awanderingmind

My personal site where I post essays about various things - generated using hakyll: https://www.awanderingmind.blog/

I found the writing, and the descriptions of the types of trisectors, strangely poignant. Most of us do not attempt to trisect an angle with straightedge and compass; but surely many of us have other irrational obsessions with which we waste our time (I have certainly been guilty of this). I hope people can find time to look up from their interactions with social media and LLMs for enough healthy introspection to avoid these traps.

This is a strange analogy. The policy saves people money on their power bill. The backyward furnaces are considered a disaster because, among other things, they produced low quality steel, and diverted labour from agriculture and other things, none of which is the case here - people pay for solar panels and install them once, and then achieve savings.

I incorrectly had a ≤ instead of =, my apologies.

There is a lot of focus in the comments on the authors' credentials and, apparently, their writing style. It is a pity, because I think their discussion of scaling is interesting, even if comparing LLMs to grid-based differential equation solvers might be unconventional (I haven't convinced myself whether the analogy is entirely apt/valid yet, but it could conceivably be).

According to the actual paper (https://arxiv.org/pdf/2506.24088), it has been an open conjecture since at least 1977. The quote:

> Unknotting number has long been conjectured to be additive under connected sum; this conjecture is implicit in the work of Wendt, in one of the first systematic studies of unknotting number [37]. It is unclear when and where this was first explicitly stated; most references to it call it an ‘old conjecture’. It can be found in the problem list of Gordon [13] from 1977 and in Kirby’s list [16].

'Additive' here means that if u(K1) is defined as the unknotting number of the knot K1, and u(K1#K2) the unknotting number of the knots K1 and K2 joined together, then u(K1#K2) = u(K1) + u(K2). It is this that has (assuming the paper is correct) been proven false. A deceptively simple property!

edit: I initially incorrectly had a ≤ sign instead of =

Cool example in that link, thanks!

Cool study. I really want to believe the results, but the effect on life extension is so large (see figure 2B) that I find it hard to. Maybe there was some uncontrolled confounding factor? It is noted in the 'Methods' section that 'Researchers were not blinded to group allocation [...]', which is unfortunate.

This perspective is addressed in the article... TLDR; that doesn't seem to be where this is going.

This is a rare type of article - a concrete analysis of different approaches to programming (that are arguably themselves reflections of different cognitive styles), that outlines the shortcomings of one approach in a specific domain, without generalising too much.

You mean something like this https://en.wikipedia.org/wiki/Bremermann%27s_limit or this https://en.wikipedia.org/wiki/Quantum_speed_limit?

I hope you are right - the point about energy efficiency is certainly spot on, and I do think it is possible that people are getting carried away by analogies when discussing the topic (I wrote something about that too, but will avoid linking to it here to avoid excessive self-promotion).

That said, the article doesn't assume such a thing will happen soon, just that it may happen at some time in the future. That could be centuries away - I would still argue the end result is something to be concerned about.

English is my second language too - ironically I am not so as proficient in my mother tongue, due to globalisation/imperialism. But it's ok, fortunately I love the language!

Well, it is still a career, but it's very niche, and more attuned to 'art' than 'documenting the world'.

Thanks for the positive feedback on my writing style! Based on feedback in this thread it seems to be a divisive topic, haha.

True!

I don't literally live in a cave, but fortunately not everyone is so allergic to whimsical language :D.

A reasonable question - another way of looking at it is that theorems are just a side effect of mathematical research. Much of the world economy depends on things like cryptography, which involves a bunch of theorems. The question is then 'what as yet undiscovered mathematical realms might models think up that could make people money'? It is hard to imagine what doesn't exist yet, but much harder to imagine that all potentially profitable mathematics has already been discovered. This could 'just' look like algorithmic improvements.

I am not suggesting models will be capable of generating 'all' proofs - that is clearly impossible. Merely that they will get better at doing so, and there is no clear reason at the moment to believe they will never reach a human level of competence. If you have one model functioning at such a level, it is presumably trivial to have a million of them, none of which will need to be paid, housed, or sleep etc.

Some people do!

In my case, when you aren't paid to do it, and don't have (much) time to study it formally anymore.

I wasn't aware of this, thanks for posting! Very amusing.

'A language is a dialect with an army and a fleet ' --> I hadn't heard this before, love it! For the curious: https://en.wikipedia.org/wiki/A_language_is_a_dialect_with_a...

Good insight about how feelings of 'superiority' or 'sophistication' can suck the joy out of life. I fell into this trap myself, and it took a long time to get out of it.

That said, there are times when a certain type of appreciation of 'sophistication' is warranted - you just shouldn't use it to believe you are therefore above other people, or beyond the simple pleasures of life.

The comparison to Pascal's Wager is apt - I came to the same conclusion when I first read about Roko's Basilisk, but the insight somehow slipped from my mind. It's like an epistemic tarpit attractor.