Theories, International Relations, AI, and Paleolithic Lights
What is a Theory?*
*To skip to Paleolithic lights, scroll all the way to the bottom.
Why do I keep buying books? And worse, why do I keep subjecting people to blog posts about them?
The good news is nobody reads this blog. As for the book: I’m blaming a student. She’s taking an international relations course and mentioned Kenneth Waltz’s Theory of International Politics multiple times — as one does — so I had to buy it. The book
…is arguably the most influential book in international relations, causing a fundamental discursive transformation and bringing the concept of anarchy to the forefront. It is the most assigned book in International Relations graduate training at U.S. universities. (Wikipedia)
The title of the book got me thinking about the word “theory,” which is very loaded, but also the word theorem, which is strictly mathematical. “Theorem” also spins a web of complication for me now that AI can fact-check proofs and ostensibly prove incredibly difficult theorems — theorems that have left world-class mathematicians scratching their heads for years.
So right. About international relations. I think what Waltz is saying is that you can’t look at a bunch of events, make connections between them, and go “this is a theory.” Because what you have is a description of the world, not an explanation of it, and this is the “inductivist illusion”:
Today’s students of politics nevertheless display a strong commitment to induction. They examine numerous cases with the hope that connections and patterns will emerge and that those connections and patterns will represent the frequently mentioned “reality that is out there.” (Waltz 1979)
In Aristotelian physics — Waltz talks about this in a book on international relations; how great — you push a cart and see that it moves. You push the cart harder and see that it moves more. Now your brain is percolating on a correlation between pushing and moving but this is misleading. It’s a description of the world, not an explanation of it.
…a paradigm such as Aristotelian physics may provide a useful system to frame known reality, but cannot be used to plan different realities, since it lacks a rigorously deductive structure. (Russo 2000)
So we see in a second, wildly unrelated source that the Aristotelian way of doing things lacks heft. Lucio Russo’s definition of scientific theory — the rigorous one he uses to examine whether there was science in Hellenistic Greece — says that scientific theories meet these criteria:
Their statements are not about concrete objects, but about specific theoretical entities.
The theory has a rigorously deductive structure.
Applications to the real world are based on correspondence rules between the entities of the theory and concrete objects.
I see how point 1 can map to what Waltz is saying about observing events in the real world and wrongly creating a theory based on associations.
I’m really curious to see how he develops his theory of international relations. Will he give us a truly scientific way to look at political things happening in the world? What is truly scientific, anyway?
Aside from theories, I wonder if he will mention paradigms. Are there political paradigms? Will the word “revolution” cross over into international crisis-contexts?
Why should a change in paradigm be called a revolution? In the face of the vast and essential differences between political and scientific development, what parallelism can justify that metaphor that finds revolutions in both? (Kuhn 1962)
Thomas Kuhn has a very famous book called The Structure of Scientific Revolutions. He posits that normal science is the day-to-day work of scientists operating under a paradigm. Once a paradigm exists, a field of study exists. A scientific revolution is a paradigm shift.
Paradigms gain their status because they are more successful than their competitors in solving a few problems that the group of practitioners has come to recognize as acute.
To be more successful is not, however, to be either completely successful with a single problem or notably successful with any large number. The success of a paradigm — whether Aristotle’s analysis of motion, Ptolemy’s computations of planetary position, Lavoisier’s application of the balance, or Maxwell’s mathematization of the electromagnetic field — is at the start largely a promise of success discoverable in selected and still incomplete examples. (Kuhn 1962)
We see Aristotle a third time as a bellwether of what we’re calling science, or a theory, or a scientific theory.
Aristotle is also mentioned many times in Pierre Duhem’s The Aim and Structure of Physical Theory but this book is so dense, truly, that I can’t get make sense of anything past the foreword. (Which was written by Nobel Prize-winning phycisist Louis de Broglie no less! Also, I have a comment in my notes that De Broglie had a “deep antipathy” toward pictorial models and…I need to find out why I wrote that.)
The Can of Worms/Math AI Stuff
Hot off the press:
There was a talk posted today (recorded yesterday) by the Museum of Mathematics called When AI Solves Major Open Problems: What Does It Mean for the Future of Mathematics—and Beyond? The panel is moderated by Fields Medalist Manjul Bhargava and includes Stephen Wolfram! October 6, 2026
See New York Times piece OpenAI Releases Findings on 377 Math Problems, Further Roiling Field. October 6, 2026.
Oh right, math! The subject this whole enterprise is about! I have a lot to add here about math and AI. A lot. I don’t think this section is anywhere near done. So you might as well skip to Paleolithic lights.
Unlike a theory, which seems to have multiple definitions, mathematical theorems are unshakably true, built brick by brick on the buttress of axioms. Which means a mathematical theorem is an objective truth, right?
I love Levain cookies.
This is not an axiom, but maybe it should be. It’s 8 in the morning and I’m knee-deep in this stuff and I need a cookie break.
I started thinking about this (like all of this) while reading “Why Mathematical Proof is a Social Compact” in Quanta Magazine:
Then there’s this notion of objectivity — of being sure that what is claimed is right, of feeling like you have an ultimate truth. But how can we know we’re being objective?
One can also ask what is objectively interesting or important in mathematics. But this is also clearly subjective. Why do we consider Shakespeare to be a good writer? Shakespeare wasn’t as popular in his own time as he is today. There are obviously social conventions around what’s interesting, what’s important. And that depends on the current paradigm. (Granville 2023)
Granville’s actual paper is here and I think it’s worth a read. In the 2023 Quanta interview he says:
Perhaps it could assist in creating a proof. Maybe in five years’ time, I’ll be saying to an AI model like ChatGPT, “I’m pretty sure I’ve seen this somewhere. Would you check it out?” And it’ll come back with a similar statement that’s correct. (Granville 2023)
Which is totally valid. It is wild to me that, in 2026, AI not only checks proofs but has purportedly solved a Millennium Problem all on its own — a problem so hard it came with a $1 million prize. As of yesterday, it solved 377 more. Which means I literally can’t stay on top of this fast enough.
For more math and AI, you can browse some of the files on the website for the 2025 Mechanization and Mathematical Research workshop in Leiden. I mention this Leiden workshop because famous mathematicians attended and gave talks. One of them is Fields Medalist Akshay Venkatash, whose talk “What do we tell our students about AI?” is on YouTube! Tim Gowers, another Fields Medalist, wrote a whole blog post about his trip to Leiden.
I will also throw Fields Medalist Terence Tao’s keynote at the July 2026 ICM here. It’s called “Mathematics in the Age of AI” and here are his slides. Giving the keynote at this conference is one of the highest honors in mathematics.
This may seem like I am dumping the work on you, dear reader, and maybe I am. You might have to go through Fields Medalist-palooza and sort this all out for yourself. At least until I semi-sort it out myself. I’m sorry. When I do sort it out, I’m writing a blog post called I’ve Sorted It Out.
If you continue down this road, you should know something about the Leiden Declaration on Artificial Intelligence and Mathematics. The Leiden Decleration is explicitly mentioned in OpenAI’s August 2026 post Ten advances in mathematics and theoretical computer science:
There are many views as to the role of AI in mathematics, and we have deep respect and understanding for those concerned with its impact, including the signers of the Leiden declaration on AI and Mathematics (OpenAI 2026)
OpenAI also has a 253-page paper elaborating on the ten advances and a more feasible 62-page paper on How the Ideas Came Together.
You should also know about Lean, a programming language that is a proof assistant or proof checker.
Now Shakespeare. The critic Northrop Frye — one of the most important and influential literary theorists of the twentieth century — argues that social conventions are often wrong about good art. Therein lies the job of the critic. In fact Frye has a whole theory of criticism and even uses the words “axioms and postulates of criticism.” Isn’t that wild? Anyway:
Whatever popularity Shakespeare and Keats have now is equally the result of the publicity of criticism. A public that tries to do without criticism, and asswerts that it knows what it wants or likes, brutalizes the arts and loses its cultural memory. (Frye 1957)
Should mathematicians play the role of Northrop Frye? Of Roberta Smith?
I forgot to talk about First Proof!
There is a cool mathematical project called First Proof. First Proof is the co-brainchild of Lauren Williams, who inspired my September Fountain Problem on the positive Grassmannian, which has nothing whatsoever to do with AI. She is incredibly original and all over the place. Which is why she won a MacArthur Genius Grant.
For the First Proof project, mathematicians crowdsource 10 interesting problems, solve them, don’t publish their results, and then sit back and see how OpenAI does. You can watch the webinars where the editors of First Proof (including Lauren Williams!) candidly discuss the results. For example, Nikhil Srivastava explains that OpenAI wrote a proof that was correct but did it using polar derivatives. Why?
This is the process that openAI used and the product of this process was an 8-page proof which was correct. It's very interesting because it's different from the human proof. And I want to highlight three differences. One is there's an important step in the proof where it uses something called a polar derivative which is something that goes back to Laguerre....It eventually yields this result in a way that's kind of magical. It wasn't clear what the overarching narrative of the proof was, but the proof works. (Srivastava 2026)
There’s also a First Proof talk at the Museum of Mathematics too moderated by Fields Medalist Manjul Bharghava. It’s dated March 14, 2026 which already feels outdated!
On First Proof, mathematician Daniel Litt writes:
What constitutes a solution? The authors write that "we consider that an AI model has answered one of our questions if it can produce in an autonomous way a proof that conforms to the levels of rigor and scholarship prevailing in the mathematics literature. In particular, the AI should not rely on human input for any mathematical idea or content, or to help it isolate the core of the problem."
It is by necessity a messy project. The problems contained in the initial batch cover a wide range of fields and difficulty levels; the problems, comments, and official solutions can be found here. Of the 10 problems, two of them (problems 9 and 10) could be solved by publicly available models "out of the box." The authors were aware of this; the lemmas were not chosen adversarially. In fact problem 10 had an answer that was more or less directly available in the literature, and problem 9 was (in my view) a pretty minor variant of previous work of one of the authors. A sketch of a solution to Problem 1 was available online, with many details omitted. (Litt 2026)
Litt gave a talk called “Working with LLMs to do high quality math” on September 9, 2026 at Harvard and I attended that talk on Zoom. He was fantastic. I took many notes and ate several cookies. You can watch the whole recording here.
I am not even close to done with this topic but I need a break.
Paleolithic Lights
Did you know there is a scientist out there studying how lighting worked in Paleolithic caves? I like ancient things like fossils and dinosaurs, and flip through Architectural Digest to gawk at the homes, but I never thought to put two and two together.
I am also a big fan of lighting. It makes or breaks a room. Or cave.
The project is called The “Archeology of the Light”: A multiproxy, interdisciplinary and experimental approach to Paleolithic subterranean activities.
I am already obsessed with this, and might have to email Ángeles Medina-Alcaide and ask her how this whole project came to light, heh:
The "Archeology of the Light" (A-LIGHT) project aims to improve our knowledge of Palaeolithic cave activities through an interdisciplinary methodology applied to rarely-studied remains: the residues of Palaeolithic light from lamps, fireplaces and torches (specially, charcoal and soot). (Medina-Alcaide 2024)
The author picked six European caves (including Altxerri cave, which is a UNESCO World Heritage site) and, I think, replicated plausible lighting scenarios based on the soot and charcoal inside.
I’m most intrigued by the fireplaces. One of my dreams, if you must know, is to have a New York apartment with an old-timey fireplace. I will probably fill it with books, or a litter box, but I still want one.
For fireplaces, the wood fuel used was thin branches of juniper and oak wood in a dry state, arranged in a tepee-shaped structure. Birch bark was used to start the fire. This was placed inside the combustion structure. The fireplace was 23 cm in diameter and 7 cm high (measured before lighting), had no boundary structure, and was lit on a clay substrate. This experiment was carried out at a distance of 80 cm from the closest wall and 1.60 m from the ceiling in order to better assess the reflection of the light. (Medina-Alcaide 2024)
In 2018 she studied the broader activity inside Paleolithic caves and maybe this is what got her interested in the whole lighting situation.
The inner zones of caves are those areas unreached by sunlight that remain in complete darkness and require artificial light if humans are to occupy them. They are characterized by a high degree of humidity and scarcely varying annual temperature. In general, such areas are inimical to prolonged and stable human settlement in comparison to areas closer to cave entrances. The latter have been used as places of more prolonged occupation, where many different activities were carried out (permanent and sporadic settlement, hunting refuges, etc.). ( (Medina-Alcaide et. al. 2018)
Duhem, Pierre. The Aim and Structure of Physical Theory. Princeton University Press (1954)
Frye, Northrop. Anatomy of Criticism. Princeton University Press (1957)
Kuhn, Thomas. The Structure of Scientific Revolutions. The University of Chicago Press. (1962)
Medina-Alcaide, Ángeles. “The “Archeology of the Light”: A multiproxy, interdisciplinary and experimental approach to Paleolithic subterranean activities.” https://open-research-europe.ec.europa.eu/articles/4-216 Note that this article had reservations and notes in the peer-review.
Russo, Lucio. The Forgotten Revolution: How Science Was Born in 300 BC and Why it Had to Be Reborn. Springer (2000)
Srivastava, Nikhil. “Dan Freed | First Proof Introduction” 28:08. https://www.youtube.com/watch?v=fNrR4lTiScQ Uploaded June 4, 2026.