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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 exactly the meaning of "abstraction" as Aristotle used it, when he provided his philosophy of mathematics.
  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 by using "quantity". If there is no difference between measurement-omission and abstraction, then what is new about her philosophy of mathematics in light of Aristotle's works on the subject? Well this is not so unusual--this is just saying that some emotions come in degrees.
  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 made to look just like the other one. What is forbidden is any tearing of the shapes, or gluing. Hence I have the shape of a sphere, and so does every other human being, since you can compress us into that shape. However, none of us have the shape of a donut, because that involves a tear in the center, or gluing your head to your foot. Likewise, a figure-eight donut is a distinct shape from a regular donut and a sphere, and so on.) As another example, set theory does not seem to measure anything. It can certain be used to count things, but it is just the study of the basic relation of set membership, and without some restriction on the nature of the sets in a particular topic, doesn't really say anything interesting or useful about the rest of the world (that's not already known to a pre-schooler). I guess there are a few obvious ways that an Objectivist could explain this: Either say that these are not genuine topics in mathematics, but perhaps their own topic, which is foundations of mathematics; or, these are genuine topics in mathematics, and the notion of "measurement" is a lot broader than is usually meant in casual conversation. Is there a third option, or a way to adjudicate between these?
  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 within the last week, and while Atlas Shrugged has been selling steadily for decades. So some Janet Evanovich book, which lasts three weeks on the best-seller if she's really done a good job, isn't going to out-sell Atlas over the course of the next 50 years. But is that really the way to quantify sales? Wouldn't things like the Bible and A Tale of Two Cities win that competition? And also, if we're talking about what's winning in the market today, the weekly best-sellers seems more appropriate. Never knew there was a difference.
  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 know of) is that it's hard to reproduce that kind of sophistication. But given enough time, I see no reason why the Linux community couldn't produce comparable software, in light of the fact that they have the most efficient and secure OS that's popularly available. Let's also just note that market popularity isn't a mark of a good product. How many people own Caterpillar dump trucks? Far fewer than those who own a Dodge, I'm sure. Does that mean Dodge trucks are superior? No, they're just targeted to a different market. Windows targets the market that generally doesn't know or care a whole lot about the details of technology, which is not a bad thing. Linux targets people who are more tech savvy. Also, let's note that Atlas Shrugged isn't on the top of the best-sellers list, either. There are some highly suspect trends in the philosophy of open-source software, but that doesn't mean that all open-source software must be associated with that philosophy. There's nothing wrong with contributing freely to a community, if the fostering of the community is rewarding. Everybody who posts on this forum recognizes that.
  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 if there is any other way in which evidence can be proffered for the existence of infinite quantities? I doubt it, but if it can be shown that infinite quantities are impossible, then I will be satisfied in the first place. Yes. Why do I accept the existence of atoms? Because the atomic theory is that which best explains and predicts behavior which I can observe. I have never observed an atom, but the atomic theory has too many theoretical virtues to justify rejecting it. It is (relatively) simple, lends to calculation, predicts events accurately, accurately suggests ways to manipulate the world around us, and so on. In fact, simplicity was the original reason for the popular adoption of heliocentrism. Geocentrism was never disproved, probably until we launched a person into space. While Ptolemaic geocentrism could accurately predict the location of celestial bodies, for an observer at a fixed parallel, when shifting parallels the calculations were no longer accurate. Trying to come up with Ptolemaic models of celestial positions and orbits became computationally nightmarish, when Copernicus's incredibly simple model lent to easy computation with just as much accuracy, and so it was adopted. I don't believe either of these theoretical changes are philosophically suspect, in spite of not having been supported by direct observation. Likewise, in general, if a theory has such theoretical virtues, then I come to believe the theory without insisting on direct observation. Thus if a theory has these virtues and makes essential claim to an actual infinity of objects, then I will believe in the theory and thereby believe in an infinity of objects. So long as the theories are not self-contradictory, yes, I take them to be possible in that sense. I take the claim that there is a person on the roof of my building to be equally arbitrary in that sense, and equally possible in that sense. Some of the arbitrary claims will sound fantastic because we have never seen or heard anything like it before, some of them will sound mundane and more plausible because we have seen similar things, but that doesn't affect the fact that they're equally unsupported by observational evidence and so--in the sense you're using the term "arbitrary" here--equally arbitrary. And fittingly, I don't believe in the existence of Zeus, and I don't believe in the existence of an infinite quantity. Now, if somebody were to claim to have evidence for the existence of Zeus, I would be more suspicious since there seems to be so much evidence against it: What's he been doing for the last couple thousand years at the least? Where's he been? Moreover, the story of Zeus seems to be too easily explained by child-like fantasies and desperate attempts by ignorant people to explain the world. That is to say, I would be more reluctant to accept the Zeus hypothesis because I already have another one that is working quite well. But if, some how--say by direct observation, or by theories which make essential use of the Zeus hypothesis to predict all events better than current theories, etc.--there were better evidence for Zeus than against him, then I would believe that Zeus exists. Infinity, on the other hand, has none of these unpleasant properties. Current theories, as far as I can tell, are completely agnostic on this account and so there is no evidence against the existence of infinite quantities. Moreover, it doesn't seem like some infantile attempt at explaining anything. So far, it's no explanation at all in the first place--there is no attempt at explaining any fact by reference to infinite quantities. The hypothesis of an actual infinity does not claim to actually solve anything, just yet. But if it did, and if there were good evidence and theoretical virtues, then it would have a case worth listening to. And as we seem to have established, there is no reason why this theory would violate the Principle of Identity, and so there would no reason to reject it out of hand. Why is that problematic? There wouldn't be an end-point. So what? Reference? And why can you not talk about "greater than"? Sure, the natural numbers are infinite, but you can still pick some two natural numbers and compare whether they're greater, less, or equal to each other. Comparing some natural number to the number infinity itself? Well, here you'd have to somewhat expand your notion of "greater" and "equinumerous", or you'd be right, there would be no sense in comparing them. But if your notion of "greater" is redefined by the notion of mappings, as we started with, then infinite sets will be greater in quantity than any finite set. Well then, what's the axiom? You can't just say it's an axiom without naming what axiom you're appealing to. And if you say "identity", then you have to say how identity precludes infinity. Because as far as I can tell, there is no connection whatsoever between the axiom of infinity and identity. If you bring up the point about boundaries, then you again have to say why it is that lacking a boundary on quantity is impossible. I.e., you have not said anything new here, and only suggest a rehearsal of a long conversation that has already played out, and ended with no good argument against the possibility of an infinite quantity. This is intellectually irresponsible, since it attempts to suggest that the case has been made to satisfaction, for the impossibility of an infinite quantity, when no such thing is even remotely true. This point has been made before, and refuted before, unless you can come up with a counter-counter-argument. By saying that the natural numbers are infinite, I am merely taking this to be a piece of our mathematical rules. It's not circularity, but statement of definition. To have the size of the set of natural numbers is just to stand in an bijection relation to the natural numbers, which has the properties you mention above. But sets do not grow by thinking up new members--sets are determinate collections of objects. So in sets, there is no such thing as the possibly infinite. There is only what is actually in the set and what is not. However, as stated a few posts into the discussion, even if we amend our mathematics to accommodate a notion of possible infinity, I may simply re-phrase my question: Is it possible or impossible for there to be, for every set of natural numbers, an injective map from that set to some set of disjoint physical objects? If so, then I would take this to be my definition for there being an infinite quantity. I've responded to all this before. He argues that the universe was caused, from the assumption that it has a beginning time. I don't know if he argues, or how he argues, the acceptance of the assumption, but it's not the same as arguing against an actual infinity. If you know of some such argument that he does employ, please reproduce it here because it is not an article of common knowledge, or even of widely known philosophical knowledge.
  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 cannot be right, though. Would two objects be indistinguishable simply in virtue of their being the same distance from a third object? Just because they're indistinguishable with respect to one property, doesn't imply that they're indistinguishable simpliciter.
  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 say it, and I've said it in these ways before many times. Take the hypothesis that there is such an injection, and without assuming that every possible set of disjoint physical objects is finite, without assuming that the law of identity implies finitism, without assuming that "specific" is contrary to having an infinite quantity, provide a contradiction. This may perhaps be done by proving that the law of identity implies finitism, but it cannot be done by assuming it. That is a logical fallacy, called circularity. It may be proved by proving that being a "specific" entity implies being bounded, but it cannot be prove by assuming it, since this is the very thing you hope to prove. I feel like I'm talking to Christians who have been shown their logical fallacy, but without addressing it, continue to employ the fallacious argument, as if they have some need for the argument to be valid, which is more important to them than logic.
  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 as though he is causing things to change over time. To my mind, this causal notion of a First Mover is rejected in Objectivism (I think rightly so) precisely because there is no reason to accept it. There is no obvious reason why the universe could not have infinitely many causes. Any reason you might have for rejecting infinite causes, it cannot be based on any finitism, since we now agree that causes are not things and so there is no problem with supposing that they are infinite--since this would not violate the finiteness of all things. You assume that there must have been a point at which everything was motionless, which I don't see a reason to accept. Without the theory of the Big Bang (and modulo issues of certain quantities of elements existing, indicating some passage of time), it seems a perfectly reasonable hypothesis about the universe, that it simply has always been pretty much like it is now and always will be. But beyond that, we could have gone through infinitely many Big Bangs. I see nothing inconsistent about any of this. My argument above was against precisely this, unless you have some additional reason for thinking that an infinity of causes really doesn't constitute a cause.
  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 root of two is not located anywhere, naturally, no. If you mean, "can we build a correspondence which is a bijection between the real numbers and physical reality?" I don't know, in the same way that I don't know if there is a simple countable infinity. But I don't think it's self-contradictory, so it at least makes sense that we could form a bijection where the Dedekind cut which we identify with the square-root of two being assigned to some physical object. That obviously would not mean that the object would exhibit any behavior or properties that I can imagine, which would be shared by the square-root of two since, obviously, the map is not a homomorphism. I somehow sense this is supposed to be an objection, but I'm grasping at straws about how you might conceive this to be a challenge.
  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 that there can be no finite quantities. I am relatively confident that there is no (strong) proof of an infinite quantity.
  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 am not arguing that there is an infinite quantity. In the first case, I was supposing a scientific theory which essentially claims that there are infinite quantities. In the second, I was providing a slightly different scenario that built on some things that we actually do know about the Big Bang.
  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 physical theories which claim it; if no, then people on the forums have been right. And again, just because I cannot produce an explicit mapping doesn't mean that one doesn't exist, so this argument has no force in the question I'm posing. Now if I were trying to positively prove that there actually is an infinite quantity, you might respond this way, but that is not my intent. Now beyond this, here is a scenario in which one could prove (insofar as the theory of atoms has been proved) an infinite quantity: A hypothesis about physics has, as an essential component, the assumption that there are infinite quantities. It explains all of past recorded events and facts, and it perfectly predicts all future events, with accuracy as close as it is possible for us to measure. Rather than reject the hypothesis because it makes claim to an infinite quantity, I would accept the hypothesis as proven theory. Alternately, suppose that we found that the Cosmic Background Radiation were constant everywhere, and emanating from a single direction. Since our best hypothesis is that this radiation is a product of the Big Bang, and so indicates the location of the Big Bang, then it would make sense to think that for any distance traveled toward the Big Bang, one will not reach its origin (the constant frequency of the radiation indicating that, for any given point, that point is just as distant from the origin of the Big Bang as any other point to which we might travel, in principle) and continue to encounter this radiation. Thus one could count an infinity of this radiation, whatever it is. I'm rather certain that the frequency of the Cosmic Background Radiation is not constant, but the point being that such facts could indicate an infinite quantity. In any case, if you concede that your only objection is the lack of evidence, then you agree with me. It is not a self-contradictory hypothesis to suppose that there is an infinite quantity.
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