used to give an intelligible account of it.
12/5/03 2:06 PM
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Dogbert
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Edited: 05-Dec-03
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"This is one reason why I think modern math is unitelligible." And that was just counting... "It will lead one to claim that one infinite series is larger than another; you might as well say A is not-A. It is just non-sensical." What´s wrong with defining an order on sequences, other than saying A is non-A, there is no reason to believe it´s not perfectly consistent. Of course modern math is hard. All sciences are hard on a higher level, that´s to be expected. "I just mean that it cannot be used to say anything meaningful about the world." I don´t think that´s what math is made for. Mathematics is more a kind of language for talking about the world. WHAT we say is not part of mathematics.
12/5/03 3:12 PM
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Socrates
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Edited: 05-Dec-03
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I never made any claims about consistency. I made a claim about intelligibility. And I think you have completely misunderstood my claim if you think it has anything to do with the difficulty of higher level mathamatics. By un-intelligible I do not mean confusing. "I don´t think that´s what math is made for. Mathematics is more a kind of language for talking about the world. WHAT we say is not part of mathematics." I believe that you are quite mistaken. Isn't a huge part of math devoted to trying to understand what the math TELLS us about nature. Many mathamaticians have seemed to think so. Isn't this why we have all this Quantum Mechanics gibberish about something being BOTH a wave and a particle, and people walking around saying things like "space is curved". That all comes from mathamaticians trying to put math into language. All those little symbols are supposed to have some correspondance to reality. My whole point is that one cannot translate math into language, and that is what makes math unintelligible. But people WANT to use math to say things about reality. Thus, they end up speaking nonsense, but nonsense that is verified by math, and thus they become OK with speaking nonsense. And with that we arrive at my original claim, modern math is not concerned with being intelligible. Have I clarified my original position?
12/6/03 11:45 AM
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Dogbert
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Edited: 06-Dec-03
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"sn't this why we have all this Quantum Mechanics gibberish about something being BOTH a wave and a particle, and people walking around saying things like "space is curved". That all comes from mathamaticians trying to put math into language. All those little symbols are supposed to have some correspondance to reality." No that´s what physicist do. Wrong department. "My whole point is that one cannot translate math into language, and that is what makes math unintelligible. " So how have mathematicians (they do exist) have learned their trade if not by language? "But people WANT to use math to say things about reality. Thus, they end up speaking nonsense, but nonsense that is verified by math, and thus they become OK with speaking nonsense." Math doesn´t verify anything said about the world. "And with that we arrive at my original claim, modern math is not concerned with being intelligible. Have I clarified my original position?" I think so. But I don´t think math folks can be blamed for modern physics.
12/6/03 5:33 PM
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Socrates
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Edited: 06-Dec-03 05:39 PM
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Fair enough, I'll try to seperate math from physics (although I do think that there is a strong connection). Math folk do speak of actual infinities, do they not? (You yourself did a little while ago, although you claimed you weren't going to). And don't math folk say some infinties are greater than others? Aren't they trying to say something about something? In addtion, if math doesn't verify ANYTHING about the world, isn't it completely pointless? Why would anyone study it?
12/6/03 6:56 PM
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DonnaTroy
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Edited: 06-Dec-03
Member Since: 22-Sep-02
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In my opinion, math is the best language to describe rythms, relations, proportions, patterns and things like this. See the Fibonacci sequence, for example. Explains a lot of things that happens on Nature.
12/6/03 7:04 PM
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Dogbert
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Edited: 06-Dec-03
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"Math folk do speak of actual infinities, do they not? (You yourself did a little while ago, although you claimed you weren't going to)." I admit it. "And don't math folk say some infinties are greater than others? Aren't they trying to say something about something?" Well they have a notion of the size of sets that makes it possible to distinguish between infinite sets of different "size". But usually they don´t talk about infinitely big or small objects in geometry, with a few exceptions. Either way, you don´t need actual infinities to do calculus and to resolve Zeno´s paradox. It´s just that the standard method in math is to define everything via set, and then you have to talk about actual infinities. "In addtion, if math doesn't verify ANYTHING about the world, isn't it completely pointless? Why would anyone study it?" For the same reason why people study grammar, it makes it easier to communicate things about the actual world. It´s a language, as I said.
12/6/03 8:13 PM
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Socrates
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Edited: 06-Dec-03
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You admit that modern math speaks about actual infinity. I believe that speaking of actual infinity is nonsense (as did Aristotle, I believe). Infinite sets of different sizes certainly seems to be nonsense. Modern math is thus not concerned with being intelligible. "Either way, you don´t need actual infinities to do calculus and to resolve Zeno´s paradox. It´s just that the standard method in math is to define everything via set, and then you have to talk about actual infinities" Are you sure? I mean, when you first spoke of sets, you didn't seem to realize that they necessitated speaking about actual infinites. As a matter of fact, you implied that they were a way of AVOIDING actual infinities. Were you simply mistaken then, or were you trying to trick me? I suspect that actual infinity is essential to all calculas-it's just hidden in phrases like sets, summation, etc...
12/7/03 3:01 PM
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Dogbert
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Edited: 07-Dec-03
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"You admit that modern math speaks about actual infinity. I believe that speaking of actual infinity is nonsense (as did Aristotle, I believe). Infinite sets of different sizes certainly seems to be nonsense." Maybe the meaning of "obvious nonsense" lies in the eye of the beholder. I for one think assuming the existence of potential infnites but not actual infinites is nonsense. "Modern math is thus not concerned with being intelligible." I don´t agree with your presumption. "Are you sure? I mean, when you first spoke of sets, you didn't seem to realize that they necessitated speaking about actual infinites." Well, the problem is that people didn´t use clear definitions at the times of Aristotle, so how else should I say what the numbers 1,2,3,4.. are in a rigorous fashion? "As a matter of fact, you implied that they were a way of AVOIDING actual infinities. Were you simply mistaken then, or were you trying to trick me?" No, I used sets for personal convenience in a part where they don´t matter. "I suspect that actual infinity is essential to all calculas-it's just hidden in phrases like sets, summation, etc..." Well, I give you the completely actual infinite free definition of a limit. A sequence is a Formula f(n), where you can use the number 1 or any number of the form 1+1+1+...+1 as n. The sequence is said to converge to a or ,similar, the limit of f(n) is a if for every rational number r (a quotient) there is a number m of the 1+1+1+1...+1 kind such that whenever n is bigger or equal m, the distance between f(n) and a, given by |f(n)-a| is smaller than r. Defining the 1, 1+1, 1+1+1..numberts and the (a/b) numbers isn´t my job, but they were used at the time of Aristotle.
12/7/03 5:48 PM
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FudoMyoo
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Edited: 07-Dec-03
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I just hate when the discussions get to technical with formulas and shit. It reminds me of my lack of mathematical knowledge. ;-) Doesn´t the forum guidelines say that english should be the spoken language here.. ;-)
12/7/03 5:57 PM
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Dogbert
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Edited: 07-Dec-03
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"Doesn´t the forum guidelines say that english should be the spoken language here.. ;-)" Yeah, but the written language is math. :-P
12/7/03 6:40 PM
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Dogbert
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Edited: 07-Dec-03 06:47 PM
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Background on the "Actual Infinite" in mahematics: www.mathacademy.com/pr/minitext/infinity/index.asp I try to explain the concept of limit´s in a clear and understandable way, by using the much more intelligible notion of actual infinity: Assume you have a sequence of numbers, a1,a2,a3,a4,a5,a6,a7... and so on forever. For example one could take the sequence a1=1, a2=1/2, a3=1/4, a4=1/8, a5=1/16, a6=1/32 and so on. We see that they become smaller and smaller. The bigger the n, the smaller the a_n. For every positive number (from the real line) there is a N, such that from N on all an´s are smaller than that real number, it get´s closer and closer to 0. We can say that the difference |an-0| between every an and 0 becomes smaller than every positive number. More general: A sequence a1, a2, a3,... converges if there is a number "a" such that for every positive number r there is a natural number N such that from N on all an´s become smaller than a. In other words you can make the difference as small as you want by proceeding on the sequence.
12/7/03 6:58 PM
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Dogbert
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Edited: 07-Dec-03
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This may be more understandable: www.mathacademy.com/pr/minitext/infinity/index.asp
12/8/03 5:50 AM
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Socrates
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Edited: 08-Dec-03 06:05 AM
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"Maybe the meaning of "obvious nonsense" lies in the eye of the beholder." I do not think so. Like the law of non-contradiction, I doubt it is a matter of perspective. "I for one think assuming the existence of potential infnites but not actual infinites is nonsense." I will dispense with pleasentries and be blunt. This comment makes me think that you have not thought much about this question, and that your responses are knee-jerk reactions in defense of modern math (most likely because you've invested time in it). "Well, the problem is that people didn´t use clear definitions at the times of Aristotle" This makes me think that you've never read either Aristolte or Plato. "No, I used sets for personal convenience in a part where they don´t matter." Now you are being dishonest, either to me or to yourself (or both). Look at what you actually said. "...since you are apposed to actual infinities, let´s look at the work of Cauchy/Weierstrass, the standard approach today." This CLEARLY implies that what will follow will not use actual infinities. That was your WHOLE POINT, that you could avoid using actual infinities. It defeats your whole purpose to then talk of sets, which entail actual infinity. So, it is in NO WAY convenient nor does it thus it make any sense... UNLESS you did not know that sets entail actual infinity OR you were trying to trick me.
12/8/03 6:03 AM
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Socrates
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Edited: 08-Dec-03 06:07 AM
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"Well, I give you the completely actual infinite free definition of a limit." Are you trying to trick me again? I never asked for a actual infinite free definition of a limit. This can be found in Newton's Principia. I don't dispute the existence of limits nor do I have any problem understanding what they are. My problem is with the USE of limits in calculus. Look at what I said LONG ago: "On a conceptual level, Newton relies on "taking things to the limit" (which he also says can never be reached). The point beyond which a process cannot go is treated as the end of the process; and this entails positing actual infinities." You see, it is not the limit itself that I have a problem with, but the use of limits as the END of an INFINITE series. In other words, Summation. But now we are going in circles. I think that this should give you pause.
12/8/03 6:06 AM
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Socrates
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Edited: 08-Dec-03
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"Assume you have a sequence of numbers, a1,a2,a3,a4,a5,a6,a7... and so on forever." Wait, give me a second... (HA! I hope you realize how funny that is).
12/10/03 9:11 AM
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Dogbert
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Edited: 10-Dec-03
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"I will dispense with pleasentries and be blunt. This comment makes me think that you have not thought much about this question, and that your responses are knee-jerk reactions in defense of modern math (most likely because you've invested time in it)." Well, if you want to talk about numbers in general, you have to talk about actual infinities. "This makes me think that you've never read either Aristolte or Plato." This makes me think that ou never read modern math texts. Compare the geometric axiomatics of Hilbert and Euclid. Big difference in rigor. "Now you are being dishonest, either to me or to yourself (or both). Look at what you actually said. "...since you are apposed to actual infinities, let´s look at the work of Cauchy/Weierstrass, the standard approach today." This CLEARLY implies that what will follow will not use actual infinities. That was your WHOLE POINT, that you could avoid using actual infinities. It defeats your whole purpose to then talk of sets, which entail actual infinity. So, it is in NO WAY convenient nor does it thus it make any sense... UNLESS you did not know that sets entail actual infinity OR you were trying to trick me." Well, I gave you the ugly potential infinite version above: '' sequence is a Formula f(n), where you can use the number 1 or any number of the form 1+1+1+...+1 as n. The sequence is said to converge to a or ,similar, the limit of f(n) is a if for every rational number r (a quotient) there is a number m of the 1+1+1+1...+1 kind such that whenever n is bigger or equal m, the distance between f(n) and a, given by |f(n)-a| is smaller than r. Defining the 1, 1+1, 1+1+1..numberts and the (a/b) numbers isn´t my job, but they were used at the time of Aristotle. '' "Are you trying to trick me again? I never asked for a actual infinite free definition of a limit. This can be found in Newton's Principia. I don't dispute the existence of limits nor do I have any problem understanding what they are." You obviously don´t know what Newton wrote. His fluxions are actually (sic!) infinitely small. "My problem is with the USE of limits in calculus. Look at what I said LONG ago: "On a conceptual level, Newton relies on "taking things to the limit" (which he also says can never be reached). The point beyond which a process cannot go is treated as the end of the process; and this entails positing actual infinities."" Nobody forces you to interpret the infinity symbol in calculus as a symbol for the actual infinite. Wether you use actual infinities or not de facto doesn´t depend on wether you use phrases like "as n goes to infinity". The limit of a series is just the limit of the sequence given by the partial sums. "Wait, give me a second... (HA! I hope you realize how funny that is)." Can you explain counting without any mentioning of ANY infinite. Please substitute your own way of counting for the one used in real mathematics. As I wrote: ''Defining the 1, 1+1, 1+1+1..numberts and the (a/b) numbers isn´t my job, but they were used at the time of Aristotle.''
12/10/03 5:05 PM
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Rastus
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Edited: 10-Dec-03
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Your problem is that of definitions, as Dogbert suggests. You need to re-examine the role of an infinite set of zero dimensional points as well. "length" is a mathematical construct, called a "metric". You aren't "adding points", per se. However, consider a+a+a...+a, or N*a, where N is the number of repetitions, and "a" is some length less than 1. Consider the integer "A", where A is the first integer greater than |1/a|. So long as we choose N = A, the value of N*a ( or N/(1/a) ) will always be greater than 1. Take the limit as a -> zero, and we see that so long as we retain this convention of naming "N" as greater than (1/a)...we will always have a non-zero result of the sum.
12/11/03 10:36 AM
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Dogbert
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Edited: 11-Dec-03
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Rastus is correct.
12/13/03 4:41 AM
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MasterDebater
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Edited: 13-Dec-03
Member Since: 14-Apr-03
Posts: 207
Yeah, you have created a fallacy in your incorrect definition it seems. A fallacy of ambiguity caused by a translation from mathematical language to english.
12/14/03 1:43 PM
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Socrates
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Edited: 14-Dec-03
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"Well, if you want to talk about numbers in general, you have to talk about actual infinities." Not true. "This makes me think that ou never read modern math texts. Compare the geometric axiomatics of Hilbert and Euclid. Big difference in rigor." I disagree. "Well, I gave you the ugly potential infinite version above:" This is completely irrelevent, and you are OBVIOUSLY dodging my DIRECT question. This post came one week after the post which CLAIMED to give me infinite free calculas (not simply a defintion of a limit, by the way), but I showed that it contain an actaul infinite. Now, my question is regarding that exchange, did you know that your example actually contained infinity (and thus you were trying to decieve me) or were you simply mistaken? "You obviously don´t know what Newton wrote. His fluxions are actually (sic!) infinitely small." Obviously, hey? He may have used infinitely small fluxions, but my claim was that he didn't use actual infinite to define a limit, or to give examples of limits, for that matter. If memory serves, he defined it as the first number which is never reached by a series, or something to that effect. His example was determining the area under a curve using squares. I am fine, and quite impressed, with this. He goes on, however, to abuse the use of a limit by treating it as the end of an infintie sequence, which is VERY DIFFERENT than saying treating it as the forst number NOT REACHED by a series. "Nobody forces you to interpret the infinity symbol in calculus as a symbol for the actual infinite. Wether you use actual infinities or not de facto doesn´t depend on wether you use phrases like "as n goes to infinity". The limit of a series is just the limit of the sequence given by the partial sums." The phrase "at the limit" DOES necessitate actual infinity. I never said I had a problem with the limit. You are creating a stawman, and I think you have misunderstood my whole point. What is the process of summation? How is the limit USED in modern math? I'm under the impression that they it is treated in a way as an end of an infinite series, which it is not (for example, a parabola intercepts its asymtopes "at the limit" according to Newton). Isn't this the case? Isn't this how modern math derives the instantanious velocity of an object? What exactly was my incorrect definition and how is it flawed?
12/14/03 2:20 PM
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Dogbert
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Edited: 14-Dec-03
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"Not true." Proof me wrong. "I disagree." Which isn´t a argument. "This is completely irrelevent, and you are OBVIOUSLY dodging my DIRECT question. This post came one week after the post which CLAIMED to give me infinite free calculas (not simply a defintion of a limit, by the way), but I showed that it contain an actaul infinite." The actual content was actual infinity free. That only came in in the definition of number. "Now, my question is regarding that exchange, did you know that your example actually contained infinity (and thus you were trying to decieve me) or were you simply mistaken?" I considered it to be unimportant in the part it came in. Maybe you can help me by explaining how you do define numbers. "Obviously, hey? He may have used infinitely small fluxions, but my claim was that he didn't use actual infinite to define a limit, or to give examples of limits, for that matter." Yeah, because he didn´t use limits to devine derivatives and integrals. He relied on th actual infinite to do that. "If memory serves, he defined it as the first number which is never reached by a series, or something to that effect. His example was determining the area under a curve using squares. I am fine, and quite impressed, with this." And how do you define a series without actual infinities? This is as clearly actual infinity as possible.
12/15/03 2:48 PM
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Socrates
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Edited: 15-Dec-03
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"Proof me wrong... Which isn´t a argument." You apparently did not get my joke. My mere assertions were intended to highlight in a humorous way your mere assertions: "Well, if you want to talk about numbers in general, you have to talk about actual infinities... This makes me think that ou never read modern math texts. Compare the geometric axiomatics of Hilbert and Euclid. Big difference in rigor" These are not arguments. They are asserions. I thought I would respond in kind. I will respond to the rest later. It's finals week and I have very little free time right now. Take care.
12/16/03 2:23 PM
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FudoMyoo
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Edited: 16-Dec-03
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I´m enjoying reading this discussion, I think it´s very interesting, so please continue. :)
12/18/03 3:06 PM
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Socrates
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Edited: 18-Dec-03
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"The actual content was actual infinity free. That only came in in the definition of number." The actual content was a set, which entails actual infinity. Look at what you said: "Solid foundations came with the work of Cauchy and Weierstrass, who don´t use any "actual infinities"... since you are apposed to actual infinities, let´s look at the work of Cauchy/Weierstrass, the standard approach today. A sequence is a set of numbers, ordered by the natural numbers." You might as well have said, "I won't use any actual infinites. Now, let's start with an actual infinity". I have much more to say; I think you have missed my point. I'll be back later.

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