Hacker News: U4E4

New comment by U4E4 in "AI outperforms law professors in Stanford Law study"

U4E4 — Wed, 03 Jun 2026 02:51:31 +0000

This is true in the naive case.

There are however LLM context building techniques that anchor completions in data structures that persist the structure of claims that support the conclusion contained in a completion. Lots of different patterns exist —organizing logic in language is a rich domain— but the one I’ve liked the most is something called a Claim Dependency Graph that models the relationships between atomic claims as graph edges.

There’s a whole suite of operations you can perform on these structures, and “reconstruct how you came to this conclusion” is absolutely one of them.

New comment by U4E4 in "Daily pill can double survival time for deadliest cancer, trial shows"

U4E4 — Sun, 31 May 2026 20:39:33 +0000

I find both the honesty and tact of your comment a generous gift. After watching the Sasse interview, reading the parent comment and reading your comment, I’m reminded abstractly how much of the emotional and psychological work of reconciling biological mortality is built on personal cognitive context that a mind-body builds over its cycles living in the world. So much about mortality is shared. But so much of the context for interpreting mortality is radically personal.

New comment by U4E4 in "Underwater robot tracks sperm whale conversations in real time"

U4E4 — Wed, 06 May 2026 01:17:05 +0000

I’ve never quite appreciated the peak hippie bizarreness of a space probe checking in to see if the whales are still home Star Trek IV: The Voyage Home. Great San Francisco based movie tho.

New comment by U4E4 in "Category Theory Illustrated – Orders"

U4E4 — Sat, 18 Apr 2026 08:27:01 +0000

A thing is its relationships. (Yoneda lemma.) Keep track of how an object connects to everything else, and you’ve recovered the object itself, up to isomorphism. It’s why mathematicians study things by probing them: a group by its actions, a space by the maps into it, a scheme in algebraic geometry defined as the rule for what maps into it look like. (You do need the full pattern of connections, not just a list — two different rings can have the same modules, for instance.) [0]

Writing a program and proving a theorem are the same act. (Curry–Howard–Lambek.) For well-behaved programs, every program is a proof of something and every proof is a program. The match is exact for simple typed languages and leaks a bit once you add general recursion (an infinite loop “proves” anything in Haskell), but the underlying identity is real. Lambek added the third leg: these are also morphisms in a category. [1]

Algebra and geometry are one thing wearing different costumes. (Stone duality and cousins.) A system of equations and the shape it cuts out aren’t related, they’re the same object seen from opposite sides. Grothendieck rebuilt algebraic geometry on this idea, with schemes (so you can do geometry on the integers themselves) and étale cohomology (topological invariants for shapes with no actual topology). His student Deligne used that machinery to settle the Weil conjectures in 1974. Wiles’s Fermat proof lives in the same world, though it leans on much more than the categorical foundations. [2]

[0] https://en.wikipedia.org/wiki/Yoneda_lemma

[1] https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...

[2] https://en.wikipedia.org/wiki/Stone_duality