uncool

Because the property (a + b)*c = a*c + b*c requires it. Or, to slightly modify what John Cuthber said: if you cancel someone else's debt, you are giving them money.

So how is that n proven to exist in the Actor system? What you're saying reminds me of the fact that e.g. any strictly decreasing sequence of ordinals is, in fact, finite, even though (if the initial value is infinite) it can be arbitrarily long. However, I still don't see what it is about the Actor system that forces the "stop" message to be eventually acted on.

I guess my question is, if the "stop" message can be postponed for an unbounded time, why it couldn't be postponed forever, analogous to the Turing machine algorithm.

Why is one of the computations guaranteed to stop while the other isn't?

That said, curvature of a (near-)circle is supposed to be 1/radius, which in the case of the Earth is (if I haven't made a mistake) 16 in/mi^2. I'm guessing that the source thinks that the deviation from flat is supposed to be curvature*distance^2, rather than half that (similar to how the displacement after constant acceleration from a standstill is not a*t^2, but (1/2) a*t^2).

Um. What? Don't get me wrong, I think the OP is wrong, but this almost sounds like the objection to acceleration in the form of meters per seconds squared because there's no such thing as a square second. "8 inches per miles squared" just means that each mile, the rate at which the Earth "drops away" changes by 8 inches per mile (using an approximation where the surface looks like a parabola). The units of curvature are inverse distance, and 8 inches/miles^2 is inverse distance.

Sorry, I didn't quite read closely enough; replace j in the above with k. When you have an answer for that: I think that you have not understood what is so essential about the complex numbers. They are algebraically closed, that is, any nonconstant polynomial with complex coefficients will have a complex root. For example, the complex polynomial x^2 - i will have a complex root. As such, any "addition" of the type you have constructed will, inevitably, result in some kind of redundancy. In most cases, all that happens is that you essentially come up with something equivalent to "multiple copies of the complex numbers". In this case: any "recomplex" number can be expressed as: a + b*(1/2 + j*sqrt(2)/4 - k*sqrt(2)/4) for some complex numbers a and b. I'll call the constant in the above C. Then (a1 + b1*C) + (a2 + b2*C) = (a1 + a2) + (b1 + b2)*C, and (a1 + b1*C)*(a2 + b2*C) = (a1*a2) + (b1*b2*C), if I haven't made any arithmetic errors. You can see some effects of this, e.g. if you try to find (sqrt(2)*j - i + 1)/(sqrt(2)*j + i - 1). Depending how close you approximate sqrt(2), you will get absurdly large values, which are an artifact of the fact that you are, in essence, dividing by 0.

plus or minus (sqrt(2)/2 - i*sqrt(2)/2). Alternatively: if I've skim-understood correctly, you've defined j such that j^2 = -i. In that case, what do you get when you multiply (sqrt(2)*j - i + 1) and (sqrt(2)*j + i - 1)?

I accept that this is an impression, and that I probably overstated with "blatantly". My point isn't simply that Maxine Waters called for violence; my point is that the method in which Pelosi analyzed Waters's statement is very different from the method in which she analyzed Trump's statements. And yes, there are reasons to do so - Trump has been and was blatantly dishonest, and blatantly pandered to white supremacists and conspiracy theorists. But I don't think that someone who can analyze Trump's statements and see beyond the perfunctory "Peacefully protest!" can think that Waters's statement was only about "confrontation in the manner of the civil rights movement".

Before I do, I want to make clear: I do not think of any of this as disqualifying, or even that important. I think of Nancy Pelosi as, for the most part, a pretty typical politician; I think that nearly every politician at her level has similar hypocrisies. For a recent example (that is on-topic): Pelosi's response to Maxine Waters's statement that protesters should get "more confrontational". Maxine Waters said "[...] and we've got to get more active, we've got to get more confrontational, we've got to make sure that they know that we mean business" in response to a question of what protesters should do if the Chauvin verdict wasn't 3 guilties (manslaughter, murder 3, murder 2 - though she may have only heard "what should protesters do?" without the specific circumstances). The phrase "more confrontational" got pushback, which led Pelosi to say "No, Maxine talked about confrontation in the manner of the civil rights movement." I don't think that she can believe this is as simple as that, given her support for the impeachment of Trump for the riot at the capitol. She appropriately noted there that even if Trump said to protest peacefully, that the context easily allowed people to interpret it as support for intimidation. The same is true, if on a much smaller scale (and with less direct import), for Waters's comment. Now, I agree that this is arguable, and not only that, but that there are many, many worse cases. But there is reason to dislike her beyond "liberal".

Mehh. I'm pretty strongly against Republicans, but there is valid reason to dislike her. She's a blatantly pandering politician who is willing to say things, not because they are true, not even because she believes them, but because they are convenient to the current narrative she wants. In other words, a typical politician. And, in my opinion, not as bad as many Republicans (see: Ted Cruz, Lindsey Graham), but one of the more blatant on the Democratic side of the aisle, if only because of her prominence.

Keep in mind that in one sense, most mathematicians don't ascribe an enormous amount of importance to mathematical foundations, but in another they very much do: The standard place where most mathematicians place the foundations of mathematics is set theory, most commonly in the form of Zermelo-Fraenkel with the axiom of Choice (or ZFC). I say standard because there is quite a bit of work on nonstandard foundations - simply removing the axiom of choice is common; some replace ZFC with Homotopy Type Theory (which is beyond me entirely). I say most because there are some mathematicians that don't work in ZFC at all - some even reject the consistency of ZFC (some even reject the consistency of ZF, though that's rare). The sense in which it's not important is that for the most part, mathematicians don't need to know exactly how every symbol works; as long as certain things can be done, most mathematicians don't need to care what underlies it. The sense in which it is very important is that it's still necessary to know (or at least, strongly believe) that it can be done consistently; a failure in the foundation would be a failure of mathematics as a whole. So imagine it like the foundation of a building: it's not important that you know the details of the foundation, but it's important that you know the foundation is there doing its job.

I don't see how "base 10" makes x^2 = x/x.

1) That is not a polynomial equation; note the division. 2) Basic manipulation can turn it into the equation x^4 = 0, which means that it has no solutions (since when x = 0, the RHS involves division by 0). 3) What, precisely, do you mean by "solve"? Numerically? Using radicals? Finding a minimal polynomial?

Depends on what you mean. For any angle A, sin(A) = (e^(i*A) - e^(-i*A))/2i, which is an expression of the sine using exponents, but I'm guessing that's not what you're going for (in part, because you seem to have an objection to numerically finding the value). It sounds like you are trying to define sines in terms of radicals of rational numbers. There are uncountably many angles, and only countably many expressions using radicals. So we'd have to restrict ourselves to some countable subset of angles. You chose to look at pi/4; that suggests using only rational multiples of pi. And with that restriction, the answer is yes - sin(pi*(a/b)) can always be expressed in terms of radicals. In fact, the expression I gave above can count: sin(pi*(a/b)) = (1^(a/2b) - 1^(-a/2b))/(2*(-1^(1/2))). That expression...isn't really helpful in finding the value of sin, but I'm pretty sure it can be converted to an expression that is.

I found several more by googling <bijection 0 1 1 infinity> (without the braces), so...

Gravity does not "always act[...] as an opposing force to inertia", which you'd know if you ever went skydiving. Gravity is a conservative force, and in a universe with just gravity, perpetual motion machines of the second kind are possible. The force that "opposes inertia" (more precisely, that equalizes velocities) is friction.

It is a scientific statement, which (as you have said) has been falsified. In this terminology, "scientific" does not mean "confirmed". It's approximately equivalent to saying that a sentence is grammatically correct - the content may be true or false, but it satisfies certain rules about the sentence itself.

The statement that the function is discontinuous is calculus. Additionally, this reverses the usual order of definitions. In calculus, continuity is defined in terms of limits, not the other way around. So this begs the question: how do you know the function is discontinuous?

taeto - the function I described does exist; it's not hard to construct using the axioms. In nonstandard analysis (specifically, the version using Internal Set Theory), it isn't standard, and nonstandard analysis defines limits and derivatives for standard functions (as I understand it).

Dasnulium: I don't think you have ever answered this question. In the system you favor, what would the limit of f(x) be in this case?

I think you have either used standard calculus or assumed your conclusion without realizing it in your paper, by assuming that "r" (which you should more explicitly define than "the two sets of terms as a ratio") can always be expressed in the form "+- b epsilon^2 +- c epsilon^3 +-.../+- a epsilon". More generally: let's say we have the function f(x) = 0 if x is neither infinitesimal nor 0, and 1 if x is 0 or infinitesimal (in other words, if x is smaller than any rational number). What is the limit as x approaches 0 of f(x)? (More on this after an answer)

Just because this definition doesn't show the beauty or power doesn't make it a bad definition, though. A definition doesn't have to describe every aspect of something.

The way I'd put it is everything that can be proven. Specifically, this includes statements of the form "If we make these assumptions, this is what we can prove".