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I think your rather over-egging the cake here. A new nephew joined our extended family nearly a year ago, and it's given me the opportunity to see how his reaction develops to not only the only non-African face in the compound, but quite possibly the only white face in a local community of maybe 25,000. Whenever I emerge from my man=cave, his attention is immediately drawn in a way that's immediately obvious to the rest of the family and a source of great amusement as his eyes follow me around the room. He obviously perceives me as different to other family members, and although there's undeniably some element of anxiety there, curiosity dominates. I've seen similar trends with his elder siblings and I've no doubt that in two or three years he too will be regularly visiting the 'troll's lair' to watch my latest Minecraft creations take shape, and ask me for the umpteenth time how to build Iron Golems. A simple common interest on which to bond. He will not be harbouring deep-seated primitive hatreds against fair-skinned people since these only develop if given cause and we as a family shall not give them cause. Indeed, in the near quarter century I've spent here, I've experienced no significant racially-motivated antagonism from anyone in the community. In stark contrast to the frequent abuse directed at my wife by both individuals and bureaucracies when she spends time in the UK. So no. I don't believe you can partially excuse racial hatred as being a natural urge to be overcome by reason..It's embryonic source in nature will only manifest and deepen if nurtured to do so. I've only to watch a Youtube political news clip from the UK or US to see that nurturing in action.

For the love god can we send this to the Trash Can?!!

Some outlets have reported the NYPD has been requested to alter the color of their orange prisoner jumpsuits so as not to match the former President's skin tone.

Are you stupid? I've just told you we did not know what the wine was until after we had commented on it.

@Genady: Stephani’s Exact Solutions to the Einstein Field Equations is an absolute must if you want to go above and beyond the basics - not only does it classify the known exact solutions according to different schemes, it also explains the general features of these classes of spacetimes, and their mathematical treatment. The book also gives an overview over which general methods are available to find solutions to the equations, and what forms these solutions may take. So it’s definitely much more than “just” a reference catalogue. But be warned - this is definitely not a beginner’s text, it’s mathematically fairly demanding in places.

Yes, good point Yes, with “necessarily” being the key term. It does work sometimes and gives the right intuition; but at other times it doesn’t work. The problem is that generally speaking there is no easy way to tell which is which - that’s why I think it’s dangerous to mix Newton with GR. That’s a difficult to answer question, unfortunately. My own GR understanding came about as the sum total of a large number of different texts, and there’s really no single book that has all the subtleties in it. One text that was instrumental for me personally is one that is unfortunately not available in English (to the best of my knowledge), which is T Fliessbach, Allgemeine Relativitätstheorie. It’s the only text I have seen that explicitly and step-by-step goes through the entire procedure of solving the Einstein equations for Schwarzschild and FLRW from scratch - you can see exactly where sources come into this, how boundary conditions appear, where things like the mass term in the metric originate etc. It found that very illuminating, because it shows how many of our Newtonian intuitions about gravity just don’t hold. But like I said, unfortunately there seems to be no English version. Then there’s of course MTW - ignoring the introductory parts, it’s the only text I know of that goes into the question of why the field equations look like they do. It goes into the topological considerations and conservation laws that underlie the entire machinery of GR, and explains the actual meaning of things like the Bianchi identities very well. I have not seen some of this material in any other text I know of. This is a 1500 pages tome, and evenly divided between introductory topics and more advanced stuff, but I think anyone interested in GR will take something away from this book, irrespective of what level they are on. For the formal maths the gold standard is of course Wald, though to be honest as being an amateur I found it to be above my pay grade. I can see this would be of great benefit to someone who actually has a background in maths, or at least has studied the entire subject matter at university level. It’s a great reference for theorems, proofs etc though. If you can give me some time, I will have to think about your question some more, and have a look through my personal library. It’s been years since I’ve really gone through these books - I’ve taken the key points out and compiled extensive notes for myself, which is what I am mainly working off these days. I might add some more recommendations here later

This is more to do with the 'instability' that DrmDoc mentioned, rather than strictly racism. The biggest motivator seems to be fear. Someone like INow, or Ken, are very worried about climate change and global warming. It will not cause the extinction of the human race, but it has the potential to massively change how we live our lives. Others, like a redneck MAGA supporters, are worried about immigration, because immigrants will take the manual labor jobs and they'll have to re-train or be unemployed. Either way, it is a massive change to how they live their lives. fear of change seems to be the common denominator.

For the most part, we’re mostly hairless apes throwing feces at each other. That said, even single celled organisms tend to distinguish between “ouch” and “not ouch” or between “food” and “not food” and likewise between dark and light or hot and cold. Life and chemistry itself seem to be classifying and reacting machines. The categories and distinctions really aren’t the problem though, IMO. It’s how we respond and react to them that matters. For example in another currently relevant context, do we treat trans kids as subhuman scary predators here to slice away the innocence of your own children, or just a different kind of human still deserving of the same love, respect, and protection as everyone else… It’s okay to recognize real differences from the mean, but not okay turning that into a permission structure for BEING mean.

Our institutions of governance and law work to moderate our problematic behaviors, because we know people can't be trusted to be honest or fair minded or without prejudice. The larger and more complex the society the more important to have those institutions, which may well build preferential prejudice into the system and sustain it by force but do offer routes to less prejudicial institutions - which we do see. I suspect much of the success at making peaceful societies out of disparate groups comes from having independent rule of law that (ideally) doesn't base it's judgements on the race or religion of the accused, but on evidence and testimony. Even sustaining an appearance of independence and fairness from turning to police can help moderate tensions. Where that is not the case I expect more taking matters into their own hands, with retaliatory revenge and more framing of conflicts as about ethnic or other differences - which can see blaming of groups for the actions of individuals, so the revenge may be taken out on the wrong people and make inter-group conflicts worse. At worst the groups have their own police and authorities who approve or participate in those conflicts.

I think this statement is both false and ultimately meaningless. First of all, the distinction between “simple” and “complex” is not objective, but very much contextual - what’s simple to me might be complex for you, and vice versa. It depends on our respective backgrounds. Secondly, even if the distinction was objective, the statement that you haven’t really understood something unless you can explain it to your child/grandmother etc is clearly misleading - it was never meant to be taken this literally. In my native language, we have a reasonably complicated system of declining nouns and conjugating verbs - according to grammatical case, gender, number, aspect, conditionality, tense etc etc we add prefixes, suffixes, prepositions and postpositions, and sometimes we change the middle part of the word as well, depending on what type of word it is and how it is used. There is also a large number of personal pronouns, never even mind all the irregular verbs, plural forms and so on. There is no way I could quickly and easily explain these things to someone who comes from a different language background, irrespective of what age they are, because some of the basic concepts (e.g. grammatical gender of nouns, or explicit declination by case) simply may not exist in other languages. Of course anyone can - at least in principle - learn the language, but it is generally going to take years of study and practice (unless you are lucky enough to have a mother tongue that is closely related), and not so many foreigners ever become completely fluent in it. It’s just a grammatically complex language, to the extent that even native speakers occasionally have trouble with complex grammatical structures. Does my inability to easily explain these things mean I do not understand the concepts of my native language? Of course not - I understand them perfectly well, to the extent that they are intuitive and self-evident to me. But that doesn’t mean I can easily explain to someone from a different linguistic background why “girl” should be of neutral gender, or why the “me” in “give me it”<>”give it to me” requires different pronouns. Some of these things don’t evenhave explanations, they are just conventions that have organically grown over time, so they have to be acquired, not understood. It’s no different in science - e.g. I understand the concepts of differentiation and integration well enough, I consider these basic operations just like addition and subtraction. But could I explain them to a 5-year old, who has no concept of curves, functions, variables, tangents, limits etc etc, in a way that he’ll actually end up understanding it all? Probably not, unless I’m dealing with an unusually gifted child. But my inability to explain does not mean that I don’t understand, it just means that the child doesn’t have the necessary prerequisites yet to benefit from my explanations. Crucially, my inability to explain it also doesn’t mean that the 5-year old cannot acquire this understanding over time - if you teach him the requisite concepts, there’s no reason why he couldn’t understand differentiation and integration once he’s a bit older. You just start with the basics, and then build onto them. That’s how people acquire new skills. How do you define “fundamental” and “derived”? To me, the most fundamental and most broadly-applicable principle, which underlies a guess-timated 70% or so of all know physics from the Standard Model right up to General Relativity, is the principle of extremal action: \[\delta S=0\] This statement is as simple in form and function as it is powerful, and as close to a “theory of everything” as we have at this point in time. It also does not rely on any particular choice of units or spacetime embedding. To me this is pretty much the bedrock of much of currently known physics, though my feeling is that you probably wouldn’t consider it “fundamental”.

Because the transfinite ordinals and cardinals are not real numbers. The subject of the thread is "The geometry of the real number line." That's YOUR topic, right? So we are discussing the real numbers. The transfinite ordinals and cardinals are fascinating in their own right, but have nothing to do with the real numbers. In fact the transfinite ordinals (and the cardinals, which are technically a proper subclass of the ordinals) do not intersect the real numbers at all. The transfinite ordinals and cardinals are neither subclasses nor superclasses of the real numbers. They're just a completely different subject. If you are trying to understand whether there's a largest number in [0,1), it's no help to think about transfinite numbers, since transfinite numbers are not in that interval at all. Does that make sense? Besides, haven't you already said you are only interested in the standard real numbers? Why are you suddenly interested in mathematical objects that are NOT standard real numbers? This para does not make sense. First, we can't use transfinite numbers in a discussion of the reals, because transfinite numbers are not members of the real numbers. We can of course use transfinite numbers to talk about the cardinality of various subsets of the reals, but that's not what we're talking about here. Ah. Well, the length of a single point is zero. The length of [0,1] is 1, and so is the length of [0,1). The addition or deletion of a single point makes no difference when we're calculating the length of an interval. I'll keep my clothes on if it's all the same to you, thanks. The length of a point is zero, so adding or deleting a point can not make any difference in the length of a line segment. I think what you are saying is that the intervals [0,5] and [5,6] both contain the number 5, and that is correct. I don't see how that helps you to find a largest number in [0,1). So the two intervals would "bang into each other" at the point 5. You are correct that 5 is an element of both intervals. But as a point, the number 5 has length 0. The two facts are both true. 5 is a point on the real number line and it has length 0. So yes, two points of zero length can still bang into each other, if you want to put it that way. Remember, Newton showed that you can reduce gravitational calculations to "point masses." So if it helps, you can think of them that way. They are points with zero dimensions, zero length, and zero volume, but they still pack a punch. I'm not saying that's any kind of mathematical argument, but if it helps you to resolve this particular objection, I'm ok with it. In your imagination. But the proof that there is no largest number in [0,1) should cause you to realize that your intuition is flawed. It should give you a better intuition. Now there is nothing wrong with having such a faulty intuition. Pretty much everyone has faulty intuitions about the real numbers before they see these technical discussions. But now that you've seen a formal proof that there is no largest number in [0,1), you should be willing to realize that your intuition is faulty, pre-mathematical as it were, and you should update your intuition. What exactly about the proof are you still unconvinced about? I asked you that in my previous post. It's no good for you to say you're unconvinced, without saying exactly what aspect of the proof you are unsure about. If you could focus on the proof we could discuss that. I can't discuss onions or bodies of water or vague pre-mathematical notions of the real numbers. It's more helpful to focus on the actual math. We are not going to discover a largest number in [0,1), because we already proved a few posts back that there is no such thing. Now I'm confused. Didn't you see and more or less agree with the proof I already posted? There is no largest number in [0,1). Proof: Suppose you claim that x∈[0,1) is the largest number in that set. Take half the distance between x and 1 , namely 1−x2 and add it to x , giving: x+1−x2 You can see that we have the strict inequality x<x+1−x2<1 so that x is not the largest number in [0,1) after all. Since x is entirely arbitrary, we have just shown that there is no largest number in [0,1). You have already seen this proof, and more or less said you agree with it. But now you are saying "Are you sure there are proofs?" You already saw the proof. Yes, I'm sure there's a proof, I've now stated it twice. And you've agreed to it. So I have no idea what you mean by asking if I'm sure there's a proof. Ah ... ps ... you said, am I sure THEY are proofs. Are you asking if the proof I gave is actually a proof? Yes, I'm sure. If there is any part of it you are unsure of, I wish you would ask about it or say which part you find unconvincing, so that we can focus on that.

Eh, the Omicron variant has a very high transmission rate, which is why its presence is ubiquitious now.

I hope not, but you were in a group and that sort of question, tends to reduce ones sensitivity too, stupid...