cohenmacaulay

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cohenmacaulay — LessWrong

Replying toBad at Arithmetic, Promising at Math

cohenmacaulay3y

Bad at Arithmetic, Promising at Math

Thanks for the pointer to davinci-003! I am certainly not interested in ChatGPT specifically, it just happens to be the case that ChatGPT is the easiest to pop open up and start using for a non-expert (like myself). It was fun enough to tinker with, so I look forward to checking out davinci.
I had not heard of GPT-f - appreciate the link to the paper! I've seen some lean demonstrations, and they were pretty cool. It did well with some very elementary topology problems (reasoning around the definition of "continuous"), and struggled with analysis in interesting ways. There was some particular theorem (maybe the extreme value theorem? I could be forgetting) that

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Replying toBad at Arithmetic, Promising at Math

cohenmacaulay3y

Bad at Arithmetic, Promising at Math

I think I largely agree with this, but I think it's also pretty hard to put to practice in training an AI proof system.

Fermat's theorem is actually a great example. Imagine my goal is to "align" a mathematician to solve FLT. Imagine a bizarre counterfactual where we have access to advanced AI models and computational power but, for some reason, we don't have algebraic number theory or algebraic geometry -- they were just never invented for some reason. If you want something less far-fetched: imagine the abc conjecture today if you're of a camp that believes the inter-universal Teichmuller theory stuff is not going to work out, or imagine perhaps the Collatz... (read 745 more words →)

Replying toBad at Arithmetic, Promising at Math

cohenmacaulay3y

Bad at Arithmetic, Promising at Math

I think what I mean by "promising at math" is that I would assign a reasonably confident belief that a scaled up model with the same basic architecture as this one, with maybe a more mathematical training set, would be able to accomplish tasks that a mathematician might find helpful to a nonzero degree. For example, I could imagine a variant that is good at 'concept-linking,' better than google scholar at least, where you ask it something like "What are some ideas similar to valuations rings that people have explored?" or "Where do Gorenstein rings come up in combinatorics?" and getting a list of some interesting leads, some nonsense, and some lies... (read more)

Replying toBad at Arithmetic, Promising at Math

cohenmacaulay3y

Bad at Arithmetic, Promising at Math

Thanks for the link, this is great!

Replying toBad at Arithmetic, Promising at Math

cohenmacaulay3y

Bad at Arithmetic, Promising at Math

If anyone knows where to find an equivalent concept, definitely let me know! It's hard to prove that a concept has not been investigated before, so I'll just give my reasoning on why I think it's unlikely:

I believe that the name " $n$ -cohesive" is untethered to the literature, even if the underlying concept exists, since it was specifically told to change the name to something new.
The original name, " $n$ -smooth," appears in number theory, but the number theory concept is genuinely unrelated. The word "smooth" is heavily overloaded in algebra, but generally points towards a "main" concept called "smoothness of a morphism," and smoothness of a morphism is incapable of caring about which power

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Bad at Arithmetic, Promising at Math

cohenmacaulay

$n$ -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 $S$ , $p^{n}$ divides the order of the multiplicative group $S^{\times}$ . 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: $Z / 2^{5}$ is a $4$ -cohesive ring. The multiplicative group $R^{\times}$ 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 $2^{5}$ . The order of the multiplicative group $R^{\times}$ is $16$ , which is divisible by $2^{4}$ , 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 $2^{2}$ , which means that $I$ is a $2$ -cohesive ideal.

The notion of an $n$ -cohesive ring, and the dual notion of $n$ -cohesive ideals,... (read 5768 more words →)

102

Replying toWhy square errors?

cohenmacaulay3y

Why square errors?

Here is an additional perspective, if it's helpful. Suppose you live in some measure space X, perhaps $R^{n}$ for concreteness. For any $p \in [1, \infty]$ can make sense of a space $L^{p} (X)$ of functions $f : X \to R$ for which the integral $\int_{X} | f |^{p}$ is defined (this technicality can be ignored at a first pass), and we use the norm $| | f | | := {(\int_{X} | f |^{p})}_{X}^{1 / p}$ . Of course, $L^{1}$ and $L^{2}$ are options, and when X is a finite set, these norms give the sum of absolute values as the norm, and the square root of the sum of squares of absolute values, respectively.

Fun exercise: If X is two points, then $L^{p} (X)$ is just $R^{2}$ with the norm $(| a^{p} | + | b^{p} |)^{1 / p}$ . Plot the unit circle in this norm (i.e, the set of all points at a distance of $1$ from the origin). If $p = 1$ , you get a square... (read 1108 more words →)

cohenmacaulay

Bad at Arithmetic, Promising at Math

cohenmacaulay

cohenmacaulay

Bad at Arithmetic, Promising at Math

n-Cohesive Rings

$n$ -Cohesive Rings