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.

you could think about it this way: one sphere and two spheres have the same “number” of points. (in the same way that there are just as many real numbers as there are real numbers in the interval (0,1).)

so, it becomes “”plausible”” that you could use one sphere to construct two spheres (because in some sense, you aren’t “adding any new points”).

but in the real world, “spheres” only have a finite number of atoms. so if we regard atoms as “points”, then it’s no longer true that one sphere and two spheres have the same number of “points”. and in some sense, this is why the sphere duplication trick doesn’t work in the real world.

it’s also worth mentioning that you have to do some pretty fucked up and unusual things in order to actually duplicate the sphere, and if you don’t allow such weird things to be done to the sphere, then it’s no longer possible to duplicate it, even with the axiom of choice.

yeah this is true. i should have clarified a bit better that a well ordering wouldn’t give you a “least gay” person in that sense of the word. it would be more correct to say there is a well ordering ⊰, and so there is a “⊰”-least gay person. but of course a “⊰”-least gay person could be in the middle of that spectrum.

but the number of people on earth is finite, so in fact the usual ordering is a well-ordering in this case. so i guess those two mistakes i made cancel each other out, and the axiom of choice isn’t even needed here.

oops. you’re completely right. i forgot there are only a finite number of people on earth. there is a gayest person

a consequence of the axiom of choice is that every set can be given a well ordering. and well orderings always have smallest elements, but they may not have largest elements.

so there is someone who is the least gay, but there may not be a single person who is the most gay.

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 (*v _{1}, v_{2}, ..., v_{n}*), which can be thought of as a function {1, 2, ..., n} → ℝ, where

*k*∊ {1, ..., n} gets sent to

*v*. So, an

_{k}*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 (*v _{1}, v_{2}, ...*) 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.)

The default allocation for Recall on a device with 256 GB will be 25 GB, which can store approximately 3 months of snapshots.

this comes out to about 2 GB / week. it’s honestly terrifying they could be generating 2 GB of activity data for just a weeks worth of computer use. it’s both a privacy nightmare and an optimization nightmare

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

from a topological perspective, wraps and tacos are two different beasts.

in a wrap, the bread completely surrounds (and encloses) the other ingredients, so theres a 2-dimensional hole involved (which basically means the inside is hollow).

in a taco, no such wholes are present.

you can also distinguish sandwiches from tacos and wraps (since sandwiches involve two pieces of bread, like you said). but unfortunately, you can’t topologically distinguish a burger from a sandwich

it is possible to rigorously say that 1/0 = ∞. this is commonly occurs in complex analysis when you look at things as being defined on the Riemann sphere instead of the complex plane. thinking of things as taking place on a sphere also helps to avoid the "positive"/"negative" problem: as |x| shrinks, 1 / |x| increases, so you eventually reach the top of the sphere, which is the point at infinity.

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

being a prompt engineer is so much more than typing words. you also have to sometimes delete the words and then type new ones

i think this is a fairly reasonable gut reaction to first hearing about the "unnatural" numbers, especially considering the ways they're (typically) presented at first. it seems like kids tend to be introduced to the negative numbers by people saying things like "hey we can talk about numbers that are less 0, heres how you do arithmetic on them, be sure to remember all these rules". and when presented like that, it just seems like a bunch of new arbitrary rules that need to be memorized, for seemingly no reason.

i think there would be a lot less resistance if it was explained in a more narrative way that explained why the new numbers are useful and worth learning about. e.g.,

- negative numbers were invented to make it possible to subtract any two whole numbers (so that it's possible to consistently undo addition).
- rational numbers were invented to make it possible to divide any two whole numbers (so that it's possible to consistently undo multiplication, with 0 being a weird edge-case).
- real numbers were invented to facilitate handling geometrical problems (hypotenuse of a triangle, and π for dealing with circles), and to facilitate the study of calculus (i.e. so that you can take supremums, limits, etc)
- complex numbers were invented to make it possible to consistently solve polynomial equations (fundamental theorem of algebra), and to better handle rotations in 2d space (stuff like Euler's formula)

i think the approach above makes the addition of these new types of numbers seem a lot more reasonable, because it justifies the creation of all the various types of numbers by basically saying "there weren't enough numbers in the last number system we were using, and that made it a lot harder to do certain things"