Comment by abeppu
1 year ago
I haven't actually dug up the source, so I'm not sure if the gaps here are related, but GC Rota apparently claimed that one reason the umbral calculus remained mysterious for as long as it did was about ambiguities with `=`.
> Rota later stated that much confusion resulted from the failure to distinguish between three equivalence relations that occur frequently in this topic, all of which were denoted by "=".
https://en.wikipedia.org/wiki/Umbral_calculus#The_modern_umb...
There's so much ambiguous context that goes into your average '=', even when just talking about your standard functions on real numbers. You'll see it being used for anything from:
- We've found these to be equal
- We're hypothesizing this to be equal
- These are approximately equal
- These are defined to be equal
- This is a way to calculate something else, whether it's equal is up to your philosophy (a^2+b^2=c^2)
- I'm transforming my function into something else that looks different but is exactly the same
- I'm transforming my function into something else that is the same for some of the stuff I care about (but for example does not work anymore for negative numbers, complex nrs, etc.)
- I'm transforming my function into something else, but it's actually a trapdoor, and you can't convert it back.
- This is kind of on average true within an extremely simplified context or we know it's not true at all, but we'll pretend for simplification (looking at you physics)
- We are trying to check if these two are equal
- This is equal, but only within a context where these variables follow some constraints mentioned somewhere else entirely
- This is equal, but, we're not going to say whether you can or can't replace the variables with functions or whether it supports complex nrs, negative nrs, non-integers, etc.
A lot of this is usually kind of clear from context, but some of these differences are a nightmare if you want to code it out