tag:blogger.com,1999:blog-8666091.post7749511801785392098..comments2019-08-16T18:26:26.975+02:00Comments on The Reference Frame: Cardinality of bases doesn't matter for Hilbert spacesLuboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.comBlogger35125tag:blogger.com,1999:blog-8666091.post-74881414807279702942014-02-13T12:14:26.261+01:002014-02-13T12:14:26.261+01:00Thanks for your reply. Ok, let's start from Ca...Thanks for your reply. Ok, let's start from Cantor theorem, agreed. Now the fact that e.g. computer files are countable does not imply that (for instance) the halting problem is unphysical... Or in other words we can think that the "physical numbers" or the "physical Hilbert space" are separable and that's it, why should we apply Cantor's diagonal tool here ? The tool is valid (like any other math theorem), just that usage seems unphysical in that specific context.Giulionoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-36557689542806178342014-02-13T09:00:53.194+01:002014-02-13T09:00:53.194+01:00Once your ivory tower ego has recovered, please fe...Once your ivory tower ego has recovered, please feel free to substantiate your claims ;).Eelco Hoogendoornnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-81120159593306127302014-02-13T08:54:29.613+01:002014-02-13T08:54:29.613+01:00"Whether this is “physical” or not is not my ..."Whether this is “physical” or not is not my concern or that of almost any mathematician."<br /><br />This is exactly the ivory tower mentality which is at the root of this discord. Synthetic knowledge without analytical insight is just plain data; but analytical knowledge without a connection to the body of our synthetic knowledge, is devoid of meaning. Quine 101.<br /><br />A fiercely disagree with the epistemology of people like Hilbert whom feel mathematics is just a game with symbols. Sure, one can always ask 'what is the consequences of these axioms in the abstract', but as a physicist, at some point you want to know which axioms are 'true'; axioms which support downstream conclusions which are physical, and which best integrate with your body of synthetic knowledge.<br /><br />If you want to rely on a bunch of guys navelgazing inside an ivory tower for you to find out which axioms are true, you are doing yourself a big disservice as a physicist.Eelco Hoogendoornnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-56350578620915271252014-02-13T08:52:43.617+01:002014-02-13T08:52:43.617+01:00No, Giulio, you're wrong. All the insights abo...No, Giulio, you're wrong. All the insights about Gödel theorems and similar things would be impossible without the basic observations and arguments by Cantor, and already this starting point is unphysical.<br /><br /><br />Gödel's theorems are surely different from Russell's paradox etc. but the reason why both of these things are unphysical has nothing whatever to do with the minor technical differences between these mathematical results which is why these technical features of the individual theorems would be entirely off-topic in a discussion of their unphysical character.Luboš Motlhttp://motls.blogspot.com/noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-51403817253288056032014-02-13T07:43:25.968+01:002014-02-13T07:43:25.968+01:00Your article is saying nothing about Gödel's t...Your article is saying nothing about Gödel's theorem (it is different from Cantor's one), so your claim of its irrelevance for physics is completely unjustified. The core part of your post is about cardinality of Hilbert space (nothing to do with Gödel). Even here the fact is that most physics is based on separable Hilbert space and, that holds true beside the use of Lebesgue integrability as an extension of Riemann ... So a lot of nice unrelated things put togetherGiulionoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-50026476753063759822014-02-13T03:33:41.730+01:002014-02-13T03:33:41.730+01:00Hmmm, we agree yet again :)
Eelco is as confused a...Hmmm, we agree yet again :)<br />Eelco is as confused as the straw man formalist ivory tower Laputan mathematicians he seems to resent so much.Gordonnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-83992836972050634972014-02-13T02:01:54.166+01:002014-02-13T02:01:54.166+01:00Hahaha - about a meager diet of maths making one b...Hahaha - about a meager diet of maths making one believe in witches, ESPecially evil ones <br />%-DPeter F.noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-9204806025654308542014-02-13T01:58:06.030+01:002014-02-13T01:58:06.030+01:00I generally try to avoid commenting on mathematica...I generally try to avoid commenting on mathematical issues here, but as this comment seems to me a harmful piece of misinformation( and matters are not improved by the author calling himself a mathematician) I have decided to make make an exception to this rule. <br /><br /><br /><br />Whether the Banach-Tarski paradox is directly relevant to physics or not matters not one iota to mathematicians; the important thing is that the paradox had a huge impact on mathematics (and some of that has relevance to physics). Among other things, the paradox had a big impact on group theory leading von Neumann to introduce the concept of amenable groups (http://en.wikipedia.org/wiki/Amenable_group ), which are basically groups that do not admit analogous paradoxes. In particular, the groups of isometries of R^1 and R^2 are amenable, so there are no paradoxes in these dimensions. The group of isometries of R^3 is not amenable, of course. <br /><br /><br /><br />The Banach-Tarski paradox has been used to justify the rejection of the axiom of choice, since indeed it can be shown that without it non-measurable sets do not arise. Yet, today there are practically no mathematicians who reject the AC. The reason is that AC allows one to prove many theorems that are very had to prove without it. Many of them do not require the AC and in many cases it is not known if the AC is actually needed, or whether a proof not using the AC has simply not been found yet. In particular, when Banach and Tarski obtained their paradox, they also proved that it could not occur in R^1 and R^2. Their original proof of this fact (in 1924) used the AC and for quite a long time it was not known whether the non-existence of the paradox in R^1 and R^2 could be proved without AC. Finally such a proof was found by Morse in 1949. So now it is known that measure-theoretic paradoxes do not arise without AC. But this does not mean that other, similar paradoxes do not. First of all, there is the very simple and purely constructive Sierpiński-Mazurkiewicz paradox:<br /><br /><br /><br />http://www.math.hmc.edu/funfacts/ffiles/30001.1-2-8.shtml<br /><br /><br /><br />This involves countable sets so there is no “measure theoretic” paradox, but there is a paradox nevertheless. Another example, is even more interesting because you can actually see it in action: <br /><br /><br /><br />http://demonstrations.wolfram.com/TheBanachTarskiParadox/<br /><br /><br /><br />(You need to install the free Wolfram CDF Player to view this). This time the paradox takes place in the hyperbolic plane (unit disc), the Axiom of Choice plays no role and only Borel sets are needed. Since the hyperbolic plane has infinite measure there is no “measure theoretic” paradox (we simply see that 2 infinity = 3 infinity = infinity) but certainly it does look paradoxical to our eyes. Whether this is “physical” or not is not my concern or that of almost any mathematician. As for the reference to "bullshit", I see it as a kind of boomerang that missed the target and hit the thrower.lucretiusnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-71500571304184254812014-02-13T01:38:09.102+01:002014-02-13T01:38:09.102+01:00Thanks, Lumo, for gifting me a personally new and ...Thanks, Lumo, for gifting me a personally new and satisfying - and not even completely intuitive :> - insight about the qualitative difference between maths and physics.Peter F.noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-6751504998400537522014-02-13T01:24:15.938+01:002014-02-13T01:24:15.938+01:00I am not a strict finitist at all; I use 'real...I am not a strict finitist at all; I use 'real' (funny name given no one has ever observed them) numbers and related computational hacks in my work on a daily basis.<br /><br />What I reject is the notion that someone has given a satisfactory and non-contradictory axiomatization of these concepts.<br /><br />David is entitled to his opinions, but these are my observations:<br /><br />1: Infinity is not an internally consistent concept. Chopping away at your axioms without regard for what your terms mean until they appear to be consistent, should abhor a physicist (and the epistemology of guys like cantor and Hilbert should make them sick to their stomachs, generally). All physicists not thinking about how to perform experiments to double spheres, implicitly agree.<br />2: This does not mean infinity cant be a useful tool. What it does mean is that explosive logics are long overdue for being banished to history books. No logic with well defined semantics for symbol lookup is explosive. Infinity may not be consistent; but so what?<br />3: Its lack of internal consistency may be of little pragmatic concern, but makes my inner physicist a little skeptical that it should be involved in a fundamental description of reality. But that's just my metaphysical preconceptions, so feel free to agree to disagree.Eelco Hoogendoornnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-60202882593960091092014-02-13T00:28:49.622+01:002014-02-13T00:28:49.622+01:00Just why would it be "nice" if we could ...Just why would it be "nice" if we could do away with infinity, or the axiom of infinity?<br />Are you a strict or ultra-finitist? Do you reject real numbers or the Cauchy and Dedekind constrution of them? Induction?<br />Why constrain math to be less than what it is?<br />" No one shall expel us from the paradise that Cantor has created for us."--David HilbertGordonnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-72210182409695583562014-02-12T19:25:32.640+01:002014-02-12T19:25:32.640+01:00Indeed mathematicians (I am one myself) mean thing...Indeed mathematicians (I am one myself) mean things in a particular way; but that does not mean you shouldn't call them out on their bullshit.<br /><br />An infinitude of points being unable to span a line, but an infinitude of lines being able to span a plane, is in no way or shape a profound insight; it is a spurious consequence of ivory tower mathematicians picking the axioms they feel are most elegant, rather than those axioms which best integrate with out body of synthetic knowledge.<br /><br />One may dismiss things like the Tarski paradox as curiosities with no relevance to any experiment; but to have to do so is a sad state of affairs. Ideally, our abstract reasoning should allow us to generalize to previously unobserved situations, and allow us to correctly predict them. That we don't go around looking for actual spheres to cut up and reassemble into two identical copies, is an indication that something is subtly yet horribly broken. Ideally, we would want our deductive results to be both true AND meaningful. Mächtigkeit my ass, Cantor. Stop avoiding the question: how big is that set?Eelco Hoogendoornnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-35454689205737463152014-02-12T16:00:36.642+01:002014-02-12T16:00:36.642+01:00Correct.
2^ℵ₀ is the cardinality of the continuum....Correct.<br />2^ℵ₀ is the cardinality of the continuum. It is the smallest cardinality<br /> that ZFC can prove to be bigger than ℵ₀, but ZFC can't prove that <br />there are none in between either. The CH is that there isn't, and <br />therefore ℵ₁ = 2^ℵ₀Ralph Hartleynoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-42391065831553980702014-02-12T15:40:47.483+01:002014-02-12T15:40:47.483+01:00The mathematical insight by Gödel has no relations...The mathematical insight by Gödel has no relationship to the Heisenberg uncertainty principle and none of the two imply that the laws of Nature cannot be pinpointed precisely, anyway.<br /><br />The second statement is correct, but there is a sense in which the incompleteness theorem is similar to the uncertainty principle.<br /><br />Once upon a time, physicists believed that they "physics is about what is out there", but the uncertainty principle made it clear that "physics is about what we can say", which is different (as you explained very clearly in the post immediately before this one).<br /><br />Mathematicians once believed something like "mathematics is about what is necessarily true", but the Gödel showed that "Mathematics is about what you can prove", which is different.<br /><br /><br />Does that mean that the incompleteness theorem is relevant to physics? No, of course not. Is it now? No. Will it ever be? Damned if I know.Ralph Hartleynoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-71824229799978543042014-02-12T15:38:53.427+01:002014-02-12T15:38:53.427+01:00Alejandro,
Astute point. Constructivist mathemat...Alejandro,<br /><br />Astute point. Constructivist mathematicians usually put the number of theorems relying on the axiom of choice (or its equivalents Zorn’s lemma, Tukey’s lemma, etc.), either directly or indirectly, in a subject like operators on a Hilbert space at about 50%. A foundation stone of functional analysis is the Hahn-Banach theorem and it relies on Zorn’s lemma, and the existence of a Hilbert space orthogonal basis in cases involving a continuous spectrum likely does also.Tomnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-83267477748292725602014-02-12T15:09:41.512+01:002014-02-12T15:09:41.512+01:00"morally correct" = wrong.
Yes, for *mo..."morally correct" = wrong.<br /><br />Yes, for *most* purposes the "position basis" works just like a basis, but determining the dimensionality of the Hilbert space is not one of them. For a "true" basis each basis vector corresponds to a degree of freedom, so the dimension of the space equals the size of the basis. But in an L2 space, individual points don't "count". Functions that differ on a set of measure zero are identified. It takes (uncountably)infinitely many "basis" vectors to make a degree of freedom. So it shouldn't be surprising that the dimension of the space is less than the number of points.<br /><br />There *are* continuous maps from R to R^n (they need not be one-to-one to show that the number of points is the same because there are also continuous maps from R^n to R). Space filling curves are not "heavily discontinuous" (whatever that means, being continuous is like being pregnant). Of course the functions are not smooth, but physics is full of non-smooth functions (Brownian motion, most of the paths in a path integral). <br /><br />You may not feel like you need to make these "fine" distinctions (and perhaps the universe doesn't either), but the *mathematicians* that figured out the different ways that divergent sums can have values, including 1+2+3+4+5+...=1/12, most certainly did.Ralph Hartleynoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-1769952077596749222014-02-12T07:17:58.621+01:002014-02-12T07:17:58.621+01:00Hello fellow blogger! Your posts are amazing; I lo...Hello fellow blogger! Your posts are amazing; I love reading them. You are such a great writer, and character of the places you visit so well. Keep up the great work! You have surprised after visit this site -<br />http://www.bookfari.com/Category/10/Education-and-ReferenceAakashnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-79207723271574758452014-02-12T05:36:13.134+01:002014-02-12T05:36:13.134+01:00O.K. this post is sociopolitical. I had a question...O.K. this post is sociopolitical. I had a question today, though. Is there beauty in Maxwell's equations? Is there an intuitive opera of delight within them?<br /><br /><br />Or...what?<br /><br /><br />Or...math?<br /><br /><br />Spirals in time, I figure.<br /><br /><br />Coupled to what?<br /><br /><br />Redefinition, here and there?<br /><br /><br />Are there really vortexes?<br /><br /><br />What is free space?<br /><br /><br />How can a wire carry ten horses, or a thousand horses along its bendable length without utterly exploding?<br /><br /> <br />Is it because human scale energy intuition is but a drop in a bucket?<br /><br /><br />I mean how *can* ten thousand watts just casually FLOW though a few wires in the air?<br /><br /><br />How can the wire not explode?<br /><br /><br />How can so much "human" scale energy spit out of a wire? <br /><br /><br /><br />-=Nik=-NikFromNYCnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-79819553492376348602014-02-12T03:38:28.253+01:002014-02-12T03:38:28.253+01:00About set theory, I am not sure about how far can ...About set theory, I am not sure about how far can Hilbert Spaces go without Zorn's lemma.Alejandro Riveronoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-26666127986324165652014-02-12T01:11:00.230+01:002014-02-12T01:11:00.230+01:00Cynholt,
I have been saying exactly, down to the ...Cynholt,<br /><br />I have been saying exactly, down to the wording, what you say in your first two sentences for many years. Unfortunately, hoping the Obama administration will do anything coherent or useful is on par with wishing for visits from the tooth fairy.<br /><br />Yeah, neocons and their wars suck.Tomnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-89221252278340969502014-02-12T00:59:46.002+01:002014-02-12T00:59:46.002+01:00Robinson,
If you are not familiar with Eugene Wig...Robinson,<br /><br />If you are not familiar with Eugene Wigner’s thoughts in this area, you will likely find them interesting: "The Unreasonable Effectiveness of Mathematics in the Natural Sciences"<br /><br />http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.htmlTomnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-10586059488763834242014-02-12T00:16:39.529+01:002014-02-12T00:16:39.529+01:00Short remark, Lubos:
I think in
\int_{-\infty}^{...Short remark, Lubos:<br /><br />I think in<br /><br />\int_{-\infty}^{+\infty} \dd x\,\varphi(x)\ket{x_0}<br /><br />\ket{x_0} should really be \ket{x}Mikaelnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-28170302478344128472014-02-11T22:06:53.208+01:002014-02-11T22:06:53.208+01:00Of course when mathematicians say that R^2 and R a...Of course when mathematicians say that R^2 and R are "equally big" they mean it in a particular sense which is not relevant for physical considerations. But the fact that R^2 is actually "bigger" than R is also well understood by mathematicians. You just need to appreciate that R^2 and R carry a natural topology, which makes it impossible to fit on inside the other. I feel like some of your critiques are like this. Mathematicians study many structures, but only a subset are relevant for thinking about the physical world, hardly a surprise.<br /><br />That being said, there are some results in mathematics, especially concerned with infinite sets, <br />which make you wonder if mathematicians have forgotten what it means to make a meaningful statement, even about mathematics (much less physics). My favorite example of this is the so-called Hamel basis for Hilbert spaces. Take for example the Hilbert space of wavefunctions in an infinite potential well. Or if you like, the space of square integrable functions on a line segment. There is a theorem which says that the Hilbert space has a basis, called a "Hamel basis," such that every wavefunction can be written as a superposition of a finite(!) number of basis elements. Stop to think about how implausible that sounds. At least to me it seems obvious that there is no such basis, and mathematicians have lost track of what it means for an object to exist.TEnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-33488889094542233042014-02-11T20:59:57.117+01:002014-02-11T20:59:57.117+01:00Glad to see you treating confused formalist ivory ...Glad to see you treating confused formalist ivory tower mathematicians for what they are; people whom we shouldn't expect to have anything profound to say about physics; or how the mathematics they use should be structured.<br /><br />Now when you realize that there is nothing profound about the diagonal argument, but that this is simply a direct result of the (completely unmotivated) choice to give one-to-one mapping axiomatic status, over the equally begging-to-be-axiomatized fact that things which are constructed from eachother by adding or substracting stuff, cant be equally big in any meaningful sense of the word (hello tarski paradox and related unphysical downstream conclusions of denying axiomatic status to this fact), you are well on your way to realizing the logical ugliness of infinity and the continuum.<br /><br />I know you don't like to hear it, as you feel it eats away at the justification for your beloved string theory, but I will remind you anyway: you have never seen an infinity, and you will never see or otherwise demonstrate the infinite divisibility of anything. That does not disprove the concept any more than it disproves russels teapot, but this is something to pause on as a supposed empiricist. It would sure be conceptually nice if we could do entirely without infinity, strings or not. <br /><br />Infinity-crusaders should do well to read up on the attitudes of late 19th century physicists whom got famous working on continuum mechanics, and who were sure that this whole atomism bullshit was a stupid fad. Surely, their empirically impeccably validated equations on the propagation of shock waves and such were not just approximations to an underlying discrete mechanism? Burn the heretics!Eelco Hoogendoornnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-25166961271535340272014-02-11T20:55:46.732+01:002014-02-11T20:55:46.732+01:00Great article, Lubos.
"mathematicians and t...Great article, Lubos.<br /><br /><br />"mathematicians and their collaborationists..." *LOL*Mikaelnoreply@blogger.com