uncool

that's...not how it works at all

It doesn't need to be a solid; the "solid sphere" version of the theorem is proven based on the "hollow sphere" version of it, as shown in the Wiki page:

Calabi-Yau manifolds are Kahler manifolds. Kahler manifolds are symplectic manifolds. Symplectic manifolds are always even-dimensional. So no.

Sure. Instead of saying "the product of 1+2i and 2+5i is -8 + 9i", you could instead say "the product of (1, 2) and (2, 5) under the operation (a, b)*(c, d) = (ac - bd, ad - bc) is (-8, 9)". Yes, it's a trivial renaming of the complex numbers; that's part of my point. OK, here are two real operators (as I understand it; I never received an answer as to what you meant) that fail to commute: 0 1 0 0 0 0 1 0 Possibly more relevant to QM, here are some real operators that act like the spin operators, if I haven't made a stupid mistake: 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 -1 0 0 -1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 -1 where multiplication by i is replaced by multiplication by the matrix 0 -1 0 0 1 0 0 0 0 0 0 -1 0 0 1 0 All of the components of each operator are now real. Yes I am aware of extension algebras and arithmetic and I know tha there are some pretty exotic creations floating around there, but I am not very adept at them. I was always given to understand that for every gain in one direction you tend to loose something in another when you add to algebraic complexity. You seem to have quoted yourself; I'm not sure I see how this responds to my post.

I'd need you to first define "real operators". As I said earlier, anything that can be defined in terms of complex numbers can instead be defined in terms of pairs of real numbers, etc. I meant that I have no idea what connection you are trying to draw between unique factorization and quantum physics. Why not? Why should √5 be considered "integer", while (1+√5)/2 can't be? How are you defining "integer" outside of the actual integers? It turns out that (1+√5)/2 actually is quite integer-like, in that it can solve monic polynomials; such numbers are actually called "algebraic integers", and act somewhat like integers do for rational numbers.

Err, what? Complex numbers are entirely commutative, and any way in which noncommutativity appears in QM can appear in a "real" explanation just as much as for a "complex" one. Um. Two things. First, there are plenty of subrings of the real numbers that fail to satisfy unique factorization. Consider Z[√5], and factor 4. That can be fixed in a somewhat natural way by adding (1+√5)/2; however, the "natural" fix doesn't work for Z[√7] (and this failure - among others - is a reason for the modern study of algebraic number theory, especially ideal class groups). Failure of unique factorization is not inherently related to complexity. Second, I see no reason why a failure of unique factorization is "essentially a quantum behavior".

Strictly speaking, no, you do not "need" the complex numbers for anything. You can write everything you do in term of pairs of complex numbers, which just so happen to have the properties of the complex numbers. You could go even further; real numbers are equivalence classes of sequences of rational numbers (under an equivalence relation of, approximately, "limits to the same number"). And further still; rational numbers are equivalence classes of pairs of integers (under an equivalence relation of "same ratio"). And yet further; integers are equivalence classes of pairs of rational numbers (under an equivalence relation of "same difference"). But it's far more convenient to not constantly be going through this entire hierarchy to make even simple statements. It's much, much easier to simply define a set of numbers once, and then deal with all of the ramifications by repeatedly using that definition. And the complex numbers turn out to be useful enough to do so - much more than you might expect, in fact.

Funny enough, for Catholics it apparently doesn't necessarily have to be pure for the purposes of baptism. https://christianity.stackexchange.com/questions/65635/how-much-can-holy-water-be-diluted

No. They are, in fact, in bijection, and both are in bijection with the set of all real numbers, and both are in bijection with the set of all complex numbers. Again, what is your point? Again: there is nothing inherently uncountable about individual transcendental numbers. You keep bringing up that some of these numbers are transcendental as if that, by itself, implies uncountability. It does not. First: what is your argument for uncountability? Second: that set is very clearly countable. It is very clearly the image of a countable set, namely the set of pairs (x, y) where x and y are both in i', under the map of exponentiation. Once again: emphasizing that you are getting results which are transcendental numbers does not tell you anything about whether the output you get is uncountable. At the absolute best, it removes a single, overly-simplistic proof of countability; that does not constitute a proof of uncountability.

It was almost certainly the tensor product. The statement "these two particles are entangled" is, more formally, "the combined state of these two particles is not a pure tensor element in the tensor product of the state spaces of each element separately". While I don't know the history itself, entanglement itself is a rather natural aspect of the fact that quantum physics is fundamentally linear, and that the tensor product is the "natural" way of dealing with bilinearity.

Bit more than that; don't forget the inner product. I'd say that that's the central aspect of Hilbert spaces. Depends on what you mean by "aspect" or "attribute"; I'd also be careful about what you mean when you say "something like its own dimension". If you mean that each particle (and simplifying to quantum mechanics, where there is no particle creation or annihilation), then kind-of, but not really. To explain further, and in some formality: the state space for a combination of particles comes from the (completion of the) tensor product of the state spaces of the individual particles, not a direct sum. Sorry, but I don't really have a less technical explanation for what that means. I'd say that each eigenstate a QM "entity" can be in can be thought of somewhat separately, yes (until you get some time evolution that mixes between eigenstates). To my recollection (which is admittedly rusty), the use of "state" can be somewhat ambiguous; it can either be used to mean a single eigenstate (of an unspecified operator), or can mean an arbitrary mixture. In the latter case, I'd say that the answer to your question is "no"; for example, the the states |a>, (3/5) |a> + (4/5) |b>, and |b> cannot really be thought of as in "different dimensions".

On what level do you want an answer? Because on one level, it was a bunch of wars and abolition movements. But that doesn't answer how all those wars and abolition movements came together. On another, it was a bunch of philosophical movements that led to ideas of liberty. But that doesn't answer why those movements came to the forefront. On another, it was a result of economic forces giving rise to systems that outcompeted slavery. But that doesn't answer why they hadn't happened earlier. And so on and so forth.

Yes, that makes sense; I had read your statement as "So he could determine in principle that Alice has performed a measurement [instantly], but he wouldn't be able to tell which outcome Alice got until he made the measurement", because the second half of the statement was qualified by timing while the first half was unqualified. Sorry about that.

...if Bob could determine whether Alice has made a measurement, then Alice could send a message faster than light purely by choosing whether to perform measurements, if I'm understanding what you're saying correctly.

..."protected class" is a designation based on various US laws, most prominently the Civil Rights Act of 1964. And in that law, the classes are not restricted to the "oppressed" classes. Both "men" and "women" are classes that are protected, all religions (including Christianity) are protected, and so on. Explicitly: "All persons shall be entitled to the full and equal enjoyment of the goods, services, facilities, privileges, advantages, and accommodations of any place of public accommodation, as defined in this section, without discrimination or segregation on the ground of race, color, religion, or national origin.[...] All persons shall be entitled to be free, at any establishment or place, from discriniination or segregation of any kind on the ground of race, color, religion, or national origin, if such discrimination or segregation is or purports to be required by any law, statute, ordinance, regulation, rule, or order of a State or any agency or political subdivision thereof."

...or until it failed under internal stress, or until it was shattered by random space debris, or ... it would be an absurd undertaking on logistical terms alone, even before considering actual use. the largest artificial satellite that we have ever launched is the international space station, which is 73 meters by 109 meters. you're talking about a mile wide object. that's 1609 meters. then you have to consider the feasibility of using a mile-wide mirror. that is, you need a properly curved, smooth, reflective surface that achieves a specified position in orbit above earth. and I'm pretty sure swansont meant 6000 degrees as an absurd upper limit, not a reasonable estimate, although i'm not sure that all of the necessary assumptions for that theorem are met (i haven't checked thoroughly, though, so i'd default to trusting him!)

https://www.law.cornell.edu/uscode/text/18/1512#c_2 "Corruptly" isn't statutorily defined, as far as I can tell; there is some case law on the definition. https://www.ce9.uscourts.gov/jury-instructions/node/732 If he urged people to the Capitol in an attempt to impede the counting of the electoral votes (which multiple courts have ruled is an official proceeding), and if he did so with some consciousness of wrongdoing (which is something that might be inferred), then yes, that is enough, if I've read it right.

I read the question as a general question of whether Congress has any power to bring charges and punish under its own authority. But even with the interpretation of charges being specifically about the Jan 6 insurrection, I think a case could be made for Congress having the power to punish the insurrectionists itself. Not that that is what is happening, or even that it is likely to happen, but that that is within its power.

While they are not the same, part of my point with the link is that Congress's power is not limited to finding someone in contempt, but may include the ability to prosecute those charges itself. The link I provided gives examples of both houses of Congress themself trying cases where someone was accused of attempting to bribe a member. From the link: "Ultimately, Mr. Anderson was found in contempt of Congress and was ordered to be reprimanded by the Speaker for the “outrage he committed” and discharged into the custody of the Sergeant-at-Arms. Mr. Anderson subsequently filed suit against Mr. Thomas Dunn, the Sergeant-at-Arms of the House, alleging assault, battery, and false imprisonment. [...] The Anderson decision [by the Supreme Court] indicates that Congress’s contempt power is centered on those actions committed in its presence that obstruct its deliberative proceedings. " Unless you are making a distinction between "bringing criminal charges" and imprisoning someone for contempt, I don't see how the context changes my point - not that this is likely to happen, but that Congress may have the power to do the latter.

As an explanation of what is likely to happen, I think this is correct; as an explanation of what can happen, I think this is incorrect. Congres has the inherent power to hold people in contempt, and can do so without involving the executive branch whatsoever, and this can include fining or imprisoning the accused contemnor for the length of that session of Congress. And this can be for either a coercive purpose - trying to force the accused contemnor to do something - or a punitive one - punishing them for failing to do so. https://sgp.fas.org/crs/misc/RL34097.pdf With that said, it apparently hasn't happened since 1935, and was rare before then as well.

I don't think that anyone really answered the question, because the argument the question is about uses assumptions that aren't that obvious. The statement "Force is proportional to mass" is somewhat imprecise; the fully precise statement is "Force is proportional to mass when acceleration is held constant." That is, if you have two masses which are accelerating in the same way, then the force on one divided by its mass will equal the force on the other divided by its mass. Alternatively, for any situation with forces and masses, F/m is a function of the acceleration. Similarly, the statement "Force is proportional to acceleration" is imprecise, with a more precise version being "Force is proportional to acceleration when mass is held constant." If you have two objects with the same mass, the force on one divided by its acceleration will equal the force on the other divided by its acceleration. F/a is a function of the mass. So we have that F/m = f(a) for some function f, and F/a = g(m) for some function g. Then F/ma = f(a)/a = g(m)/m. f(a)/a is also a function of acceleration independent of mass, and g(m)/m is also a function of mass independent of acceleration. And the only way that can happen is if f(a)/a = g(m)/m is constant - is independent of both mass and acceleration. Call that constant C. Then F = C*ma - in other words, force is proportional to the product of mass and acceleration.

…no, NTuft, that is not why e^(i pi) = -1. Your blind substitution is not correct. You are claiming that e^(x*i*pi) = x cos(pi) + i x sin(pi) = -x, if I’ve remade your equations correctly. This simply isn’t true. It has nothing to do with the actual justification of the original equation (which has to do with McLaurin series), and is self-contradictory with a bit of thought.

…none of this is accurate, and this can be checked directly by e.g. wolfram alpha. e^(i’ pi) = e^ (i sqrt(2) pi) = cos(sqrt(2) pi) + i sin(sqrt(2) pi) ~ -0.266 - 0.963 i

…I have to admit, I’m…uneasy with this reasoning. It suggests that there is policy interest in who gets to reproduce based on speculation on future criminality, which has some connections with eugenics. Those connections strengthen when you add in the conservative framing of abortion being connected to the right of the fetus to life - under that framing, this argument suggests that some people don’t have a right to live because of speculative criminal activity. I’m not suggesting that this is what you meant to suggest. Just that it may have unintended implications, especially among people with other framings.

Worth noting it was used on both “sides”: by slavery supporters to say that their “rights” to own slaves shouldn’t end when they visited free states (or when slaves successfully ran away to free states) and by abolitionists to say that African Americans should have the same right to be free in all states