counting_is_hard

counting_is_hard, 28 days ago to random

Time to write the code: 5 minutes.
Time to figure out the types: 50 minutes.

counting_is_hard, 1 month ago to random

In his famous paper, "on proof and progress in mathematics", Thurston lists 8 (and implies 29 other) ways to think of the derivative.

I was bored waiting for a bus, so I tried listing the different ways I could think of what a category is (see below). Please feel free to help me add more!

A Category is...

1. The usual definition (omitted for space)
2. an abstract theory of functions / arrows (or as Awodey would say "archery")
3. a monoidoid
4. a poset with evidence (wording stolen from Alex Kavvos)
5. a set-enriched category
6. an object in CAT
7. a syntax for a programming language
8. a maze of twisted arrows all alike
9. a "path-complete" digraph (if there is a path x -> y there is an edge x -> y)
10. a multicategory where every arrow has arity 1
11. a polynomial comonad (spivak et al)

#categorytheory

johncarlosbaez, 1 month ago (edited 1 month ago) to random

Mathematicians describe points on the plane with coordinates (x,y). The first coordinate says how far 𝑎𝑐𝑟𝑜𝑠𝑠 you go and the second says how far 𝑢𝑝 you go. Then they describe entries of a matrix Mᵢⱼ with indices where the first says how far 𝑑𝑜𝑤𝑛 you go and the second says how far 𝑎𝑐𝑟𝑜𝑠𝑠 you go.

In each case I've had teachers who insinuate that this is the only reasonable thing to do and you'd have to be nuts to dream of doing anything else.

I think some teachers don't distinguish between facts and conventions. Actual facts are always worth thinking about - understanding them more and more deeply is a never-ending quest. But for arbitrary conventions, you should just memorize them and move on.

plcon, 1 month ago to random

As our field becomes more competitive, researchers are starting earlier and earlier. This year, PLCon is introducing the pre-school track. How such a junior researcher could be substantially involved in state-of-the-art methods, techniques, background knowledge, etc is frankly beyond us but. We invite pre-schoolers to submit their research paper on the topic of programming languages for social impact. Papers should be 4 pages. LLMs should make this pretty easy; go nuts.

ColinTheMathmo, 11 months ago (edited 11 months ago) to random

Seen on Twitter, I think this is a brilliant question (and you can use that to calibrate your beliefs about what I will find interesting):

Given that (p(x)) is a real polynomial of degree (\le4) such that one can find five distinct solutions to the equation (p(x)=5), what is the value of (p(5))?

(A) 0
(B) 1
(C) 4
(D) 5
(E) Cannot be determined from the information given.

Please PLEASE do not give it away in the replies ...

Please ...