joigus

Gravity would exist, but only as a continual bubbling of virtual states. Gravitons would pop up from the vacuum and die down without having found any real particle in their way. In terms of QFT, it would be only loop Feynman diagrams. Assuming a cogent theory of quantum gravity is found some day.

At several points on his lectures, Lenny Susskind makes a very emphatic case for that. I'm guessing it's in his book, The Theoretical Minimum? Any introductory book that explains to you that the most relevant thing about GR is equating geometry to energy-momentum, will do the job. Mass is not a central concept of GR, to put it mildly. I'm sorry I don't remember in what particular lecture he delves into that a little bit more, due to a question from the audience. But you should be alright if you follow his lectures on GR.

John Wheeler came up with this wonderful phrase to summarise what GR is about. But buzzwords can only get you so far. If you have a situation in which a small object moves in the vicinity of a stellar object that heavily distorts space-time around it, then it's fair to say that the star tells space-time around how to bend, while the relatively small stuff moving close is told how to move. However, consider the collision of two black holes. In that case, both the motion of the objects and the warping of space-time are very difficult to tell apart. For those cases, the only alternative is to appeal to the equations and have a computer solve them for you. The equations are highly non-linear, which means that ultimately it's impossible to express the warping as the sum of contributions of this and that piece of matter. Gravity itself gravitates. See my point? Another aspect I would like to point out is that mass is not the source of the gravitational field. It's energy-momentum that plays that role. The "mental operation" that you're proposing here, if I've understood you correctly, is to remove the sources and be left with an empty space-time, and then you ask yourself what shape does that space-time have. Well, think about this: Einstein's field equations have many solutions corresponding to an empty space-time. Gravitational waves are a particular example of solutions to the Einstein vacuum equations. So I guess my answer is: No, you can't figure out what space-time is like with nothing in it. Not a priori. You have to make a guess.

Whether torque is a vector or a pseudovector is a question that has nothing to do with energy being a scalar that comes from time-translation invariance. I'm not sure how relevant this is to the discussion, as I'm not sure of what's being discussed, and I cannot formulate what the claimed speculation is. This distinction vector/pseudovector has nothing to do with energy stemming from time-translation invariance. Noether's theorem is concerned with continuous symmetries, while the "pseudo" in "pseudovector" has to do with rotation-related vectors (angular momentum, torques, angular velocities) being perfect vectors with respect to rotations (also called SO(3), but behaving differently to normal vectors (position, velocity, etc.) with respect to so-called improper rotations; that are the product of rotations by inversions of coordinates. Normal vectors (SO(3) vectors; polar vectors being another name) change the sign of their components when we invert the sign of the coordinates. Pseudo-vectors do not, because they are really "external products of vectors". I hope that helps, but some parts of this thread sound like dadaist poetry to me. And I don't mean either Swansont of exchemist's contributions. I don't mean to be facetious. I really would like to understand what point is being made to be of any help at all.

Wouldn't you be satisfied with the regular lemmas and theorems, like L'Hôpital and such? Finding an N(epsilon) is a pain in the neck.

(My emphasis.) The question, of course, is how rare "exceedingly rare" actually is. I'm working on reading all the comments. I tend to agree with people who have pointed out that ants are interesting from many points of view and to many other organisms --Airbrush, Charon Y. Humans have been busy with ants probably for hundreds of thousands of years. The interest of hypothetical civilisations on us ants would have to be weighed against the Herculean task of overcoming the enormous difficulties of finding us in any useful way, as exchemist and swansont have implied. I don't know what that says about the argument Greene is trying to make. It's possible that the analogy does not efficiently bear out the point, even if the point is a good one.

Is what you want anywhere at the level of rigorousness as proving that, \[ \forall\varepsilon>0\:\exists N>0\:/\:x>N\Rightarrow\left|\left(1+2^{-x}\right)^{x}-1\right|<\varepsilon \] ?

Hi. I did not address the second question, because I figured you would solve it as soon as you applied yourself to it. Apparently you've solved the 1st one but are unsure of its validity? The first question is best addressed with assymptotics. Here's a sketch: \[\left(1+2^{-x}\right)^{x}\sim\left(1+2^{-x}\right)^{2^{x}x2^{-x}}\] Now, \[\left(1+2^{-x}\right)^{2^{x}}\sim e\] And, \[x2^{-x}\sim0\] So, \[\left(1+2^{-x}\right)^{x}\sim e^{0}=1\] Now, making it rigorous is another matter. In fact, I went dangerously off-bounds when I said \(x2^{-x}\sim0\), as you're not supposed to ever say anything is ∼0 in assymptotics.

Sorry. I did the derivative. You need the limit. The limit is of the kind 1^(infinity), which are solved by relating them to the definition of e=lim(1+f)^(1/f) if f-> 0 when x->infinity. The result is 1, if I'm not mistaken. I'll take another look tomorrow. Tell me what you get.

This is what you want: https://en.wikipedia.org/wiki/Logarithmic_differentiation#:~:text=In calculus%2C logarithmic differentiation or,rather than the function itself. Plus changing exponential bases: 2^(-x)=e^(-xln2)

Yes, Pinker's quote is what's in boldface. Pinker's adjectivation is unmistakable. They used atomic randomness as a random-number generator. You read the numbers of decay times and use those numbers as input for another variable that has biological relevance by setting this other variable to take those values. That's what I understand. I hope that helps answer the question, but some feedback would be very useful.

The Road to Reality is a very good book. Any recommendation from Genady is probably worth considering, OTOH. It's not a book to actually learn physics though. It's more of a whirlwind tour of the exciting topics of modern physics. As to popular books with emphasis on experiments, I recommend, Weinberg's The Discovery of Subatomic Particles. Then you can try the Feynman Lectures on Physics. If you want to learn physics in earnest, you probably can't do much better than Landau & Lifshitz's Course of Theoretical Physics. Encyclopedic (10 volumes.) A bit old, but will take you a long way in understanding the deepest principles of physics and how they're applied. There are many other books, I'm sure, and with a more modern focus. A good rule of thumb is: The more unassuming the title is, the more likely it will take you to the nuts and bolts. I think you get the idea. Although we could hardly be more off-topic.

This is no coincidence. The Laplace equation must monitor the spatial/vacuum factor of any physical equation worth its salt, as Einstein observed in his famous popular book on the theory of relativity. Physics does deal with many 3-dimensional problems, of course. When it comes to proposed generalisations of the standard model, it may even consider 26 dimensions. Why do you say that?

Yes, you're right. It's me who brought it up. I forget what it had to do with the primes, though. Or how the question surfaced. I'm no expert on number theory. You may be on to something when you say that differentiable functions of a complex variable may have to do with the prime-counting problem. A function being differentiable in a complex variable z=x+iy is a much more restrictive condition than the corresponding condition on a real-variable function. But in mathematics it's important to define anambiguously your concepts, and the state clearly what your assumptions are, and proving rigorously whatever you say about them. Sorry, I can't contribute much more at this point. I hope that was helpful.

Exactly. God of the gaps is what I meant. Filling in the gaps in your knowledge with answers that are emotionally satisfactory seems to bode well with the needs of a brain that has a frontal cortex as sophisticated as ours, with so many neural connections. And with a FOXP2 gene that accumulates comparatively so many mutations as ours does. The idea is that having a representational scaffolding for all those things we feel unable to solve by reasoning may have some kind of evolutionary advantage. From what I know, paleoanthropologists are toying with this idea. Whether such is the case, or it's simply a spandrel, we may some day learn. It could be just a spandrel that comes with the territory. I watched this talk some time ago. Very interesting. Everything Thomson says is compatible with religion being a psicological spandrel though.

What's the speculation here?

As suggested by other members, gods were invented (not created, I would say) by humans. God is a place holder for everything we don't understand. Patterns can be found in the way different communities of humans fashion their deities. It is no coincidence that monotheisms have been developed by desert peoples. Forest peoples are more prone to animism. The first gods have been identified with powerful animals, ancestors, most basic needs (fertility, good luck,...) Sun and stars. All things that were important to those societies. I think it's a natural byproduct of the activity of a brain that desperately needs to plan ahead. Mind your infinite regressions, BTW: Who created the one who created god? And so on.

I did?

I don't understand what you're getting at. Take, eg., the sentence, ¿i being used as a spanner in the works? What does that mean? The imaginary unit i does not come from a choice, as the vector (0,1) does, for example. Complex numbers are much more constrained than vectors, and obey different definitions. There is no such thing as rotational invariance for complex numbers, for example. Quaternions are very different from complex numbers also. E.g., they're non-commutative. Serious maths are not about "this looks like that" and such. So, while I don't understand what you mean, I see many problems with some of the things you say. Too loose connections in what I can understand from what you say. Harmonic functions are the real and imaginary parts of differentiable functions of a complex variable when expressed in terms of x and y. They're used to represent 2-dimensional problems having to do with the Laplace equation (electrostatics, laminar fluids in absence of eddies, etc.) I don't know what that has to do with the Riemann hypothesis.

Is this homework?

You are conflating several things here that are actually different. Surface of last scattering: The surface in the night sky beyond which we cannot see because at times older than that the universe was opaque to radiation. The distance to us of this layer of the universe changes with time, and is being pushed by expansion towards the kinematical horizon --see below. Kinematical horizon: In a De Sitter universe, every observer has a limit to how far away he can see the galaxies. The distance to us of this layer of the universe is fixed in a universe with a constant rate of expansion, and so doesn't change with time. It doesn't depend on technology either, as it is given by a natural limit. Namely, receding velocities greater than the speed of light don't allow you to see anything there due to redshift. Why the surface of last scattering is "about to disappear" (give or take a couple of billion years) from view in this time of the history of the universe, and whether that is a coincidence, is not known. But it appears to be so.

I bet we haven't heard the end of it.

That's why we have idioms as 'blessing in disguise.'

Define good (vs. bad) use of science. Define making things better. Define what is good. It's clear to me that just good is better than just better. A little better is certainly not as satisfying a lot better. I have clear ideas about "better than nothing". Agreed, but... Are all possible directions equally good?

You're right. There's no conflict. Google search term: God of the gaps.