and they’re always whispering all the time too. it drives me crazy. nobody whispers that much in real life. it doesn’t make people sound more mysterious
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, …, vn), which can be thought of as a function {1, 2, …, n} → ℝ, where k ∊ {1, …, n} gets sent to vk. 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 vk.)
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 new “Recall” feature really does look good on paper, but the taking in mind that it catalogues almost everything you do on your computer, it could turn out to be a privacy nightmare. “logging things you do in apps, tracking communications in live meetings, remembering all websites you’ve visited for research, and...
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
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.
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
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
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”
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.
it’s mathematically provable that the shortest path between any two points on a sphere will be given by a so-called “great circle”. (a great circle is basically something like the equator: one of the biggest (greatest) circles that you can draw on the surface of a sphere.) i think this is pretty unintuitive, especially because this sort of non-euclidean geometry doesn’t really come up very frequently in day to day life. but one way to think about this that on the sphere, “great circles” are the analogues of straight lines, although you’d need a bit more mathematical machinery to make that more precise.
although in practice, some airlines might choose flight paths that aren’t great circles because of various real world factors, like wind patterns and temperature changes, etc.
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New “Recall” feature in Windows 11 is a privacy nightmare (www.theverge.com)
The new “Recall” feature really does look good on paper, but the taking in mind that it catalogues almost everything you do on your computer, it could turn out to be a privacy nightmare. “logging things you do in apps, tracking communications in live meetings, remembering all websites you’ve visited for research, and...
Brazil counts cost of worst-ever floods with little hope of waters receding soon (www.theguardian.com)
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Indiana judge rules tacos, burritos are sandwiches (www.wishtv.com)
Pythagorean Theorem Found On Clay Tablet 1,000 Years Older Than Pythagoras (www.iflscience.com)
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Account required to use dark mode in 2024. (lemmy.world)
54% of young Americans say food costs are the biggest strain on their finances (www.cnbc.com)
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