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Nate T.

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Everything posted by Nate T.

  1. Where does insight come into this? Often I focus on a problem for a long period of time without success, only to have the solution occur to me later without consciously trying.
  2. And I certainly don't want to keep anyone from using the best adapted tools to solve a problem. "Nature" don't "compute" anything; when a soap bubble forms no one need be computing anything. Soap bubbles act according to their nature, and our task to try to figure out how to predict it in the best way possible.
  3. What I claim is that, to the extent that you want to use calculus (which means, to the extent that you want to apply continuum models to things) you can't really avoid some kind of real analysis. And because you're right that we can only measure with rationals, this means we need some assurance that our continuum model corresponds to reality. Now you may (and I suspect do) reject the need for such continuum models. If you argue that since we're really only solving approximate problems anyway we should just discretize everything, then you need to find a way to give assurances that these nu
  4. This ties in to what I was saying in the other thread (and it does have a philosophical point about measurement, too!) It's not just a "deficit" of computers-- we (computers and humans) have nothing but rational numbers to measure with directly, and most measurements (except counting) come with an error attached. If you haven't read IToE yet, there's a nice discussion of this in the appendix called "Exact Measurement and Continuity" that you might like. Anyway, one reason that analysis on a continuum is necessary is that without some kind of theoretical assurance of well-posedness, you
  5. Aha! This objection I understand. You should have picked the moniker Pythagoras, or maybe Kronecker. That's because people are generally taught all of these extensions of numbers (like negative, real, and complex numbers) as a system of rules apart from the context from which they arose. If they actually taught people what real numbers *are*, things would be different. If you handle only rational numbers, how are you supposed to solve x^2 = 2? Do you content yourself with saying that you can find a sequence of rationals that solves the equation as precisely as you'd like?
  6. Ah, I see, icosahedron. Your argument is that we have nothing but (positive) rationals to measure with in reality (which is true), and therefore we should have no need of real (or negative) numbers in mathematics?
  7. Closed, bounded subset, you mean. It is certainly not true that compact spaces are "continuous", since finite spaces are compact (indeed, compact sets can be regarded as generalizations of finite sets) and many other wild sets like the Cantor set are compact, too. It seems that you're talking about such discrete spaces, so this word doesn't distinguish the two models of space that you're talking about. I would stick with something like "continuous" or "continuum" rather than compact. I have indeed thought about PDEs in the context of finite spaces-- anyone who has ever nume
  8. This word, "compact" ... you keep using this word. I do not think it means what you think it means.
  9. So, if I understand you correctly, no macroscopic object (to which classical mechanics applies) is a "real" entity, since the elementary particles of which they are composed are the only "real" entities, and classical objects can be at rest in some inertial frame. So is a car not a "real" entity? Or can I not identify it? I'm just trying to get some kind of philosophical question out of here, rather than just speculative physics that has nothing to do with the purpose of this board.
  10. 1. Being isomorphic to the usual number system is not enough: two systems can be isomorphic without being conceptually similar. For instance, how would you rather deal with an object like a parabola-- as the graph of a quadratic function, or as intersections of certain planes with double cones, like Apollonius? Can you imagine doing trajectories being limited to such a framework? (Incidentally, how does this system of yours affect coordinate geometry?) 2. It is true that if you restrict the plane to the union of the positive x- and y-axes, the usual Euclidean inner product says that tw
  11. 1. The fact that algebraic sign encodes the very notion of direction that you're trying to capture with these ordered pairs, and does so in an algebraically unified way, seems quite conceptually and computationally beneficial to me. You just have to keep in mind what negative numbers mean in whatever context you're in. 2. The reason for my opening barb is that you can't use the standard Euclidean inner product on these pairs, since they are really equivalence classes of pairs, as you yourself use. So while (0, 1).(1, 0) = 0, you could also say of the same two integers that (1, 2).(2, 1)
  12. Wow. This certainly is a thread about how to handle mathematical concepts that have no relation to reality, all right. icosahedron, just for the hell of it, since you're representing integers as ordered pairs of natural numbers, just what is this inner product you're defining on the integers? The one that makes 1 and -1 orthogonal?
  13. Lines are closer to concepts of methods. I agree with the posters who say that lines are abstracted units of length. They are idealized units of length whose only property is their length (and direction), since those are the only properties we care about in the context of plane geometry. As for not being able to physically realize a line, that's right, we can't. But just as one needn't produce -2i cows in a pen to use complex numbers, we need not produce an infinitesimally thin ruler in order to apply plane geometry. The fact that objects can subtend some (straight) length and be a re
  14. Conservapedia is a wiki, right? Is it like Encyclopedia Dramatica in that it's one huge troll site, or are there people who take this seriously? I suppose there's no reason it couldn't have started out as one and become the other, by Poe's Law.
  15. Aleph_0: What I was trying to get at is this: right now you're holding out hope that a future scientific theory may provide indirect evidence for an infinite quantity. If it can be shown that scientific theories by their nature cannot justify the existence of infinite quantity, you have no grounds to suppose them except to reify them from advanced mathematics. However, scientific theories are created in a finite context, by observing finite relationships between measurable phenomena in a specific range justified by the data (one of the points made by Harriman's new book). I can't see h
  16. The phrase "rearranging the deck chairs on the Titanic" comes to mind for some reason ...
  17. Since the thread appears to be devolving into insults at the moment, maybe I'll jump back in and ask some substantive questions. Aleph_0: We've established (your point 4 in the OP) that any positive claim of a countably infinite set of physical objects is arbitrary, unless you can provide some sort of very indirect evidence. I want to try to pin you down as to exactly how arbitrary you think it is. While you seem to accept the higher-level hypothesis that there may exist a scientific theory (as in your post #20) that demands the existence of such a collection, I also take it you ar
  18. Thanks for the discussion, Aleph_0, but I think we're arguing in circles. I'll bow out and let others continue, if need be.
  19. Good dictionaries define dragons as *fictional* animals like big lizards that breath fire, etc. These refer to the imagination, which exists as a mental entity, not literal dragons existing somewhere. If you want to abstract the notion of one-to-one correspondence between mathematical sets and physical objects, you need one example of a completed infinity of physical objects to abstract from. Otherwise, like the dragon, your referent is imaginary, which as I've mentioned I have no problem with. Similarly, merely supposing a scientific theory or modifying an existing one to fit your arg
  20. Okay, I think we both agree that to make a positive claim of an infinite number of electrons would be an arbitrary claim. It is your opinion that, if we can imagine any kind of phenomena, stipulate a definition describing it, and find no logical contradiction in its terms of definition, we can therefore define a new concept based on such imaginings? This would seem to be an appeal to the analytic/synthetic dichotomy-- and it is is not how concepts work. First you need referents, then you form concepts, then you form definitions to capture the essentials. The lack of a self-contradiction
  21. I think you can accept *as a hypothetical* that an infinite number of electrons exist, in that you can form the words: "What if there were a mapping from the natural numbers to the electrons?", or even imagine in your head a bunch of electrons being labeled with various numbers without end. It's just that, being an arbitrary assertion, it won't tell you anything about anything, being based on a notion of completed infinity ripped from its context as a concept of method, seeing as you manifestly cannot produce an example of an infinite collection of objects. If not being able to positively di
  22. (i) If you want to use the concept "bijective map" as it is used in mathematics to establish quantities of collections which are not sets in mathematics, you must justify this usage in the broader context of physical objects. For finite sets (even very large ones) this is done by enumeration, and is uncontroversial. Since we do not have any referents of literally infinite collections of physical objects (for reasons already mentioned) we cannot apply this concept to physical objects. In this sense, you do need to construct such a correspondence, or you literally are talking about nothing.
  23. Yes, I probably stated things in a misleading way, now that I look at it. I didn't mean to suggest by the term "mental construct" that the quantity of a collection wasn't an actual property of that collection. After all, concepts are mental entities that your mind constructs (which abstract properties from entities in an objective way, of course), and I was just looking for a term that distinguishes concepts like numbers and such from physical objects. If you have a finite collection of entities, then of course you can assign natural numbers to them in any way you like by explicitly const
  24. Despite the fact that you're a fellow mathematical traveler, Aleph_0, I'm going to have to (in a sense) side with the finite-ists on this one. I think the difference is the fact that you want to pair off natural numbers with physical entities instead of with mental constructs like numbers and such. My argument is closest to your no. 4, but it differs a little from what you had said. We can talk about sets of infinite cardinality in mathematics because we have an unambiguous characterization of all of the sets we often consider in mathematics, such as the set of all perfect squares. Your
  25. I second Dummit and Foote, that is, if you can afford it.
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