Nate T.
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Posts posted by Nate T.
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And I certainly don't want to keep anyone from using the best adapted tools to solve a problem.
And, computing macro-level phenomenon from micro-states is only difficult because the computational framework is inefficient -- the given seems to navigate these computations with ease, eh? Like, where does Nature choose to round off PI in forming a soap bubble?
"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.
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So you are claiming that real analysis provides THE objective standard for the correctness of computations?
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.
The assumption of continuity of the physical substrate is precisely such an outmoded, inefficient idea -- and the formulation of physics in terms of PDE's is suspect in terms of conceptual efficiency, if not practical results.
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.
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I guess it's simpler anyhow to show that continuous spaces do not correspond to the discretion of measurement, especially if, as you claim, my space is compact. You're probably right, but I will have to verify for myself. Now where is that confounded analysis text?
As far as PDE's, I too have used computers to "solve" them numerically. Interestingly, computer proofs used to have a big stigma attached to them, which demonstrates the inversion of fact versus invention in mathematics.
Of course, a computer cannot represent irrationals, as it operates with finite sequences of bits. Which only emboldens me ...
Thanks,
- David
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.
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Aha! This objection I understand. You should have picked the moniker Pythagoras, or maybe Kronecker.
Now, if one chooses to identify limits of non-terminating sequences of rationals with special symbols, fine. Not sure what the point is of identifying them for metrical purposes, however.
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.
On the other hand, if one discovers solutions to equations in algebra, and then insists on conflating them with rational measurements from reality, to form the so-called real algebraic numbers, then one will have to drop some context to get the scheme to work in practice. And that's not good, long range -- it makes the math muddy at best, harder to learn and teach, and harder to use, forcing humans to "fly by instruments" instead of using their conceptual peepers.
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.
The context dropped is whether or not the idea measurably exists in reality. As usual in any invalid compromise, it is the reality basis that gets cut off in such compromise.
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.
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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?
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A space is compact if, whenever a collection of open sets covers the space, then so does (at least) one of the available, finite sub-collections. Such spaces are very nice, analytically, and serves to generalize the notion of a bounded, open subset of the real line to higher order topological spaces.
Closed, bounded subset, you mean.
Is that the notion of compact you had in mind? Compact spaces are continuous at any level of focus, and modern analytic methods rely on the fact that spaces are compact to derive all sorts of useful proofs.
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.
In particular, modern physics uses the machinery of partial differential equations in its root formulations of physical law. Have you ever tried to think about PDE's outside the context of a compact space?
I have indeed thought about PDEs in the context of finite spaces-- anyone who has ever numerically approximated solutions to PDEs has.
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I wonder, when did the compact perspective on reality take hold?
This word, "compact" ... you keep using this word. I do not think it means what you think it means.
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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.
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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.
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Look, its a perfectly good mathematical invention. It may seem strange to describe a line this way, but it's consistent with reality and linear algebra, and when you go up to planes and volumes, the benefit only increases, conceptually speaking, in terms of simplicity of representation and computation (no negatives to worry about).
I don't expect anyone to take this on faith, but can anybody else see that integers are vectors (have magnitude and direction)? That is all I ask that you grant; the rest of my framework follows logically from that.
Cheers.
- David
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).
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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?
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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.
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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.
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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.
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The phrase "rearranging the deck chairs on the Titanic" comes to mind for some reason ...
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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?
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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.
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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.
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.
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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.
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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.
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).
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... my point is not to prove that there are infinite sets of disjoint physical objects, but to ask what is contradictory about supposing it. Thus, I do not need to prove the axiom of infinity for physical objects, but to ask what is contradictory about it's hypothetically being true. So the burden of proof is actually not on me, since I am merely denying the claim that people on this forum make, that there can be no infinite quantity. I refuse to accept this because the case has not been made to satisfaction, and so I am demanding a more thorough argument.
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A theory with absolutely no evidence is to be ignored, not dismissed as demonstrably untrue. My question is how can people state that it is impossible (not just not-yet-proven) for there to be infinite quantities.
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".
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(i) Again I emphasize that correspondences exist whether people are aware of them or not. Thus, a pairing that some human being could not, in finite time, represent or think of, is still a pairing. The pairing does not need to be given as a computational procedure, for any arbitrary input--it can be an arbitrary association of some integers with some objects. That no person will ever know all of the correspondences is unimportant for the same reason that it is unimportant (to the question of quantity) that no person will ever think of the correspondence between the number of atoms in my computer and the set of that many natural numbers.
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(ii) If this is the ultimate response, then I think you concede my point, because a scientific theory which posits gremlins cannot be rejected out of hand--it can only be rejected on the basis of lack of evidence, but if evidence were ever presented one would then accept the hypothesis. I have said elsewhere how I think such evidence might conceivably be given, but that's not even important. The point that I want to establish is that such a notion is not self-contradictory or unintelligible. I just want to establish that one cannot dismiss as meaningless, a theory of the universe which incorporates some claim of an infinite quantity.
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(iii) That we could not finish such a construction does not entail that such a correspondence wouldn't exist. Again, the correspondences are not purely mental, and they're not contingent upon us in any way. That we could not, for instance, ever actually count out e^(e^(e^79)) of anything does not thereby entail that the number is not a quantity.
(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
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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.
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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.
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I second Dummit and Foote, that is, if you can afford it.
My Process of Thought
in Metaphysics and Epistemology
Posted
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.