Points as Higher-Order Constructs

Euclid’s definition of a point as “that which has no part” has been a major source of controversy in relation to the epistemological and ontological presuppositions of classical geometry, from the medieval and modern disputes on indivisibilism to the full development of point-free geometries in the 20th century. Such theories stem from the general idea that all talk of points as putative lower-dimensional entities must and can be recovered in terms of suitable higher-order constructs involving only extended regions (or bodies). This chapter focuses on what is arguably the first thorough proposal of this sort, Whitehead’s theory of “extensive abstraction”, offering a critical reconstruction of the theory through its successive installments: from the purely mereological version of ‘La théorie relationniste de l’espace’ (1916) to the refined versions presented in An Enquiry Concerning the Principles of Natural Knowledge (1919) and in The Concept of Nature (1920) to the last, mereotopological version of Process and Reality (1929).

Dedekind on Continuity

In this chapter we present Richard Dedekind’s conception of continuity and his various approaches to continuous domains in a historical context. In addition to his seminal work on foundations of irrational numbers (Stetigkeit und irrationale Zahlen, 1872), we also include a discussion of more mathematical texts (both published and unpublished) in which Dedekind also treats other continuous domains, such as Riemann surfaces, spaces, and multiply extended continuous domains. Dedekind’s reflections on these matters illustrate the wide range and general coherence of his thoughts. In particular, while Dedekind’s approach to mathematics can be characterized as being axiomatic, mapping-based, structuralist, and increasingly abstract, we argue that there is also a more general outlook underlying his methodology, which can be described as being, broadly understood, arithmetical.

Continuity and Intuition in Eighteenth-Century Analysis and in Kant

This chapter considers the status of geometrical and kinematic representations in the foundations of 18th century analysis and in Kant’s understanding of those foundations. It has two aims. First, relying on relatively recent reassessments of the history of analysis, it will attempt to bring forward a more accurate account of intuitive representation in 18th century analysis and the relation between British and Continental mathematics. Second, it will give a better account of Kant’s place in that history. The result shows that although Kant did no better at navigating the labyrinth of the continuum than his contemporaries, he had a more interesting and reasonable account of the foundations of analysis than an easy reading of either Kant or that history provides. It also permits a more accurate and interesting account of how and when a conception of foundations of analysis without intuitive representations emerged, and how that paved the way for Bolzano and Cauchy.

The Indivisibles of the Continuum

This chapter considers some significant developments in seventeenth-century mathematics which are part of the pre-history of the infinitesimal calculus. In particular, I examine the “method of indivisibles” proposed by Bonaventura Cavalieri and various developments of this method by Evangelista Torricelli, Gilles Personne de Roberval, and John Wallis. From the beginning, the method of indivisibles faced objections that aimed to show that it was either conceptually ill-founded (in supposing that the continuum could be composed of dimensionless points) or that its application would lead to error. I show that Cavalieri’s original formulation of the method attempted to sidestep the question of whether a continuous magnitude could be composed of indivisibles, while Torricelli proposed to avoid paradox by taking indivisibles to have both non-zero (yet infinitesimal) magnitude and internal structure. In contrast, Roberval and Wallis showed significantly less interest in addressing foundational issues and were content to maintain that the method could (at least in principle) be reduced to Archimedean exhaustion techniques.

Continuous Extension and Indivisibles in Galileo

This chapter studies Galileo’s account of continuous extension and his pioneering use of indivisibles in the analysis of paradoxes of infinity and continuity. Galileo’s treatment presents continuous magnitudes as composed of finite ever-divisible parts as well as infinitely many indivisible elements he calls partes non quanti. Special scrutiny is given to Galileo’s analysis of the paradox of Aristotle’s wheel as well as to Galileo’s account of continuous uniform and accelerated motion as it occurs in the Dialogo (1632) and the Discorsi (1638). Galileo’s discussion is illuminated by comparison with the techniques of Cavalieri and Huygens.

Divisibility or Indivisibility

This chapter aims to show that the earliest discussion about continuity in Western thought is a debate within metaphysics and natural philosophy about homogeneity and divisibility. All parties to this dispute – the main proponents are Parmenides, Zeno, and Aristotle – agree that magnitudes which are continuous (suneches) are homogenous and without any gaps. They disagree, however, on which inferences to draw from this for the possibility of divisibility – whether it implies indivisibility, as Parmenides and Zeno assumed, or divisibility, as Aristotle claimed, building on a mathematical understanding of extended magnitudes. Furthermore, while a modern understanding of continuity may seem to be essentially anti-Aristotelian, Aristotle is shown to prepare many of the crucial features of a modern account of continuity in this debate.

Contemporary Infinitesimalist Theories of Continua and Their Late Nineteenth- and Early Twentieth-Century Forerunners

The purpose of this chapter is to provide a historical overview of some of the contemporary infinitesimalist alternatives to the Cantor-Dedekind theory of continua. Among the theories we will consider are those that emerge from nonstandard analysis, nilpotent infinitesimalist approaches to portions of differential geometry and the theory of surreal numbers. Since these theories have roots in the algebraic, geometric and analytic infinitesimalist theories of the late nineteenth and early twentieth centuries, we will also provide overviews of the latter theories and some of their relations to the contemporary ones. We will find that the contemporary theories, while offering novel and possible alternative visions of continua, need not be (and in many cases are not) regarded as replacements for the Cantor-Dedekind theory and its corresponding theories of analysis and differential geometry.

Intuitionistic/Constructive Accounts of the Continuum Today

In this chapter we describe the properties of the continuum as it is conceived in the intuitionistic and constructive senses. The chapter ends with an account of the continuum as it is conceived in Smooth Infinitesimal Analysis, a recently developed approach to mathematical analysis based on nilpotent infinitesimals.

The Predicative Conception of the Continuum

This chapter discusses the predicative conception of the continuum. It has three parts. The first part gives a historical account of the origins of the concept of predicativity, from its birth in the general logical setting of Russell’s work on the paradoxes, to its relocation to a more specific mathematical setting in Weyl’s work on his limitative conception of analysis. The second part traces the development of the subsequent analysis of the concept of predicativity, and culminates in Feferman’s celebrated analysis. The third part addresses—in the light of these developments —the question of what the continuum looks like from the predicative point of view. Three predicative conceptions of the continuum are distinguished—one associated with Weyl, and two associated with Feferman—and it is argued that each predicative conception involves, in one way or another, a radical departure from our default conception of the continuum.

Continuity in Intuitionism

The chapter features, first, a critical presentation of Brouwer’s intuitionistic doctrines concerning logic, the real numbers, and continuity in the real number system, including his Principle for Numbers and Continuity Theorem. This is followed by a parallel examination of Hermann Weyl’s quasi-intuitionistic views on logic, continuity, and the real number system, views inspired by (but grossly misrepresenting) ideas of Brouwer. The whole business wraps up with an attempt to place Brouwer’s and Weyl’s efforts within the trajectory of informed thinking, during the late 19th and early 20th centuries, on the subjects of continua, magnitudes, and quantities.