Same in Python, Rust, Haskell and probably many others.

But apparently JS does work that way, that is its filter always iterates over everything and returns a new array and not some iterator object.

Hasn’t Google already made advances through its Alpha Geometry AI?? Admittedly, that’s a geometry setting which may be easier to code than other parts of Math and there isn’t yet a clear indication AI will ever be able to reach a certain level of creativity that the human mind has, but at the same time it might get there by sheer volume of attempts.

Wanted to focus a bit on this. The thing with AlphaGeometry and AlphaProof is that they really treat doing math as a game, not unlike chess. For example, AlphaGeometry has a basic set of rules, it can apply them and it knows when it is done. And when it is done, you can be 100% sure that the solution is correct, because the rules of the game are known; the 28/42 score reported in the article is really four perfect scores and three zeros. Those systems do use LLMs, but they really are only there to suggest to the system what to try doing next. There is a very enlightening picture in the AlphaGeometry paper here: https://www.nature.com/articles/s41586-023-06747-5#Fig1

You can automatically verify correctness of code the same way. For example Lean, the language AlphaProof uses internally, can be used for general programming. In general, we call similar programming techniques formal methods. But most people don’t do this, since this is more time-consuming than normal programming, and in many cases we don’t even know how to define the goal of our code (how to define correct rendering in a game?). So this is only really done when the correctness of the program is critical, like famously they verified the code of the automatic metro in Paris this way. And so most people don’t try to make programming AI work this way.

I really need to try out Mercury one day. When we did a project in Prolog at uni, it felt cool, but also incredibly dynamic in a bad way. There were a few times when we misspelled some clause, which normally would be an error, but in our case it just meant falsehood. We then spent waaay to much time searching for these. I can’t help but think that Mercury would be as fun as Prolog, but less annoying.

I actually use from time to time the Bower email client, which is written in Mercury.

Edit: Actually, I thought about it, and I don’t think clang’s behavior is wrong in the examples he cites. Basically, you’re using an uninitialized variable, and choosing to use compiler settings which make that legal, and the compiler is saying “Okay, you didn’t give me a value for this variable, so I’m just going to pick one that’s convenient for me and do my optimizations according to the value I picked.” Is that the best thing for it to do? Maybe not; it certainly violates the principle of least surprise. But, it’s hard for me to say it’s the compiler’s fault that you constructed a program that does something surprising when uninitialized variables you’re using happen to have certain values.

You got it correct in this edit. But the important part is that gcc will also do this, and they both are kinda expected to do so. The article cites some standard committee discussions: somebody suggested ensuring that signed integer overflow in C++20 will not UB, and the committee decided against it. Also, somebody suggested not allowing to optimize out the infinite loops like 13 years ago, and then the committee decided that it should be allowed. Therefore, these optimisations are clearly seen as features.

And these are not theoretical issues by any means, there has been this vulnerability in the kernel for instance: https://lwn.net/Articles/342330/ which happened because the compiler just removed a null pointer check.

You might also like https://github.com/nvim-neorg/neorg which is not meant to be compatible with Emacs org-mode, but rather something new that’s built around similar ideas but for Neovim. Hadn’t used it myself though, only heard about it.

It seems that I can’t see the link from 0.18.3 instances somehow. Maybe one of these will work: https://math.stackexchange.com/a/18347 https://math.stackexchange.com/a/18347 https://math.stackexchange.com/a/18347

Imagine a soccer ball. The most traditional design consists of white hexagons and black pentagons. If you count them, you will find that there are 12 pentagons and 20 hexagons.

Now imagine you tried to cover the entire Earth in the same way, using similar size hexagons and pentagons (hopefully the rules are intuitive). How many pentagons would be there? Intuitively, you would think that the number of both shapes would be similar, just like on the soccer ball. So, there would be a lot of hexagons and a lot of pentagons. But actually, along with many hexagons, you would still have exactly 12 pentagons, not one less, not one more. This comes from the Euler’s formula, and there is a nice sketch of the proof here: .

Every prime larger than 3 is either of form 6k+1, or 6k+5; the other four possibilities are either divisible by 2 or by 3 (or by both). Now (6k+1)² − 1 = 6k(6k+2) = 12k(3k+1) and at least one of k and 3k+1 must be even. Also (6k+5)² − 1 = (6k+4)(6k+6) = 12(3k+2)(k+1) and at least one of 3k+2 and k+1 must be even.