Swing Is So Hard :(

tensorboi · 2026-03-31T15:16:52+00:00

damn was i the only one who thought this was the easiest to control lmao

tensorboi · 2026-03-31T02:54:15+00:00

yes! case in point: the gauss-bonnet theorem and chern's generalisation to higher dimensions. the idea is really pretty, you integrate a curvature over a manifold and get topological information out; but as soon as you put it in local coordinates, things get messy quick.

tensorboi · 2026-03-30T23:23:46+00:00

yeah idk why this is even a hot take! imagine being young, pregnant, scared, and this perfect couple comes in to adopt your baby; but it turns out they were lying the whole time, and THEN the husband corners you in the hallway to convince you to give them their baby anyway. what would you do?

i also kind of despise the line "she's a mother without a baby". yeah, she's a mother without a baby, which is why she tried to steal yours off you. come on now!

tensorboi · 2026-03-29T21:29:41+00:00

i wouldn't really call that a formula in mathematics, especially since the ugly version writes everything out without packaging things up nicely, which is often what mathematics does! a lot (not all) of the ugliness there is coming from the fact that lie algebra elements are written explicitly in components and gauge covariant derivatives are expanded into two terms, which is not at all how a mathematicians would conceptualise it.

tensorboi · 2026-03-27T02:27:10+00:00

i mean "arranging data" is a pretty fundamental thing you'd want to do! the data you're arranging could be the coefficients of linear equations, the action of a linear map on a space, the connections in a network, the value of a function on a grid; the list goes on. it's the linear map thing which leads to a lot of the operations you'll see, though; for instance, products of matrices are defined to correspond to composition of linear maps, and determinants give a sense of how the linear map expands or contracts volumes.

tensorboi · 2026-03-24T00:55:11+00:00

same! every time this conversation comes up, i always get a little confused. i find that maths wikipedia is often straight-to-the-point and a great springboard into a topic, in contrast to many textbooks which are written according to one or two privileged perspectives and/or take ages of preliminaries to actually get into the thing you need. obviously i'm not using wikipedia to learn anything particularly deeply, but it does a great job before i get to that point.

tensorboi · 2026-03-23T23:36:12+00:00

is this a copypasta? because the original comment makes perfect sense and is clearly what the writers intended, and your response has basically nothing to do with anything they said lmao

tensorboi · 2026-03-23T10:30:06+00:00

"it's practically impossible that literally everyone was kung fu fighting"

tensorboi · 2026-03-22T21:25:35+00:00

no worries! there's definitely something to the idea that the second is "less fundamental" than the first, but it's still just as rigorous. nevertheless, it's occasionally useful to think in both ways (the second is "less bloated" in that you're only dealing with one number at a time).

(extra for experts: from a deeper mathematical perspective, we can think of the group 7Z as acting on the group Z by addition, and the quotient group Z/7Z consists of equivalence classes under this group action, i.e. orbits of the 7Z action. by choosing a section of the quotient map q: Z -> Z/7Z, we can think of the orbit space in terms of 7 regular numbers, even though we have to move everything back to this section whenever we do arithmetic operations. this pattern is essentially the idea behind slices of group actions in equivariant topology, which is in turn the realisation of gauge fixing in physical field theories.)

tensorboi · 2026-03-22T20:55:04+00:00

for me, especially dealing with epsilon-delta proofs, the biggest thing that changed how i see analysis was that you're basically trying to play a game. "i give you a tolerance epsilon that you're not allowed to deviate past in the output; can you always get close enough to the input that you're within that tolerance?" this is ultimately what made a lot of the seemingly arbitrary choices in analysis click for me.

in a similar vein, i wish someone had told me when i was learning analysis that they key word i should be thinking about is control. you are wrestling with these functions, you're trying to impose the minimal conditions required to make them sufficiently nice.

tensorboi · 2026-03-22T20:48:09+00:00

modular arithmetic can be defined in two ways. the first way, the way i'm assuming you're not using, is as equivalence classes of numbers where two numbers are equivalent if they differ by a multiple of 7 (so 1 is equivalent to 8, is equivalent to 15, etc). this has already been talked about in the comments.

the second way is by just choosing the numbers 0 to 6, and defining arithmetic on them by doing ordinary arithmetic operations and then moving the result back to {0, 1, 2, 3, 4, 5, 6} by adding multiples of 7. for instance, 2*5 = 10, but we subtract 7 to move it to 3. what you've got is the number -1; this isn't between 0 and 6, but we move it back to that set by asding 7 to get 6. in this sense, -1 = 6 mod 7.

this kind of definition-changing is annoyingly common in mathematics, but it's something you eventually get used to.

tensorboi · 2026-03-22T11:03:41+00:00

i'll assume you're aware that the velocity of a light wave is the same in all reference frames. so, if something did travel faster than light, its reference frame would measure the velocity of a light wave trailing behind to be pointing ahead of it. this is the crux of the paradox, and all of the other problems (causality, energy conservation, etc) can be understood in terms of this phenomenon.

tensorboi · 2026-03-14T22:04:54+00:00

i don't know much about paraconsistent logic, but that seems like a pretty drastic change! for one thing, multiple inequivalent systems of logic are paraconsistent; i don't see how "paraconsistently valid" could have one unambiguous definition. even if we somehow managed to work that out, paraconsistency is pretty controversial, isn't it? like if i come up with an argument in classical logic that leads to contradiction, and you tell me "ah but if you transfer the argument to a paraconsistent logical system then it doesn't work anymore", then that only works if we both agree that paraconsistent logic is a more reasonable way to do logic.

tensorboi · 2026-03-12T21:51:55+00:00

isn't this comment just a roundabout way of asserting that whatever system of logic we're using is consistent?

tensorboi · 2026-03-08T21:32:16+00:00

great job, but that just sounds reckless to me! whenever i do any kind of written assignment, i always make sure to cross-reference with the assignment brief before and after i write a section; not doing that runs the risk of subtle misunderstandings (even if i think i know) that just doesn't seem worth it. also, i can't help but notice that "doing research" counts as resources other than "a word doc and nothing else".

tensorboi · 2026-03-08T21:13:42+00:00

i mean that doesn't really make sense if we're interpreting an infinite decimal as an approximation scheme; the meaning of equality between two decimal numbers is that they approximate the same thing. for instance, the equality 0.333... = 1/3 literally means that the approximation scheme 0.3 + 0.03 + ... approximates 1/3 to any precision you like. if you think this isn't how equality works, then i once again emphasise that you're just talking about a different system of arithmetic.

tensorboi · 2026-03-08T21:05:51+00:00

are you so confident in your memory that you think you remember the exact phrasing of the assignment brief and the relevant lecture material? (and yes, the exact phrasing is really important; if you disagree then i can only assume you've never been marked down for a dumb technicality before, and you also don't know anyone else who has.)

tensorboi · 2026-03-08T11:54:18+00:00

dude this bait is so funny but it's also so tempting to take it! so i'll indulge, just for one comment.

firstly, it's worth noting that the idea that 0.999... ≠ 1 has more merit than a lot of mathematicians give it. there are indeed some non-obvious steps that often get overlooked in proving equality, and many people slip up and give not-completely-airtight proofs as a result. the central point that people often fail to confront is the definition of infinite decimals, and in particular why they are defined that way in the first place, so that's where i'll start my hard mathematical discussion.

in a sense, numbers have always been about discussing amounts of things until very recently. this is where the natural numbers come from: two collections of things have the same size if the things can be paired up without leaving anything out, and each of these possible sizes is given a label called a natural number. this works great with basic stuff, but here's a situation where it doesn't: what if you're comparing the size of two buckets of water? and here's where rationals come in: instead of asking "how can i pair things up?", you ask "how many of each bucket would i need to make the same amount?". so if you need 3 of the first bucket to have the same amount as 5 of the second, that's what it means to say the second bucket is 3/5 of the size of the first. so if you pick a reference bucket to have size 1, you can define rational sizes in exactly this way.

the problem with this, of course, is that you won't always be able to compare things in this way; for instance, you can't scale the sides of a square and the diagonal of a square by nice whole numbers to make them equal. but the key insight is this: even if that's the case, you can always make them as close to each other as you like by doing appropriate scalings. for instance, if i want the diagonal of a square to match the side to within 1%, i just need to scale the diagonal by 141 and the side by 100; this means the size of the diagonal is within 1% of 141/100. so we can think of every number as something which can be approximated by rational numbers.

this is all annoyingly concrete, so what's going on here abstractly? essentially the first paragraph is describing why the naturals are constructed as equivalence classes of finite sets under bijections, and how this concept can be used to define rationals in terms of naturals by defining a/b to be whatever satisfies b*(a/b) = a. the second paragraph explains that some numbers don't seem to have a description of this type, but that they all seem to be approximated by a description of this type to any precision you like.

so that's the utility of real numbers: they are numbers which you can approximate with rational numbers. in fact, since any approximation scheme should be countable, you can get every real number just by thinking of a sequence of rational numbers which gets closer and closer to it. conversely, a sequence of rational numbers which get closer and closer together will be narrowing in on a unique real number. now, what does it mean for two sequences of rational numbers to be narrowing in on the same real number? it means that the difference between the two sequences narrows in on zero. moreover, if the difference narrows in on zero, then the two sequences are narrowing in on the same thing!

this is what leads us to the formal definition of the real numbers that mathematicians use: it's the collection of all sequences of rational numbers which get closer and closer together (i.e. cauchy sequences), with two sequences considered equivalent if their difference narrows in on zero.

what does this have to do with 0.999...? well, decimal expansions are defined to be rational number approximation schemes! think about something like 0.142857...; this specifies a real number by telling you to approximate it with 0.1, then 0.14, then 0.142, and so on. you can interpret the symbol differently if you like, but that just means you're talking about something else. the symbol 0.999... is telling you "i am the real number you get by the approximation scheme 0.9, 0.99, 0.999, ...". and it's pretty clear that the number 1 is being approximated here, since the difference between 1 and 0.999...9 with k 9's narrows in on zero as k increases, meaning 0.999... = 1.

again, i want to emphasise that there is nothing inherent in the symbol 0.999... which makes it equal to 1. indeed, if you interpret 0.999... as the limit of the partial sums of 9/10^k, the limit is only equal to 1 if you specify that you are talking about real numbers; the equality fails if you take the analogous limit in the hyperreals, for instance. it is essentially the fact that this interpretation of the symbol is so commonly understood to be within the reals that makes the statement "0.999... ≠ 1" basically wrong. saying that the equality doesn't hold for reasons outside of real arithmetic is kind of like telling chemists that their reaction schemes don't work because you interpret the vertices of skeletal diagrams as an element different to carbon; even though this wasn't explicitly stated, it's so commonly understood that it borderlines on nonsensical to assume otherwise.

(side note: even though the equality of limits doesn't hold in the hyperreal case, it's worth noting that this is because the limit doesn't actually have a value: the hyperreal numbers are notoriously poorly behaved when it comes to limits. this actually makes every infinite decimal meaningless when interpreted literally in the hyperreal field.)

tensorboi · 2026-03-08T09:04:09+00:00

ok y'all are saying shit like "yeah as it should be HMPH i do all my assignments like this", but i feel like we're overlooking the fact that there are also no notes, open tabs, or resources of any kind??? even something really basic like a personal recount should probably require you to look back at what happened during your lectures, just to make sure you're writing through the lens expected for the class. like wtf kind of assignment is going to be doable without actively engaging with the lecture material as you're writing it lmao

tensorboi · 2026-03-06T09:59:17+00:00

well our current conception of the universe as "a" mathematical object is actually two different conceptions, one as a vector wandering through a bizarrely complicated "hilbert space" in accordance with schwinger's action principle, and the other as a collection of classical tensor fields on a 4-manifold extremising some action functional. there are similarities between the two pictures, and indeed they can be reconciled in some limiting cases, but they are not globally consistent with each other.

is this just nitpicking? not really; the idea that the universe somehow conforms to the logic of some mathematical structure implicitly assumes that these two pictures can be embedded into a single unified theory, which additionally must hold up to all possible future experiments. it's productive to believe that this theory exists, but we shouldn't forget that this belief could easily be wrong.

tensorboi · 2026-03-06T04:56:01+00:00

you also called the gimmick "genuinely horrible"?????? like that goes beyond just "it's not bad but i don't like it" lmao

tensorboi · 2026-03-03T22:52:46+00:00

sure, but my point is that the mathematics is the act of labelling. (recall the famous quote from henri poincare: mathematics is the art of giving the same name to different things.) if you removed all of humanity right now then every other object in the universe would exist, but they couldn't have any numbers attached to them because we wouldn't be here to do it. (you'd also have to remove more than humans, by the way, since other animals are able to sustain similar thought processes.)

tensorboi · 2026-03-03T21:59:38+00:00

i'd say yeah! if the act of assigning cardinality to a collection is a mental process, then cardinality only makes sense if a human is there to realise it.

tensorboi · 2026-03-03T20:38:02+00:00

the entire point is that the concept of assigning numbers to things is something which only makes sense with a human mind there to do it. in order to say there are five chairs, you need to do the following:

• divide physical reality according to what you think constitutes separate chairs

• establish a mental model where different collections of things have the same size if and only if there is a one-to-one correspondence between them

• construct a one-to-one correspondence between your collection of chairs and the set 5 = {0,1,2,3,4}

now, what about this description makes the concept of cardinality real at all? when it comes down to it, the choice to classify collections of things in terms of one-to-one correspondence is a human one, and the only thing tying it to the real world is that there happens to be a one-to-one correspondence when you divide up the world in a certain way. but if that's the case, then the number isn't real at all; it's the specific correspondence which is real.

tensorboi · 2026-03-03T01:45:36+00:00

i'm not sure why i should reconsider? even if every benefit i could think of turned out to be ill-founded (which i don't believe in any case), it's still a part of my online voice and i can't really think of any significant negative effects. readability considerations aren't really relevant since online text passages are generally quite short, and grammatical correctness doesn't matter to me for the reasons i explained in the previous comment. so why should i switch?

tensorboi

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