covid has been going around

the bottom picture would look pretty cool in a shaun video

i bet you i could predict it with 100% accuracy if you give me another 4 months

same tbh

if they use an LLM to make the suggestions then it’s possible it ends up suggesting websites that don’t even exist. or it could accidentally suggest a malware website, or make a typo, etc.

this could be dangerous if they aren’t very careful

maybe it’s being developed in the arctic circle and there really are only four nights per year

Anyone can pay $150 to become a dues-paying member and rub elbows with the court’s nine justices at events like the dinner where Windsor spoke with Alito. (Tickets for the dinner were an extra $500.)

this is all it took for him to admit this stuff? anybody with 650$ could have walked in and asked him a couple prodding questions? these guys really arent even trying to hide it anymore

where’s the creamy filling

it was over as soon they casted kevin hart

personally i’m a fan of tearing off everything except a small corner of a napkin

go for it

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.

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.

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

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

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”

booooooooooo