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.

The phrase "rearranging the deck chairs on the Titanic" comes to mind for some reason ...

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 are not claiming to have produced an example of such a theory. Such a theory would not simply rely on infinity as a concept of method (such as Newton's infinitesimal division of the Earth to prove gravitational attraction emanates from the Earth's center of mass) but must positively demand an infinite collection of objects. There are two cases. (1a) Suppose it is shown that no such scientific theory is possible. Is there then any other way in which evidence can be proffered for the existence of an infinite number of physical objects? (1b) If, for the sake of argument, such a scientific theory was proposed, do you think it would be sufficient to justify the literal existence of such a set of objects? Next, regarding the acceptance of arbitrary claims: (2a) If you seriously accept arbitrary claims such as infinite numbers of objects as a possibility due to a lack of self-contradiction, do you regard as equally admissible physical accounts appealing to the existence of Zeus, Ra, etc., as there is equal evidence for both? (2b) If you object to Zeus, Ra, etc., as being ignoreable by virtue of being unscientific, suppose one hypothesizes a scientific theory mandating the existence of such a god-like being. Would the existence of such a theory in turn demonstrate the existence of our friendly scientific deity? If you reject the possibility of such a theory, in what way is a theory postulating the existence of an infinite collection of objects different in principle?

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.

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 argument says nothing as to the fact of the matter. If you do find an actual example of such a thing I'd be interested to see it, though.

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 in your stipulated definition doesn't make the concepts involved any less vacuous. Now there *are* referents of "completed infinity", but all of them are concepts of method taken from advanced mathematics, not physical objects as in physics, which is why I say the notion of "one-to-one mapping" between infinite collections is being taken out of context. Thus if it is not your intent to exhibit an infinite collection of physical objects, you are using a floating abstraction and thus aren't really saying anything. Of course you're still free to imagine it, if you like, but it isn't a serious construction. Regarding infinities that may arise in the application of scientific theories such as the Big bang, these are artifacts of the modeling equations, and I think you'd be hard pressed to find scientists arguing for the existence of actual infinities based on the mathematical structure of their governing equations (although I bet there are some, such as general relativity theorists and black holes).

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 disprove the existence of the arbitrary (infinite numbers of electrons or God) was your point, that's true, I guess. Anyway, I didn't say theories with no evidence should be dismissed as demonstrably untrue (in that assertions about various constructs posited, etc., must not exist), I just said they should be rejected, or ignored, if you like. It doesn't stand in a positive correspondence with reality nor does it contradict any known facts, it's simply as though nothing has been said. Same as any assertions about infinitely many electrons-- which has the further disadvantage of being impossible in principle to check, not just "not-yet-proven".

(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. That you would go ahead and stipulate such a pairing using these concepts outside the mathematical context is really what is arbitrary. Put another way, you're smuggling in the premise that collections of physical objects satisfy the axioms of set theory. Unless you want to do mathematics only with finite sets (which you certainly don't), you need to show that the Axiom of Infinity Holds, which is precisely the point in question. So far, though, this is an arbitrary assertion on your part. So you'd either need to exhibit a literally infinite collection of entities for us, or start from another foundation of mathematics besides ZFC. The onus, either way, remains on you. (ii) A theory with absolutely no evidence is to be dismissed out of hand, and precisely for that reason-- that's the theory/practice unity. But (and especially in light of my challenge in (i)) I'd be interested to see how you might provide such evidence. (iii) It doesn't for mathematical objects since we have induction, but it does for physical objects. This is the meaning of the qualifier "in principle." I can whimsically say that e^(e^47) pairs to e^(2e^47), though

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 constructing some list-- there need not even be any property or attribute of the finite collection of entities that suggests an ordering of the list, since as you note only the cardinality is important in this context. However, there is a natural ordering to the collection of Presidents (ordinal by term) that makes such a correspondence obvious at once. In the case of a hypothetically infinite set of objects, in order to have a correspondence that works once and for all, you are required to use the knowledge you have of all of the entities you're trying to pair off (such as that they happen to be generated from the natural numbers by the squaring function) otherwise you're never done constructing the correspondence, not even in principle. It's also not enough to say that "well there might be lots of such correspondences that haven't been identified", since the same can be said for those gremlins on Venus, as others have said. Anyway, your reformulated question is easier to answer: you now ask whether the weaker construction I mentioned in my first post of a sequence of nested correspondences actually exists between the first few natural numbers and physical objects. It's precisely this that has to be regarded as arbitrary, since we could never finish constructing such a sequence of correspondences! I suppose I can agree, as Grames has, that one could hypothesize an infinite number of electrons, but I don't see any way of demonstrating the positive claim that the number of electrons is in fact infinite.

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 justification for having set up, once and for all, a one-to-one correspondence between the natural numbers and the perfect squares is that you've exhibited a function f(n) = n^2 which *in principle* matches every natural number to a perfect square. This works great for concepts like numbers, but how would you go about constructing such a general function pairing natural numbers with electrons? You'd have to actually sit down and actually label electrons as you find them-- there's no general formula stipulating that, hey, that electron over there is the 675,598,982nd one! So you're resorting to finding as many electrons as you can and adding them to the list post hoc, making up your one-to-one correspondence as you go along. That is, instead of having *one* rule making the correspondence between your two sets, you have a sequence of correspondences between finite subsets of those sets-- and if you ever finished, you'd have a standard one-to-one correspondence between finite sets, which even Mindy would agree is uncontroversial if you could explicitly exhibit it. Of course you can *imagine* a one-to-one correspondence between natural numbers and electrons, but this would just be you saying, in your head, "Hey, if you give me any number, I can picture the electron it would pair to, why not?" But at this point we've left the realm of reality and gone over to the arbitrary, which is an Objectivism no-no.

I second Dummit and Foote, that is, if you can afford it.