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.)

people joined a cult because of this theorem. that must be awkward

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

the standard (set theoretic) construction of the natural numbers starts with 0 (the empty set) and then builds up the other numbers from there. so to me it seems “natural” to include it in the set of natural numbers.

what if you just attach a second magnet to the car so that it pulls the first magnet forwards?

what were they doing for all that time

unfortunately this would only work on tanks that shoot lava

the teacher is the one doing the real lab experiment here

but can he grind on the topologists sine curve:

under an appropriate level of abstraction you can make lots of types of cylinders. in topology you can define a cylinder of a (topological space) X to just be X × [0,1]. this kind of definition comes up pretty frequently, and is used to create mapping cylinders, which i suppose are another type of cylinder.

the “normal” kind of cylinder is then just (circle) × [0,1], or (filled in circle) × [0,1], depending on whether you want it to look like an empty paper towel roll or a (full) can of beans

this condition makes “planetness” into a local condition. so theoretically, we can throw enough junk into space and stop anything we want from being a planet.

pluto just got unlucky in terms of the amount of trash it has in its way. its not fair :(

what if it’s not euclidean but only locally euclidean?

what’s the appeal of haskell? (this is a genuine question.) i’ve been a bit curious about it for a while but haven’t really found the motivation to take a closer look at it.

i think this is a really clean explanation of why (-3) * (-3) should equal 9. i wanted to point out that with a little more work, it’s possible to see why (-3) * (-3) must equal 9. and this is basically a consequence of the distributive law:

0  = 0 * (-3)
   = (3 + -3) * (-3)
   = 3 * (-3) + (-3) * (-3)
   = -9 + (-3) * (-3).

the first equality uses 0 * anything = 0. the second equality uses (3 + -3) = 0. the third equality uses the distribute law, and the fourth equality uses 3 * (-3) = -9, which was shown in the previous comment.

so, by adding 9 to both sides, we get:

9 = 9 - 9 + (-3) * (-3).

in other words, 9 = (-3) * (-3). this basically says that if we want the distribute law to be true, then we need to have (-3) * (-3) = 9.

it’s also worth mentioning that this is a specific instance of a proof that shows (-a) * (-b) = a * b is true for arbitrary rings. (a ring is basically a fancy name for a structure with addition and distribute multiplication.) so, any time you want to have any kind of multiplication that satisfies the distribute law, you need (-a) * (-b) = a * b.

in particular, (-A) * (-B) = A * B is also true when A and B are matrices. and you can prove this using the same argument that was used above.