wonderings

were the actions of the simulators in the talos principle ethical?

most people do believe that animals should be afforded at least some of the ethical considerations we apply to humans, generally more the more intelligent the animal is. dolphins, elephants, high primates and certain parrots have rudimentary cultures and civilizations, as well as being capable of logically solving problems, feeling empathy towards members of their own species, their caretakers, and others, and feeling emotion and pain.
if this is the case, isn't what we're doing to these species through hunting and habitat loss comparable to how europeans abused african tribes in the late 19th century?

okay.

if you perfectly simulated, atom by atom, a human being in a room, would they be conscious? sure, they could calculate and have internal thoughts. they could feel emotion and have opinions. they would THINK they were just as conscious as everybody else. but would they really have a sort of "point of view" the same way we do? or would their existence be like that of a calculation machine, grinding gears and sending signals but not actually, you know, seeing out of their own eyes?

your own self and brain is just atoms making meat and chemicals and blood and shit, just a meat computer calculating and sending signals. so the theoretical 1:1 simulation or replica is just as "alive" as any human, unless you have some notion of a soul

consider some function that is zero at every point. now set the output of some input x₀ as f(x₀), where f is some differentiable function (say, f(x) = x²), so that the function is zero everywhere except at that point. now set some other point x₁ to equal f(x₁). repeat this process uncountably infinitely many times, until this new function equals f at every point.

is this function continuous and differentiable?

it seems like it's not, because at every point in its creation, it's just a string of points. then again, it seems like it is, because it equals f at every point, and f is both continuous and differentiable. then again, maybe making a function like this is impossible, because you have to do uncountably many definitions in order to cover the reals, and the whole point of uncountable numbers is that even with infinite time, you can't do uncountably many things.

but if you can't construct a function like this, how can you construct any function? what does continuity even mean, anyway? now that i think about it, the ideas of limits and continuity seem even more shaky than the infinitesimals they were meant to replace. they make sense, but they don't seem like they could ever be rigorously justified.

even harder question: why am i pondering the nature of continuity and the real line when i should be doing my overdue history homework? it's a mystery.

dear god, there's nothing scarier than the interval (0, 1).

(with the exception, perhaps, of 5-page essays on the civil war)

I think that provided there is some open interval (x_n-δ,x_n+δ) in which every point had been mapped to the continuous and differentiable function after uncountably many steps, then the function at x_n would be continuous and differentiable.
The problem is that any function f_n(x) does not fit this criterion, because you are going one point at a time, so it's a countably infinite number, so we can do a diagonal proof to find a point however close to x_n that is still 0, and so at no time would it be continuous.

continuity????????

related: https://www.youtube.com/#/watch?v=D2xYjiL8yyE

If asked to find a random integer on (0, ∞), the only sensible answers are 0 and ∞.

suppose n, a, and b are integers, a and b > 0, a != b, and n = a^2 + b^2

sqrt(n) = sqrt(a^2 + b^2)
sqrt(n) = ||a + bi|| by the definition of the complex modulus
n = ||(a + bi)^2|| by the fact that the square of a complex number has a modulus equal to the square of the modulus of the original number
n = ||a^2 - b^2 + 2abi||
n = sqrt((a^2 - b^2)^2 + (2ab)^2) definition of complex modulus
n^2 = (a^2 - b^2)^2 + (2ab)^2

conclusion: any number that can be written as the sum of two squares has a square that can also be written as the sum of two squares. by induction, this holds for all higher n^2^k, so long as the corresponding a's and b's are never equal and never equal zero.

and in fact, you can multiply a+bi with c+di to show that the product of any two numbers that can be written as a sum of two squares can also be written as a sum of two squares

moreover we may exactly classify all numbers that can be written as a sum of two squares

lemma: Let p = 3 mod 4 and prime

if a^2+b^2 = 0 mod p then either a,b = 0 mod p (this is one solution) or we can divide by b (b is coprime to p so we can do this):

(a/b)^2=-1 mod p. So (a/b)^4=1 mod p, and thus the order of a/b in Z/pZ is 4. but wait, the cardinality of Z/pZ is p-1. and this is 2 mod 4. but by Lagrange's Theorem all element's order divide |Z/pZ| = p-1 so this is impossible!

therefore a,b = 0 mod p is the only solution.

(yeah ok we didn't really need group theory there, quadratic reciprocity does the trick too.)

anyway the punchline is that numbers with an odd number of any 3 mod 4 prime factor cannot be written as the sum of two squares. some minor details left out but you get the idea

(this is a necessary, and as it turns out, sufficient condition - what remains is to be able to show that all primes 1 mod 4 can be expressed as the sum of two squares, and you can use quadratic reciprocity)

things i hate:
string theory
the cosmological constant
quantum entanglement
the fact that all attempts to get around quantum entanglement still end up with some kind of superluminal communication (looking at you, de broglie bohm theory)

why do you hate superluminal communication?

why do you hate superluminal communication?

it breaks relatively. even if you manage to somehow mush it into behaving, it still just seems wrong.

why is upwards infinity so much easier to swallow than downwards infinity? if you told somebody that the universe will exist forever and that it has already existed for an infinite amount of time, they would almost surely find the first statement much easier to accept. why?

here's something that's been bothering me for a long time:

suppose you have two balls a and b. ball a weighs one kilogram and is moving two meters per second. ball b weighs two kilos and is stationary. if ball a strikes ball b and stops completely, how fast does ball b move?

easy, right? it's just conservation of momentum. momentum a = momentum b, momentum = mass * velocity, 1 * 2 = 2 * b velocity, so ball b moves at 1 meter per second.

but isn't energy supposed to be conserved as well? and since energy = mass * velocity squared, 1 * 2^2 = 2 * v^2, ball b velocity should be root(2), which is about 1.4. a lot faster than conservation of momentum predicts. what happened to all that energy? if conservation of momentum is strictly correct and ball b will move at 1 m/s, then we're losing exactly 2 joules, half of ball a's energy, to the aether.

"oh, that's because the ball loses a bit of energy in the collision to the air"

but what if it's in a vacuum?

"then it just sends out a few photons"

what if the the collision happened in reverse, with the heavier ball hitting the lighter one? then the system would appear to gain energy, with conservation of momentum predicting higher speeds than conservation of energy.

and most importantly, if energy isn't really conserved, at least in a meaningful, measurable way, why should momentum be conserved? is momentum somehow more important than energy? what's stopping the ball from losing momentum to the air or to friction or radiation? if we have to introduce external, unrelated factors like air resistance or heat radiation to our model, if it only works if you sort of fudge it around until everything sort of aligns, it seems like something is very wrong with our model -- or at the very least, its predictions shouldn't ever be taken fully seriously.

you are forgetting that all objects have a certain inertia, and you will lose some energy in the form of heat when one objects hits another.

of course rolling friction is way lower than the friction of like, a rubber Block so what do i know

you are forgetting that all objects have a certain inertia, and you will lose some energy in the form of heat when one objects hits another.

of course rolling friction is way lower than the friction of like, a rubber Block so what do i know

no, i'm not forgetting anything. that's the point. if energy can be lost, why can't momentum?

edit: i think i came off a bit too standoffish here. sorry about that

so like-signed electric charges repel one another, right? and an electron isn't actually a point charge, but rather a big bundle of charge spread over a region. why doesn't the electron repel... itself?

why doesn't all that charge just sort of spread out into space, like a sphere of pressurized water would disperse in lower pressure water? what's holding it together? and wouldn't the energy required to hold that bundle of charge together grow to infinity as we went towards the center and the "pressure" and "density" of the charge increased?

or does the electric field part of the electron really spread out like that, and the electron field just keeps pouring charge into the center? is that why an electron can influence things far away, at the speed of light, without photons--because its electric field is constantly growing outwards?

why is the electron field a probability wave, while the electromagnetic field is not?

why do we treat the reynold's number as constant, when it depends on the average speed of the flow in question? the average speed can change, and in fact depends in the reynold's number. it seems like the real constant should just be the viscosity and length scale parts of the reynold's number, while the speed part varies.

can i eat me ass

can i eat me ass

y

why does the electron field have inertial mass? if it's just bubbles and waves like the em field, why can't it travel the speed of light? is it because it sheds energy to the em field whenever it's moving?

what parts of the em field are quantized? just the values at any given point? the local extrema? the amplitudes of the fourier transform?

how can two fields interact with one another?

in the copenhagen interpretation, how are measurements defined? why don't the two slits collapse the wave, and why doesn't the measuring device cause interference?

why is the weak force considered a force, and not just a collection of interactions? it doesn't really impart much momentum, it interacts mainly through particles rather than fields, it doesn't really have its own associated property (like charge, gravitational mass, etc), and its force carrying particles have mass.

how does the pauli exclusion principle work on electrons that aren't surrounding an atom? does it have some radius of effect? how does it work when you think of electrons as field bubbles rather than particles?

the multiplicative group over the positive rational numbers is just the direct group product of infinitely many multiplicative groups of powers of some (essentially arbitrary) prime. until you include addition, no one prime can be said to be bigger than another, the "value" of each prime is completely arbitrary/even meaningless, and you can add or remove primes freely without inducing holes.

for example:

the powers of two under multiplication/division are a closed group -- if you multiply any one of them by any other you will end up with another power of two. so, since we always know our number will look like 2^x, let's just drop the 2^, and write (x). now, (x) * (y) = 2^x * 2^y = 2^(x + y) = (x + y). so we can redefine the product operation (x) * (y) as just (x+y).

but wait a minute. if all we have now is the operation (x) * (y) = (x + y), there's absolutely nothing in there forcing the power at the bottom to be two. it might as well be "apples", for all we know or care.

and, as an experiment, what happens when we let every number have two values, instead of just one? let's define (x, y) * (z, w) = (x + z, y + w). what we have now is a multiplication system with two primes. we've come up with a way to represent all composite numbers with these two primes as their only prime factors, as a function of the number of each prime factor in a number. but we don't know what factors are. maybe (x, y) = 2^x * 3^y, but it could just as easily be apples and oranges because we don't actually know what those two primes are, and the two columns never interact in any way, it's meaningless to talk about which is "bigger", and how they are ordered within the number system.

in conclusion:

all the weirdness with primes isn't really anything to do with primes themselves, it only comes about when you combine addition with an infinite-prime multiplicative group. and interestingly, the rules for addition seem very similar to the rules for a single-prime multiplication group.