Wikipedia:Reference desk/Archives/Mathematics/2010 November 12

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November 12

Riemann and Spivak

How do you pronounce these names (IPA please; since you're mathematicians you might not be familiar with it, here's a handy chart WP:IPA)? I know the Riemann article has a pronounciation but it is the German pronounciation, I want to know how it would be pronounced in a university/classroom setting. As for Spivak, is it ['spiː.væk], or ['spiː.vak] or something else? —Preceding unsigned comment added by 24.92.78.167 (talk) 01:10, 12 November 2010 (UTC)

I pronounce Riemann [ˈɹiːmɑːn], I think. I'm not too sure about the differences between those various A symbols in IPA. It's the same vowel I use in "father." (I speak with a Nebraska accent, though, which seems to have fewer differentiated vowels than some accents; for example, "cat" and "can't" have the same vowel for me, as do "caught" and "cot." But it's definitely not an [æ] or an [ə].) —Bkell (talk) 04:21, 12 November 2010 (UTC)

You might try the language reference desk. —Anonymous Dissident^Talk 09:44, 12 November 2010 (UTC)

I say [ˈriːmən] and [ˈspɪvæk]. Algebraist 10:33, 12 November 2010 (UTC)

I've always prounounced Riemann with an "ah" sound in the unstressed second syllable, rather than a schwa, and the first syllable rhyming with "be" as in "To be or not to be....". Pretty much the German pronunciation except that in German one might pronounce the initial "R" differently, and that difference is not essential to either language. Michael Hardy (talk) 16:54, 12 November 2010 (UTC)

I agree with Michael Hardy for Riemann, which is [ˈɹiːmaːn] as I understand IPA. I say [ˈspɪvæk] as Algebraist. I have not heard much variation on these from native speakers of English specializing in math at American universities. SemanticMantis (talk) 17:02, 12 November 2010 (UTC)

Mortality Rates India

Where can I obtain the mortality rates (or table) for India? —Preceding unsigned comment added by Aneelr (talk • contribs) 03:22, 12 November 2010 (UTC)

http://www.unicef.org/infobycountry/india_statistics.html Ginger Conspiracy (talk) 23:47, 16 November 2010 (UTC)

Godel

If we consider Physics to be an axiomized system, what would the implications of Godel's incompleteness theorem be? 76.68.247.201 (talk) 06:14, 12 November 2010 (UTC)

There's a story about Lincoln. Supposedly he once posed the question, "If we consider a tail to be a leg, how many legs has a dog?". The answer was "four", because considering a tail to be a leg does not make it one.

Physics is not an axiomatic system. --Trovatore (talk) 06:16, 12 November 2010 (UTC)

Didn't von Neumann make quantum mechanics axiomatic? 76.68.247.201 (talk) 06:52, 12 November 2010 (UTC)

I don't know just what he axiomatized. But I'm quite certain it wasn't all of physics. I don't think they even knew about, say, gluons when von Neumann was alive. --Trovatore (talk) 10:10, 12 November 2010 (UTC)

I think it is a good question, (meaning that I do not know the answer). Physics can be axiomized. A theory of everything is supposedly a set of axioms for all of physics. Newton's laws constitute axioms for celestial mechanics. Euclid's postulates constitute axioms for geometry, considered a physical theory of space. Bo Jacoby (talk) 09:39, 12 November 2010 (UTC).

I agree with Trovatore. Physics is an empirical discipline. Not matter how elegant the underlying mathematics / axioms are, if the theory predicts one thing different from experiments, we must discard the theory. Money is tight (talk) 10:07, 12 November 2010 (UTC)

I also agree with Troavtore. You can develop an axiomatic model of some part of the real world, but that is not the same as axiomatising physics, which is an empirical science. When observations or experimental results conflict with the deductions from your model, then it is time to find a new model. Newtonian gravity is a model that works pretty well, but does not completely account for the motion of Mercury; gravitational lenses show that space is not exactly Euclidean etc. Gandalf61 (talk) 10:59, 12 November 2010 (UTC)

Now, it should be said that some pretty respectable people (e.g. Hawking) have made the argument that the Goedel theorems exclude a theory of everything. I think it's a pretty silly claim myself, or rather, the sort of "theory of everything" that it excludes is the sort that no one should ever have thought might exist in the first place. For my money, a theory counts as a "theory of everything" in a physics context if it completely describes all elementary physical interactions. The possibility that you might be able to encode, in terms of those interactions, a mathematical question to which the theory might not give you an answer, is totally irrelevant.

To put it another way, if the theory describes all physical reality once given an oracle for mathematical truth, that's more than enough to make it a physics "theory of everything". --Trovatore (talk) 10:16, 12 November 2010 (UTC)

Obviously Hawking does not agree with Trovatore that Physics is not an axiomatic system. Bo Jacoby (talk) 12:31, 12 November 2010 (UTC).

What Hawking says in his lecture Gödel and the end of physics is:

"Up to now, most people have implicitly assumed that there is an ultimate theory that we will eventually discover. Indeed, I myself have suggested we might find it quite soon. However, M-theory has made me wonder if this is true. Maybe it is not possible to formulate the theory of the universe in a finite number of statements. This is very reminiscent of Godel's theorem. This says that any finite system of axioms is not sufficient to prove every result in mathematics."

Gandalf61 (talk) 13:17, 12 November 2010 (UTC)

Not to be an ass, but reality seems to have no probelm with Godel's theorems (I am being metaphorical, although there is merit to the statement about the universe computing...) More to the point though, theories in physics are not like theories in mathematics; Suppose we want to know if "x's can ever become y's", if our theory can't answer this, then we observe nature: if we find x's becoming y's, we add it to our theory; if we don't see this, then either some x will become a y at some future point, or not. If one will, eventually, then the theory with "x's can become y's, sometimes." will be stronger, thus, making our orginial not a theory of everything. On the other hand, if it never happens, then our theory isn't a theory of everything, since adding "no x's become y's" is stronger. The real question is if the list of all physically relevant such statements is infinite, or not. Finally, on an aside, if I have a theory in which X is undecidable, but X is not a physically verifiable part of the theory, does this mean anything physically? And, are we talking about theories with physical axioms, or mathematical models of them (which to me seem to be more akin to the inner(hidden) workings of a computer program, not mathematical theories, since they are built to calculate results.) —Preceding unsigned comment added by 71.61.7.220 (talk) 19:20, 12 November 2010 (UTC)

Okay, I`m not sure people are understanding my question, although I could just be misunderstanding your posts. First, I don`t see why you couldn`t axiomatize physics. Obviously physics is based on empirical results, but these empirical results would ultimately form the basis of the axioms which govern the mathematics of physics, no? At any rate, as I mentioned earlier, von Neuman was able to make axioms which could completely describe quantum mechanics, and I don't see why I would be unreasonable to suppose that quantum field theory + quantum gravity (or whatever is the eventual Theory of Everything) could be treated similarly.

So if physics is eventually axiomatized, what would it mean to say that not every result can be proved by the axioms? Would that mean that for such results we could defer to experiment, or would it be even outside the range of experimental testability? 76.68.247.201 (talk) 20:50, 12 November 2010 (UTC)

Put it this way. Take any collection of silicon atoms, at least four, that can be divided into two pieces with an equal number of atoms. Can it necessarily be divided into two groups, neither of which can be arranged into a rectangular grid (no matter the spacing between the grid points, to avoid questions about gravitation and such) in which both sides have at least two atoms?

This is the Goldbach conjecture (which is true, of course, but no one knows a proof). If you think it's legitimately a question about silicon atoms, which a true theory of everything ought to answer, then you have a case that the Goedel theorems have something to say about theories of everything.

But I think that's nonsense; it's not a question about silicon atoms at all. You could know everything about how silicon atoms interact, and still not know how to answer it, because it's really a mathematical question, not a physical one. --Trovatore (talk) 21:12, 12 November 2010 (UTC)

I'm sorry, I don't follow. If an axiom of physics is that counting objects can always be represented by natural numbers, then doesn't Goldbach's conjecture (assuming it's true) follow directly? 76.68.247.201 (talk) 21:53, 12 November 2010 (UTC)

This doesn't seem terribly complex to me... perhaps I'm oversimplifying. Accepted physical theories currently take the form of mathematical models (as opposed to using, say, metaphysical constructs). Mathematical models by definition are axiomatizable with rigid rules of inference. This is not always done for physical theories, but it must in principle be possible, otherwise math simply doesn't apply. One key is that your rules of inference must be strictly specified: how do you translate from experiments to proofs? If your theory's experimental proof axioms are strong enough, they may not be recursively enumerable and the incompleteness theorem may not apply. If the theorem does apply, the experimental proof axioms won't prove everything, so that for a given true statement, sometimes you won't be able to prove it by experiment. As a trivial example, I could take the axioms "if ice cream exists, then the Goldbach conjecture is true" and "any human can judge the existence of physical objects", in which case this theory proves the Goldbach conjecture since I say ice cream exists. (More axioms are needed and this system is almost certainly inconsistent, but that's not the point.) A complete theory of everything would be a mathematical model for every interaction in the observable universe, together with rules of inference governing the correctness of experimental proofs. Depending on if these rules of inference meet Godel's requirements, you may not be able to experimentally prove some statements--or the (assumed correct) theory of everything, and hence the universe, is fundamentally inconsistent. 67.158.43.41 (talk) 23:33, 12 November 2010 (UTC)

I'm sorry, I can't make enough sense of this to respond to it in detail. I'll just call out a few points:

This is not always done for physical theories, but it must in principle be possible, otherwise math simply doesn't apply.

That's an extraordinary statement. Can you back it up in any way? I'm afraid I think you're simply wrong here.

The rest of it uses a lot of words in ways that do not appear to be their standard meanings. It is not clear at all, for example, what an "experimental proof" is, or its rules of inference. --Trovatore (talk) 08:06, 13 November 2010 (UTC)

By "experimental proof", I meant taking the results of physically performing an experiment and turning them into a formal proof of a mathematical result. For instance, take a loop of string and make it approximately into a circle. Measure the length of the string and the radius of the circle. With the right axioms, this experiment rigorously proves bounds on the value of pi. The application of axioms to physical results which proves these bounds is what I meant by "experimental proof". I'm sorry I wasn't clearer. My point is that experimental results (even hypothetical experiments) must be interpreted in a formal manner to be a rigorous proof of any mathematical result. The rules of the translation from experiment to proof govern what is experimentally provable, even in principle, in a given physical theory.

I'm not sure what you mean by "backing up" the statement you quoted. In what way is it incorrect? Perhaps our disagreement lies in the definitions of "math", "physical theory", and "theory of everything" which are admittedly quite vague.

Overall, my point was to address the original question, "If we consider Physics to be an axiomized system, what would the implications of Godel's incompleteness theorem be?" by saying "if your axiomatized system's experiments can be translated to mathematical statements using axioms to which the incompleteness theorem applies, some statements won't be experimentally provable." I also believe the "if" in the original question is unnecessary in that physics must be in principle an axiomatized system, though it's interesting that this part of the discussion wasn't part of the original question. 67.158.43.41 (talk) 01:46, 14 November 2010 (UTC)

None of the above makes enough sense for me to address it. I think you have some very fundamental misconceptions about the nature of proof and formal systems. As to the claim that "physics must be in principle an axiomatized system", I think you're just wrong. --Trovatore (talk) 02:50, 14 November 2010 (UTC)

You aren't specific enough for me to get anything out of this comment. For instance, you simply repeat "I think you're just wrong". Ah well, I suppose; it's not worth arguing, to me. 67.158.43.41 (talk) 19:53, 14 November 2010 (UTC)

You made the original statement, so the burden is on you. You haven't supported it in any way. For that matter you haven't made it clear what you even mean. --Trovatore (talk) 20:12, 14 November 2010 (UTC)

Does the finiteness of the universe come into play? All the examples in List of statements undecidable in ZFC seem to involve infinite sets. If we restrict ourselves to a universe which only has a finite number of particles, in a finite volume which has only existed for a finite time might not all these problems simply go away?--Salix (talk): 07:53, 13 November 2010 (UTC)

Well, first of all, it is not known whether the universe is finite or infinite. However even if it is finite, while that would (probably) mean that in principle there are only finitely many statements that can be coded up in the physical universe we need to be more specific about what sort of coding we have in mind! I'm uncomfortable putting things this way but let's leave that aside for the moment, and that therefore some theoretical being living outside the universe could in principle make a list of all the ones that are true and axiomatize that list finitely (the axioms could simply be the list itself), there is no reason to think that anything like this could be done inside the physical universe.

I want to be clear that I am not saying there is a correct theory of everything. I am saying only that the Goedel theorems per se have very little to say about the question. --Trovatore (talk) 08:06, 13 November 2010 (UTC)

The OP did not require 'physics' to be some theory of everything, but perhaps rather any axiomatic theory having a physical interpretation, such as Hilbert's axioms for Euclidean geometry. It is interesting to know if there are undecidable propositions in euclidean geometry. Bo Jacoby (talk) 12:58, 13 November 2010 (UTC).

The problem with all of this is that the "axioms" of a physical theory seem to be more about a correspondance between physics and mathematics. For example, saying quantum systems can be modeled by hilbert spaces is not a mathematical axiom. If something in the mathematical model's theory, say the theory of hilbert spaces, is undecidable, that doesn't meant the physics is. If an experiment did determine something that mathematics couldnt, it would just mean that the physics doesn't perfectly correspond with the mathematics. 24.3.88.182 (talk) 18:38, 13 November 2010 (UTC)

I agree. Nothing says a given physical theory's axioms cannot be stronger than some arbitrary set of mathematical axioms. If my physical theory is inconsistent, it's stronger or equivalent to every mathematical theory trivially. 67.158.43.41 (talk) 01:54, 14 November 2010 (UTC)

Experimental outcomes are not necessarily computable using the postulates of the theory. A well known result is the following. In a universe exactly described by classical mechanics, a computer can be constructed such that executing a clock cycle takes half the time it took the execute the previous clock cycle. Memory space can also be expanded by a factor of two in each cycle. This means that in such a universe, an infinite number of computations can be performed in a finite amount of time. You can then e.g. verify the Riemann hypothesis by brute force verification that all the nontrivial roots are on the real line, regardless of whether or not a proof of this fact exists. Count Iblis (talk) 15:58, 14 November 2010 (UTC)

I would suggest the experimental outcome you describe would in fact be a rigorous proof of the Riemann hypothesis. The axioms applied in the proof would be those of classical mechanics and a correspondence between physical results and the complex numbers. Ignoring axioms translating from physical results to complex numbers (i.e. limiting yourself to pure mathematical reasoning), there may not be a proof, but that's simply because the resulting system would be weaker. 67.158.43.41 (talk) 20:06, 14 November 2010 (UTC)

OK, maybe here's an opening to nail something down and have a discussion with actual content. You do understand that, while people have made various extensions to other things, the Gödel theorems per se apply only to first-order logic? As an extreme example, the original (second-order) Peano axioms are categorical in second-order logic. All models of them (in the sense of full second-order logic) are isomorphic, so they completely determine arithmetical truth (and not just to first order!), and are presumably consistent.

The "proof" you are talking about above does not correspond to a proof in first-order logic. It could maybe be formalized in infinitary logic if you allow the omega-rule. But the Gödel theorems are then not applicable, or at least not in their usual form. --Trovatore (talk) 20:42, 14 November 2010 (UTC)

I never said the Godel theorems were necessarily applicable in the above example. I've been careful not to specify the rules of inference or types of axioms allowed. I believe you've missed almost every point I've made and have no desire to argue with you, so I will stop responding to this thread. 67.158.43.41 (talk) 04:28, 15 November 2010 (UTC)

In other words, you have refused to say what you mean. Please refrain from making contributions where you are not willing to say what you mean. --Trovatore (talk) 04:44, 15 November 2010 (UTC)

Are we talking about mathematics or physics? Something like "Quantum mechanical systems can be modeled using Hilpert spaces in such and such a way." is not an axiom of mathematics, nor is it mathematical. In short, the types of "axioms" that define the correspondance between mathematics and physics are not mathematical axioms, thus, Godels theorem doesn't really apply to them.66.202.66.78 (talk) 07:01, 15 November 2010 (UTC)

Polynomials given by a recurrence relation

Let W_k(x) be a monic polynomial of the degree k satisfying the follwoing recurrence relation:

W_k+1(x) = x W_k(x) +(a k² + b k + c) W_k-1(x) ,

where a, b, and c are real numbers.

My question is: what is known about such polynomials (name, generating function, integral representation, roots, differential equations, etc.)? Obviously, those are none of the well-known orthogonal polynomials.

Thanks in advance!

Physicist —Preceding unsigned comment added by 212.14.57.130 (talk) 11:25, 12 November 2010 (UTC)

Assuming W₋₁=0 (which does not really have degree −1), and W₀=1, the trick is to solve the recurrence relation for k≥1 to get a closed representation for W_k. Study recurrence relation. Bo Jacoby (talk) 16:01, 12 November 2010 (UTC).

With those initial conditions I get the following explicit formula:

$W_{n}(x)=S(0,n-1,2)x^{n}+S(1,n-1,2)x^{n-2}+...+S(k,n-1,2)x^{n-2k}+...$

where

$S(i,n,k)=\sum \prod _{j=1}^{i}A(m_{j})$

with the sum being taken over all $1\leq m_{1}<...<m_{i}\leq n$ satisfying $m_{j+1}-m_{j}\geq k$ and

where S(0, n, k) is taken to be 1. Also,

$A(k)=ak^{2}+bk+c$

In words, S(i, n, k) is the polynomial whose terms are all the monomials formed by evaluating A at i (integer) values and multiplying those evaluations, where those values are at least k apart and are between 1 and n. In the case of n=0, this is just S(0, -1, 2) x^0 = 1. For n=1, this is S(0, 0, 2) x^1 = x. For n=2, this is S(0, 1, 2) x^2 + S(1, 1, 2) x^0 = x^2 + A(1). Less trivially, for n=6, this is

$W_{6}(x)=S(0,5,2)x^{6}+S(1,5,2)x^{4}+S(2,5,2)x^{2}+S(3,5,2)x^{0}$

$=x^{6}+(A(1)+A(2)+A(3)+A(4)+A(5))x^{4}+(A(1)A(3)+A(1)A(4)+A(1)A(5)+A(2)A(4)+A(2)A(5)+A(3)A(5))x^{2}+A(1)A(3)A(5)$

The polynomials S(i, n, k) are similar to the elementary symmetric polynomials. The term S(1, n, k) simplifies in your case significantly; the others might as well. Perhaps someone else knows more. Also, if I were you, I would double-check my algebra. 67.158.43.41 (talk) 17:45, 12 November 2010 (UTC)

vocabulary of curly things

Euler's spiral is the curve in which κ′, the first derivative of curvature (=second derivative of the tangent angle) with respect to arc-length, is constant. Is there a more concise name for κ′?

Another word for that spiral is clothoid. Is this word also applied to other curves in which κ′ is continuous (e.g. polynomial) but not constant? If not, is there a more general word? (I'm experimenting with such curves.) —Tamfang (talk) 20:05, 12 November 2010 (UTC)

Possibly Tortuosity but I don't think there is any well established term.--Salix (talk): 20:53, 12 November 2010 (UTC)

Tortuosity — that's me byword. --Trovatore (talk) 07:36, 14 November 2010 (UTC)

To the second question, [1] has "polynomial spiral". —Tamfang (talk) 22:29, 12 November 2010 (UTC)

In some engineering applications (vibration analysis in particular), strain is equivalent to the first derivative of curvature. Though this is a bit specialised, you could use the word for conciseness, not forgetting to define it if communicating with someone else.→86.132.164.178 (talk) 14:34, 13 November 2010 (UTC)