Other Planets of Intelligence: What AI in Mathematics Indicates for Other Fields

August 5, 2026

marcel blattner | august 26

Introduction

In March 2026, the mathematician Terence Tao and Tanya Klowden, whose background is in the study of art, posted a manuscript titled "Mathematical Methods and Human Thought in the Age of AI". The essay examines what AI does to human thought, using mathematics as its main case.

The authors present mathematics as a sandbox: a field whose results can be verified objectively, allowing experimentation with AI at low stakes, with lessons that may transfer to other domains. This post takes the framing one step further. Mathematics is the discipline with the strongest verification machinery available to any field. The problems the essay documents there occur under the best possible conditions. The same problems can be expected in other fields in stronger form, because the protections are weaker. The sections below take the essay's main findings and state their general versions.

The Certificate Problem

For most of history, a finished artifact certified an invisible process. A proof indicated that its author understood the problem. A legal brief indicated hours of reasoning about precedent. A working codebase indicated that a programmer understood the system. The process behind an artifact was rarely audited, because the artifact could not exist without it.

The essay's diagnosis for mathematics is that this connection has broken. Frontier models produce proofs that can be verified without reproducing the reasoning practices that were previously required to find them. The form of the output has become decoupled from the thought process behind it. This condition is not specific to mathematics. It now applies to every field whose products consist of text, images, code, or structured argument.

Professional judgment fails in a corresponding way. The authors describe the "smell" of a mathematical argument: the impression of credibility an experienced mathematician forms before checking any individual step, a metaphor taken from "code smell" in software engineering. Comparable heuristics exist in most professions: editors develop an ear for invented quotes, clinicians for records that do not fit the patient, engineers for designs without visible trade-offs. These heuristics were calibrated on human failure, which has a consistent signature: humans who lack understanding produce work that looks wrong. AI systems that lack understanding produce work that looks right. The authors call the result in mathematics "odorless proofs": formally correct and lacking insight. The corresponding artifacts in other fields are briefs, diagnoses, and reports that read as competent and carry no underlying reasoning.

The Verification Gradient

Mathematics is better positioned for this development than any other field, partly because of tools such as LEAN, a proof assistant that certifies correctness without trusting the author, and partly because of its culture of consensus. The essay recounts an example. In 2011, the logician Edward Nelson claimed that the axioms of arithmetic are inconsistent. A flaw in the argument was identified, Nelson accepted it, and the claim was withdrawn. Few disciplines resolve disputes this way.

Other fields can be ordered by the same criterion: how independently their outputs can be checked. Software has compilers and test suites, which verify what was specified rather than what was intended. Empirical science has replication, which is slow, expensive, and rare. Law has adversarial review, which corrects errors at high cost and after the fact. Medicine has patient outcomes, which arrive late, contain noise, and are difficult to attribute. Journalism, scholarship in the humanities, and the arts are verified by community attention, the resource that machine-generated volume consumes.

The essay's practical rule, taken from cybersecurity, is to keep AI on the "red team": reviewing, testing, and critiquing human work. Using it on the "blue team", to generate content, is safe only up to what the field's verification methods can check. Stated generally: the amount of AI generation a field can absorb is limited by the strength of its verifiers. In mathematics, that limit is high. In journalism, it is close to zero.

Two points in the essay show that even the strongest verification has gaps. The first is a thought experiment. An AI asked about Fermat's last theorem that assumes the natural numbers include zero can produce a formally certified "disproof" of the theorem. The verifier certifies the formal statement, not the intended one. Outside mathematics, this gap appears as the metric, the benchmark, or the test suite: every verifier checks a proxy, and AI systems optimize proxies systematically. The second point is an event the essay reports. Tao helped organize an effort to use automated literature-search tools to document known results on a catalogue of open problems. The reports were accurate and useful. The same class of tools then began citing these reports as authoritative sources, and subsequent searches stopped producing new findings. Randall Munroe named this loop "citogenesis" in an xkcd comic. It formed in the field with the strongest verification infrastructure; in fields with weaker verification, it can be expected to form faster.

The Artifact Was the Exercise

The essay builds on an observation by William Thurston: a good proof does not only establish a claim, it creates understanding in its readers and in its author. The general version of this observation is that in most professions, the artifact was the exercise.

A student essay does not exist to inform the teacher. The first-year associate's brief, the medical resident's notes, and the junior developer's pull request served two functions at once: they were deliverables, and they were training. Because both functions were carried by the same object, training did not have to be financed separately. AI separates the two functions. The deliverable can now be produced without the development.

The essay raises two concerns that follow from this: students who complete coursework with AI acquire credentials without skills, and entry-level positions are eliminated by automation. These are the same process at two career stages. AI takes over the tasks that were simple enough to learn from and valuable enough to be paid for, and those tasks constituted the training path of most professions. The essay refers to the Luddites in this context: the Nottingham textile workers of the early nineteenth century opposed automation because it removed their livelihoods, not because they opposed machinery as such. Organizations that require experts in fifteen years will need to finance training deliberately, where it was previously a by-product of production.

Recommendations for Use

For current use, the authors propose a culinary comparison. Vanilla extract is unpleasant on its own, improves most desserts in small amounts, and ruins a dish above an upper limit that is not precisely known. (A footnote cites a Tumblr thought experiment: a cake consisting of 44% vanilla extract is inedible.) The corresponding practice is to use AI for limited passes such as grammar and structure, and not as the main component of the work.

For the longer term, the essay points to chess. Engines have outplayed the best human players for decades, and chess has not declined. Players train with engines, analyze with them, and have developed new forms of competition around them. The question of what the game is for did not disappear; it received new answers. This serves as an existence proof that a field can be surpassed at its own product and continue as a human practice.

A Final Perspective

The essay closes with an analogy from astronomy. Over several centuries, the Earth lost its assumed position at the center of the universe: first shown to orbit the Sun, then located in one galaxy among many, in a universe without a spatial center. This did not reduce human attachment to the planet. Klowden and Tao propose the same shift for intelligence. Human cognition is losing its assumed position at the center, because other forms of intelligence are being discovered or built, comparable in some respects and different in others. Both kinds can be placed in the same ontological category. Attachment to the human sphere remains justified, and an external perspective on human cognition, previously unavailable, becomes possible. This is consistent with the argument in an earlier post on this blog: complementary forms of intelligence on different substrates, rather than a contest for a single center.

The authors state three conditions for further development: the tools should benefit most humans rather than a small group, should address actual human needs, and their harms should be measured against their benefits and reduced where possible. They acknowledge that many of these conditions will not be met in practice and argue that stating them is the necessary first step. Mathematics will encounter these questions with the strongest verification available. Other fields will encounter the same questions with less. This is the main reason to follow the mathematical case closely.