Gamma, a student in Imre Lakatos’s drama, Proofs and Refutations, suggests: ‘Why not have mathematical critics just as you have literary critics, to develop mathematical taste by public criticism?’footnote¹ Lakatos’s drama is set in a mathematics classroom. The students are debating the proof of the Euler characteristic for polyhedra. Proofs and Refutations follows the journey of Euler’s Theorem from its birth as naive conjecture through the mistakes and revolutions of nineteenth-century mathematics, to adulthood.footnote² As they speed through a century of mathematical history, the students live Lakatos’s lesson: rigour and proof are historically variable values and practices. Proofs and Refutations offers, too, an education in the value of error: mathematical knowledge develops by dialectical criticism.footnote³

Lakatos’s stated enemy was the ‘formalist’ philosophy of mathematics. In particular, he objected to the formalist image of mathematics, which equated mathematics with ‘its formal axiomatic abstraction’ and the philosophy of mathematics with metamathematics. In Lakatos’s opinion, formalism was disconnecting mathematics not just from its philosophy but also from its history:

According to the formalist concept of mathematics, there is no history of mathematics proper. Any formalist would basically agree with Russell’s ‘romantically’ put but seriously meant remark, according to which Boole’s Laws of Thought (1854) was ‘the first book ever written on mathematics’.footnote⁴

Mathematics further exempts itself from history, in Lakatos’s view, by forced adherence to a particular style of writing. This ‘Euclidean’ or ‘deductivist’ style imposes a fixed structure on the presentation of mathematics: the text begins with a list of axioms, lemmas and/or definitions, this list is followed by the theorem, and the theorem is followed by the proof. Readers of mathematics are watching a ‘conjuring act’: the deductivist style enforces the dogma that ‘all propositions are true and all inferences valid’ and presents mathematics ‘as an ever-increasing set of eternal, immutable truths’.footnote⁵By tearing the results from their heuristic context and hiding the mathematician’s initial conjectures, the counter-examples and the work of proof-analysis, deductivist style enforces a sense of finality and steels itself against criticism. ‘Deductivist style’, Lakatos writes, ‘hides the struggle, hides the adventure. The whole story vanishes, the successive tentative formulations of the theorem in the course of the proof-procedure are doomed to oblivion while the end result is exalted into sacred infallibility’. Lakatos advocates, instead, the adoption of a heuristic style, in which the text would tell the story of its own emergence: the adventure and struggle of conjecture, counter-examples, criticism and proof-analysis.footnote⁶

Gamma’s suggested new genre—‘mathematical criticism’—can be found in the form of mathematical manifestos, prefaces, parables and essays that mediate and frame the discipline’s entanglement with that which it understands to be other than itself—the liminal genres of modern mathematics. These works of prescriptive and performative disciplinary criticism seek to shape mathematical ‘taste’ by ‘public criticism’. Mathematical manifestos reinforce and reconfigure the links between the disciplinary self-construction of mathematics, the repertoire of cultural images of mathematics and the social structure in which mathematical knowledge is embedded. The metaphors and rhetorical strategies deployed in ‘peri-mathematical’ or threshold texts are mediators: translating mathematics out of the formal language of proof and into a network of historical and rhetorical entanglements. At the same time, mathematical manifestos mobilise the political conditions and cultural assumptions of the historical moments in which they were written. Traces of the discipline’s social and cultural history—of the making of mathematical values like rigour and exactness—are inscribed in the manifesto’s mathematical criticism.

Manifestos mark crucial moments in the nineteenth and twentieth-century history of mathematics. The introduction to Augustin-Louis Cauchy’s 1821 textbook, the Cours d’analyse, is seen by many historians of mathematics as marking a disjuncture between the hugely productive but informal development of the calculus over the previous 150 years and the start of its formalisation. Hermann Weyl’s 1921 ‘Über die neue Grundlagenkrise der Mathematik’ was the single most trenchant response to the late nineteenth and early twentieth century ‘foundations crisis’, a crisis that was in a sense the consequence of a series of mathematical results that showed contradictions in, or inherent limits of, efforts at formalization. N. Bourbaki’s ‘L’Architecture des Mathématiques’ (1948) was an enormously influential mid-twentieth century effort to re-found mathematics. ‘Nicolas Bourbaki’ was the collective pseudonym of a group of predominantly French mathematicians whose Éléments de mathématique was designed to be a self-contained reconstruction of the core elements of modern mathematics in largely formalized language.

Mathematical manifestos are works of polemic and performative disciplinary criticism that announce a new foundational programme in mathematics, break with the previous order and promote a certain image of mathematics and its history. There are, certainly, broad parallels between ‘modernist mathematics’ and other forms of cultural modernism: a rupture with tradition, a turn toward formalism, and a heightened self-reflexivity. As such, mathematical manifestos may be read alongside other examples of the genre—literary and artistic manifestos such as Marinetti’s ‘Technical Manifesto of Futurist Literature’ (1912), Pound’s ‘Vorticism’ (1914), Khlebnikov’s ‘To the Artists of the World!’ (1919) and the Oulipo manifestos (1960–73) of François Le Lionnais, Raymond Queneau and Jacques Roubaud.