This week there was a story that made me think perhaps the leap was here, and I am still not completely sure, but I don’t think so. On July 19 the mathematician and Anthropic employee Levent Alpöge posted on X that Fable had found a counterexample to the Jacobian Conjecture, first proposed in 1939, which asks whether a certain kind of polynomial transformation that is reversible locally must also be reversible globally. The counterexample means it is not true, even though mathematicians had mostly thought it was. Legit mathematicians have tried and failed to find such a counterexample over the years, including Yitang Zhang, who spent seven years on it during his PhD before becoming world-famous for his breakthrough on the twin prime conjecture in 2013. So finding one is a pretty big deal. Finding one with AI is a potentially huge deal, depending on how it was done.

So the question is: Is this the leap? Did Alpöge say, ‘Hey Fable, give me a counterexample to the Jacobian Conjecture’ and out it popped? Irritatingly, Alpöge has not released the conversation he had with Fable (my guess is there’s all sorts of other random stuff in there; he talked about doing this during the World Cup final, which, in his defense, was stultifyingly boring). But Alpöge is a heavyweight mathematician in his own right, and he did his PhD under an even more heavyweight mathematician, Manjul Bhargava, who won the Fields Medal for work on the geometry of numbers, including counting number fields and bounding the ranks of elliptic curves. I must admit I don’t know what that means, but according to Claude the counterexample is very much in Alpöge’s mathematical wheelhouse. So I think it’s likely that Alpöge had some kind of insight into where the counterexample might be found, gave it a rip on Fable while waiting for Argentina to consider venturing a shot at goal, and he just happened to strike gold. Claude agrees with this insights-from-human-mechanistic-work-from-AI hypothesis (with some hedging of course), and in Claude’s view the work has Alpöge’s ‘fingerprints’ on it. I would love to see evidence to the contrary, and hopefully Alpöge will tell his story in more detail at some point, but right now I don’t think AI has made the leap yet.

The counterexample was important enough that even mathematical mega-mind Terence Tao wrote about it, and he also posted the conversation he had with ChatGPT where he dug into its details. The nuances of that discussion are not accessible to anyone without a pretty significant mathematical background, but what is fascinating to me is how he just asked question after question, 53 turns in all, probing and interrogating and suggesting things to do until finally rebuilding the counterexample from scratch out of his own understanding by the end. I believe this is how AI will be used by master practitioners in every domain -- asking good questions and using the AI to carry out the work of answering them, until they arrive at some new insight or decision.

So then the question becomes: How do you ask good questions? And the answer is unsatisfying: understand the domain, understand other people, understand yourself, have a deep store of experience to pattern-match against, and use all of that to somehow make the ‘leap’ to the right question. Be the right kind of person, in other words. Fortune magazine in its coverage said: ‘Beyond calculating, beyond even logical reasoning, “understanding,” at the bottom, is knowing what to ask, what Silicon Valley has taken to calling “taste.”’ Then they quote mathematician Kevin Buzzard as saying: “People have tried to get machines to ask questions, and they’re abysmal … All the questions they ask are either boring or obviously true or obviously false.”

So clearly, he does not think AI has made the leap yet, either. I don’t want to tempt fate, but long may it last.

Even computer science godhead Donal Knuth (it’s pronounced kuh-NOOTH, you philistine) is surprised by AI capabilities: https://www-cs-faculty.stanford.edu/%7Eknuth/papers/claude-cycles.pdf

If you want to get a glimpse of the future, use-cases with a lot of compute are a decent guide. NotebookLM cinematic overviews from Google, only for top tier subscribers: https://x.com/NotebookLM/status/2029240601334436080

I generally avoid updates that include the word ‘insane’, but worth looking past it this time: agents pushing against security barriers as reported by Alibaba. Agents are not conscious (I think…) but then neither are ants, and the anthills are hard to deny. https://x.com/AlexanderLong/status/2030022884979028435

Today: attention. Information and AI may be going exponential, but your brain isn’t.

Nobel-prize-winning scientist Niels Bohr believed physical activity was essential for intellectual freshness and persistent attention. British publisher Felix Dennis used his awareness of death as a daily filter for how he spent his time and attention. Inventer of ‘flow’ Mihaly Csikszentmihalyi calculated that humans can process only 126 bits of information per second, making attention the fundamental constraint on experience.

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Bohr maintained lifelong athletic habits—football in youth, skiing and sailing later—believing physical activity was essential for intellectual work. He observed that insights often came during or after physical recreation, when his conscious mind was occupied elsewhere. Bohr encouraged students to maintain physical practices as professional necessity.

Felix Dennis cultivated daily awareness of mortality as a decision-making tool. He would ask whether conflicts, meetings, and projects would matter with six months to live. This filter helped him walk away from meaningless fights, pursue genuine desires, and focus on what he actually valued rather than obligatory concerns.

Csikszentmihalyi calculated that humans process only 126 bits of information per second. This cognitive bottleneck means attention is the ultimate scarce resource. Every moment spent on anxiety or distraction is unavailable for meaning. A focused craftsman experiences more richness than a distracted billionaire. Quality of life equals quality of attention. Guard yours accordingly.

I’m looking for a handful of users to test advanced features of the tool and give me feedback — if that’s something you’re intereste in, drop me a reply or DM.

Stephen

This is what it looks like:

It’s a map of what it takes to be successful.

On this Substack I’ll be sharing outcomes from the system and what I’m learning by building it. And as the very first cluster I’ve shared publicly, this one is pretty fitting.

Jacques Cousteau couldn’t explore the ocean because diving technology limited humans to minutes underwater. So in 1943, he partnered with an engineer named Émile Gagnan to invent the Aqua-Lung. Gagnan’s regulator was originally designed for cars running on cooking gas during wartime fuel rationing. Cousteau looked at it and saw the ocean floor. That invention created a $4 billion recreational diving industry and made possible every underwater documentary you’ve ever watched. Without the tool, the exploration couldn’t happen.

Forty years later, James Cameron faced the same problem in a completely different domain: No existing camera system could capture what he envisioned for Avatar. So he co-invented the Fusion Camera System and Simulcam technology, spending millions of dollars on the project. The system became the industry standard for 3D filmmaking. The tool outlived the project.

And decades before either of them, Marvin Minsky, one of the founders of artificial intelligence, was building programming languages, inventing the confocal scanning microscope, and designing theoretical frameworks, all because existing tools constrained what he could think.

The lesson is this: Your imagination can reach much further than the tooling, but you’d be surprised how quickly the tooling can catch up.

My AI didn’t know this cluster would emerge. I didn’t design it -- I loaded thousands of lessons into a semantic space and asked: What connects? The math of semantic embeddings found this pattern on its own. There are thousands more like it, and literally countless lessons now hiding in plain sight.

If this catches your interest, I’m looking for a small number of pre-release testers -- just reply to this post, or send me a DM.

What helped shove computing into the mainstream was the rise of the graphical user interface, famously developed at Xerox PARC, then ‘perfected’ by Apple, then ‘re-perfected’ by Microsoft. No more typing of commands, just point and click, and then later, just touch.

But if you’ve ever gotten extremely familiar with a system that can accept keyboard input, you’ll know how fast and smooth operations can be when you don’t need to take your hands off the keyboard. You can find a very deep state of flow, and having to ‘interrupt’ that state to use the mouse and then visually find something and click on it can feel like a major irritation. When I worked as an editor at various newspapers in Ireland they all used the same publishing platform, and I knew it so well I could manipulate the menus with the keyboard faster than the eye could read them. I’ve seen other people use Photoshop and Excel the same way.

The trouble is that it takes a LOT of time to get to that level of capability. You need to learn all the product’s little quirks and ways of doing things.

But AI, as you may have noticed, is taking us in strange new directions. To get an AI-driven system to do something, you don’t need to learn any commands any more -- you just tell it what you want in natural language. And soon enough, you won’t need to learn any product quirks any more, either, because you will just describe what you want to happen and the system will do it.

I have previously written about the magical feeling of using Claude Code, and the sense of power and delight it gives me to be able to tell the system to generate something in code, and have it generate that thing. Watching it use tools and fix errors and reason through things to achieve my aims makes me ‘feel the AGI’, as we said in the early days of AI, about six weeks ago.

I think that feeling is so powerful that it’s going to be addictive for people. Plus, there is going to be an explosion of capability in those systems. First, they’re going to get connected to a multitude of MCP-enabled agents to do a vast range of new tasks. And second, if the precise thing that you want for your personal needs does not exist, the system will just create it for you. ChatGPT and its companions have already shown us that people are much more text-friendly than the rise of TikTok et al might have led you to believe.

The result of all this could well be that the command line makes a surprise resurgence, because people will be able to just quickly type things into it one after the other, and get both the sense of uninterrupted flow and unlimited power. Email John and ask him about tomorrow’s presentation. Check my Slack, did Anne message? What time is my flight next Monday? Where is [my wife / friend / coworker lover] right now? Ping them and ask. What’s the weather at the weekend? Add a to-do to check in with Dan on the thing. Open that doc I started before lunch. What did I email Alice that I would include in Project Z?

And on and on. Answers popping up as text responses (‘I’ve emailed John, I’ll let you know when he responds’) or visual UX (for the weather, the doc) as needed. And most of all: glorious, uninterrupted focus -- no pop-ups, no banner ads, no requests to sign up for newsletters, no moving the mouse and reading UIs and looking for things on the screen to click on like a caveperson. Pretty much every webpage in existence has multiple things on it all clamouring for your attention, but the command line doesn’t. It just helps you do one thing at a time quickly and efficiently.

I think it’s time for the return of a very old friend.

I spent a lot of yesterday evening after the kids were in bed playing with Claude Code, running on the newly-released Opus 4. The experiences was extraordinary. I felt I was speaking to an intelligence that understood me. In a few hours I was able to use it to build out some ideas that have been on my mind for years. One was this: I want to understand all the ‘named' ideas on Wikipedia, from Occam’s Razor to Murphy’s Law. I’ve had a few attempts at this before. Last night, just before bed, I explained my desire to Claude. It wrote a Python script for me, but I never even looked at the code. Even as I write this, I have still not seen it. Once we agreed on the max number of API calls, and once I had set my OpenAI API account-level budget to something reasonable in case of disaster, I asked Claude to run it. It said (and I am so, so tempted to write ‘he said’ ):

Sweet dreams! The script will work through the night finding all those fascinating named ideas from Wikipedia! 🌙

This morning I asked: ‘Hi -- is the script still running?’ (The weird thing is, I am not even sure where it was running. It wasn’t in its own terminal window. It was just… on my machine somewhere. Claude was handling it.)

Claude’s response:

The results are fascinating. It did exactly what I hoped. I’ll figure out something interesting to do with the data. And again — I never saw the code. Claude just did it all for me. By itself. Mmm.

The old saw is that ideas are cheap but execution is expensive. But now, the price of execution is dropping. How low will it go? We’re still in the very early stages — it’s still hard to build a phone app, for example, and in my professional life I know first hand how hard it is to build high-quality prodution software for actual users with the irritatingly inflexible demands of actual reality. But the direction of travel is towards almost anyone being able to build almost anything. Larry Page used to say at Google that it was unwise to bet against the internet. And of course he’d say that — the internet made him $150B dollars. But he was right. And now it seems unwise to bet against AI. A lot of things that used to be valuable will no longer be valuable, and a lot of completely new valuable things will come into being.

I wonder what those things are.