I linked to a pdf of Sethna’s book, which should allow you to evaluate it to see if it’s the kind of thing you want to buy. There’s no (legal) online version of Kardar’s book, but the book is an expansion of his lecture notes available from MIT OpenCourseWare: here and here. The notes give a good sense of the technical level of his discussion in the textbook and the topics he covers.
pragmatist
If you don’t say that only conscious agents can collapse waveforms, then you have to agree that something in the box collapses the waveform as seen from inside the box, while it’s still uncollapsed to Schroedinger.
I don’t understand this claim. If I think collapse can occur without conscious observation (say by the interaction of microscopic and macroscopic systems, however that distinction may be drawn, or by some spontaneous dynamical process) why would I have to agree that the waveform is still uncollapsed to Schroedinger? You seem to be assuming here that Schroedinger’s epistemic state is relevant to whether or not the waveform is collapsed to him, but I thought the whole point of this option was to render Schroedinger’s epistemic state (and, indeed, his status as a conscious agent) irrelevant to the situation.
If interacting with another subatomic particle were sufficient to collapse a waveform, then you couldn’t prepare two particles in an entangled state. But entangled particles are regularly prepared in the lab, so your conjecture seems to have been refuted.
I think the best bet for a collapse interpretation of quantum mechanics is some form of spontaneous collapse theory, like GRW.
Here is how I interpret your claim here: The only way the Copenhagen interpretation could be an absolute state theory—i.e. a theory where the quantum state of a system is absolute, not relative to some other system—is for collapse to be caused by conscious agents. Am I misinterpreting you?
If I’m not, I don’t see why you believe this. The Copenhagen interpretation does say that a certain class of interactions—measurement interactions—produce collapse. And I acknowledge that it cannot maintain that all physical interactions are measurement interactions. That view has been conclusively refuted empirically. However, why think that the only alternative is that measurement interactions must involve conscious observation? Bohr, as far as I can tell from his mysterious proclamations on the topic, seemed to think that any interaction with a macroscopic system is a measurement interaction. He didn’t think that consciousness played any essential role in his interpretation. I think Wigner was the one who emphasized consciousness.
Now you could say that Bohr’s interpretation is untenable, since microscopic/macroscopic is a continuum, not a binary distinction. Also, macroscopic systems are just built out of microscopic systems, so why think the measurement problem doesn’t apply to them? I agree! But the exact same criticisms can be raised about consciousness, so Wigner’s interpretation is not on sounder footing here. So I guess I’m not seeing why you think a Wigner-type delineation of measurement interactions is the only way to avoid Copenhagen collapsing into Everett.
I’m assuming the ‘X’ is supposed to be a cross, the traditional academic symbol for an answer being wrong (as opposed to check for right answers). The ‘<’ is the ‘less than’ symbol.
Good point. I guess ‘<’ is meant to be read “less”, rather than the standard “less than”.
I like issues (2) and (3) in your breakdown, but I don’t think (1) captures an important aspect of the Bayesian/frequentist debate. I don’t really associate frequentism with a denial of probabilism (the claim that the degrees of belief of a rational agent obey the probability calculus). I do think there is an interesting disagreement in the vicinity of (1) about how degrees of belief should be set.
My model of a frequentist is someone who thinks relative frequency should be treated as an expert function: If rf(X) is the relative frequency with which propositions like X are true in some appropriate reference class, then P(X | rf(X) = x) = x. This seems to me the most natural interpretation of the claim that probabilities are just relative frequencies. My frequentist doesn’t answer “no” to (1). She does think that subjective anticipations obey the probability calculus, and this is because relative frequencies obey the calculus and subjective anticipations should be guided by knowledge of relative frequencies. So she treats relative frequency as an expert function, which means she tries to maximize her calibration.
The Bayesian does not think the rational agent should always try to maximize calibration. There are situations where one should be willing to sacrifice calibration for discrimination. Eliezer has a good example of this in A Technical Explanation of Technical Explanation. Here’s my understanding of the difference: The Bayesian treats the truth function (the function that assigns 1 to truths and 0 to falsehoods) as an expert function, and this is is incompatible with treating relative frequency as an expert function. Trying to estimate truth can lead you to intentionally sacrifice calibration for discrimination; trying to maximize calibration cannot.
So maybe (1) should be supplemented with something like this:
(1′) If the answer to (1) is “yes”, whether subjective anticipations should always be guided by beliefs about relative frequencies.
I use MySpeed to speed up online lectures. I’m finding the lectures quite easy to follow at twice the original speed.
I just tried, and the software player does seem to do a better job than the web player. It certainly sounds different. I don’t have much trouble following Ng at 150% on the web player either, however, so it’s hard for me to make a useful comparison.
I should say, though, that the machine learning videos don’t seem to be loading fast enough on my connection for smooth sped-up streaming using the software. This makes the software pretty useless unless you pre-load the videos or are willing to put up with the frustration of intermittent buffering. I don’t have this problem with the AI videos (or youtube videos in general).
Discrete particles vs. continuous wave functions is a red herring, I think. It’s true that in a simulation of QM one would have to approximate amplitudes up to some finite precision (and approximate infinite dimensional Hilbert spaces using finite dimensional Hilbert spaces). But this is not a problem that is unique to QM. Simulating classical mechanics also requires approximating the positions and momenta of particles to finite precision.
You are right, though, that computing quantum mechanics is harder than computing classical mechanics. This is true even if we completely discretize both theories. The length of a vector representing the state of a classical system is linear in the size of the system, but the length of a vector representing the state of a quantum system is exponential in the size of the system.
The magnitude is variable as well. A wavefunction is a map from configuration space to the entire complex plane, not just the unit circle on the complex plane.
Perhaps I am misreading you, but I think your gloss is incorrect. Eliezer’s point is about his map, not the territory. He is describing circumstances under which he would be convinced that 2 + 2 = 3, not circumstances under which 2 + 2 would actually be 3. I do not take him to be arguing (as you suggest) that math is physical, whatever that would mean. He is arguing that beliefs about math are physically instantiated, and subject to alteration by some possible physical process.
I’m afraid you lose me completely in the second part of your comment. Why is physics definitely not objective? And what does the similarity of math to modus ponens and its dissimilarity from empirical statements have to do with the subjective/objective distinction?
In short, Eliezer isn’t describing how he could come to belief 2 + 2 = 3, but how new evidence might show 2 + 2 would truly equaled 3.
From the beginning of his post:
I admit, I cannot conceive of a “situation” that would make 2 + 2 = 4 false. (There are redefinitions, but those are not “situations”, and then you’re no longer talking about 2, 4, =, or +.) But that doesn’t make my belief unconditional. I find it quite easy to imagine a situation which would convince me that 2 + 2 = 3.
So on the point of interpretation, I’m pretty sure you are wrong.
On the substantive point, I think reliance on traditional philosophical distinctions (a priori/a posteriori, analytic/synthetic) is a recipe for confusion. In my opinion (and I am far from the first to point this out) these distinctions are poorly articulated, if not downright incoherent. If you are going to employ these concepts, however, an important thing to keep in mind is the hard-won philosophical realization, stemming from a tradition stretching from Kant to Kripke, that the a priori/a posteriori distinction is orthogonal to the necessary/contingent distinction. The former is an epistemological distinction (propositions are justifiable a priori or a posteriori), and the latter is a metaphysical distinction (propositions are true/false necessarily or contingently).
My position (and, I believe, Eliezer’s) is that mathematical truths are necessarily true. A world in which 2 + 2 = 3 is impossible. This does not, however, entail that it is impossible to convince me that 2 + 2 = 3. Nor does it entail that empirical considerations are irrelevant to the justification of my belief that 2 + 2 = 4.
I am sure there is some proposition (perhaps some complicated mathematical truth) that you believe is necessarily true, but you are not certain that it is true. Maybe you are fairly confident but not entirely sure that you got the proof right. So even though you believe this proposition cannot possibly be false, you admit the possibility of evidence that would convince you it is false.
I would not even say that physics is different in a black hole. In fact, I would strongly bet against it. What is true is that our best physical theories fail to give coherent results at the center of a black hole, but of course this does not mean the actual physics is different. One must not confuse the map and the territory.
A proof of the Pythagorean Theorem is not (empirical) evidence that the theorem is true. The proof (metaphysically) is the truth of the theorem.
Consider the four color theorem. We have a proof by computer of this theorem, but it is far too complex for any human to verify. Would you agree that the fact that a computer built and programmed in a certain way claims to have proven the theorem is empirical evidence for the truth of the theorem? If yes, then why treat a proof computed by a human brain differently?
Here’s a quick overview of the necessity/contingency distinction in philosophy. For a deeper overview, try Kripke’s Naming and Necessity.
I would switch the order of DH1 and DH2. A tone argument is very rarely relevant to the substantive dispute. In most cases, the tone of an article shouldn’t lead you to update your belief in the conclusion. An ad hominem argument, on the other hand, is often substantively relevant, especially given the power of motivated reasoning. It is entirely reasonable to lower your credence in the conclusion of an article arguing that senators are underpaid once you discover that the author of an article is a senator. Of course, if you have already evaluated the argument itself, and are fairly confident in your evaluation, then learning the identity of the author shouldn’t significantly impact your belief in the conclusion (kind of like argument screens off authority), but that is true of tone arguments as well.
The justfication for placing tone above ad hominem in the hierarchy is that the former at least responds to the writing, not the writer. But surely this isn’t adequate justification. One might respond to the writing in many ways that are entirely irrelevant to the disagreement, e.g. by reproducing the written piece in reverse order. The question should be, which of these responses is more often relevant to a proper assessment of the truth of the conclusion.
OK, that helps. Perhaps this should be made explicit in the post, Luke?
Agreed. A good elementary exposition of relativity along these lines is Bob Geroch’s General Relativity from A to B.
EDIT: Actually, I realize I’m only in partial agreement with Vladimir. While I do think that many pop-sci explanations of theoretical physics are fairly worthless and often actively misleading, I do not think that it is impossible to gain real insight into (say) the general theory of relativity without mastering differential geometry. Geroch’s book presupposes only high school mathematics, but it provides a genuinely deep insight into relativity.
Two excellent recent textbooks on statistical mechanics (well, three really, but two of them are better considered different volumes of the same text):
Statistical Mechanics: Entropy, Order Parameters and Complexity by James Sethna. Available for free download here.
Statistical Physics of Particles and Statistical Physics of Fields by Mehran Kardar.
Sethna gives a very unorthodox presentation of the material, focusing more attention on exciting new developments in condensed matter physics than other texts at the same level and (I think correctly) de-emphasizing thermodynamics. He treats statistical mechanics as an interdisciplinary field of study rather than just a branch of physics, and this is an ideal text for people interested in the application of statistical mechanical tools to economics, computer science, cosmology, population biology, and many other sciences given short shrift in more traditional treatments. Sethna is an excellent writer and the book is not too techical, so it is great for self-study. The book also has the most impressive set of exercises I have ever seen in a physics text. The exercises are detailed, explore a huge variety of topics, and are a joy just to read through even if you don’t intend to solve them. The text does sacrifice depth for breadth. While most important topics in statistical mechanics are discussed somewhere in the book, they are often not treated with a high degree of rigor. The book is extremely enlightening if you want a broad overview of the techniques and achievements of contemporary stat. mech., but if you are looking for, say, a careful derivation of Boltzmann’s equation, this is not the place to look. The text should be accessible if you have a solid grasp of calculus, basic statistics and intermediate classical mechanics. Some sections presuppose knowledge of quantum mechanics.
Kardar’s text is more technically demanding and more of a slog to read than Sethna’s. It is, however, the most lucid presentation of the material I have yet encountered. Kardar develops the subject carefully and rigorously, filling in conceptual gaps that are traditionally left unaddressed. I am writing a dissertation on statistical physics, so I consider myself very familiar with the field, but I still learned a huge amount from reading Kardar. There were many places where he presented material I had previously encountered in a novel manner that finally made things click. The first volume of his text covers thermodynamics and traditional single-ensemble topics in stat. mech. The second volume discusses scaling and renormalization, and is by far the best introduction to these topics I have ever read. Kardar includes a number of very interesting problems (although not quite as interesting as Sethna’s), and many of them are solved. The text is at the graduate level, but I think it can be tackled by a motivated upper-level undergraduate.
Which one of these books would I recommend? It depends on what you’re looking for. If you’re not a physicist and want a fairly advanced introduction to statistical mechanics, Sethna is the way to go. He covers a broader range of topics, is less focused on training physicists, and is a more entertaining read. If you have studied stat. mech. at an undergraduate level and want a more rigorous understanding of the fundamentals, or if you’re really interested in the foundations of the discipline rather than its myriad applications, I would recommend Kardar. If you’re fairly new to stat. mech. and want to develop an understanding of the field that will allow you to tackle the professional literature, read Sethna first, then Kardar.