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aleph_0

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Everything posted by aleph_0

  1. Yes, that sounds fine enough. But that's basically the kind of picture Aristotle had, and he just used the term "abstraction" to describe it, without the more loaded term "measurement". So was there a reason for departing from his language, or was she just repeating Aristotle's view with her own jargon?
  2. So let's take this as the claim: Mathematics is the science of measurement. Presumably this means that, in a very broad reading of the term "measurement", an element being in a set is a measurement of the set, or possibly of the element. I'm not really sure. And I suppose the claim would be that the study of sets omits every other possible measurement of the elements or sets. But why cast it in this language, rather than abstraction? How is this distinct from Aristotle? I think I'm missing the point that you're making. What is it that you're arguing? This seems like exac
  3. We can forget trying to define mathematics, since I don't have the interest to pick through ITOE to find relevant quotes. I just want to know why she chose this phrase rather than "abstraction" to describe the mathematician's activity, and how this account differs from Aristotle's.
  4. If this is the correct interpretation, then I suppose this is why I find her claims so unsatisfying. Why use the phrase "measurement-omission" rather than "abstraction" if all she claims is that mathematics is the science of reasoning about certain features of object(s), while omitting others? Measurement usually denotes a property of objects which can be quantified by rational numbers. Even quantity is not a measurement in the ordinary sense of the English word since it is only described by natural numbers, so if she wanted a very broad term she would have been a little bit better-served b
  5. I generally understand the point about measurement-omission as the claim that mathematics omits particular measurements, but that it is the science of measurements. If that's a misconception--it's been a while since I read ITOE--then the question is misguided. That's my understanding as well. So the consensus thus far seems to be that the notion of "measurement" is much broader than is used in colloquial conversation.
  6. Here is a relatively short, and possibly simple, question that has just occurred to me. The fundamental concept necessary to understand mathematics, Rand has claimed, is measurement-omission. However, at least prima facie there are disciplines in mathematics which do not measure or claim to be able to measure anything. For instance, topology lacks a distance metric. It doesn't measure anything, in any obvious sense, but studies the shapes of objects. (In topology, two objects are said to have the same shape if one of them can be stretched, bent, enlarged, or shrunk, such that it can be ma
  7. I doubt it for a couple of reasons. One, it wouldn't make sense for that to be the way they organize the best-sellers. The idea of the best-seller list is to encourage book sales by making consumers aware of what everyone else is reading, so that they feel like they can talk to other readers. I can't imagine a good reason for eliminating a classic from that list. Also, I once worked in a bookstore and remembered seeing some best-sellers stay on the shelf for more than a year, so I'd have to wonder when that cut-off period would be. Now it's true that the best-seller list measures sales
  8. Linux isn't popular because, in addition to reasons mentioned above, it's not compatible with popular software. That's partly because some of the high-demand software is licensed and so it would be illegal for the Linux creators to produce and distribute the OS with that kind of software--most notably, to my mind, is software that plays most DVDs, and also Photoshop. However, at least in the former case, most Linux users just find illegal software that plays DVDs. Problems like the lack of Photoshop are more serious, and I imagine the reason there isn't a popular illegal version (that I kno
  9. Same here. Anyone down for philosophical reading, or reading the classics?
  10. Three times. The first time in high school, about ten years ago, and I didn't really absorb much of it. The second time about three years later, when I really understood the idea of the intellectual strike. The third time about two years ago.
  11. I'm moving back down to the Sebastian area, wondering if there are any Objectivists around... Or even just signs of intellectual life. Melbourne? Palm Bay? Vero Beach? Fellsmere? Fort Pierce? Stuart?
  12. Not at all. I may produce scenarios in which a theory that makes use of infinite quantities is the most explanatory, but I don't claim that they're actual. I'm asking a metaphysical rather than a factual question. But to have it said: If we are no longer disputing the impossibility of infinite quantities, and it is recognized that no satisfying argument has been provided to that effect, then I am happily unconcerned with other issues. However, I'll still respond to the questions below. You mean, suppose that it is impossible for there to be infinite quantities, and then you ask
  13. I was pointing out that neither you nor anybody HAS presented a valid argument, and yet you're cavalier.
  14. That sure proves your point. I'm convinced. Good argument.
  15. How is that problematic or relevant? Sure, there are infinitely many transfinite cardinalities. What's the problem?
  16. First, just because you will be no closer to an end does not mean that there cannot be discrete spaces since there can be an infinity of discrete points. As an analogy, think ofthe integers which stretch to infinity, though you can cordon off finite intervals which contain discrete points. Second, I'm not convinced that space and material objects are discrete. However, from the rest of what you wrote, it seems you don't mean "discrete" but "distinguishable". But this seems to assume that the only way to distinguish two objects is by their distance from an end-point. This principle cann
  17. Note also that if you are to repeat some previous argument to this effect, like the one about boundedness implying no definition, then you need to have some substantial counterargument to my post pointing out that this is an insufficient proof (in that particular case, due to an equivocation of the use of the word "bound"). I can just foresee this kind of thing coming, so as to distract from the lack of any real, working argument.
  18. Absolutely nobody has at any point disputed the law of identity. I accept it. The very question is whether infinity is contrary to it, and what reason we have to suppose that it is. The point is to give an argument showing how an injective map as I described above implies the failure of identity, i.e. it is to show that such a mapping implies that there is something which lacks definition, where definition does not just mean "finite or bounded", i.e. the point is to show, in a non-circular way, that the hypothesis of the existence of such a map is self-contradictory. I don't know how else to s
  19. Note that Aristotle's view of a Prime Mover is not your view, not just in the matter of consciousness, but also in the entire argument for its existence. Aristotle's First Mover is not a causal agent--it does not physically push and pull stuff, I believe it was Aquinas who made this argument. Aristotle's First Mover is a logical first-cause. In the series of questions, "Why is a plant green? Because of its leaves. Why are the leaves green? Because of the chlorophyll. Why is chlorophyll green?..." Aristotle's First Mover is the logical cause which grounds all of these facts. It is not a
  20. Moreso--it's only natural that I should start the conversation. Again, I don't know what "no definite multiplicity" could mean. I understand what it is for a set to be not well-defined, but this notion has no meaning to me. If you just mean "not infinite" then that is patently false. The correspondence would be simple: Take any arbitrary natural number, the correspondence associates it to a unique object (in the exact same sense as, for some three objects, there is a correspondence with {1, 2, 3}. What is impossible about this? Since the Dedekind cut for the square r
  21. I just found, in iTunes U, a free course on radical capitalism titled "Radical Capitalism" taught by Dr. William Kline, in which it seems that more than half of the classes are spent discussing Atlas Shrugged. I haven't listened to any of it yet, but it seems like it should be interesting. Note that iTunes U is accessible by iTunes or an iPhone/iTouch. Maybe it's also accessible by other browsers, but if not, iTunes is free for download and compatible with Windows.
  22. I still have the concept of a dragon, fictional or no. If I saw one, I could identify it as one, and I wouldn't protest about calling it a "dragon" because the term only applies to fictional characters. Why would I need an example of an infinite set? All I need is the notion of infinity in mathematics, then abstracting from sets of empty sets, to the more general setting of sets in general. And yet again (I think this is the fifth time or more), I am not trying to state a fact of the matter. I am discussing the logic of physical theories and hypotheses, namely this unfounded claim t
  23. Can I not form the concept of a dragon? I have no referent, but surely I understand the concept. Of course, I have abstracted this concept from other concepts, like lizard-like features, but that can't be the point at issue since my correspondence between infinite sets of numbers and sets of physical objects is just an abstraction from its use in pure mathematics. As for your last paragraph, it seems to confuse two distinct scenarios I provided. I wasn't appealing to their use of mathematical equations which employ infinity in order to argue that there is an infinite quantity--again, I a
  24. I would expect no less a post from a hybrid between tensor and man. Though I'm not quite so certain that such a thing is immune to proof. We have some pretty awesome techniques for discovering things that seem impossible to discover. Whatever. An accompanying voice of reason is welcome.
  25. I mean more than this. For instance, sure, I can form the words, "There is a round square at the bottom of the ocean," but this is an outright contradiction and so we can reject any such hypothesis without further empirical investigation--and here I mean actual rejecting, not just ignoring. Can we do the same with the hypothesis of an injective mapping from natural numbers to disjoint physical objects? If yes, then we can take this notion to define the phrase "infinite quantity", and people on this forum have no justification for the claim that all quantities are finite, and cannot dismiss
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