my experience studying math was the same, so many concepts clicked when i was in the shower. a bit later on, a lot of the eureka moments in research projects happened in the shower too. i’ve wondered how common it is in other fields to be doing a considerable chunk of the research in the shower.

i thought this was pretty weird too when i found out about it. i’m not entirely sure why it’s done this way but i think it has to do with conventions on where to start indexing. most programming languages start their indexing at 0 while much of the time in math the indexing starts at 1, so i=0 to n-1 becomes i=1 to n.

yeah thats a good example and it shows weird the number 0 is compared to the positive integers. it seems like a lot of the time things are first “defined” for the positive integers and then afterwards the definition is extended to 0 in a “consistent way”. for example, the idea of taking exponents aⁿ makes sense when n is a positive integer, but its not immediately clear how to define a⁰. so, we do some digging and see that a^m+n = a^maⁿ when m and n are positive integers. this observation makes defining a⁰=1 “consistent” with the definition on positive integers, since it makes a^m+n = a^maⁿ true when n=0.

i think this sort of thing makes mathematicians think of 0 as a weird index and its why they tend to prefer starting at 1, and then making 0 the index for the “weird” term when it’s included (like the displacement vector in affine space or the constant term in a taylor series).

if you take a modular approach and allow different measures to be used, it also lets the integral sign be a discrete for-loop

i completely agree. this sentiment was echoed pretty well in a (nontechnical and accessible) paper i read a few years ago. he says the current approach is like forcing people to learn music, but only teaching them how to read sheet music and not letting them touch any instruments. it hides the creativity and problem-solving of the discipline and reduces it to memorizing formulas.

i’ve seen a few people leave more algebraic/technical explanations so i thought i would try to give a more handwavy explanation. there are three things we need:

1. the sum of two numbers doesn’t depend on how those numbers are written. (for example, 1/2 + 1/2 = 0.5 + 0.5.)
2. 1/3 = 0.33…
3. 1/3 + 1/3 + 1/3 = 1.

combining these three things, we get 0.99… = 0.33… + 0.33… + 0.33… = 1/3 + 1/3 + 1/3 = 1.

it’s worth mentioning the above argument could be refined into an actual proof, but it would require messing around with a formal construction of the real numbers. so it does actually explain “why” 0.99… = 1.

it’s pretty useful for explaining high level math concepts, or at least it used to be. before chatgpt 4 launched, it was able to give intuitive descriptions of stuff in algebraic topology and even prove some properties of the structures involved.

man that really sucks. i had to get rabies shots a couple years ago and it was not fun. for my second follow up shot i thought about skipping the wait in the ER and going to a pharmacy 40 minutes away that offered rabies vaccines, only to find out there’s a difference between pre-exposure and post-exposure shots and having to go to the ER anyway. if you end up also getting tired of the 3 hour waits in the ER and look for an alternative, i would strongly recommend making sure they offer post-exposure vaccines.

what if the number of people on the tracks is “infinite in both directions”? eg the trolley “starts at” -∞?

yeah that may cause a few problems. there would also need to be an infinite amount of trolley track which may pose some infrastructure challenges