Algebraic generality versus arithmetic generality in the 1874 controversy between C. Jordan and L. Kronecker
This article revisits the 1874 controversy between Camille Jordan and Leopold Kronecker over two theorems, namely Jordan’s canonical forms and Karl Weierstrass’s elementary divisors theorem. In particular, it compares the perspectives of Jordan and Kronecker on generality and how their debate turned into an opposition over the algebraic or arithmetic nature of the ‘theory of forms’. It also examines the ways in which the various actors used the the categories of algebraic generality and arithmetic generality. After providing a background on the Jordan-Kronecker controversy, the article explains Jordan’s canonical reduction and Kronecker’s invariant computations in greater detail. It argues that Jordan and Kronecker aimed to ground the ‘theory of forms’ on new forms of generality, but could not agree on the types of generality and on the treatments of the general they were advocating.
Jean- Gaël Barbara
This article examines generality in biology by focusing on two French schools of anatomy: the discipline of anatomie générale that was founded in France in 1800 by Xavier Bichat and the one developed in the 1870s by Louis Ranvier at the Collège de France by means of microscopy. The works of Bichat and Ranvier involved the disciplines of anatomy and physiology. Bichat’s work, especially his research on tissues, is of interest for understanding which kind of concept of generality gained favor in the life sciences at the start of the nineteenth century. Ranvier’s later career sheds light on the ways that generality was searched for at the microscopic level and its significance in the discovery of real and minute biological objects. Following a discussion of Bichat and Ranvier’s anatomie générale, this article explores the two men’s interests in generality as an actor’s category.
Elaboration of a statement on the degree of generality of a property: Poincaré’s work on the recurrence theorem
This article examines the statement that a property is true for ‘almost all’ considered objects, in a precise mathematical sense, by referring to Henri Poincaré’s reflections on the generality of recurring trajectories. In 1890, Poincaré introduces a statement of a new type in which he formulates mathematically the remark that he had previously made in vague terms: ‘the trajectories that have this property [of stability, AR] are more general than those that do not’. This article first considers how Poincaré adapts the calculus of probability to show that the non-recurring trajectories are exceptional before analyzing the proofs of the recurrence theorem and the corollary that Poincaré added to the theorem. It also discusses the change of status of the recurrence theorem between 1889 and 1891 and suggests that the confinement inside the trajectory surfaces seemed to be the key property for the definition of stability.
This article examines Stefan Banach’s contributions to the field of functional analysis based on the concept of structure and the multiply-flvored expression of generality that arises in his work on linear operations. More specifically, it discusses the two stages in the process by which Banach elaborated a new framework for functional analysis where structures were bound to play an essential role. It considers whether Banach spaces, or complete normed vector spaces, were born in Banach’s first paper, the 1922 doctoral dissertation On operations on abstract spaces and their application to integral equations. It also analyzes what appears to be the core of Banach’s 1922 article and the transformation into a general setting that it represents. The main achievements of Banach’s dissertation, as well as all the essential features that bear witness to the birth of a new theory, are concentrated in the study of linear operations.
This article discusses generality in Gottfried Leibniz’s mathematics. In principle, Leibnizian mathematics has a philosophical-theological basis. From the beginning everything that exists is to be found in an orderly relation. The general and inviolable laws of the world are an ontological a priori. The universal harmony of the world consists in the largest possible variety being given the largest possible order so that the largest possible perfection is involved. After considering the relationship between the value of generality and the harmonies that are at the center of Leibniz’s concern, this article explores his view that generality implies beauty as well as conciseness and simplicity. It also examines how the interest in generality relates to notations, taking the examples of determinants and sums of powers, and to utility and fecundity. Finally, it demonstrates how generality is connected with laws of formation.
This article examines Henri Poincaré’s philosophical conceptions of generality in mathematics and physics, and more specifically his claim that induction in experimental physics does not consist in extending the domain of a predicate. It first considers Poincaré’s view that generalization is not a means to reach generality and that the issue of infinity is related to the theme of generality. It then shows how generality in mathematics and physics is construed by Poincaré in a very specific way and how he analyzes empirical induction in physics. It also analyzes the distinction suggested by Poincaré between generalizations used in mathematical physics and generalizations used by ‘naturalists’. In particular, it explains the distinction between mathematical generality and the so-called predicative generality. Finally, it compares Poincaré’s concern regarding empirical induction with Nelson Goodman’s ‘new riddle of induction’, arguing that ‘the new riddle of induction’ was originally formulated by Poincaré half a century earlier.
This article examines how the concept of homology is used as an expression of generality in the life sciences. Throughout its long history, homology expressed a quest for generality in the understanding of animal anatomy by suggesting that a diversity of forms resulted from modifications of a single ‘primitive’ structure. However, the meaning of this quest as well as the practices associated with it changed considerably with the different theoretical context of the life sciences. Thus, homology was an element of continuity in the history of biology and played a central role in some developments, particularly the emergence of evolutionary theory. This article first considers the use of homology in pre-transformist comparative anatomy and how it paved the way for the conceptualization of evolutionary theory before discussing the rise of new meanings of homology in genetics.
This article examines how the genus category was perceived and conceived in zoology (with occasional references to botany), in reference to species on the one hand and to higher categories on the other hand. In systematic zoology and botany, animals and plants are classified and named according to their species, genera, and higher categories (family, order, etc.). Linguistic relationships between the words ‘genus’ and ‘general, generality’ might have played a role in some intuitive meaning of the genus. This article traces the evolution of the concept of genus as used in systematic zoology from antiquity to the present time, focusing on the contributions of Plato, Aristotle, Carl Linnaeus, Georges-Louis Leclerc de Buffon, Jean-Baptiste Lamarck, Georges Cuvier, and Charles Darwin. It also considers the introduction of a new, rank-free system called the PhyloCode to replace Linnaean ranking—and especially the genus level.
This article examines Gottfried Leibniz’s notion of analysis by focusing on his investigation of transcendental curves. It argues that Leibnizian analysis can be understood as an art of both discovery and justification in mathematics that aims for generalization rather than abstraction, and explanation rather than formal proof. The article first considers Leibniz’s work on the catenary before discussing some of his pronouncements on analysis as the search for conditions of intelligibility. It also evaluates some modern accounts of Leibniz’s notion of analysis by contemporary philosophers, including Carlo Cellucci, Herbert Breger, and Nancy Cartwright. It argues that concrete terms can be used to say something true only when they are combined with more abstract locutions that express the conditions of intelligibility of the thing denoted, the formal causes that make the thing what it is and so make its resemblance to other things possible.
This article examines the gradual development of James Clerk Maxwell’s electromagnetic theory, arguing that he aimed at general structures through his models, illustrations, formal analogies, and scientific metaphors. It also considers a few texts in which Maxwell expounds his conception of physical theories and their relation to mathematics. Following a discussion of Maxwell’s extension of an analogy invented by William Thomson in 1842, the article analyzes Maxwell’s geometrical expression of Michael Faraday’s notion of lines of force. It then revisits Maxwell’s honeycomb model that he used to obtain his system of equations and the concomitant unification of electricity, magnetism, and optics. It also explores Maxwell’s view about the Lagrangian form of the fundamental equations of a physical theory. It shows that Maxwell was guided by general structural requirements that were inspired by partial and temporary models; these requirements were systematically detailed in Maxwell’s 1873 Treatise on electricity and magnetism.
Evelyn Fox Keller
This article examines the different meanings as well as values associated with the mathematical and the biological sciences in relation to the kinds of generality favored by the practitioners of biological evolution. It first considers the joke of the ‘spherical cow’ told by physicists about experimental biologists, which reflects the difference in epistemological cultures that, in turn, reflects (at least in part) differences between the two fields. It then explores what these differences can tell us about practices of generalization in mathematical physics, in biology, and even in strategies of biological evolution. It also discusses generality in the physical sciences and in the life sciences, focusing on Nicolas Rashevsky’s dispute with the cell biologists that arose at the 1934 meeting of the Cold Spring Harbor Symposia on Quantitative Biology. Finally, it looks at biologists’ lack of interest in universality and universal laws.
This article examines the so-called problem of generality in Euclid’s Elements. More specifically, it asks whether there is a ‘more general’ point of view from which magnitudes and numbers might be treated without distinction, and if so, whether Euclid violated the Aristotelian prescription by treating general features (typically the operative core of “proportions”) in specific cases in Book V (for magnitudes) and in Book VII (for numbers). The article first considers the general context of the problem of generality, taking into account the object domains in Euclidean mathematics, and some of the modern strategies proposed to solve it. These strategies rely on historical reconstructions and often make use of evidence taken from Aristotle, but tend to have a purely utilitarian relationship to such evidence. The article also discusses Aristotle’s conception of scientific knowledge, the problem of the katholou, and his view of generality in mathematics.
This article examines Ernst Kummer’s creation of ideal factors, which provides an interesting example of generalization within the set of complex numbers. Kummer developed a theory of ideal numbers in order to generalize arithmetical properties of natural numbers by extending these properties to certain complex numbers. His goal was to make complex numbers analogous to natural ones. This article first considers Kummer’s use of several analogies, primarily with arithmetic and chemistry, to come up with ideal factors of complex numbers. It then situates Kummer’s investigations on complex numbers with respect to Carl Friedrich Gauss’s work and compares his theory of ideal factors with Richard Dedekind’s ideals theory. It shows that Kummer’s method of generalization is premised on the distinction he articulated between ‘permanent’ and ‘accidental’ properties of complex numbers. This distinction draws from Kummer’s conception of mathematics, which was essentially different from those espoused by Gauss and Dedekind.
Karine Chemla, Renaud Chorlay, and David Rabouin
This book examines generality in mathematics and the sciences and how it has been shaped by actors, in part by introducing specific terminologies to distinguish between different levels or forms of generality. Focusing on early modern and modern Europe, it investigates how actors from Gottfried Leibniz and Henri Poincaré to René Descartes and James Clerk Maxwell worked out what the meaningful types of generality were for them, in relation to their project, and the issues they chose to deal with. Such a view implies that there are different ways of understanding the general in different contexts. Accordingly, it suggests a nonlinear pattern for a history of generality. The book considers actors’ historiography of generality and their reflections upon its epistemological value, the historicity of the statements used by actors to formulate the general, and the ways that actors tackle the general using specific practices.
This article examines ways of expressing generality and epistemic configurations in which generality issues became intertwined with epistemological topics, such as rigor, or mathematical topics, such as point-set theory. In this regard, three very specific configurations are discussed: the first evolving from Niels Henrik Abel to Karl Weierstrass, the second in Joseph-Louis Lagrange’s treatises on analytic functions, and the third in Emile Borel. Using questions of generality, the article first compares two major treatises on function theory, one by Lagrange and one by Augustin Louis Cauchy. It then explores how some mathematicians adopted the sophisticated point-set theoretic tools provided for by the advocates of rigor to show that, in some way, Lagrange and Cauchy had been right all along. It also introduces the concept of embedded generality for capturing an approach to generality issues that is specific to mathematics.
This article examines the role of genericity in the development of dynamical systems theory. In his memoir ‘Sur les courbes définies par une équation différentielle’, published in four parts between 1881 and 1886, Henri Poincaré studied the behavior of curves that are solutions for certain types of differential equations. He successfully classified them by focusing on singular points, described the trajectories’ behavior in important particular cases and provided new methods that proved to be extremely useful. This article begins with a discussion of singularity theory and its influence on the first definitions of genericity, along with the application of the notions of structural stability and genericity to understand dynamical systems. It also analyzes the Smale conjecture and how it was proven wrong and concludes with an overview of changes in the definitions of genericity meant to describe the ‘dark realm of dynamics’.
Universality versus generality: an interpretation of the dispute over tangents between Descartes and Fermat
This article examines the ‘dispute over tangents’ between René Descartes and Pierre de Fermat, which began in 1638, by comparing their methods of constructing tangents. Descartes’s method aims at a universality, whereas Fermat’s method aims at a generality. The article begins with a discussion of Descartes’s method of tangents, which he proposed in 1637 as a way of constructing ‘all the problems of geometry’. It then considers Fermat’s method, in which he introduces a rule for finding a maximum or minimum. It also looks at Descartes’s critiques of Fermat’s method as applied to the ellipse and the hyperbola, and how Fermat arrived at the question of the ‘specific property’ of a curve by constructing the tangents to non-geometric curves.
This article discusses the value of generality in Michel Chasles’s historiography of geometry, articulated in his 1837 book Aperçu historique sur l’origine et le développement des méthodes en géométrie, particulièrement de celles qui se rapportent à la géométrie moderne, suivi d’un mémoire de géométrie sur deux principes généraux de la science: la dualité et l’homographie. In the book, mathematics and history were combined as two kinds of tools useful in achieving a single aim: to show how the recent advances in geometry allowed the strictly geometrical methods in this domain to rival the analytical approach to geometry. Generality, and the related property of simplicity, are Leitmotive in Aperçu historique. This article first examines Chasles’s historical analysis of geometry and his methods related to generality in geometry before considering some aspects of the problem of generality in geometry.