The above poster are talking about special relativity only. It's "special" because it's only specific to inertia frame. So for the sake of completeness I will talk about general relativity, which follows the principle of general covariance, in which all frame of references are valid; in other word, all time and distance measurement are relative. This should show you the real crux of the problem: it's not that "acceleration is absolute and speed is relative", but rather "physical constants are differed between accelerating frame".

To achieve general covariance, general relativity comes with metric tensor. The metric tensor measure proper time and proper length, and this way we unshackle the concept of length and time from the coordinate. The metric tensor is obviously needed since an arbitrary coordinate system means you can pick system in which directions are distorted (e.g. 2 units to the right has equal length as 10 units to the front).

So you're now free to rotate your coordinate system at will. These distance galaxy will move at insane speed in your system of coordinate, but if you look at the metric their speed is completely normal.

So what's the lesson from this? Coordinate system is a completely arbitrary construct that has nothing to do with actual physic. There is nothing wrong with an object moving fast according to the coordinate system. If you have a rotating coordinate system, it's as strange as having a scaling coordinate system where the length of the axes changes over time; in either case, far away objects will appear to move too fast according to the coordinate system, but nothing actually physically change.

How did bitcoin ended up gaining any values at all initially before it had enough momentum to get going?

How widespread are the practice of using multiple wallets to manipulate the market, and is it possible to deal with that problem through cryptography?

Other people had mentioned various standards that exist.

But I just want to point out that color perception is extremely subjective even when you account for the physical and biomechanical factors (ie. what light come into your eye, what cone cells you have). Even if 2 people see the exact same light they can still perceive different colors. It's not just abut the actual lightning condition, it's also about what they believe the lightning condition to be. And then there is also a matter of optical illusion. For example, human eye has edge detecting capability, and we will perceive colors next to an edge with more contrast.

It's "just a convention", but it's something so ubiquitous that pretty much many mathematician independently came to the same thing.

Historically, it used to be context-dependent, sometimes you add first. For example, look at this picture: https://en.wikipedia.org/wiki/File:JakobBernoulliSummaePotestatum.png . Can you guess which line you add first and which line you multiply first? If you understand the math, you can figure it out, but if not, this can be confusing.

Eventually, various mathematicians made up various different conventions on how to write and interpret formulas. Even though they produce different convention, one thing are shared among them all: multiplication takes priority over addition (unless superseded by other stuff like brackets and lines). And there is a good reason for that. Multiplication distributes over addition, so it's a lot easier to write an expression as a sum of product than a product of sum. In other word, a lot more formulas would end up requiring you to do all the multiplications first to get a bunch of products, and then doing addition to add up those products. So it is a lot simpler to make that the default: by requiring that multiplication takes priority over addition, you don't have to put brackets everywhere.

So even though it's "just a convention", there is a mathematical reason behind the convention.

Chaotic is different from stochastic. Stochastic means there are randomness involved in the evolution of the state. 2-body and 3-body problems are deterministic, not stochastic. The state always evolve the same given the exact same initial condition.

But for 3-body problem, it's chaotic. If you don't have exact values for the initial condition, the error became exponentially large as time go by, so after a certain amount of time the state became essentially unpredictable (but there are special exceptions). If you do have exact values for initial condition, then you can make arbitrarily accurate prediction for arbitrary long period of time, but you will need to perform a lot more calculations compare to non-chaotic case to control this exponentially growing error (you always acquire error due to numerical imprecision). Chaotic implies a few properties. One is the butterfly effect, as I mentioned above. Another one is mixing: it's not merely that you can't predict precisely if time is long enough and you don't have the exact initial condition, you can't even make a vague estimate that carry any useful information at all.

Why is 2-body problem not chaotic? Essentially, it has too few variables compared to the amount of symmetry. It's known that if you have at most 2 free variables you can't be chaotic. A 2-body problem has 12 variables (position and velocity for each body), but standard physics 10 symmetries gives you 10 constant of motions (center of mass, linear momentum, angular momentum, energy) so the problem is reduced to 2 dimensions. Actually, there is a special 11th constant of motion specific to this problem: the LRL vector, but only its direction on the plane of motion matter, because the plane of motion and length is determined by angular momentum and energy, so the problem is reduced to just 1 dimension. 1 dimensional system is very easily solvable explicitly.

For 3-body problems, you start with 18 variables, but you only have the usual 10 symmetry. It's proven that there are no other algebraic constant of motions, so at most you can reduce this to an 8 dimensional problem using this technique. Actually showing that this problem is chaotic (and hence you really can't reduce further) is harder.

Something people had neglected to mention. For many file formats (especially image file), inside the file, there will be a header, a piece of data that tells you information about the file, including explicitly what format it is. So the application can tell easily what file format it is in despite the wrong extension name.

I think the issue is deeper than just that we want to write the solution in the specific form of using only elementary functions. The real problem is that the solution is chaotic, which is an inherent mathematical fact that has nothing to do with what functions we want to use. The reason why we want to solve with elementary functions in the first place is because they have very predictable behavior; for example, you can make long range predictions without much difficulty without increasingly large errors. We could introduce new functions (and people do, these are often taught under special functions), but other "nice" functions won't solve the problem, and the one that does solve the problem would be chaotic and hard to analyze. Ultimately, the main issue is that the general solutions to the 3-body problem is just too chaotic that it is resistance to analysis.

Just wondering, if only 1 X is ever activated, why does 48, XXXX individuals suffer from many serious health conditions?

That is an interesting effect, thank you.

>But contextual information can.

I guess, for the purpose of my question, the issue is whether the contextual information must be delivered visually. For example, let's say the test subject look through a small hole into monochrome grey wall with no other contextual information, then the experimenter tells the subject that the wall is being illuminated by red/green/blue lighting (or maybe the 2 events occur in the opposite order). But the subject never see the light bulb itself or any other objects that would allow them to deduce the actual color of the lighting. Would that affect color perception?

My analogy is with perception of sound direction. Our perception of where the sound is coming from can be overridden by visual information, that's how ventriloquism works. So it seems plausible that visual perception can be overridden by non-visual information as well. This does happen for certain kind of visual perception, but color perception seems to be the most impervious to these effects, which is why I wonder.