Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Bad at Arithmetic, Promising at Math, published by cohenmacaulay on December 18, 2022 on LessWrong.
n-Cohesive Rings
Definition: Let n be a positive integer. We define an n-cohesive ring to be a commutative ring S such that, for every prime p dividing the characteristic of S, pn divides the order of the multiplicative group S×. We define an n-cohesive ideal of a ring R to be an ideal I of R such that the quotient ring R/I is an n-cohesive ring.
Example: Z/25 is a 4-cohesive ring. The multiplicative group R× is the set {1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31}, which consists of the 16 elements of R that are relatively prime to 25. The order of the multiplicative group R× is 16, which is divisible by 24, so R is an n-cohesive ring for n=4.
Example: Consider the ideal (8) of the ring Z. The multiplicative group of Z/I is {1,3,5,7}, whose order is 4. The highest power of 2 that divides the order of this group is 22, which means that I is a 2-cohesive ideal.
The notion of an n-cohesive ring, and the dual notion of n-cohesive ideals, do not, to the best of my knowledge, appear in the mathematical literature. I know of no definitions off the top of my head that are equivalent to n-cohesiveness. The definition is rigorous, logically sound, and there exist nontrivial examples of n-cohesive ideals. A problem like "classify all 5-cohesive ideals of Z" strikes me as not completely trivial. A problem like "classify all 5-cohesive ideals of [insert number ring here]" strikes me as potentially very difficult (though I am not a number theorist). If someone came along and proved a strong classification result about n-cohesive ideals in number rings, they could probably publish that result in a mid-tier algebra or number theory journal. I could easily imagine handing it off as a research project to an undergraduate learning about unit groups, or maybe even a grad student who was particularly bored.
The most interesting thing about the concept of n-cohesive ideals, however, is that it was not invented by a human.
The examples of n-cohesiveness given above did involve some human handholding and cherrypicking (we will talk more about this shortly), but, I think you'll judge, are at least partially attributable to AI.
Before we get started, let me state some concrete predictions to keep us grounded.
By 2030, there will exist a paper whose topic was chosen by an AI, with at least some examples and theorems suggested by the AI (possibly after significant human cherrypicking), whose proofs are mainly human-written (possibly with some AI contribution, involving significant handholding), published in a pure mathematics journal of reasonable quality: 95%
By 2030, there will exist an a correct proof primarily written by an AI, with at most minor human editing and corrections, published in a pure mathematics journal of reasonable quality: 30%.
By 2030, there will exist a correct, original, wholly AI-written paper, whose topic was chosen by the AI, published in a pure mathematics journal of reasonable quality: <1%.
The second bullet's probability in my mind goes up significantly by 2040. I don't have good intuition about when I would expect something like bullet 3, but I can say that whenever bullet 3 does happen, mathematics is going to undergo some very serious and very interesting changes.
We're getting a bit ahead of ourselves, though. Let's talk about n-cohesive rings.
Formal and Natural Mathematical Languages
At this point, it is well-known that ChatGPT is terrible at arithmetic. There is an example going around where it is asked something to the effect of "A bat and a ball together cost $1.10, and the bat costs $1 more than the ball, how much does the ball cost?" and it often says something like $0.10. It is safe to say that nobody is going to be using ChatGPT as their pocket calculator...