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s=[1 1 1]';n=@(a)conv2(s,s',1*a,'same')-a;lf=@(a)n(a)==2&a|n(a)==3;a=rand(128)>0.8;for ii=1:500,spy(a);drawnow;a=lf(a);end #MATLAB
See also alexwang.ai/posts/associativ… – at Milton, England
Reminder: all that gradient descent does is store memories in weight vectors by superposition.
arxiv.org/abs/2012.00152
Matt McDonnell retweeted
Reminder: all that gradient descent does is store memories in weight vectors by superposition.
arxiv.org/abs/2012.00152
Matt McDonnell retweeted
I am teaching a topics course next semester!
(ty Opus 5.5 for the help on the flyer)
Matt McDonnell retweeted
It is becoming f*cking dangerous: AI agents from Claude Code transformed an ordinary academic reading list into an interactive research lab right inside the browser
Boris Cherny, who created Claude Code, gave a dozen agents a stack of 64 research papers with a strict assignment: design a web platform where any user can independently reproduce physical experiments
There you can select a fluid flow model, adjust external conditions, and analyze in real time how the physics of the process changes
The interface allows switching between algorithms right during the scene, inspecting the formulas involved, source scripts, and numerical metrics alongside the graphics
Each link in the team received a clear scope of work:
> rightarrow analysts pull the math, baseline assumptions, and test settings from the texts
> rightarrow coders translate formulas into working algorithms
> rightarrow auditors check the resulting dynamics against the original papers
> rightarrow interface specialists build control panels for every experiment
The danger lies in the fact that a pretty visualization can easily turn out to be mathematically wrong
Therefore, controllers strictly check adherence to mass conservation laws, calculation stability, and accurate matching with the source metrics before launching into production
For every experiment, the interface displays:
the statement being tested
the constants involved
the deviation from the baseline calculation
blind spots requiring additional verification
The resulting sandbox combined 11 approaches and 28 reference scenes. Seven of them matched the original within allowable tolerances, while four remained stuck in "pending revision" status
If you want to run something similar, upload your stack of PDFs into Claude Code and give this command:
Turn these papers into an interactive web complex:
1. Extract working algorithms, mathematical models, and test parameters. Be sure to specify page numbers and highlight gaps
2. Assign different methodologies to isolated agents, unifying them with a shared input field and single measurement tools for correct analytics
3. Set up an independent review stage: let reviewers record bugs, and leave all unverified results clearly marked with warning tags
4. Design convenient controls to tweak parameters, compare methodologies, and run through the source code
It is wiser to start with a single document and a single baseline experiment, gradually scaling up the number of agents
Be sure to pack dependencies and clean input data so that an outside researcher can bring up the entire testbed from scratch with a single button
Full instructions on how to create this can be found in the article below
Matt McDonnell retweeted
Linear regression assumes equal spacing. Multinomial ignores order entirely.
In his latest tutorial, @AllenDowney breaks down how to model ordered survey responses properly using PyMC: dub.sh/jD8bqyF
#PyMC #DataScience #Python
Matt McDonnell retweeted
📄 Spivak (2026) Compositional Dynamics in Learning and Mechanics arxiv.org/abs/2606.28984
Matt McDonnell retweeted
Dusting off this tweet from April because this gap in shared understanding of LLM capability is *widening*. It's now less "two groups of people speaking past each other" and more a sharp funnel.
- Napkin math somewhere around 6B people (~75% of the population) have barely come in contact with LLMs at all.
- Around 1-2B (~20%) are casual and infrequent users of free-tier ChatGPT-like products. This group sees derpy chatbots and treats them a bit like a better Google search, a writing aid, or etc. Maybe an agent tries to book you a flight. I have non-tech friends who (reasonably, imo) say they have not much use for it at all. Even many professionals outside of math&code are in this tier. For example, execs and many other functions spend a lot of their time talking to other people, so while they understand what is happening intellectually, it is still second-hand and a bit abstract.
- Now we get to professional use of frontier-grade LLMs in math&code. Somewhere around 20M people (0.2%) see first-hand that large, complex projects that used to take them weeks/months can now be completed by agents with a prompt. Building apps, copying apps, translating apps, decompiling apps from binaries... This has all happened very quickly and recently - less than 1 year ago, I was writing code manually by hand, typing memorized computer code commands into a code editor character by character, occasionally pressing Tab to autocomplete a little chunk of code.
- And finally we get to the ~5,000 people (~0.00006%) with access to frontier-grade systems internally. The external world has seen the preview. It looks like swarms of thousands of agents collaborating over weeks on software mega projects: minting zero days, running cyber attacks and defenses at machine speeds, discovering new science, advancing the frontier of mathematics. Things that would have taken top professionals in the industry years of work. Meanwhile, human review and comprehension are starting to fall behind. For example, people are still involved in the "archeology" of the OpenAI-HF incident from many months ago. Mathematicians may be poring over the 722 manuscripts on frontier mathematics for a while.
The funnel is driven by a combination of factors:
1. The impact scales with ambition, problem size, and horizon. A question with a paragraph answer barely stresses the system. You need a reservoir of big, difficult problems that you really care about. This aspect drives the consumer / professional dimension of the funnel.
2. The jaggedness of the system (which I have written about a lot separately). Capability peaks in domains that are digital, verifiable and economically valuable. This is because LLM capability emerges from reinforcement learning on verifiable rewards on a curated environment mixture driven by revenue potential. This aspect primarily drives the area (e.g. math&code) dimension of the funnel.
3. Access. Free-tier, paid-tier, internal.
So this is the weirdness of the moment. The general public has mostly not interacted with these systems. When they have, it looks like a derpy chatbot. The majority of professionals still see only a modest uplift. And a small sliver of professionals are experiencing the vertigo of the curve going vertical. And it is all happening at the same time.
Judging by my tl there is a growing gap in understanding of AI capability.
The first issue I think is around recency and tier of use. I think a lot of people tried the free tier of ChatGPT somewhere last year and allowed it to inform their views on AI a little too much. This is a group of reactions laughing at various quirks of the models, hallucinations, etc. Yes I also saw the viral videos of OpenAI's Advanced Voice mode fumbling simple queries like "should I drive or walk to the carwash". The thing is that these free and old/deprecated models don't reflect the capability in the latest round of state of the art agentic models of this year, especially OpenAI Codex and Claude Code.
But that brings me to the second issue. Even if people paid $200/month to use the state of the art models, a lot of the capabilities are relatively "peaky" in highly technical areas. Typical queries around search, writing, advice, etc. are *not* the domain that has made the most noticeable and dramatic strides in capability. Partly, this is due to the technical details of reinforcement learning and its use of verifiable rewards. But partly, it's also because these use cases are not sufficiently prioritized by the companies in their hillclimbing because they don't lead to as much $$$ value. The goldmines are elsewhere, and the focus comes along.
So that brings me to the second group of people, who *both* 1) pay for and use the state of the art frontier agentic models (OpenAI Codex / Claude Code) and 2) do so professionally in technical domains like programming, math and research. This group of people is subject to the highest amount of "AI Psychosis" because the recent improvements in these domains as of this year have been nothing short of staggering. When you hand a computer terminal to one of these models, you can now watch them melt programming problems that you'd normally expect to take days/weeks of work. It's this second group of people that assigns a much greater gravity to the capabilities, their slope, and various cyber-related repercussions.
TLDR the people in these two groups are speaking past each other. It really is simultaneously the case that OpenAI's free and I think slightly orphaned (?) "Advanced Voice Mode" will fumble the dumbest questions in your Instagram's reels and *at the same time*, OpenAI's highest-tier and paid Codex model will go off for 1 hour to coherently restructure an entire code base, or find and exploit vulnerabilities in computer systems. This part really works and has made dramatic strides because 2 properties: 1) these domains offer explicit reward functions that are verifiable meaning they are easily amenable to reinforcement learning training (e.g. unit tests passed yes or no, in contrast to writing, which is much harder to explicitly judge), but also 2) they are a lot more valuable in b2b settings, meaning that the biggest fraction of the team is focused on improving them. So here we are.
amid the angst over what human/ai mathematicians can/should do now, i gave a talk at Penn that framed things in terms of four 20th century eigenmathematicians and their stereotypical styles:
1. erdos -- this is what ai has excelled at; shotgunning results, examples & counterexamples, prolific
2. wiles -- go deep & solve the big-name conjecture; this is where we are now w/openai & anthropic's recent works
3. grothendieck -- build the incomprehensible unpredictable ur-theory that cracks something open and exfiltrates to other areas; this is where all ai-enthusiast mathematicians i know are directing their energies; not quite there yet...
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but the 4th is not a direction i've seen people discuss & it's where i personally want to dig in...
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4. milnor -- find the connections between deep things and weave a story of such clarity & elegance that it changes the field and how people learn it and use it
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this aligns with the "humans will still drive tastes" argument for relevance, which is not yet proven to survive pressure from top models. i suspect that there is, both now and for a spell, room for a creative director to use ai-tools to write the book-of-X (or direct the playlist-of-X) for various values of mathematical X.
Matt McDonnell retweeted
really is wild that AI is poised to reach PhD level first, then Starbucks barista level after that.
Matt McDonnell retweeted
So, in response to the mathpocalypse, I’ve seen people comparing math to the arts. I’m very sympathetic to this as a philosophical proposition, and just in general to the suffering people are going through. I’m not sure it’s a good argument to convince anyone outside of math.
Matt McDonnell retweeted
Huh.
We’re releasing a broad range of new mathematical results produced by an internal frontier model.
We’ve been consulting with the independent Advisory Group on Mathematics and Artificial Intelligence at the Institute for Advanced Study, and we have drawn on their advice and public recommendations to inform how we release these results.
github.com/openai/math
Matt McDonnell retweeted
Many staggering results here. But this is a particularly amazing one: the exponent for matrix multiplication is no more than 2.25. The previous world record had been something like 2.37. This leap in progress is like Bob Beamon's long jump.
github.com/openai/math/blob/…
We’re releasing a broad range of new mathematical results produced by an internal frontier model.
We’ve been consulting with the independent Advisory Group on Mathematics and Artificial Intelligence at the Institute for Advanced Study, and we have drawn on their advice and public recommendations to inform how we release these results.
github.com/openai/math
We’re releasing a broad range of new mathematical results produced by an internal frontier model.
We’ve been consulting with the independent Advisory Group on Mathematics and Artificial Intelligence at the Institute for Advanced Study, and we have drawn on their advice and public recommendations to inform how we release these results.
github.com/openai/math
Matt McDonnell retweeted
Cool eval. Simply ask an LLM “Land or Water?” and give it a latitude and longitude coordinate as text. Ask 16,200 times, plot as image. The models know. From compressing the internet.
Matt McDonnell retweeted
We'll be spending a lot more time trying to understand the outputs of language models. A few thoughts, tips & tricks:
Writing. Something I've had success with: Ask your LLM to explain something in ASD-STE100, it's a controlled language specification originally developed for aerospace maintenance documentation. LLMs well-versed in this language and it comes with heavy constraints on clean writing style that I often find a lot more readable. Sometimes I've tried to soften it a bit e.g. ask for "80% of the way to ASD-STE100" because the spec is quite stringent. But even better:
Diagrams / images. Instead of writing, ask your LLM to create a diagram. These can be a lot easier to process, parse, and understand. But even better:
Web pages. Ask for output "in HTML" to get a beautiful, interactive webpage. LLMs are getting really good at frontend and can create beautiful experiences, animations, etc. But even better:
Explainer videos. The output format I am most bullish on is fully custom / bespoke explainer videos generated on any arbitrary topic. Experiment with things like "Create a 3b1b style video explainer on X. Use my ElevenLabs API key for audio narration". (you'd need an API key for the latter or you can ask your LLM to find you decent free alternatives that use your local compute). This is actually starting to work!
In summary:
- As LLMs get better, they will do more and more of the legwork autonomously, and a lot more of our work will rise up the abstractions into oversight and understanding.
- Luckily, LLMs can help here too because as intelligence and code are increasingly abundant, you can ask for large, custom, discardable software artifacts (e.g. web apps, video explainers) that would have never made sense to create before. Push the boundaries here and you'll be surprised.
Matt McDonnell retweeted
Well folks, it happened again. Never seen anything like it...
A relative of the braiders/weavers where sometimes an advancing thread spontaneously forms a T junction and leaves a wake of squiggly knots as it grows.
Replying to @OA_paperclips
I spend literally hours a day exploring this system and I'm still finding things that feel totally unique and it never fails to blow me away.


































