CONTINUITY

In the decades bracketing the turn of the twentieth century, the real number system was dubbed the arithmetic continuum because it was held that this number system is completely adequate for the analytic representation of all types of continuous phenomena. In accordance with this view, the geometric linear continuum is assumed to be isomorphic with the arithmetic continuum, the axioms of geometry being so selected to ensure this would be the case. In honor of Georg Cantor (1845–1918) and Richard Dedekind (1831–1916), who first proposed this mathematico-philosophical thesis, the presumed correspondence between the two structures is sometimes called theCantor-Dedekind axiom.

Since their appearance, the late nineteenth-century constructions of the real numbers have undergone set-theoretical and logical refinement, and the systems of rational and integer numbers on which they are based have themselves been given a set-theoretic foundation. During this period the Cantor-Dedekind philosophy of the continuum also emerged as a pillar of standard mathematical philosophy that underlies the standard formulation of analysis, the standard analytic and synthetic theories of the geometrical linear continuum, and the standard axiomatic theories of continuous magnitude more generally.

Since its inception, however, there has never been a time at which the Cantor-Dedekind philosophy has either met with universal acceptance or has been without competitors. The period that has transpired since its emergence as the standard philosophy has been especially fruitful in this regard, having witnessed the rise of a variety of constructivist and predicativist theories of real numbers and corresponding theories of analysis as well as the emergence of a number of alternative theories that make use of infinitesimals. Whereas the constructivist and predicativist theories have their roots in the early twentieth-century debates on the foundations of mathematics and were born from critiques of the Cantor-Dedekind theory, the infinitesimalist theories were intended to either provide intuitively satisfying (and,in some cases, historically rooted) alternatives to the Cantor-Dedekind conception that have the power to meet the needs of analysis or differential geometry, or to situate the Cantor-Dedekind system of real numbers in a grander conception of an arithmetic continuum.

Speculation regarding the nature and structure of continua and of continuous phenomena more generally therefore naturally falls into three periods: the period of the emergence and eventual domination of the Cantor-Dedekind philosophy, and the periods before and after. These three periods are considered in this entry in historical turn.

The Aristotelian Conception

Before the Cantor-Dedekind philosophy the idea of the continuum stood in direct opposition to the discrete and was generally thought to be grounded in our intuition of extensive magnitude, in particular of spatial or temporal magnitude, and of the motion of bodies through space. Some of the essential characteristics of what emerged as the standard ancient conception were already described by Anaxagoras of Clazomenae (c. 500–428 BCE) when he observed that "Neither is there a smallest part of what is small, but there is always a smaller (for it is impossible that what is should cease be [no matter how far it is being subdivided])" (Kirk, Raven, and Schofield 1983, p. 360). Thus, not only is the continuum infinitely divisible, but through the process of division it cannot be reduced to discrete indivisible elements that are, as Anaxagoras picturesquely put it, "separated from one another as if cut off with an axe (ibid. p. 371)." However, while ingredients of the standard ancient conceptionare already found in the writings of some of the pre-Socratics, it was Aristotle (384–322 BCE), inspired by the writings of the geometers of his day, who provided its earliest systematic philosophical treatment.

Central to Aristotle's analysis is the distinction between discrete and continuous quantity; whereas the former lack, the latter have, a common boundary at which the parts join to form a unity. For Aristotle, number—by which he meant the positive integers greater than or equal to two—is discrete, whereas measurable magnitude—lines, surfaces, bodies, time, and place—are continuous. Lines, in particular, are continuous because "it is possible to find a common boundary at which its parts join together, a point" (Categories 6, 5^a1–2, in Aristotle 1984, p. 8); in the cases of surfaces and bodies, the common boundaries are lines, and lines or surfaces, respectively, and in the case of time they are moments.

Motion for Aristotle is also continuous, its continuity being a reflection of spatial and temporal continuity (Physics IV.11, 219^a10–13, in Aristotle 1984, p. 371). It is this reflection or isomorphism, for Aristotle, that endows continuous motion with its familiar characteristic properties such as the absence of spatial jumps and the absence of temporal pauses.

Aristotle's characterization of the continuous emerges as the culmination of the following series of definitions he offers in the third chapter of Book V of his Physics :

Things are said to be together in place when they are in one primary place. … Things are said to be in contact when their extremities are together. … A thing is in succession when it is after the beginning in position or in form or in some other respect in which it is definitely so regarded, and when further there is nothing of the same kind as itself between it and that to which it is in succession. … A thing that is in succession and touches [i.e., is in contact] is contiguous. … The continuous is a subdivision of the contiguous: things are called continuous when the touching limits become one and the same … (Physics V.3, 226^b22–227^a12, in Aristotle 1984, pp. 383–384)

Aristotle maintains that the previous definition implies that

nothing that is continuous can be composed of indivisibles: e.g. a line cannot be composed of points, the line being continuous and the point indivisible. For the extremities of two points can neither be one (since of an indivisible there can be no extremity as distinct from some other part) nor together (since that which has no parts can have no extremity, the extremity and the thing of which it is the extremity being distinct). … Moreover, it is plain that everything continuous is divisible into divisibles that are always divisible; for if it were divisible into indivisibles, we should have an indivisible in contact with an indivisible, since the extremities of things that are continuous with one another are one and in contact. (Physics VI.1, 231^a24–29, 231^b15–19, in Aristotle 1984, pp. 390–391)

For Aristotle, the infinite divisibility of the continuous—a property, which on occasion, he appears to take to define the continuous—is a potential infinite. Indeed, for Aristotle, the infinite, which is a property of a process rather than of a collection or of a substance, is always potential as opposed to actual or completed; that is, no matter which finite stage of the process has been completed, in principle another such stage can be completed. Processes may be infinite with respect to addition or division. Moreover, in the case of spatial continua, in particular, it is the very process of division from which points arise. Thus, while a line segment contains an infinite number of points and an infinite number of parts, for Aristotle it does so only potentially. It is the infinite divisibility of the continuum in this sense that Aristotle appeals to in his treatment of the various paradoxes of Zeno of Elea that are intended to challenge the coherence of the continuity of space, time, matter, and motion. It was also this conception of the continuum that was the dominate conception among philosophers, scientists, and mathematicians alike until the time of Cantor and Dedekind.

Nonstandard Ancient Conceptions

However, while Aristotle's theory was the dominate theory until well into the nineteenth century, it never achieved hegemony. Among the ancients, in particular, there were a number of alternative conceptions of continua, including a variety of atomistic conceptions (Furley 1967, Sorabji 1983, White 1992) and the nonatomic conception of the Stoics (Sambursky 1959, White 1992). While atomic theories tended to apply solely to the physical realm, there appear to have been atomistic conceptions of geometrical continua as well. Democritus, for example, apparently held that a cone was made up of an infinite number of parallel sections, each of the same indivisible thickness; some who sought to square the circle, including Antiphon, also appear to have embraced atomic theories of geometrical objects. The Stoics, on the other hand, while continuing to adhere to the Aristotelian conception in the mathematical realm, and even to infinite divisibility in the physical realm, may well have distanced themselves from the standard conception in an important respect. According to the interpretation introduced by Shmuel Sambursky (1959) and championed by Michael J. White (1992), the Stoics maintained that there are no points, edges, and surfaces serving as sharp boundaries in physical continua, but rather regions of indeterminacy in which the parts of bodies and adjacent bodies blend. Sambursky (1959, p. 98) likens the physical continuum of the Stoics to the fluid intuitionistic conception of L. E. J. Brouwer, and White proposes instead that "[p]erhaps the best place to look for contemporary elucidation of the Stoic idea is the nonstandard mathematics based on L. A. Zadeh's fuzzy-set theory" (1992, p. 288).

Unlike the Stoics, Aristotle maintains that the physical continuum is a reflection of the geometrical con-tinuum. Indeed, according to Aristotle, "geometry investigates natural [ie. physical] lines, but not qua natural" (Physics II.2, 194^a9–10, in Aristotle 1984, p. 331). It was this widely held ancient view that the physical mirrors the geometrical that bequeathed to the geometers and their ideas regarding continua an influence far beyond the mathematical domain.

Ancient Geometrical Conceptions

Aspects of the distinction between discrete number and continuous magnitude are conspicuous in Euclid's Elements. Whether Euclid (flourished c. 300 BCE) was directly influenced by the Aristotelian corpus or by earlier geometric practice, however, is the subject of dispute. In any case, reminiscent of Aristotle's characterization, Euclid regards number as a multitude composed of units and believing that one is not itself a number but the unit of number, he appears to identify the numbers with the positive integers greater than one. Geometrical magnitude, on the contrary, for Euclid, is infinitely divisible. Line segments, in particular, can not only be bisected (Book 1, Proposition 10) ad infinitum, they can be divided into n congruent segments for each positive integer n (Book 6, Proposition 9).

Other ingredients of the Euclidean synthesis that shed important light on the nature of the classical conception of the geometrical continuum are the theories of proportions and incommensurable magnitudes presented in Books 5 and 10, respectively, and the so-called method of exhaustion that is developed in Book 12.

Though arguably the result of an evolutionary process (Knorr 1975, 1978), the theory of proportions developed in Book 5 is usually attributed in its entirety to Eudoxus (c. 400–c. 350 BCE), who, like his contemporaries Plato and Aristotle, lived about half a century before Euclid's Elements were compiled. Central to the theory is the concept of a ratio :

A ratio [says Euclid] is a sort of relation in respect of size between two magnitudes of the same kind. … Magnitudes are said to have a ratio to one another which are capable, when multiplied [by a positive integer], of exceeding one another. … Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are equal to, or alike fall short of the latter equimultiples respectively taken in corresponding order. (Book 5, Definitions 3–5 in Heath 1956, Volume 2, p. 114)

While Euclid never contends that two magnitudes of the same kind necessarily have a ratio to one another, his geometry (with the one possible exception of his treatment of horn angles [Book 3, Proposition 16]) is limited to systems of magnitudes for which this is the case. Following Otto Stolz (1842–1905), such systems are said to satisfy the axiom of Archimedes (although it is Eudoxus to whom Archimedes (c. 287–212 BCE) attributes the proposition). In contemporary parlance, if A and B are members of a given system of magnitudes, A is said to be infinitesimal relative to B if A and B do not have a ratio to one another and A is smaller than B. Collaterally, if A is infinitesimal relative to B, B is said to be infinite relative to A. Thus, in Euclid's geometry, no line segment is either infinitesimal or infinite relative to another segment, and analogous results hold for planer and solid figures as well. Moreover, as in the case of line segments, where there is a well-defined means of subtracting the smaller of two magnitudes of the same kind from the larger, the absence of infinitesimal magnitudes of a given kind precludes the existence, more generally, of magnitudes of a given kind that differ by an infinitesimal amount.

Among the virtues of the theory of proportions of Book 5 is that, unlike the older Pythagorean theory that was based on ratios of integers, it is applicable to both commensurable and incommensurable magnitudes. Following Euclid, "Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure" (Book 10, Definition 1 in Heath 1956, Volume 3, p. 10). The commensurable-incommensurable dichotomy is as close as the ancients came to the modern dichotomy of rational and irrational numbers, a dichotomy that is central to the Cantor-Dedekind conception of the continuum. The discovery of the existence of incommensurable magnitudes, which is usually attributed to the fifth-century Pythagoreans, was significant because it showed that not every pair of magnitudes of the same kind (straight line segments, rectilinear plane figures bounded by such segments, and so on) has a common measure that divides each an exact integral number of times. Thus, for example, since the side and diagonal of a square are incommensurable, it was not possible (given the ancients' conception of measure) to measure all the sides of even so simple a figure as an isosceles right triangle employing a common unit of measure. These and related discoveries, coupled with their conception of number as a multitude of units, convinced the ancients that it was impossible to bridge the gap between the discrete domain of number and the continuous domain of geometry.

Guided by their intuitions about geometrical continua, the Greeks assumed that simple curvilinear planer figures such as circles, ellipses, and segments of parabolas have areas and perimeters of the same kinds as those of polygons, and they made analogous assumptions about the surface areas and volumes of solids such as spheres, cylinders, and cones. The misleadingly term exhaustion was introduced by Gregory of St. Vincent (1584–1667) to describe the method devised by Eudoxus, incorporated into the Elements by Euclid, and later extended by Archimedes to measure these and other lengths, areas, and volumes in a rigorous fashion without appealing to either the infinitesimal techniques of the Newtonian and Leibnizian calculi or the passage to the limit concept that has been characteristic of the standard approach to calculus since its arithmetization during the latter part of the nineteenth century.

As early as 430 BCE Hippocrates of Chios established that the area of a lune (that is, a curvilinear area bounded by circular arcs) of a particular kind is equal to the area of a square. Soon thereafter, Antiphon contended it was possible to obtain a rectilinear figure having the same area as a circle by beginning with an inscribed regular polygon, say a square, and constructing successively more inclusive inscribed regular polygons until the area of the circle was exhausted. Surprisingly, however, he held that the area of the circle would be exhausted after a sufficiently large finite number of steps (perhaps believing that the side of the polygon would coincide with a small arc of the circumference of the circle). Bryson (c. 420 BCE) later developed an alternative account where he considered a circle C sandwiched between a finite series of successively more inclusive inscribed regular polygons, on the one hand, and a finite series of successively less inclusive circumscribed regular polygons, on the other. He maintained that for some positive integer n there is an n -sided regular polygon P, whose area equals the area of C, that properly contains and is properly contained within the aforementioned inscribed and the circumscribed polygons, respectively. To reach his conclusion he appears to have invoked a continuity principle to the effect that a magnitude passes from a smaller to a greater value solely through values of magnitudes of the same kind. The reliance on this principle, which was criticized by Proclus, John Philoponus, and others, was later obviated by the Eudoxean approach.

Central to the exhaustion approach is an alternative continuity principle—the so-called bisection principle —that, following Euclid, may be stated as follows:

Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than or equal to its half, and from that which is left a magnitude greater than or equal to its half, and if this process is repeated continuously, there will be left some magnitude which will be less than the lesser magnitude set out. (Book 10, Proposition 1 in Heath 1956, Volume 3, p. 14)

Using the Archimedean axiom, Euclid proves the bisection principle for the case where the magnitudes subtracted are greater than half the given magnitude, and immediately thereafter observes that "the theorem can be similarly proved even if the parts subtracted be half" (Book 10, Proposition 1 in Heath 1956, Volume 3, p. 15).

The exhaustion method essentially consists of showing that the magnitude (or, more often, ratio of magnitudes) M in question is equal to another magnitude (or ratio of magnitudes) M′ one already knows how to determine by showing—using a pair of reductio ad absurdum arguments, each of which employs the principle of bisection—that neither M < M′ nor M′ < M. To draw the conclusion that M′ = M, a tacit appeal is made the presupposition, alluded to earlier, that either M < M′, M′ < M, or M′ = M. In the version developed by Archimedes, by evoking the geometrical properties of the geometrical object whose perimeter, area, or volume is to be determined, two sequences I ₁, I ₂, …, I_n, and C ₁, C ₂, …, C_n, consisting respectively of inscribed and circumscribed lines, polygons, or polyhedrons are constructed whose corresponding perimeters, areas, or volumes are such that I _n < M < C _n and I _n < M′ < C _n for all n. Using the principle of bisection, it is then either shown that, given ℇ > 0, C _n − I _n < ℇ for n sufficiently large or that, given α > 1, C_n /I_n < α for n sufficiently large.

Archimedes's version of the method of exhaustion resembles and, to some extent, inspired the technique of integration later employed in the calculus. Before the development of the calculus, however, a variety of the concepts and techniques inherited from the ancient geometers would undergo marked change. Of these perhaps none has had a more profound or lasting impact on theories of continua than the rethinking of the number concept and its relation to the geometrical continuum.

Early Modern Theory of Real Numbers

The early modern theory of real numbers began to emerge when mathematicians such as Simon Stevin (1548–1620) argued that not only is 1 also a number but that there is also a complete correspondence between (positive) number and continuous magnitude, as well as a parallelism between certain geometrical constructions and the now familiar arithmetic operations on numbers. In a number of his works, including his influential L'Arithmétique (1585), Stevin expresses the matter thus:

I consider the relation between number and magnitude to be such that what can be done by the one can be done by the other. …

… these two quantities cannot be distinguished by continuity or discontinuity.

To a continuous magnitude corresponds the continuous number to which it is attributed. (Strong 1976, p. 105; Klein 1968, p. 195; Stevin 1585/1958, p. 501)

This viewpoint soon led to, and was implicit in, the analytic geometry of René Descartes (1596–1650), and was made explicit by John Wallis (1616–1703) and Isaac Newton (1643–1727) in their arithmetizations thereof. Influenced by work of Wallis, Isaac Barrow (1630–1677), and others, the (positive) numbers came to be associated with Eudoxian-Euclidean ratios that were assumed to exist between the magnitudes of a given kind and a selected unit magnitude of the same kind (compare Klein 1968, Pycior 1997). In accordance with the Eudoxian-Euclidean framework, no two such magnitudes of the same kind could differ by an infinitesimal amount. In his popular and influential Arithmetica Universalis (1707/1728), Newton extends the correspondence between numbers and ratios to include negative numbers and zero, but whereas some of his predecessors had identified the positive numbers with the symbolic representations of the ratios, Newton identifies numbers with the abstracted ratios themselves. Emphasizing his sharp break with the ancient conception of number, says Newton:

By Number we understand not so much a Multitude of Unities, as the abstracted Ratio of any Quantity, to another Quantity of the same Kind, which we take for Unity. And this is threefold; integer, fracted, and surd: An Integer is what is measured by Unity, a Fraction, that which a submultiple Part or Unity measures, and a Surd, to which Unity is incommensurable. (Newton 1728/1967, p. 2)

That zero could not be a number in accordance with this definition did not preclude Newton from asserting it was, and the careful treatment that late nineteenth-century mathematicians would recognize to be required to handle ratios involving negative quantities is nowhere to be found.

The Calculus of Newton and Leibniz

During the sixteenth century the works of Archime-des were widely studied by Western mathematiciansand served as the chief source of inspiration for theseventeenth-century development of the infinitesimal calculus, the branch of mathematics erected by Newton and Gottfried Wilhelm Leibniz (1646–1716) for the study of continuously varying magnitudes or quantities. The conception of the continuum embraced by most mathematicians of the period was geometrical or kinematical by nature and grounded in intuition. It was commonplace to consider a curve as a path of a moving point, the curve being continuous insofar as motion itself was presumed to be continuous. Moreover, perhaps as a result of the medieval speculations on the infinite and the continuum, the mathematicians of the day, unlike their mainstream Greek counterparts, were not adverse to employing infinitesimal techniques and appeals to the actual infinite in these and related works. Some authors, such as Galilei Galileo (1564–1642), following in the footsteps of such fourteenth-century thinkers as Henry of Harclay, Nicholas Bonet, Gerard of Odo, Nicholas of Autrecourt, and John Wyclif (Murdoch 1982), maintained that line segments, surfaces, and solids are made up of an actual infinite number of indivisible or infinitesimal elements. And similar ideas were employed by Johannes Kepler (1571–1630), Bonaventura Cavalieri (1598–1647), and others in their determinations of areas and volumes and by Barrow in his determinations of tangents to curves, determinations that would be the focus of the unifying algorithmic frameworks that would come to be called the calculus.

Following in the footsteps of their just-cited forerunners, infinitesimal techniques were employed by Newton and Leibniz in their treatments of the calculus, but unlike some of their predecessors neither of them attributed ontological status to either the actual infinite or the actual infinitesimal. Both regarded infinitesimals—or incomparables as Leibniz sometimes called them—as varying quantities in a state of approaching zero that serve as useful fictions to abbreviate their mathematical proofs. The abbreviated proofs in turn, they contended, could be replaced by limit-based proofs the latter of which not only constitute the rigorous formulation of calculus but are a direct version of the indirect method of exhaustion due to Archimedes. Newton and Leibniz also agreed that the justification for the limit-based proofs lay in the concept of continuity but they differed on the justification itself. Whereas Newton sought it in terms of one's intuition of continuous flow, Leibniz sought it in terms of his law of continuity, a philosophical principle to the effect that "[n]o transition is made through a leap" or that "nothing takes place suddenly." The natures of their respective attempts at justification, however, only begin to intimate that the limit-based proofs envisioned by Newton and Leibniz, while akin to, are by no means identical with the limit-based proofs that emerged during the nineteenth century.

Unlike the calculus of today, the calculi of Newton and Leibniz were not concerned with functions but with variable quantities, their rates of change, and so on. However, whereas Newton regarded these quantities as varying at finite rates with respect to time, Leibniz envisioned them as ranging over discrete sequences of values that successively differ by infinitesimal amounts. Underlying this difference was a difference in fundamental concepts: For Newton it was the fluxion or finite rate of change of the variable with respect to time, and for Leibniz it was the just-cited infinitely small differences or the differential. As a result, in Leibniz's treatment of the calculus the limit concept was suppressed or at least disguised, whereas it was explicit in Newton's formulation. In the case of differentiation, for example, since for Leibniz it is the distinct differentials dx and dy that are fundamental, their ratio dy /dx is of principal significance, whereas for Newton, especially in his later treatment, it is the derivative itself—as a ratio of fluxions or an ultimate ratio of evanescent quantities—that is of central importance.

That there were foundational difficulties with the science of continuously varying quantities was well known among seventeenth- and eighteenth-century mathematicians including Newton and Leibniz themselves. These difficulties were brought into sharp focus by George Berkeley (1685–1753) in his stinging critique of the logical and ontological foundations of the calculus titled The Analyst (1734/1992). According to Berkeley, there is no justification for attributing existence to either limits or infinitesimals: A limit of a ratio is either a limit of finite quantities, and therefore not an ultimate ratio, as Newton contended, or it is a mysterious indeterminate ratio 0/0; and if the infinitesimal quantities dx and dy are not equal to zero, one has the problem of explaining how it is possible that x + dx = x, and if they are equal to 0, once again one has the problem of explaining the meaning of the indeterminate 0/0. It was with these and related quandaries that mathematicians concerned with the study of continuously varying magnitude grappled well into the nineteenth century. Some eighteenth-century mathematicians, such as Colin Maclaurin (1698–1746) and Jean Le Rond d'Alembert (1717–1783), attempted to address the foundational worries with refinements of the limit approach of Newton, but it was the infinitesimal approach of Leibniz that emerged as the dominant approach of the day. Moreover, the remaining puzzlement over infinitesimals no longer applied solely to the fictional infinitesimals of Leibniz, but to the actual infinitely large and the actual infinitely small numbers and magnitudes employed to great effect throughout the eighteenth and early nineteenth century by a host of distinguished analysts working in the Leibnizian tradition including Jakob Bernoulli (1654–1705), Johann Bernoulli (1667–1748), Daniel Bernoulli (1700–1782), and Leonhard Euler (1707–1783), to name only a few.

The Arithmetization of Analysis

During the nineteenth century, building on the work begun in 1821 by Augustin-Louis Cauchy (1789–1857) in his Cours d'analyse, the calculus was given a rigorous foundation that is still accepted today. By the middle of the century, developments in subject persuaded many mathematicians that the traditional concepts of the calculus were too imprecise, unreliable, and ineffective to provide such a basis. It was held that the traditional relation between real quantities and intuitively given continuous magnitudes such as straight lines was more of a hindrance than an aid in achieving that end as was the then familiar reliance on infinitesimals. In response, the modern arithmetico-set-theoretic conception of a real number emerged when a number of mathemati-cians including Cantor (1872/1939) and Dedekind (1872/1996) introduced systems of real numbers that were designed to dispense with the former and provide a basis for the calculus that made superfluous the latter.

Cantor's system is based on Cauchy sequences of rational numbers. A sequence {r _n } of rational numbers (indexed over the natural numbers) is said to be a Cauchy sequence (or fundamental sequence, as Cantor called it) if for every rational number ε > 0 there is a natural number k such that |r _m − r _n | < ε for all m, n > k. Two such sequences {r _n } and {s _n } are said to be equivalent if for every rational number ε > 0 there is a natural number k such that |r _n − s _n | < ε for all n > k. In modern parlance, Cantor's construction amounts to identifying the set ℝ of real numbers with the set of all equivalence classes of Cauchy sequences of rational numbers. If an equivalence class contains the Cauchy sequence {r _n } where r _n = r for all n, the equivalence class corresponds to the rational number r, otherwise it corresponds to an irrational number. Each irrational number is associated with the equivalence class containing the Cauchy sequence consisting of the initial segments of its unique nonperiodic decimal representation. For example, √2 is associated with the equivalence class containing the Cauchy sequence
r ₀ = 1; r ₁ = 1.4; r ₂ = 1.41; r ₃ 1.414; …
If α and β are real numbers represented by the Cauchy sequences {r _n } and {s _n }, then the sum and product of α and β are represented by the Cauchy sequences {r _n + s _n } and {r _n s _n }, respectively; and it is said that α > β if there is an a > 0 such that r _n ≥ s _n + a for sufficiently large n.

Dedekind's system, by contrast, is based on cuts of the ordered set ℚ of rational numbers. By a cut (A ₁, A ₂) of ℚ Dedekind means a partition of ℚ into two nonempty sets A ₁ and A ₂ in which every member of A ₁ precedes every member of A ₂. Dedekind identities ℝ with the set of all cuts of ℚ; if A ₁ has a greatest member or A ₂ has a least member, say, r, the cut (A ₁, A ₂) is associated with the rational number r ; otherwise it defines an irrational number. √2, in particular, is defined by the cut
({a ∈ ℚ:a < 0 ∨ a ² ≤ 2}, {a ∈ ℚ:a ² > 2}).
Given two real numbers α = (A ₁, A ₂) and β = (B ₁, B ₂), Dedekind stipulates that α < β if A ₁ is a proper subset of A ₂. He also defines α + β = (C ₁, C ₂) where C ₂ = ℚ − C ₁ and C ₁ is the set of all c ∈ ℚ such that c ≤ a ₁ + b ₁ for some a ₁ ∈ A ₁ and some b ₁ ∈ B ₁, and further observes that the remaining familiar arithmetic operations on real numbers can likewise be defined.

On the basis of Cantor's and Dedekind's systems of real numbers, whose equivalence (modulo the then tacitly emerging underlying set-theoretic assumptions) would soon be established, the classical concepts of the calculus, including Cauchy's and Bernhard Riemann's (1826–1866) modern definitions of the derivative and the integral, were reformulated in a rigorous fashion using the now familiar δ, ε -techniques, as were the concepts of convergence, sum of an infinite series, and continuity to name only a few. Central to this development was Karl Weierstrass's (1815–1897) replacement of Cauchy's dynamic conception of the limit concept, together with its Newtonian connotations of continuous motion, with a static purely arithmetical formulation. Instead of setting lim_{x →ξ}  f (x ) = L provided that f (x ) approaches L as x approaches ξ, Weierstrass set lim_{x →ξ}  f (x ) = L provided that, given a positive real number ε, there is a positive real number δ such that |f (x ) − L | < ε if 0 < |x − ξ | < δ. With the host of limit dependent concepts so reformulated, the calculus assumed the form that one still finds in the standard textbooks of the early twenty-first century.

Continuous Functions

As was already noted, the calculus of Newton and Leibniz was not a calculus of functions. It was Euler in the middle of the eighteenth century who placed the concept of function and, in particular, the concept of continuous function at the center of analysis, and it was Cauchy in 1821, and independently Bernard Bolzano (1781–1848) in 1817, who gave the concept its now standard meaning. Following these authors, a function f (x ), defined in a neighborhood of a point ξ, is said to be continuous at ξ if lim_{x →ξ}  f (x ) = f (ξ ); and f (x ) is said to be continuous in a closed interval [a, b ] if it is continuous at each point of the interval, it being understood that the limits corresponding to the endpoints a and b are right-sided and left-sided limits, respectively. Thus, within the Weierstrassian framework, f (x ) is continuous in a closed interval iff for each member ξ of the interval and for each positive real number ε there is a positive real number δ such that |f (x ) − f (ξ )| < ε whenever |x − ξ | < δ.

The Cauchy-Bolzano conception of continuity accords nicely with the intuition that the values of a continuous function f differ slightly when its arguments differ slightly and, hence, with its geometric analog that the graph of f does not have a break or jump in the interval in question. Indeed, using the Cauchy-Bolzano definition nineteenth-century mathematicians were able to show that formal replacements of a number of the familiar intuitions about continuous functions and curves could be established as theorems including the following two:

The Intermediate Value Theorem. If f is a continuous function on a closed interval [a, b ] of ℝ and f (a ) < ξ < f (b ) for some ξ, then there is a c ∈ (a, b ) with f (c ) = ξ ;

Extreme Value Theorem. If f is a continuous function on a closed interval [a, b ] of ℝ, then f has a maximum value at some c ∈ [a, b ] and f has a minimum value at some d ∈ [a, b ].

The Cauchy-Bolzano conception of continuity is local by nature, referring to the behavior of the function in the neighborhood of a point. Even the notion of continuity in a closed interval is defined in terms of continuity at every point of the interval. A more global conception of continuity that gradually emerged during the process of rigorization that implies but is not implied by the Cauchy-Bolzano conception is that of uniform continuity. Following Heinrich Heine (1821–1881), who carefully separated the two notions that had apparently been conflated by Cauchy, a function f is said to be uniformly continuous on a set A of real numbers if, for each positive real number ε, there is a positive real number δ such that for each pair of members ξ and ξ′ of A, |f (ξ ) − f (ξ′ )| < ε whenever |ξ − ξ′ | < δ. Essentially, this asserts that for a given ε, the same δ for the continuity condition works for all members of A. The following result, which is of central importance in both standard analysis and a number of the nonstandard alternatives that will be discussed later on, is also due to Heine: f is uniformly continuous on a closed interval [a, b ] of ℝ, whenever f is continuous on [a, b ].

It is important to emphasize that the class of functions that are deemed to be continuous by standard analysts are more inclusive than those envisioned implicitly or explicitly by their seventeenth-, eighteenth-, and even early nineteenth-century predecessors. Basically, all the curves treated by seventeenth-century analysts were expressed everywhere by one and the same algebraic or transcendental equation and were, accordingly, continuous in the now standard sense. In the mid-eighteenth century so-called discontinuous functions were introduced into analysis by Euler, though they would not be recognized as such today. According to Euler's distinction, which was used up to the time it was replaced by that of Bolzano and Cauchy, a function is continuous if it is characterized by a single analytic expression, and it is discontinuous if it lacks any analytic expression, as in the case of freehand curves, or, if it is defined by different analytic expressions in a finite number of different intervals, the points at which the analytical expressions change being the points of discontinuity.

Euler's points of discontinuity correspond to points of the curve having no well-defined derivative. Accordingly, if one thinks of a continuous curve as the path of a moving point—an intuition that played an important heuristic role in the development of the calculus—Euler's points of discontinuity correspond to points at which the moving point has no well-defined direction. With this in mind it is not difficult to understand why during much of the nineteenth century it was widely believed that functions that are continuous in the Cauchy-Bolzano sense may fail to have derivatives at no more than a finite number of points. In fact, a number of mathematicians including Bolzano himself attempted to prove just this. Mathematicians were therefore surprised when in 1861 Weierstrass provided an example of a continuous function that is nowhere differentiable. A similar such function was constructed by Bolzano in 1834, but like the remainder of his work, it did not become known to the mathematical community till after the work of Weierstrass.

The Cauchy-Bolzano conception of discontinuity, by contrast with Euler's, is closer to that of discontiguity, an extreme case being P. G. Lejeune Dirichlet's (1805–1859) so-called monster function—the nowhere continuous function defined on the real line by the condition

Euler apparently was aware of the existence of discontinuous functions in the modern sense (Youschkevitch 1976, pp. 64–65), but they did not play a fundamental role in the calculus of his time. With the work of Riemann on the convergence of Fourier series during the middle of the nineteenth century, however, this all changed, and they have come to enjoy widespread application not only in analysis, but also in empirical science from where they came. Indeed, referring to their early nineteenth-century roots in the work of Joseph Fourier, the great philosopher-mathematician Jules Henri Poincaré (1854–1912) musingly observes:

Fourier's series is a precious instrument of which analysis makes continual use, it is by this means that it has been able to represent discontinuous functions; Fourier invented it to solve a problem of physics relative to the propagation of heat. If this problem had not come up naturally, we should never have dared to give discontinuity its rights; we should still long have regarded continuous functions as the only true functions. (Poincaré 1913, p. 286)

The Cantor-Dedekind Continuum

Central to Cantor's and Dedekind's constructions of the real number system was the underlying belief that only after providing a precise definition of a continuum based on the science of number would it be possible to lend precision to the idea of the continuity of the Euclidean straight line and of continuous magnitude more generally. For this purpose they turned to properties of ℝ whose continuity they assumed as a mathematico-philosophical thesis. According to Cantor, the continuity of ℝ consists in its being both connected and Cauchy complete ; it is connected because whenever a and a′ are elements of the system such that a < a′, then for any positive element ε of the system, there is a finite sequence a ₁, …, a _n of elements of the system where a < a ₁ < … < a _n < a′ such that a ₁ − a, …, a′ − a _n < ε ; and it is Cauchy complete since every convergent sequence of elements of the system has a limit in the system. Dedekind, by contrast, identified the continuity of ℝ with its being a totally ordered system having what is today called the Dedekind continuity property; that is, whenever the system is partitioned into two nonempty subsets X and Y such that every member of X precedes every member of Y, then either X has a greatest member or Y has a least member, but not both.

Connectivity, in Cantor's sense, was soon recognized to be equivalent the Archimedean axiom for a large class of structures including ordered fields, the latter of whose roots lie in analytic geometry. Indeed, since the time that Wallis and Newton incorporated directed segments into Cartesian geometry, it had been loosely understood that given a unit segment AB of a line L of a classical Euclidean space, the collection of directed segments of L emanating from A including the degenerate segment AA itself constitutes an Archimedean ordered field with AA and AB the additive and multiplicative identities of the field and addition and multiplication of segments suitably defined. These ideas were made precise by Giuseppe Veronese (1854–1917) in his Fondamenti di geometria (1891) and by David Hilbert (1862–1943) in his Grundlagen der Geometrie (1899), works on the foundations of geometry from which the modern conceptions of Archimedean and non-Archimedean ordered fields emerged. It was also these and related works on the foundations of geometry that gave rise to the following familiar characterizations of the arithmetic continuum whose continuity properties are associated with Cantor, Dedekind, Bolzano, and Hilbert, respectively:

(1) ℝ is (up to isomorphism) the unique Archimedean ordered field that is Cauchy complete

(2) ℝ is (up to isomorphism) the unique Dedekind continuous ordered field

(3) ℝ is (up to isomorphism) the unique ordered field having the least upper bound property (that is, every subset of the system that is bounded above has a least upper bound)

(4) ℝ is (up to isomorphism) the unique Archimedean ordered field that admits no proper extension to an Archimedean ordered field

Each of these characterizations of ℝ makes use of metrical conceptions. However, in 1895 Cantor demonstrated that it is possible to provide a categorical characterization of the ordered set of real numbers and, hence, of a Cantor-Dedekind linear continuum, using order-theoretic concepts alone. Another such characterization that emerged soon thereafter is the following one given by Edward V. Huntington (1874–1952) that indicates what besides simple density—the set-theoretic analog of infinite divisibility—is required to characterize the order type of ℝ (1917):

(1) 〈K, <〉 is a totally ordered set having neither a first element nor a last element;

(2) 〈K, <〉 is dense, that is, if a and b are elements of K such that a < b there is an element x in K such that a < x < b ;

(3) 〈K, <〉 is Dedekind complete, that is, if K ₁ and K ₂ are any two nonempty subsets of K, such that every element belongs either to K ₁ or K ₂ and every element of K ₁ precedes every element of K ₂, then there is at least one element x in K such that any element that precedes x belongs to K ₁, and every element that follows x belongs to K ₂;

(4) the class K contains a denumerable subset K′ in such a way that between any two elements of K there is an element of K′.

Reflecting on the then newly developed order-theoretic conception of the mathematical continuum, the great philosopher-mathematician Jules Henri Poincaré perceptively remarks that:

[t]he continuum so conceived is only a collection of individuals ranged in a certain order, infinite in number, it is true, but exterior to one another. This is not the ordinary conception, wherein is supposed between the elements of a continuum a sort of intimate bond which makes of them a whole, where the point does not exist before the line, but the line before the point. Of the celebrated formula, "the continuum is unity in multiplicity," only the multiplicity remains, the unity has disappeared. The analysts are none the less right in defining their continuum as they do, for they always reason on just this as soon as they pique themselves on rigor. But this is enough to apprise us that the veritable mathematical continuum is a very different thing from that of the physicist and that of the metaphysician. (1913, p. 43)

To some extent these views are a reflection of those of Cantor and Dedekind themselves. For example, distancing himself from a long line of metaphysicians, Cantor writes:

The concept of the "continuum" has … always evoked the greatest differences of opinion and even vehement quarrels. This lies perhaps in the fact that, because the exact and complete definition of the concept has not been bequeathed to the dissentients, the underlying idea has taken on different meanings; but it may also be (and this seems to me the most probable) that the idea of the continuum had not been thought out by the Greeks (who may have been the first to conceive it) with the clarity and completeness which would have been required to exclude the possibility of different opinions among their posterity. Thus we see Leucippus, Democritus, and Aristotle consider the continuum as a composite which consists from parts divisible without end, but Epicurus and Lucretius construct it out of their atoms considered as finite things. Out of this a great quarrel arose among the philosophers, of whom some followed Aristotle, others Epicurus; still others, in order to remain aloof from this quarrel, declared with Thomas Aquinas that the continuum consisted neither of infinitely many nor of a finite Anzahl [number] of parts, but of absolutely no parts. … Here we see the medieval-scholastic origin of a point of view which we still find represented today, in which the continuum is thought to be an unanalysable concept, or as others express themselves, a pure a priori intuition which is scarcely susceptible to a determination through concepts. Every arithmetical attempt at determination of this mysterium is looked on as a forbidden encroachment and repulsed with due vigor. Timid natures thereby get the impression that with the "continuum" it is not a matter of a mathematically logical concept but rather of religious dogma. (1883/1996, p. 903)

Moreover, as to the necessity of even conceiving space as continuous, Dedekind remarks, "If space has at all a real existence it is not necessary for it to be continuous" (1872/1996, p. 772). Indeed,

If anyone should say that we cannot conceive of space as anything else than continuous, I should venture to doubt it and call attention to the fact that a far advanced, refined scientific training is demanded in order to perceive clearly the essence of continuity and to understand that besides rational quantitative relations, also irrational, and besides algebraic, also transcenden-tal quantitative relations are conceivable. (1888/1996, p. 793)

Bertrand Russell, who played the leading role in introducing the ideas of Cantor and Dedekind to the English-speaking philosophical world, goes even further when he remarks, "Whether the axiom of continuity be true as regards our actual space, is a question I see no means of deciding. For any such question must be empirical, and it would be quite impossible to distinguish empirically what may be called a rational space from what might be called a continuous space" (1903, p. 440).

However, despite these and other such pronouncements made in the years bracketing the turn of the twentieth century, and despite the ongoing speculation about quantized space and time that emerged soon thereafter with the advent of the quantum theory (compare Cepek 1961, pp. 223–243; Sorabji 1983, pp. 381–383, 447) and that was redirected toward overcoming the difficulties of harmonizing the quantum theory with the relativistic theory of gravity and space-time (compare Markopoulou 2004; Smolin 2001, 2004), it became and remains commonplace among philosophers and physicists to assume not only that space and time are continuous, as most of their modern predecessors had supposed, but also that they are continuous in the sense of Cantor and Dedekind. Whether this assumption should be construed instrumentally or realistically there is a multiplicity of views (compare Earman 1989, chapters 8 and 9; Maddy 1997, chapter 6; Hellman 1998), as are the views regarding the testable status of the thesis itself (compare Forrest 1995, Markopoulou 2004).

Modern Euclidean Geometry and the Continuum

At least as far back as the seventeenth century there were thinkers who observed that there are places in the Elements where Euclid tacitly employs continuity assumptions that are not warranted by the axioms and common notions he assumes. For example, in his proof that given any segment, there is an equilateral triangle having the given segment as one of its sides (Book 1, Proposition 1), Euclid assumes

The Circular Continuity Principle : If a circle C has one point inside and one point outside another circle C′, then the two circles intersect in two points.

And in his proof that through a point outside a given line there is a line perpendicular to the given line (Book 1, Proposition 12), he assumes

The Line-Circle Continuity Principle : If one endpoint of a segment is inside a circle and the other outside, then the segment intersects the circle at one point.

Among the thinkers who thus criticized Euclidwas Leibniz in his Specimen geometriae luciferae (c. 1695/1962, p. 284). Such criticisms were significant not only because they drew attention to gaps in Euclid's reasoning but also because they intimated that, contrary to the ancient and the then still standard view, infinite divisibility is not sufficient for continuity. In fact, in the just-cited paper as Ernst Cassirer (1902, p. 183) importantly observes, Leibniz departed from his usual acceptance of the standard view and explicitly stated just this.

During the late nineteenth century geometers showed that by supplementing the then newly emerging refinements of Euclid's system of axioms with continuity axioms that ensured the satisfaction of the Cantor-Dedekind axiom, one could establish the circular continuity principle and the line-circle continuity principle and with these all of Euclid's continuity needs (compare Heath 1956, pp. 1:234–240; Greenberg 1993, pp. 93-101). However, as Cantor and Dedekind were aware, whereas the Cantor-Dedekind axiom suffices for the continuity needs of Euclid, it goes beyond those needs. Cantor makes this point, albeit only implicitly, when (following his proof-sketch that to each point of the Euclidean line there corresponds a real number) he maintains:

In order to complete the connection of numerical quantity with the geometry of the straight line, one must only add an axiom which simply says that conversely every numerical quantity also has a determined point on the straight line, whose coordinate is equal to that quantity. … I call this proposition an axiom because by its nature it cannot be generally proved. (Cantor 1872/1932, p. 96)

And Dedekind makes the point more forcefully and explicitly when he revealingly observes:

If we select three non-collinear points A, B, and C at pleasure, with the single limitation that the ratios of the distances AB, AC, BC are algebraic numbers, and regard as existing in space only those points M, for which the ratios of AM, BM, CM to AB are likewise algebraic numbers, then it is easy to see that the space made up of the points M is everywhere discontinuous. But in spite of this discontinuity, and despite the existence of gaps in this space, all constructions that occur in Euclid's Elements, can, so far as I can see, be just as accurately effected here as in a perfectly continuous space; the discontinuity of this space would thus not be noticed in Euclid's science, would not be felt at all. (1888/1996, p. 793)

The ordered field of real algebraic numbers to which Dedekind is referring is an instance of a Euclidean ordered field, that is, an ordered field in which every positive element is the square of some element of the field. Besides being Euclidean, the ordered field of real algebraic numbers is both countable and Archimedean. During the twentieth century it emerged that a model of all the axioms of (a modern refined version of) Euclidean geometry less the Archimedean axiom and the Cantor-Dedekind axiom satisfies the circular continuity principle iff it satisfies the line-circle continuity principle iff a line of the model is modeled by a Euclidean ordered field (compare Tarski 1959/1986; Hartshorne 2000, pp. 104–112, 144–145; Greenberg 1993, pp. 143–144). If, following Euclid, the Archimedean axiom is also assumed, the Euclidean ordered fields must be Archimedean as well. It is essentially for this reason that Euclidean ordered fields have been so named. For historically important examples of modern refined versions of Euclidean geometry, see David Hilbert (1899/1971) and Alfred Tarski (1959/1986); and for examples of non-Archimedean Euclidean ordered fields and their corresponding non-Archimedean Euclidean geometries, see Philip Ehrlich (1997a).

Set Theory and the Continuum

The Cantor-Dedekind theory of the continuum was originally formulated in a naïve set-theoretic framework, grounded in intuitions about sets that included the then radical assumption that infinitely many entities could be collected together in a set. Within this framework Cantor established the existence of an exhaustive hierarchy ℵ₀, ℵ₁, ℵ₂, … of increasingly large infinite cardinals, proved that the cardinality of the set of rational numbers is ℵ₀, that the cardinality of ℝ is 2^ℵ₀, and that 2^ℵ₀ is greater than ℵ₀. In the early decades of the twentieth century Cantorian set theory was placed on an axiomatic basis that sidestepped a medley of paradoxes that had befallen the naïve theory. In honor of two of its principal architects, Ernst Zermelo (1871–1953) and Abraham A. Fraenkel (1891–1965), the theory has come to be designated ZFC, where ZF indicates the body of axioms outside the axiom of choice. Of the open problems inherited from the naïve theory none was regarded more important than that of deciding the veracity of the continuum hypothesis (CH)—Cantor's conjecture that the cardinality of the continuum is ℵ₁. However, in 1938 Kurt Gödel showed that ZFC + CH is consistent if ZFC is, and twenty-five years later Paul Cohen demonstrated if ZFC is consistent so is ZFC + ¬CH, thereby establishing the independence of CH from standard set theory. Since the work of Cohen there has been a good deal of speculation on the part of philosophers and logicians whether or not the axioms of ZFC should be supplemented with one or more additional axioms that would settle the matter one way or another (Kanamori 2003). Gödel (1947/1983), who held a platonist view of sets, maintained that CH is either objectively true or objectively false and, accordingly, promoted the search for additional axioms to settle the matter. Many set theorists, including Cohen (1990), however, believe that there is nothing in the intuitive concept of set that would recommend the adoption of an additional axiom that would conclusively settle CH one way or another. The views of Gödel and Cohen, however, only begin to indicate the range of opinion on the matter as is evident from the debate between Soloman Feferman et al. (2000), the writings of Donald Martin and H. G. Dales (Dales and Oliveri 1998), and the intriguing though controversial views of W. Hugh Woodin (2001, 2002, 2004). Both models of ZFC + CH and ZFC + ¬CH are being explored by set theorists with perhaps a bit more attention being devoted to the latter. Mathematicians who are not set theorists, however, tend to use CH freely for the purpose of theorem proving, their reliance on CH being indicated in the statement of the theorem.

Nonstandard Theories of Continua

Although the Cantor-Dedekind theory of real numbers and philosophy of the continuum have occupied privileged positions in standard mathematical philosophy since the decades following the turn of the twentieth century, it has never enjoyed the complete allegiance of either philosophers or mathematicians. Early opponents such as Hermann Hankel (1839–1873) and Paul du Bois-Reymond (1831–1889) were critical of the attempts by Cantor, Dedekind, and others to treat irrational numbers formally and without the concept of continuous magnitude, and others such as Leopold Kronecker (1823–1891) complained, on the contrary, that the arithmetization had not gone far enough. Still others, including Emil Borel (1871–1956) and a young L. E. J. Brouwer (1881–1966), continued to regard the continuum as a primitive concept given to one directly by geometric intuition that was not amenable to analysis (compare Troelstra 1982); and others, including Hermann Weyl (1885–1955) (1918) as well as a more mature Brouwer (compare 1918, 1924, 1952), while embracing an analytical approach questioned one or another aspect of the logico-set-theoretic underpinnings of the Cantor-Dedekind theory. Another complaint that was, and to some extent still is, a stumbling block to the acceptance of the Cantor-Dedekind theory is the contention that the Cantor-Dedekind philosophy of the continuum is committed to the reduction of the continuous to the discrete, a program whose philosophical cogency, and even logical consistency, had been called into question over the centuries. Inspired by arguments originating with Aristotle, and reiterated and further developed by Immanuel Kant (1724–1804), Franz Brentano (1838–1917), Edmund Husserl (1859–1938), and others, a string of late nineteenth- and twentieth-century mathematicians and philosophers beginning with du Bois-Reymond (1882) maintained that unlike the unextended points that, by their lights, compose the Cantor-Dedekind continuum, the elements of a genuine continuum must themselves have extension if the continuum itself is to have extension. This view led Charles Sanders Peirce (1939–1914) to sketch a nonarithmetic theory of the continuum based on infinitesimals (1898/1992, 1900), and it played important contributing roles in the development of Brouwer's and Weyl's aforementioned intuitionist and predicativist alternatives to the standard conception as well.

Veronese (1889, 1891, 1894), however, while agreeing that the parts of a continuum must be intervals as opposed to points, held that for the sake of geometry the geometer may treat the line as an ordered collection of unextended points; moreover, holding that one's intuitive conception of the continuum is independent of the Archimedean axiom, he developed a general axiomatic theory of continua that was not only satisfied by the standard arithmetic continuum but by certain structures with infinitesimals as well, and he illustrated the latter by means of a synthetic construction of a non-Archimedean ordered field that is continuous in his sense. Veronese's non-Archimedean continuum was placed on a logically sound arithmetic foundation by Tullio Levi-Civita (1873–1941), who therewith provided the first analytic constructions of non-Archimedean ordered fields (1892–1993/1954, 1898/1954).

Building on the work of Levi-Civita, Hans Hahn (1879–1934) constructed non-Archimedean number systems (1907) having properties that generalize the aforementioned continuity properties of Dedekind and Hilbert, and he demonstrated that his number systems collectively provide a panorama of the finite, infinite, and infinitesimal numbers that can enter into a non-Archimedean theory of continua based on the concept of an ordered field (compare Ehrlich 1995, 1997, 1997a). Throughout the remaining first half of the twentieth century there continued to be important contributions to the theory of the continuum including the algebraic (Artin and Schreier 1926/1965) and logical (Tarski 1939/1986, 1948/1986) versions of the theory of elementary continuity. During the 1950s, under the influence of A. A. Markov (1856–1922), a Russian school of constructive analysis was developed based on a continuum consisting of real numbers with a recursive Cauchy sequence (compare Kushner 1984), and during the 1960s Errett Bishop (1928–1983) introduced an alternative constructive approach to analysis (1967) based on a model of the continuum whose theorems, unlike those of Brouwer and the Russian school, are all provable in classical mathematics.

Also during roughly the same period, interest in Weyl's predicative theory was rekindled by Feferman (1998), who developed his own predicative approach to analysis. In addition, since the late 1950s a number of nonstandard theories of continua have appeared that make use of infinitesimals including those arising from Abraham Robinson's (1918–1974) nonstandard approach to analysis (1961/1969, 1966/1974), those arising from F. W. Lawvere (1979, 1980), and Anders Kock's (1981) ideas on smooth infinitesimal analysis and synthetic differential geometry, the partially ordered continuum of Curt Schmieden and Detlef Laugwitz (1958), and Ehrlich's so-called absolute arithmetic continuum (1987, 1989, 1992) based on J. H. Conway's (1976) theory of surreal numbers. Still another theory that arose during this period is the theory of fuzzy real numbers based on Lofti Asker Zadeh's (1987) theory of fuzzy sets.

Of the nonstandard theories mentioned earlier, the intuitionist, Bishop-style constructivist, predicativist, and the Robinsonian and post-Robinsonian infinitesimalist theories have been given the most attention by philosophers. In the subsequent three sections these will be considered in turn with some attention paid to the Russian constructivist theory as well.

Constructivist Theories

Constructivism is a rubric that has come to designate a family of approaches to the foundations of mathematics that are loosely united by their opposition to certain forms of mathematical reasoning employed in the mainstream mathematical community. However, as the term loosely suggests, there are significant differences between the various schools and substantial differences in attitude can be found even among the representatives of a given school or a single representative over time. However, whether the result of their rejection of actual infinities or the universal applicability of certain principles of classical logic, or their insistence on the use of algorithmic constructions of one form or another, constructivists have always found themselves at odds with the Cantor-Dedekind theory of the continuum and have sought to provide alternatives that are constructively sanctioned by their own particular lights. To distinguish the resulting intuitionist, Russian constructivist, and Bishop-style constructivist arithmetic continua from ℝ, from now on they will be denoted as ℝ_I, ℝ_R, and ℝ_C, respectively.

Before the late 1960s the constructivist theory of the continuum that attracted the greatest attention is the intuitionistic theory of Brouwer. Until 1914, Brouwer, like Borel before him, embraced a holistic view in which the continuum is regarded as a primitive notion given directly by intuition that cannot be understood as the totality of its individually definable elements. Thereafter, while still clinging to certain aspects of the irreducible conception, Brouwer adopted a more analytic view in which the continuum, while not a completed nondeumerable totality, can be more and more completely specified in a never-ending fashion with one's increasing knowledge as a medium of free development.

The basis of Brouwer's conception of the continuum is the concept of a choice sequence, a concept not acceptable to classical mathematics. According to Brouwer, the construction of such a potentially infinite sequence is always incomplete in the sense that at any given instant of its construction the sequence is limited to a finite number of terms. A choice sequence α is given by a fixed initial segment α ₁, α ₂, …, α _n of mathematical objects along with a corresponding set of restrictions R ₁, R ₂, …, R _n , where R _n restricts the range of possible choices for α _{n + 1}. In particular, real numbers are introduced by Brouwer as choice sequences that are Cauchy sequences of rational numbers. Of course, being a choice sequence, the notion of a Cauchy sequence of rational numbers must be appropriately understood. More specifically, according to the intuitionist one can assert that a potentially infinite sequence r ₁, r ₂, … of rational numbers is a Cauchy sequence only if one knows there is a procedure that, given any positive integer k, effectively produces a natural number N along with a proof that N has the specified Cauchy property, for example, |r _m − r _n | < 1/k for all m, n < N.

For Brouwer, such a choice sequence is not a technique for approximating some preexisting real number—it is the choice sequence itself, growing in time, that is the real number. Some such real numbers are introduced by letting the choices be prescribed by a fixed algorithm. These so-called lawlike sequences lead to the existence of real numbers such as e, π, and √2. Other real numbers arise from forbidding any restriction on the rational numbers one chooses outside of assuring that the choice sequence is Cauchy. Between these two extremes, however, there is a wide range of possibilities for introducing real numbers. Until 1927 Brouwer did not place any restrictions on choice sequences—having regarded them as sufficiently explained by the freedom of a supposed ideal mathematician generating them—but thereafter he became more specific and continued to revise his conception of permissible choice sequences until the early 1950s (compare Troelstra 1982, pp. 472–474).

The adoption of choice sequences forces a nonclassical logic on intuitionists that rejects the universal validity of the law of excluded middle. For example, given the incomplete nature of choice sequences the intuitionist has no right to assume for an arbitrary pair of choice sequences α and β having identical initial segments that α = β ∨ α ≠ β. The logic that has come to be embraced by intuitionists is a subtheory of classical logic called intuitionistic logic (compare Heyting 1971, chapter 7). For the intuitionist, logic does not provide a foundation for mathematics but emerges from one's mathematical practice. The adoption of intuitionistic logic leads intuitionists to a theory of real numbers and corresponding theory of continuity that differs markedly from their classical counterparts. For example, besides the just-cited deviation, it is not possible for intuitionists to prove precise analogs of the following classical results for their own system ℝ_I of real numbers: ∀x, y ∈ ℝ(xy = 0→(x = 0 ∨ y = 0)); ∀x, y ∈ ℝ(x ≤ y ∨ x ≥ y ); ∀x, y ∈ ℝ(x > y ∨ x = y ∨ x > y ); every subset of ℝ that is bounded above has a least upper bound; ∀x ∈ ℝ(x is rational ∨ x is irrational). Nor can they prove the intermediate value theorem or the extreme value theorem. From Brouwer's perspective, this inability is not a limitation since each of the previous assertions implies an instance of the law of excluded middle that is not intuitionistically sanctioned (compare Bridges and Reeves 1999, pp. 72–73). This attitude, and the embrace of intuitionistic logic more generally (with the aforementioned implications for their own theories of real numbers and corresponding theories of continuity) is a common thread that binds constructive mathematicians. From the constructivist point of view, accepting the law of excluded middle as a universal principle would mean the existence of a universal procedure for generating for each proposition P, either a proof of P or a proof of ¬P, where a procedure for generating a proof of ¬P is understood to be a method for generating a contradiction from a hypothetical proof of P. However, such a procedure is not available; if it were one could decide a proposition P —such as Goldbach's conjecture—the truth of which has not been decided.

Brouwer's concept of a choice sequence that is Cauchy corresponds, as Heyting notes, "to the intuitive concept of the continuum as a possibility of a gradual determination of points" (1971, p. 34). To develop an adequate theory of continuity and of analysis more generally, however, the mathematician must be able to talk about classes of such real numbers and functions. From the standpoint of the intuitionist, however, one cannot collect them together into a Cantorian set—there are simply too many of them. Rather, for the intuitionist, they are held together in a spread —roughly speaking, a growing tree, whose emerging paths through the tree correspond to the various ways an initial segment of a choice sequence can be continued (compare Heyting 1971, chapter 3).

Moreover, to obtain the central continuity theorems concerning such classes of real numbers and functions, Brouwer introduced two fundamental ideas governing the mathematical treatment of choice sequences: the weak continuity principle for numbers (compare Veldman 2001, Atten and Dalen 2002, Troelstra and Dalen 1988, chapter 5) and the principle of bar induction (compare Kushner 2001). The continuity principle makes choice sequences serviceable by contending that a total function from choice sequences to natural numbers never requires more input than an initial segment (of a choice sequence) to generate its output; and the induction principle ensures, among other things, that the entire intuitionistic continuum can be treated in a constructively manageable fashion (compare Atten 2004, chapters 3–4).

The adoption of the continuity principle and the induction principle gives the intuitionistic theory of continuity its own distinctive constructivist flavor and leads to even more striking deviations from the classical theory than those listed earlier. For example, in virtue of the continuity principle, the analogs of the aforementioned classical results that cannot be established as a result of the adoption of intuitionistic logic are now provably false (compare Bridges and Richman 1987, pp. 4–5; Troelstra and Dalen 1988, pp. 257–258). What is perhaps the most notorious such deviation, however, is that all functions from ℝ_I to ℝ_I are continuous, and uniformly continuous at that. This apparent absurdity arises in part from the fact that the contention "f is a function defined on all of ℝ_I" is substantially stronger construed intuitionistically than is the contention "f is a function defined on all of ℝ" construed classically. Consider, for example, the classical discontinuous function f defined by

From Brouwer's perspective, f is not a function at all since one cannot prove ∀x ∃y (f (x ) = y ) by intuitionistically sanctioned means. In particular, one cannot prove ∀x (x < 0 ∨ x ≥ 0) insofar as the definition of f does not tell one how to compute f (x ) if x is a number for which one cannot assert either x < 0 or x ≥ 0. Closely related to this is still another striking deviation, the so-called unsplittability of the intuitionistic continuum; that is, unlike the Cantor-Dedekind arithmetic continuum, there do not exist two disjoint nonempty subsets of an interval of the intuitionistic continuum whose union is the given interval, nor are there such partitions of the intuitionistic continuum whose union is the continuum itself. Accordingly, for the intuitionist, as for Anaxagoras and Aristotle before them, it is not possible to separate out a point from their continuum or from an interval thereof.

While Brouwer's theory has attracted a good deal of attention from philosophers and logicians, it has received comparatively little attention from standard mathematicians. Whether this is because of the philosophical precepts underlying it, the highly nonclassical nature of the mathematical arguments it employs, the belief that the resulting mathematics is too impoverished, or simply the absence of a perceived need for it, is difficult to say. In 1967, however, Brouwer's theory was given an especially stinging critique, not by standard mathematicians, but by Bishop, whose treatise Foundations of Constructive Analysis is widely credited with having breathed new life into constructive mathematics. In the treatise's polemical opening chapter Bishop describes the construction and motivation underlying Brouwer's theory of the continuum in the following biting terms:

Brouwer became involved in metaphysical speculation by his desire to improve the theory of the continuum. A bugaboo of both Brouwer and the logicians has been compulsive speculation about the nature of the continuum. In the case of the logicians this leads to contortions in which various formal systems, all detached from reality, are interpreted within one another in the hope that the nature of the continuum will somehow emerge. In Brouwer's case there seems to have been a nagging suspicion that unless he personally intervened to prevent it, the continuum would turn out to be discrete. He therefore introduced the method of free-choice sequences for constructing the continuum, a consequence of which the continuum cannot be discrete because it is not well enough defined. This makes mathematics so bizarre it becomes unpalatable to mathematicians, and foredooms the whole of Brouwer's program. This is a pity, because Brouwer had a remarkable insight into the defects of classical mathematics, and he made a heroic effort to set things right. (1967,p. 6)

Bishop sought to place analysis on a constructivist foundation that is free of the perceived difficulties mentioned earlier, a project that has been extended by a number of other mathematicians including Douglas Bridges and Fred Richman. In Bishop's arithmetic continuum ℝ_C a real number is simply defined as a sequence {x _n } of rational numbers that satisfies the condition |x _m − x _n | ≤m ⁻¹ + n ⁻¹ (for all integers m, n ≥ 1); though some authors, following Troelstra and Dirk van Dalen (1988, pp. 253–254), prefer to use equivalence classes of certain sequences of rational numbers in their place. Thus, for Bishop, as for Markov, every real number is a lawlike Cauchy sequence. Using a system of axioms for constructive ordered fields with a formulation of the Archimedean axiom and a constructive formulation of the least upper-bound principle, Bridges (1999) shows that ℝ_C can be characterized in a manner that closely resembles one of the aforementioned standard axiomatizations of ℝ. Working independently, Alberto Ciaffaglione and Pietro Di Gianantonio (2002) and Herman Geuvers and Milad Niqui (2002) provide equivalent axiomatizations of ℝ_C that employ constructive versions of Cauchy completeness in place of Bridges's least upper-bound condition. Assuming the axiom of countable choice (compare Troelstra and Dalen 1988, pp. 189–190)—an axiom that is frequently adopted by constructive mathematicians—Geuvers and Niqui (2002) further establish the categoricity of the axiomatizations. Absent the choice axiom (or some equivalent thereof), there are models of the axioms that are not isomorphic to ℝ_C—in particular, Troelstra and Dalen's version of the constructive continuum based on Dedekind cuts (1988, pp. 270–274).

Bishop tended to distinguish his theory of analysis from the classical theory by emphasizing the former's demand for algorithmic constructions. Following in the footsteps of Brouwer, Bishop took the concept of an algorithm as a primitive, undefined notion and was led to reject the universal validity of the law of excluded middle by interpreting mathematical existence strictly in terms of computability or constructivity. Bridges and, especially, Richman speculate that the theory of analysis that emerges from Bishop's approach may be regarded as the subtheory of the classical theory that is obtainable employing intuitionistic logic as opposed to classical logic as the underlying logic (Bridges and Richman 1987; Richman 1990, 1996; Bridges 1999). Since (in accordance with Heyting's axiomatization of intuitionistic logic [1971]) one passes from intuitionistic logic to classical logic by embracing the universal validity of the law of excluded middle, on their view Bishop's theory may be regarded as the subtheory of the classical theory that can be obtained without appealing to the instances of this classical law that are not intuitionistically sanctioned (Bridges and Richman 1987, p. 120).

Besides being a subtheory of the classical theory, Bishop's theory may be regarded as a subtheory of the intuitionist and Russian constructivist theories as well (Bridges and Richman 1987, chapter 6). Whereas, in accordance with Bridges and Richman's view, one moves from Bishop's theory to the classical theory by embracing the universal validity of the law of excluded middle, to move from Bishop's theory to the intuitionistic theory one introduces Brouwer's weak continuity principle along with a seminal consequence of Brouwer's principle of bar induction called the fan theorem, and to pass from Bishop's theory to the Russian constructivist theory one adds a consequence of Church's thesis that all computable sequences of natural numbers are recursive (Bridges and Richman 1987, chapter 5). Talk of such passages, however, applies solely to theories in the logician's narrow sense of the term; it ignores the divergent philosophical motivations and mathematical trappings of the four theories. With respect to the Russian constructivist theory, for example, it ignores that every real number is a recursive real number, that algorithms are Markov algorithms, that functions are Gödel numbers of algorithms that compute them, and so on.

The differences between the classical, Russian constructivist, and intuitionist theories, however, go beyond their respective philosophical motivations and mathematical trappings; they have different theories of continuity, as is evident from the following theorems of the three respective theories:

(Classical): Some functions f :[0, 1]→ℝ are not continuous and, hence, not uniformly continuous

(Russian Constructivist): Whereas all functions f :[0, 1]→ℝ_R are continuous, some are not uniformly continuous

(Intuitionist): All functions f :[0, 1]→ℝ_I are not only continuous, they are uniformly continuous

Despite these differences, Bishop's theory manages to lie comfortably within the common core of the three theories in part because in Bishop's theory attention is restricted from the outset to functions that are assumed to be uniformly continuous on each closed interval of ℝ_C.

Restricting attention to the just-said functions, Bishop managed to obtain a surprisingly robust theory of analysis that includes among a wide range of other theorems constructive analogs of the intermediate value theorem and the extreme value theorem. Like many of the theorems of constructive analysis, the latter two theorems differ from their classical counterparts by having weakened conclusions or strengthened hypotheses. One constructive version of the intermediate value theorem asserts that a uniformly continuous function f :[a, b ]→ℝ_C takes on a value as close to the given intermediate value as one pleases, and the constructive version of the extreme value theorem asserts that a uniformly continuous function f :[a, b ]→ℝ_C does have a maximum (minimum), though the maximum (minimum) is not necessarily assumed. There is also a constructive version of the intermediate value theorem that asserts that in a particular class of cases (which includes all real-analytic functions), the intermediate value in question is in fact realized (compare Bishop and Bridges 1985, pp. 40–41; Troelstra and Dalen 1988, pp. 292–295).

However, despite the strength of Bishop's analysis and its compatibility with classical mathematics, Bishop's theory, like its intuitionist and Russian constructivist forerunner's, has not attracted the kind of attention from classical mathematicians its practitioners had hoped for (compare Bridges and Reeves 1999, p. 67). Moreover, while praising the significance of Bishop's accomplishment, some devotees of Brouwer's theory have questioned the adequacy of the analysis of the continuum that emerges from Bishop's approach. In particular, they are concerned that in Bishop's theory, like Markov's before it, the continuum of real numbers is restricted to those real numbers introduced by lawlike Cauchy sequences. For example, in his monograph devoted to Brouwer's theory, Mark van Atten remarks:

One may, like Markov and Bishop, settle for just the lawlike sequences … but while practical, that also amounts to ducking the issue of how to model the full continuum. Brouwer's achievement is to have found a way to analyze the continuum that does not let it fall apart into discrete elements … and it is constructive to boot. (2004, p. 33)

Predicative Theories

Between classical mathematics, in which arbitrary sets are embraced, and Bishop's constructive mathematics, in which only algorithmically constructed objects are permissible, there is an intermediate approach called predicativism, in which only definable sets are considered, and in which quantifiers over sets are interpreted as ranging only over sets that have previously been defined. Although having its roots in Russell's and Poincaré's attempts to lay the blame for the paradoxes of set theory on definitions that define entities in terms of classes to which they belong—so-called impredicative definitions—it was Weyl, in his monograph Das Kontinuum (1918/1994), who first undertook the development of a theory of the arithmetic continuum and of analysis on it in a predicatively acceptable fashion.

Central to Weyl's critique of the classical theory is its dependence on the proposition that every nonempty set of real numbers that is bounded above has a least upper bound, the definition of the least upper bound of a set being inextricably impredicative. Weyl proposed overcoming this by employing the predicatively sanctioned proposition: Every nonempty sequence of reals having an upper bound has a least upper bound. Using this idea, Weyl constructed a restricted set of real numbers containing all reals that are expressible as Dedekind cuts definable in his system. Although the set of standard real numbers not definable in Weyl's system is everywhere dense, Weyl showed that on the basis of his continuum most, if not all, of the nineteenth-century analysis of piece-wise continuous functions can be carried out predicatively. On the other hand, as Weyl conceded, substantial and significant portions of modern analysis are not obtainable in his system including "the more far-reaching integration- and measure theories of Riemann, Darboux, Cantor, Jordan, Lebesque and Carathéodory" (Weyl 1918/1994, p. 86).

In the years following the publication of Das Kontinuum Weyl abandoned his own approach in favor of Brouwer's intuitionistic framework. Soon thereafter, however, he returned to the standard mathematical fold and distanced himself from Brouwer's school and from foundations work more generally. In the ensuing years, Weyl's predicative theory lay largely dormant until the 1960s, when a number of authors including, most prominently, Feferman undertook a formalization and systematic analysis of Weyl's system as well as the development of a variety of predicative extensions thereof (compare Feferman 1964, 1988/1998, 1993/1998). Unlike Weyl, who worried about the purported vicious circles associated with impredicative definitions, Feferman was motivated in part by the concern that the unbridled use of such definitions presupposed a strong form of platonic realism regarding sets, a view he found philosophically objectionable; he was also interested in providing an analysis of predicativity itself, as well as with the purely logical question of the extent to which analysis can be carried out by predicative means. One of Feferman's extensions, called W in honor of Weyl, has been proven to be sufficiently strong to permit the reconstruction of almost all of classical analysis as well as important portions of modern analysis that are not obtainable in Weyl's original system. Feferman maintains that:

While there are clearly parts of theoretical analysis that cannot be carried out in W because they make essential use of the 1.u.b. axiom applied to sets rather than sequences, or because they make essential use of transfinite ordinals or cardinals, or because they deal with nonseparable spaces, the working hypothesis that all of scientifically applicable analysis can be developed in W has been verified in its core parts. What remains to be done is to examine results closer to the margin to see whether this hypothesis indeed holds in full generality. (1993/1998, p. 294)

Infinitesimalist Approaches

Following Emil Artin (1898–1962) and Otto Schreier (1901–1929), an ordered field K may be said to be real-closed if it admits no extension to a more inclusive ordered field that results from supplementing K with solutions to polynomial equations with coefficients in K (1926/1965). Intuitively speaking, real-closed ordered fields are precisely those ordered fields having no holes that can be filled by algebraic means alone. Tarski demonstrated that real-closed ordered fields are precisely the ordered fields that are first-order indistinguishable from ℝ or, to put this another way, they are precisely the ordered fields that satisfy the elementary (i.e., first-order) content of the Dedekind continuity axiom (1939/1986, 1948/1986). For this reason they are called elementary continua. While ℝ is the best-known elementary continuum, it is hardly the only one.

Some elementary continua, like ℝ, are Archimedean, though most are non-Archimedean; and among the latter many are extensions of ℝ. In the early 1960s Robinson (1961, 1966) made the momentous discovery that among the real-closed extensions of the reals there are number systems that can provide the basis for a consistent and entirely satisfactory nonstandard approach to analysis based on infinitesimals. Robinson motivated his work with the following words:

It is our main purpose to show that these models [i.e. number systems] provide a natural approach to the age old problem of producing a calculus involving infinitesimal (infinitely small) and infinitely large quantities. As is well known, the use of infinitesimals, strongly advocated by Leibnitz and unhesitatingly accepted by Euler fell into disrepute after the advent of Cauchy's methods which put Mathematical Analysis on a firm foundation. Accepting Cauchy's standards of rigor, later figures in the domain of nonarchimedean quantities concerned themselves only with small fragments of the edifice of Mathematical Analysis. We mention only du Bois-Reymond's Calculus of In-finites [1875] and Hahn's work on nonarchimedean fields [1907] which in turn were followed by the theories of Artin-Schreier [1926] and, returning to analysis, of Hewitt [1948] and Erdös, Gillman and Henriksen [1955]. Finally, a recent and rather successful effort at developing a calculus of infinitesimals is due to Schmieden and Laugwitz [1958] whose number system consists of infinite sequences of rational numbers. The drawback of this system is that it includes zero-divisors and that it is only partially ordered. In consequence, many classical results of the Differential and Integral calculus have to be modified to meet the changed circumstance. (1961/1979, p. 4)

Being elementary continua, Robinson's number systems do not have the just-cited drawbacks of the number system of Schmieden and Laugwitz. By analogy with Thoralf Skolem's (1934) nonstandard model of arithmetic, a number system from which Robinson drew inspiration, Robinson called his totally ordered number systems nonstandard models of analysis. These number systems, which are often called hyperreal number systems (Keisler 1976, 1994), may be characterized as follows: Let 〈ℝ, S : S ∈ 𝔉〉 be a relational structure where 𝔉 is the set of all finitary relations defined on ℝ (including all functions). Furthermore, let *ℝ be a proper extension of ℝ and for each n -ary relation S ∈ 𝔉 let *S be an n -ary relation on *ℝ that is an extension of S. The structure 〈*ℝ, ℝ, *S :S ∈ 𝔉〉 is said to be a hyperreal number system if it satisfies the Transfer Principle : Every n -tuple of real numbers satisfies the same first-order formulas in 〈ℝ, S :S ∈ 𝔉〉 as it satisfies in 〈*ℝ, ℝ, *S :S ∈ 𝔉〉.

The existence of hyperreal number systems is a consequence of the compactness theorem of first-order logic and there are a number of algebraic techniques that can be employed to construct such a system. One commonly used technique is the ultapower construction (Keisler 1976, pp. 48–57; Goldblatt 1998, chapter 3), though not all hyperreal number systems can be obtained this way. By results of H. Jerome Keisler (1963; 1976, pp. 58–59), however, every hyperreal number system must be (isomorphic to) a limit ultapower.

Using the transfer principle, one can develop satisfactory nonstandard conceptions and treatments of all of the basic concepts and theorems of the calculus including those from the theories of integration, differentiation, and continuity to name only a few (compare Keisler 1986, Goldblatt 1998, Loeb 2000). For example, it follows from the transfer principle that a real-valued function f is continuous at a ∈ ℝ iff *f (x ) is infinitesimally close to *f (a ) whenever x is infinitesimally close to a, for all x ∈ *ℝ. On the basis of this result one may prove various classical properties governing the continuity of real-valued functions including the intermediate and extreme valuetheorems (Goldblatt 1998, pp. 79–80). It should be emphasized, however, that Robinson's discoveries do not provide vindication of the Leibnizian formalism or of the seventeenth- and eighteenth-century preanalytic formalisms more generally. For example, whereas Leibniz conceived of differentiation and integration in terms of ratios of and infinite sums of infinitesimals, respectively, for Robinson they are real numbers that are infinitesimally close to such ratios and sums. On the other hand, nonstandard analysis not only demonstrates that the branch of mathematics erected for the study of continuously varying magnitude can be fully developed using infinitely large and infinitely small numbers as Leibniz and his followers had envisioned but it also provides one with an intuitively satisfying alternative to the standard picture of a continuum and of continuous phenomena more generally that is mathematically adequate and logically sound relative to classical mathematics.

Modern analysis, however, goes far beyond the traditional province of the calculus, dealing with arbitrary sets of reals, sets of sets of reals, sets of functions from sets of reals to sets of reals, and the like. Importantly, nonstandard analysis is entirely applicable to this expanded arena as well. However, the methods of superstructures (Robinson 1966) and internal set theory (Nelson 1977) that are most usually employed for this purpose are of little relevance here (compare Chang and Keisler 1990, §4.4; Robert 1988).

Unlike ℝ, the structures that may play the role of *ℝ in a hyperreal number system are far from being unique up to isomorphism. From a purely mathematical point of view this causes no difficulty and from the standpoint of varying applications can even be advantageous (compare Keisler 1994, p. 229). On the other hand, if one takes *ℝ to be a model of the continuous straight line of geometry—something practitioners of nonstandard analysis tend not to do—the absence of uniqueness is a bit disconcerting. Still, as several nonstandard analysts including Tom Lindstrøm (1988, p. 82) and Keisler (1994, p. 229) emphasize, even ℝ is not as unique as one would like to think since its uniqueness up to isomorphism is in fact relative to the underlying set theory. In particular, by retaining the construction of ℝ and supplementing the set theory with additional axioms, one can change the second-order theory of the real line. This leads Keisler (1994) to suggest that not only is ZFC not the appropriate underlying set theory for the hyperreal number system but also that set theory might have developed differently had it been developed with the hyperreal numbers rather than the real numbers in mind. According to Keisler, an appropriate set theory "should have the power set operation to insure the unique existence of the real number system, and another operation which insures the unique existence of the pair consisting of the real and hyperreal number systems" (p. 230).

Consistent with the previous observation, one type of axiom that is used to secure categoricity is a saturation axiom (Keisler 1976, pp. 57–60). As the name suggests, saturation axioms ensure that the line is extremely rich. A hyperreal number system 〈*ℝ, ℝ, *S : S ∈ 𝔉〉 is said to be κ-saturated if any set of formulas with constants from *ℝ of power less than κ is satisfiable whenever it is finitely satisfiable. If κ is the power of *ℝ, the hyperreal number system is said to be saturated. Although there is a wide range of hyperreal number systems in ZFC that are saturated to varying degrees of power less than the power of *ℝ, saturated hyperreal number systems do not exist in ZFC. In virtue of classical results from the theory of saturated models, however, there is (up to isomorphism) a unique saturated hyperreal number system of power κ whenever κ > 2^ℵ₀ and either κ is (strongly) inaccessible or the generalized continuum hypothesis (GCH) holds at κ (i.e., κ = ℵ_{α + 1} = 2^ℵ_α for some α ). So, for example, by supplementing ZFC with the assumption of the existence of an uncountable inaccessible cardinal, one can obtain uniqueness (up to isomorphism) by limiting attention to saturated hyperreal lines having the least such power (Keisler 1976, p. 60).

However, as Ehrlich (2002, 2004) emphasizes, perhaps the most remarkable of all elementary continua that may play the role of *ℝ in a hyperreal number system (and bring categoricity to the hyperreal line to boot) is Conway's ordered field of surreal numbers (1976 2001), a system that was not created with nonstandard analysis in mind. This would correspond (to within isomorphism) of adopting a hyperreal number system that is the union of an elementary chain of ω _α -saturated hyperreal number systems where α ranges over the class On of all ordinals. Though such models do not exist in ZFC, they can be suitably characterized and shown to exist (up to isomorphism) in von Neumann–Bernays–Gödel (NBG) set theory with the axiom of global choice (Ehrlich 1989). Since NBG is a conservative extension of ZFC, its sets have the same properties as those of standard set theory (compare Fraenkel, Bar-Hillel, and Levy 1973). The idea of employing such a hyperreal number system to establish the categoricity of the hyperreal line appears to be due (at least implicitly) to Keisler (1976, p. 59; 1994, p. 233; theorem 3 of addendum to Ehrlich 1989), but guided by reasons of simplicity and convenience he chooses the least uncountable inaccessible cardinal approach instead.

The ordered field of surreal numbers, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered (number) fields—be individually definable in terms of sets of NBG, it may be said to contain "All Numbers Great and Small." In this respect, No bears much the same relation to ordered fields as the system of real numbers bears to Archimedean ordered fields. Ehrlich (1987, 1989a, 1992, 2002) suggests that whereas the real number system may be regarded as an arithmetic continuum modulo the Archimedean axiom, the system of surreal numbers may be regarded as a sort of absolute arithmetic continuum modulo NBG. To lend credence to this thesis, Ehrlich provides a variety of categorical axiomatizations of ℝ making use of novel continuity axioms (that are equivalent to any of the familiar continuity axioms) with axioms for Archimedean ordered fields (or Archimedean real-closed ordered fields) and shows that by simply deleting the Archimedean axiom one obtains categorical axiomatizations of No (Ehrlich 1992, theorems 1, 4, and 6). Ehrlich also introduces a natural generalization of Dedekind's conception of a gap, called a set-gap, and provides further evidence for the thesis by showing that whereas ℝ is (up to isomorphism) the unique elementary continuum having no set-gaps that satisfies the Archimedean condition, No is (up to isomorphism) the unique elementary continuum having no set-gaps that satisfies the On -Archimedean condition, the latter being a natural generalization of the Archimedean condition that is appropriate for No (Ehrlich 1992, Lemma 1, Theorem 7; 2001, pp. 1255–1256). Critical to the proof of the latter result is Ehrlich's (1988, 1989, 2001) further characterization of No (up to isomorphism) as the unique elementary continuum such that for all subsets X and Y of the field where every member of X precedes every member of Y there is a member of the field lying strictly between those of X and those of Y. Intuitively, this characterizes No (up to isomorphism) as the unique ordered field having neither algebraic limitations nor order-theoretic limitations that are definable in terms of sets of standard set theory.

Besides its distinguished structure as an ordered field, No has a rich hierarchical structure that emerges from the recursive clauses in terms of which it is defined. This algebraico-tree-theoretic structure, or simplicity hierarchy, as Ehrlich (1994, 2001) calls it, depends on No 's structure as a lexicographically ordered binary tree and arises from the fact that the sums and products of any two members of the tree are the simplest possible elements of the tree consistent with No 's structure as an ordered group and an ordered field, respectively, it being understood that x is simpler than y just in case x is a predecessor of y in the tree. Among the remarkable consequences of this algebraico-tree-theoretic structure is that much as the surreal numbers emerge from the empty set of surreal numbers by a transfinite recursion that yields chains of increasingly less and less simpler numbers, the recursive process of defining No 's arithmetic in turn gives rise to chains of increasingly richer and richer numbers systems with the result that an isomorphic copy of every elementary continuum emerges in No as the union of a chain of elementary continua each of which is an initial subtree of No (Ehrlich 2001).

Conway (1976 2001) shows that besides the reals No contains a natural isomorphic copy of Cantor's ordinals, and hence, by virtue of the axiom of choice, the cardinals as well (Ehrlich 2001, pp. 1253–1256). Ehrlich (1988, 2001, 2002, 2004) notes that No also provides a natural setting for the non-Cantorian theories of the infinite (and infinitesimal) pioneered by Veronese, Levi-Civita, Hilbert, and Hahn in connection with their work on non-Archimedean-ordered algebraic and geometric systems and by du Bois-Reymond, Stolz, G. H. Hardy (1877–1947), and Felix Hausdorff (1868–1942) in connection with their work on the rate of growth of real functions (compare Ehrlich 1994, 1995, 2005; Fisher 1981). This, together with the observation about the relationship between No and hyperreal number systems, leads Ehrlich (2002, 2004) to observe that over and above providing a panorama of the entire set-theoretic spectrum of numbers great and small (modulo NBG), the purported absolute arithmetic continuum provides a unifying framework for many of the most important totally ordered systems of finite, infinite, and infinitesimal numbers that have played and continue to play prominent roles in mathematics since the days of Cantor and Dedekind.

Within a decade of the development of nonstandard analysis, Lawvere proposed a profound and novel approach to differential geometry based on infinitesimals. Unlike Robinson, who was stimulated by Leibniz's idea that the properties of infinitesimals should reflect the properties of the reals, Lawvere's ideas more closely mirror the heuristic ideas of geometers who envisioned a vector tangent to a surface at a point as a tiny arc of a curve having the vector tangent to it. Building on Lawvere's ideas, Kock (1981) presents a systematic treatment of the theory under the rubric synthetic differential geometry (SDG).

Unlike the nonzero infinitesimals employed in nonstandard analysis, the nonzero infinitesimal elements of SDG are nilpotent, that is, each such infinitesimal d satisfies the condition d ² = 0. Nilpotent infinitesimals are not invertible (in the sense that they have no multiplicative inverses) and as such a line in SDG in not modeled by a field or a portion thereof. Rather, in SDG a line is modeled by a ring ℜ containing a subset D = {d ∈ ℜ:d ² = 0} which satisfies the
Kock-Lawvere axiom: For every mapping f : D →ℜ, there is precisely one b ∈ ℜ, such that for all d ∈ D, f (d ) = f (0) + d · b.
Geometrically speaking, the Kock-Lawvere axiom asserts that the graph of every function f :D →ℜ is a piece of the unique straight line through (0, f (0)) with slope b. It is a consequence of this assumption that in SDG a tangent vector to a curve C at a point p is a nondegenerate infinitesimal line segment around p coincident with C.

Another consequence of the Kock-Lawvere axiom is that in SDG, unlike in Euclidean geometry, there are pairs of points in the plane that are not connected by a unique straight line. In this regard, SDG resembles Johannes Hjelmslev's (1873–1950) natural geometry (compare Kock 2003), a geometry that was designed to mirror real (as opposed to ideal) sense experience and that also employs nilpotent infinitesimals. However, unlike in natural geometry, in SDG there are pairs of points in a plane that are not connected by any line at all. This arises in part from the fact that whereas the nilpotent infinitesimals in natural geometry have "a quantitative (linear ordered) character," those employed in SDG do not (Kock 2003, pp. 226–228). For an axiomatization of "Euclidean Geometry with Infinitesimals" inspired by SDG, see Succi Cruciani (1989).

A space X in SDG is said to be indecomposable if no proper nonempty part U of X is detachable in the sense that there is a part V such that U ∪ V = X where U ∩ V = ∅. There are models of SDG in which a classical space ℝⁿ has a counterpart X that is indecomposable if X is connected. John Bell takes this to imply that "the connected continua of SDG are true continua in something like the Anaxagoran sense" (1995, p. 56). In this respect, they are also reminiscent of the unsplittable continuum of Brouwer; however, the similarity is not perfect and varies depending on the axioms adopted for SDG (Bell 2001).

Another respect in which SDG is similar to Brouwer's theory is the failure of the intermediate value theorem in its underlying theory of analysis. In fact, in SDG, unlike in Brouwer's system, the theorem even fails for some polynomials (Moerdijk and Reyes 1991, pp. 317–318), a failure that runs contrary to the thinking of Leibniz and Euler let alone Bolzano, Cauchy, and Weierstrass. Accordingly, while SDG may provide a viable alternative for differential geometry, its underlying analysis may not be as well suited to provide a natural alternative for classical analysis, at least not if it hopes to mirror the latter's most central ideas regarding continuity.

Unlike nonstandard analysis, which is developed in a set-theoretic setting, SDG is developed in a category-theoretic framework. Moreover, whereas the underlying logic employed in nonstandard analysis is classical logic, in SDG the underlying logic is intuitionistic logic. In SDG every function f :ℜ→ℜ is differentiable and, hence, infinitely differentiable (i.e. smooth ) as well as continuous in the sense that it sends neighboring points to neighboring points. It is sometimes maintained (compare Bell 1995, p. 56) that it is the ubiquitous nature of continuity within SDG that forces the change from classical to intuitionistic logic. This, however, is apt to be misleading since it is possible to develop a theory of continua in which the continuity of functions from the continuum to the continuum is likewise ubiquitous though the underlying logic is classical (compare the so-called Cauchy continuum due to Schmieden and Laugwitz [1958; Laugwitz 2001, p. 134]). Rather, it is the Kock-Lawvere axiom that underlies the incompatibility of SDG with classical logic (compare Lavendhomme 1996, pp. 2–5). It is therefore interesting to note that Paolo Giordano (2001), by suitably modifying the axiom, presents a variation of SDG based on nilpotent infinitesimals in which the underlying logic is entirely classical, and he observes that the nilpotent infinitesimals could be supplemented with invertible infinitesimals as well. Earlier, Ieke Moerdijk and Gonzalo E. Reyes (1991), while retaining the underlying intuitionistic logic, also introduced an alternative approach in which invertible as well as nilpotent infinitesimals are employed. The work of Moerdijk, Reyes, and Giordano, much like the pioneering work of Lawvere and Kock, provides still other models of mathematical continua.

Concluding Remarks

"Bridging the gap between the domains of discretenessand of continuity, or between arithmetic and geometry, is a central, presumably even the central, problem of the foundations of mathematics." So write Fraenkel, Yehoshua Bar-Hillel, and Azriel Levy in their mathematico-philosophical classic Foundations of Set Theory (1973, p. 212). Cantor and Dedekind of course believed they had bridged the gap with the creation of their arithmetico-set theoretic continuum of real numbers, and it remains a well-entrenched tenet of standard mathematical philosophy that indeed they had. At the same time, Cantor was overly sanguine when in 1883 he seemed to suggest, or at least implied, that his theory of the continuum, unlike that of the ancients, had "been thought out … with the clarity and completeness … required to exclude the possibility of different opinions among [its] posterity" (Cantor 1883/1996, p. 903). Indeed, while Cantor and Dedekind had succeeded in replacing the vague ancient conception with a clear and complete arithmetico-set-theoretic conception, a conception that was adequate for the needs of analysis, differential geometry, and the empirical sciences, they did not, nor could not, free their theory of its logical, theoretical, and philosophical presuppositions, nor could they preclude the possibility that other adequate conceptual schemes, each self-consistent, could be devised offering alternative visions of the continuum.

However, it was critiques of the former that gave rise to some of its competitors and the realization of the logical possibility of the latter that gave rise to others. To some extent, the architects of each of its competitors were motivated by the belief, or at least the hope, that their respective theories are or with time would be adequate for the needs of analysis (or differential geometry), though in the cases of the constructivist and predicativist architects analysis was equated with legitimate analysis constructively and predicatively construed. Outside of the overarching question of the historical needs of analysis, the question of whether legitimate analysis thus understood is adequate for the needs of the empirical sciences and the physical sciences, in particular, is the subject of dispute (compare Fletcher 2002; Hellman 1993, 1993a, 1997, 1998; Bridges 1995; Billinge 2000; Bridges and Ishihara 2001; Feferman 1988/1998, postscript). Nonstandard analysis has bypassed all of these questions since from the standpoint of the standard domain it is as strong as or even stronger than standard analysis depending on what one assumes (Henson and Keisler 1986). Moreover, like their late nineteenth- and early twentieth-century non-Archimedean geometric forerunners, nonstandard analysis and the infinitesimalist approaches more generally have drawn attention to the possibility of physical continua whose logical cogency let alone physical possibility had long been in doubt. Whether empirical science will require such a theory, as some already contend (compare Fenstad 1987, 1988) and others, following Veronese (compare 1909/1994, p. 180), will not rule out, only time will tell. Nevertheless, while showing little sign of displacing the standard theory, the constructivist, predicativist, and infinitesimalist alternatives have performed, and continue to perform, important logical and philosophical service. Nonstandard analysis has also had real success in shedding important light on and establishing significant new results in various areas of analysis, theoretical physics, and economics (compare Albeverio, Luxemburg, and Wolff 1995; Arkeryd, Cutland, and Henson 1997; Loeb and Wolff 2000). However, whether nonstandard analysis or any of the other nonstandard theories canvassed earlier, with its corresponding theory of the continuum, will eventually assume the status of the standard theory (or even stand alongside the standard theory as a co-standard theory) remains to be seen.

See also Infinity in Mathematics and Logic.

Bibliography

ancient and medieval philosophical and cosmological theories

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ancient mathematical theories

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the early modern theory of real numbers

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history of the calculus

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leibniz's views

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newton views

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general set theory

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the cantor-dedekind theory of the continuum

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Cantor, Georg. "Über die Ausdehnung eines Satzes der Theorie der trigonometrischen Reihen" (1872). In Georg Cantor Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, edited by Ernest Zermelo. Berlin: J. Springer, 1932.

Dedekind, Richard. "Stetigkeit und irrationale Zahlen" (1872). Translated by W. W. Beman as "Continuity and Irrational Numbers." In From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Vol. 2, edited by William B. Ewald (Oxford, U.K.: Clarendon Press, 1996).

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Ferreirós, José. Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Berlin: Birkhäuser Verlag, 1999.

Hilbert, David. Foundations of Geometry (1899). 10th ed. Translated by Leo Unger. LaSalle, IL: Open Court, 1971.

Hilbert, David. Grundlagen der Geometrie. Leipzig, Germany: Teubner, 1899.

Huntington, Edward V. The Continuum and Other Types of Serial Order: With an Introduction to Cantor's Transfinite Numbers. 2nd ed. Cambridge, MA: Harvard University Press, 1917.

Poincaré, Henri. Science and Hypothesis. In The Foundations of Science. Translated by George Bruce Halsted. New York: Science Press, 1913. "Mathematical Magnitude and Experience" is a translation of a reprinting in modified form of "Le Continu Mathématique," Revue de Métaphysique et de Morale (1893).

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the continuum hypothesis

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constructivist approaches: general background

Beeson, Michael J. Foundations of Constructive Mathematics: Metamathematical Studies. Berlin: Springer-Verlag, 1985.

Bridges, Douglas, and Fred Richman. Varieties of Constructive Mathematics. New York: Cambridge University Press, 1987.

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brouwer's intuitionistic approach

Atten, Mark van. On Brouwer. Belmont, CA: Wadsworth, 2004.

Atten, Mark van, and Dirk van Dalen. "Arguments for the Continuity Principle." Bulletin of Symbolic Logic 8 (3) (2002): 329–347.

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Dalen, Dirk van, ed. Brower's Cambridge Lectures on Intuitionism. New York: Cambridge University Press, 1981.

Dummett, Michael. Elements of Intuitionism. 2nd ed. Oxford, U.K.: Clarendon Press, 2000.

Fraenkel, Abraham A., Yehoshua Bar-Hillel, and Azriel Levy. Foundations of Set Theory. 2nd ed. Amsterdam, Netherlands: North-Holland, 1973.

Heyting, Arend. Intuitionism: An Introduction. 3rd rev. ed. Amsterdam: North-Holland, 1971.

Kushner, Boris. "On Brouwerian Bar Induction." In Reuniting the Antipodes: Constructive and Nonstandard Views of the Continuum, edited by Peter Schuster, Ulrich Berger, and Horst Osswald. Dordrecht, Netherlands: Kluwer Academic, 2001.

McCarthy, David. "Intuitionism in Mathematics." In The Oxford Handbook of Philosophy of Mathematics and Logic, edited by Stewart Shapiro. New York: Oxford University Press, 2005.

Posy, Carl. "Intuitionism and Philosophy." In The Oxford Handbook of Philosophy of Mathematics and Logic, edited by Stewart Shapiro. New York: Oxford University Press, 2005.

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Troelstra, A. S. "On the Origin and Development of Brouwer's Concept of Choice Sequence." In The L.E.J. Brouwer Centenary Symposium, edited by A. S. Troelstra and Dirk van Dalen. Amsterdam, Netherlands: North-Holland, 1982.

Veldman, Wim. "Understanding and Using Brouwer's Continuity Principle." In Reuniting the Antipodes: Constructive and Nonstandard Views of the Continuum, edited by Peter Schuster, Ulrich Berger, and Horst Osswald. Dordrecht, Netherlands: Kluwer Academic, 2001.

bishop's constructive approach

Bishop, Errett. Foundations of Constructive Analysis. New York: McGraw-Hill, 1967.

Bishop, Errett, and Douglas Bridges. Constructive Analysis. New York: Springer-Verlag, 1985.

Bridges, Douglas. "A Constructive Look at the Real Number Line." In Real Numbers, Generalizations of the Reals, and Theories of Continua, edited by Philip Ehrlich. Dordrecht, Netherlands: Kluwer Academic, 1994.

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Bridges, Douglas, and Steve Reeves. "Constructive Mathematics in Theory and Programming Practice." Philosophia Mathematica (3) 7 (1999): 65–104.

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Ciaffaglione, Alberto, and Pietro Di Gianantonio. "A Tour with Constructive Real Numbers." In Types for Proofs and Programs (Durham, 2000), edited by Paul Callaghan et al., 41–52. Berlin: Springer-Verlag, 2002.

Geuvers, Herman, and Milad Niqui. "Constructive Reals in Coq: Axioms and Categoricity." In Types for Proofs and Programs (Durham, 2000), edited by Paul Callaghan et al., 79–95. Berlin: Springer-Verlag, 2002.

Richman, Fred. "Interview with a Constructive Mathematician." Modern Logic (3) 6 (1996): 247–271.

Richman, Fred. "Intuitionism as Generalization." Philosophia Mathematica 5 (1990): 124–128.

russian constructive approach

Beeson, Michael J. Foundations of Constructive Mathematics: Metamathematical Studies. Berlin: Springer-Verlag, 1985.

Kushner, Boris A. Lectures on Constructive Mathematical Analysis. Translated by Elliott Mendelson. Providence, RI: American Mathematical Society, 1984.

continuity and mathematical physics

Billinge, Helen. "Applied Constructive Mathematics: On G. Hellman's 'Mathematical Constructivism in Spacetime.'" British Journal for the Philosophy Science 51 (2) (2000): 299–318.

Bridges, Douglas. "Constructive Mathematics and Unbounded Operators. Reply to: 'Constructive Mathematics and Quantum Mechanics: Unbounded Operators and the Spectral Theorem.'" Journal of Philosophical Logic 24 (5) (1995): 549–561.

Bridges, Douglas, and Hajime Ishihara. "Constructive Unbounded Operators." In Reuniting the Antipodes: Constructive and Nonstandard Views of the Continuum, edited by Peter Schuster, Ulrich Berger, and Horst Osswald. Dordrecht, Netherlands: Kluwer Academic, 2001.

Capek, Milic. The Philosophical Impact of Contemporary Physics. Princeton, NJ: D. Van Nostrand Company, Inc., 1961.

Earman, John. World Enough and Space-Time. Cambridge, MA: MIT Press, 1989.

Fletcher, Peter. "A Constructivist Perspective on Physics." Philosophia Mathematica 10 (1) (2002): 26–42.

Forrest, Peter. "Is Space-Time Discrete or Continuous? An Empirical Question." Synthese 103 (3) (1995): 327–354.

Hellman, Geoffrey. "Constructive Mathematics and Quantum Mechanics: Unbounded Operators and the Spectral Theorem." Journal of Philosophical Logic 22 (3) (1993): 221–248.

Hellman, Geoffrey. "Gleason's Theorem Is Not Constructively Provable." Journal of Philosophical Logic 22 (2) (1993a): 193–203.

Hellman, Geoffrey. "Mathematical Constructivism in Spacetime." British Journal for the Philosophy Science 49 (3) (1998): 425–450.

Hellman, Geoffrey. "Quantum Mechanical Unbounded Operators and Constructive Mathematics—A Rejoinder to Bridges." Journal of Philosophical Logic 26 (2) (1997): 121–127.

Markopoulou, Fotini. "Planck-scale Models of the Universe." In Science and Ultimate Reality. Quantum Theory, Cosmology, and Complexity. In Honor of the 90th Birthday of John Archibald Wheeler, edited by John D. Barrow, Paul C. W. Davies, and Charles L. Harper, Jr. Cambridge, U.K.: Cambridge University Press, 2004.

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Smolin, Lee. Three Roads to Quantum Gravity. New York: Basic Books, 2001.

Smolin, Lee. "Quantum Theories of Gravity: Results and Prospects." In Science and Ultimate Reality. Quantum Theory, Cosmology, and Complexity. In Honor of the 90th Birthday of John Archibald Wheeler, edited by John D. Barrow, Paul C. W. Davies, and Charles L. Harper, Jr. Cambridge, U.K.: Cambridge University Press, 2004.

weyl's predicative approach and its aftermath

Bell, John L. "Hermann Weyl on Intuition and the Continuum." Philosophia Mathematica (3) 8 (2000): 259–273.

Feferman, Solomon. "Predicativity." In The Oxford Handbook of Philosophy of Mathematics and Logic, edited by Stewart Shapiro. New York: Oxford University Press, 2005.

Feferman, Solomon. "Systems of Predicative Analysis." Journal of Symbolic Logic 29 (1964): 1–30.

Feferman, Solomon. "Weyl Vindicated: Das Kontinuum 70 Years Later (1988)." In In the Light of Logic, edited by Solomon Feferman. New York: Oxford University Press, 1998.

Feferman, Solomon. "Why a Little Bit Goes a Long Way: Logical Foundations of Scientifically Applicable Mathematics (1993)." In In the Light of Logic, edited by Solomon Feferman. New York: Oxford University Press, 1998.

Parsons, Charles. "Realism and the Debate on Impredicativity, 1917—1944." In Reflections on the Foundations of Mathematics: Essays in Honor of Solomon Feferman, edited by Wilfried Sieg, Richard Sommer, and Carolyn Talcott. Natick, MA: A. K. Peters, 2002.

Scholz, Erhard. "Herman Weyl on the Concept of Continuum." In Proof Theory: History and Philosophical Significance, edited by Vincent F. Hendricks, Stig Andur Pedersen, and Klaus Frovin Jorgensen. Dordrecht, Netherlands: Kluwer Academic, 2000.

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Weyl, Hermann, et al. "Part II: H. Weyl." In From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s, edited by Paolo Mancosu. New York: Oxford University Press, 1998.

elementary continuity

Artin, Emil, and Otto Schreier. "Algebraische Konstruktion reeller Körper" (1926). In The Collected Papers of Emil Artin, edited by Serge Lang and John T. Tate. Reading, MA: Addison-Wesley, 1965.

Tarski, Alfred. Completeness of Elementary Algebra and Geometry (1939). In Collected Papers of Alfred Tarski, Vol. 2, 1935–1944, edited by Steven R. Givant and Ralph N. McKenzie. Basel, Switzerland: Birkhäuser Verlag, 1986.

Tarski, Alfred. A Decision Method for Elementary Algebra and Geometry (1948). In Collected Papers of Alfred Tarski, Vol. 3, 1945–1957, edited by Steven R. Givant and Ralph N. McKenzie. Basel, Switzerland: Birkhäuser Verlag, 1986.

Tarski, Alfred. "What Is Elementary Geometry?" (1959). In Collected Papers of Alfred Tarski, Vol. 4, 1958–1979, edited by Steven R. Givant and Ralph N. McKenzie. Basel, Switzerland: Birkhäuser Verlag, 1986.

Tarski, Alfred, and Steven Givant. "Tarski's System of Geometry." Bulletin of Symbolic Logic 5 (2) (1999): 175–214.

Sinaceur, Hourya. "Calculation, Order, and Continuity." In Real Numbers, Generalizations of the Reals, and Theories of Continua, edited by Philip Ehrlich. Dordrecht, Netherlands: Kluwer Academic, 1994.

Sinaceur, Hourya Benis. Fields and Models: From Sturm to Tarski and Robinson. New York: Springer-Verlag, 2006.

continuity and euclidean geometry

Ehrlich, Philip. "From Completeness to Archimedean Completeness: An Essay in the Foundations of Euclidean Geometry." In A Symposium on David Hilbert, edited by Alfred Tauber and Akihiro Kanamori. Synthese 110(1) (1997a): 57–76.

Greenberg, Marvin J. Euclidean and Non-Euclidean Geometries: Development and History. 3rd ed. New York: Freeman, 1993.

Hartshorne, Robin. Geometry: Euclid and Beyond. New York: Springer-Verlag, 2000.

Heath, Sir Thomas L. ed. The Thirteen Books of Euclid's Elements: Translated from the Text of Heiberg, Volumes 1–3. New York: Dover Publications, Inc., 1956.

Tarski, Alfred. "What is Elementary Geometry?" (1959). In Collected Papers of Alfred Tarski, Vol. 4, 1958–1979, edited by Steven R. Givant and Ralph N. McKenzie. Basel: Birkhäuser Verlag, 1986.

Tarski, Alfred, and Givant, Steven. "Tarski's System of Geometry." Bulletin of Symbolic Logic 5(2) (1999): 175–214.

infinitesimalist approaches: historical background

Ehrlich, Philip. "Hahn's Über die nichtarchimedischen Grössensysteme and the Development of the Modern Theory of Magnitudes and Numbers to Measure Them." In From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics, edited by Jaakko Hintikka. Dordrecht, Netherlands: Kluwer Academic, 1995.

Ehrlich, Philip. "The Rise of Non-Archimedean Mathematics and the Roots of a Misconception I: The Emergence of Non-Archimedean Systems of Magnitudes." Archive for History of Exact Sciences, (Springer-Verlag Online) (2005).

Fisher, Gordon. "The Infinite and Infinitesimal Quantities of du Bois-Reymond and Their Reception." Archive for History of Exact Sciences 24 (2) (1981): 101–164.

du bois-reymond's conception

Du Bois-Reymond, Paul. Die allgemine Functionentheorie. Tübingen, Germany: Verlag der H. Laupp'schen Buchhandlung, 1882.

Du Bois-Reymond, Paul. "Sur la grandeur relative des infinis des fonctions." Annali di matematica pura ed applicata 4 (1870–1871): 338–353.

Du Bois-Reymond, Paul. "Ueber die Paradoxen des Infinitärcalcüls." Mathematische Annalen 11 (1877): 149–167.

veronese's non-archimedean continuum

Ehrlich, Philip. "Dedekind Cuts of Archimedean Complete Ordered Abelian Groups." Algebra Universalis 37 (2) (1997): 223–234.

Ehrlich, Philip. "General Introduction." In Real Numbers, Generalizations of the Reals, and Theories of Continua, edited by Philip Ehrlich. Dordrecht, Netherlands: Kluwer Academic, 1994.

Fisher, Gordon. "Veronese's Non-Archimedean Linear Continuum." In Real Numbers, Generalizations of the Reals, and Theories of Continua, edited by Philip Ehrlich. Dordrecht, Netherlands: Kluwer Academic, 1994.

Veronese, Giuseppe. Fondamenti di geometria a più dimensioni e a più specie di unità rettilinee esposti in forma elementare. Padua, Italy: Tipografia del Seminario, 1891.

Veronese, Giuseppe. Grundzüge der Geometrie von mehreren Dimensionen und mehreren Arten gradliniger Einheiten in elementarer Form entwickelt. Mit Genehmigung des Verfassers nach einer neuen Bearbeitung des Originals übersetzt von Adolf Schepp. Leipzig: Teubner, 1894.

Veronese, Giuseppe. "La geometria non-Archimedea" (1909). Translated by Mathieu Marion as "On Non-Archimedean Geometry." In Real Numbers, Generalizations of the Reals, and Theories of Continua, edited by Philip Ehrlich. Dordrecht, Netherlands: Kluwer Academic, 1994.

Veronese, Giuseppe. "Il continuo rettilineo e l'assioma V di Archimede." Memorie della Reale Accademia dei Lincei, Atti della Classe di scienze naturali, fisiche e matematiche (4) 6 (1889): 603–624.

the non-archimedean contributions of levi-civita and hahn

Ehrlich, Philip. "From Completeness to Archimedean Completeness: An Essay in the Foundations of Euclidean Geometry." In A Symposium on David Hilbert, edited by Alfred Tauber and Akihiro Kanamori. Synthese 110 (1) (1997a): 57–76.

Ehrlich, Philip. "Hahn's Über die nichtarchimedischen Grössensysteme and the Development of the Modern Theory of Magnitudes and Numbers to Measure Them." In From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics, edited by Jaakko Hintikka. Dordrecht, Netherlands: Kluwer Academic, 1995.

Hahn, Hans. "Über die nichtarchimedischen Grössensysteme." Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien, Mathematisch-Naturwissenschaftliche Klasse 116 (Abteilung IIa) (1907): 601–655.

Laugwitz, Detlef. "Tullio Levi-Civita's Work on Non-Archimedean Structures (with an Appendix: Properties of Levi-Civita Fields)." Tullio Levi-Civita Convegno Internazionale Celebrativo Del Centenario Della Nascita, Accademia Nazionale Dei Lincei Atti Dei Convegni Lincei, Rome 8 (1975): 297–312.

Levi-Civita, Tullio. "Sugli infiniti ed infinitesimi attuali quali elementi analitici" (1892–1893). In Tullio Levi-Civita, Opere Matematiche, Memorie e Note, Volume primo 1893–1900. Bologna, Italy: Nicola Zanichelli, 1954.

Levi-Civita, Tullio. "Sui Numeri Transfiniti" (1898). In Tullio Levi-Civita, Opere Matematiche, Memorie e Note, Volume primo 1893–1900. Bologna, Italy: Nicola Zanichelli, 1954.

nonstandard analysis

Albeverio, Sergio A., Wilhelm A. J. Luxemburg, and Manfred P. H. Wolff, eds. Advances in Analysis, Probability, and Mathematical Physics: Contributions of Nonstandard Analysis. Dordrecht, Netherlands: Kluwer Academic, 1995.

Arkeryd, Leif O., Nigel J. Cutland, and C. Ward Henson, eds. Nonstandard Analysis: Theory and Applications. Dordrecht, Netherlands: Kluwer Academic, 1997.

Chang, C. C., and H. Jerome Keisler. Model Theory. 3rd ed. Amsterdam, Netherlands: North-Holland, 1990.

Di Nasso, Mauro, and Marco Forti. "On the Ordering of the Nonstandard Real Line." In Logic and Algebra, edited by Yi Zhang. Providence, RI: American Mathematical Society, 2002.

Fenstad, Jens Erik. "The Discrete and the Continuous in Mathematics and the Natural Sciences." In L'Infinito Nella Scienza, edited by Giuliano Toraldo di Francia. Rome: Istituto Della Enciclopedia Italiana, Fondata Di G. Treccani, 1987.

Fenstad, Jens Erik. "Infinities in Mathematics and the Natural Sciences." In Methods and Applications of Mathematical Logic: Contemporary Mathematics. Vol. 69, edited by Walter A. Carnielli and Luiz Paulo de Alcantara. Providence, RI: American Mathematical Society, 1988.

Goldblatt, Robert. Lectures on the Hyperreals: An Introduction to Nonstandard Analysis. New York: Springer-Verlag, 1998.

Henson, C. W., and H. Jerome Keisler. "On the Strength of Nonstandard Analysis." Journal of Symbolic Logic 51 (2) (1986): 377–1386.

Keisler, H. Jerome. Elementary Calculus. 2nd ed. Boston: Prindle, Weber and Schmidt, 1986.

Keisler, H. Jerome. Foundations of Infinitesimal Calculus. Boston: Prindle, Weber and Schmidt, 1976.

Keisler, H. Jerome. "The Hyperreal Line." In Real Numbers, Generalizations of the Reals, and Theories of Continua, edited by Philip Ehrlich. Dordrecht, Netherlands: Kluwer Academic, 1994.

Keisler, H. Jerome. "Limit Ultrapowers." Transactions of the American Mathematical Society 107 (1963): 383–408.

Lindstrøm, Tom. "An Invitation to Nonstandard Analysis." In Nonstandard Analysis and Its Applications, edited by Nigel Cutland. New York: Cambridge University Press, 1988.

Loeb, Peter A. "An Introduction to Nonstandard Analysis." In Nonstandard Analysis for the Working Mathematician, edited by Peter A. Loeb and Manfred Wolff. Dordrecht, Netherlands: Kluwer Academic, 2000.

Loeb, Peter A., and Manfred Wolff, eds. Nonstandard Analysis for the Working Mathematician. Dordrecht, Netherlands: Kluwer Academic, 2000.

Nelson, Edward. "Internal Set Theory: A New Approach to Nonstandard Analysis." Bulletin of the American Mathematical Society 83 (1977): 1165–1198.

Robert, Alain. Nonstandard Analysis. New York: Wiley, 1988.

Robinson, Abraham. "The Metaphysics of the Calculus" (1967). In Abraham Robinson Selected Papers. Vol. 2, Nonstandard Analysis and Philosophy, edited by W. A. J. Luxemburg and Stephan Körner. New Haven, CT: Yale University Press, 1979.

Robinson, Abraham. "Non-standard Analysis" (1961). In Abraham Robinson Selected Papers. Vol. 2, Nonstandard Analysis and Philosophy, edited by W. A. J. Luxemburg and Stephan Körner. New Haven, CT: Yale University Press, 1979.

Robinson, Abraham. Non-standard Analysis. Amsterdam, Netherlands: North-Holland, 1966.

Robinson, Abraham. Non-standard Analysis. Rev. ed. Amsterdam: North-Holland Publishing Company, 1974.

Skolem, Thoralf. "Über die Nichtcharakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen." Fundamenta Mathematica 23 (1934): 150–161.

Zakon, Elias. "Remarks on the Nonstandard Real Axis." In Applications of Model Theory to Algebra, Analysis, and Probability, edited by W. A. J. Luxemburg. New York: Holt, Rinehart and Winston, 1969.

smooth infinitesimal analysis

Bell, John L. "The Continuum in Smooth Infinitesimal Analysis." In Reuniting the Antipodes: Constructive and Nonstandard Views of the Continuum, edited by Peter Schuster, Ulrich Berger, and Horst Osswald. Dordrecht, Netherlands: Kluwer Academic, 2001.

Bell, John L. "Infinitesimals and the Continuum." The Mathematical Intelligencer 17 (2) (1995): 55–57.

Bell, John L. A Primer of Infinitesimal Analysis. New York: Cambridge University Press, 1998.

Giordano, Paolo. "Nilpotent Infinitesimals and Synthetic Differential Geometry in Classical Logic." In Reuniting the Antipodes: Constructive and Nonstandard Views of the Continuum, edited by Peter Schuster, Ulrich Berger, and Horst Osswald. Dordrecht, Netherlands: Kluwer Academic, 2001.

Kock, Anders. "Differential Calculus and Nilpotent Real Numbers." Bulletin of Symbolic Logic 9 (2) (2003): 225–230.

Kock, Anders. Synthetic Differential Geometry. New York: Cambridge University Press, 1981.

Lavendhomme, René. Basic Concepts of Synthetic Differential Geometry. Dordrecht, Netherlands: Kluwer Academic, 1996.

Lawvere, F. William. "Categorical Dynamics." In Topos Theoretic Methods in Geometry, edited by Anders Kock. Aarhus Matematisk Institut: Various Publication Series, 1979.

Lawvere, F. William. "Toward the Description in a Smooth Topos of the Dynamically Possible Motions and Deformations of a Continuous Body." Cahiers de Topologie et Géométrie Différentielle Catégoriques 21(4) (1980): 377–392.

McLarty, Colin. "Defining Sets as Sets of Points." Journal of Philosophical Logic 17 (1988): 75–90.

Moerdijk, Ieke, and Gonzalo E. Reyes. Models for Smooth Infinitesimal Analysis. New York: Springer-Verlag, 1991.

Succi Cruciani, Rosanna. "Euclidean Geometry with Infinitesimals." Rendiconti di Matematica e delle sue Applicazioni, Serie (Serie VII) 8 (4) (1989): 557–578.

the work of schmieden and laugwitz

Laugwitz, Detlef. "Kurt Schmieden's Approach to Infinitesimals: An Eye-Opener to the Historiography of Analysis." In Reuniting the Antipodes: Constructive and Nonstandard Views of the Continuum, edited by Peter Schuster, Ulrich Berger, and Horst Osswald. Dordrecht, Netherlands: Kluwer Academic, 2001.

Laugwitz, Detlef. "Leibniz's Principle and Omega Calculus." In Le Continu Mathematique, Colloque de Cerisy, edited by Hourya Sinaceur and Jean-Michel Salanskis. Paris: Springer-Verlag, 1992.

Laugwitz, Detlef. "Ω-Calculus as a Generalization of Field Extension—An Alternative Approach to Nonstandard Analysis." In Nonstandard Analysis: Recent Developments, edited by Albert Emerson Hurd. Berlin: Springer-Verlag, 1983.

Schmieden, Curt, and Detlef Laugwitz. "Eine Erweiterung der Infinitesimalrechnung." Mathematische Zeitschrift 69 (1958): 1–39.

surreal numbers

Conway, J. H. On Numbers and Games. London: Academic Press, 1976.

Conway, J. H. On Numbers and Games. 2nd ed. Natick, MA: A.K. Peters, 2001.

Conway, J. H. "The Surreals and Reals." In Real Numbers, Generalizations of the Reals, and Theories of Continua, edited by Philip Ehrlich. Dordrecht, Netherlands: Kluwer Academic, 1994.

Ehrlich, Philip. "The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small." In Philosophical Insights into Logic and Mathematics (Abstracts). Nancy, France: Université de Nancy Laboratoire de Philosophie et d'Histoire des Sciences, Archive Henri Poincaré (2002): 41–43.

Ehrlich, Philip. "The Absolute Arithmetic and Geometric Continua." In PSA 1986. Vol. 2, edited by Arthur Fine and Peter Machamer. Lansing, MI: Philosophy of Science Association, 1987.

Ehrlich, Philip. "Absolutely Saturated Models." Fundamenta Mathematicae 133 (1) (1989): 39–46.

Ehrlich, Philip. "All Number Great and Small." In Real Numbers, Generalizations of the Reals, and Theories of Continua, edited by Philip Ehrlich. Dordrecht, Netherlands: Kluwer Academic, 1994.

Ehrlich, Philip. "An Alternative Construction of Conway's Ordered Field No." Algebra Universalis 25 (1988): 7–16. Errata, Algebra Universalis 25 (1988): 233.

Ehrlich, Philip. "Number Systems with Simplicity Hierarchies: A Generalization of Conway's Theory of Surreal Numbers." Journal of Symbolic Logic 66 (3) (2001): 1231–1258. Corrigendum, 70 (3) (2005): 1022.

Ehrlich, Philip. "Surreal Numbers: An Alternative Construction (Abstract)." Bulletin of Symbolic Logic 8 (3) (2002a): 448.

Ehrlich, Philip. "Surreal Numbers and the Unification of All Numbers Great and Small." Bulletin of Symbolic Logic 10 (2) (2004): 253.

Ehrlich, Philip. "Universally Extending Arithmetic Continua." In Le Continu Mathematique, Colloque de Cerisy, edited by Hourya Sinaceur and Jean-Michel Salanskis. Paris: Springer-Verlag, 1992.

Ehrlich, Philip. "Universally Extendeding Continua." Abstracts of Papers Presented to the American Mathematical Society 10 (1989a): 15.

cauchy's continuum

Cleave, J. P. "Cauchy, Convergence and Conitinuity." British Journal for the Philosophy of Science 22 (1971): 27–37.

Grabiner, Judith V. The Origins of Cauchy's Rigorous Calculus. Cambridge, MA.: MIT Press, 1981.

Lakatos, Imre. "Cauchy and the Continuum." The Mathematical Intelligencer (1) (1978): 151–161.

Laugwitz, Detlef. "Definite Values of Infinite Sums: Aspects of the Foundations of Infinitesimal Analysis around 1820." Archive for History of Exact Sciences 39 (1989): 195–245.

Laugwitz, Detlef. "Infinitely Small Quantities in Cauchy's Textbook." Historia Mathematica (14) (1987): 258–274.

Lützen, Jesper. "The Foundations of Analysis in the 19th Century." In A History of Analysis, edited by Hans Niels Jahnke. Providence, RI: American Mathematical Society, 2003.

Robinson, Abraham. "The Metaphysics of the Calculus" (1967). In Abraham Robinson Selected Papers. Vol. 2, Nonstandard Analysis and Philosophy, edited by W. A. J. Luxemburg and Stephan Körner. New Haven, CT: Yale University Press, 1979.

Robinson, Abraham. Non-standard Analysis. Rev. ed. Amsterdam, Netherlands: North-Holland, 1974.

Spalt, Detlef D. "Cauchys Kontinuum. Eine historiografische Annäherung via Cauchys Summensatz." Archive for History of Exact Sciences 56 (4) (2002): 285–338.

bernard bolzano

Rusnock, Paul. Bolzano's Philosophy and the Emergence of Modern Mathematics. Amsterdam, Netherlands: Editions Rodophi B. V., 2000.

Russ, Steve, ed. The Mathematical Works of Bernard Bolzano. New York: Oxford University Press, 2004.

continuous functions

Youschkevitch, A. P. "The Concept of Function up to the Middle of the 19th Century." Archive for History of Exact Sciences 16 (1) (1976): 37–85.

charles sanders peirce

Ehrlich, Philip. "The Peircean Linear Continuum: A Surreal Model (Abstract)." The Bulletin of Symbolic Logic 11(4) (2005).

Herron, Timothy. "Charles Sanders Peirce's Theories of Infinitesimals." Transactions of the Charles S. Peirce Society 33 (3) (1997): 590–645.

Myrvold, Wayne. "Peirce on Cantor's Paradox and the Continuum." Transactions of the Charles S. Peirce Society 31 (3) (1995): 508–541.

Noble, N. A. Brian "Peirce's Definitions of Continuity and the Concept of Possibility." Transactions of the Charles S. Peirce Society 25 (2) (1989): 149–174.

Peirce, Charles Sanders. "Infinitesimals" (1900). In Collected Papers of Charles Sanders Peirce. Vol. 3, edited by Charles Hartshone and Paul Weiss. Cambridge, MA: Harvard University Press, 1935.

Peirce, Charles Sanders. "The Logic of Continuity." In Reasoning and the Logic of Things: The Cambridge Conferences Lectures of 1898, edited by Kenneth Laine Ketner. Cambridge, MA: Harvard University Press, 1992.

Potter, Vincent, and Paul Schields. "Peirce's Definitions of Continuity." Transactions of the Charles S. Peirce Society 13 (1) (1977): 20–34.

Putnam, Hilary. "Peirce's Continuum." In Reasoning and the Logic of Things: The Cambridge Conferences Lectures of 1898, edited by Kenneth Laine Ketner, with an Introduction by Kenneth Laine Ketner and Hilary Putnam. Cambridge, MA.: Harvard University Press, 1992: 37–54.

fuzzy real numbers

Lowen, Robert. Fuzzy Set Theory: Basic Concepts, Techniques, and Bibliography. Dordrecht, Netherlands: Kluwer Academic, 1996.

Zadeh, Lotfi Asker. Fuzzy Sets and Applications: Selected Papers. Edited by R. R. Yager et al. New York: Wiley, 1987.

Philip Ehrlich (2005)

Continuity

Resources

Continuity is the property of being uninterrupted. Intuitively, a continuous line or mathematical function is one that can be graphed without having to lift the pencil from the paper; there are no missing points, no skipped segments, no sudden jumps or disconnects. This intuitive notion of continuity goes back to ancient Greece, where many mathematicians and philosophers believed that reality was a reflection of number. Since numbers are infinitely divisible, space and time must also be infinitely divisible. In the fifth century BC, however, the Greek mathematician Zeno pointed out that a number of logical inconsistencies arise when assuming that space is infinitely divisible; he stated his findings in the form of paradoxes. For example, in one paradox Zeno argued that the infinite divisibility of space actually meant that all motion was impossible. His argument went approximately as follows: before reaching any destination a traveler must first complete one-half of his journey, and before completing one-half he must complete one-fourth, and before completing one-fourth he must complete one-eighth, and so on indefinitely. Any trip requires an infinite number of steps, so ultimately, Zeno argued,

no journey could ever begin, and all motion was impossible. Zeno’s paradoxes had a disturbing effect on Greek mathematicians, and the ultimate resolution of his paradoxes did not occur until the intuitive notion of continuity was finally dealt with logically. (The reason that travelers do complete their journeys is not that space is discontinuous, but that—as mathematicians of Zeno’s time did not know—an infinite number of quantities can, depending on their nature, sum to a finite quantity. Thus, the infinite number of time intervals that can be imagined between the beginning and end of a journey can, and do, sum to a non-infinite time interval.)

As late as the seventeenth century, mathematicians continued to believe, as the ancient Greeks had, that continuity was a simple result of density, meaning that between any two points, no matter how close together, there is always another. This is true, for example, of the rational numbers. However, the rational numbers do not form a true continuum, since irrational numbers like √2 are missing, leaving infinitesimal holes or discontinuities. The irrational numbers are required to complete the continuum. Together, the rational and irrational numbers do form a continuous set—the set of real numbers. Thus, the continuity of points on a line is ultimately linked to the continuity of the set of real numbers by establishing a one-to-one correspondence between the two. This approach to continuity was first established in the 1820s by Augustin-Louis Cauchy, who finally began to solve the problem of handling continuity logically. In Cauchy’s view, any line corresponding to the graph of a function is continuous at a point if the value of the function at x, denoted by f (x), gets arbitrarily close to f (p), when x gets close to a real number p. If f (x) is continuous for all real numbers x contained in a finite interval, then the function is continuous in that interval. If f (x) is continuous for every real number x, then the function is continuous everywhere.

Cauchy’s definition of continuity is essentially the one we use today, though somewhat more refined versions were developed in the 1850s and later in the nineteenth century. For example, the concept of continuity is often described in relation to limits. The condition for a function to be continuous is equivalent to the requirement that the limit of the function at the point p be equal to f (p), that is:

lim f (x) = f (p).

x → P

In this version, there are two conditions that must be met for a function to be continuous at a point. First, the limit must exist at the point in question, and, second, it must be numerically equal to the value of the function at that point. For instance, polynomial functions are continuous everywhere, because the value of the function f (x) approaches f (p) smoothly, as x gets close to p, for all values of p (Figure 1).

However, a polynomial function with a single point redefined is not continuous at the point x = p if the limit of the function as x approaches p is L, and not f (p) (Figure 2). This is a somewhat artificial example, but it makes the point that when the limit of f (x) as x approaches p is not f (p) then the function is not continuous at x = p. More realistic examples of discontinuous functions include the square wave (Figure 3), which illustrates the existence of right and left hand limits that differ; and functions with infinite discontinuities, that is, with limits that do not exist (Figure 4).

These examples serve to illustrate the close connection between the limiting value of a function at a point and continuity at a point.

There are two important properties of continuous functions. First, if a function is continuous in a closed interval, then the function has a maximum value and a minimum value in that interval. Since continuity implies that f (x) cannot be infinite for any x in the interval, the function must have both a maximum and a minimum value, though the two values may be equal. Second, the fact that there can be no holes in a continuous curve implies that a function, continuous on a closed interval [a, b], takes on every value between f (a) and f (b) at least once. The concept of continuity is central to isolating points for which the derivative of a function does not exist. The derivative of a function is equal to the slope of the tangent to the graph of the function. For some functions it is not possible to draw

KEY TERMS

Function— A set of ordered pairs, defined by a rule or mathematical statement, describing the relationship between the first element of each pair, called the independent variable, and the second element of each pair, called the dependent variable or value of the function.

Interval— An interval is a subset of the real numbers corresponding to a line segment of finite length and including all the real numbers between its end points. An interval is closed if the endpoints are included and open if they are not.

Limit— The limit (L) of a function f (x) is defined for any point p to be the value that f (x) approaches when x gets infinitely close to p. If the value of the function becomes infinite, the limit does not exist.

a unique tangent at a particular point on the graph, such as any endpoint of a step function segment. When this is the case, it is not possible to determine the value of the derivative at that point. Today, the meaning of continuity is settled within the mathematics community, though it continues to present problems for philosophers and physicists.

Resources

BOOKS

Bridger, Mark. Real Analysis: A Constructive Approach. New York: Wiley-Interscience, 2006.

Larson, Ron, et al. Calculus with Analytic Geometry. Boston: Houghton Mifflin Company, 2005.

Verberg, Dale, et al. Calculus. 9th ed. Upper Saddle River, NJ: Prentice Hall, 2006.

J. R. Maddocks

Continuity

Continuity expresses the property of being uninterrupted. Intuitively, a continuous line or function is one that can be graphed without having to lift the pencil from the paper ; there are no missing points, no skipped segments and no disconnections. This intuitive notion of continuity goes back to ancient Greece, where many mathematicians and philosophers believed that reality was a reflection of number. Thus, they thought, since numbers are infinitely divisible, space and time must also be infinitely divisible. In the fifth century b.c., however, the Greek mathematician Zeno pointed out that a number of logical inconsistencies arise when assuming that space is infinitely divisible, and stated his findings in the form of paradoxes. For example, in one paradox Zeno argued that the infinite divisibility of space actually meant that all motion was impossible. His argument went approximately as follows: before reaching any destination a traveler must first complete one-half of his journey, and before completing one-half he must complete one-fourth, and before completing one-fourth he must complete one-eighth, and so on indefinitely. Any trip requires an infinite number of steps, so ultimately, Zeno argued, no journey could ever begin, and all motion was impossible. Zeno's paradoxes had a disturbing effect on Greek mathematicians, and the ultimate resolution of his paradoxes did not occur until the intuitive notion of continuity was finally dealt with logically.

The continuity of space or time, considered by Zeno and others, is represented in mathematics by the continuity of points on a line. As late as the seventeenth century, mathematicians continued to believe, as the ancient Greeks had, that this continuity of points was a simple result of density , meaning that between any two points, no matter how close together, there is always another. This is true, for example, of the rational numbers. However, the rational numbers do not form a continuum, since irrational numbers like √ 2 are missing, leaving holes or discontinuities. The irrational numbers are required to complete the continuum. Together, the rational and irrational numbers do form a continuous set, the set of real numbers . Thus, the continuity of points on a line is ultimately linked to the continuity of the set of real numbers, by establishing a one-to-one correspondence between the two. This approach to continuity was first established in the 1820s, by Augustin-Louis Cauchy, who finally began to solve the problem of handling continuity logically. In Cauchy's view, any line corresponding to the graph of a function is continuous at a point , if the value of the function at x, denoted by f(x), gets arbitrarily close to f(p), when x gets close to a real number p. If f(x) is continuous for all real numbers x contained in a finite interval , then the function is continuous in that interval. If f(x) is continuous for every real number x, then the function is continuous everywhere.

Cauchy's definition of continuity is essentially the one we use today, though somewhat more refined versions were developed in the 1850s, and later in the nineteenth century. For example, the concept of continuity is often described in relation to limits. The condition for a function to be continuous, is equivalent to the requirement that the limit of the function at the point p be equal to f(p), that is:

In this version, there are two conditions that must be met for a function to be continuous at a point. First, the limit must exist at the point in question, and, second, it must be numerically equal to the value of the function at that point. For instance, polynomial functions are continuous everywhere, because the value of the function f(x) approaches f(p) smoothly, as x gets close to p, for all values of p.

However, a polynomial function with a single point redefined is not continuous at the point x = p if the limit of the function as x approaches p is L, and not f(p). This is a somewhat artificial example, but it makes the point that when the limit of f(x) as x approaches p is not f(p) then the function is not continuous at x = p. More realistic examples of discontinuous functions include the square wave, which illustrates the existence of right and left hand limits that differ; and functions with infinite discontinuities, that is, with limits that do not exist.

These examples serve to illustrate the close connection between the limiting value of a function at a point, and continuity at a point.

There are two important properties of continuous functions. First, if a function is continuous in a closed interval, then the function has a maximum value and a minimum value in that interval. Since continuity implies that f(x) cannot be infinite for any x in the interval, the function must have both a maximum and a minimum value, though the two values may be equal. Second, the fact that there can be no holes in a continuous curve implies that a function, continuous on a closed interval [a,b], takes on every value between f(a) and f(b) at least once. The concept of continuity is central to isolating points for which the derivative of a function does not exist. The derivative of a function is equal to the slope of the tangent to the graph of the function. For some functions it is not possible to draw a unique tangent at a particular point on the graph, such as any endpoint of a step function segment. When this is the case, it is not possible to determine the value of the derivative at that point. Today, the meaning of continuity is settled within the mathematics community, though it continues to present problems for philosophers and physicists.

Resources

books

Allen, G.D., C. Chui, and B. Perry. Elements of Calculus. 2nd ed. Pacific Grove, CA: Brooks/Cole Publishing Co, 1989.

Boyer, Carl B. A History of Mathematics. 2nd ed. Revised by Uta C. Merzbach. New York: John Wiley and Sons, 1991.

Larson, Ron. Calculus With Analytic Geometry. Boston: Houghton Mifflin College, 2002.

Paulos, John Allen. Beyond Numeracy, Ruminations of a Numbers Man. New York: Alfred A. Knopf, 1991.

Silverman, Richard A. Essential Calculus With Applications. New York: Dover, 1989.

periodicals

McLaughlin, William I. "Resolving Zeno's Paradoxes." Scientific American 271 (1994): 84-89.

J. R. Maddocks

KEY TERMS

Function

—A set of ordered pairs, defined by a rule, or mathematical statement, describing the relationship between the first element of each pair, called the independent variable, and the second element of each pair, called the dependent variable or value of the function.

Interval

—An interval is a subset of the real numbers corresponding to a line segment of finite length, and including all the real numbers between its end points. An interval is closed if the endpoints are included and open if they are not.

Limit

—The limit (L) of a function f(x) is defined for any point p to be the value that f(x) approaches when x gets infinitely close to p. If the value of the function becomes infinite, the limit does not exist.

con·ti·nu·i·ty / ˌkäntnˈ(y)oōətē/ • n. (pl. -ties) 1. the unbroken and consistent existence or operation of something over a period of time. ∎  a state of stability and the absence of disruption. ∎  (often continuity between/with) a connection or line of development with no sharp breaks: the Church stands in direct continuity with the Old Testament people of God.2. the maintenance of continuous action and self-consistent detail in the various scenes of a movie or broadcast. ∎  the linking of broadcast items, esp. by a spoken commentary.