Nate T.

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.

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.

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 numerical schemes yield results sufficiently close to the correct answer in a way you can control. If you think that space-time by nature is discrete so the problem is moot, this does not invalidate the fact that computing macro-level phenomenon from their constituent particles is not computationally feasible. I don't think that the differential equation is as hapless a tool for describing reality as you seem to think. The advances in the industrial revolution and more are in a large part a consequence of these continuum models, so the practical results speak for themselves.

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 can't really be sure that your numerics are giving you good answers, or if they are, you can't say given some step size how close they're getting to the real solution. So there is a distinction between numerically solving a problem with a computer and proving that such a numerical solution approximates the true solution well-- and the former, while possibly being some evidence for a solution, isn't a proof.

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? Because when you unpack the definition of the real number \sqrt(2), that's what it amounts to. I agree that "reifying" real or negative numbers (such as claiming a stick has length exactly \sqrt(2), or that there be -1 cows in a field) is silly. But in fact I don't think people generally make the kind of gross category errors you subscribe to them, and I think it's possible to keep in mind what these concepts represent in whatever context you're applying them to. For example, reifying real numbers is generally harmless in applications since people pass to approximations at the end anyway, and I've never met a student so thoroughly confused by negative numbers to seriously think that the use of "-1" implies the existence of negative numbers of objects in reality.

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?

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 numerically approximated solutions to PDEs has.

This word, "compact" ... you keep using this word. I do not think it means what you think it means.

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.

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 two nonzero numbers in that set are of "opposite sign" if and only if they are orthogonal. But in that case, you've essentially invented new notation for the negative sign that takes 3 to 4 times as long to write. 3. You gave your model a notion of positive the second you associated one of the slots in your ordered pair with 1 and the other with -1. Just because you don't use the word doesn't mean that isn't exactly what's going on. In any case, we should probably return the thread back to its original topic. You can have the last word if you want.

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) = 4. So your operation is not well-defined since you don't get a unique answer for the inner product of two integers. 3. What you are talking about might be better described using positive linear combination. In that case, -1 and 1 are positively linearly independent, as you claim. You might consider checking the notion out on wikipedia (c.f., here).

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?

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 reasonable approximation to a line in some context is all that we need to abstract to the idea of a line.

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.

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 how, from this, you can ever extrapolate to justifying an infinite number of objects. The analogy to atoms is inappropriate, since you claim that infinite quantities are justified by scientific theories, not just inperceptibly large (or small) quantities.