math.HO — History and Overview

The Rising-Sea Theses: Twelve Theses on Mathematics and Machine Reasoning

Contributed by Vincent Liu

The Grothendieckian Programme is a set of seven works on machine reasoning about mathematics (34 volumes, 26,021 pages), released openly in October 2026 as first drafts that have not been peer-reviewed. This paper draws from them twelve theses on how mathematics and machine reasoning should be combined. Each thesis is stated in one sentence, argued, tied to the passages of the seven works on which it rests, related to existing work, and paired with an observation that would refute it. The theses concern the unit of mathematical knowledge for a reasoning system, the place of epistemic status, the latent space of mathematical reasoning, attention, failure, authority, and the method of discovery. The paper adds no mathematical result to the seven works; none of them claims a proof of the Riemann Hypothesis, the Birch and Swinnerton-Dyer conjecture, the Hodge conjecture, the abc conjecture or the Langlands correspondence.

Author comments: 26 pages, 5 figures. Version 1.0 of 28 October 2026; the same text is deposited at Zenodo, DOI 10.5281/zenodo.23734424 (all versions: 10.5281/zenodo.23734423). This copy was typeset with pdfLaTeX for Hexagon: its text is identical to version 1.0 at Zenodo, whose sealed PDF was typeset with LuaLaTeX, except that a broken cross-reference on page 3 ("igref{works}") is corrected to "Figure 1"; a few line and page breaks differ. The twelve theses are also set out at https://github.com/AlienMuse/grothendieckian-problems/blob/main/THESES.md; the seven works on which they rest, each with its DOI, are listed at https://github.com/AlienMuse/grothendieckian-programme. Corrections recorded since version 1.0 to the seven works and to the papers: https://github.com/AlienMuse/grothendieckian-problems/blob/main/ERRATA.md. Companion papers: The Grothendieckian Programme: An Overview (DOI 10.5281/zenodo.23734421) and The 57 Grothendieckian Problems (DOI 10.5281/zenodo.23734426).