Recent content on Rado's Radical Reflections
AI Disclosure: Draft was written fully by a human. Heavy use of AI for editing. In April 2026, Kevin Buzzard posed the following challenge for Lean AI autoformalization: can an AI system both define the Jacobian variety in Lean and prove basic properties about it? ( zulip thread , github gist ). The motivation behind this challenge, as far as I can tell, was to point the autoformalization…
AI Disclosure: Draft was written fully by a human. Heavy use of AI for editing. See On AI usage for more. At this point no one doubts that AI is a tremendous capability boost for all human intellectual activities. There are interesting questions about what happens next - at what point this stops being a human operating an AI tool and becomes a fully autonomous AI. However, in this post I will…
Recent advances in general models — ChatGPT and Claude — have started to autonomously solve open mathematical problems. For example, Erdős 1196 , Tim Gowers’s PhD student problems , OpenAI’s Ramsey numbers result . There are a lot of caveats — were the problems widely enough studied, could the solutions be coming from some past human result, etc. But if one zooms out and considers the…
Note: AI was used to edit this post. As a proof-of-human thought and input, I am also publishing the original draft which was written fully before asking AI to edit the post with me. This post is aimed at Lean language beginners that are interested in writing proofs in Lean, but still feel lost when reading Lean code. A very simplified mental model of Lean is that at the core there are two…
This post is aimed at a general programmer and no prior knowledge of math or CS is assumed. I got nerd-sniped to write this after reading Simon’s excellent post Code proven to work . As someone who works in the software industry that is adapting to using AI tools, the post was very timely and very well put. I agree with everything said, but I just can’t resist being that guy - bringing…
Disclaimer - I wrote the core ideas; Claude helped flesh out and polish the article. See appendix for more on this. This is a follow up to my previous post on leaning on Claude for Lean. I’ve now worked up to chapter 8.3 in Tao’s companion. The speed is great and Claude’s capabilities continue to impress (autoformalization is possible, but not my goal). I haven’t been stuck…
Preface: Why All Mathematicians Should Learn Lean LLMs can generate plausible-sounding proofs at unprecedented speed and scale. Some are correct, many are not, and LLMs themselves cannot reliably tell the difference. Humans can — but they don’t scale. Formal proof assistants like Lean do: they verify correctness mechanically and can serve as a backstop to the torrent of AI-generated…
It’s been five months since my last dedicated Lean post and as usual I have started to lose steam on Lean projects. After the thrill of discovering the world of formalized mathematics started to wear off, I did not find motivation to push as hard as before. The SF Math with Lean work group kept me vaguely connected (at least in the one hour a week we meet (see retro) ) but other than that I…
I started learning formalized mathematics with Lean last year 1 , 2 , 3 , 4 . As I progressed beyond basics, it dawned on me that I need to start interacting with the broader community as Lean is rapidly evolving and beyond basic getting started guides there aren’t as many high-quality resources to learn solo. It could also be a great way to meet folks with similar interests and build more…
In 2019, as functional programming was making the last inroads dethroning OOP, I kept hearing the mantra of “just use pure functions” in JS. Something didn’t sit right with me when talking very deterministically about pure functions in a large unpure language like JS. Especially, after seeing JS tooling perform optimizations based on a pure annotation (like webpack) . While…
I read a good post by one of the authors of the Isabelle theorem prover, that got me thinking. The author, Lawrence Paulson, observed that most math proofs are trivial, but writing them (preferably with a proof assistant) is a worthwhile activity, for reasons similar to safety checklists - “Not every obvious statement is true.” As I have been a bit obsessed with doing formalized…
It’s been 3 months since my previous post about learning Lean part 3 , so it’s time to write another one. I have mostly continued to work through Tao’s Analysis book through his excellent companion - which means all proofs from the paper textbook are fully formalized in Lean, while examples and exercises are stated in Lean but left with a sorry for me to fill in. Altogether the…
I am continuing to learn Lean (see part 1 and part 2 ). I lost some steam around March-April, but in the last two months I picked it up again. In a way it was a nice spaced repetition for relearning some of the basics. This summary was supposed to be my notes from finishing the rest of Mathematics in Lean and finishing chapters 2 and 3 of exercises from Terence Tao’s Analysis textbook , but…
Around 5-8 years ago when working on Angular and TypeScript rules in Bazel, I got very interested in exploring the core ideas of incremental computation, which seemed like the right way to abstractly (but still precisely) describe the similarities between build systems and UI reactivity. As an aside, incremental dataflow is a third view into the same concept, but I am least familiar with it. I…
I am continuing to learn Lean (see part 1 ) by going through Mathematics in Lean . These are my notes as I just finished chapters one through five. Mathematics in Lean The online book is well-paced and well-written to connect with what an average student of mathematics already knows. The focus is on learning how to do basic mathematical proofs in Lean, and the underlying language is taught as…
Motivation I’ve been captivated by the recent movement to popularize mathematics formalization through the Lean theorem prover, and this year I’m diving deeper into learning it. For those unfamiliar with this revolution, I highly recommend watching Kevin Buzzard’s talks on YouTube for an overview of why formal mathematics is generating such excitement in the mathematical…
Ticket to Ride: First Journey simulation authored with AI After playing Ticket to Ride: First Journey with my family recently, I got the urge to analyze some statistical properties of the game. This seemed like an excellent opportunity to experiment with Cursor AI and explore more automated approaches to coding. Here’s what I discovered. Results After a couple of hours the AI and I produced…
As the year draws to a close, I want to share my favorite puzzle games from 2024. This isn’t meant to be an exhaustive review or ranking of all puzzle games released this year, but rather a personal reflection on the games that captured my attention. If you’re new to the puzzle genre, I recommend starting with the essential thinky games list - all of those titles are excellent.…
Advent of Code AoC is an annual programming competition that releases daily coding puzzles throughout December. For the past four years, I’ve tackled these challenges from the West Coast, where the 9 PM PST release time perfectly suits my schedule. While I compete for points on the global leaderboard (with modest success), the real joy comes from the community around it. Through…
Why TypeScript? Maybe you wondering why did I use TypeScript for this post. As it happens I have been working in the frond-end community in the last 8 years. My interest in incremental computation started by observing the similarities between some of the work I have done inside Angular’s “change detection” algorithms, and work I have done around integrating TypeScript’s…
We have been talking about general computation, but so far our language was very limited. We only used function calls, numbers, and simple variable binding. We will slowly add more language primitives and see how to still have full incrementality of the computation. Conditional statements Let’s add a single conditional statement first. function cond ( b : boolean , x : number , y : number )…
Incremental computation is a way of performing computations, with the expectation of future changes in the inputs to the computiation. When those changes occur the output can be updated efficiently, at minimum faster than redoing the whole computation from scratch. This problem occurs in many programming domains - UI programming, data flow, build systems, compilers, code editors. Likely you have…
Hello! I’m Rado Kirov, a front-end engineer at Stripe where I maintain stripe.js . Before this, I worked at Google on front-end infrastructure, helping build core technologies like Angular, TypeScript tooling, and the Closure Compiler. I started my career in academia as a mathematician, completing a postdoc before transitioning to software engineering.