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aleph_0

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aleph_0 last won the day on August 3 2010

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  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?
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