A reflection on understanding —with AI

Milton Lin

Why?

Why do I care?

Because I want to participate in the production of knowledge.

Production of knowledge looks a little different.

A few examples, since 2023

Wang et al., PRL 130 (2023) 244002 · Romera-Paredes et al., Nature 625 (2024) 468 · Hubert et al., Nature 651 (2026) · Novikov et al., arXiv:2506.13131 · DeepMind (21 Jul 2025) · arXiv:2509.14185 · Math Inc (2026) · arXiv:2605.00301 · Anthropic (4 Sep 2026) · github.com/openai/math (Lean catalogue). Hexagon (hexagonmath.org) is different: not a result but a new venue for results, a repository without peer review where depositors declare their AI use and how much the human contributors understand. Hover a node for details.

One click per event, in date order. Solid line: the later work says it builds on the earlier one. Dashed: an earlier case of the same kind, not a stated influence. FLT is not out of nowhere: Anthropic's proof "adapts pieces from the Imperial College London FLT project led by Kevin Buzzard and the flt-regular project", and Math Inc's Gauss had already finished sphere packing (Feb 2026; before that, Strong PNT, Sep 2025, 25,000 lines). The OpenAI release has an earlier case in Erdős #1196: a model-found proof of an open problem. Roots with no parent here (Euler, FunSearch, Erdős, Hexagon) have older ancestors outside the frame: physics-informed neural networks (Karniadakis et al. 2021), program search, the erdosproblems.com list, arXiv itself. The previous version of this slide (who found / checked / understands each result) is appendix A0. Unstable singularities: numerics, not a theorem. Erdős #1196: eight authors rewrote and reviewed; not yet refereed. OpenAI: who understands the 722 papers is not stated. Hexagon: 80 deposits by 7 Oct 2026, text primarily AI in 44.

A very large proof of a theorem we already believed

“Note that mathematically this work of anthropic tells us essentially nothing […] What this work does tell us, however, is what is possible in the field of autoformalization.” K. Buzzard, “FLT: Anthropic has beaten me to it,” Xena blog, 4 Sep 2026

Lines: Buzzard (“over 13.4 million”; Anthropic’s post says 13 million). Mathlib: mathlib stats, 7 Oct 2026 (2.47M lines). Lines measure size, not content.

Has something similar happened in the past?

Has information management been a problem in the past?

Antiquity, 3rd c. BCE – 1st c. CE

Flow diagram. After 331 BCE, accumulation: the Library of Alexandria sought copies of all works ever composed in Greek, about 500,000 papyrus rolls. 4th c. BCE? to 1st c. CE, too many books: Ecclesiastes, of making books there is no end; Seneca, the abundance of books is distraction; felt then by only a narrow elite. Two responses: restrict (Seneca, always read the standard authors) or master the mass (Callimachus's Pinakes, probably the first large-scale reference tool). A dashed arrow returns from the Pinakes to accumulation: the catalog was itself about 120 more rolls.

A torn, light-brown papyrus fragment with eight lines of Greek uncial script and a small square diagram divided into rectangles.

A papyrus roll, ca. 75–125 CE: Euclid, Elements II.5, statement and diagram, no proof.

Two responses to abundance, a canon or tools to master the mass, are already in place.

A. Blair, Too Much to Know: Managing Scholarly Information before the Modern Age (Yale UP, 2010), ch. 1, pp. 14–17; “narrow elite”: Blair, “A Multitude of Books” (2021). Papyrus: P.Oxy. I 29, Penn Museum E 2748, public domain.

Blair: "only once writing spread beyond utilitarian texts can we glean evidence of attitudes toward the process of managing information, and these attitudes were often contradictory" (p. 14). Plato's critique of writing is "an early example of the ambivalence that has often accompanied the adoption of a new technology" (p. 14). Seneca, "distringit librorum multitudo": his rich contemporaries wasted time and money accumulating too many books; his remedy, a canon, "exemplifies an effective and often dominant method of information management—to limit the quantity and nature of information to an established canon of works deemed acceptable, usually on moral grounds" (p. 15). Ecclesiastes is "probably the oldest condemnation of overabundance", date disputed (p. 15). Alexandria: the Ptolemies confiscated manuscripts from incoming ships and returned hasty copies (p. 16). Pinakes: title, incipit and number of lines per work; not a shelf catalog, so finding a book still meant asking the librarian; attributed to Callimachus but "no doubt involved the collective labor of many" (pp. 16–17). Irony: Callimachus coined "a big book is a great evil" (mega biblion, mega kakon). "In Alexandria, far from complaining about overload, scholars seem to have thrived on the challenges of mastering increasing numbers of books" (p. 17). Accumulators later also cited Pliny's Natural History: 20,000 things from 2,000 volumes, with a table of contents as book 1 (pp. 17–18). Diagram: the arrows are logical, not chronological (the Pinakes predates Seneca); the dashed loop is the parallel to printing, where the remedies were themselves more books. Do not use Callimachus's "a big book is a great evil" as irony about the Pinakes: Blair says he likely coined it in defense of short poems (p. 17). "Narrow elite" is from Blair 2021 (Lapham's). Picture: P.Oxy. I 29, a fragment of a papyrus roll of Euclid, Elements II, ca. 75–125 CE, found at Oxyrhynchus by Grenfell and Hunt (1897); Penn Museum E 2748. It gives the statement of II.5 and its diagram, but none of the proof. It survived because it was buried in dry sand, not because anyone recopied it.

Printing, 1450–1700 CE

Flow diagram. 1450s–1600, production explodes: 27,000 editions by 1500, more than 320,000 in the 1500s, over 300 million books versus 6–7 million manuscripts available in 1500. After c. 1550, information overload: too many books for an individual to read and to master. 1500s–late 1600s, new tools for reading: note-taking advice, reviews in learned periodicals, bibliographies, reference works. A dashed arrow returns from the tools to production: the remedies were printed books too.

Engraving of a tall cabinet with its door open: rows of vertical bars with hooks holding paper slips, lettered headings along the top, drawers below.

A new tool: the note closet, slips on hooks under 3,000+ headings (Placcius, 1689).

6 Oct 2026: 722 AI papers in one release → open problems “harvested … in an unsustainable fashion” (Tao) → new tools for understanding?

A. Blair, “A Multitude of Books,” Lapham’s Quarterly, 27 Jan 2021 (laphamsquarterly.org); excerpted from Blair, Duguid, Goeing & Grafton (eds.), Information: A Historical Companion (Princeton UP, 2021). “after c. 1550” = Blair’s “after the first century of printing”. Closet: Blair, Too Much to Know (2010), fig. 2.1.

Verbatim (Blair): "at least 27,000 editions of incunabula were printed by 1500, followed by more than 320,000 editions in the sixteenth century. The number of printed books that resulted (likely more than three hundred million, excluding the many ephemera that have been completely lost) dwarfed not only the estimated number of European manuscripts surviving from the sixth to the sixteenth century (1.3 million) but also the number of manuscripts estimated to have been available in 1500 (six to seven million)." Overload: "While only a narrow elite of scholars had experienced it in earlier periods, after the first century of printing a broad cross section of the educated articulated a concern about what we would call 'information overload'—the sense that there were too many books for an individual to read and to master." Complaints included "railing about the base commercial motives of printers who produced whatever sold without a care for quality or intellectual merit" and "worrying about the end of civilization from too many people writing (bad) books and no one reading good ones anymore". Remedies: "advice on how to read well, for example by taking good notes; judgments and reviews of books to aid in selecting them (a prime content of learned periodicals starting in the late seventeenth century); bibliographies to identify existing books ...; and reference works designed to collect the best parts of the best books". The loop: "These complaints about the consequences of printing were of course made in printed books, which added to the abundance of which they complained". Complaints predate print (Ecclesiastes 12:12; Vincent of Beauvais, 13th c.), but became widespread only after it. Parallels: production ↔ the OpenAI release and Hexagon (slide 4); overload ↔ Tao, 6 Oct 2026, "harvested at large scale in an unsustainable fashion"; tools ↔ blueprints, interactive explainers, verification (Act III). The analogy is mine, not Blair's. Box dates for tools are approximate; only the periodicals date is Blair's. Picture: Blair, Too Much to Know (2010), fig. 2.1, p. 95: Vincent Placcius, De arte excerpendi (1689), tab. IV, after a manuscript by Thomas Harrison (1640s); Houghton Library, Harvard, *GC6.P6904.689d. Caption: "Notes taken on slips of paper were stored on the hooks attached to the metal bars; each hook was associated with a topical heading inscribed on the front of the bar. The closet could accommodate 3,000 to 3,300 headings. At least two such closets were built—Placcius and Leibniz each owned one—but no examples survive today."

A reflection on understanding —in the modern society

understanding involves participating

A first form of participation: history

SKILL.md · § 2. history.html

“Required, read first after the hub: it explains why the target is shaped the way it is. A timeline with as many events as the story needs, in order. Each event: the obstacle, the advance, what stayed open, its key examples, the pictures people used, sources (linked to references.html), and links to the pages that develop it. Later pages link back to the event where their idea first arose […]”

.agents/skills/explain-math-html/SKILL.md, lines 81–89 (this repository)

History page of the orbit-closures explainer: a timeline graph from Satake 1963 through Serre's trees, geometric Satake, Gaitsgory–Nadler, Sakellaridis–Venkatesh norm balls and Chen–Yi orbits to the current draft

“Understanding is to be thought of less as a subjective act than as participating in an event of tradition, a process of transmission in which past and present are constantly mediated.” H.-G. Gadamer, Truth and Method (1960), 2nd rev. ed., trans. J. Weinsheimer & D. G. Marshall (Sheed & Ward, 1989), p. 290

Screenshot: history page of the explainer Hecke translates of orbit closures (this site), built with the explain-math-html skill. German: “Einrücken in ein Überlieferungsgeschehen”, Wahrheit und Methode, Part II.

Participation comes in degrees. The first degree is joining a community that exists across time: the people who posed the problem, were blocked, and reshaped it (Satake 1963 → Serre's trees → geometric Satake → Gaitsgory–Nadler → Sakellaridis–Venkatesh → Chen–Yi → this draft). The skill puts this page first after the hub, before any background or proof (the excerpt is verbatim). Gadamer's point: understanding is not a private act of the reader but taking one's place in a tradition, which is exactly this slide's sense of participation; the sentence continues, "This is what must be validated by hermeneutic theory, which is far too dominated by the idea of a procedure, a method." The same idea for teaching mathematics: Poincaré, "La logique et l'intuition dans la science mathématique et dans l'enseignement," L'Enseignement mathématique 1 (1899): "La tâche de l'éducateur est de faire repasser l'esprit de l'enfant par où a passé celui de ses pères, en passant rapidement par certaines étapes mais en n'en supprimant aucune. À ce compte, l'histoire de la science doit être notre guide." (wording checked only against Wikiquote, not the 1899 original). Also Lave and Wenger's legitimate peripheral participation, with the historical literature as the community. The skill makes this cheap to build, but every reference is checked against its source, so the history is not hallucinated.

Microworlds: a place where mathematics answers back

Photo: Matematicamente.it, CC BY-SA 3.0, via Wikimedia Commons

Understanding mathematics in a “Mathland”: a world where mathematical actions have consequences you can explore. Adapted from S. Papert, Mindstorms (Basic Books, 1980), p. 6

Turtle: REPEAT 36 [FORWARD 10 RIGHT 10] → a circle you can debug by walking it.

Understanding through constructing a public object: the constructionist perspective — Papert & Harel, “Situating constructionism” (1991).

What is a \(K\)-orbit?

K=\mathrm{GL}_n(\mathcal O)\curvearrowright F^n,\qquad K\cdot v=\{kv:k\in K\}

Why do we want to know the orbits?

f(kv)=f(v)\quad\Longrightarrow\quad\text{one value of }f\text{ per orbit}

Here the nonzero orbits are shells

X_m=\varpi^m\mathcal O^n\setminus\varpi^{m+1}\mathcal O^n,\qquad a_m:=f(v)\ (v\in X_m)

\varphi_0=\mathbf1_{\mathcal O^n},\qquad \varphi_0|_{X_m}=\begin{cases}0&m<0,\\1&m\ge0.\end{cases}

\(F\): non-archimedean local field; \(\mathcal O\): its integers; \(\varpi\): a uniformizer; \(\mathbf1_A\): indicator of \(A\). We study \(C_K=C_c^\infty(F^n)^K\), the locally constant, compactly supported \(K\)-invariant functions.

What can we learn by moving a point?

\(F^2\), \(K=\mathrm{GL}_2(\mathcal O)\); \(I\subset K\): upper triangular modulo \(\varpi\); \(t=\varpi\). Widget from the basic-function geometry page. ★ denotes the zero vector.

Part I · Recording the shell values

\(y\) records a shift of one shell

Take \(y\) as a formal variable and store the values of \(f\) as coefficients:

f\longmapsto F_f(y):=\sum_{m\in\mathbb Z}a_m y^m

Where \(F_f\) lives

F_f(y)\in\frac1{1-y}\,\mathbb C[y^{\pm1}]\subset\mathbb C((y))

\(\mathbb C((y))\): formal Laurent series with only finitely many negative powers.

a_m=0\ (m\ll0),\qquad a_m=f(0)\ (m\gg0)

The basic function gives a geometric series

F_{\varphi_0}(y)=\sum_{m=0}^{\infty}y^m=\frac1{1-y}

As an analytic series, \(F_f(y)\) converges on \(0<|y|<1\).

(Tf)(v):=f(\varpi^{-1}v)\qquad\Longrightarrow\qquad F_{Tf}(y)=yF_f(y)

C_K=\operatorname{span}_{\mathbb C}\{T^m\varphi_0:m\in\mathbb Z\},\qquad T^m\varphi_0=\mathbf1_{\varpi^m\mathcal O^n}

Compact support gives the zero tail as \(m\to-\infty\), local constancy at \(0\) gives the constant tail as \(m\to+\infty\). Details: the shell model.

The field F and the transform F_f are different objects: F is the local field, while F_f is a complex generating series. Initially y is a formal indeterminate. The target (1-y)^{-1} C[y^{±1}] is a vector space of rational functions, embedded in formal Laurent series by expansion at y = 0. It is not being asserted to be a subring. To verify the target, choose integers L and M with a_m = 0 for m < L and a_m = c = f(0) for m ≥ M. Then F_f(y) = sum_{L ≤ m < M} a_m y^m + c y^M/(1-y). Conversely, every P(y)/(1-y), with P a Laurent polynomial, is a finite linear combination of the transforms y^m/(1-y) of lattice indicators. Thus this is the exact image. The formal series also converges at every complex y with 0 < |y| < 1. Its rational continuation can have a simple pole at 1, whose tail coefficient is f(0): lim_{y→1}(1-y)F_f(y) = f(0). The shift T sends the coefficient a_m to a_{m-1}. With vol(K) = 1, T is the central Hecke operator 1_{varpi K}, so the displayed span also gives spherical generation by the basic function. An alternative coordinate map is Phi(f) = (1-y)F_f(y), taking values in C[y^{±1}] and sending varphi_0 to 1. This explains why the lattice and shell descriptions give different images of the same basic function. No L-function interpretation is needed for either description.

Where does this appear in the literature?

Tate’s thesis (1950; published 1967): the case \(n=1\)

\(\chi:F^\times\to\mathbb C^\times\) unramified, \(q=|\mathcal O/\varpi\mathcal O|\), \(\operatorname{vol}(\mathcal O^\times)=1\).

F^\times=\bigsqcup_{m\in\mathbb Z}\varpi^m\mathcal O^\times,\qquad y=\chi(\varpi)q^{-s}

\int_{F^\times}\mathbf1_{\mathcal O}(x)\chi(x)|x|^s\,d^\times x =\sum_{m\ge0}y^m=\frac1{1-y}=L(s,\chi)

The orbit decomposition turns the integral into the shell sum, initially for \(|y|<1\).

Sakellaridis (2012), §1.3 and §3.3.3: basic functions on spherical varieties

For smooth \(X\), the basic function is \(\mathbf1_{X(\mathcal O)}\); Tate’s \(X=\mathbb A^1\) is the simplest example.

The broader continuation conjecture is Conjecture 3.2.2.

J. Tate, “Fourier analysis in number fields and Hecke’s zeta-functions,” in Algebraic Number Theory (1967), pp. 305–347. Y. Sakellaridis, “Spherical varieties and integral representations of L-functions”, Algebra & Number Theory 6 (2012), 611–667, §1.3, §3.3.3.

The displayed calculation is Tate's unramified local zeta integral, not a new transform asserted without a reference. The multiplicative Haar measure gives every shell volume 1; chi is constant on a shell because it is unramified, and |varpi| = q^{-1}. Taking the indicator of O keeps exactly m ≥ 0. The generating series from the preceding slide becomes this local L-factor only after setting y = chi(varpi)q^{-s}. For n > 1 the preceding shell generating series alone is not a claim about an arbitrary automorphic L-function. Sakellaridis, §1.3, identifies the smooth-X basic function with the indicator of integral points and explicitly relates this to Tate and Godement–Jacquet. Section 3.3.3 gives the geometric basic-function construction; the conjectural Schwartz-space and continuation framework is Conjecture 3.2.2. Source checked: author's paper, https://arxiv.org/html/0905.4245, §§1.3, 3.2, 3.3.3. This separates the proved rank-one calculation from the general conjectural framework.

A third participation: collaborative participation?

Level 3 · contribute: pose the next question; judge what is interesting. This takes other people.

Table 1 · Two forms of mathematical collaboration.
	who took part	who checked	what machines did
Polymath1 2009 · Gowers’s blog	27 people, ≈ 800 comments in 37 days	participants, then referees (Annals 2012)	blog and wiki only
Equational Theories Project Sep 2024 – Apr 2025 · Tao et al.	over fifty contributors, “diverse academic and professional backgrounds”	Lean: all 22 028 942 implications	automated provers: the bulk; LLMs: little; ≈ two dozen cases by hand

T. Gowers & M. Nielsen, “Massively collaborative mathematics,” Nature 461 (2009) 879–881, doi:10.1038/461879a; D. H. J. Polymath, Ann. of Math. 175 (2012) 1283–1327. M. Bolan et al., “The Equational Theories Project,” arXiv:2512.07087 (2025), pp. 2, 7, 12.

Polymath1 (from 27 Jan 2009) found an elementary proof of the density Hales–Jewett theorem for k = 3, which generalized to all k; the record of the discussion is public, so one can see how the result was found. ETP: launched Sep 2024, primary goal reached 14 Apr 2025 (p. 2); "only 10 657 (0.13%) of the positive implications needed a direct proof" (p. 6); "the vast majority of negative implications could be quickly resolved ... either with human input or in a completely automated fashion", but about two dozen needed "quite delicate and sui generis constructions" (p. 7); the hardest, E1729 ⊭ E817, took several months. One LLM success: ChatGPT guessed a complete rewriting system for E3523 (p. 57). Conclusions (p. 64): the project "largely moved away from the use of such LLMs". Can an AI be a fellow participant, or only a tool inside a human community? Hexagon's answer (slide 4): "Artificial-intelligence systems cannot be contributors." Lean blueprints (FLT, sphere packing) are a third form: a dependency graph that lets strangers claim one lemma.

A question stays attached to its passage

Scroll, open a proof or source, or highlight a passage to comment. Open the full page ↗ · Comment dashboard ↗

What do the comments tell us about a reader?

The relevant part of Toan’s personalization profile.

Repeated questions reveal where the explanation needs more detail.

For Toan: show where each hypothesis enters, with examples and counterexamples.

An inferred preference, open to correction by the reader.

Screenshot: User preferences · Toan, 21 October 2026. Personalization plan: retain requests and their evidence across changing proposition statements.

The review returns to the reader

Gmail screenshot of Milton’s sent email to Toan, Trees update: Five questions on strata and generation, linking the review and summarizing its five answers. — Sent to Toan, 21 October 2026 at 11:46 CDT: the review link and its five questions.

A reader’s question leads to a review, then a concrete invitation to continue the discussion.

Actual sent message, “Trees update: Five questions on strata and generation.” The linked review contains the proof breakdowns, examples and sources. Click the image for the full-resolution PNG.

A reflection on understanding —did all that help?

1. I showed you the tools. Did they help me understand better?

2. We benchmark AI all the time;
we should benchmark human–AI interaction.

3. I should have shown you the process of how this all came to be.
How do I demonstrate this?

A reflection on understanding —how would you want to understand?

How would you want to participate?

result	who found the proof?	who checked it?	who understands it?	what is new?
FLT in Lean Sep 2026	Claude agents, 11 days, following the 1990s human proof	Lean kernel; build checked by Buzzard	experts, since 1995; the 13M lines, no one	nothing mathematical: a known theorem, machine-checked
Erdős #1196 May 2026	first proof from a GPT-5.4 Pro run	humans eight authors rewrote and reviewed; not yet refereed	the eight authors	a new theorem
Unstable singularities 2025	no proof: neural solvers designed by the authors	numerics proofs “in preparation”	the authors	evidence, not a theorem
OpenAI `math` 6 Oct 2026 · 722 papers	an unreleased OpenAI model, ~4,000 problems posed; selection by significance (process not described)	Lean main result for 162/722, statements “unchecked”; the rest none yet	not stated	claimed new: conjectures, bounds, counterexamples
Hexagon since Sep 2026 · 80 deposits	declared per deposit: text primarily AI in 44/80, mixed 27, human 9	none no peer review; acceptance “not a judgment about its correctness”	declared: all 31 · some 39 · none 9	not judged: depositors vouch “in good faith”

compare	two methods side by side, with prompts to compare them and judge which is more efficient
sequential	the same methods one at a time, reflecting on each

gain at posttest	compare vs. sequential
procedural knowledge	greater
procedural flexibility	greater
conceptual knowledge	comparable

A reflection on understanding —with AI

A few examples, since 2023

A very large proof of a theorem we already believed

Antiquity, 3rd c. BCE – 1st c. CE

Printing, 1450–1700 CE

A reflection on understanding —in the modern society

A first form of participation: history

Microworlds: a place where mathematics answers back

What is a \(K\)-orbit?

What can we learn by moving a point?

(a) \(K\)-orbits

(b) \(I\)-orbits

\(y\) records a shift of one shell

Where does this appear in the literature?

A third participation: collaborative participation?

A question stays attached to its passage

What do the comments tell us about a reader?

The review returns to the reader

A reflection on understanding —did all that help?

A reflection on understanding —how would you want to understand?

Participating in the production of knowledge

interprets

poses, checks, judges, explains

theorems, proofs, examples

Few Examples

Compare two cases, or read them one at a time?

Same problem, two methods: side by side or one at a time