John Bell

The rapid development of mathematical analysis in the eighteenth century had not concealed the fact that its underlying concepts not only lacked rigorous definition but were even (e.g. in the case of differentials and infinitesimals) of doubtful logical character. The lack of precision in the notion of continuous function—still vaguely understood as one which could be represented by a formula and whose associated curve could be smoothly drawn—had led to doubts concerning the validity of a number of procedures in which that concept figured. For example when Lagrange had formulated his method for “algebraizing” the calculus he had implicitly assumed that every continuous function could be expressed as an infinite series by means of Taylor’s theorem. Early in the nineteenth century this and other assumptions began to be questioned, thereby initiating an inquiry into what was meant by a function in general and by a continuous function in particular.

The leading practitioner of the calculus, indeed the leading mathematician of the eighteenth century, was Leonhard Euler (1707–83). While Euler’s genius has been described as being of “equal strength in both of the main currents of mathematics, the continuous and the discrete”, philosophically he was a thoroughgoing synechist. Rejecting Leibnizian monadism, he favoured the Cartesian doctrine that the universe is filled with a continuous ethereal fluid and upheld the wave theory of light over the corpuscular theory propounded by Newton.

The early modern period saw the spread of knowledge in Europe of ancient geometry, particularly that of Archimedes, and a loosening of the Aristotelian grip on thinking. In regard to the problem of the continuum, the focus shifted away from metaphysics to technique, from the problem of “what indivisibles were, or whether they composed magnitudes” to “the new marvels one could accomplish with them” through the emerging calculus and mathematical analysis. Indeed, tracing the development of the continuum concept during this period is tantamount to charting the rise of the calculus.

Category theory is a framework for the investigation of mathematical form and structure in their most general manifestations. Central to it is the concept of structure-preserving map, or transformation. While the importance of this notion was long recognized in geometry (witness, for example, Klein’s Erlanger Programm of 1872), its pervasiveness in mathematics did not really begin to be appreciated until the rise of abstract algebra in the 1920s and 30s, where, in the form of homomorphism, the idea had been central from the beginning. So emerged the view that the essence of a mathematical structure lies not in its internal constitution as a set-theoretical construct, but rather in the nature of its relationship with other structures of the same kind through the network of transformations between them.

In the late nineteenth and early twentieth century the investigation of continuity led to the creation of topology, a major new branch of mathematics conferring on the idea of the continuous a vast generality. The origins of topology lie both in Cantor’s theory of sets of points as well as the idea, which had first emerged in the calculus of variations, of treating functions as points of a space. Central to topology is the concept of topological space. A topological space is a domain equipped with sufficient structure to enable functions between such spaces to be identified as continuous. Now intuitively, a continuous function is one with no “jumps”, that is, it always sends “neighbouring” points of its domain to “neighbouring” points of its range. In order to support the idea of a continuous function in this intuitive sense, the concept of topological space must accordingly embody some notion of neighbourhood. It was in terms of the neighbourhood concept that Felix Hausdorff (1868–1942) first introduced, in 1914, the concept of topological space.

Once the continuum had been provided with a set-theoretic foundation, the use of the infinitesimal in mathematical analysis was largely abandoned. And so the situation remained for a number of years. The first signs of a revival of the infinitesimal approach to analysis surfaced in 1958 with a paper by A. H. Laugwitz and C. Schmieden. But the major breakthrough came in 1960 when it occurred to the mathematical logician Abraham Robinson (1918–1974) that “the concepts and methods of contemporary Mathematical Logic are capable of providing a suitable framework for the development of the Differential and Integral Calculus by means of infinitely small and infinitely large numbers.” This insight led to the creation of nonstandard analysis (NSA), which Robinson regarded as realizing Leibniz’s conception of infinitesimals and infinities as ideal numbers possessing the same properties as ordinary real numbers. In the introduction to his book on the subject he writes.

In constructive mathematics, a problem is counted as solved only if an explicit solution can, in principle at least, be produced. Thus, for example, “There is an x such that P(x)” means that, in principle at least, we can explicitly produce an x such that P(x). If the solution to the problem involves parameters, we must be able to present the solution explicitly by means of some algorithm or rule when given values of the parameters. That is, “for every x there is a y such that P(x, y) means that, we possess an explicit method of determining, for any given x, a y for which P(x, y). This leads us to examine what it means for a mathematical object to be explicitly given.

Traditionally, there have been two methods of deriving the theorems of geometry: the analytic and the synthetic. While the analytical method is based on the introduction of numerical coordinates, and so on the theory of real numbers, the idea behind the synthetic approach is to furnish the subject of geometry with a purely geometric foundation in which the theorems are then deduced by purely logical means from an initial body of postulates. The most familiar examples of the synthetic geometry are classical Euclidean geometry and the synthetic projective geometry introduced by Desargues in the 17th century and revived and developed by Carnot, Poncelet, Steiner and others during the 19th century. The power of analytic geometry derives very largely from the fact that it permits the methods of the calculus, and, more generally, of mathematical analysis, to be introduced into geometry, leading in particular to differential geometry (a term, by the way, introduced in 1894 by the Italian geometer Luigi Bianchi). That being the case, the idea of a “synthetic” differential geometry seems elusive: how can differential geometry be placed on a “purely geometric” or “axiomatic” foundation when the apparatus of the calculus seems inextricably involved? To my knowledge there have been two attempts to develop a synthetic differential geometry. The first was initiated by Herbert Busemann in the 1940s, building on earlier work of Paul Finsler. Here the idea was to build a differential geometry that, in its author’s words, “requires no derivatives”: the basic objects in Busemann’s approach are not differentiable manifolds, but metric spaces of a certain type in which the notion of a geodesic can be defined in an intrinsic manner. I shall not have anything more to say about this approach. The second approach, that with which I shall be concerned here, was originally proposed in the 1960s by F. W. Lawvere, who was in fact striving to fashion a decisive axiomatic framework for continuum mechanics..

We are all familiar with the idea of a set, also called a class or collection. As examples, we may consider the set of all coins in one's pocket, the set of all human beings, the set of all planets in the solar system, etc. These are all concrete sets in the sense that the objects constituting them—their elements or members—are material things. In mathematics and logic we wish also to consider abstract sets whose members are not necessarily material things, but abstract entities such as numbers, lines, ideas, names, etc. We shall use the term set to cover concrete and abstract sets, as well as sets which contain a mixture of material and abstract elements.

Consider the following well-known result from the theory of normed linear spaces ([2], p. 80, 4(b)): (g) the unit ball of the (continuous) dual of a normed linear space over the reals has an extreme point. The standard proof of (~) uses the axiom of choice (AG); thus the implication AC~(w) can be proved in set theory. In this paper we show that this implication can be reversed, so that (*) is actually eq7I2valent to the axiom of choice. From this we derive various corollaries, for example: the conjunction of the Boolean prime ideal theorem and the Krein-Milman theorem implies the axiom of choice, and the Krein-Milman theorem is not derivable from the Boolean prime ideal theorem.

This essay is an attempt to sketch the evolution of type theory from its beginnings early in the last century to the present day. Central to the development of the type concept has been its close relationship with set theory to begin with and later its even more intimate relationship with category theory. Since it is effectively impossible to describe these relationships (especially in regard to the latter) with any pretensions to completeness within the space of a comparatively short article, I have elected to offer detailed technical presentations of just a few important instances.

I describe two approaches to modelling the universe, the one having its origin in topos theory and differential geometry, the other in set theory. The first is synthetic differential geometry. Traditionally, there have been two methods of deriving the theorems of geometry: the analytic and the synthetic. While the analytical method is based on the introduction of numerical coordinates, and so on the theory of real numbers, the idea behind the synthetic approach is to furnish the subject of geometry with a purely geometric foundation in which the theorems are then deduced by purely logical means from an initial body of postulates. The most familiar examples of the synthetic geometry are classical Euclidean geometry and the synthetic projective geometry introduced by Desargues in the 17th century and revived and developed by Carnot, Poncelet, Steiner and others during the 19th century. The power of analytic geometry derives very largely from the fact that it permits the methods of the calculus, and, more generally, of mathematical analysis, to be introduced into geometry, leading in particular to differential geometry (a term, by the way, introduced in 1894 by the Italian geometer Luigi Bianchi). That being the case, the idea of a “synthetic” differential geometry seems elusive: how can differential geometry be placed on a “purely geometric” or “axiomatic” foundation when the apparatus of the calculus seems inextricably involved? To my knowledge there have been two attempts to develop a synthetic differential geometry. The first was initiated by Herbert Busemann in the 1940s, building on earlier work of Paul Finsler. Here the idea was to build a differential geometry that, in its author’s words, “requires no derivatives”: the basic objects in Busemann’s approach are not differentiable manifolds, but metric spaces of a certain type in which the notion of a geodesic can be defined in an intrinsic manner. I shall not have anything more to say about this approach. The second approach, that with which I shall be concerned here, was originally proposed in the 1960s by F..

One of the most familiar uses of the Russell paradox, or, at least, of the idea underlying it, is in proving Cantor's theorem that the cardinality of any set is strictly less than that of its power set. The other method of proving Cantor's theorem — employed by Cantor himself in showing that the set of real numbers is uncountable — is that of diagonalization. Typically, diagonalization arguments are used to show that function spaces are "large" in a suitable sense. Classically, these two methods are equivalent. But constructively they are not: while the argument for Russell's paradox is perfectly constructive, (i.e., employs intuitionistically acceptable principles of logic) the method of diagonalization fails to be so. I describe the ways in which these two methods..

This book is written for those who are in sympathy with its spirit. This spirit is different from the one which informs the vast stream of European and American civilization in which all of us stand. That spirit expresses itself in an onwards movement, in building ever larger and more complicated structures; the other in striving in clarity and perspicuity in no matter what structure. The first tries to grasp the world by way of its periphery—in its variety; the second at its centre—in its essence. And so the first adds one construction to another, moving on and up, as it were, from one thing to the next, while the other remains where it is and what it tries to grasp is always the same.

Axioms for the continuum, or smooth real line R. These include the usual axioms for a commutative ring with unit expressed in terms of two operations + and i, and two distinguished elements 0 ≠ 1. In addition we stipulate that R is a local ring, i.e., the following axiom: ∃y x i y = 1 ∨ ∃y (1 – x) i y = 1. Axioms for the strict order relation < on R. These are: 1. a < b and b < c implies a < c. 2. ¬(a < a) 3. a < b implies a + c < b + c for any c. ≤ 4. a < b and 0 < c implies acbc..

This book explores and articulates the concepts of the continuous and the infinitesimal from two points of view: the philosophical and the mathematical. The first section covers the history of these ideas in philosophy. Chapter one, entitled ‘The continuous and the discrete in Ancient Greece, the Orient and the European Middle Ages,’ reviews the work of Plato, Aristotle, Epicurus, and other Ancient Greeks; the elements of early Chinese, Indian and Islamic thought; and early Europeans including Henry of Harclay, Nicholas of Autrecourt, Duns Scotus, William of Ockham, Thomas Bradwardine and Nicolas Oreme. The second chapter of the book covers European thinkers of the sixteenth and seventeenth centuries: Galileo, Newton, Leibniz, Descartes, Arnauld, Fermat, and more. Chapter three, 'The age of continuity,’ discusses eighteenth century mathematicians including Euler and Carnot, and philosophers, among them Hume, Kant and Hegel. Examining the nineteenth and early twentieth centuries, the fourth chapter describes the reduction of the continuous to the discrete, citing the contributions of Bolzano, Cauchy and Reimann. Part one of the book concludes with a chapter on divergent conceptions of the continuum, with the work of nineteenth and early twentieth century philosophers and mathematicians, including Veronese, Poincaré, Brouwer, and Weyl. Part two of this book covers contemporary mathematics, discussing topology and manifolds, categories, and functors, Grothendieck topologies, sheaves, and elementary topoi. Among the theories presented in detail are non-standard analysis, constructive and intuitionist analysis, and smooth infinitesimal analysis/synthetic differential geometry. No other book so thoroughly covers the history and development of the concepts of the continuous and the infinitesimal.