Each field derived it from first principles. Each named it differently. Minimal cross-citation. The affiliated scientific paper traces this convergent discovery and asks: if the same structure keeps emerging, what does that tell us about how we organize knowledge?
https://academia.stackexchange.com/questions/9602/rediscover...
I think I found it in that other world that is the past on Slashdot - which was a Hacker News from another era https://m.slashdot.org/story/144664
Here's the manuscript at any rate, somewhat hard to find on the webpage:
Convergent Discovery of Critical Phenomena Mathematics Across Disciplines: A Cross-Domain Analysis https://arxiv.org/abs/2601.22389
Thanks for pulling out the direct link. I'll change the site to make it more prominent. This is my first serious attempt at social media engagement. Thanks for pointing out flaws and where there's room for improvment.
I view it as nice that you’ve got someone serious who thinks the work is worth posting to arXiv, but the endorsement bar is generally quite low. I’d encourage you to send it to a journal (Didier might be able to recommend an appropriate venue) and really engage with the process and community. I’ve found that process to be extremely valuable (and humbling).
I read that as the paper being reformatted LLM output.
[0] I haven't actually tried this, but I'm pretty sure that even just telling the robot "please write tersely, follow the typical style for HN comments" would make the output less annoying.
Does this apply to that cool chem trick where a solution goes from black to transparent and back again a few times? I don't know enough to know if that's relevant or not, but I remember seeing that and be puzzled about how "sudden" the reaction appears.
What surprised us was how many fields derived this independently. The superheated water intuition you describe maps directly to what ecologists call "critical slowing down" and what financial engineers call "increased autocorrelation near instability." Same math, three different names, minimal cross-citation.
https://youtu.be/itRV2jEtV8Q?si=qm51bvuo-ZIT_Pjk
Veritasium has a good video on how criticality applies to other things too,
Phase transitions and statistical mechanics have a long history in physics. Over time, physicists and applied mathematicians began applying these techniques to other domains under the banner of "complex systems" (see, for example, https://complexsystemstheory.net/murray-gell-mann/).
Rather than independent reinvention, it seems much more likely that these fields adopted existing physics machinery. It wouldn't be the first time authors claimed novelty for applied concepts; if they tried this within physics, they’d be eaten alive. However, in other fields, reviewers might accept these techniques as novel simply because they lack the background in statistical mechanics.
The part of this that could totally be true is that a clinical application somewhere along the way "independently" "reinvented" it. There's a hilarious collection of peer-reviewed journal articles out there inventing a "new" method of calculating the sizes of shapes and areas under the curve. The method involves adding up really small rectangles. (I think a top comment already mentioned the Tai article [2])
[1] source: my doctoral advisor was a really really old theoretical neuroscientist who trained as an electrical engineer and mathematician. If you want a more concrete example, the work of Bard Ermentrout on neural criticality starting in the 70's or 80's. He read a lot of physics textbooks.
[2] https://science.slashdot.org/story/10/12/06/0416250/medical-...
Where I'd push back: even after physicists brought the tools into neuroscience, the receiving field didn't connect it back to the parallel work in ecology or cardiology. Ermentrout's neural work and Goldberger's cardiac work used the same underlying math but didn't cross-cite. The silos reformed around the imported tools.
You're correct that "none of them knew" is too strong. Fair point. "Most of them didn't talk to each other even after import" is closer to what the citation data actually shows.
I'm not sure if you're being entirely serious with that remark, but clearly citing the earlier work would have bolstered their credibility: interdisciplinary research is a plus and hardly something to hide. If it's something that's taught in physics class, you can cite a common textbook.
Imagine if every graphics paper had to cite every concept they use from arithmetic, trigonometry, and linear algebra textbooks...
Some specific cases: Wissel (1984) derived critical slowing down for ecology independently and was ignored for 20 years. The actual import to ecology came via economist Buz Brock, not a physicist. Nolasco & Dahlen (1968) derived period-doubling for cardiac tissue before Feigenbaum's universality result. Jaeger (2001) derived the edge-of-chaos condition for recurrent neural networks without citing Bak, Kauffman, or Langton.
The complex systems movement you reference existed. The paper documents that it didn't actually solve the transfer problem. The cross-citation analysis shows the gaps persisted through the 2000s and 2010s.
You're right that some domains imported rather than reinvented. The paper maps where each transfer was independent, where it was imported, and where it was partial. That's the point — the pattern is messier and more interesting than either "all independent" or "all imported."
Note that phase transitions are 100 years old or so. If someone genuinely does not know statistical mechanics, they still may know a lot of tools derived from it (a famous one - Shannon entropy).
I am not saying it is impossible to independently discover something (it happens all the time), but if discoveries are not (more or less) as the same time, likely there was some knowledge diffusion before.
> You're right that…
> That's the point —
I think this whole operation just completely violates HN rules.
You don't get to claim you invented it, but a lot of progress happens by finding connections between things that are individually well known.
Re-inventing the wheel is completely in order, so long as one makes the wheel more round.
Good math is universal, which means it's probably been discovered millions of times across the universe.
That's not normal diffusion. Those are 30-year gaps for math with direct life-safety applications. The paper asks why, and finds structural explanations in how we organize knowledge.
Consider during the cold war, that the U.S. created fake nuclear designs, then allowed them to fall into the hands of the KGB. The KGB and Russian nuclear engineers then wasted significant time trying to build nuclear devices that failed to work, and could have been dangerous.
The bar scene of "A Beautiful Mind" comes to mind.
Re. the title, I started with a boring conservative title and got precisely zero engagement, so I changed the title to be a bit more clickbaitish. Just like most of the other titles in New. Did I do wrong?
As I said, this is my first serious attempt at social media engagement and I'm just learning how it works.
And hey, I know everyone's doing it, but it's still annoying.
On HN specifically, you're supposed to avoid clickbait, avoid excessive reposts, and avoid using the site only for self-promotion[0]. This helps to create a community that promotes curiosity, instead of chasing growth hacks and engagement like many other social media platforms.
Otherwise, you’ve just described yet another synthetic model that exhibits criticality (without proof no less). Which is not particularly interesting, unless your model subsumes other phenomena.
Generative AI may be just the type of thing to connect these types of previously solved problems across disciplines.
Anyway, none of this is that surprising since deduction takes higher level ideas and tests them on lower level to prove the hypothesis.
If anyone wants to read Karl Popper, this will seem significantly less noteworthy.
Do you think this is something that should be taught generally? In which class would it fit? It feels generally diffeq-ish.
If you've done diffeq and linear algebra you have the prerequisites. Appendix B (page 17 of the paper) is our attempt at making it practical — worked examples rather than proofs. Would be curious if it lands for someone with your background.
We plan to do a follow-up paper that provides a standard format for this math that could be taught across domains. That doesn't belong in this first paper. First priority was to show the pattern and get people thinking about it.
I'll explain how we got to this point. I had previously mentored my friend, Robin Macomber, in math & physics for several years. Robin Macomber independently discovered a variation of criticality math and asked me to evaluate. After due consideration I recognized a pattern: his work echoed that of Kenneth Wilson's renormalization group theory, which I'd previously studied. I then conducted a detailed survey of all academic fields that touched on criticality (using an LLM!) and found, to my great surprise, that this same math had been independently discovered many times in many domains. So I wrote a paper about it.
I thought Taleb won (complex system outcomes, in the sociopolitical realm, cannot be predicted). But then I'm a Taleb fanboy.
Sornette (my first and last exposure to him) came across as a relic from a different age. Pitifully out of touch.
Its almost like the math came first, then the problem later.
You might want to read about induction vs deduction, this is deduction. I don't totally agree with Karl Popper, but at least he can explain why we see this math in multiple places.
