it’s rough when the math gets so complicated that you have to break your finger in order to be able to visualize things

i wish they would do this in math instead of the boring system where it’s always alphabetical

that would be a lot clearer. i’ve just been burned in the past by notation in analysis.

my two most painful memories are:

• in the (baby) rudin textbook, he uses f(x+) to denote the limit of _f _from the right, and f(x-) to denote the limit of f from the left.
• in friedman analysis textbook, he writes the direct sum of vector spaces as M + N instead of using the standard notation M ⊕ N. to make matters worse, he uses M ⊕ N to mean M is orthogonal to N.

there’s the usual “null spaces” instead of “kernel” nonsense. ive also seen lots of analysis books use the → symbol to define functions when they really should have been using the ↦ symbol.

at this point, i wouldn’t put anything past them.

unless f(x₀ ± δ) is some kind of funky shorthand for the set f(x) : x ∈ ℝ, x - x₀ | < δ . in that case, the definition would be “correct”.

it’s much more likely that it’s a typo, but analysts have been known to cook up some pretty bizarre notation from time to time, so it’s not totally out of the question.

i think the ε-δ approach leads to way more cumbersome and long proofs, and it leads to a good amount of separation between the “idea being proved” and the proof itself.

it’s especially rough when you’re chasing around multiple “limit variables” that depend on different things. i still have flashbacks to my second measure theory course where we would spend an entire two hour lecture on one theorem, chasing around ε and η throughout different parts of the proof.

best to nip it in the bud id say

i still feel like this whole ε-δ thing could have been avoided if we had just put more effort into the “infinitesimals” approach, which is a bit more intuitive anyways.

but on the other hand, you need a lot of heavy tools to make infinitesimals work in a rigorous setting, and shortcuts can be nice sometimes

the “categorical” way of defining tensor products is essentially “that thing that lets you turn multi-linear maps into linear maps”, and linear maps (of finite dimensional vector spaces) are basically matrices anyways. so i don’t see it as much of a stretch to say tensors are matrices.

(can you tell that i never took a physics class?)

a tensor is a multi-linear map V × … × V × V^* × … × V^* → F, and a multi-linear map V × … × V × V^* × … × V^* → F is the same as a linear map V ⊗ … ⊗ V ⊗ V^* ⊗ … ⊗ V^* → F. and a linear map is ““the same thing as”” a matrix. so in this way, you can associate matrices to tensors. (but the matrices are formed in the tensor space V ⊗ … ⊗ V ⊗ V^* ⊗ … ⊗ V^*, not in the vector space V.)

but imagine you’ve just gotten use to living on a moss planet over the past 40 million years, and now all of a sudden you walk outside and all the moss is gone

life becomes joyous again when “a vector space is a module over a field”

i think it depends on what you mean by “accurately”.

from the perspective of someone living on the sphere, a geodesic looks like a straight line, in the sense that if you walk along a geodesic you’ll always be facing the “same direction”. (e.g., if you walk across the equator you’ll end up where you started, facing the exact same direction.)

but you’re right that from the perspective of euclidean geometry, (i.e. if you’re looking at the earth from a satellite), then it’s not a straight line.

one other thing to note is that you can make the “perspective of someone living on the sphere” thing into a rigorous argument. it’s possible to use some advanced tricks to cook up a definition of something that’s basically like “what someone living on the sphere thinks the derivative is”. and from the perspective of someone on the sphere, the “derivative” of a geodesic is 0. so in this sense, the geodesics do have “constant slope”. but there is a ton of hand waving here since the details are super complicated and messy.

this definition of the “derivative” that i mentioned is something that turns out to be very important in things like the theory of general relativity, so it’s not entirely just an arbitrary construction. the relevant concepts are “affine connection” and “parallel transport”, and they’re discussed a little bit on the wikipedia page for geodesics.

it’s a bit of a “spirit of the law vs letter of the law” kind of thing.

technically speaking, you can’t have a straight line on a sphere. but, a very important property of straight lines is that they serve as the shortest paths between two points. (i.e., the shortest path between A and B is given by the line from A to B.) while it doesn’t make sense to talk about “straight lines” on a sphere, it does make sense to talk about “shortest paths” on a sphere, and that’s the “spirit of the law” approach.

the “shortest paths” are called geodesics, and on the sphere, these correspond to the largest circles that can be drawn on the surface of the sphere. (e.g., the equator is a geodesic.)

i’m not really sure if the line in question is a geodesic, though

as someone who studied both computer science and “higher level” math, i think that the use of “higher level” does kind of loosely match the computer science meaning. “higher level” math is all about abstracting away the details, to focus on the “big picture” of how things work. e.g., measure theory focuses on looking at integration from a very abstract perspective, and this abstract perspective lets you treat summation and Riemannian integration as “the same thing”. you can draw a parallel to how in programming, a higher level perspective lets you treat various operating systems/pieces of hardware as “the same thing”.

another example would be how abstract algebra lets you treat various algebraic structures as “the same thing”, e.g. just about anything is a group, lots of things are modules, etc. and then there’s category theory.

probably the biggest difference is that higher level math tends to be more challenging than lower level math, while lower level computer science tends to be more challenging than higher level computer science. (at least in my experience)

even bigger sloths

mmmm cookies and cream

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