<p>I was just wondering if any of you ever felt that maybe we're missing something huge in mathematics and/or physics. It has happened before. I invite anyone who ever heard something in math/physics/CS/etc. that left a bad taste in their mouth to post their doubts here, and we can discuss them. I'll go first.</p>
<p>I sometimes doubt that irrational numbers have a physical manifestation in our world. I'm no doctorate, but I've never seen a single convincing proof of this assumption. It seems like somebody woke up one day and said "time, space, matter, and energy are all continuous like the reals." Well, that's not true of charge, right? Who's to say it's true of any of the others? Perhaps they're all rationals... which means they're really all discrete. All of calculus would have to be reformulated, but who knows? Maybe it could work.</p>
<p>I don't like the ball and vase problem. For those of you who haven't heard of it before, it goes like this: you start off with a vase which contains 10 balls, numbered 1-10. At noon, you remove the first ball and add 10 new balls, numbered 11-20. After this, you wait 1 hour and do the same, removing ball #2 and adding 10 new balls. You repeat this process after another half-hour, then 15 minutes, etc., each time decreasing the interval by a factor of 2. How many balls are in the vase at 2pm? Well, the procedure is repeated an infinite number of times, and since you remove the balls in order, you end up removing them all. So mathematicians would tell you that you have 0 balls left at 2pm. I find this absurd on several levels, and can provide what appear to be fool-proof logical arguments against this result.</p>
<p>I'll start off with these and see if anybody wants to discuss/debate/add to the thread. I invite anyone to argue with me over these first two points, as well... if you're convinced I'm just not paying close enough attention, a bit of intellectual correspondence would certainly keep me from playing video games non-stop.</p>
<p>No mathematician in his right mind would claim complex numbers to be true (in an absolute sense) because mathematicians are very aware of the relativity of what they are doing. Mathematics does not try to explain the world (in contrast to physics) but it is axiomatic: you start out with a set of assumptions and use a given logic to study the structure that is implied by those assumptions. The implications are only as true as the initial assumptions on which they rest. While complex numbers are true relative to the standard axioms of algebra, they need not be "absolutely true" (if absolute quantities exist at all).</p>
<p>Analysis (/Calculus) was invented centuries before any physicist knew what to do with it, and so have prime numbers been studied for a long time before someone had the idea use these findings for data encryption.</p>
<p>Back to your complaints:</p>
<p>Complex numbers are a handy way to represent two-dimensional vectors (do you think those exist?) and they have further applications in quantum mechanics and relativity. By the way, mathematically speaking there is no reason to limit yourself to two-dimensional numbers. Ever heard of quaternions? Think 4-dimensional numbers.</p>
<p>The point of the vase and balls problem is to demonstrate some anti-intuitive qualities of infinity. At any point shortly before 2 pm, there would be "close to infinitely" many balls in the vase, and as you get infinitely close to 2 pm there are infinitely many balls in the vase. The problem is that the series gets infinitely close to 2 but never actually reaches 2 (it's discontinuous). If instead of asking "how many balls are in the vase just prior to 2" you ask "how many balls are in the vase AT 2" the answer is none. You name the number of a ball and I will tell you the time at which it was removed from the vase. Since you cannot name a ball, the vase must be empty.</p>
<p>The vase problem sounds like the person who cannot cross a bridge because he/she must go halfway before reaching the end, and there is always a halfway point.</p>
<p>Since you mentioned CS, you may want to look into "computable reals". The idea is, since the real numbers are uncountable, but there are only a countable number of computer programs (aka turing machines), therefore almost all real numbers are "uncomputable." </p>
<p>That is, only for a very small, select subset of real numbers, can we actually obtain the digits by any known method of computation. Some "constructionists" have questioned the reality or existence of noncomputable reals.</p>
<p>As for the "vase" problem, and other seeming "paradoxes" of mathematics (look up the Banach-Tarski Paradox for an excellent example), consider that we are not talking about reality here in the same sense as, say, physics. What we are doing is finding the logical consequences of assuming sets of axioms, like b@r!um said above.</p>
<p>I guess if I didn't understand the basic concept of a limit approaching zero, the ball and vase problem/zeno's paradox would confuse me too. </p>
<p>It's true there are some fundamental precepts of math/physics that end up failing under very specific circumstances and at certain scales, but we're working on figuring it all out.</p>
<p>b@r!um: the OP said irrational numbers, not complex numbers.</p>
<p>OP: Calculus would not have to be rewritten at all because math is independent of the physical world. It has applications to the real world, to be sure, but you'd be hard pressed to find mathematicians gleefully talking about all the real world applications their research on 5-dimensional topology has. For example, take the area of a circle. Its impossible for an area to be irrational in the real world, simply because of the laws of physics. However, for our purposes, approximating the area of a circle with pi*r^2 is sufficient because typically the amount of area that it would differ by would only be seen at the atomic and subatomic levels. It should also be noted that concepts such as volume and area are fabricated by humans. Atoms can't really said to have an "edge" because electrons are always moving and besides that, atoms are mostly empty space. Hell, according to the standard model of physics, particles are points. There is no such thing as volume, just the illusion of it created by the fundamental forces.</p>
<p>With things like the Ball and Vase problem and Zeno's paradox the issue is that things that are purely mathematical nature are attempting to be described using real world terms. With the ball and vase problem its obviously impossible to have an infinitely large vase to being with. With Zeno's paradox (which is actually several paradoxes but I'm referring to the halving distance one) you can't divide distance forever. The planck length is the minimum unit of distance.</p>
<p>Barium:
The irrational number question was really exactly targeted at physical existence. While it's clear the imaginary ones don't have a physical existence (I assume you agree with this), do any numbers have a physical manifestation? It would appear that the natural numbers correspond to physical quantities (1 apple, 2 apples, etc). Whether zero is a number or not is an interesting discussion. As far as the integers go, it seems like it may as well be the same thing as the natural numbers, just looking at it funny (the integers seem superfluous). The rationals make decent sense to me, but maybe I've just not thought about that long enough either. The irrationals are where it starts seeming like we might be getting too far away from physical reality to apply mathematics in describing the physical reality. Now, mathematically speaking, we can talk about the real line. I'm just not sure that we're justified in applying the real numbers to the study of natural phenomena. I've been contemplating writing a paper entitled "Rational Universe" or something, based on similar arguments. So I guess in a sense, I don't think your first part addresses my exact concern, although your thoughts are certainly interesting.</p>
<p>As to the ball and vase problem, I understand the argument but reject its conclusion on the basis that the conclusion is inconsistent. Consider the following experiment:</p>
<p>Do everything as before, but remove random balls and write the numbers 1, 2, 3, ..., n on them. Therefore, you end up with balls 1 to infinity in the "removed" pile. However, it is uncertain whether you ever remove the ball initially marked "2" from the vase. You could consciously perform the experiment to never remove 2, in fact. Therefore the question isn't of whether infinity > 10 x infinity (both infinities countable). They are. My problem is that I don't think the simple application of set theory to this problem is justified. I believe that the sets generated by taking all natural numbers 1 at a time and the set made by taking 10 at a time are equal; what I don't believe is that this model corresponds to the situation modeled by the problem precisely as it is stated. However, I don't really have a better way of stating it...</p>
<p>Excellent post, and I agree with everything you say. And when I said Calculus would have to be redone, I misspoke, as you claim. Our understanding of the application of calculus would have to change; but then again not really, as math (as you correctly said) only approximately describes reality.</p>
<p>I suppose my intention with this thread was to question the extent to which mathematics corresponds to reality. It's clear to me that it does correspond on some level (I could be wrong, but I do believe deep down that mathematics is the language of the Universe), but the question is: which kinds of mathematics, and to what extent, can be used to describe the physical world? Is there a way to ascribe mathematical truth to the physical world? I know this isn't the aim of science, but is proving nature even possible? If so, can mathematics do it, or do we need something else? I suppose my question is really more like philosophy than math or physics, but... yeah.</p>
<p>Yes, I understand that there are several solutions to Zeno's paradox, and I've made peace with that one. And I also know that some of the best people in the world are working on making what we already know make sense and patching the holes of knowledge, but that doesn't mean we have to ignore the imperfections while they're working on it.</p>
<p>jbusc:
I have encountered the idea of the computable reals and it makes me very happy indeed. Numbers generally make me a very happy panda. The whole idea of whether numbers are rational or irrational, whether they are computable or have ready-made-names, etc. is just about the most fascinating thing in all of mathematics to me. Perhaps my favorite book is Conway's on Numbers and Games. Fantastic reading for anyone who likes numbers... Conway will make them all for you, and then some.</p>
<p>And thanks everyone for their input! Any other ideas from anybody? Any philosophy of math/physics/CS they want to discuss? Here's another one of mine, just as food for thought.</p>
<p>I was tutoring some freshman in physics the other day, and we got as our answer that the time needed for something to happen was proportional to PI (it was some oscillation problem, or something). This left a bad taste in my mouth... it seems to me that a measure of time shouldn't contain an explicit dependence on PI. Or the square root of 2, 3, etc. Even if irrationals do have a concrete physical basis, why should time depend on them? I always pictured time as more or less "linear-esque", so why should it depend on "non-linear-esque" quantities? I've discussed this with several people, and only a very few have managed to follow, so I don't expect this to make incredible sense. I guess I was wondering if this had happened to anyone else... do you ever calculate a perfectly acceptable numerical answer, do a double take, and ask "why on Earth would it be that number?". Not that the number is too big or too small, but that it looks weird... like sin(sqrt(2)), or e^pi, e+pi, e-pi, etc. Just odd things, you know.</p>
<p>And just because I'm throwing in this new topic does not mean I want to kill discussion of the others. Please feel free to add to the discussion however any of you see fit.</p>
<p>Do you mean the scientific constant π? It's an abstraction, but it IS a "number" representing the ratio of circumference to diameter in a perfect circle. I don't exactly understand what your problem is with it, and it's certainly "linear"... What do you mean by time "depending" on "non-linear-esque" quantities? You mean non-integers?</p>
<p>Numbers look weird because we evolved with ten fingers and thus calculate our numbers in base 10, which is a handy enough system but doesn't represent many things in the natural universe well. So we need things like natural logs and pi that are messy - they're all human creations that follow human rules, though, so it's not like there's anything "unnatural" about them.</p>
<p>Sorry, I misread irrational for imaginary. But just for the record, I believe in the "existence" of both of them (in the same way that natural numbers "exist").</p>
<p>On an abstract level I don't see how the number 2 is existentially different from the number pi or the square root of 2. We just picked a random base for our number system (10) and it happened to turn out that way. We might have well picked another number (say 1.5 or e or radical 10). Why does it make a difference whether the "exact" answer according to our formulas is e^2 or 10? Why does it make a difference how "weird" the "exact" answer looks like?</p>
<p>Would you ming explaining what you mean with "linear-esque"?
Why do you think that time is "linear-esque" and irrational numbers are not ?</p>
<p>The difference between labeling and not labeling the balls in the vase reminds me of Schroedinger's cat. Why does it make a difference whether I look into the box or not?</p>
<p>The issue with objecting to numbers like sqrt(2) is that such arguments would apply equally well to integers as well. Negative numbers, where do those come from? Or even, where does "2" itself come from? "2" is an abstract concept to which we assign a particular symbol and apply to other things. What is "2" besides just a mathematical definition?</p>
<p>In fact, going back to CS, consider the class of "definable" numbers, that is, those numbers that we can map one-to-one into a sets of arbitrary symbols. Since that's still countable (using any reasonable interpretation) we see that the vast majority of real numbers are not "definable" by any reasonable concept of definition, and I can certainly understand an objection to non-definable numbers moreso than an objection to irrational numbers...</p>
<p>
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The difference between labeling and not labeling the balls in the vase reminds me of Schroedinger's cat. Why does it make a difference whether I look into the box or not?
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</p>
<p>It doesn't. In reality, the Schroedinger's cat experiment turns out exactly as you might expect - either the cat is alive or not - and there's never a "superposition", and doesn't really matter if you look or not.</p>
<p>There's two parts to it: first is a misinterpretation of what physicists mean by "observation" or "measurement", which work in the "schroedinger's cat" sense when applied to, say, electrons, but don't work at all when applied to cats. Second is that Schroedinger himself didn't think that's what actually happened, and just came up with the Schroedinger's cat experiment to show what absurd conclusions could be drawn from quantum theory (since then, more modern quantum theories of decoherence can explain why it applies to electrons and not cats)</p>
<p>CS - You can enumerate all words over an alphabet. (infinitely many words)
I read Hegel which basically says infinity is a qualitative concept which basically means 2 is no closer to it than 1. I imagined the set as existing but that the time (infinite in quantity) as not really existing and so I never really go to accepting this. Sure you can say at time x I can enumerate that word, but we experience time in now, in finite magnitude, where as before I imagined the set as already existing.</p>
<p>Although I don't have time to respond fully to all of your insightful posts, I will say this.</p>
<p>I do believe in the technical <em>existence</em> of PI, e, sqrt(2), etc. Why not? I even believe that the irrationals without ready-made names, and even the non-computable numbers have some <em>reality</em>. Still, I'm not sure that these sorts of numbers can be appropriately applied to the real world... and it's not an issue of practicality, but of theoretical possibility.</p>
<p>For example: I don't believe that <em>perfect</em> right triangles can exist unless their sides form a Pythagorean triple. I don't believe it is possible to make a <em>perfect</em> square, at all. I don't even think that as an object is travelling through space at a constant velocity (along the X axis) that its center of mass ever occupies the point x=sqrt(2), x=PI, x=e, etc. at any instant in time. Now, practically this is moot, since we can't measure it anyway. But I do wonder if the limit is our ability to perceive things, or whether things are actually intrinsically limited.</p>
<p><em>when I say perfect, warning lights should go off. Nothing real is *perfect</em>. I hope that my argument still sort of makes sense even with this nonsensical liberty.</p>
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I don't believe that <em>perfect</em> right triangles can exist unless their sides form a Pythagorean triple.
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<p>Let me rephrase that: you only believe in right triangles with sides of integer length but not in right triangles with sides of rational or irrational length? Would you mind explaining that??? (e.g. you believe that the triangle 3-4-5 is a right triangle but not 1.5-2-2.5?)</p>