joigus

You must be a very good teacher, Mordred. Thanks for the tips. The boy has no problem visualizing things, though. He takes brief spells of time processing the explanation in which he seems to be lost in his mind, I hear him mumbling something, and ends up shouting "now I understand!!!" It makes my day. He meant h with no bar for linearized gravity. For some reason h is universally used as the first order corrections to the metric tensor. I've never seen them written with any other symbol.

I don't know about that, but it wouldn't surprise me either. I don't feel very strongly really about defending Christianity or any other monotheism. The Romans played the same propaganda trick against the Phoenicians, so I see no reason why they wouldn't have played it again. I simply don't know enough about it to be sure one way or the other. But what you're saying makes a lot of sense to me.

Overall, ADHD attitudes (I'm not sure they're as much a congenital condition, though it may well be) are much more of a problem for a good teacher or a teacher than intends to be a good one than neurodivergence in the form of AS. Very detailed and interesting analysis. Thank you. Yes, I think I'm in the clear in this respect. I've developed an attitude not to expect anything to go according to plan. It's nice if it happens to go, though. Ha ha. I understand. I'm getting used to his "You got it wrong!!!" or "follow me?" . If you understand where it's coming from, it's quite funny and endearing in a way. I must say: I do not always follow him! I suppose those extreme cases have to do with environmental factors aggravating the situation... Look at it this way, Markus: I sometimes can't find a good reason to remove myself from social situations. You, on the contrary, can go back to your equations and your meditation in order to protect yourself. I know it's easy to say, but looked upon the right way, it's a gift!

Yeah, I'm generally optimistic, which doesn't mean I'm starry-eyed or overconfident. It may take time, but the course of change is well defined in one direction. Resistance is met, I know, primitive drives die hard. I hope I'm not wrong. Although I'm an atheist too, I agree that Christianity probably played a positive role during the first centuries when it entered the Roman Empire, by substituting human sacrifice and similar horrours by something more humane. Less damaging myths at that particular point in history if you wish. Some of my atheist friends do not agree with me. But obviously, after that, pretty much Christianity by itself with its different internal divides and war against other religions is involved in every major conflict that Europe and the Middle East have been involved in through the Middle Ages and after that. Both race (as a convenient political construct to argue in the past for the rights of your tribe to dominate over other tribes, to incorporate @iNow's last point) and religion, have played a major role in arguments for political power.

I think a basic, down-to-earth, practical-for-life, science education for everybody is important because we're living times when religions are loosening their grip on people's minds in the West and other countries with similar tendency. As a consequence, it's only too obvious that some religions are tightening their grip --as a defensive move-- on what's left of their flocks (or desperately trying to) in many developing countries, while swathes of people in the West are turning their heads towards the occult, and fresh-from-the-nonsense-factory myths. The reason why I imply that "race" is a blurry concept at best is because: 1st) Humans have a minimal genetic dispersion among primates 2nd) Africans have a maximal genetic dispersion among themselves in the human family So take, e.g., people of African origin. They have been some of the most discriminated-against for centuries, based on the colour of their skin. Let's leave aside for a moment the most emotional factors like those related to sheer cruelty, injustice, pigeonholing, etc., important thought they are. Does it make any sense at all to separate Africans based on a secondary characteristic (catch anything you can by Nina Jablonski on the role that vitamin D played on it) once we learn from science that humans are extraordinarily homogeneous from a genetic POV (Toba catastrophe being pointed at as the main culprit) and that, even that being very likely, Africans are among the most genetically disperse among themselves? Race, as a concept to divide humans, or even classify them, especially if or when one of the proposed categories is "Africans" vs "non-Africans" would be at least about as silly as trying to study animals as separated into "jellyfish" and "non-jellyfish." That completes my point about race. And I do hope nobody out there (I'm sure not you, @koti) misunderstands my use of the jellyfish example. Agreed. That's why it's so important to study mechanisms in people's minds as such, and not as "they do it just because they want to." I'm optimistic though. I think humans have an uncanny ability to collaborate that's not present in other species. We just need to stimulate it and render the irrational self-defence mechanisms dormant when they become more of an obstacle or bring about any kind of injustice.

+1. These are confusing times. Science education is more important than it has ever been. Just one more thing: The concept of race no longer is useful in any sense that I can think of.

Thank you, Studiot +1. I also have students with ADHD BTW, which is probably worth another thread --very different topic that I'm just mentioning. A crash course on these matters seems in order.

Thank you, @Phi for All. OK. I've been looking for "puffer fish+asperger" on the internet, and I think I may be starting to get a faint idea of what you mean. It may have to do with a day that he was overly defensive because he had been talking with friends on his social network and seemed to have felt "under attack." Is that what you mean? Thank you, @naitche. I just hope I don't let him down. The heads-up about the pressure on understanding and the one about expectations have been very valuable. Maybe the topics on which I'm a bit more faltering and I normally tend to try to wing it, I have to rehearse in advance so that the lessons convey an impression of easiness and smoothness. That's something about @Markus Hanke that strikes me. He "sounds" on his writing pieces as very articulate indeed. So maybe he's practiced on language to the point of completely compensating for the verbal handicap.

Bingo!

Does that help? v = 0 is when the velocity graph crosses the axis. Just "one instant." Let's say the particle is at rest but doesn't have time to rest. Words are tricky in physics. They sometimes project "mirages" in our minds...

Ok. I think it's what MigL was trying to tell you. It doesn't come to rest. It is not at rest. It goes through rest for an instant, so to speak. I'm still looking for a graph. Maybe Studiot can help with the graph, which is after all something you were asking for. Here:

I will have to go in bits and pieces, because it's taking me a while to get up to date with everything. By reductionism (not shallow, nor naive reductionism) I mean the contention that in general small (many in their instances, simple in structure, and few in their categories) parts determine what the big (much fewer in their instances, very complex in structure, many in their categories) self-organizing systems of matter do, and not the other way about. "Determine," for me, is a physical causal connection; not --repeat, not-- a contingency in our mechanisms of explanation. So it doesn't really matter that much whether the explanation is easy or convenient or how many variables you have to use to describe the causal connection. By emergent phenomena I mean those patterns of regularities that do not belong to the simpler parts on their own, but can be deduced in principle from the fact of there being many of them and how the different categories interrelate. A relatively small-scale example would be phospholipids in their role of forming membranes. You can't see a membrane in the properties of one individual molecule, but if you consider many of them in aqueous solution, it's relatively easy to understand how they tend to group together by hiding their fatty hydrophobic part towards the interior of a bubble, showing the polar part towards the exterior, and thus forming an isolating unit. It is because the questions of reductionism and emergence are important for this matter of free will or whether it makes any sense at all, or is just about words that I put it out here. I wish to correct myself. I think the panoply of states that determines decisions at least must include: wish belief fear revulsion reckoning ... More coming.

At maximum height \[\boldsymbol{v}=0\] but \[\boldsymbol{a}\neq0\] Is that what's confusing you?

Thank you. Yes, exactly. Yesterday night, while I was watching a documentary on the topic, I started thinking about something very similar, to what you're saying, although with respect to the signal sending/detecting problem, instead of the interstellar travel. It doesn't seem to occur to anyone to invest effort and resources to send signals around in all directions in order to increase the chances for possible ET civilizations to detect us. Would we want to do that? My instinct is that life anywhere is bound to be very cautious and tend to listen, probe and grope in the dark rather than be loud and proclaim "Hey, we're here!" My best guess is that there are probably some other ears in the universe listening in the silence and eyes watching in the dark. Maybe not many, but none very interested in being detected themselves.

I had you in mind, Markus, but I didn't want to press you for information, so I didn't mention you. Thanks a lot for your input. This totally checks with what I'm experiencing regularly. Sometimes I explain something, give him an example. He seems to have zero problems understanding very difficult things, like transcendent operations, limits, and the like. Geometry is a piece of cake for him once he pictures in his mind what's to be done. Only once it was a bit painful when dealing with projection of one vector on another, because he was stuck in trying to solve the triangle a, b and a-b, when it was about the right triangle. It took me a while to realise what he was trying to do. It's language what seems to be more of a problem sometimes. Especially when I require him to tell me in words what the idea is. I try not to press too much about it. I just try to rephrase slowly and deliver more clearly. I do have the feeling that something more abstract and quite independent from language is going on in his mind. If you just knew how many times I've thought I may have suffered from some kind of very mild form of the autistic spectrum myself. I certainly had many problems with the way most people used innuendo, double meaning, and the like. Yet I was very skillful with language. My way of seeing it today is that what we consider autistic-spectrum diseases may be more usefully considered as a different or non-overlapping spectrum of cognitive abilities that render you inefficient at dealing with certain situations that pose no problem to other people, precisely because your brain is trying to weave a more complex and powerful structure than standard. I love comedy. It's helped me a lot in understanding language, its limits and its flexibility. So that's going to be part of my homework for the Summer. You're welcome, and thanks back to you. I recognize this as a once in a lifetime opportunity. There are more comments I would like to make about these wonderful minds. I've met some of them while I was studying physics, but this is the closest encounter so far, and I'm loving every minute of it. Thank you. More comments coming, I promise. Something I've noticed (and I'm a novice in this) is that this boy does suffer! It's just that his way of suffering is different. He suffers when he notices that other people don't take things the way he does, and don't understand him. Please, hang around, because your comments always prove very valuable. To be honest, I want to help this kid almost as much as I want to learn from him. More information and observations coming in case you're interested.

@MigL is right. More to the point. Almost infinite is nonsense. If someone gives you an "almost infinite" number, multiply it by 10^10^10...^10 and it's "almost nothing" no matter how big it looked.

Thanks a lot to both of you. Expectations are very important is the main idea that I wasn't aware of. I forgot to mention that the classes are taking place as teleconference, due to Covid-19. As to the big things is seems I've somehow got it right. Maybe it was intuition. As to the little things it seems that his family already are taking care of everything that's not under my control. I wasn't aware of this. I see. This particular boy is so keen on understanding things, that I don't need to press him at all. He often says, "don't tell me; I want to figure it out by myself!" That's not always a good idea... Yes, he might. He's very intelligent. I just hope he doesn't ask me what time is...

With the exceptions of occasional crackpots, the selfish, the unkind, and the uncouth; and on rare occasions, the sheer lunatics that come and go, most of the members of this community are giving me an enormously valuable environment to exchange ideas. Teaching has been a great pleasure for me for many years. Now, for the first time, I'm tutoring a boy that suffers from Asperger's syndrome. After a month now, I must say that it's been one of the most gratifying teaching experiences I've had so far. Things are going pretty well, good results keep coming, and everyone involved seems to be happy. We've started with maths. Every time he shouts “now I understand it!” is priceless. But he's emotionally vulnerable, and also gets quite anxious when he misunderstands something and gets embroiled in the wrong calculation. The verbal feedback is somewhat wanting, because he stumbles over words and speaks too quickly for me. So sometimes it takes me a while to realise what he really means. Any experience that any of you may have to share with me, any tips and directions, will be greatly appreciated. Especially heads-up when it comes to physics, which I'm kind of dreading.

I personally don't abide by what maybe @Eise would call a shallow reductionism. I consider myself a reductionist in the sense that I think that what molecules do in people's brains, however complex, determines what they think, feel, believe, wish, and finally do. And those things must have been set in motion by what they have seen, heard, thought, felt, believed, wished, and finally done, in full circle, before that, reflected in the behaviour of the molecules in their brains, which have registered somehow in their states the previous experiences. I don't think that saying that what people wish, or think, feel, believe, wish, and finally do determines what the molecules in their brains do. There is a fundamental asymmetry in the explanation, if nothing else. I think that it's the workings of the molecules what gives rise to actions, feelings and decisions. It's very awkward to me, to say the least, that it's the other way around. I don't think that's what @Eise is saying though, I must clarify. One thing is what molecules determine with their behaviour and a very different thing is the system of concepts that we need to describe it (rest potentials and activation potentials, membrane processes, homeostasis, and so on.) An atom alone is not engaged in homeostasis. It wouldn't "know" it is. It's the fact that there are many atoms, that they are what they are in the proportions that they are, and that they are combined in the precise way the are, that determines my wishes, beliefs, and course of action. My wishes determine nothing in what an atom does. Or maybe concepts can be twisted so that you can express things as if they do, but it's the most uneconomic way of describing things. I just hope we don't agree just because we've had similar upbringings, @vexspits. I've always envied the Protestant/Jewish/atheist upbringing because of the tradition of rational criticism. But it's been a long time since I freed myself from my Catholic upbringing. At a high price family-wise, of course. But when you dislodge yourself from such amount of nonsense, the momentum you gain is truly amazing. The truth is that the more we talk about this, the more points of agreement I find with you, at least as to a working definition of free will. Being able to act according to your wishes and beliefs is good enough for me. I would throw in things like fear, or reckoning, etc. But... It's possible that our sticking point is just about the role that emergence plays in all this. It seems to me that you contemplate emergence as something that stands on its own, independently of what @iNow calls "the components." I find it very difficult to see, as you seem to do, a mechanism levitating logically on top of other substructures that we know to be there and be sub in a meaningful way.

My guess would be that they wouldn't, because R3 is special in that covectors are isomorphic to vectors. In Minkowski I've seen them used, but that's because you've got spacelike and timelike, and the derivative by the invariant parameter must preserve the genus, so you would have the timelike element plus a Frenet-Serret spacelike triad. Spacelike go their own way, while timelike keep the isomorphy between 3-vectors and 3-covectors, so that Frenet-Serret would still be useful. But that would be just my guess...

My guess would be cerebrorum malleus.

3Blue1Brown is a very high quality channel for maths, IMO. And Mathologer. Mathologer is especially rigorous while intuitive at the same time. I don't remember if they have anything on infinitesimals or geometry, but I'm sure they have.

True. But that is in GR. In Newtonian gravity fields can be ascribed to point sources or densities of mass, and studied separately to a certain extent. Also true. In GR frames of reference are local, not global ("global" is preferable to "non-local," for what I think you mean) "non-local" being generally reserved for other concept in field theories. Namely, couplings or interactions of the form, \[\varphi_{1}\left(x\right)\varphi_{2}\left(x+d\right)\] with \varphi_1 \varphi_2 being any field variables. Fields are better looked upon as distortions of space-time, rather than as a property of matter. "Matter" in modern field theory (QFT) rather being considered as excitations of fields. Most fields (Yang-Mills fields) requiring an "internal variable" (independent of space-time) and gravity being peculiar in the sense that these "distortions of space-time" are felt the same by all matter and radiation (equivalence principle.) That's why gravity can be considered as a distortion of space-time itself, while Yang-Mills fields are excitations of these internal variables that can be studied in terms of their space-time projections. All are local fields, and none are strictly linear. Even electromagnetism in QFT (quantum electrodynamics) has non-linear effects at sufficiently high energies.

My interpretation of the question was more along the lines of @swansont, rather than @Markus Hanke's. Although I think Markus is totally right in his qualifications. And the reason is that the user said "how do gravitational fields interact," using the plural for "gravitational fields." The moment you consider different sources of gravitational field and ask yourself how they combine, it seems to me that you must be taking a Newtonian approach. Something like, \[\boldsymbol{g}=\boldsymbol{g}_{1}+\boldsymbol{g}_{2}\] In GR there is no useful way that I'm aware of in which you consider the gravitational field as made up of individual gravitational fields that can be tagged.

You're both welcome. A couple of things more. First, I just wanted to add that Fermi was no idiot, of course. But the assumptions he made date back to 1950. We know much more about planets now. Something we know now, for example, is that the Earth-Moon system is far from typical. The Moon is an unusually large satellite and has a rather bigger than normal stabilizing effect on the Earth's tilt. To the point that astronomers are starting to look upon the Earth-Moon system as a binary planetary system, rather than a standard planet and its small satellite or group of satellites. The huge tidal effect that the Moon has on the Earth is believed to have played a major part in the origin of life at least during the first billion years, stirring the chemicals dissolved in water and thus triggering volume reactions (much quicker and efficient) rather than surface effects. https://www.space.com/12464-earth-moon-unique-solar-system-universe.html Another factor is the presence of outer giants like Jupiter and Saturn, for billions of years playing the role of shuttles for asteroids from the Kuiper belt, etc. Water and amino acids in the asteroids are also thought to have been very important. In case any of these factors were found to be essential to the appearance of life, it could be a basis to estimate the number of solar systems in the Milky Way that satisfy similar conditions. Would other different sets of conditions be just as good, or maybe even better? I don't know. I don't know if anybody knows. Drake's equation came later than Fermi's argument (in 1961). Actually, I think Drake's equation is a more promising ground for estimating the chance of there being intelligent life forms, among other things, precisely because, although ambitious, it's a much less assuming parametrization of the probability, rather than an equation or a "closed" calculation. There is room for re-estimating the factors as we learn more about the phenomenology of galactic (or extra-galactic) solar systems. Plus the last factor is, if I'm not mistaken, the probability that a civilization will be able to send signals, rather than travel to Earth, which significantly increases the odds. Fermi was concerned with interstellar travel, AFAIK. The detection of signals with a message in them will probably be the first evidence, if there ever is one, of some form of intelligent life besides us in the universe, rather than the flight of UFOs. But here's the bad news, IMO: Take a look at this table with time gaps separating the appearance of new levels of organization: First prokaryotes (from Earth's formation): 1 billion years First eukaryotes (from prokaryotes): 800 million years First multicellular eukaryotic organisms (from single-celled eukaryotes): 2 billion years First intelligent life (from multicellular eukaryotes): 700 million years Average for the appearance of a new level of organization: 1.125 billion years Now suppose there's a planet out there with something like eukaryotes (cells with a nucleus). You're going to have to sit there waiting for 1.1 billion years for you to see anything interesting to happen if the above table is anything to go by. That's the problem.