I just finished auditing a graduate mathematics class at a university that shall remain nameless, but I did not enjoy the experience at all. I have never taken a math class where everything was so pre-digested for me. At the beginning of each section, the professor assigned problems from the book as exercises, standard operating procedure. A few lectures later, he would select a couple of the more challenging ones (translation: the only interesting ones) and provide "hints" so comprehensive that 90% of the proof had been worked out. Before the exam, he handed out a graded homework assignment that contained the same proof again (not a similar proof, the same proof). And then on exam day, the same proof would show up a third time. A shame.
Thursday, November 30, 2006
Monday, October 09, 2006
Fraleigh Exercise 4.39
Prove that a nonempty set G together with a binary operation * on G such that ax=b and ya=b have [unique] solutions for all a and b in G is a group.
(1) The binary operation is associative.
(2) Fix a in G. Then the equation ax=a has a unique solution, call it e. Then ae=a. Now pick an arbitrary b in G. Then the equation ya=b has a unique solution, call it c. Then ca=b. Multiplying ae=a by c on the left, cae=ca. Thus, be=b. Therefore, e is a right identity on G.
(3) Again pick an arbitrary b in G. The equation bx=e has a unique solution, call it b'. Therefore, every element b as a right inverse in G.
This suffices to show that G is a group.
(1) The binary operation is associative.
(2) Fix a in G. Then the equation ax=a has a unique solution, call it e. Then ae=a. Now pick an arbitrary b in G. Then the equation ya=b has a unique solution, call it c. Then ca=b. Multiplying ae=a by c on the left, cae=ca. Thus, be=b. Therefore, e is a right identity on G.
(3) Again pick an arbitrary b in G. The equation bx=e has a unique solution, call it b'. Therefore, every element b as a right inverse in G.
This suffices to show that G is a group.
Tuesday, October 03, 2006
2006 Nobel Prize in Physics
For their discovery of the blackbody form and anisotropy of the cosmic microwave background radiation
Goddard Space Flight Center
Goddard Space Flight Center
Tuesday, September 12, 2006
The Bogdanov Affair
To think that all these years I have tried to master the field of theoretical physics and possibly make a contribution. It turns out that I didn't need to learn anything and could have just made up my contribution. Has the field of physics been reduced to the level of "social criticism" where any idea no matter how nonsensical can be published? How can this happen?
What, you may ask, am I talking about. I'm speaking about the Bogdanov affair. First, let's start with just the facts (from Wikipedia)...
The Bogdanov Affair is an academic dispute regarding a series of theoretical physics papers written by French twin brothers Igor and Grichka Bogdanov (or Bogdanoff). These papers were published in reputable scientific journals, and culminated in a proposed theory for describing what occurred at the Big Bang. While the Bogdanovs defend the veracity of their work, many physicists have alleged that the papers are nonsense, emphasizing the fallibility of the peer review system used to approve papers. The Bogdanovs obtained Ph.D. degrees from the University of Bourgogne; Grichka Bogdanov received his degree in mathematics in 1999, and Igor Bogdanov received his in theoretical physics in 2002. Both were given the low but passing grade of "honorable", in Igor's case only after publishing in respected physics journals to establish the acceptability of his work.
Before moving on to other topics, let's consider briefly the appalling intellectual laziness of allowing journals to establish the "acceptability" of the work. If you don't understand the paper, you have no business recommending the awarding of a doctorate.
Inside Higher Education said it best: "A PhD is a license to reproduce and an obligation to maintain the quality of your intellectual descendants. [...] If you vote to pass someone who is marginal or worse they, in turn, have the same privilege. If they are not up to standard, it is likely that some of their descendants will also not be."
Leaving this shameful aspect of the affair, let's now look at some actual text from the papers in question, courtesy of John Baez's blog...
We consider inertia as a topological field, linked to the topological charge Q = 1 of the "singular zero size gravitational instanton" which can be identified with the initial singularity of space-time in the standard model.
Huh?
Baez: "The paper goes on to discuss the supposed connection between N = 2 supergravity, Donaldson theory, KMS states and the Foucault pendulum experiment, which he claims 'cannot be explained satisfactorily in either classical or relativistic mechanics.' If you know some physics you'll find this statement odd. The Foucault pendulum behaves exactly the way classical mechanics predicts: it is a standard textbook exercise."
We draw from the above that whatever the orientation, the plane of oscillation of Foucault's pendulum is necessarily aligned with the initial singularity marking the origin of physical space S3, that of Euclidean space E4, and, finally, that of Lorentzian space-time M4.
Baez: "How in the world could the plane of oscillation of a Foucault's pendulum be 'aligned with the initial singularity', i.e. the big bang?"
We simply suggest that at 0 scale, the observables must be replaced by the homology cycles in the moduli space of gravitational instantons. We then get a deep correspondence - a symmetry of duality - between physical theory and topological field theory.
Baez: "(1) Could you please define "at 0 scale"? (2) "Observables" in which theory? (3) You say "must be replaced". Why? How? (4) You speak of a "deep correspondence" between some unspecified physical theory and some unspecified topological field theory. Which theories are you talking about here? How does the correspondence go?"
Lee Smolin: "They were at best wrong, and most likely just throwing around words with no calculations or proofs to back them up."
Jacques Distler: "The papers consist of buzzwords from various fields of mathematical physics, string theory and quantum gravity, strung together into syntactically correct, but semantically meaningless, prose."
Eli Hawkins: "This paper is built around the idea that 'at the Planck scale, the space-time system is in a themodynamical equilibrium state.' It is not quite clear what the author means by this. He may mean that when the matter is at the Planck temperature, it is in thermodymanic equilibrium with the geometry. He does not explain why there should not be thermal equilibrium at all temperatures. It may be simply that the author does not know what he is talking about. The main result of this paper is that this thermodynamic equilibrium should be a KMS state. This almost goes without saying; for a quantum system, the KMS condition is just the concrete definition of thermodynamic equilibrium. The hard part is identifying the quantum system to which the condition should be applied. It is difficult to describe what is wrong in Section 4, since almost nothing is right. The author seems to believe that just because an analytic continuation of a function exists, the argument must be considered a complex number. He also makes the rather obvious claims that complex numbers should be the sums of real and imaginary parts. The remainder of the paper is a jumble of misquoted results from math and physics. It would take up too much space to enumerate all the mistakes: indeed it is difficult to say where one error ends and the next begins."
So the papers are clearly recognizable as nonsense to anybody with a basic understanding of physics. Now we get to the more depressing elements of the story. Not only did these con artists get their theses by ignorant committee members, but they got them published in some of the most prestigious physics journals (Classical and Quantum Gravity, Nuovo Cimento, Annals of Physics). Here are some of the comments regarding the quality of the papers by what I would have thought were strong physicists...
Roman Jackiw (MIT) (According to most sources, the thesis had "many" things Dr. Jackiw didn't understand, but I have been unable to find a direct quote to this effect.): "The author proposes a novel, speculative solution to the problem of the pre-Big-Bang initial singularity ... the thesis and the published papers provide an excellent introduction to these ideas, and can serve as a useful springboard for further research in this area. All these were ideas that could possibly make sense. It showed some originality and some familiarity with the jargon. That's all I ask."
That's all you ask?!?! WTF?!?! I have some familiarity with the jargon. I'm sure I can write something original (especially if these Bogdanov papers are setting the bar for "originality"). Where the hell is my Ph.D.?!?!
Jack Morava (Johns Hopkins University): "The thesis work is of great interest, dominated by new ideas with fundamental physical implications in cosmology and in many other fields connected with gravitation."
Lubos Motl (Harvard): "They are proposing something that has, speculatively, the potential to be an alternative story about quantum gravity. What they are proposing is a potential new calculational framework for gravity."
So, there you have it. Physicists at some of the best universities in the world have been taken in by this meaningless pile of crap. Amazing! One almost hopes that these folks were bribed or have some other ulterior motive in stating that these papers constitute good physics. Because the alternative explanation - that they are fools - is too terrible to contemplate. However, deep down I have become convinced that this is in fact the case. I have to accept that the field that I have loved for over 20 years is turning into a pathetic joke. I will finish this post with two quotes that most clearly explain the situation...
Jackiw: "One person looks at a piece of art and says it is gibberish; another person looks and says it's wonderful."
When did physics become comparable to modern art in deciding what was of value? I seem to recall learning about something called "the scientific method" in high school (oh, so many years ago). I will give the last word to Frank Wilczek (MIT) who provides the best explanation of why we find ourselves in this sorry state.
"This says something profound about what happens to theoretical physics in the absence of the discipline of experiment."
What, you may ask, am I talking about. I'm speaking about the Bogdanov affair. First, let's start with just the facts (from Wikipedia)...
The Bogdanov Affair is an academic dispute regarding a series of theoretical physics papers written by French twin brothers Igor and Grichka Bogdanov (or Bogdanoff). These papers were published in reputable scientific journals, and culminated in a proposed theory for describing what occurred at the Big Bang. While the Bogdanovs defend the veracity of their work, many physicists have alleged that the papers are nonsense, emphasizing the fallibility of the peer review system used to approve papers. The Bogdanovs obtained Ph.D. degrees from the University of Bourgogne; Grichka Bogdanov received his degree in mathematics in 1999, and Igor Bogdanov received his in theoretical physics in 2002. Both were given the low but passing grade of "honorable", in Igor's case only after publishing in respected physics journals to establish the acceptability of his work.
Before moving on to other topics, let's consider briefly the appalling intellectual laziness of allowing journals to establish the "acceptability" of the work. If you don't understand the paper, you have no business recommending the awarding of a doctorate.
Inside Higher Education said it best: "A PhD is a license to reproduce and an obligation to maintain the quality of your intellectual descendants. [...] If you vote to pass someone who is marginal or worse they, in turn, have the same privilege. If they are not up to standard, it is likely that some of their descendants will also not be."
Leaving this shameful aspect of the affair, let's now look at some actual text from the papers in question, courtesy of John Baez's blog...
We consider inertia as a topological field, linked to the topological charge Q = 1 of the "singular zero size gravitational instanton" which can be identified with the initial singularity of space-time in the standard model.
Huh?
Baez: "The paper goes on to discuss the supposed connection between N = 2 supergravity, Donaldson theory, KMS states and the Foucault pendulum experiment, which he claims 'cannot be explained satisfactorily in either classical or relativistic mechanics.' If you know some physics you'll find this statement odd. The Foucault pendulum behaves exactly the way classical mechanics predicts: it is a standard textbook exercise."
We draw from the above that whatever the orientation, the plane of oscillation of Foucault's pendulum is necessarily aligned with the initial singularity marking the origin of physical space S3, that of Euclidean space E4, and, finally, that of Lorentzian space-time M4.
Baez: "How in the world could the plane of oscillation of a Foucault's pendulum be 'aligned with the initial singularity', i.e. the big bang?"
We simply suggest that at 0 scale, the observables must be replaced by the homology cycles in the moduli space of gravitational instantons. We then get a deep correspondence - a symmetry of duality - between physical theory and topological field theory.
Baez: "(1) Could you please define "at 0 scale"? (2) "Observables" in which theory? (3) You say "must be replaced". Why? How? (4) You speak of a "deep correspondence" between some unspecified physical theory and some unspecified topological field theory. Which theories are you talking about here? How does the correspondence go?"
Lee Smolin: "They were at best wrong, and most likely just throwing around words with no calculations or proofs to back them up."
Jacques Distler: "The papers consist of buzzwords from various fields of mathematical physics, string theory and quantum gravity, strung together into syntactically correct, but semantically meaningless, prose."
Eli Hawkins: "This paper is built around the idea that 'at the Planck scale, the space-time system is in a themodynamical equilibrium state.' It is not quite clear what the author means by this. He may mean that when the matter is at the Planck temperature, it is in thermodymanic equilibrium with the geometry. He does not explain why there should not be thermal equilibrium at all temperatures. It may be simply that the author does not know what he is talking about. The main result of this paper is that this thermodynamic equilibrium should be a KMS state. This almost goes without saying; for a quantum system, the KMS condition is just the concrete definition of thermodynamic equilibrium. The hard part is identifying the quantum system to which the condition should be applied. It is difficult to describe what is wrong in Section 4, since almost nothing is right. The author seems to believe that just because an analytic continuation of a function exists, the argument must be considered a complex number. He also makes the rather obvious claims that complex numbers should be the sums of real and imaginary parts. The remainder of the paper is a jumble of misquoted results from math and physics. It would take up too much space to enumerate all the mistakes: indeed it is difficult to say where one error ends and the next begins."
So the papers are clearly recognizable as nonsense to anybody with a basic understanding of physics. Now we get to the more depressing elements of the story. Not only did these con artists get their theses by ignorant committee members, but they got them published in some of the most prestigious physics journals (Classical and Quantum Gravity, Nuovo Cimento, Annals of Physics). Here are some of the comments regarding the quality of the papers by what I would have thought were strong physicists...
Roman Jackiw (MIT) (According to most sources, the thesis had "many" things Dr. Jackiw didn't understand, but I have been unable to find a direct quote to this effect.): "The author proposes a novel, speculative solution to the problem of the pre-Big-Bang initial singularity ... the thesis and the published papers provide an excellent introduction to these ideas, and can serve as a useful springboard for further research in this area. All these were ideas that could possibly make sense. It showed some originality and some familiarity with the jargon. That's all I ask."
That's all you ask?!?! WTF?!?! I have some familiarity with the jargon. I'm sure I can write something original (especially if these Bogdanov papers are setting the bar for "originality"). Where the hell is my Ph.D.?!?!
Jack Morava (Johns Hopkins University): "The thesis work is of great interest, dominated by new ideas with fundamental physical implications in cosmology and in many other fields connected with gravitation."
Lubos Motl (Harvard): "They are proposing something that has, speculatively, the potential to be an alternative story about quantum gravity. What they are proposing is a potential new calculational framework for gravity."
So, there you have it. Physicists at some of the best universities in the world have been taken in by this meaningless pile of crap. Amazing! One almost hopes that these folks were bribed or have some other ulterior motive in stating that these papers constitute good physics. Because the alternative explanation - that they are fools - is too terrible to contemplate. However, deep down I have become convinced that this is in fact the case. I have to accept that the field that I have loved for over 20 years is turning into a pathetic joke. I will finish this post with two quotes that most clearly explain the situation...
Jackiw: "One person looks at a piece of art and says it is gibberish; another person looks and says it's wonderful."
When did physics become comparable to modern art in deciding what was of value? I seem to recall learning about something called "the scientific method" in high school (oh, so many years ago). I will give the last word to Frank Wilczek (MIT) who provides the best explanation of why we find ourselves in this sorry state.
"This says something profound about what happens to theoretical physics in the absence of the discipline of experiment."
Monday, September 04, 2006
Dark Matter, Yes?
Experimental evidence for the existence of dark matter explained by Sean Carroll in his blog.
Here is a picture of the distribution of the matter based on gravitational lensing. (Of course, labeling it Dark Matter assumes the very thing we are trying to prove, so that is a little disappointing.)
Here is a picture of the distribution of matter based on X-ray observations of the hot gas.
The previous two images superimposed:
The gravitational field, as reconstructed from lensing observations, is not pointing toward the ordinary matter. That’s exactly what you’d expect if you believed in dark matter.
There is no denying the conclusion from the third picture that there is definitely "dark matter" there in the sense that the hot gas does not constitute the bulk of the gravitational mass. However, the conclusion that there is Dark Matter (meaning nonbaryonic cold dark matter, in the sense that the term is usually used in cosmology) depends on the assumption that the hot gas is the bulk of the regular matter and there cannot be nonvisible regular matter causing this effect.
Whatever matter is in the blue areas in the pictures above cannot interact heavily with regular matter (else, it would have been dragged along with the pink areas). This tends to support the hypothesis that it is Dark Matter. However, it is my opinion that our imperfect understanding of the universe simply does not allow us at this time the luxury of treating it as the only possible hypothesis to the exclusion of all others. True Dark Matter (Higgs, Superpartners, WIMPs, etc)? Baryons? Neutrinos? Who knows!
One response to Sean Carroll's post seems worth reproducing here: The history of the Vulcan hypothesis shows the way: the strange movement of Mercury was not caused by something which was huge and strangely impossible to see; it was caused by a small relativistic effect which was hidden in normal situations by the very fact of its weakness. When people start saying that an effect is caused by something three times the mass of the visible universe but which has escaped notice for the whole of history, is it really unreasonable to doubt them? I don’t think so. Dark matter is a fudge; a placeholder until we come up with something sensible.
Here is a picture of the distribution of the matter based on gravitational lensing. (Of course, labeling it Dark Matter assumes the very thing we are trying to prove, so that is a little disappointing.)
Here is a picture of the distribution of matter based on X-ray observations of the hot gas.
The previous two images superimposed:
The gravitational field, as reconstructed from lensing observations, is not pointing toward the ordinary matter. That’s exactly what you’d expect if you believed in dark matter.
There is no denying the conclusion from the third picture that there is definitely "dark matter" there in the sense that the hot gas does not constitute the bulk of the gravitational mass. However, the conclusion that there is Dark Matter (meaning nonbaryonic cold dark matter, in the sense that the term is usually used in cosmology) depends on the assumption that the hot gas is the bulk of the regular matter and there cannot be nonvisible regular matter causing this effect.
Whatever matter is in the blue areas in the pictures above cannot interact heavily with regular matter (else, it would have been dragged along with the pink areas). This tends to support the hypothesis that it is Dark Matter. However, it is my opinion that our imperfect understanding of the universe simply does not allow us at this time the luxury of treating it as the only possible hypothesis to the exclusion of all others. True Dark Matter (Higgs, Superpartners, WIMPs, etc)? Baryons? Neutrinos? Who knows!
One response to Sean Carroll's post seems worth reproducing here: The history of the Vulcan hypothesis shows the way: the strange movement of Mercury was not caused by something which was huge and strangely impossible to see; it was caused by a small relativistic effect which was hidden in normal situations by the very fact of its weakness. When people start saying that an effect is caused by something three times the mass of the visible universe but which has escaped notice for the whole of history, is it really unreasonable to doubt them? I don’t think so. Dark matter is a fudge; a placeholder until we come up with something sensible.
No Dark Matter?
First they invented dark matter. However, so far there is no evidence that such a thing exists, although the Higgs particle does turn out to have precisely the right properties to be the missing dark matter. Then they invented dark energy. And as far as I know, there's not even a candidate for what this mysterious substance might actually be.
This is all very radical, although I suppose that Planck's quantum hypothesis and the Bohr hypothesis about electronic orbits were just a radical in their day. But now there's another radical theory, although perhaps somewhat less radical than dark matter and dark energy.
According to Discover, Mordehai Milgrom proposed an alternative to dark matter and dark energy named MOND (MOdified Newtonian Dynamics), which consists of the idea that when the accelerations become very small the well-known formula F=ma changes to F=ma^2/a_0. Of course, there is no more theoretical justification for this formula than for the introduction of dark matter. Turns out Milgrom wrote a Scientific American article back in 2002 titled Does Dark Matter Really Exist?
The story is back in the pages of the popular science media is because in 2004 Jacob Bekenstein published a relativistic extension of MOND called TeVeS (Tensor, Vector, Scalar) which can correctly explain gravitational lensing and should be applicable to the problem of galaxy formation.
Also, here's a link to a webpage with many resources on Modified Newtonian Dynamics: http://www.astro.umd.edu/~ssm/mond/litsub.html.
This is all very radical, although I suppose that Planck's quantum hypothesis and the Bohr hypothesis about electronic orbits were just a radical in their day. But now there's another radical theory, although perhaps somewhat less radical than dark matter and dark energy.
According to Discover, Mordehai Milgrom proposed an alternative to dark matter and dark energy named MOND (MOdified Newtonian Dynamics), which consists of the idea that when the accelerations become very small the well-known formula F=ma changes to F=ma^2/a_0. Of course, there is no more theoretical justification for this formula than for the introduction of dark matter. Turns out Milgrom wrote a Scientific American article back in 2002 titled Does Dark Matter Really Exist?
The story is back in the pages of the popular science media is because in 2004 Jacob Bekenstein published a relativistic extension of MOND called TeVeS (Tensor, Vector, Scalar) which can correctly explain gravitational lensing and should be applicable to the problem of galaxy formation.
Also, here's a link to a webpage with many resources on Modified Newtonian Dynamics: http://www.astro.umd.edu/~ssm/mond/litsub.html.
This is what a critique of Woit should look like
http://zippy.ph.utexas.edu/~abergman/Review.pdf
I will focus on one passage for a minute...
There is a seemingly robust description of zillions and zillions of vacua, each of which seems to describe consistent physics completely unlike our world. Faced with this proliferation, one immediately begins to worry about the predictivity of the theory. With so many vacua it might be possible to explain any experimental result one can imagine. If this were true, it would not mean that string theory was wrong, but it would mean that it would be completely useless as it could never make a prediction. To avoid this unhappy state of affairs, a fair number of senior people have decided to take a radically new approach to predictivity. The idea is that one should determine all the possible vacua that are consistent with the existence of intelligent life and imagine that it is equally likely that we could be in any of them. Then, while we could not precisely predict anything about our world, we could assign probabilities to the results of future experiments.
Philosophically, the bolded phrase is indicative of the major problem that critics have with supersymmetry and string theory.
I will focus on one passage for a minute...
There is a seemingly robust description of zillions and zillions of vacua, each of which seems to describe consistent physics completely unlike our world. Faced with this proliferation, one immediately begins to worry about the predictivity of the theory. With so many vacua it might be possible to explain any experimental result one can imagine. If this were true, it would not mean that string theory was wrong, but it would mean that it would be completely useless as it could never make a prediction. To avoid this unhappy state of affairs, a fair number of senior people have decided to take a radically new approach to predictivity. The idea is that one should determine all the possible vacua that are consistent with the existence of intelligent life and imagine that it is equally likely that we could be in any of them. Then, while we could not precisely predict anything about our world, we could assign probabilities to the results of future experiments.
Philosophically, the bolded phrase is indicative of the major problem that critics have with supersymmetry and string theory.
- Supersymmetry solves the Higgs problem by introducing a supersymmetric partner to each known particle - never mind that nobody has ever observed a supersymmetric particle
- Superstring theory requires more than four dimensions - no problem, compactify the superfluous ones
- M Theory has no predictability - no problem, redefine predictability
Another article on Smolin and Woit
Scientific American joined the fray...
[Critics claim] string theory is not just untested but untestable, incapable of ever making predictions that can be experimentally checked. [...] There are, in fact, some 10^500 perfectly good M theories, each describing a different physics. The theory of everything, as Smolin puts it, has become a theory of anything.
I have no idea which side of this debate is correct. However, I would prefer to see reasonable debate on the questions as befits proper science. Instead I see the following (to be fair, these comments are all from Motl, but I do not see anything to the contrary from his string theory colleagues) ...
[Critics claim] string theory is not just untested but untestable, incapable of ever making predictions that can be experimentally checked. [...] There are, in fact, some 10^500 perfectly good M theories, each describing a different physics. The theory of everything, as Smolin puts it, has become a theory of anything.
I have no idea which side of this debate is correct. However, I would prefer to see reasonable debate on the questions as befits proper science. Instead I see the following (to be fair, these comments are all from Motl, but I do not see anything to the contrary from his string theory colleagues) ...
- Sean Carroll has essentially joined Peter Woit and others in their irrational attack against existing high-energy theoretical physics (Carroll actually wrote that he did not agree with Woit, but apparently in Motl's world even allowing that Woit's book has a right to exist is an attack not on him personally but on the entire enterprise of theoretical physics)
- semi-official unholy alliance between Cosmic Variance and the anti-science activists (another reference to Carroll)
- They're orders of magnitude and decades of education from being able to do something like [presenting an alternative to superstrings]. The only thing that Woit and others offer and share is their hatred against science and a very poor knowledge of the actual problems that are being discussed.
- [Greene's victory in the debate was] because of Brian Greene's virtues, based not only on his superior communication skills and psychological balance but also on his active knowledge of the topics and clean conscience of a real contributor to science. (So anyone with whom Motl disagrees is not only not "a real contributor to science" but is psychologically unbalanced? It is also worth noting that Motl's description of the debate bears VERY little resemblance to the audio.)
- a scientific microbe (referring to Woit)
- the attack is led by people with poor quantitative skills who dislike a careful and detailed technical analysis of the scientific issues [...] This description is primarily about Peter Woit
- Another postmodern diatribe against modern physics and scientific method (about Smolin's book) and bitter emotions and obsolete understanding of high-energy physics (about Woit)
Sunday, September 03, 2006
Exchange with Professor Motl
I posted the following response to Motl's comments about the Time magazine article on superstring theory.
The article hardly seems to warrant your "physics is a sin" response. Other than engaging in ad hominem attacks, you haven't really explained why you disagree with the article. That hardly seems like science to me. What's up?He gave me the following reply, which I thought I'd reproduce here.
Dear Alberto,
I won't be explaining why I disagree with the article because this is not a matter of agreement or disagreement. The article is a pile of vitriolic nonsense and conspiratory theories written by rather incompetent people and, in my opinion, there is nothing in the article that merits a scientific discussion. I apologize but if you need someone who will be commenting every sentence of every vitriolic author with high-school physics education, you will be disappointed.
It is up to you whether you believe that I/we know what we're saying or not. Be aware that if you choose to take the Time magazine more seriously than my fast comment, I won't think that you are an intelligent person.
Best
Lubos
The Unraveling of String Theory?
Ever since I was an undergraduate, superstring theory has made me uneasy philosophically. Green,Schwartz,Witten came out in 1987 (I was a junior), and I remember discussing with classmates and professors my concerns that "physics was still an experimental science." The results of the theory are probably mathematically valid (although the lack of rigor with which some results are obtained would probably bring tears to the eyes of most mathematicians). However, I question the validity (as physics). The field has always seemed to me to be an exercise in mathematics, pursued with little or no concern regarding its possible relation to experimental proof. As the field currently stands, it isn't even experimentalists or the experimental equipment that are the limiting factor. It is the theory itself that cannot even be formulated in a way that allows it to be applied to the universe around us.
With the recent publication of Not Even Wrong by Peter Woit and The Trouble with Physics by Lee Smolin, this point of view is starting to be debated among top physicists (sometimes rudely so as in the blog wars between Woit and Harvard's Lubos Motl) and this month even made the pages of Time magazine.
[I attempted to add a trackback to this post from Motl's blog, but because I just transferred my own blog to the beta version of Blogger I am unable to do so at this time.]
With the recent publication of Not Even Wrong by Peter Woit and The Trouble with Physics by Lee Smolin, this point of view is starting to be debated among top physicists (sometimes rudely so as in the blog wars between Woit and Harvard's Lubos Motl) and this month even made the pages of Time magazine.
Now, it seems [to critics], at least some superstring advocates are ready to abandon the essential definition of science itself on the basis that string theory is too important to be hampered by old-fashioned notions of experimental proof.The story does seem a little one-sided in that the opinions of physicists who support string theory are never solicited (it's sad that the media screw up basic things like this even when political bias is not a consideration). However Motl's hysterical response describing the article as a "dirty, immoral, dishonest, and anti-scientific enterprise" seems extreme. What ever happened to logical scientific debate? I find his review of Smolin's book on Amazon particularly incoherent, as he calls it "a post-modern diatribe" and illogically claims that Smolin posits that "it is wrong for mathematics to play a crucial role in theoretical physics." He isn't helping his side in the debate with his idiocy.
[I attempted to add a trackback to this post from Motl's blog, but because I just transferred my own blog to the beta version of Blogger I am unable to do so at this time.]
Thursday, August 31, 2006
The Sokal Affair
I won't bore the reader by describing the well-known hoax, but I did want to provide a link to a website that has collected a lot of documentation related to it.
http://www.physics.nyu.edu/faculty/sokal/
http://www.physics.nyu.edu/faculty/sokal/
Tuesday, August 29, 2006
Set Theory Proof
Here's a proof that I have always found fascinating.
Theorem: Given any set A, the power set of A has strictly larger cardinality than A.
For two sets to have the same cardinality, there must exist a bijection between the two sets. So, let us assume that there exists such a bijection between A and P(A), denoted f:A->P(A).
For each x in A, either x is in f(x) or x is not in f(x). Consider the subset S of A defined as
S = {x in A | x is not in f(x)}.
Since we have assumed that f:A->P(A) is a bijection and S is in P(A), then there exists some t in A such that f(t) = S.
Either t is in S or t is not in S. Suppose t is in S. Then t satisfies the condition that t is not in f(t) which is S. Therefore t is not in S. Suppose t is not in S. Then t does not satisfy the condition, and t is in f(t)=S. Therefore t is in S.
So in either case, we arrive at the contradiction that t is in S and simultaneously t is not in S. The source of the contradiction is the assumption that there exists a bijection f:A->P(A). Therefore, there exists no such bijection and the cardinality of P(A) is not the same as the cardinality of A. Since the cardinality of P(A) cannot be smaller than the cardinality of A, the conclusion is that P(A) has strictly larger cardinality than A. (Q.E.D.)
Theorem: Given any set A, the power set of A has strictly larger cardinality than A.
For two sets to have the same cardinality, there must exist a bijection between the two sets. So, let us assume that there exists such a bijection between A and P(A), denoted f:A->P(A).
For each x in A, either x is in f(x) or x is not in f(x). Consider the subset S of A defined as
S = {x in A | x is not in f(x)}.
Since we have assumed that f:A->P(A) is a bijection and S is in P(A), then there exists some t in A such that f(t) = S.
Either t is in S or t is not in S. Suppose t is in S. Then t satisfies the condition that t is not in f(t) which is S. Therefore t is not in S. Suppose t is not in S. Then t does not satisfy the condition, and t is in f(t)=S. Therefore t is in S.
So in either case, we arrive at the contradiction that t is in S and simultaneously t is not in S. The source of the contradiction is the assumption that there exists a bijection f:A->P(A). Therefore, there exists no such bijection and the cardinality of P(A) is not the same as the cardinality of A. Since the cardinality of P(A) cannot be smaller than the cardinality of A, the conclusion is that P(A) has strictly larger cardinality than A. (Q.E.D.)
Saturday, August 26, 2006
Finite Simple Group (of Order Two)
The lyrics of this song can be found here. More information on the group (from Northwestern University) can be found at http://www.kleinfour.com/.
[Thanks to TMB for sending me this video.]
Friday, August 25, 2006
Fields Medals 2006
Awarded once every 4 years by the International Mathematical Union, the Fields Medal recognizes outstanding contributions to mathematics.
This year's winners...
Andrei Okounkov “for his contributions briding probability; representation theory and algebraic geometry.”
Grigori Perelman “for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow.” (Perelman's work may prove the more general Thurston Geometrization Conjecture from which the Poincare Conjecture would then follow. Perelman is quite the recluse and actually turned down the Fields Medal.)
Terence Tao “for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory.” (At 31, Tao is one of the youngest mathematicians to be awarded the Fields Medal.)
Wendelin Werner “for his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory.”
This year's winners...
Andrei Okounkov “for his contributions briding probability; representation theory and algebraic geometry.”
Grigori Perelman “for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow.” (Perelman's work may prove the more general Thurston Geometrization Conjecture from which the Poincare Conjecture would then follow. Perelman is quite the recluse and actually turned down the Fields Medal.)
Terence Tao “for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory.” (At 31, Tao is one of the youngest mathematicians to be awarded the Fields Medal.)
Wendelin Werner “for his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory.”
Thursday, August 24, 2006
Busy year for Pluto
The International Astronomical Union defined the term "planet" for the first time. This definition excluded Pluto from planethood, so Pluto was reclassified under the new category of "dwarf planet" along with and Ceres and Eris.
Tuesday, August 15, 2006
At Berkeley
On a bulletin board in the mathematics department at UC Berkeley, somebody had left a flier offering a $300 reward for a solution to the equation x^5 + ax^4 + bx^3 + cx^2 + dx = e in terms of a, b, c, d and e. If you studied college level math, you know this problem is unsolvable. The insolubility of the quintic is proved as part of Galois theory in the second semester of undergraduate abstract algebra. I am not sure if this was a hoax or exactly what the purpose of this flier might have been, but this being Berkeley several people had scribbled sarcastic offers to square the circle (another problem that has been proven to be unsolvable) or to prove that pi is a natural number.
Wednesday, June 21, 2006
Nix and Hydra
Two additional moons of Pluto (in addition to Charon) were imaged by astronomers working with the Hubble Space Telescope on 5/15/2005. The International Astronomical Union officially named the moons Nix and Hydra. These moons orbit Pluto at approximately two and three times the distance of Charon: Nix at 48,700 kilometres and Hydra at 64,800 kilometres from the barycenter of the system.
Sunday, May 21, 2006
Geeky Math Humor
Q: What's the value of a contour integral around Western Europe?
A: Zero, because all the Poles are in Eastern Europe.
A: Zero, because all the Poles are in Eastern Europe.
Sunday, May 14, 2006
Wednesday, February 22, 2006
Ridiculous
On today's Law & Order Criminal Intent, I learned that elementary age kids without any training in advanced mathematics or supervision by a knowledgeable mathematician nevertheless have a reasonable chance of solving the Riemann Hypothesis just because they are geniuses.
A couple of weeks ago, I learned that in theoretical physics "theories" are formulated without any appeal to mathematics, and then mathematics is used to prove those theories.
A couple of weeks ago, I learned that in theoretical physics "theories" are formulated without any appeal to mathematics, and then mathematics is used to prove those theories.
Monday, January 23, 2006
Ridiculous
Having a bad day? No wonder. According to one scientist, today, January 23, is the gloomiest day of 2006. Cliff Arnall, a health psychologist at the University of Cardiff in Wales, has devised a formula that combines personal and seasonal factors to calculate the year's emotional low point.
Caught this story in today's paper. The same story made the rounds last year. It's utterly ridiculous. According to this "scientist," the variables are weather (W), debt (D), monthly salary (S), time since Christmas (T), time since failure to quit a bad habit (Q), low motivational levels (M) and the need to take action (N). Sounds reasonable enough. But now let's look at his "formula."
Gloominess = [W + (D-S)xTxQ] / [M x N]
So we take time (since Christmas) times time (since failure) times money. So we have a quantity that is measured in $ x days squared. Then we add this to a weather variable, which is measured in . . . ?
Then we take this already meaningless concoction and further divide it by "low motivational levels" and "the need to take action." I'm unclear on how either of these can be measured well enough to allow any mathematical comparisons to be made between January 23 and January 20 or January 26 ... or July or November or any other random date I pull out of a hat. Furthermore, why do we divide by them; why not multiply?
I am appalled that some idiot can get this "research" published. What a freakin' joke.
Caught this story in today's paper. The same story made the rounds last year. It's utterly ridiculous. According to this "scientist," the variables are weather (W), debt (D), monthly salary (S), time since Christmas (T), time since failure to quit a bad habit (Q), low motivational levels (M) and the need to take action (N). Sounds reasonable enough. But now let's look at his "formula."
Gloominess = [W + (D-S)xTxQ] / [M x N]
So we take time (since Christmas) times time (since failure) times money. So we have a quantity that is measured in $ x days squared. Then we add this to a weather variable, which is measured in . . . ?
Then we take this already meaningless concoction and further divide it by "low motivational levels" and "the need to take action." I'm unclear on how either of these can be measured well enough to allow any mathematical comparisons to be made between January 23 and January 20 or January 26 ... or July or November or any other random date I pull out of a hat. Furthermore, why do we divide by them; why not multiply?
I am appalled that some idiot can get this "research" published. What a freakin' joke.
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