## Naturalness as an Epistemic Virtue: Case Studies from the History of Mathematics

Sponsor

Forum for the History of the Mathematical Sciences (FoHoMS)

Organizer

Nicolas Michel

Université de Paris, SPHERE

Chair

Jemma Lorenat

Pitzer College

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Session Abstract

Naturalness as an epistemic virtue has been appealed to throughout the history of mathematics, in a variety of contexts. In a 1704 eulogy of the mathematician L'Hôpital, the Enlightenment savant Fontenelle singled out certain methods as "simple and natural," adding in an apparent paradox that "the most natural methods or ideas are not those that arise most naturally." In mid-19th-century Germany, mathematicians such as Riemann and Dedekind put forth a methodology for the introduction of 'natural' definitions in geometry and algebra. More recently, Grothendieck's 1986 autobiography is rife with descriptions of his own 'childlike' ability, in various areas of mathematics, to find the concepts that are 'natural' to it and from which important theorems flow effortlessly. While many mathematicians invoke naturalness, the notion has many different but often overlapping meanings. For some, it meant a strongly personal virtue, characteristic of the mathematical genius; for others, an epistemological value crucial for the shaping and production of scientific knowledge; for others still, the theoretical horizon to all proper mathematical research. However, this epistemic virtue has received little attention in its own right. Drawing upon the contemporary literature on epistemic virtues and epistemological values, the contributions to this panel study the normative role the notion of naturalness played in shaping textual practices, epistemologies, and scientific identities. A study of naturalness opens a promising line of inquiry that shifts the focus of the history of mathematics away from more standard values such as rigor and certainty and towards understanding the role of values such as fruitfulness, simplicity, or clarity. Additionally, the longer commentary and discussion period in this session will explore the rich and largely uncharted history of circulations, transformations and reinterpretations of naturalness between the case studies presented therein and beyond.

Presenter 1

Nature in Mathematics after Newton: The Epistemic Priority of Mathematics in The Analyst Controversy

Julia C. Tomasson

Columbia University

Abstract

This paper offers a case study in the history of naturalness as a mathematical virtue. Drawing its empirical content from eighteenth-century British science and mathematics, I focus on the Analyst Controversy as an episode of scientific controversy which illuminates competing ideas of the epistemic priority of mathematics after Newton's death in 1727. The Analyst Controversy refers to the public and private debates following the publication of Bishop George Berkeley's (1685-1753) incendiary tract condemning the foundations of calculus The Analyst; Or, A Discourse Addressed to an Infidel Mathematician (1734). While this text lives on in the history of mathematics in infamy and ridicule, the controversy itself is dismissed as philosophically, historically, and mathematically inconsequential. From the perspective of epistemic virtues and vices, however, attention to this debate makes clear that no one debated the truth or falsity of Newton's proof of the "product rule" (Book II, Lemma II), but rather, the epistemological, ontological, and social grounds for its truth. I claim that three types of "nature" and their interrelation are at stake in this controversy: 1) the nature of the world, 2) the nature of mathematics, and 3) the nature of mathematicians. Focusing in particular on the value of "naturalness" we see how all sides in this controversy blurred extra-mathematical and mathematical concerns in ways which exclusive attention to ideals like "rigor" foreclose.

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Presenter 2

Clever Artifice or Signpost to the Hidden Nature of Things? The Circulation of Leibniz's Analogy of Powers and Differences

David Waszek

McGill University

Abstract

This paper is about a notation and a closely related "analogy," about the strikingly divergent appreciation of them by different mathematicians, and about the epistemic values - naturalness, fruitfulness, simplicity, generality - that these divergences bring to light. The setting of my case study is the differential calculus in continental Europe in the late 17th and 18th centuries. The analogy in question relates powers, usually written with exponents (like x³, x⁻¹), with differentials and integrals, and is closely bound up with Leibniz's decision to write the latter with exponents as well (e.g., writing d²x for ddx, d⁻¹x for ∫x, etc.). Throughout the period I study, this analogy could not be put on a 'rigorous' footing, at least not in its full generality; yet some mathematicians valued it highly. Leibniz himself, upon discovering that the analogy allowed writing simple and very general formulas valid for powers and differences as well as for sums, enthusiastically declared that there was something "real" there and rashly concluded that the differential calculus and the integral calculus were in fact one and the same. Joseph-Louis Lagrange too was later enthralled by the highly general formulas the analogy yielded, despite admitting that "the principles of this analogy are not evident in and of themselves" and that its consequences had to be checked on a case-by-case basis; he even used the analogy as a crucial signpost in his efforts to refound the differential calculus on a theory of series. Others saw matters in a different light: Johann Bernoulli seemingly considered the analogy as an "artifice" - a useful computation trick - and Leonard Euler as no more than a curiosity. This case study thus illustrates how concerns of naturalness can play out in mathematics, both at the level of theory organization and at the level of writing practices or notations.

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Presenter 3

Sweet Is the Lore Which Nature Brings: Clarity, Simplicity, and Naturalness in Chasles’ and Poinsot’s Geometrical Mechanics

Nicolas Michel

Université de Paris, SPHERE

Abstract

All mathematical truths can become equally simple and intuitive once the proper and natural path to them has been found, wrote French mathematician Michel Chasles (1793-1880) in his 1837 survey of the historical development of geometry. Through this creed he echoed a contrast routinely expressed by his close friend and institutional ally Louis Poinsot (1777-1859), between the difficulty and obscurity of propositions obtained by cryptic series of algebraic computations, and the clear knowledge derived from proper methods. Such natural methods, they conceded, might well be slower than those in fashion, but they proceed at a gait which allows for complete knowledge of the very object they aim to illuminate. To walk these natural paths for Chasles and Poinsot meant to reject the use of auxiliary, extrinsic quantities, such as the coordinate systems of Cartesian geometry. Instead, they claimed to construct and handle symbols and concepts intrinsic to the figures and phenomena at play. This did not mean to return to old Euclidean or diagram-based practices - which can only show concrete, individual cases of more general truths. Rather, they set out to invent novel ways of writing mathematical propositions and proofs which depicted before the mathematician's eye the abstract and natural truths. In so doing, Chasles and Poinsot constructed a counter-model of the mathematician who lets truth reveal itself, fully and on its own terms. Drawing from Chasles' and Poinsot's work on mechanics, this paper explores their shared epistemic ideal and the value of naturalness therein. In particular, it argues that this ideal must be located against the backdrop of a scientific culture of 'ingénieurs-savants' shaped and transmitted at the Ecole Polytechnique; and explores the status of naturalness in the moral economy of Chasles' and Poinsot's mathematics.

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Commentator

Michael J. Barany

University of Edinburgh