Happy, Those Able to Know the Causes of Things: Reading LLMs as Cultural Technologies Rather Than Agents That Replace Mathematicians
Synopsis
This essay by Nestor Guillen argues that large language models (LLMs) should be understood as a kind of cultural and social technology, like markets, bureaucracies, and the scientific literature, which aggregates and compresses human accumulated information, so that when an LLM output contains a new mathematical idea it should be received as the fruit of mathematics' shared heritage rather than as a defeat for mathematicians; drawing on Farrell, Gopnik, Shalizi, and Evans's claim that large models are cultural technologies, the author proposes the metaphor of a 'convex hull of ideas,' suggesting that whether an LLM can solve a given mathematical problem depends largely on the state of the mathematical literature at the time rather than merely on model scale, illustrating this with the Kryl
Interpretation
Understanding LLMs as cultural and social technologies rather than anthropomorphized agents, thereby attributing mathematical ideas in LLM outputs to humanity's accumulated mathematical heritage. The author quotes Farrell, Gopnik, Shalizi, and Evans's central assertion that 'Large models should not be viewed primarily as intelligent agents but as a new kind of cultural and social technology,' and applies it to mathematical practice, describing LLMs as in a way 'wrappers' around an older cultural technology, the scientific literature. This is a conceptual and argumentative essay grounded in citation of prior work and the author's own teaching and research experience rather than experiments or statistics.
Proposing the metaphor of a 'convex hull of ideas': whether an LLM can solve a given mathematical problem is largely a function of that problem's position relative to the mathematical literature of its time. The author explicitly states he cannot give a more concrete definition, but uses the contrast between the infinitude of primes and the infinitude of primes in arithmetic progressions before Euler and Dirichlet, and the example of the Krylov-Safonov theorem's dependence on the Aleksandrov estimate from convex geometry, to illustrate that some ideas 'stand squarely outside' the existing literature while others lie closer to its 'convex hull.' This is a heuristic metaphor and case-based argument; the author writes 'I will not give a more concrete definition for this, because I am not able to,' and develops its implications through a parallel-universe thought experiment.
Using the history of professional skateboarding as an analogy for the turbulence currently facing mathematics, showing that creative communities can reorganize by changing their media of communication and evaluation. The author recounts that in 1984 Kevin Harris observed street skaters with only 'six months of experience' earning more than Rodney Mullen, and that the recession around 1990 caused the collapse of contests and sponsorships, after which skate videos became the new medium and a 'part' came to play the role of a journal article. This is historical narrative and analogical argument; the author himself acknowledges the transition 'turned out to be for the best for skateboarding as a discipline, while it also turned out for the worse for many individual skateboarders.'
Calling on the mathematical community to separate LLM technology from the product vision of large AI companies, and to promote open LLMs, public funding for science, and new professional arrangements. The author explicitly separates 'generative AI, the technology' from 'generative AI, the specific product and vision from the large AI companies,' and proposes exploring 'lighter, mathematics-specific LLMs with well curated data' along with policy advocacy. This is a normative claim and policy proposal grounded in the author's observations of AI company public relations and his judgment that open-weight models are 'a seed for this world.'
Perspective
The essay is addressed to members of the mathematical community and to readers interested in the relationship between AI and science, and it applies to discussions of LLMs' role in mathematics, as well as to policy and professional-arrangement questions in the field; its cultural-technologies framework and 'convex hull' metaphor are meant to help readers reinterpret the provenance and meaning of LLM outputs rather than to provide testable predictions or technical solutions.
The essay offers the 'convex hull of ideas' only as a metaphor without a concrete definition, and the author states he is unable to give a more precise characterization; the rapid pace of LLM improvement makes 'what can LLMs do?' a moving target; the discussion of Euler and an eighteenth-century LLM is a hypothetical thought experiment; the account of skateboarding history relies on episodes selected by the author; moreover, as a commentary essay it contains no reproducible experiments or formal proofs, so readers seeking to assess specific LLM capabilities on mathematical problems will still need to consult the relevant technical reports and mathematical literature.