Hacker News: lg5689

New comment by lg5689 in "An OpenAI model has disproved a central conjecture in discrete geometry"

lg5689 — Thu, 21 May 2026 01:33:06 +0000

The problem was pretty well known, and had many human attempts. There's some room to argue that the right humans hadn't attempted it, as the solution used advanced methods from another field of math. But imho, whereas many prior AI victories could be explained by not enough human attention, there is no such excuse in this case, and one should acknowledge this is a notable achievement.

New comment by lg5689 in "Frontier AI has broken the open CTF format"

lg5689 — Sat, 16 May 2026 17:33:33 +0000

This is happening to other forms of competitive programming too. The most recent AIs have problem solving skills rivaling top humans, and so if AI can't be easily banned, the competition is dominated by AI agents.

I thought code golf would take longer for AIs because there's so little training data (it's more niche), but we're seeing AIs starting to match expert humans there too. Sucks because golf has been my favorite type of programming puzzle.

It's crazy how far AIs have come in problem solving ability.

New comment by lg5689 in "Mathematicians disagree on the essential structure of the complex numbers (2024)"

lg5689 — Wed, 11 Feb 2026 03:45:26 +0000

You can't do this for general functions, but it's fine to do in cases where the definition of f naturally embeds into the rationals. For example, a polynomial over Z is also a polynomial over Q or C.

New comment by lg5689 in "Mathematicians disagree on the essential structure of the complex numbers (2024)"

lg5689 — Wed, 11 Feb 2026 03:30:13 +0000

The movement from R to C can be done rigorously. It gets hand-waved away in more application-oriented math courses, but it's done properly in higher level theoretically-focused courses. Lifting from a smaller field (or other algebraic structure) to a larger one is a very powerful idea because it often reveals more structure that is not visible in the smaller field. Some good examples are using complex eigenvalues to understand real matrices, or using complex analysis to evaluate integrals over R.

New comment by lg5689 in "Mathematicians disagree on the essential structure of the complex numbers (2024)"

lg5689 — Wed, 11 Feb 2026 03:16:04 +0000

You can go farther and say that you can't even construct real numbers without strong enough axioms. Theories of first order arithmetic, like Peano arithmetic, can talk about computable reals but not reals in general.