Ok after going through the proof of the chain rule, here is my sense for what is going on. Please correct me if I am mistaken.
An approximately-shortest program outputting
We get
However I don’t think this necessarily means that there is an approximately-shortest program producing
I’m not talking about the spirit of the problem, I’m talking about the actual program corresponding to the derivation in this article. I’m not super familiar with the field though, so I could be wrong.
The math you’re doing here implies that if 0 ≈ K(A) then 0 ≈ K(A) + K(A) = 2K(A), so constant multipliers seem to be allowed in your approximations.
It seems to me that the “approximately-shortest program” here could just be generating S_1 and S_2, then throwing away S_2 and generating the data from S_1 (or vice versa)?
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I’ve been particularly impressed by 3.1 Pro’s ability to do math problems. I have 3 problems that I like to pose to AIs in increasing levels of difficulty (all requiring or greatly aided by a postgraduate-level knowledge of mathematics).
Gemini 3.1 Pro and Opus 4.6 are the first models that could solve the first one, or even come close to a correct solution. Opus was unnecessarily verbose and appealed to some advanced mathematics jargon, while Gemini gave a much simpler, far more readable solution.
The second problem was eventually solved by Opus after a couple of false claims and some strong hints (and the final solution still had some inaccuracies), but Gemini just breezed through it and gave a solution that was both more general and more elegant than the one that I came up with. The problem and a solution sketch could be in the training data as a singular Reddit comment, but that didn’t seem to help Opus and Gemini’s solution appears to be novel.
The third problem takes a long time to solve and requires several indirect steps—I suspect asking an LLM to one-shot the solution is simply the wrong format and that a more Ralph-loop style approach might be appropriate. Opus was hopeless and I couldn’t even get it to reason well about the problem even with direct hints. Gemini, however, got the first important insight then got lost from there, but some strong hinting about what to look for eventually led it to a correct solution.
I’ve noticed that while Opus’ mathematical output is often vague, filled with jargon, and difficult to understand, 3.1 Pro is much easier to read and it seems to prefer to directly use elementary techniques rather than appealing to advanced theorems. Even when it is wrong, it is quite easy to see where the specific incorrect step is. It makes the output much more potentially useful overall. I could see it legitimately helping with advanced mathematical work.
I also wasn’t a heavy user, it’s just something that I noticed from a few conversations with Sonnet 4.5, then I started noticing it in writing that other people co-wrote with Claude. It wouldn’t surprise me if Opus uses it even more but I’m not really sure.
As a data point, I did notice the overuse of “genuinely” before the constitution was added to Claude (at least publicly). So I think it would have been introduced somehow during training.
It refers to the existing norm. The author is saying that a recently developed norm is likely less load-bearing than a long-existing one, so the attempt to abolish it is less likely to be flawed.
Sometimes you are a bad friend in ways that you don’t realise; everyone has their blind spots. Telling people before taking actions that affect them can let you adjust your expectations before doing something you don’t realise is harmful.
As someone who recently got too wrapped up in “just doing things” at the expense of a friend whose permission I did not ask, this post strongly resonates with me right now. Seems like a good word of caution.
On the other hand, things will probably be fine in the long term and as @Bastiaan said, I probably have a more accurate sense of where the boundaries are than I would have if I just avoided doing things. So it’s maybe a grey area. But you’ll probably always be better off asking permission from the people you might be affecting, as long as you care about their opinion and they are in good faith.
My experience of a Goenka retreat was basically the exact same as yours, with the added complication that I got sick during the retreat, which was very much not fun but also didn’t seem to stop my body from eventually dissolving into waves of vibration (so to speak).
Cool and interesting, but it didn’t seem much more than that and enough of the retreat made me feel skeptical/put off that I didn’t go again.
I suspect that “enlightenment” is probably a bundle of different things rather than one discrete thing, and maybe what it means depends on the culture and even how an individual relates to the world. This is based on the heuristic that when you dig into the nature of mental states, they tend to not fall into neat categories that are the same from person to person.
However, there are people existing today who claim to be “awakened” who were certainly self-aware, and still describe a dramatic change in their perception of the world. The descriptions tend to fall along similar lines, and include:
A dissolving of the boundary between “self” and “other”.
A sense of fundamental peace/ok-ness that in independent of current thoughts and emotions.
The ability to rest in some space that is “beyond thought” (or something along those lines, it sounds like a sazen).
A natural and automatic removal of anxiety, fear and other negative emotions.
Unlearning something about thought and the self-model that most people implicitly take to be true without realising it.
This sounds like there’s something more going on than gaining consciousness, and in some ways points in the opposite direction. It is often described as more of an “unlearning” than a learning.
I’m getting the impression that “consciousness” is inherently not well defined; that is, there is no singular thing we can point to that will meaningfully determine whether or not something is “conscious”.
In this sense, consciousness might be a red herring. A similar but more concrete question worth asking: what behaviours would an AI agent have to exhibit for you to want it to be granted fundamental rights/autonomy? Or otherwise for it to be intrinsically unethical to create and run an instance of it?
That makes sense—everything in context. I wouldn’t want to go around assuming that I can just tease anyone who is experiencing psychological distress, but I think I do have a sense of specific circumstances where it feels appropriate. And hey, I cannot remember the last time I looked like an asshole, so I’m probably overdue anyway.
Reading this has made something click for me, I think.
The other day a friend of mine had what he felt was an extremely embarrassing moment—although really it was not nearly as bad as he felt like it was. I kind of had this blog series in mind when we were assuring him that it was fine, and it didn’t quite connect with him, and I knew it wouldn’t, but I also didn’t really know what to do so I felt kind of awkward even though I worried that feeling awkward would make things worse.
Part of it is that I was hiding information, in that I actually found the situation interesting and slightly fun but I didn’t feel secure in demonstrating that because I was afraid of standing out, failing the bid and making myself look like an asshole. But now I’m realising that going all-in on how I really felt and approaching it with a sense of playfulness would have both been more honest and probably would have defused the situation better.
I have frequently had the experience of wanting to console someone who is experiencing an emotional difficulty, but something about my attempt feels performative and effortful even though I do actually care. In hindsight I think that I am hiding some of my authentic experience because I feel like I’m supposed to Take Their Emotions Seriously, and that any positivity or playfulness would come across as being dismissive. I think I’m starting to understand where the disconnect is and how I could better handle these situations.
Another excellent post. This particular post has clarified the framework for me enough that I could imagine it impacting my interactions with people.
It seems like this is formalising things that people tend to gain a partial intuition for through social interaction.
How much of this is perscriptive vs descriptive? I could use music theory to explain why a song sounds good, but in most cases music theory works better as a post-facto explanation than an instruction for how to write good music. Do you think this framework is useful for learning how to change people’s expectations/beliefs/attention/etc. or is it a description of something that could be learnt just as well without the framework?
I very much enjoyed this short story.
Before I read the spoiler text at the end, I was confused about the postscript. While the main story has a very clear metaphor and intention, the postscript completely diverges from that and instead sets up the intro to a cliche YA science fantasy action novel; it could have been written by James Patterson. I wonder if modern LLMs would do any better.
Edit: I tried it with ChatGPT. It gave a more realistic opener that matches the text better, but it was too explicit in calling back to phrases directly used in the text, like someone trying to show off how much they remember. Plausibly this could be fixed with the right prompting.
I’m very much enjoying the series so far.
I find this topic very interesting, but it’s hard for me to tell what these techniques would look like in practice. It might help if you have more examples of what these looks like. Or maybe it’s the kind of thing that I would need to experience personally to get it?
How do you know how much respect the other person is giving you, so that you can successfully bid for attention? Is this just a matter of experience?
Very interesting post. I’ll be interested to see how this fits in with other psychological frameworks that have been posited on this forum, like Chipmonk’s and Steven Byrnes’.
Some of what you’ve said so far resonates with me—I have had the experience of a particular instance of suffering dissolving pretty quickly once I noticed that the thing I was observing and the suffering I felt from it are distinct and can be separated from each other. Some of this seems unlike anything I’ve heard before (like the Attention-Respect-Security model) and I’m curious to see how this works in practice.
I think it’s also that the public doesn’t understand maths well enough to judge the importance of these new developments. What’s most visible is the AI companies themselves pushing these results hard, which they would clearly do for marketing purposes regardless of whether it is actually sensational.
Given that the mathematics community as a whole is not freaking out (or at least the reactions are very mixed), caution is probably an appropriate response if you don’t have enough experience to form your own opinion.
I think it would be different if AI solved the Riemann Hypothesis (with a positive proof rather than a counterexample), mathematicians would not shut up about it and that would gather some very wide attention.