The Work of Digestion: On Terence Tao's ICM 2026 Lecture

When machines can produce verifiable proofs on demand for a few dollars of compute, why spend years writing a six-hundred-page mathematics book by hand?

The Shift from Truth to Values

Late tonight, as I finished adjusting the interior geometry and print signatures for Mathematics for AI and Machine Learning, the page count settled squarely at exactly 600 pages.

During a brief pause, I read Terence Tao's public lecture paper delivered at the 2026 International Congress of Mathematicians: Mathematics in the age of AI (arXiv:2608.16753).

Tao opens his essay with a striking historical comparison:

At the turn of the twentieth century, Russell's paradox and Gödel's incompleteness theorems triggered a thirty-year "crisis in foundations." That turbulence forced mathematicians to interrogate the boundaries of naive set theory, truth, and consistency, ultimately producing the rigorous, formal axiomatic scaffolding that makes modern mathematics verifiable—and mechanizable.

Today, Tao argues, mathematics is entering a period of comparable turbulence.

This time, however, what is being stress-tested is not our definition of logical truth, but rather our implicit values, goals, and culture:

  • What constitutes a meaningful mathematical contribution?
  • What achievements do we actually reward?
  • What does it truly mean for a human mind to "understand" a result?
  • When automated tools can generate lengthy derivations, who—or what—has done the real work?

Rather than joining the debate over how quickly AI systems will master research mathematics, Tao adopts an elegant Working Hypothesis: assume that AI tools will, in the near future, accomplish a significant fraction of research-level mathematical tasks reliably, semi-autonomously, and at low cost.

Conditioning on that premise forces an immediate realization: we are transitioning from an era of proof scarcity into an era of proof abundance.

Proof Abundance and Proof Indigestion

For centuries, mathematical institutions—journals, hiring committees, priority conventions, Clay prizes—were designed around scarcity. Resolving a major open problem required decades of solitary struggle by rare mathematical minds.

Because proofs were scarce, our community adopted a convenient proxy metric: solve as many unsolved problems as possible.

Yet, as Goodhart's law warns: when a measure becomes a target, it ceases to be a good measure.

When generative AI and autoformalization engines (Lean, Rocq, Isabelle) accelerate proof production, the pipeline encounters what Tao calls severe proof indigestion:

  1. Machine-generated derivations accumulate far faster than human experts can referee or verify them.
  2. Even when verified, formal proofs pile up far faster than they can be translated into lucid, explanatory mathematical prose.
  3. The most disquieting failure mode of all: what happens when an AI generates a 500-page verified proof of a landmark theorem, but no human alive—not even the author who prompted the system—understands the intuition well enough to explain it?

Here Tao quotes William Thurston's classic 1994 essay:

"We are not trying to meet some abstract production quota of definitions, theorems and proofs. The measure of our success is whether what we do enables people to understand and think more clearly and effectively about math."

A proof that sits in a machine registry, untouched by human understanding, is logically valid but culturally inert. It cannot fertilize adjacent fields; it cannot be taught to graduate students; it cannot inspire an engineer to build a more resilient system.

The Slowness of Canonicalization

In Tao's breakdown of the problem-solving pipeline, he identifies five distinct stages:

$$\text{Problem Solving} \longrightarrow \text{Verification} \longrightarrow \text{Communication} \longrightarrow \text{Publication} \longrightarrow \text{Digestion} \longrightarrow \text{Canonicalization}$$

Automated systems will drastically accelerate most of this chain. But the final stage—canonicalization—remains uniquely, irreducibly human.

Canonicalization is the process by which a community digests a result: stripping away the ad-hoc scaffolding of its first discovery, recasting it in its natural coordinate-free generality, discovering the right proof rather than the first proof, and organizing it into textbooks that cultivate the next generation.

It is the slowest stage of all. And, as Tao reminds us, it is also the foundation upon which AI itself rests:

"The very success of AI tools in mathematics depends crucially on the canonical theories that human mathematicians have painstakingly built and rebuilt over the centuries: the training data for these tools is, quite literally, the output of the canonicalization process."

Impermanence and "Structure as Memory"

Reading this, something clicked with a resonance that reached far beyond mathematics. Tao's essay was describing the exact same dilemma I had explored in Chapter 12 ("Emptiness") of my memoir, Above The Clouds.

In that chapter, meditating on why art and human struggle exist in the first place, I wrote:

when life has no answer,
when you know you will lose in the end,
you stop.

When rendering that passage into English, the suspended phrase "lose in the end" had held two meanings: one was the sadness of impermanence (无常)—the inevitable loss of time, youth, loved ones, and our own fleeting existence in the cosmos. The other was the quiet despair of defeat—sitting behind the smartest student in the class, or watching an AI system render human labor obsolete.

In 2026, that sense of defeat is no longer just individual psychology; it is the universal human condition. If we measure ourselves by the raw speed of symbol manipulation or the volume of code generated, humans will inevitably lose in the end.

Faced with that inescapable impermanence, how does a human respond?

In Above The Clouds, the answer was simple:

You stop. You make a mark in the dirt. A programmer signs their name in the source code. A monkey leaves behind three extra bananas.
Structure is memory.

Raw bits of information scatter into noise. Ephemeral algorithms are superseded. Ephemeral proofs generated by machines, if ungrounded by human comprehension, are merely shifting sand.

What Tao calls canonicalization and digestion is nothing other than structure as memory applied to the collective human intellect.

Machines can spit out ten thousand lemmas overnight. But until a human mind penetrates them, discerns the underlying geometric symmetries, and knits them into a coherent tapestry, those lemmas are impermanent flotsam. The structures we build—curated, digested, and passed down—are the only lasting memory our civilization possesses.

Why Write Twenty-One Chapters?

Looking back at my screen—at the twenty-one chapters across four coherent parts, and the two hundred MathIcon vector diagrams glowing in the dark—the lingering doubt evaporated.

Over the past few years, friends have occasionally asked me:

"Modern LLMs can write PyTorch and explain backprop on demand. Why spend years deriving linear spaces, matrix calculus, Bellman operators, SDEs, and Fokker–Planck PDEs from scratch? Who is going to read a 600-page mathematical tome in the age of AI?"

Now I can answer them with complete clarity:

An AI tool gives you fragments. Ask it for a gradient or a projection, and it will give you a local derivation. Ask it for a diffusion sampling loop, and it will emit working code.

What it cannot give you is the overarching mental architecture.

It will not show you how Gilbert Strang's four fundamental subspaces naturally give birth to projection operators, which then structure causal self-attention and the block-partitioned efficiency of FlashAttention.
It will not show you why variational inference, policy iteration in reinforcement learning, and continuous optimal control in deep networks are all manifestations of the same underlying information flow.
And it will not give you that sudden aesthetic clarity when you realize that score-based diffusion is nothing more than Langevin dynamics reversing entropy along a continuous probability current in phase space.

That feeling of seeing disparate methods suddenly rhyme is what digestion feels like.

The more prolific automated generators become, the more valuable human integration becomes. Without a structured mathematical core, an engineer drowning in AI-generated papers will suffer perpetual proof indigestion, reduced to tweaking hyperparameters on a black box.

These twenty-one chapters, these six hundred pages, are simply the mark we made in the dirt—our defense of human understanding against the impermanence of the machine age.

Closing

Tonight, I added a quiet tribute to Tao's paper in the book's Preface:

This kind of unification matters more, not less, as AI tools grow capable of generating proofs, derivations, and code on demand. The hard part of mathematical work was never just producing results, but digesting them—turning scattered derivations into forms that a human mind can hold, connect, and build on.

Code will rot. Models will be replaced. Frameworks will be deprecated.
But the mathematical structures that human minds have digested and turned into intuition remain, quiet and luminous, across every generation.