Ron Maimon

@Adam: I hope you will come to understand that the interpretation of "dV" and "dP" as "coordinate projecting one forms" is 20th century dipshits taking a dump on the great classical work of Cavalieri and Leibnitz, that it is motivated by snobbery and stupidity (because Cavalieri was not an academic) and that absent the genius of Abraham Robinson, the issue could have festered indefinitely. I hope you take the time to learn Ito calculus to the point where you will see that my example is apropos, and this is not smoke and mirrors.

@Adam: The point of the Ito calculus is not obscurantism, just the opposite! The Ito calculus makes it clear that you need to use infinitesimal increments, and not falsely pretend that the concept is captured by linear algebra on tangent spaces. The Ito calculus, expressed correctly, is as transparent as ordinary calculus, it just requires you to note that $dn^2$ is an infinitesimal of the same order of magnitude as $dt$, and this doesn't happen in the realm of differentiable stuff. I wish there were a differentiable example of cases where differential orders don't match. But there can't be.

@Lubos: Perhaps I misunderstood your comment. Are you saying that differentials are always infinitesimal? That is not so. The answer above gives a conceptually different definition, namely as the dual space of the space of tangent vectors. There is nothing infinitesimal, nor displaced, about duals to vectors. Dual vectors are a normal workaday finite mathematical object, with no limiting connetations. The surprise is that one can partially identify these objects with the infinitesimal objects of calculus. It requires explanantion. The infinitsimals come first, esp. in Ito, see below.

@Lubos: You are just wrong about this. The notion of infinitsimal does not coincide with the notion of differential, as can be seen from the case in my comment above, where $dn^2 = dt$. How do you interpret a second-order infinitesimal coinciding with a first order? You just never studied Ito calculus. It is important to learn this, because it can be interpreted as the first rigorous path integral construction.

@Nikolaj: The right form is $dn= \eta(t) dt$, where $\eta(t)$ is a random quantity which has the defining property that the integral from a to b of $\eta$ is a Gaussian random number of variance $a-b$. The need to deal with square-root infinitesimals comes when you need to show that $n(t+dt)*{dn\over dt}|_t - n(t-dt){dn\over dt}|_t $ is not zero, but constant for infinitesimal dt. This is standard stochastic calculus, but it is considered "advanced" because of the presence of square roots of infinitesimals (although they don't call them that in formal mathematics, of course).

@wmnorth: I learned calculus in a tradition limits way from Serge Lang's calculus book. I can't honestly recommend another book, because the material is too elementary, so I wouldn't know what makes one book better than another anymore, they all look the same now. But I do remember being annoyed by Lang that I was forced to rediscover the infinitesimal interpretation for myself. Lang has an epsilon-delta appendix. I don't think a nonstandard analysis textbook is what you want, they tend to introduce unnecessary complications, like "standard part" which are only useful for logicians.

@Nikolaj: In the last comment adressed at you, I should have said "the displacement in a time $dt$ is $\sqrt{dt}$". The mathematician's version of this omits the differential square-roots from the notation, and is called "Ito's lemma". The Feynman path integral is the same as this. The analogs for Levy flights are still not worked out anywhere, because of the avoidance of fractional roots of infinitesimals.

@wmnorth: The idea of an infinitesimal is just a tiny number which is so small, it's square is negligible. Then you ask, what is the change in, say, (x^2) when you go to (x+dx)^2, and this is 2x dx, which gives the definition of derivative. The infinitesimals are the main idea in calculus, and I don't understand how a book can omit them. If you look at old enough calculus books (the best books are around the mid to late 19th century), you can find a discussion. Feynman also discusses infinitesimals in vol 1. They are commonplace.

@Nikolaj: I don't like giving references for obvious things. When you have a random walk, the distance squared is linear in t, so that the displacement in a time $dt$ is $\sqrt{t}$. This is well known in stochastic calculus, but usually stated extremely opaquely because of the drive to avoid infinitesimals in this field too.

@Nikolaj: I shouldn't have downvoted, the answer is not wrong, but I don't like that to understand something as obvious as an infinitesimal relation, the student is diverted to the superficially intimidating ideas of duals of tangent spaces. The two concepts are only obliquely related. For an example of something infinitesimal which is not interpretable as a differential which arises in gasses, consider the number fluctuations in a finite region over an infinitesimal time dt, these scale as $\sqrt{dt}$. You can't express this infinitesimal relation using differentials.

-1: This answer is no good. Fermi doesn't mean differentials, he means infinitesimals. The two concepts are different, only coinciding to first order, and the mathematical obscurantism regarding this is intolerable.

@user1247: The problem is that CTC theory the way you envision it is not a real deterministic theory, there are constraints on the data which require self-consistency. It is a little silly, for example, if you take a thermodynamic system around a CTC, there must be a maximum entropy point, and entropy goes down on both sides of this. It's really not the way physics works, and it would be worse than QM philosophically, it would require constrained computations to determine the future, and it would have miraculous entropy reducing moments along CTC's. It would be magic.

@user1247: My skepticism is based on the fact that the linear superposition aspects of QM are not reproduced from any classical theory so far, even when I say "go make one up, no constraints, do whatever!" This makes it doubly implausible when someone comes up and says "maybe it's classical fields". There are no CTC's produced exterior to black holes in classical GR, and the interior and exterior of black holes are causally disconnected in classical GR. But even if you make up laws that say they are produced, you still can't reproduce QM from this, because it's a tall order to reproduce QM.

@user1247: You are repeating the same annoying points--- a wormhole is when you go in one black hole and out another disconnected one. This is much less plausible than going in one black hole and coming out the same black hole later. QM just doesn't emerge from GR or any other classical field thery, it is a different kind of theory, it has amplitudes for superpositions of different GR fields, it has a global notion of wavefunction that has no relation to classical fields. I don't know why you think that GR has to explain QM, other than that Einstein hoped this was so. It doesn't work.

@user1247: The "CTC"'s in traversable black holes (going in and out) are completely unphysical (this is the only reason one should take this picture seriously)--- it's just a mathematical CTC in a tiny unphysical skin near the horizon. It's not a real CTC, it's something that's washed out by the proper quantum definition of horizon.

@user1247: I didn't say anything about wormholes.

@lurscher: The point is there are no CTC's in GR outside of some ridiculous black hole interiors, so this is a game of make-believe. There are no actual CTC's even in the case I gave, where you come out of a black hole in this universe, since the only closure happens in un unphysical trans-Planckian skin around the horizon. The OP was asking about CTC's, and there aren't any.

@user1247: Ok, then, the pure classical GR solution doesn't allow things to enter and leave in this universe, it requires the black hole to link to another universe. This picture doesn't make sense at all, and you get endless paradoxes, like "what happens if I throw positive charge into a neutral black hole to make it near extremal, fly in, come out, then throw negative charge behind me to neutralize the hole again?" These questions only become clear if the classical in-out is an actual in-out in one universe, not in the classical way, between separate universes.

@user1247: I meant the classical solution has geodsics that go in and out in r, going in and out of identical black holes. In classical physics, these entry and exit events are disconnected, but they are linked quantum mechanically, so that the coming out is only a finite time later than the coming in. In strict classical mechanics, the coming in is not linked to going out.

@LubošMotl: Yes, I know. I was also intrigued by Einstein's program, but perhaps more skeptical, as I never seriously believed it would work to reproduce QM. I also halfheartedly looked for solitons early on, but never found anything, then I read Skyrme and Coleman, and gave up. What was your soliton? Was it GR? I think Einstein had a trick for getting Bohr Sommerfeld quantization from an action condition in GR (it is reproduced in a GR book without attribution, but I suspect it's Einstein's claw). Your biography can be turned into mine, if you substitute East for West and stop in 1995.

@user1247: If Wikipedia says there are CTC's on the exterior, they are totally completely and unequivocally wrong. There are no CTC's except inside the horizon, the exterior is always causal. Why can't CTC's reproduce Bell's inequality violations? Because the CTC's are always inside one black hole, and don't link two distant points. To get Bell's violation, you need to go back in time a long ways and fiddle with the hidden variables in the far past.

@user1247: Your hypothesis as explained already, is ridiculous--- quantum effects have nothing to do with GR. I am not quite talking about quantum effects, only sort of. If the singularity has closed timelike loops, they have to be noncontractible and you then have to unwrap it to a simply connected causal thing, and I don't know how to do that. There are no CTC's in the exterior, nor are there actual CTC's in exterior plus interior in-out traversal, although there are phony CTC's. The CTC's don't give quantum mechanics, they can't do anything like reproduce Bell's inequality violations.

@user1247: The CTC's in Kerr metric are near the "ring singularity", and are in a region where you can't trust the classical theory completely. I am talking about exterior observers which never get close to the singularity. These, if they go in and out, as I believe they do, in a technical sense return to their "past" (not really, this is not their past, except in an unwarranted and known-to-be-false near-horizon extrapolation). This is supportable by classical calculations and precise quantum models. What happens near the Kerr singularity, I can't say for sure. I don't know.

But quantum gravity, with the holographic skin, allows (and requires) near extremal black holes to emit classical stuff. It's not complicated, and it doesn't change much of what is known. CTC's of the usual sort are still not allowed.

@user1247: What's a "gold star"? Classically, you get a closed timelike curve whenever you exit the same black hole you enter any time later, because the geodesics which cross the horizon going in get frozen forever in t, while geodesics going out also get frozen in t, so they meet somewhere impossibly close to the horizon. This is a well-known artifact, discovered by 't Hooft, of horizon descriptions, it is also the reason black hole entropy is naively infinite, and the resolution is to say that you exclude a brick wall. There are no CTC's in classical GR, so people assumed BH's don't emit.

@user1247: Lubos's answer, condescending or not, is correct. The thing I was talking about is not an acausal thing with closed timelike loops, but a case where the closed timelike loops only superficially appear, in reality, they are only time-like close in a skin too close to the horizon to be physical--- within a tiny part of a Planck length of the horizon itself. In quantum gravity (string theory) you get rid of this region as unphysical, and this is why you need quantum mechanics to make full sense of this behavior. It is not acausal at all.

But it is additive in parallel. There is no contradiction with energy conservation, because the displacement in the direction of the force is additive in serial and not additive in parallel. This is the intuitive puzzle for Alraxite--- serial vs. parallel springs, and it's not completely trivial in my opinion, and his arguments were over this thing, not over the notion of tension.

I sorted out the intuition failure--- it might be just mine and Alraxite (but I doubt it). When the two astronauts are pulling on the same rope, the momentum flow is constricted, and the forces are half as what they are when you are pulling on different ropes. This is what was counterintuitive to me and also to Alraxite, for similar reasons--- why should the flow of a conserved quantity care about the details of what particular region the conserved quantity is going through? But it does care, because when momentum is flowing through two force-generators, the net flow is not additive in serial,

-1: Alraxite makes reasonable arguments (he confused me), and the responses to his questions are flippant and wrong. If you have two astronauts carrying nearly infinitely heavy backpacks, with ropes, one attached from A to B's chest and the other from B to A's chest, you wouldn't get confused that the pulls of A and B add up to determine how fast they approach each other. So why is it confusing when they are both pulling on the same rope? To be honest, I agree that it is confusing, but I haven't sorted out why. There is some failure of intuition here among the trained, not among the untrained.

Dennett is almost a positivist, in that he accepts their conclusions, but not their principle. The founding principle is "two ideas which are identical in sense impressions are identical period". That's incompatible with the logical coherence of the zombie concept, something Dennett accepts, but does not use the basic positivist tenet to refute. Your characterization of Carnap et al, is naive, they thought exactly like I do, to the tee, except for the God bit.

It's not the whole world, but it's something, and even this little bit has been taken away in the philosophy department.

And it stops you from arguing over pointless distinctions that don't make sense.

It just lets you see what makes them sophisticated.

It doesn't let you speak about sophisticated notions better.

I am explaining what the foundation gives you.

What?

Or at least, solved as well as grandmasters understand it.

But, for example, chess evaluation is solved.

The algorithm is pretty simple.

The reason is that grammar is done automatically by "cheap circuits" that don't bother the parts of our mind that does the ideas.

Language grammar is not that complicated. Semantics is.

The grammar business? That's not hard.... it's just done incompetently by linguists.

To understand the mind, you need an AI.

For example, to understand language grammar, you need a computer program that understands grammar.

It does happen, but extremely slowly.

It allows you to speak about the meaning, to know what is vague and what is precise, and to know when you have completed the project of understanding something.

Because the computer implements the reduction of the imprecise to the precise.

What do you mean?

The computer is precise, but its internal computations produce contradictions.

The ambiguous meanings, even with its contradictions, can be represented in a precise foundation, using an intermediate computer to simulate the imprecise thinking.