Software Engineer | Mathematician | Catholic | "There are 3 certainties in life: death, taxes and software updates"
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Joined April 2012
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Tobi Lehman retweeted
@axiommathai should address these comments ASAP.
Many are quick-fixes:
1. Mathlib duplication
2. Incomplete formalizations
3. Outdated dependencies
4. Library build warnings
5. Quality audit, with unnecessary hypotheses and verbose proofs
kim-em.github.io/blog/2026-1…
In my view, autoformalization won't be Mathlib-level quality for another few months, but should ultimately be the goal.
Haven't seen this discussed elsewhere on X: Kim Morrison wrote a fantastic review of Axiom's Stacks
She identifies places where Stacks is not fully formalized; and lots of duplication of Mathlib
"I want to talk about ways in which we can do even better"
kim-em.github.io/blog/2026-1…
Tobi Lehman retweeted
Norway now has more than 2,000 underwater AI laser robots shooting lice off its salmon
they're made by a company called Stingray, and they already guard more than 50 million salmon and trout 24/7
the reason is sea lice, tiny parasites that latch onto salmon and feed on them
as few as 11 can kill a small fish, and one pen can hold 150,000 salmon
the usual fix is pumping the fish out and blasting them with warm water or water jets to knock the lice off
but the handling is so rough that Norway's fish health report ranks it the top cause of death in full-grown farmed salmon
so instead, Stingray puts a robot inside the pen that hunts lice one at a time, all day:
> two cameras watch every salmon that swims past
> AI checks two frames to make sure it's really a louse
> then a green laser fires from up to 2 meters away
> the salmon's scales reflect the beam, so the fish swims off fine and the louse is dead
we now live in a world where AI lasers do pest control on your dinner lmao
Tobi Lehman retweeted
I would also like to address fathers and grandfathers. Young people need to see that you pray, receive the sacraments, treat your wives with respect, ask for forgiveness and live with integrity, even when doing so requires sacrifice. Children learn a great deal by seeing that their father can kneel before God without feeling that he is any less of a man for doing it.
Tobi Lehman retweeted
A magical corollary: EVERY FINITE SIMPLE GROUP IS GENERATED BY 2 ELEMENTS
here we go: every finite simple group is generated by 2 elements! github.com/AxiomMath/CFSGCor…
Tobi Lehman retweeted
Category theory for the sciences math.mit.edu/~dspivak/
Tobi Lehman retweeted
I’m seeing a lot of mathematicians making software and a lot of software developers learning math.
I think this may be the most insane lock-tf-in-winter of all time for humanity.
Tobi Lehman retweeted
For nearly 75 years, these two knots sat in the tables as different knots.
In 1973, Kenneth Perko, a New York lawyer trained in knot theory at Princeton showed they are the same.
Today, AI can explain it in seconds.
Tobi Lehman retweeted
Six degrees of Kevin Bacon, Paul Erdos, and Wikipedia johndcook.com/blog/2018/12/0…
Tobi Lehman retweeted
Where we started (left, 1979) vs. where we are now (right, 2026)
ALT Social Processes paper arguing formal proof will never be important to CS and softwate engineering
ALT Conclusion from the Anthropic post from yesterday arguing verification is about to become as practical as fuzzing
Tobi Lehman retweeted
LLM: Here is a Lean proof of Navier–Stokes Problem.
Humans: There’s a mismatch between the Lean proof and the problem statement.
LLM: Thank you for pushing back. You are absolutely right.
👀👀 "[…] we provide several examples of AI mistranslations of natural language (NL) statements and proofs into Lean in practice, resulting in mismatches between NL proofs and their Lean 'verifications'. These include OpenAI's announced Navier-Stokes proof. In particular, we show that the formalised Lean proof does not correspond to the NL proof of blow-up of solutions to the Navier-Stokes equations."
arxiv.org/abs/2610.08144
Tobi Lehman retweeted
Major announcement in mathematical formalization!
Finally, after 3 months of very intensive, nonstop work by several AI agents (Codex and Claude Code), we have settled a classification of ALL 15,973 semigroups of order 6. Every one of the 15,969 that has a finite basis now has it written down explicitly, and the remaining 4 are proved to have none at all.
The whole list of bases is produced for the first time in history. Until now, for most of these semigroups, mathematicians only knew that a basis exists; nobody had ever written one down. The formalization in Lean, orchestrated with a multi-agent approach, reached more than 5 MILLION lines of verified Lean code, one of the largest auto-formalization projects to date.
It's been an amazing joint work with @JanotaMikolas and @Jan_Hula, accompanied by senior experts in semigroup theory, João Araújo and Edmond W. H. Lee.
Our paper describing this project, "Proving at Scale for Universal Algebra", has been accepted at the MATH-AI workshop, NeurIPS 2026!
When we launched this project, we were a bit pessimistic: producing proofs for 15,973 semigroups seemed out of scope for the current technology. But with a careful setup (agents communicating through a mailbox we arranged, orchestrating the effort with a custom method of bootstrapping and auto-research), it has finally come to the finish line. The last 390 semigroups took 23 more days. The very last one needed a whole structural analysis before its proof could even be written.
Tobi Lehman retweeted
We’ve been using tons of TLA+ for Tardigrade, very bullish on it.
Engineers should get into math more than ever because the best specification for a system is math. Not some text in a markdown file.
Specifying systems by Leslie Lamport (author of TLA) is a great book for those getting started.
lamport.azurewebsites.net/tl…
I used Opus 5.5 to formally verify the Claude Agent SDK using Lean. A couple short prompts = 16 PRs fixing various bugs and race conditions. Video attached.
TLA+ also works well. I sometimes combine Lean and TLA+ to look for issues around data flow, concurrency, and state mgmt.
I don't know either language well, but Claude is excellent at both. This approach is super useful for formally modeling your code and finding bugs that a human probably wouldn't have spotted.
Is formal verification the future of coding (or at least, bug finding)?
Tobi Lehman retweeted
Pithy version: if you can’t make phones, you can’t make drones
Tobi Lehman retweeted
con-leche, a CONsistent LEan CHEcker: an external Lean checker proven (in Lean) to be consistent, meaning it does not accept a proof of False. A project conceived by Joachim Breitner (@nomeata): github.com/leanprover/con-le…
#leanprover #leanlang #lean4
Tobi Lehman retweeted
A message I recently sent to some guy friends:
This is the same thing I’ve said for years, to all men, and it’s this: the fastest way to destroy feminism is for men to control their sexual faculties. Saying “no” to the diabolical temptations we all experience is the predicate to saying “no” to all this diabolical insanity in the world. We defeat the evil city within ourselves, then the evil city outside. This helps produce courage and fortitude within the soul, and both are required for all of this to end. Whenever we choose sin, we become weak. Our courage dissipates, because we’ve battered down the wall to the city of our own soul.
Women will always struggle with the curse of Eve. That will always be there. But Adam saying “no” stops much of it in its tracks. We must constantly fight against our own curse that makes us prone to cowardice and weakness, none of which we can do without God’s help obtained through the sacraments, prayer, and penance.
Tobi Lehman retweeted
An internal version of Astra, @OpenAI’s next major model family, solved 10 major open problems in mathematics, quantum complexity, and theoretical computer science.
We believe it will be a major step for scientific reasoning. openai.com/index/ten-advance…
Tobi Lehman retweeted
When your accountant asks why you haven't put any money in your super since 2024 and you have to explain what the Omega Point hyperobject acausally building itself backwards through time is.
Tobi Lehman retweeted
England beat Norway on the 1099th anniversary of England defeating the Vikings.
