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[silverspawn]

on some level I still find it incredible that this is a thing that humans can just do without any help or substances, and most people have no idea. The world just continues to look crazier and weirder.

[silverspawn]

Today I finally learned the reason behind

e^ix = cos(x) + i * sin(x)

[silverspawn]

Whenever I obtain a new math insight, I'm like "why didn't people explain it this way initially? "

At least you could have pointed out that the behavior wrt taking derivatives match! It's super easy to verify!

(cos(x) + i*sin(x))' = -sin(x) + i*cos(x) = i*cos(x) + i^2 * sin(x) = i * [cos(x) + i*sin(x)]
(e^ix)' = i*[e^ix]

[faust]

Whenever I obtain a new math insight, I'm like "why didn't people explain it this way initially? "

Well different explanations work for different people. You need to find the right fit for the right person.

You insight seems to have to do with uniqueness of solutions to differential equations, but that won't help a person who doesn't intuitively understand that differential equations should have unique solutions. The power series proof for Euler's identity will appeal to someone with a more formal mindset, etc.

That said, current mathematical education favors formalism pretty hard, to the detriment of other approaches. It's understandable because from a teaching perspective it is the easiest to do, but there's a lot of room for improvement.

[silverspawn]

You insight seems to have to do with uniqueness of solutions to differential equations, but that won't help a person who doesn't intuitively understand that differential equations should have unique solutions. The power series proof for Euler's identity will appeal to someone with a more formal mindset, etc.

It was actually both the differentiation and the power series, and I'd heard neither of them before!

[silverspawn]

Three things that are common wisdom in media but seem blatantly false

- there are people with photographic memory (Does not exist according to Wikipedia)
- you can knock someone out in a few seconds by making them breath in or drink certain substances (I've briefly researched this; no such substance seems to exist, at least not one that you inhale or drink. There's stuff that will knock people out, but it'll take a while.)
- bottles break when you hit people with them

Someone should make a longer list

[silverspawn]

Ok one day break was enough. I'm ready. Hit me with another heroin trip I mean jhana.

To enter the jhana, one should not go into meditation with the goal of entering the jhana, but I want to enter the jhana so who are we kidding

[Awaclus]

- you can knock someone out in a few seconds by making them breath in or drink certain substances (I've briefly researched this

...why

[silverspawn]

...why

because the main character in my webnovel needed to knock some people out safely...

[silverspawn]

Ok one day break was enough. I'm ready. Hit me with another heroin trip I mean jhana.

To enter the jhana, one should not go into meditation with the goal of entering the jhana, but I want to enter the jhana so who are we kidding

Well that was a bad idea. This time it was too intense too quickly and it was lacking on the pleasurableness side. Which is also something I've heard can happen; if it's just intense bodily feelings that are not particularly high valence, it can even be unpleasant. So for the first time ever, had to stop a meditation because the effects were too strong.

... Years of meditating so-and-so, one 10 day retreat, and a few weeks of intense concentration meditation, and nothing that powerful ever happened. But one week of attempted Metta and things are blown way out of proportion. I did not expect this.

[silverspawn]

One of the annoying things about my new world view is that I can no longer get away with just computer stuff. Like, under the functionalist assumption, AI is the big story, AI is about computer science and math, so it's enough to be good at that.

But Alas. the brain doesn't work that way. So now, instead of just math and CS, I now also need to know physics, biology, and philosophy (I wouldn't say they have accumulated a body of knowledge, but one should be familiar with the arguments nonetheless).

Rn I'm working through two lectures on kinetics and oscillations. The most fun thing so far: while simple oscillations are a 1d movement, instead of describing them in 1d, we're going to describe them with complex numbers in 2d and then throw out the complex components afterward. Why??  Apparently it makes it easier. e^ix and all that. You can now express them as exponentials.

Also it's interesting; physically, an oscillatory 1d motion like a pendulum swing is now (the 1 axis projection of) a circular motion. So you can imagine that every pendulum actually goes around in a circle, except only the x dimension is visible. Does it actually go around in a circle in any real way? Probably not, but who knows?

[silverspawn]

I'm definitely vibing with this physics stuff. It's beautiful in its own way.

[silverspawn]

Taylor Series is another thing that I feel like was criminally poorly explained to me, though I don't remember where I first learned it. But I remember thinking of them as weird and unintuitive, whereas now I feel like they're one of the most amazing results in all of mathematics, and also make perfect intuitive sense.

[silverspawn]

So I've lost my phone

- on a little table in a train when getting up
- - But a fellow passenger notified me in time, so I could grab it
- on a little farm in the mountains while hiking
- - But I noticed it during the same trip, went back, and got it
- in a bus
- - But I noticed it, caught the same bus when it returned an hour later, and the bus driver had it

The phone is currently still safely in my possession.

Conclusion: There's no reason to be careful with your phone; if you lose it, you'll get it back anyway. That definitely checks out.

[silverspawn]

I've never understood as much about physics as I do now!

[silverspawn]

there really are people with amazing chess intuition who cannot rub two logical sticks together. I know that the correlation is weak, but intuitively this is still baffling to me.

[Awaclus]

there really are people with amazing chess intuition who cannot rub two logical sticks together. I know that the correlation is weak, but intuitively this is still baffling to me.

There really are people who can rub two logical sticks together with bad correlation intuition.

[silverspawn]

https://www.wsj.com/articles/chess-cheating-hans-niemann-report-magnus-carlsen-11664911524

[silverspawn]

[silverspawn]

[MiX]

Congratulations silver!

[silverspawn]

So there was this story where someone was caught jerking off in a business meeting or something, thinking his web cam was off.

But why's that so bad? Let's analyze the situation a bit.

The jerk, of course, is the derivative of the derivative of the derivative of position, i.e., the derivative of the derivative of velocity, i.e., the derivative of acceleration. But it's unclear what jerking off means. Did he "jerk of" in the sense of having a constant large jerk? Or an increasing jerk?

Either one may be unusual. Like, I can imagine different kinds motion. The one where position is a constant. Easy. The one where the derivative (i.e., velocity) is a constant. Sorta easy. And the one where the derivative of the derivative (i.e., acceleration) is a constant. Still doable.

But a constant jerk? Very strange. I don't know if humans have the capacity to differentiate constant-jerk motions and constant-acceleration motions. So perhaps it's understandable if people found it strange to see someone jerking off. And if it means an increasing jerk, that's even weirder.

At the same time, one should have some sense of proportion. Imagine if instead he was seen popping off. The pop, of course, is the derivative of the derivative of the derivative of the derivative of the derivative of the derivative of position, i.e., the derivative of the derivative of the derivative of the derivative of the derivative of velocity, i.e., the derivative of the derivative of the derivative of the derivative of acceleration, i.e., the derivative of the derivative of the derivative of jerk, i.e., the derivative of the derivative of snap, i.e., the derivative of crackle. So if popping off means a constant large pop, seeing someone popping off would be bizarre indeed. Not to speak of an increasing pop.

Something I've wondered for a long time is if real motion always has continuous functions of position, velocity, acceleration, jerk, snap, crackle, pop, et cetera. Intuitively, it seems like it may be the case that all of those, and infinitely more, are required to be strictly continuous. But it also may be that you can go from 0 to a specific jerk in an instant.

And now that I understand physics a bit better, I think the latter may be true! Because force is proportional to acceleration, so if you drop an object in mid-air, its acceleration should immediately jump to a certain value, thus creating a discontinuous jerk; it's 0 - infinity - 0. In other words, normal objects tend not to jerk off, I believe.

Then again, Newtonian physics is an approximation. Perhaps if forces are treated as themselves moving only with the speed of light, things change.

[silverspawn]

But I think it's safe to say that everyone who claims to be popping off in any area is probably lying

[silverspawn]

Full Report: https://drive.google.com/file/d/11IokKgTVSXdpYEzAuyViIleSZ_2wl0ag/view

[silverspawn]

So let's see if I can explain physics.

Newton's equations tell us two things. One is that forces are determined solely by the positions of objects. The other is that forces determine how objects accelerate. In particular, F = ma, where F is force, m is the mass of an object, and a is acceleration.

What does this mean? Well, consider any system of N objects with particular masses. Suppose we know the position and velocity (first derivative of position) of each object. Because we know the position, we know the forces. Because we know the forces, we can use F = ma to compute the acceleration.

What about the derivative of the acceleration (the "jerk?") Well, we have

F = ma

Thus, if we take derivative wrt time on both sides...

dF/dt = dma/dt = m * da/dt = m * "jerk"

We know m (the mass), and the jerk is what we want to compute. Using the chain rule, the left side is

dF/dx * dx/dt = dF/dx * v

I.e., force with respect to time is just force with respect to position times position with respect to time. Which is velocity. So it's force with respect to position times velocity. We know velocity by assumption, and we know force with respect to position because Newton's laws tell us how forces change depending on position.

So we can compute the derivative of the acceleration; the full equation is dF/dx * v =m * jerk, or jerk = dF/dx * v/m. The same is true for arbitrary further derivatives. Just repeat the process above.

In other words, we can predict how the system evolves into the future indefinitely. And what did we need to do this? The position and velocity of each particle. So each system of N objects requires precisely 2N pieces of information: the position of each particle, and the velocity of each particle. Those are enough to predict the system's behavior indefinitely into the future. Concretely, this means computing, for each object, a function for how this object moves over time.

Alternatively, if we have a parametrized system of N objects, we can compute all of the forces based on those parameters, and then (using F=ma and solving the resulting differential equations), obtain functions for the position of each object. This will yield a solution with 2N unknowns, or variables. Each of the 2N pieces of information (initial position or velocity of one of the objects) will yield one equation; with all 2N equations, we can obtain the values of the 2N unknowns.