European futbol has spent the past fifteen years solving the game like chess. A generation of coaches optimized for pass completion, pressing triggers, territorial control, rest defense, and positional occupation. The problem is that they optimized for what is measurable. Attacking space early, attempting the difficult pass, dribbling past a defender, deliberately creating chaos: these are…
The man in the painting at the top of this page is doing the oldest verification job there is. He is weighing each coin on a balance, one at a time, because the face stamped on a coin is a claim and its weight is the evidence, and a money changer who confused the two went broke. He does not trust the mint. He trusts the scale. Five hundred years later, almost all of our software asks us to trust…
I want Ethereum to have more formal verification, not less. That’s why I’m writing this.
The last year and a half, but particularly the last six months, were incredible but also very very tough. I went through difficult personal problems and had to expand my tolerance for pain to extremes I didn’t know existed. You don’t fight pain by toughening up. You fight it by controlling what you let near you. I left alcohol, lowered my consumption of caffeine to bare minimums, deleted my…
Series note: this article is part of the Concrete series and responds to Dmitri Sotnikov’s Giving LLMs a Formal Reasoning Engine for Code Analysis . Related: When the Compiler Is the Oracle and Why Concrete Exists . When an AI agent explores a codebase, it usually greps for names, reads a few matches, searches for callers, reads those, and tries to piece together a mental model of the program from…
You send a prompt to an LLM API. The provider says it ran Llama 70B. Maybe it did. Maybe it served a smaller model to save money, changed the quantization, altered the decode settings, or patched the answer after generation. Today you usually cannot tell. You get text back, an invoice, and a promise. For casual use, a promise is often enough. For enterprise procurement, regulated systems,…
Series note: this is the tradeoffs entry in the Concrete series. For the foundation, start with Why Concrete Exists . For the most practical demo, read When the Compiler Is the Oracle . The previous articles in this series argued that Concrete’s design constraints are worth it. Explicit capabilities make code auditable. Linear types prevent resource leaks at compile time. No hidden behavior means…
Series note: this is the most practical entry point in the Concrete series. If you want the shorter manifesto first, read Why Concrete Exists . If you want the language reference behind this article, use Concrete Spec . I have been building Concrete for a while now. Something happened this week that I did not plan for, and it may end up mattering more than the things I set out to build on purpose.…
The first article in this series explained the Curry-Howard correspondence: propositions are types, proofs are programs. The second built a tiny theorem prover from scratch in Python. The third embedded the same ideas inside Julia’s type system. Now we use the real tool. This article takes the exact same theorems you proved by hand in Python and shows them in Lean 4. You will see what changes and…
This article is based on Guillermo Angeris’s talk “Programming a (mini-)Lean in Julia’s type system” . A theorem prover, stripped to its engine, is a small trusted kernel, a type checker, and a boundary between the two. Guillermo Angeris answers this by live-coding a toy theorem-proving kernel inside Julia that illustrates how Lean works architecturally. The result is a tiny kernel that makes the…
The first article in this series explained the Curry-Howard correspondence: propositions are types, proofs are programs. That tells you why theorem proving fits so naturally with programming languages. It does not yet tell you what the machine looks like. A theorem prover, concretely, is smaller than most people expect. A tiny theorem prover is just: a language for terms a language for types…
Most programmers think Turing completeness is the interesting threshold for a computational system. It gets all the attention. But a lower, stranger threshold matters more for the origin of complex behavior: self-replication. A recent paper by Agüera y Arcas et al. shows that self-replicating programs spontaneously emerge from soups of random code. No one designs them. No fitness function selects…
In the 1930s, Haskell Curry noticed something strange. He was working on combinatory logic, a system for manipulating abstract functions, and he realized that the rules governing his combinators looked identical to the rules of a logical system called intuitionistic propositional logic. It was as if he’d found two different maps of the same territory. Three decades later, William Howard found the…
When I was young I loved science and engineering. Like most nerds, I thought thinking was the only thing that mattered. Working out seemed like a vanity project, something for people who cared about how they looked and not much else. I didn’t understand the body-mind connection at all. I was a skinny kid who spent all day reading, tinkering with computers, and hanging out with friends. The idea…
Series note: this is the proof-roadmap entry in the Concrete series. For the language overview, start with Why Concrete Exists and Concrete Spec . For the most practical compiler-report demo, read When the Compiler Is the Oracle . I wrote the first version of this essay when proving Concrete programs in Lean was still mostly a plan. The question was simple and a little risky: can we take a…
Series note: this article assumes the basic Concrete frame is already in place and asks a narrower question about AI-era language adoption. For the series foundation, read Why Concrete Exists . For the main Rust comparison, read The Rust Effects Debate and Concrete’s Case for a Smaller Language . Edgar Luque recently wrote about how AI creates a new adoption barrier for programming languages . His…
Series note: this is the main Rust-comparison entry in the Concrete series. If you are new here, start with Why Concrete Exists . If you want the language reference, use Concrete Spec . Yosh Wuyts recently wrote about his “grand vision” for Rust , outlining three directions he thinks the language should pursue: effects, stronger substructural types, and refinement types. The Hacker News thread…
I have been circling the same problem for a while now. Friction produces value. Legibility destroys what it measures. Formation requires lived time. The modern self is dissolving through redundancy. Previous technological shifts gave people decades to adapt, and this one might give them months. These kept feeling like separate observations. I no longer think they are. # The Speed Mismatch Every…
In the 18th century, German foresters invented scientific forestry. They looked at a messy, diverse forest and saw inefficiency. Old trees, young trees, deadwood, underbrush, species with no commercial value. They cleared it all and planted Norway spruce in straight rows, evenly spaced, same age, same species. The forest became legible. You could measure it, manage it, predict its yield with…
Never cross a river that is on average four feet deep. If the river is eight feet deep in the middle and dry on the sides, the average tells you nothing about whether you will drown. You will drown in the middle, or you won’t. There is no averaging across parallel universes where you both survive and die. The same asymmetry shows up wherever outcomes compound. Lose 50% of your wealth and you need…
Stock markets crash . The S&P 500 price index fell about 57% from October 9, 2007 to March 9, 2009, and about 34% from February 19, 2020 to March 23, 2020. A put option is a contract that pays you when the market falls below a certain price (the “strike”). If you hold stocks and also hold puts, the puts can offset some of your losses during a crash. The question is whether the cost of buying puts…
Financial markets don’t follow normal distributions . That is a claim about frequency, not just theory: it tells you how often catastrophic events happen. Under a naive Gaussian model, a crisis on the scale of 2008 lands so deep in the tails that standard risk models treat it as effectively impossible. It happened on a Tuesday. The problem is that we keep using tools designed for thin-tailed…
In 1252, Florence minted the gold florin. Within decades it became the dominant trade currency of medieval Europe. Merchants in Bruges, Venice, and Constantinople quoted prices against it. By the 1400s, the florin’s dominance had faded, replaced by the Venetian ducat. Then the Spanish real. Then the Dutch guilder. Then sterling. Then the dollar. Each transition involved devaluations, defaults, and…
# The Last Scarcity Most discussions of AGI focus on distribution: who gets access, who profits, who loses their job, who controls the infrastructure. Those are real problems, but they’re not the deepest one. The deeper problem is what happens to desire. I do not mean ambition in the generic sense. I mean the capacity to want something at a distance, to stay oriented toward something you do not…
China is trying to win by commoditizing the complement. The strategy is working, though not without friction. This is a structural challenge the West should take seriously instead of dismissing. For the last two decades, the West exported cognition because it owned the platforms, the cloud, the software distribution, and the talent concentration. If the cognitive engine becomes cheap, portable,…
Every type error you’ve ever cursed at was a bug caught before production. Type systems reject nonsense at compile time so you don’t discover it at 3 AM. But they vary wildly in what they can express and what guarantees they provide.
This is the foundation piece for the Concrete series. If you want the most practical demonstration first, start with When the Compiler Is the Oracle . If you want the living language reference, use Concrete Spec . Systems programming has a recurring problem. We want to write code close to the machine, but we also want to make strong claims about what that code does. Does it allocate? Does it touch…
The core argument is simple: many features of human life that appear stable and natural are historically produced. As society accelerates, a number of these features begin to lose their function and their permanence. I believe consciousness as we know it is one of them. # Individuality as technology Life is organized around information that replicates under constraint. Computation generalizes this…
Some systems improve the longer you stay with them. Repetition sharpens execution, experience carries forward, and judgment builds on itself. At Hermès, a leather worker trains for two years before touching a bag. One artisan makes one bag start to finish, every stitch by hand, fifteen to twenty-four hours of work per piece. This is the opposite of speed at all costs . It is also one of the most…
By Federico Carrone and Roberto Catalan The previous article argued that the internet left a gap in institutional infrastructure: it moved information but not ownership. Ethereum fills that gap by embedding ownership, transfer, and enforcement into shared software. Financial institutions today spend enormous resources on authorization, accounting, reconciliation, and compliance. Ethereum…
By Federico Carrone and Roberto Catalan The internet made copying and sending information almost free. It did not answer the harder question: who owns something online, who can transfer it, and who enforces the answer when two people disagree? For a long time the practical answer was simple: the platform decides. Facebook decides what happens to your page. Apple decides what ships in its store.…
# Crypto and the accelerated and chaotic 21st Century Crypto has been most useful where trust is weakest. In practice, it has found product-market fit in two places: In countries where inflation, capital controls, or censorship are ordinary constraints, crypto gives people and companies tools they actually need. In internet-native communities, crypto provides a financial layer that lets people…
Markets are the cleanest laboratory for reflexive dynamics. Prices are not determined by beliefs alone. Cash flows, constraints, inventories, regulation, leverage, and flows all matter. But beliefs about prices become trades, trades become prices, and prices update beliefs. That loop is the object. Biology reaches critical-like organization through selection, feedback, and development over…
The ladder so far was built from abstract objects: maps, distributions, operators, proofs, programs, and types. Biology is where the abstraction becomes physical. A living organism does more than occupy a state. It is a dynamical system that holds itself inside a viable region of state space while carrying a description that helps reproduce the system itself. That is why the tempting sentence is:…
The previous essay followed the first engine, iteration, to its end. A rule applied again and again produces fixed points, attractors, invariant measures, scaling laws, and finally the invariant tori of KAM theory, held together or torn apart by the arithmetic of a single frequency. Through all of it, arithmetic stayed outside the system. Integers counted returns. Continued fractions measured…
The first three essays were about one engine: iteration. A rule is applied again and again, and invariant objects appear. Fixed points. Attractors. Invariant sets. Invariant measures. Scaling laws. None of those systems knew anything about themselves. The logistic map does not contain its own formula. It is a rule being turned, and we watched what the turning left behind. This essay is still about…
The previous essay ended with fractals: objects whose structure survives repeated zooming. This essay turns that geometric statement into a statistical one. If an object has no characteristic length scale, then counting it often produces a power law. If a distribution has a power-law tail, then extremes stop being negligible. If extremes stop being negligible, then averages become unstable. If the…
The previous essay built the local picture. Repeated rules can converge to fixed points, and fixed points can organize nearby trajectories into attractors. But the real subject of this series is not only convergence. It is what happens when convergence starts to fail. The logistic map gives the cleanest path into the failure: $$x_{n+1}=r x_n(1-x_n).$$ At small parameter values, the map settles.…
Start with something small: a number, a rule, and the patience to apply the rule again. What happens when you repeatedly apply a rule? That question is small enough to fit into a first course in analysis, but it opens into a surprising amount of modern mathematics. A repeated rule can settle into a fixed point. A fixed point can become an attractor. An attractor can split. Repeated splitting can…
The last essay ended with Taleb’s objection. The tail is where the risk is, and the tail is exactly the part of the distribution the data refuses to pin down. If you cannot know the probability of ruin, stop pretending you can. Change your exposure instead. Jón Daníelsson adds a colder institutional version of the same argument. He says something stronger than “risk models are inaccurate”: in…
Six essays in, we have a toolkit. Bouchaud’s branching ratio, Sornette’s critical time, Gabaix and Koijen’s flow multiplier, Scheffer’s slowing-down: different instruments, all pointed at the same condition, a system loaded near its edge. This essay is about the catch that has been sitting in the room the whole time. That condition, the distance to the edge, is the single hardest thing to pin down…
So far the series has been about why crashes happen and why you cannot read them off their triggers. This one is about the rare practical payoff: the chance of seeing a break coming. A system heading for a tipping point often gives off warning signs, and the odd thing is where they hide. They hide in the calm, in a precise, measurable sense: the quietest system can be the one closest to breaking.
The last essay put a number on reflexivity. A branching ratio near one says the market spends most of its time reacting to itself, with one trade setting off the next. That is already a hard blow to the tidy picture where prices mostly digest outside news. But there is an even cleaner number, and it comes from mainstream economics rather than econophysics. Xavier Gabaix and Ralph Koijen asked a…
Three essays in, the same idea keeps surfacing in different words: markets react to themselves. Prices move because prices moved, and the big moves arrive with no outside cause. George Soros built a whole investing philosophy on this and gave it a name, reflexivity. Markets, he said, act on their own reflection, and that reflection feeds back into the world. It is a good story, and the trouble…
If you believe the last two essays, prediction is mostly a fool’s errand. Crashes come from inside, the biggest moves have no cause worth the name, and looking for the grain that set off the avalanche means looking for something that was never special. Take that seriously and the only sane move is defense: carry slack, cut leverage, expect the slide. Didier Sornette spent a career arguing that…
The last essay ended with a puzzle it did not solve. Markets, it argued, sit near a critical point, the knife-edge where a tiny shock can cascade into a huge move. But phase transitions are fussy. Water turns to ice at one exact temperature. A magnet loses its magnetism at one exact spot on the dial. To sit near that kind of edge, something usually has to hold the dial there with great care. So…
On October 19, 1987, the S&P 500 lost more than 20 percent. The size of the fall was only part of the strangeness. The usual story never arrived. No bank failed that morning. No war started. No earnings report, rate decision, or political announcement came close to explaining a one-day move of that size. People searched for the cause because markets are supposed to need causes. Black Monday mostly…