joigus

Where does this anti-thermodynamics of yours hold? Not in this universe. Sorry. It wasn't @michel123456 who said this. It was @Edgard Neuman. The quote function (or my misuse of it) did that.

OK. Give me a list of books, papers, and authors I'd better read to meet your standards. I really would like to improve my knowledge in the part. Another victim of the peer-review system! No specifics? The referees didn't get past the generalities! May I ask what generalities were those that the experts found so unpalatable? Or is this not relevant to the present discussion, like talking about energy and Noether's theorem?

How do you convert seconds to bits? That is only too obviously not the case. Fixed set of rules do not in general determine behaviour. Nor true in literature in particular. OTOH, literature obeys patterns rather than strict rules. Where does this anti-thermodynamics of yours hold? Not in this universe.

I think you're living the "deductive only" delusion. Either that, or completely missing the essential connection between induction and deduction. Plus, may I say, everything most people are telling you here. There's a common understanding in science and rational thinking that apparently you're not privy to: All thinking starts with induction/observation (that comes first) Then: --> inference of patterns --> proposing definitions and laws --> deduction of both seen and previously unforeseen consequences --> Testing --> Refinement of induction --> confirmation/rejection of theory --> formulation of new theory or refinement of previous one. Something like that. It's long, it's arduous; it takes time, effort, money, and many brains working together. That is the process. You need to get over the axiomatic dream, or the illusion that induction and deduction occur in completely separate levels that don't talk to each other. That's not how it works. And the absolutely essential piece that closes the circle is experiment. As to connecting experiment, religion, and common sense as different motifs for "belief"... I think you've really lost your bearings there. Evidence and belief are not the same thing. It is true that the evidence is always affected by the theory as to its format; the language, if you will, in which the answer is presented. But the process by which we acquire evidence, and the one by which we acquire belief; plus the degree of certainty of both, the objectivity the achieve... It could hardly be more different.

I do remember: It's not possible. It is completely on-topic, as we're talking about symmetries and conservation laws in dynamics, which is relevant here. You haven't answered any of my questions/objections, or anybody else's really, which can only mean one thing: You have no answers but just insist on foisting your idea on everybody by means of a war-of-attrition tactics. Repeating your initial point over and over and not paying heed to anything anybody tells you won't get your idea very far. And I honestly think I'm being very generous by saying "your idea". If you're so sure about it, why don't you send it to a peer-reviewed journal?

They do. I know, and you know. And I know you know. And you know I know you know. Can we stick to the topic, please? It's algebra. Group theory. That's why we are @ Linear Algebra and Group Theory normal subgroup problem Ah, OK. I'm sorry if I misunderstood your question in any sense.

Also, by "normal" (in this context) I understand: \[gHg^{-1}\subseteq H\] Not "perpendicular". Any more questions?

Bingo!! "Is a group" refers to algebraic properties. \[g_{1},g_{2}\in H\Rightarrow g_{1}g_{2}\in H\] We're not talking topological groups. (I'm not aware that anybody mentioned a basis of neighbourhoods). Welcome to page 1. Neither have I read anything about a metric space.

To tell you the truth, this has caught me a bit unawares, because I've noticed that my comments on philosophy go largely unnoticed. And there must be an objective reason for that. I think people here (serious people like yourself, I mean) instinctively perceive that I'm a bit flabby when it comes to philosophy. Not that I'm complaining; I was just kind of assuming that I'm probably not up to the standards of the community in this particular front. I think I can make more significant contributions on the interface of mathematics and natural science, helping with insights in problem-solving, physical concepts, etc. Also learning about them. My attitude towards philosophy is more like that of a tinkerer. I do have a problem with philosophers, I must say. I tend to get enamoured of different tidbits of philosophical thinking. I see them as cute little thinking gadgets. Some examples are Wittgenstein's critique of classical categories, or Kant's "cardinal" partition of categories of judgement into a priori, a posteriori, analytic and synthetic. What I've never been able to feel comfortable with is their systems of thinking as a whole, for lack of a better phrasing. I am very suspicious of the claims for generality and encompassing power of almost every philosopher, or philosophical school. Even the usefulness of big chucks of it. "Their big picture", I think are the right words. I suppose the same happens to me with respect to Dennett. For some reason I find his criterion of free will as a hasty one. I think he's succumbed to the pressure of the need for a practical criterion, rather than facing up front the hairier aspects of it.* It's only natural for every great thinker. They all have their fuzzier areas, I suppose. And I do agree that Dennett is one of the most brilliant philosophical minds of our time. In the particular arena of free will, I would feel more comfortable with a graded concept, rather than a switch to decide when people enjoy free will and when they don't. I think a concept like that would be far more powerful, amenable to the usual hypothesis-prediction-falsifiability sequence of science. The backcloth would be: 1) Nobody is really free, because, in the end, we're all physical systems. But, having said that: 2) There is a working definition of free will which is of great practical, social importance. You'd better have something like that going on in your life or you're going to have a very difficult life and be a problem for others and for yourself. And, 3) This practical definition, you can fall short of achieving for many different reasons: neurological, biochemical, societal, educational, etc. (necessary qualifications and criteria for measurability, predictability, falsifiability would go here). Something like that would make me much, much happier. I don't like conceptual switches. * Edit: Although I agree that his dissociation of determinism and the problem of free will is very enlightening

Take a guess.

Please, be patient. It's been a long time since my last homework assignment.

No mercy, please! +1

Far fewer. I hate to make grammatical mistakes!

Good tip.+1. Is this homework, @Prasant36? Edit: You also need Abelian character for showing closure

OK. Give me the dynamics of the Solar System with only integrals of motion derived from continuous symmetries. You can't. And the reason is that the system is not integrable (the number of integrals of motion is insufficient, far less than the number of degrees of freedom.) This is how: You wrote down Einstein's equations in the vacuum from the Einstein-Hilbert action. I just assumed you were talking about GR. Were you not? I know (and I've told you so) that there are problems that you can solve with an energy-like integral of motion. Example: Using the Schwarzschild solution to derive the anomalous motion of Mercury. But this is not generally valid. If you think it is, give me the integral of the energy for the FRWL metric. You can't. And the reason is that the metric does not have time-translation symmetry. Need I say more? Something tells me I do.

Yeah, I don't know what you did with the quoting function there, @ruibin.niu. Be careful. I think there are some merits to your gedanken, but you got the analysis wrong. Careful application of SR formulas (Einstein's velocity transformation law) gives you that, \[\theta_{i}<\theta_{r}\] While you're assuming that Snell's law, \[\theta_{i}=\theta_{r}\] is correct. It is not for moving mirrors. You can find the derivation in Landau-Lifshitz; Classical Field Theory, Vol. II. You have a larger reflection angle than incident angle, which accounts for the relative direction of light rays in the \[v_{\parallel}\] for both observers to coincide and the whole picture to be consistent.

If you're not a supporter of GR, don't ever use a GPS navigating system; because GPS-monitoring satellites routinely use GR in order to find your location. And they seem to do a good job of it. You see, "supporting" a theory (your words) is not like supporting a football team. You need to go back to page one on the book of science and start all over again. There's something very basic you haven't understood. https://www.physicscentral.com/explore/writers/will.cfm#:~:text=GPS accounts for relativity by,functions within about 2 minutes.

Pre-Clovis artifacts have been popping up for quite a while. Very interesting topic. +1 There's even a hypothesis whose main proponent is Dennis Stanford, from Smithsonian, that puts forward the idea that some Solutreans populated parts of the Americas following the track of seals across the North Atlantic. The last thing I've read is that DNA analysis has shown that no relevant traces of proto-European DNA have been found in native Americans of non-mixed descent. But there's more information on the Wikipedia article.

Oh, you're sooo right. +1. I will rephrase the flaw in the OP's argument: Snell's law, as is, does not hold for moving mirrors. https://physics.stackexchange.com/questions/405904/deriving-the-law-of-reflection-for-a-moving-mirror-in-relativity

Very good point. +1 While it is true that you've show an intuition of something unexpected, we know how all kinds of very similar naive intuitions fail one by one when you apply the formalism. Can you provide equations, showing the velocities and the angles in both inertial frames, or are you expecting everybody else to believe it only based on intuition? Do you want somebody else to do the analysis for you? IOW: Why should anyone bother with this?

You make some very good points here. +1. You also introduce an element which must have slipped my mind, which is the one of usefulness. This element of usefulness actually had more to do with the original meaning of my question. The ethical question, important though it is, was not what I had in mind. That doesn't mean I don't welcome any other aspects that other members may have in mind. I do. Also, I'm not saying that the ethical question is out of reach for rational thinking, which I think it is. Concepts such as common sense, equanimity and the like are quite useful (again) to get to working standards for ethics. Golden rule is the best example. I agree that potential for harm should not be the criterion upon which we base what we ought to know and what we oughtn't.

And a take-home lesson about diet. Maybe bad science is science unchecked by philosophical concerns?

The technique that Markus describes is actually something I've seen done in theories of gravity when people try to discuss emergent time. But I think you're right that the sequencing aspect is lost or not entirely obvious. That's why I suggested the introduction of a proper length parameter as the most likely candidate to where, mathematically, time comes from. Suppose you have a physical system with variables (x1,x2,...,xN). If you had N-1 implicit equations or "constraints": f1(x1,...xN)=0 f2(x1,...xN)=0 ... fN-1(x1,...,xN)=0 This would amount to having a common history for all the x's. The complete deterministic solution to a particular system's evolution deployed before your eyes. Giving values to all of them except one allows you to infer the value of the remaining one. Although that value may not be unique. For example. At a certain point in your life you asked your sweetheart to marry you. That's x7=1. But there are two values for the answer that she gave you: x8=1 and x8=0. That's because at some point she said "no", you asked again a week later, and she said "yes". Assigning values to the rest of the variables of the universe, you can get "back" to the event you want to picture. This gives you a bird's eye view of the history. If you have a metric for the x's, you can do something quite impressive in principle if your metric is positive definite and based on a quadratic form. You build the infinitesimal interval along your implicit curve: ds2=gijdxidxj and define your sequencing (primitive time) as either, dt=+sqr(gijdxidxj) or dt=-sqr(gijdxidxj) I know @Markus Hanke has some exceptions/criticism for this procedure. But I think it's a mathematical language that very much fits what we know about the mathematics of the world, has many of the essential features of time in it, and leads to a quite clear-cut setting (if not solution) of the problem: Why is it that we only see the world evolve in one of the two orientations that the metric allows you to pick? The mathematics seem to confirm that: You chose one or the other, but once you're fixed on one, no valid re-parametrization can give you the other. I have an intuition that this may have to do with the Michel/Studiot argument about the cup of tea. But I must go over their arguments again, I must say. Edit: x-posted with Eise Edit 2: The metric should be defined on the f's, not the x's. So it'd be, ds2=gijdfidfj

Trying to use reason to understand the internal arguments of religion is as hopeless as trying to thread a needle through a loop.

Good point. +1 Why did god do it? Many scientists in the past, feeling very uncomfortable about making statements that denied god, chose to phrase the question as, Why did god choose to make the world like this? One answer for every question "God did it" is clearly not good enough.