akhil

Gray describes the unsatisfactory state of calculus that led to the development of what we now call "analysis".

hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3, has jacobian determinant -2, and sends (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0)

🚨 [Ackman-esque long post. You should prob skip. Don't tell you it's too long in the replies. I used AI to correct typos and run-on sentences.] This is a thoughtful write-up. I think I understand and sympathize with its core intuition, but I disagree with it. I don't think we've overestimated intelligence or hit a ceiling on what IQ can do. On the contrary, I think we don't have remotely enough intelligence, for long enough and cheap enough. Let me expand. Cliff's argument is two-fold: you cannot make genuinely useful output from doing first principles reasoning alone. To solve genuinely economically valuable problems, AI must directly interact with the world, collect data, leverage tools, and iterate. But I think that’s exactly the direction AI is currently traveling. And that’s why it was so important that we’ve had the agentic AI (re)evolution — AI directly using existing tools and analytics, basically being able to browse the internet, use a computer, call analytics functions instead of doing it from scratch. I think a telling example is that today few people doubt that AI will eventually be able to solve management consulting problems like McKinsey, or do legal research and prepare litigation strategies and court arguments. It’s already able to do tax saving optimizations or offer more accurate medical diagnoses. The funny thing is I actually think it’s the first principles stuff that AI has yet to show it is truly capable of, such as new scientific discoveries in maths, physics, biology, and medicine etc. I think it’s excellent at empirical analysis and will likely become very good at devising empirical analysis strategies — deciding what experiments to run and what data to collect. It’s not quite able to autonomously collect physical world data by itself yet, but with automation in drones and robotics maybe eventually it will. I think the reason we haven’t had an economic takeoff is exactly because we haven’t had enough “thinking.” On the one hand, “enterprise” adoption remains abysmally low. It’s a brand-new technology that’s changing extremely rapidly. It’s very costly and difficult to learn and adopt into workflows, especially when the cost of tokens is still very high. That’s why it’s actually a very good thing that frontier intelligence continues to deflate with further scaling. If you look at the problems I cited above, these are all actual concrete problems that human workers do to generate enormous billable value. With adoption of AI, we’ll be able to do all of those tasks more quickly and more cheaply. That brings us to the second reason. I think the lack of takeoff continues to derive from the fact that the “thinking” or model just isn’t good enough yet. We are a long way from hitting diminishing returns, much less the “ceiling” of intelligence or “good thinking.” As smart as the models have gotten, they are still pretty “dumb” in predictable ways. They can’t yet be fully trusted to complete the above-mentioned tasks without close human supervision — at least not in a way that makes us feel comfortable. The models are now able to think about much harder problems and work for a very long time to code up very impressive software. These are progresses that were unthinkable two years ago. The models only just recently — maybe this year — became genuinely and reliably useful. Most of the “multi-hour tasks” that AI is able to do are still in the coding domain. You wouldn’t really expect a good outcome from a multi-hour AI session on a more qualitative/judgement-heavy use case like consulting, medicine, or law. The last time I checked, ChatGPT still sucked at making good PPT presentations. I’m sure someone will jump up and tell me that I’m out of touch with AI progress and it’s already capable of doing such things. But that already tells us why enterprise adoption/takeoff is low. The tech is just moving too fast and is all too new. Adoption takes time and some stability of expectations. In short, I think there’s a crucial difference between the existence of certain frontier capabilities and the existence of such capabilities in cheap abundance. There was a long gap between the invention of electricity and its wide adoption and productivity boost. Similarly with the inventions of combustion engines and the internet. So the fundamental problem in my opinion is actually converse to Cliff's: the models are still simply not good enough both in capabilities and cost per intelligence. Like that famous line from the Jiang Wen movie: we have to let the bullets fly a while longer.

It’s interesting how a lot is determined by your learning rate. If you have sufficiently low, barred from certain fields. You can learn anything given enough time. The impressive pet is can you learn things fast enough to either keep up in one field or multiple fields at once. A real scenario is it takes someone 1hr to learn something others need 1week to learn (learning system, raw hardware, env, friends all contribute). Imagine the capability gap over time tho. If you have a sufficiently high learning rate imagine either the depth or the breadth you can master

my spicy theory is that chinese ai labs keep winning because of culture, not talent or resources. chinese labs still have a deeply hands-on engineering culture. deepseek and kimi are flat organizations where science and engineering are fused: the same people move between algorithms, data, and infra, doing whatever it takes to make the model work. but sf tech bros have decided that “researcher” is the high-status title while infra is merely support work. every new sf ai startup calls itself a “lab,” every ambitious engineer quietly upgrades their title to “research engineer,” and the infra work is left to whoever failed to escape it. but at frontier scale, infra determines experiment velocity, and experiment velocity determines research output. at frontier scale, infra **is** research.

hot take: mathematics is just programs mathematicians write and run in their minds, so programming is actually the more fundamental discipline.

Scaling as a strategy works as long as the transition to victory is a clear line you cross. If it's a fractal, you will get closer but never cross. As an exercise, consider how this changes the game theory.

My advice to every dude in college: Don’t seriously date in college. You’re trying to find a wife while competing with: • Frat parties every Friday • Spring Break in Cancun • Miami weekends • Nashville girls trips • “The girls booked Europe for 3 weeks” • Study abroad in Italy College is built around freedom, novelty, and “once-in-a-lifetime experiences.” Graduate. Build your career. Then start looking for your wife while you move to a big city. (After 25) You’ll save yourself a lot of headaches.