it was over as soon they casted kevin hart

mmmm cookies and cream

applied mathematics can get very messy: it requires performing a bunch of computations, optimizing the crap out of things, and solving tons of equations. you have to deal with actual numbers (the horror), and you have to worry about rounding errors and stuff like that.

whereas in theoretical math, it’s just playing. you don’t need to find “exact solutions”, you just need to show that one exists. or you can show a solution doesn’t exist. sometimes you can even prove that it’s impossible to know if a solution exists, and that’s fine too. theoretical math is focused more on stuff like “what if we could formalize the concept of infinity plus one?”, or “how can we sidestep Russel’s paradox?”, or “can we turn a sphere inside out?”, or The Hairy Ball Theorem, or The Ham Sandwich Theorem, or The Snake Lemma.

if you want to read more about what pure math is like, i strongly recommend reading A Mathematician’s Lament by Paul Lockhart. it is extremely readable (no math background required), and i thought it was pretty entertaining too.

Infinite-dimensional vector spaces also show up in another context: functional analysis.

If you stretch your imagination a bit, then you can think of vectors as functions. A (real) n-dimensional vector is a list of numbers (v₁, v₂, …, v_n), which can be thought of as a function {1, 2, …, n} → ℝ, where k ∊ {1, …, n} gets sent to v_k. So, an n-dimensional (real) vector space is a collection of functions {1, 2, …, n} -> ℝ, where you can add two functions together and multiply functions by a real number.

Under this interpretation, the idea of “infinite-dimensional” vector spaces becomes much more reasonable (in my opinion anyway), since it’s not too hard to imagine that there are situations where you want to look at functions with an infinite domain. For example, you can think of an infinite sequence of numbers as a function with infinite domain. (i.e., an infinite sequence (v₁, v₂, …) is a function ℕ → ℝ, where k ∊ ℕ gets sent to v_k.)

and this idea works for both “countable” and “uncountable” “vectors”. i.e., you can use this framework to study a vector space where each “vector” is a function f: ℝ → ℝ. why would you want do this? because in this setting, integration and differentiation are linear maps. (e.g., if f, g: ℝ → ℝ are “vectors”, then D(f + g) = Df + Dg, and ∫*(f+g) = ∫f + ∫g, where D denotes taking the derivative.)

i forgot for a second that the winters and summers get flipped in the southern hemisphere

it will only be the strongest material in the universe until it gets boiled. trust me on this one

if they invent some new kind of fucked up math to do it then there could be far reaching consequences

“shittitest alchemist currently alive” has got to be one of the most challenging titles to hold onto for any serious length of time

you can always add an empty room without changing the total number of rooms, so there should be plenty of room for sisyphus and his boulder at the hotel

it’s worth mentioning that very rarely is baby formula better than breast milk. the contents of breast milk change depending on the what the child needs at the moment. it’s really sick that some companies market it as a better option than breast milk

source

this reminds me of what happened to the instagram cofounders when zuckerberg asked to buy their company:

Systrom [cofounder] said he feared turning down an acquisition offer from Facebook would send Zuckerberg into “destroy mode” — a concern that Cohler [early investor] affirmed.

(source)

this stuff came up in a court hearing, and then nothing happened about it

one time i had to call a company during regular 9-5 business hours to cancel a subscription after starting a free trial.

that experience was so horrible ive since sworn off free trials altogether. nowadays, if i need a free trial to use an app or website for a couple days, then i will simply not use that app or website.

can we compromise on drinking raw milk with flouride added?

i think this a really nice way of thinking of things, especially for regular everyday life.

as a mathematician though, i wanted to mention how utterly and terribly cursed square roots are. (mainly just to share some of the horrors that lurk beneath the surface.) they’ve been a problem for quite some time. even in ancient greece, people were running into trouble with √2. it was only fairly recently (around the 17th century) that they started looking at complex numbers in order to get a handle on √-1. square roots led to the invention of two different “extensions” of the standard number systems: the real numbers (e.g. for √2), and later, the complex numbers (e.g. for √-1).

at the heart of it, the problem is that there’s a fairly straightforward way to define exponentiation by whole numbers: 3ⁿ just means multiply 3 by itself a bunch of times. but square roots want us to exponentiate things by a fraction, and its not really clear what 3^1/2 is supposed to mean. it ends up being that 3^1/2 is just defined as 3^1/2 = x, where x is "“the number that satisfies x² = 3"”. and so we’re in this weird situation where exponentiating by a fraction is somehow defined differently than exponentiating by a whole number.

but this is similar to how multiplication is defined: when you multiply something by a whole number, you just add a number to itself a bunch of times; but if you want to multiply by a fraction, then you have to get a bit creative. and in a very real sense, multiplication “is the exponentiation of addition”.

from a formal perspective, division is an “”abbreviation”” for multiplying by a reciprocal. for example, you first define what 1/3 is, and then 2/3 is shorthand for 2 * (1/3). so in this sense, multiplication and division are extremely similar.

same thing goes for subtraction, but now the analogy is even stronger since you can subtract any two numbers (whereas you “can’t” divide by 0). so x - y is shorthand for x + (-y). and -y is defined “to be the number such that y + (-y) = 0”.

i think this has a lot to do with it.

and i think he’s also boosted a lot by the fact that he doesn’t really communicate that many original thoughts. instead, it seems like he tends to blindly agree with whoever he has on camera. so he simultaneously cultivates these personas of “having intellectual curiosity” while also being a stand-in for the average college dorm bro.

(i’m not trying to defend him here, he still causes serious harm by platforming bad actors and endorsing their views.)

other people have given good answers to this question but i think it’s worth saying that this isn’t a dumb question. it took a lot of smart people and thousands of years to figure out that time passes at different speeds in different parts of the universe. it’s not intuitive at all.

i haven’t driven in a while but i can’t remember really needing to control my keys that much. usually they stay put pretty well once they’re in the ignition

what happens if people get their weed from may

most wikipedia thing ever to start the joke page with a link to the article defining “humor”.