The usual meaning of the wordcontinuous is“unbroken” or “uninterrupted”: thus acontinuous entity—acontinuum—has no“gaps”. We commonly suppose that space and time arecontinuous, and certain philosophers have maintained that all naturalprocesses occur continuously: witness, for example, Leibniz’sfamous apothegmnatura non facit saltus—“naturemakes no jump”. In mathematics the word is used in the samegeneral sense, but has had to be furnished with increasingly precisedefinitions. So, for instance, in the later eighteenth centurycontinuity of a function was taken to mean that infinitesimal changesin the value of the argument induced infinitesimal changes in thevalue of the function. With the abandonment of infinitesimals in thenineteenth century this definition came to be replaced by oneemploying the more precise concept oflimit.

Traditionally, aninfinitesimal quantity is one which, whilenot necessarily coinciding with zero, is in some sense smaller thanany finite quantity. For engineers, an infinitesimal is a quantity sosmall that its square and all higher powers can be neglected. In thetheory of limits the term “infinitesimal” is sometimesapplied to any sequence whose limit is zero. Aninfinitesimalmagnitude may be regarded as what remains after a continuum hasbeen subjected to an exhaustive analysis, in other words, as acontinuum “viewed in the small”. It is in this sense thatcontinuous curves have sometimes been held to be“composed” of infinitesimal straight lines.

Infinitesimals have a long and colorful history. They make an earlyappearance in the mathematics of the Greek atomist philosopherDemocritus (c. 450 BCE), only to be banished by the mathematicianEudoxus (c. 350 BCE) in what was to become official“Euclidean” mathematics. Taking the somewhat obscure formof “indivisibles”, they reappear in the mathematics of thelate Middle Ages and later played an important role in the developmentof the calculus. Their doubtful logical status led in the nineteenthcentury to their abandonment and replacement by the limit concept. Inrecent years, however, the concept of infinitesimal has been refoundedon a rigorous basis.

• 1. Introduction: The Continuous, the Discrete, and the Infinitesimal
• 2. The Continuum and the Infinitesimal in the Ancient Period
• 3. The Continuum and the Infinitesimal in the Medieval, Renaissance, and Early Modern Periods
• 4. The Continuum and the Infinitesimal in the Seventeenth and Eighteenth Centuries
• 5. The Continuum and the Infinitesimal in the Nineteenth Century
• 6. Critical Reactions to Arithmetization
• 7. Nonstandard Analysis
• 8. The Constructive Real Line and the Intuitionistic Continuum
• 9. Smooth Infinitesimal Analysis
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Introduction: The Continuous, the Discrete, and the Infinitesimal

We are all familiar with the idea ofcontinuity. To be continuous^[1] is to constitute an unbroken or uninterrupted whole, like the oceanor the sky. A continuous entity—acontinuum—hasno “gaps”. Opposed to continuity isdiscreteness:to be discrete^[2] is to be separated, like the scattered pebbles on a beach or theleaves on a tree. Continuity connotes unity; discreteness,plurality.

While it is the fundamental nature of a continuum to beundivided, it is nevertheless generally (although notinvariably) held that any continuum admits of repeated or successivedivision without limit. This means that the process ofdividing it into ever smaller parts will never terminate in anindivisible or anatom—that is, a part which,lacking proper parts itself, cannot be further divided. In a word,continua aredivisible without limit orinfinitelydivisible. The unity of a continuum thus conceals a potentiallyinfinite plurality. In antiquity this claim met with the objectionthat, were one to carry out completely—if only inimagination—the process of dividing an extended magnitude, suchas a continuous line, then the magnitude would be reduced to amultitude of atoms—in this case, extensionless points—oreven, possibly, to nothing at all. But then, it was held, no matterhow many such points there may be—even if infinitelymany—they cannot be “reassembled” to form theoriginal magnitude, for surely a sum of extensionless elements stilllacks extension.^[3] Moreover, if indeed (as seems unavoidable) infinitely many pointsremain after the division, then, following Zeno, the magnitude may betaken to be a (finite) motion, leading to the seemingly absurdconclusion that infinitely many points can be “touched” ina finite time.

Such difficulties attended the birth, in the fifth century BCE, of theschool ofatomism. The founders of this school, Leucippus andDemocritus, claimed that matter, and, more generally, extension, isnot infinitely divisible. Not only would the successive division ofmatter ultimately terminate in atoms, that is, in discrete particlesincapable of being further divided, but matter hadinactuality to be conceived as being compounded from such atoms. Inattacking infinite divisibility the atomists were at the same timemounting a claim that the continuous is ultimately reducible to thediscrete, whether it be at the physical, theoretical, or perceptuallevel.

The eventual triumph of the atomic theory in physics and chemistry inthe nineteenth century paved the way for the idea of“atomism”, as applying to matter, at least, to becomewidely familiar: it might well be said, to adapt Sir WilliamHarcourt’s famous observation in respect of the socialists ofhis day, “We are all atomists now”. Nevertheless, only aminority of philosophers of the past espoused atomism at ametaphysical level, a fact which may explain why the analogousdoctrine upholding continuity lacks a familiar name: that which isunconsciously acknowledged requires no name. Peirce coined the termsynechism (from Greeksyneche,“continuous”) for his own philosophy—a philosophypermeated by the idea of continuity in its sense of “being connected”.^[4] In this article I shall appropriate Peirce’s term and use it ina sense shorn of its Peircean overtones, simply as a contrary toatomism. I shall also use the term “divisionism” for themore specific doctrine that continua are infinitely divisible.

Closely associated with the concept of a continuum is that ofinfinitesimal.^[5] Aninfinitesimal magnitude has been somewhat hazilyconceived as a continuum “viewed in the small”, an“ultimate part” of a continuum. In something like the samesense as a discrete entity is made up of its individual units, its“indivisibles”, so, it was maintained, a continuum is“composed” of infinitesimal magnitudes, its ultimateparts. (It is in this sense, for example, that mathematicians of theseventeenth century held that continuous curves are“composed” of infinitesimal straight lines.) Now the“coherence” of a continuum entails that each of its(connected) parts is also a continuum, and, accordingly, divisible.Since points are indivisible, it follows that no point can be part ofa continuum. Infinitesimal magnitudes, as parts of continua, cannot,of necessity, be points: they are, in a word,nonpunctiform.

Magnitudes are normally taken as beingextensive quantities,like mass or volume, which are defined over extended regions of space.By contrast, infinitesimal magnitudes have been construed asintensive magnitudes resembling locally defined intensivequantities such as temperature or density. The effect of“distributing” or “integrating” an intensivequantity over such an intensive magnitude is to convert the formerinto an infinitesimal extensive quantity: thus temperature istransformed into infinitesimal heat and density into infinitesimalmass. When the continuum is the trace of a motion, the associatedinfinitesimal/intensive magnitudes have been identified aspotential magnitudes—entities which, while notpossessing true magnitude themselves, possess atendency togenerate magnitude through motion, so manifesting“becoming” as opposed to “being”.

An infinitesimalnumber is one which, while not coincidingwith zero, is in some sense smaller than any finite number. This sensehas often been taken to be the failure to satisfy thePrinciple ofArchimedes, which amounts to saying that an infinitesimal numberis one that, no matter how many times it is added to itself, theresult remains less than any finite number. In the engineer’spractical treatment of the differential calculus, an infinitesimal isa number so small that its square and all higher powers can beneglected. In the theory of limits the term“infinitesimal” is sometimes applied to any sequence whoselimit is zero.

The concept of anindivisible is closely allied to, but to bedistinguished from, that of an infinitesimal. An indivisible is, bydefinition, something that cannot be divided, which is usuallyunderstood to mean that it has no proper parts. Now a partless, orindivisible entity does not necessarily have to be infinitesimal:souls, individual consciousnesses, and Leibnizian monads allsupposedly lack parts but are surely not infinitesimal. But these havein common the feature of being unextended; extended entities such aslines, surfaces, and volumes prove a much richer source of“indivisibles”. Indeed, if the process of dividing suchentities were to terminate, as the atomists maintained, it wouldnecessarily issue in indivisibles of a qualitatively different nature.In the case of a straight line, such indivisibles would, plausibly, bepoints; in the case of a circle, straight lines; and in the case of acylinder divided by sections parallel to its base, circles. In eachcase the indivisible in question is infinitesimal in the sense ofpossessing one fewer dimension than the figure from which it isgenerated. In the sixteenth and seventeenth centuriesindivisibles in this sense were used in the calculation of areas andvolumes of curvilinear figures, a surface or volume being thought ofas a collection, or sum, of linear, or planar indivisiblesrespectively.

The concept of infinitesimal was beset by controversy from itsbeginnings. The idea makes an early appearance in the mathematics ofthe Greek atomist philosopher Democritus c. 450 BCE, only to bebanished c. 350 BCE by Eudoxus in what was to become official“Euclidean” mathematics. We have noted their reappearanceas indivisibles in the sixteenth and seventeenth centuries: in thisform they were systematically employed by Kepler, Galileo’sstudent Cavalieri, the Bernoulli clan, and a number of othermathematicians. In the guise of the beguilingly named“linelets” and “timelets”, infinitesimalsplayed an essential role in Barrow’s “method for findingtangents by calculation” (1670 [1916: 119]), which appears inhisLectiones Geometricae of 1670. As “evanescentquantities” infinitesimals were instrumental (although laterabandoned) in Newton’s development of the calculus, and, as“inassignable quantities”, in Leibniz’s. The Marquisde L’Hôpital, who in 1696 published the first treatise onthe differential calculus (entitledAnalyse des Infiniments Petitspour l’Intelligence des Lignes Courbes), invokes theconcept in postulating that “a curved line may be regarded asbeing made up of infinitely small straight line segments” (1696:3 [Postulate II]), and that “one can take as equal twoquantities differing by an infinitely small quantity” (1696: 2[Postulate 1]).

However useful it may have been in practice, the concept ofinfinitesimal could scarcely withstand logical scrutiny. Derided byBerkeley in the eighteenth century as “Ghosts of departedQuantities” (1734: 59), in the nineteenth century execrated byCantor as “cholera-bacilli” infecting mathematics (1893[1965: 505], translated by Fisher 1981: 116), and in the twentiethroundly condemned by Bertrand Russell as “unnecessary,erroneous, and self-contradictory” (1903: 345), these useful,but logically dubious entities were believed to have been finallysupplanted in the foundations of analysis by the limit concept whichtook rigorous and final form in the latter half of the nineteenthcentury. By the beginning of the twentieth century, the concept ofinfinitesimal had become, in analysis at least, a virtual“unconcept”.

Nevertheless the proscription of infinitesimals did not succeed inextirpating them; they were, rather, driven further underground.Physicists and engineers, for example, never abandoned their use as aheuristic device for the derivation of correct results in theapplication of the calculus to physical problems. Differentialgeometers of the stature of Sophus Lie and Élie Cartan relied ontheir use in the formulation of concepts which would later be put on a“rigorous” footing. And, in a technical sense, they livedon in the algebraists’ investigations of non-Archimedeanfields.

A new phase in the long contest between the continuous and thediscrete has opened in the past few decades with the refounding of theconcept of infinitesimal on a solid basis. This has been achieved intwo essentially different ways, the one providing a rigorousformulation of the idea of infinitesimalnumber, the other ofinfinitesimalmagnitude.

First, in the 1960s Abraham Robinson, using methods of mathematicallogic, creatednonstandard analysis, an extension ofmathematical analysis embracing both “infinitely large”and infinitesimal numbers in which the usual laws of the arithmetic ofreal numbers continue to hold, an idea which, in essence, goes back toLeibniz. Here by an infinitely large number is meant one which exceedsevery positive integer; the reciprocal of any one of these isinfinitesimal in the sense that, while being nonzero, it is smallerthan every positive fraction \(1/n\). Much of the usefulness ofnonstandard analysis stems from the fact that within it everystatement of ordinary analysis involving limits has a succinct andhighly intuitive translation into the language of infinitesimals.

The second development in the refounding of the concept ofinfinitesimal took place in the 1970s with the emergence ofsynthetic differential geometry, also known assmoothinfinitesimal analysis. Based on the ideas of the Americanmathematician F. W. Lawvere, and employing the methods of categorytheory, smooth infinitesimal analysis provides an image of the worldin which the continuous is an autonomous notion, not explicable interms of the discrete. It provides a rigorous framework formathematical analysis in which every function between spaces is smooth(i.e., differentiable arbitrarily many times, and so in particularcontinuous) and in which the use of limits in defining the basicnotions of the calculus is replaced bynilpotentinfinitesimals, that is, of quantities so small (but not actuallyzero) that some power—most usefully, the square—vanishes.Smooth infinitesimal analysis embodies a concept of intensivemagnitude in the form ofinfinitesimal tangent vectors tocurves. A tangent vector to a curve at a point \(p\) on it is a shortstraight line segment \(l\) passing through the point and pointingalong the curve. In fact we may take \(l\) actually to be aninfinitesimalpart of the curve. Curves in smoothinfinitesimal analysis are “locally straight” andaccordingly may be conceived as being “composed of”infinitesimal straight lines in de L’Hôpital’ssense, or as being “generated” by an infinitesimal tangentvector.

The development of nonstandard and smooth infinitesimal analysis hasbreathed new life into the concept of infinitesimal,and—especially in connection with smooth infinitesimalanalysis—supplied novel insights into the nature of thecontinuum.

2. The Continuum and the Infinitesimal in the Ancient Period

The opposition between Continuity and Discreteness played asignificant role in ancient Greek philosophy. This probably derivedfrom the still more fundamental question concerning the One and theMany, an antithesis lying at the heart of early Greek thought (Stokes1971). The Greek debate over the continuous and the discrete seems tohave been ignited by the efforts of Eleatic philosophers such asParmenides (c. 515 BCE), and Zeno (c. 460 BCE) to establish theirdoctrine of absolute monism.^[6] They were concerned to show that the divisibility of Being into partsleads to contradiction, so forcing the conclusion that the apparentlydiverse world is a static, changeless unity.^[7] In hisWay of Truth Parmenides asserts that Being ishomogeneous andcontinuous. However in asserting thecontinuity of Being Parmenides is likely no more than underscoring itsessential unity. Parmenides seems to be claiming that Being is morethan merely continuous—that it is, in fact, a single whole,indeed anindivisible whole. The single Parmenidean existentis a continuum without parts, at once a continuum and an atom. IfParmenides was a synechist, his absolute monism precluded his being atthe same time a divisionist.

In support of Parmenides’ doctrine of changelessness Zenoformulated his famous paradoxes of motion. (see entry onZeno’s paradoxes) TheDichotomy andAchilles paradoxes both restexplicitly on the limitless divisibility of space and time.

The doctrine ofAtomism,^[8] which seems to have arisen as an attempt at escaping the Eleaticdilemma, was first and foremost a physical theory. It was mounted byLeucippus (fl. 440 BCE) and Democritus (b. 460–457 BCE) whomaintained that matter was not divisible without limit, but composedof indivisible, solid, homogeneous, spatially extended corpuscles, allbelow the level of visibility.

Atomism was challenged by Aristotle (384–322 BCE), who was thefirst to undertake the systematic analysis of continuity anddiscreteness. A thoroughgoing synechist, he maintained that physicalreality is a continuous plenum, and that the structure of a continuum,common to space, time and motion, is not reducible to anything else.His answer to the Eleatic problem was that continuous magnitudes arepotentially divisible to infinity, in the sense that they may bedividedanywhere, though they cannot be dividedeverywhere at the same time.

Aristotle identifies continuity and discreteness as attributesapplying to the category of Quantity.^[9] As examples of continuous quantities, orcontinua, he offerslines, planes, solids (i.e., solid bodies), extensions, movement, timeand space; among discrete quantities he includes number^[10] and speech.^[11] He also lays down definitions of a number of terms, includingcontinuity. In effect, Aristotle defines continuity as arelation between entities rather than as anattribute appertaining to a single entity; that is to say, hedoes not provide an explicit definition of the concept ofcontinuum. He observes that a single continuous whole can bebrought into existence by “gluing together” two thingswhich have been brought into contact, which suggests that thecontinuity of a whole should derive from the way itspartsjoin up (seePhysics V, 3). Accordingly for Aristotlequantities such as lines and planes, space and time are continuous byvirtue of the fact that their constituent parts “join togetherat some common boundary” (Catergories, VI [MOMM]). Bycontrastno constituent parts of a discrete quantity canpossess a common boundary.

One of the central theses Aristotle is at pains to defend is theirreducibility of the continuum to discreteness—that a continuumcannot be “composed” of indivisibles or atoms, parts whichcannot themselves be further divided.

Aristotle sometimes recognizesinfinitedivisibility—the property of being divisible into partswhich can themselves be further divided, the process never terminatingin an indivisible—as a consequence of continuity as hecharacterizes the notion. But on occasion he takes the property ofinfinite divisibility asdefining continuity. It is thisdefinition of continuity that figures in Aristotle’sdemonstration of what has come to be known as theisomorphismthesis, which asserts that either magnitude, time and motion areall continuous, or they are all discrete.

The question of whether magnitude is perpetually divisible intosmaller units, or divisible only down to some atomic magnitude leadsto thedilemma of divisibility (Miller 1982), a difficultythat Aristotle necessarily had to face in connection with his analysisof the continuum. In the dilemma’s first, ornihilistichorn, it is argued that, were magnitude everywhere divisible, theprocess of carrying out this division completely would reduce amagnitude to extensionless points, or perhaps even to nothingness. Thesecond, oratomistic, horn starts from the assumption thatmagnitude is not everywhere divisible and leads to the equallyunpalatable conclusion (for Aristotle, at least) that indivisiblemagnitudes must exist.

As a thoroughgoing materialist, Epicurus^[12] (341–271 BCE) could not accept the notion of potentiality onwhich Aristotle’s theory of continuity rested, and so waspropelled towards atomism in both its conceptual and physical senses.Like Leucippus and Democritus, Epicurus felt it necessary to postulatethe existence of physical atoms, but to avoid Aristotle’sstrictures he proposed that these should not be themselvesconceptually indivisible, but shouldcontain conceptuallyindivisible parts. Aristotle had shown that a continuous magnitudecould not be composed ofpoints, that is, indivisible unitslacking extension, but he had not shown that an indivisible unit mustnecessarily lack extension. Epicurus met Aristotle’s argumentthat a continuum could not be composed of such indivisibles by takingindivisibles to be partless units of magnitude possessingextension.

In opposition to the atomists, the Stoic philosophers Zeno of Cition(fl. 250 BCE) and Chrysippus (280–206 BCE) upheld theAristotelian position that space, time, matter and motion are allcontinuous (Sambursky 1954 [1956], 1959; White 1992). And, likeAristotle, they explicitly rejected any possible existence of voidwithin the cosmos. The cosmos is pervaded by a continuous invisiblesubstance which they calledpneuma (Greek:“breath”). This pneuma—which was regarded as a kindof synthesis of air and fire, two of the four basic elements, theothers being earth and water—was conceived as being an elasticmedium through which impulses are transmitted by wave motion. Allphysical occurrences were viewed as being linked through tensileforces in the pneuma, and matter itself was held to derive itsqualities from the “binding” properties of the pneuma itcontains.

3. The Continuum and the Infinitesimal in the Medieval, Renaissance, and Early Modern Periods

The scholastic philosophers of Medieval Europe, in thrall to themassive authority of Aristotle, mostly subscribed in one form oranother to the thesis, argued with great effectiveness by the Masterin Book VI of thePhysics, that continua cannot be composedof indivisibles. On the other hand, the avowed infinitude of the Deityof scholastic theology, which ran counter to Aristotle’s thesisthat the infinite existed only in a potential sense, emboldenedcertain of the Schoolmen to speculate that the actual infinite mightbe found even outside the Godhead, for instance in the assemblage ofpoints on a continuous line. A few scholars of the time, for exampleHenry of Harclay (c. 1275–1317) and Nicholas of Autrecourt (c.1300–69) chose to follow Epicurus in upholding atomismreasonable and attempted to circumvent Aristotle’scounterarguments (Pyle 1997).

This incipient atomism met with a determined synechist rebuttal,initiated by John Duns Scotus (c. 1266–1308). In his analysis ofthe problem of “whether an angel can move from place to placewith a continuous motion” (Opus Oxoniense, see Grant 1974, §52)he offers a pair of purelygeometrical arguments against the composition of a continuum out ofindivisibles. One of these arguments is that if the diagonal and theside of a square were both composed of points, then not only would thetwo be commensurable in violation of Book X of Euclid, they would evenbe equal. In the other, two unequal circles are constructed about acommon center, and from the supposition that the larger circle iscomposed of points, part of an angle is shown to be equal to thewhole, in violation of Euclid’s axiom V.

William of Ockham (c. 1280–1349) brought a considerable degreeof dialectical subtlety^[13] to his analysis of continuity; it has been the subject of muchscholarly dispute.^[14] For Ockham the principal difficulty presented by the continuous isthe infinite divisibility of space, and in general, that of anycontinuum. The treatment of continuity in the first book of hisQuodlibet of 1322–7 rests on the idea that between anytwo points on a line there is a third—perhaps the first explicitformulation of the property ofdensity—and on thedistinction between acontinuum “whose parts form aunity” (QQ, 510) from acontiguum of juxtaposedthings. Ockham recognizes that it follows from the property of densitythat on arbitrarily small stretches of a line infinitely many pointsmust lie, but resists the conclusion that lines, or indeed anycontinuum, consists of points. Concerned, rather, to determine“the sense in which the line may be said to consist or to bemade up of anything” (QQ, 507), Ockham claims that “nopart of the line is indivisible, nor is any part of a continuumindivisible” (QQ, 507). While Ockham does not assert that a lineis actually “composed” of points, he had the insight,startling in its prescience, that a punctate and yet continuous linebecomes a possibility when conceived as a dense array of points,rather than as an assemblage of points in contiguous succession.

The most ambitious and systematic attempt at refuting atomism in thefourteenth century was mounted by Thomas Bradwardine (c. 1290 –1349). The purpose of hisTractatus de Continuo (c. 1330) wasto “prove that the opinion which maintains continua to becomposed of indivisibles is false” (Murdoch 1957: 54). This wasto be achieved by setting forth a number of “firstprinciples” concerning the continuum—akin to the axiomsand postulates of Euclid’sElements—and thendemonstrating that the further assumption that a continuum is composedof indivisibles leads to absurdities (Murdoch 1957).

The views on the continuum of Nicolaus Cusanus (1401–64), achampion of the actual infinite, are of considerable interest. In hisDe Mente Idiotae of 1450, he asserts that any continuum, beit geometric, perceptual, or physical, is divisible in two senses, theone ideal, the other actual. Ideal division “progresses toinfinity”; actual division terminates in atoms after finitelymany steps (see Stones 1928: 447).

Cusanus’s realist conception of the actual infinite is reflectedin his quadrature of the circle (Boyer 1939 [1959: 91]). He took thecircle to be aninfinilateral regular polygon, that is, aregular polygon with an infinite number of (infinitesimally short)sides. By dividing it up into a correspondingly infinite number oftriangles, its area, as for any regular polygon, can be computed ashalf the product of the apothem (in this case identical with theradius of the circle), and the perimeter. The idea of considering acurve as an infinilateral polygon was employed by a number of laterthinkers, for instance, Kepler, Galileo and Leibniz.

The early modern period saw the spread of knowledge in Europe ofancient geometry, particularly that of Archimedes, and a loosening ofthe Aristotelian grip on thinking. In regard to the problem of thecontinuum, the focus shifted away from metaphysics to technique, fromthe problem of “what indivisibles were, orwhether they composed magnitudes” to “the newmarvels one couldaccomplish with them” (Murdoch 1957:325) through the emerging calculus and mathematical analysis. Indeed,tracing the development of the continuum concept during this period istantamount to charting the rise of the calculus. Traditionally,geometry is the branch of mathematics concerned with the continuousand arithmetic (or algebra) with the discrete. The infinitesimalcalculus that took form in the sixteenth and seventeenth centuries,which had as its primary subject mattercontinuous variation,may be seen as a kind of synthesis of the continuous and the discrete,with infinitesimals bridging the gap between the two. The widespreaduse of indivisibles and infinitesimals in the analysis of continuousvariation by the mathematicians of the time testifies to theaffirmation of a kind of mathematical atomism which, while logicallyquestionable, made possible the spectacular mathematical advances withwhich the calculus is associated. It was thus to be the infinitesimal,rather than the infinite, that served as the mathematical steppingstone between the continuous and the discrete.

Johann Kepler (1571–1630) made abundant use of infinitesimals inhis calculations. In hisNova Stereometria of 1615, a workactually written as an aid in calculating the volumes of wine casks,he regards curves as being infinilateral polygons, and solid bodies asbeing made up of infinitesimal cones or infinitesimally thin discs(Baron 1969 [1987: 108–116]; Boyer 1939 [1959:106–110]). Such uses are in keeping with Kepler’scustomary use of infinitesimals of the same dimension as the figuresthey constitute; but he also used indivisibles on occasion. He spoke,for example, of a cone as being composed of circles, and inhisAstronomia Nova of 1609, the work in which he states hisfamous laws of planetary motion, he takes the area of an ellipse to bethe “sum of the radii” drawn from the focus.

It seems to have been Kepler who first introduced the idea, which waslater to become a reigning principle in geometry, ofcontinuouschange of a mathematical object, in this case, of a geometricfigure. In hisAstronomiae pars Optica of 1604 Kepler notesthat all the conic sections are continuously derivable from oneanother both through focal motion and by variation of the angle withthe cone of the cutting plane.

Galileo Galilei (1564–1642) advocated a form of mathematicalatomism in which the influence of both the Democritean atomists andthe Aristotelian scholastics can be discerned. This emerges when oneturns to the First Day of Galileo’sDialogues Concerning TwoNew Sciences (1638). Salviati, Galileo’s spokesman,maintains, contrary to Bradwardine and the Aristotelians, thatcontinuous magnitude is made up of indivisibles, indeed an infinitenumber of them. Salviati/Galileo recognizes that this infinity ofindivisibles will never be produced by successive subdivision, butclaims to have a method for generating it all at once, therebyremoving it from the realm of the potential into actual realization:this “method for separating and resolving the whole of infinityat a single stroke” (1638 [NE: 92–93; 1914: 48]) turns outsimply to the act of bending a straight line into a circle. HereGalileo finds an ingenious “metaphysical” application ofthe idea of regarding the circle as an infinilateral polygon. When thestraight line has been bent into a circle Galileo seems to take itthat that the line has thereby been rendered into indivisible parts,that is, points. But if one considers that these parts are the sidesof the infinilateral polygon, they are better characterized not asindivisible points, but rather as unbendable straight lines, each atonce part of and tangent to the circle.^[15] Galileo does not mention this possibility, but nevertheless it doesnot seem fanciful to detect the germ here of the idea of considering acurve as a an assemblage of infinitesimal “unbendable”straight lines.^[16]

It was Galileo’s pupil and colleague Bonaventura Cavalieri(1598–1647) who refined the use of indivisibles into a reliablemathematical tool (Boyer 1939 [1959]); indeed the “method ofindivisibles” remains associated with his name to the presentday. Cavalieri nowhere explains precisely what he understands by theword “indivisible”, but it is apparent that he conceivedof a surface as composed of a multitude of equispaced parallel linesand of a volume as composed of equispaced parallel planes, these beingtermed the indivisibles of the surface and the volume respectively.While Cavalieri recognized that these “multitudes” ofindivisibles must be unboundedly large, indeed was prepared to regardthem as being actually infinite, he avoided following Galileo intoensnarement in the coils of infinity by grasping that, for the“method of indivisibles” to work, the precise“number” of indivisibles involveddid not matter.Indeed, the essence of Cavalieri’s method was the establishingof a correspondence between the indivisibles of two“similar” configurations, and in the cases Cavaliericonsiders it is evident that the correspondence is suggested on solelygeometric grounds, rendering it quite independent of number. The verystatement of Cavalieri’s principle embodies this idea: if planefigures are included between a pair of parallel lines, and if theirintercepts on any line parallel to the including lines are in a fixedratio, then the areas of the figures are in the same ratio. (Ananalogous principle holds for solids.) Cavalieri’s method is inessence that of reduction of dimension: solids are reduced to planeswith comparable areas and planes to lines with comparable lengths.While this method suffices for the computation of areas or volumes, itcannot be applied to rectify curves, since the reduction in this casewould be to points, and no meaning can be attached to the“ratio” of two points. For rectification a curve has, itwas later realized, to be regarded as the sum, not of indivisibles,that is, points, but rather of infinitesimal straight lines, itsmicrosegments.

René Descartes (1596–1650) employed infinitesimalisttechniques, including Cavalieri’s method of indivisibles, in hismathematical work. But he avoided the use of infinitesimals in thedetermination of tangents to curves, instead developing purelyalgebraic methods for the purpose. Some of his sharpest criticism wasdirected at those mathematicians, such as Fermat, who usedinfinitesimals in the construction of tangents.

As a philosopher Descartes may be broadly characterized as asynechist. His philosophical system rests on two fundamentalprinciples: the celebrated Cartesian dualism—the divisionbetween mind and matter—and the less familiar identification ofmatter and spatial extension. In theMeditations Descartesdistinguishes mind and matter on the grounds that the corporeal, beingspatially extended, is divisible, while the mental is partless. Theidentification of matter and spatial extension has the consequencethat matter is continuous and divisible without limit. Since extensionis the sole essential property of matter and, conversely, matteralways accompanies extension, matter must be ubiquitous.Descartes’ space is accordingly, as it was for the Stoics, aplenum pervaded by a continuous medium.

The concept of infinitesimal had arisen with problems of a geometriccharacter and infinitesimals were originally conceived as belongingsolely to the realm of continuous magnitude as opposed to that ofdiscrete number. But from the algebra and analytic geometry of thesixteenth and seventeenth centuries there issued the concept ofinfinitesimal number. The idea first appears in the work ofPierre de Fermat (1601–65) on the determination of maximum andminimum (extreme) values, published in 1638 (Boyer 1939 [1959: 155]).

Fermat’s treatment of maxima and minima contains the germ of thefertile technique of “infinitesimal variation”, that is,the investigation of the behavior of a function by subjecting itsvariables to small changes. Fermat applied this method in determiningtangents to curves and centers of gravity.

4. The Continuum and the Infinitesimal in the Seventeenth and Eighteenth Centuries

Isaac Barrow^[17] (1630–77) was one of the first mathematicians to grasp thereciprocal relation between the problem of quadrature and that offinding tangents to curves—in modern parlance, betweenintegration and differentiation. In hisLectiones Geometricaeof 1670, Barrow observes, in essence, that if the quadrature of acurve \(y = f(x)\) is known, with the area up to \(x\) given by\(F(x)\), then the subtangent to the curve \(y = F(x)\) is measured bythe ratio of its ordinate to the ordinate of the original curve.

Barrow, a thoroughgoing synechist, regarded the conflict betweendivisionism and atomism as a live issue, and presented a number ofarguments against mathematical atomism, the strongest of which is thatatomism contradicts many of the basic propositions of Euclideangeometry.

Barrow conceived of continuous magnitudes as being generated bymotions, and so necessarily dependent on time, a view that seems tohave had a strong influence on the thinking of his illustrious pupilIsaac Newton^[18] (1642–1727). Newton’s meditations during the plague year1665–66 issued in the invention of what he called the“Calculus of Fluxions”, the principles and methods ofwhich were presented in three tracts published many years after theywere written:De analysi per aequationes numero terminoruminfinitas;Methodus fluxionum et serierum infinitarum;andDe quadratura curvarum. Newton’s approach to thecalculus rests, even more firmly than did Barrow’s, on theconception of continua as being generated by motion.

But Newton’s exploitation of the kinematic conception went muchdeeper than had Barrow’s. InDe Analysi, for example,Newton introduces a notation for the “momentary increment”(moment)—evidently meant to represent a moment orinstant of time—of the abscissa or the area of a curve, with theabscissa itself representing time. This“moment”—effectively the same as the infinitesimalquantities previously introduced by Fermat and Barrow—Newtondenotes by \(o\) in the case of the abscissa, and by \(ov\) in thecase of the area. From the fact that Newton uses the letter \(v\) forthe ordinate, it may be inferred that Newton is thinking of the curveas being a graph of velocity against time. By considering the movingline, or ordinate, as the moment of the area Newton established thegenerality of and reciprocal relationship between the operations ofdifferentiation and integration, a fact that Barrow had grasped buthad not put to systematic use. Before Newton, quadrature orintegration had rested ultimately “on some process through whichelemental triangles or rectangles were added together” (Baron1969 [1987: 268]), that is, on the method of indivisibles.Newton’s explicit treatment of integration as inversedifferentiation was the key to the integral calculus.

In theMethodus fluxionum Newton makes explicit hisconception of variable quantities as generated by motions, andintroduces his characteristic notation. He calls the quantitygenerated by a motion afluent, and its rate of generation afluxion. The fluxion of a fluent \(x\) is denoted by\(\dot{x}\), and its moment, or “infinitely small incrementaccruing in an infinitely short time \(o\)”, by \(\dot{x}o\). The problem of determining a tangent to a curve is transformedinto the problem of finding the relationship between the fluxions\(\dot{x}\) and \(\dot{z}\) when presented with an equationrepresenting the relationship between the fluents \(x\) and \(z\). (Aquadrature is the inverse problem, that of determining the fluentswhen the fluxions are given.) Thus, for example, in the case of thefluent \(z = x^n\), Newton first forms \(\dot{z} + \dot{z} o =(\dot{x} + \dot{x} o)^n\), expands the right-hand side using thebinomial theorem, subtracts \(z = x^n\), divides through by \(o\),neglects all terms still containing \(o\), and so obtains \(\dot{z} =nx^{n-1} \dot{x}\).

Newton later became discontented with the undeniable presence ofinfinitesimals in his calculus, and dissatisfied with the dubiousprocedure of “neglecting” them. In the preface totheDe quadratura curvarum he remarks that there is nonecessity to introduce into the method of fluxions any argument aboutinfinitely small quantities. In their place he proposes to employ whathe calls the method ofprime and ultimate ratio. This method,in many respects an anticipation of the limit concept, receives anumber of allusions in Newton’s celebratedPrincipiamathematica philosophiae naturalis of 1687.

Newton developed three approaches for his calculus, all of which heregarded as leading to equivalent results, but which varied in theirdegree of rigor. The first employed infinitesimal quantities which,while not finite, are at the same time not exactly zero. Finding thatthese eluded precise formulation, Newton focussed instead on theirratio, which is in general a finite number. If this ratio is known,the infinitesimal quantities forming it may be replaced by anysuitable finite magnitudes—such as velocities orfluxions—having the same ratio. This is the method of fluxions.Recognizing that this method itself required a foundation, Newtonsupplied it with one in the form of the doctrine of prime and ultimateratios, a kinematic form of the theory of limits.

The philosopher-mathematician G. W. F. Leibniz^[19] (1646–1716) was greatly preoccupied with the problem of thecomposition of the continuum—the “labyrinth of thecontinuum”, as he called it. Indeed we have it on his owntestimony that his philosophicalsystem—monadism—grew from his struggle with theproblem of just how, or whether, a continuum can be built fromindivisible elements. Leibniz asked himself: if we grant that eachreal entity is either a simple unity or a multiplicity, and that amultiplicity is necessarily an aggregation of unities, then under whathead should a geometric continuum such as a line be classified? Now aline is extended and Leibniz held that extension is a form ofrepetition, so, a line, being divisible into parts, cannot be a (true)unity. It is then a multiplicity, and accordingly an aggregation ofunities. But of what sort of unities? Seemingly, the only candidatesfor geometric unities are points, but points are no more thanextremities of the extended, and in any case, as Leibniz knew, solidarguments going back to Aristotle establish that no continuum can beconstituted from points. It follows that a continuum is neither aunity nor an aggregation of unities. Leibniz concluded that continuaarenot real entities at all; as “wholes precedingtheir parts” they have instead a purely ideal character. In thisway he freed the continuum from the requirement that, as somethingintelligible, it must itself be simple or a compound of simples.

Leibniz held that space and time, as continua, are ideal, and anythingreal, in particular matter, is discrete, compounded of simple unitsubstances he termedmonads.

Among the best known of Leibniz’s doctrines is thePrinciple orLaw of Continuity. In a somewhatnebulous form this principle had been employed on occasion by a numberof Leibniz’s predecessors, including Cusanus and Kepler, but itwas Leibniz who gave to the principle

a clarity of formulation which had previously been lacking and perhapsfor this reason regarded it as his own discovery. (Boyer 1939 [1959:217])

In a letter to Bayle of 1687, Leibniz gave the following formulationof the principle:

in any supposed transition, ending in any terminus, it is permissibleto institute a general reasoning in which the final terminus may beincluded. (quoted in Boyer 1939 [1959: 217] which cites Leibniz,Early Mathematical Manuscripts, p. 147)

This would seem to indicate that Leibniz considered“transitions” of any kind as continuous. Certainly he heldthis to be the case in geometry and for natural processes, where itappears as the principleNatura non facit saltus. Accordingto Leibniz, it is the Law of Continuity that allows geometry and theevolving methods of the infinitesimal calculus to be applicable inphysics. The Principle of Continuity also furnished the chief groundsfor Leibniz’s rejection of material atomism.

The Principle of Continuity also played an important underlying rolein Leibniz’s mathematical work, especially in his development ofthe infinitesimal calculus. Leibniz’s essaysNovaMethodus of 1684 andDe Geometria Recondita of 1686 maybe said to represent the official births of the differential andintegral calculi, respectively. His approach to the calculus, in whichthe use of infinitesimals, plays a central role, has combinatorialroots, traceable to his early work on derived sequences of numbers.Given a curve determined by correlated variables \(x, y\), he wrote\(\Dx\) and \(\Dy\) for infinitesimal differences, ordifferentials, between the values \(x\) and \(y\): and\(\Dy/\Dx\) for the ratio of the two, which he then took to representthe slope of the curve at the corresponding point. This suggestive, ifhighly formal procedure led Leibniz to evolve rules for calculatingwith differentials, which was achieved by appropriate modification ofthe rules of calculation for ordinary numbers.

Although the use of infinitesimals was instrumental in Leibniz’sapproach to the calculus, in 1684 he introduced the concept ofdifferential without mentioning infinitely small quantities, almostcertainly in order to avoid foundational difficulties. He stateswithout proof the following rules of differentiation:

If \(a\) is constant, then

\[\begin{align}\D a & = 0\\\D (ax) & = a \Dx\\\D (x+y-z) & = \Dx + \Dy - \D z\\\D (xy) & = x \Dy + y \Dx \\\D (x/y) & = \frac{[- x \Dy + y \Dx ]}{y^2}\\\D (x^p) & = px^{p-1}\Dx\text{, also for fractional \(p\)}\\\end{align}\]

But behind the formal beauty of these rules—an earlymanifestation of what was later to flower into differentialalgebra—the presence of infinitesimals makes itself felt, sinceLeibniz’s definition of tangent employs both infinitely smalldistances and the conception of a curve as an infinilateralpolygon.

Leibniz conceived of differentials \(\Dx , \Dy\) as variables rangingover differences. This enabled him to take the important step ofregarding the symbol \(\D\) as anoperator acting onvariables, so paving the way for theiterated application of\(\D \), leading to the higher differentials \(\D\Dx = \D^2 x\),\(\D^3 x = \D\D^2 x\), and in general \(\D^{n+1}x = \D\D^{n}x\).Leibniz supposed that the first-order differentials \(\Dx\),\(\Dy\),…. were incomparably smaller than, or infinitesimalwith respect to, the finite quantities \(x\), \(y\),…, and, ingeneral, that an analogous relation obtained between the\((n+1)^{\textrm{th}}\)-order differentials \(\D^{n+1}x\) and the\(n^{\textrm{th}}\)-order differentials \(\D^n x\). He also assumedthat the \(n^{\textrm{th}}\) power \((\Dx )^n\) of a first-orderdifferential was of the same order of magnitude as an\(n^{\textrm{th}}\)-order differential \(\D^{n} x\), in the sense thatthe quotient \(\D^{n} x/(\Dx )^n\) is a finite quantity.

For Leibniz the incomparable smallness of infinitesimals derived fromtheir failure to satisfy Archimedes’ principle; and quantitiesdiffering only by an infinitesimal were to be considered equal. Butwhile infinitesimals were conceived by Leibniz to be incomparablysmaller than ordinary numbers, the Law of Continuity ensured that theywere governed by the same laws as the latter.

Leibniz’s attitude toward infinitesimals and differentials seemsto have been that they furnished the elements from which to fashion aformal grammar, an algebra, of the continuous. Since he regardedcontinua as purely ideal entities, it was then perfectly consistentfor him to maintain, as he did, that infinitesimal quantitiesthemselves are no less ideal—simply useful fictions, introducedto shorten arguments and aid insight.

Although Leibniz himself did not credit the infinitesimal or the(mathematical) infinite with objective existence, a number of hisfollowers did not hesitate to do so. Among the most prominent of thesewas Johann Bernoulli (1667–1748). A letter of his to Leibnizwritten in 1698 contains the forthright assertion that “inasmuchas the number of terms in nature is infinite, the infinitesimal existsipso facto” (Boyer 1939 [1959: 239], quoting Leibniz,Mathematische Schriften, III (Part 2), 555). One of hisarguments for the existence of actual infinitesimals begins with thepositing of the infinite sequence 1/2, 1/3, 1/4,…. If there areten terms, one tenth exists; if a hundred, then a hundredth exists,etc.; and so if, as postulated, the number of terms is infinite, thenthe infinitesimal exists.

Leibniz’s calculus gained a wide audience through thepublication in 1696, by Guillaume de L’Hôpital(1661–1704), of the first expository book on the subject, theAnalyse des Infiniments Petits Pour L’Intelligence desLignes Courbes. This is based on two definitions:

1. Variable quantities are those that continually increase ordecrease; and constant or standing quantities are those that continuethe same while others vary.
2. The infinitely small part whereby a variable quantity iscontinually increased or decreased is called the differential of thatquantity.

And two postulates:

1. Grant that two quantities, whose difference is an infinitely smallquantity, may be taken (or used) indifferently for each other: or(what is the same thing) that a quantity, which is increased ordecreased only by an infinitely small quantity, may be considered asremaining the same.
2. Grant that a curve line may be considered as the assemblage of aninfinite number of infinitely small right lines: or (what is the samething) as a polygon with an infinite number of sides, each of aninfinitely small length, which determine the curvature of the line bythe angles they make with each other.

Following Leibniz, L’Hôpital writes \(\Dx\) for thedifferential of a variable quantity \(x\). A typical application ofthese definitions and postulates is the determination of thedifferential of a product \(xy\):

Here the last step is justified by Postulate I, since \(\Dx \Dy\) isinfinitely small in comparison to \(y \Dx + x \Dy\).

Leibniz’s calculus of differentials, resting as it did onsomewhat insecure foundations, soon attracted criticism. The attackmounted by the Dutch physician Bernard Nieuwentijdt^[20] (1654–1718) in works of 1694–6 is of particular interest,since Nieuwentijdt offered his own account of infinitesimals whichconflicts with that of Leibniz and has striking features of its own(see Mancosu 1996: 158–160). Nieuwentijdt postulates a domainof quantities, or numbers, subject to a ordering relation of greateror less. This domain includes the ordinary finite quantities, but itis also presumed to contain infinitesimal and infinitequantities—a quantity being infinitesimal, or infinite, when itis smaller, or, respectively, greater, than any arbitrarily givenfinite quantity. The whole domain is governed by a version of theArchimedean principle to the effect that zero is the only quantityincapable of being multiplied sufficiently many times to equal anygiven quantity. Infinitesimal quantities may be characterized asquotients \(b/m\) of a finite quantity \(b\) by an infinite quantity\(m\). In contrast with Leibniz’s differentials,Nieuwentijdt’s infinitesimals have the property that the productof any pair of them vanishes; in particular each infinitesimal isnilsquare, i.e., its square and all higher powers are zero(see Mancosu 1996: 159). This fact enables Nieuwentijdt to show that,for any curve given by an algebraic equation, the hypotenuse of thedifferential triangle generated by an infinitesimal abscissalincrement \(e\) coincides with the segment of the curve between \(x\)and \(x + e\). That is, a curve trulyis an infinilateralpolygon.

The major differences between Nieuwentijdt’s and Leibniz’scalculi of infinitesimals are summed up in the following table:

In responding to Nieuwentijdt’s assertion that squares andhigher powers of infinitesimals vanish, Leibniz objected that it israther strange to posit that a segment \(\Dx\) is different from zeroand at the same time that the area of a square with side \(\Dx\) isequal to zero (Mancosu 1996: 161). Yet this oddity may be regarded asa consequence—apparently unremarked by Leibniz himself—ofone of his own key principles, namely that curves may be considered asinfinilateral polygons. Consider, for instance, the curve \(y = x^2\).Given that the curve is an infinilateral polygon, the infinitesimalstraight stretch of the curve between the abscissae 0 and \(\Dx\) mustcoincide with the tangent to the curve at the origin—in thiscase, the axis of abscissae—between these two points. But thenthe point \((\Dx , \Dx ^2)\) must lie on the axis of abscissae, whichmeans that \(\Dx ^2 = 0\).

Now Leibniz could retort that that this argument depends crucially onthe assumption that the portion of the curve between abscissae 0 and\(\Dx\) is indeed straight. If this be denied, then of course it doesnot follow that \(\Dx ^2 = 0\). But if one grants, as Leibniz does,that that there is an infinitesimal straight stretch of the curve (aside, that is, of an infinilateral polygon coinciding with the curve)between abscissae 0 and \(e\), say, which does not reduce to a singlepoint then \(e\) cannot be equated to 0 and yet the above argumentshows that \(e^2 = 0\). It follows that, if curves are infinilateralpolygons, then the “lengths” of the sides of these lattermust be nilsquare infinitesimals. Accordingly, to do full justice toLeibniz’s (as well as Nieuwentijdt’s) conception,two sorts of infinitesimals are required: first,“differentials” obeying—as laid down byLeibniz—the same algebraic laws as finite quantities; and secondthe (necessarily smaller) nilsquare infinitesimals which measure thelengths of the sides of infinilateral polygons. It may be said thatLeibniz recognized the need for the first, but not the second type ofinfinitesimal and Nieuwentijdt, vice-versa. It is of interest to notethat Leibnizian infinitesimals (differentials) are realized innonstandard analysis, and nilsquare infinitesimals in smoothinfinitesimal analysis (for both types of analysis see below). In factit has been shown to be possible to combine the two approaches, socreating an analytic framework realizing both Leibniz’s andNieuwentijdt’s conceptions of infinitesimal.

The insistence that infinitesimals obey precisely the same algebraicrules as finite quantities forced Leibniz and the defenders of hisdifferential calculus into treating infinitesimals, in the presence offinite quantities,as if they were zeros, so that, forexample, \(x + \Dx\) is treated as if it were the same as \(x\). Thiswas justified on the grounds that differentials are to be taken asvariable, not fixed quantities, decreasing continually until reachingzero. Considered only in the “moment of theirevanescence”, they were accordingly neither something norabsolute zeros.

Thus differentials (or infinitesimals) \(\Dx\) were ascribed variouslythe four following properties:

1. \(\Dx \approx 0\)
2. neither \(\Dx = 0\) nor \(\Dx \ne 0\)
3. \(\Dx ^2 = 0\)
4. \(\Dx \rightarrow 0\)

where “\(\approx\)” stands for “indistinguishablefrom”, and “\(\rightarrow 0\)” stands for“becomes vanishingly small”. Of these properties only thelast, in which a differential is considered to be a variable quantitytending to 0, survived the nineteenth century refounding of thecalculus in terms of the limit concept.^[21]

The leading practitioner of the calculus, indeed the leadingmathematician of the eighteenth century, was Leonhard Euler^[22] (1707–83). Philosophically Euler was a thoroughgoing synechist.Rejecting Leibnizian monadism, he favored the Cartesian doctrine thatthe universe is filled with a continuous ethereal fluid and upheld thewave theory of light over the corpuscular theory propounded byNewton.

Euler rejected the concept of infinitesimal in its sense as a quantityless than any assignable magnitude and yet unequal to 0, arguing: thatdifferentials must be zeros, and \(\Dy/\Dx\) the quotient \(0/0\).Since for any number \(\alpha\), \(\alpha \cdot 0 = 0\), Eulermaintained that the quotient \(0/0\) could represent any number whatsoever.^[23] For Eulerqua formalist the calculus was essentially aprocedure for determining the value of the expression \(0/0\) in themanifold situations it arises as the ratio of evanescentincrements.

But in the mathematical analysis of natural phenomena, Euler, alongwith a number of his contemporaries, did employ what amount toinfinitesimals in the form of minute, but more or less concrete“elements” of continua, treating them not as atoms ormonads in the strict sense—as parts of a continuum they must ofnecessity be divisible—but as being of sufficient minuteness topreserve their rectilinearshape under infinitesimal flow,yet allowing theirvolume to undergo infinitesimal change.This idea was to become fundamental in continuum mechanics.

While Euler treated infinitesimals as formal zeros, that is, as fixedquantities, his contemporary Jean le Rond d’Alembert(1717–83) took a different view of the matter. FollowingNewton’s lead, he conceived of infinitesimals or differentialsin terms of the limit concept, which he formulated by the assertionthat one varying quantity is the limit of another if the second canapproach the other more closely than by any given quantity.D’Alembert firmly rejected the idea of infinitesimals as fixedquantities, and saw the idea of limit as supplying the methodologicalroot of the differential calculus. For d’Alembert the languageof infinitesimals or differentials was just a convenient shorthand foravoiding the cumbrousness of expression required by the use of thelimit concept.

Infinitesimals, differentials, evanescent quantities and the likecoursed through the veins of the calculus throughout the eighteenthcentury. Although nebulous—even logically suspect—theseconcepts provided,faute de mieux, the tools for deriving thegreat wealth of results the calculus had made possible. And while,with the notable exception of Euler, many eighteenth centurymathematicians were ill-at-ease with the infinitesimal, they would notrisk killing the goose laying such a wealth of golden mathematicaleggs. Accordingly they refrained, in the main, from destructivecriticism of the ideas underlying the calculus. Philosophers, however,were not fettered by such constraints.

The philosopher George Berkeley (1685–1753), noted both for hissubjective idealist doctrine ofesse est percipi and hisdenial of general ideas, was a persistent critic of thepresuppositions underlying the mathematical practice of his day(Jesseph 1993). His most celebrated broadsides were directed at thecalculus, but in fact his conflict with the mathematicians wentdeeper. For his denial of the existence of abstract ideas of any kindwent in direct opposition with the abstractionist account ofmathematical concepts held by the majority of mathematicians andphilosophers of the day. The central tenet of this doctrine, whichgoes back to Aristotle, is that the mind creates mathematical conceptsbyabstraction, that is, by the mental suppression ofextraneous features of perceived objects so as to focus on propertiessingled out for attention. Berkeley rejected this, asserting thatmathematics as a science is ultimately concerned with objects ofsense, its admitted generality stemming from the capacity of perceptsto serve as signs for all percepts of a similar form.

At first Berkeley poured scorn on those who adhere to the concept ofinfinitesimal. maintaining that the use of infinitesimals in derivingmathematical results is illusory, and is in fact eliminable. But laterhe came to adopt a more tolerant attitude towards infinitesimals,regarding them as useful fictions in somewhat the same way as didLeibniz.

InThe Analyst of 1734 Berkeley launched his most sustainedand sophisticated critique of infinitesimals and the whole metaphysicsof the calculus. AddressedTo an Infidel Mathematician,^[24] the tract was written with the avowed purpose of defending theologyagainst the scepticism shared by many of the mathematicians andscientists of the day. Berkeley’s defense of religion amounts tothe claim that the reasoning of mathematicians in respect of thecalculus is no less flawed than that of theologians in respect of themysteries of the divine.

Berkeley’s arguments are directed chiefly against the Newtonianfluxional calculus. Typical of his objections is that in attempting toavoid infinitesimals by the employment of such devices as evanescentquantities and prime and ultimate ratios Newton has in fact violatedthe law of noncontradiction by first subjecting a quantity to anincrement and then setting the increment to 0, that is, denying thatan increment had ever been present. As for fluxions and evanescentincrements themselves, Berkeley has this to say:

And what are these fluxions? The velocities of evanescent increments?And what are these same evanescent increments? They are neither finitequantities nor quantities infinitely small, nor yet nothing. May wenot call them the ghosts of departed quantities? (1734: 59)

Nor did the Leibnizian method of differentials escape Berkeley’sstrictures.

The opposition between continuity and discreteness plays a significantrole in the philosophical thought of Immanuel Kant (1724–1804).His mature philosophy,transcendental idealism, rests on thedivision of reality into two realms. The first, thephenomenal realm, consists of appearances or objects ofpossible experience, configured by the forms of sensibility and theepistemic categories. The second, thenoumenal realm,consists of “entities of the understanding to which no objectsof experience can ever correspond” (Körner 1955: 94), thatis, things-in-themselves.

Regarded as magnitudes, appearances are spatiotemporally extended andcontinuous, that is infinitely, or at least limitlessly, divisible.Space and time constitute the underlying order of phenomena, so areultimately phenomenal themselves, and hence also continuous.

As objects of knowledge, appearances are continuousextensivemagnitudes, but as objects of sensation or perception they are,according to Kant,intensive magnitudes. By an intensivemagnitude Kant means a magnitude possessing adegree and socapable of being apprehended by the senses: for example brightness ortemperature. Intensive magnitudes are entirely free of the intuitionsof space or time, and “can only be presented asunities”. But, like extensive magnitudes, they arecontinuous. Moreover, appearances are always presented to the sensesas intensive magnitudes.

In theCritique of Pure Reason (1781) Kant brings a newsubtlety (and, it must be said, tortuousity) to the analysis of theopposition between continuity and discreteness. This may be seen inthe second of the celebrated Antinomies in that work, which concernsthe question of the mereological composition of matter, or extendedsubstance. Is it (a) discrete, that is, consists of simple orindivisible parts, or (b) continuous, that is, contains parts withinpartsad infinitum? Although (a), which Kant calls theThesis and (b) theAntithesis would seem tocontradict one another, Kant offers proofs of both assertions. Theresulting contradiction may be resolved, he asserts, by observing thatwhile the antinomy “relates to the division ofappearances”, the arguments for (a) and (b) implicitly treatmatter or substance as things-in-themselves. Kant concludes that bothThesis and Antithesis “presuppose an inadmissiblecondition” and accordingly “both fall to the ground,inasmuch as the condition, under which alone either of them can bemaintained, itself falls”.

Kant identifies the inadmissible condition as the implicit taking ofmatter as a thing-in-itself, which in turn leads to the mistake oftaking the division of matter into parts to subsist independently ofthe act of dividing. In that case, the Thesis implies that thesequence of divisions is finite; the Antithesis, that it is infinite.These cannot both be true of thecompleted (or at leastcompletable) sequence of divisions which would result from takingmatter or substance as a thing-in-itself.^[25] Now since the truth of both assertions has been shown to follow fromthat assumption,it must be false, that is, matter andextended substance are appearances only. And for appearances, Kantmaintains, divisions into parts are not completable in experience,with the result that such divisions can be considered, in a startlingphrase, “neither finite nor infinite”. It follows that,for appearances, both Thesis and Antithesis arefalse.

Later in theCritique Kant enlarges on the issue ofdivisibility, asserting that, while each part generated by a sequenceof divisions of an intuited whole is given with the whole, thesequence’s incompletability preventsit from forming awhole;a fortiori no such sequence can be claimed to beactually infinite.

5. The Continuum and the Infinitesimal in the Nineteenth Century

The rapid development of mathematical analysis in the eighteenthcentury had not concealed the fact that its underlying concepts notonly lacked rigorous definition, but were even (e.g., in the case ofdifferentials and infinitesimals) of doubtful logical character. Thelack of precision in the notion of continuous function—stillvaguely understood as one which could be represented by a formula andwhose associated curve could be smoothly drawn—had led to doubtsconcerning the validity of a number of procedures in which thatconcept figured. For example it was often assumed that everycontinuous function could be expressed as an infinite series by meansof Taylor’s theorem. Early in the nineteenth century this andother assumptions began to be questioned, thereby initiating aninquiry into what was meant by a function in general and by acontinuous function in particular.

A pioneer in the matter of clarifying the concept of continuousfunction was the Bohemian priest, philosopher and mathematicianBernard Bolzano (1781–1848). In hisRein analytischerBeweis of 1817 he defines a (real-valued) function \(f\) to becontinuous at a point \(x\) if the difference \(f(x + \omega) - f(x)\)can be made smaller than any preselected quantity once we arepermitted to take \(w\) as small as we please. This is essentially thesame as the definition of continuity in terms of the limit conceptgiven a little later by Cauchy. Bolzano also formulated a definitionof the derivative of a function free of the notion of infinitesimal(Bolzano 1851 [1950]). Bolzano repudiated Euler’s treatment ofdifferentials as formal zeros in expressions such as \(\Dy/\Dx\),suggesting instead that in determining the derivative of a function,increments \(\Delta x,\) \(\Delta y,\) …, befinallyset to zero. For Bolzano differentials have the status ofideal elements, purely formal entities such as points andlines at infinity in projective geometry, or (as Bolzano himselfmentions) imaginary numbers, whose use will never lead to falseassertions concerningreal quantities.

Although Bolzano anticipated the form that the rigorous formulation ofthe concepts of the calculus would assume, his work was largelyignored in his lifetime. The cornerstone for the rigorous developmentof the calculus was supplied by the ideas—essentially similar toBolzano’s—of the great French mathematician Augustin-LouisCauchy (1789–1857). In Cauchy’s work, as inBolzano’s, a central role is played by a purely arithmeticalconcept of limit freed of all geometric and temporal intuition. Cauchyalso formulates the condition for a sequence of real numbers toconverge to a limit, and states his familiar criterion for convergence,^[26] namely, that a sequence \(\langle s_n\rangle\) is convergent if andonly if \(s_{n+r} - s_n\) can be made less in absolute value than anypreassigned quantity for all \(r\) and sufficiently large \(n\).Cauchy proves that this is necessary for convergence, but as tosufficiency of the condition merely remarks “when the variousconditions are fulfilled, the convergence of the series isassured” (Kline 1972: 951). In making this latter assertion heis implicitly appealing to geometric intuition, since he makes noattempt to define real numbers, observing only that irrational numbersare to be regarded as the limits of sequences of rational numbers.

Cauchy chose to characterize the continuity of functions in terms of arigorized notion of infinitesimal, which he defines in theCoursd’analyse as “a variable quantity [whose value]decreases indefinitely in such a way as to converge to the limit0” (Kline 1972: 951). Here is his definition ofcontinuity. Cauchy’s definition of continuity of \(f(x)\) in theneighborhood of a value \(a\) amounts to the condition, in modernnotation, that \(\lim_{x\rightarrow a}f(x) = f(a)\). Cauchy definesthe derivative \(f '(x)\) of a function \(f(x)\) in a manneressentially identical to that of Bolzano.

The work of Cauchy (as well as that of Bolzano) represents a crucialstage in the renunciation by mathematicians—adumbrated in thework of d’Alembert—of (fixed) infinitesimals and theintuitive ideas of continuity and motion. Certain mathematicians ofthe day, such as Poisson and Cournot, who regarded the limit conceptas no more than a circuitous substitute for the use of infinitesimallysmall magnitudes—which in any case (they claimed) had a realexistence—felt that Cauchy’s reforms had been carried toofar. But traces of the traditional ideas did in fact remain inCauchy’s formulations, as evidenced by his use of suchexpressions as “variable quantities”, “infinitesimalquantities”, “approach indefinitely”, “aslittle as one wishes” (Kline 1972: 951) and the like.^[27]

Meanwhile the German mathematician Karl Weierstrass (1815–97)was completing the banishment of spatiotemporal intuition, and theinfinitesimal, from the foundations of analysis. To instill completelogical rigor Weierstrass proposed to establish mathematical analysison the basis of number alone, to “arithmetize”^[28] it—in effect, to replace the continuous by the discrete.“Arithmetization” may be seen as a form of mathematicalatomism. In pursuit of this goal Weierstrass had first to formulate arigorous “arithmetical” definition of real number. He didthis by defining a (positive) real number to be a countable set ofpositive rational numbers for which the sum of any finite subsetalways remains below some preassigned bound, and then specifying theconditions under which two such “real numbers” are to beconsidered equal, or strictly less than one another.

Weierstrass was concerned to purge the foundations of analysis of alltraces of the intuition of continuous motion—in a word, toreplace the variable by the static. For Weierstrass a variable \(x\)was simply a symbol designating an arbitrary member of a given set ofnumbers, and a continuous variable one whose corresponding set \(S\)has the property that any interval around any member \(x\) of \(S\)contains members of \(S\) other than \(x\). Weierstrass alsoformulated the familiar \((\varepsilon , \delta)\) definition ofcontinuous function:^[29] a function \(f(x)\) is continuous at \(a\) if for any \(\varepsilon\gt 0\) there is a \(\delta \gt 0\) such that \(\lvert f(x) -f(a)\rvert \lt \varepsilon\) for all \(x\) such that \(\lvert x -a\rvert \lt \delta\).^[30]

Following Weierstrass’s efforts, another attack on the problemof formulating rigorous definitions of continuity and the real numberswas mounted by Richard Dedekind (1831–1916). Dedekind focussedattention on the question: exactly what is it that distinguishes acontinuous domain from a discontinuous one? He seems to have been thefirst to recognize that the property of density, possessed by theordered set of rational numbers, is insufficient to guaranteecontinuity. InContinuity and Irrational Numbers (1872) heremarks that when the rational numbers are associated to points on astraight line, “there are infinitely many points [on the line]to which no rational number corresponds” (1872: section 3 [1999:770]) so that the rational numbers manifest “a gappiness,incompleteness, discontinuity”, in contrast with the straightline’s “absence of gaps, completeness, continuity”(1872: section 3 [1999: 771]). Dedekind regards this principle asbeing essentially indemonstrable; he ascribes to it, rather, thestatus of an axiom “by which we attribute to the line itscontinuity, by which we think continuity into the line” (1872[1999: 771–772]. It is not, Dedekind stresses, necessary forspace to be continuous in this sense, for “many of itsproperties would remain the same even if it were discontinuous”(1872 [1999: 772]).

The filling-up of gaps in the rational numbers through the“creation of new point-individuals” (1872 [1999: 772]) isthe key idea underlying Dedekind’s construction of the domain ofreal numbers. He first defines acut to be a partition\((A_1,\) \(A_2)\) of the rational numbers such that every member of\(A_1\) is less than every member of \(A_2\). After noting that eachrational number corresponds, in an evident way, to a cut, he observesthat infinitely many cuts fail to be engendered by rational numbers.The discontinuity or incompleteness of the domain of rational numbersconsists precisely in this latter fact.

It is to be noted that Dedekind does not identify irrational numberswith cuts; rather, each irrational number is newly“created” by a mental act, and remains quite distinct fromits associated cut. Dedekind goes on to show how the domain of cuts,and thereby the associated domain of real numbers, can be ordered insuch a way as to possess the property ofcontinuity, viz.

if the system \(\Re\) of all real numbers divides into two classes\(\fA_1\), \(\fA_2\) such that every number \(a_1\) of the class\(\fA_1\) is less than every number \(a_2\) of the class \(\fA_2\),then there exists one and only one number by which this separation isproduced. (1872: section 5 [1999: 776])

The most visionary “arithmetizer” of all was Georg Cantor^[31] (1845–1918). Cantor’s analysis of the continuum in termsof infinite point sets led to his theory of transfinite numbers and tothe eventual freeing of the concept of set from its geometric originsas a collection of points, so paving the way for the emergence of theconcept of general abstract set central to today’s mathematics.Like Weierstrass and Dedekind, Cantor aimed to formulate an adequatedefinition of the real numbers which avoided the presupposition oftheir prior existence, and he follows them in basing his definition onthe rational numbers. Following Cauchy, he calls a sequence \(a_1,\)\(a_2 ,\)…, \(a_n,\)… of rational numbers afundamental sequence if there exists an integer \(N\) suchthat, for any positive rational \(\varepsilon\), there exists aninteger \(N\) such that \(\lvert a_{n+m} - a_n \rvert \lt\varepsilon\) for all \(m\) and all \(n \gt N\). Any sequence\(\langle a_n\rangle\) satisfying this condition is said to have adefinite limit b. Dedekind had taken irrational numbers to bemental objects associated with cuts, so, analogously, Cantorregards these definite limits, as nothing more thanformalsymbols associated with fundamental sequences (Dauben 1979:38). The domain \(B\) of such symbols may be considered an enlargementof the domain \(A\) of rational numbers. After imposing anarithmetical structure on the domain \(B\), Cantor is emboldened torefer to its elements as (real)numbers. Nevertheless, hestill insists that these “numbers” have no existenceexcept as representatives of fundamental sequences. Cantor then showsthat each point on the line corresponds to a definite element of\(B\). Conversely, each element of \(B\) should determine a definitepoint on the line. Realizing that the intuitive nature of the linearcontinuum precludes a rigorous proof of this property, Cantor simplyassumes it as an axiom, just as Dedekind had done in regard to hisprinciple of continuity.

For Cantor, who began as a number-theorist, and throughout his careercleaved to the discrete, it was numbers, rather than geometric points,that possessed objective significance. Indeed the isomorphism betweenthe discrete numerical domain \(B\) and the linear continuum wasregarded by Cantor essentially as a device for facilitating themanipulation of numbers.

Cantor’s arithmetization of the continuum had the followingimportant consequence. It had long been recognized that the sets ofpoints of any pair of line segments, even if one of them is infinitein length, can be placed in one-one correspondence. This fact wastaken to show that such sets of points have no well-defined“size”. But Cantor’s identification of the set ofpoints on a linear continuum with a domain of numbers enabled thesizes of point sets to be compared in a definite way, usingthe well-grounded idea ofone-one correspondence between setsof numbers.

Cantor’s investigations into the properties of subsets of thelinear continuum are presented in six masterly papers published during1879–84,Über unendliche linearePunktmannichfaltigkeiten (“On infinite, linear pointmanifolds”). Remarkable in their richness of ideas, these papersprovide the first accounts of Cantor’s revolutionary theory ofinfinite sets and its application to the classification of subsets ofthe linear continuum. In the fifth of these papers, theGrundlagen of 1883, are to be found some of Cantor’smost searching observations on the nature of the continuum.

Cantor begins his examination of the continuum with a tart summary ofthe controversies that have traditionally surrounded the notion,remarking that the continuum has until recently been regarded as anessentially unanalyzable concept. It is Cantor’s concern to

develop the concept of the continuum as soberly and briefly aspossible, and only with regard to themathematical theory ofsets. (1883 [1999: §10, para. 2, p. 903])

This opens the way, he believes, to the formulation of an exactconcept of the continuum. Cantor points out that the idea of thecontinuum has heretofore merely been presupposed by mathematiciansconcerned with the analysis of continuous functions and the like, andhas “not been subjected to any more thorough inspection”(1883 [1999: §10, para. 3, p. 904]).

Repudiating any use of spatial or temporal intuition in an exactdetermination of the continuum, Cantor undertakes its precisearithmetical definition. Making reference to the definition of realnumber he has already provided (i.e., in terms of fundamentalsequences), he introduces the \(n\)-dimensional arithmetical space\(G_n\) as the set of all \(n\)-tuples of real numbers \(\langlex_1,x_2 ,\ldots ,x_n\rangle\), calling each such anarithmeticalpoint of \(G_{n.}\) The distance between two such points is givenby

Cantor defines anarithmetical point-set in \(G_n\) to be any“aggregate of points of the points of the space \(G_n\) that isgiven in a lawlike way” (1883 [1999: §10, para. 6, p.904]).

After remarking that he has previously shown that all spaces \(G_n\)have the same power as the set of real numbers in the interval (0,1),and reiterating his conviction that any infinite point sets has eitherthe power of the set of natural numbers or that of (0,1),^[32] Cantor turns to the definition of the general concept of a continuumwithin \(G_{n}.\) For this he employs the concept ofderivative orderived set of a point set introducedin a paper of 1872 on trigonometric series. Cantor had defined the derivedset of a point set \(P\) to be the set oflimit points of\(P,\) where a limit point of \(P\) is a point of \(P\) withinfinitely many points of \(P\) arbitrarily close to it. A point setis calledperfect if it coincides with its derived set.^[33] Cantor observes that this condition does not suffice to characterizea continuum, since perfect sets can be constructed in the linearcontinuum which are dense in no interval, however small: as an exampleof such a set he offers the set^[34] consisting of all real numbers in (0,1) whose ternary expansion doesnot contain a “1”.

Accordingly an additional condition is needed to define a continuum.Cantor supplies this by introducing the concept of aconnected set. A point set \(T\) is connected inCantor’s sense if for any pair of its points \(t, t'\) and anyarbitrarily small number \(\varepsilon\) there is a finite sequence ofpoints \(t_1,\) \(t_2,\)…, \(t_n\) of \(T\) for which thedistances \([tt_1],\) \([t_1 t_2],\) \([t_2 t_3],\) …, \([t_nt'],\) are all less than \(\varepsilon\). Cantor now defines acontinuum to be a perfect connected point set.

Cantor has advanced beyond his predecessors in formulating what is inessence atopological definition of continuum, one that,while still dependent on metric notions, does not involve an order relation.^[35]It is interesting to compare Cantor’s definition with thedefinition of continuum in modern general topology. In a well-knowntextbook (Hocking & Young 1961: 43) on the subject we findacontinuum defined as a compact connected subset of atopological space. Now within anybounded region of Euclideanspace it can be shown that Cantor’s continua coincide withcontinua in the sense of the modern definition. While Cantor lackedthe definition of compactness, his requirement that continua be“complete” (which led to his rejecting as continua suchnoncompact sets as open intervals or discs) is not far away from theidea.

Throughout Cantor’s mathematical career he maintained anunwavering, even dogmatic opposition to infinitesimals, attacking theefforts of mathematicians such as du Bois-Reymond andVeronese^[36]to formulate rigorous theories of actual infinitesimals. As far asCantor was concerned, the infinitesimal was beyond the realm of thepossible; infinitesimals were no more than “castles in the air,or rather just nonsense” (1893 [1965: 506], translated by Fisher1981: 118), to be classed “with circular squares and squarecircles” (1893 [1965: 507], translated by Fisher 1981: 118). Hisabhorrence of infinitesimals went so deep as to move him to outrightvilification, branding them as “Cholera-bacilli ofmathematics” (1893 [1965: 505], translated by Fisher 1981:116). Cantor’s rejection of infinitesimals stemmed from hisconviction that his own theory of transfinite ordinal and cardinalnumbers exhausted the realm of the numerable, so that no furthergeneralization of the concept of number, in particular any whichembraced infinitesimals, was admissible.

6. Critical Reactions to Arithmetization

Despite the great success of Weierstrass, Dedekind and Cantor inconstructing the continuum from arithmetical materials, a number ofthinkers of the late nineteenth and early twentieth centuries remainedopposed, in varying degrees, to the idea of explicating the continuumconcept entirely in discrete terms. These include the philosophersBrentano and Peirce and the mathematicians Poincaré, Brouwerand Weyl.

In his later years the Austrian philosopher Franz Brentano(1838–1917) became preoccupied with the nature of the continuous(Brentano [PISTC]). In its fundamentals Brentano’s accountof the continuous is akin to Aristotle’s. Brentano regardscontinuity as something given in perception, primordial in nature,rather than a mathematical construction. He held that the idea of thecontinuous is abstracted from sensible intuition. Brentano suggeststhat the continuous is brought to appearance by sensible intuition inthree phases. First, sensation presents us with objects having partsthat coincide. From such objects the concept ofboundary isabstracted in turn, and then one grasps that these objects actuallycontain coincident boundaries. Finally one sees that this isall that is required in order to have grasped the concept of acontinuum.

For Brentano the essential feature of a continuum is its inherentcapacity to engender boundaries, and the fact that such boundaries canbe grasped as coincident. Boundaries themselves possess a qualitywhich Brentano callsplerosis (“fullness”).Plerosis is the measure of the number of directions in which the givenboundary actually bounds. Thus, for example, within a temporalcontinuum the endpoint of a past episode or the starting point of afuture one bounds in a single direction, while the point marking theend of one episode and the beginning of another may be said to bounddoubly. In the case of a spatial continuum there are numerousadditional possibilities: here a boundary may bound in all thedirections of which it is capable of bounding, or it may bound in onlysome of these directions. In the former case, the boundary is said toexist infull plerosis; in the latter, inpartialplerosis. Brentano believed that the concept of plerosis enabledsense to be made of the idea that a boundary possesses“parts”, even when the boundary lacks dimensionsaltogether, as in the case of a point. Thus, while the present or“now” is, according to Brentano, temporally unextended andexists only as a boundary between past and future, it still possessestwo “parts” or aspects: it is both the end of the past andthe beginning of the future. It is worth mentioning that for Brentanoit was not just the “now” that existed only as a boundary;since, like Aristotle he held that “existence” in thestrict sense means “existencenow”, itnecessarily followed that existing things exist only as boundaries ofwhat has existed or of what will exist, or both.

Brentano took a somewhat dim view of the efforts of mathematicians toconstruct the continuum from numbers. His attitude varied fromrejecting such attempts as inadequate to according them the status of “fictions”.^[37] This is not surprising given his Aristotelian inclination to takemathematical and physical theories to be genuine descriptions ofempirical phenomena rather than idealizations: in his view, if suchtheories were to be taken as literal descriptions of experience, theywould amount to nothing better than“misrepresentations”.

Brentano’s analysis of the continuum centered on itsphenomenological and qualitative aspects, which are by their verynature incapable of reduction to the discrete. Brentano’srejection of the mathematicians’ attempts to construct it indiscrete terms is thus hardly surprising.

The American philosopher-mathematician Charles Sanders Peirce’s(1839–1914) view of the continuum^[38] was, in a sense, intermediate between that of Brentano and thearithmetizers. Like Brentano, he held that the cohesiveness of acontinuum rules out the possibility of it being a mere collection ofdiscrete individuals, or points, in the usual sense. And even beforeBrouwer, Peirce seems to have been aware that a faithful account ofthe continuum will involve questioning the law of excluded middle.Peirce also held that any continuum harbors anunboundedlylarge collection of points—in his colorful terminology, asupermultitudinous collection—what we would today callaproper class. Peirce maintained that ifenough points were to be crowded together by carrying insertion of newpoints between old to its ultimate limit they would—throughalogical “transformation of quantity intoquality”—lose their individual identity and become fusedinto a true continuum.

Peirce’s conception of the number continuum is also notable forthe presence in it of an abundance ofinfinitesimals, Peircechampioned the retention of the infinitesimal concept in thefoundations of the calculus, both because of what he saw as theefficiency of infinitesimal methods, and because he regardedinfinitesimals as constituting the “glue” causing pointson a continuous line to lose their individual identity.

The idea of continuity played a central role in the thought of thegreat French mathematician Henri Poincaré^[39] (1854–1912). While accepting the arithmetic definition of thecontinuum, he questions the fact that (as with Dedekind andCantor’s formulations) the (irrational) numbers so produced aremere symbols, detached from their origins in intuition. Unlike Cantor,Poincaré accepted the infinitesimal, even if he did not regardall of the concept’s manifestations as useful.

The Dutch mathematician L. E. J. Brouwer (1881–1966) is bestknown as the founder of the philosophy of (neo)intuitionism(Brouwer [1975]; van Dalen 1998). Brouwer’s highly idealistviews on mathematics bore some resemblance to Kant’s. ForBrouwer, mathematical concepts are admissible only if they areadequately grounded in intuition, mathematical theories aresignificant only if they concern entities which are constructed out ofsomething given immediately in intuition, and mathematicaldemonstration is a form of construction in intuition. While admittingthat the emergence of noneuclidean geometry had discreditedKant’s view of space, Brouwer held, in opposition to thelogicists (whom he called “formalists”) that arithmetic,and so all mathematics, must derive from temporal intuition.

Initially Brouwer held without qualification that the continuum is notconstructible from discrete points, but was later to modify thisdoctrine. In his mature thought, he radically transformed the conceptof point, endowing points with sufficient fluidity to enable them toserve as generators of a “true” continuum. This fluiditywas achieved by admitting as “points”, not only fullydefined discrete numbers such as \(\sqrt{2},\) \(\pi,\) \(e,\) and thelike—which have, so to speak, already achieved“being”—but also “numbers” which are ina perpetual state of becoming in that there the entries in theirdecimal (or dyadic) expansions are the result of free acts of choiceby a subject operating throughout an indefinitely extended time. Theresultingchoice sequences cannot be conceived as finished,completed objects: at any moment only an initial segment is known. Inthis way Brouwer obtained the mathematical continuum in a waycompatible with his belief in the primordial intuition oftime—that is, as an unfinished, indeed unfinishable entity in aperpetual state of growth, a “medium of free development”.In this conception, the mathematical continuum is indeed“constructed”, not, however, by initially shattering, asdid Cantor and Dedekind, an intuitive continuum into isolated points,but rather by assembling it from a complex of continually changingoverlapping parts.

The mathematical continuum as conceived by Brouwer displays a numberof features that seem bizarre to the classical eye. For example, inthe Brouwerian continuum the usual law of comparability, namely thatfor any real numbers \(a, b\) either \(a \lt b\) or \(a = b\) or \(a\gt b\), fails. Even more fundamental is the failure of the law ofexcluded middle in the form that for any real numbers \(a, b\), either\(a = b\) or \(a \ne b\). The failure of these seeminglyunquestionable principles in turn vitiates the proofs of a number ofbasic results of classical analysis, for example theBolzano-Weierstrass theorem, as well as the theorems of monotoneconvergence, intermediate value, least upper bound, and maximum valuefor continuous functions.^[40]

While the Brouwerian continuum may possess a number of negativefeatures from the standpoint of the classical mathematician, it hasthe merit of corresponding more closely to the continuum of intuitionthan does its classical counterpart. Far from being bizarre, thefailure of the law of excluded middle for points in the intuitionisticcontinuum may be seen as fitting in well with the character of theintuitive continuum.

In 1924, Brouwer showed that every function defined on a closedinterval of his continuum is uniformly continuous. As a consequencethe intuitionistic continuum isindecomposable, that is,cannot be split into two disjoint parts in any way whatsoever. Incontrast with a discrete entity, the indecomposable Brouweriancontinuum cannot be composed of its parts. Brouwer’s vision ofthe continuum has in recent years become the subject of intensivemathematical investigation.

Hermann Weyl (1885–1955), one of most versatile mathematiciansof the twentieth century, was preoccupied with the nature of thecontinuum (Bell 2000). In hisDas Kontinuum of 1918 heattempts to provide the continuum with an exact mathematicalformulation free of the set-theoretic assumptions he had come toregard as objectionable. As he saw it, there is an unbridgeable gapbetween intuitively given continua (e.g., those of space, time andmotion) on the one hand, and the discrete exact concepts ofmathematics (e.g., that of real number) on the other. For Weyl thepresence of this split meant that the construction of the mathematicalcontinuum could not simply be “read off” from intuition.Rather, he believed that the mathematical continuum must be treatedand, in the end, justified in the same way as a physical theory.However much he may have wished it, inDas Kontinuum Weyl didnot aim to provide a mathematical formulation of the continuum as itis presented to intuition, which, as the quotations above show, heregarded as an impossibility (at that time at least). Rather, his goalwas first to achieveconsistency by putting thearithmetical notion of real number on a firm logical basis,and then to show that the resulting theory isreasonable byemploying it as the foundation for a plausible account of continuousprocess in the objective physical world.

Later Weyl came to repudiate atomistic theories of the continuum,including that of his ownDas Kontinuum. He accordinglywelcomed Brouwer’s construction of the continuum by means ofsequences generated by free acts of choice, thus identifying it as a“medium of free Becoming” which “does not dissolveinto a set of real numbers as finished entities” (Weyl 1921,50). Weyl felt that Brouwer, through his doctrine of intuitionism, hadcome closer than anyone else to bridging that “unbridgeablechasm” between the intuitive and mathematical continua. Inparticular, he found compelling the fact that the Brouwerian continuumis not the union of two disjoint nonempty parts—that it isindecomposable. “A genuine continuum”, Weyl says,“cannot be divided into separate fragments”. In laterpublications he expresses this more colorfully by quoting Anaxagorasto the effect that a continuum “defies the chopping off of itsparts with a hatchet”.

7. Nonstandard Analysis

Once the continuum had been provided with a set-theoretic foundation,the use of the infinitesimal in mathematical analysis was largelyabandoned. And so the situation remained for a number of years. Thefirst signs of a revival of the infinitesimal approach to analysissurfaced in 1958 with a paper by C. Schmieden and D. Laugwitz. But themajor breakthrough came in 1960 when it occurred to the mathematicallogician Abraham Robinson (1918–1974) that

the concepts and methods of contemporary Mathematical Logic arecapable of providing a suitable framework for the development of theDifferential and Integral Calculus by means of infinitely small andinfinitely large numbers. (preface of Robinson 1966: vii [1996: xiii])

This insight led to the creation ofnonstandard analysis,^[41] which Robinson regarded as realizing Leibniz’s conception ofinfinitesimals and infinities as ideal numbers possessing the sameproperties as ordinary real numbers.

After Robinson’s initial insight, a number of ways of presentingnonstandard analysis were developed. Here is a sketch of one ofthem.

Starting with the classical real line \(\Re\), a set-theoreticuniverse—thestandard universe—is firstconstructed over it: here by such a universe is meant a set \(U\)containing \(\Re\) which is closed under the usual set-theoreticoperations of union, power set, Cartesian products and subsets. Nowwrite \(\fU\) for the structure \((U, \in)\), where \(\in\) is theusual membership relation on \(U\): associated with this is theextension \(\cL(U)\) of the first-order language of set theory toinclude a name \(\boldsymbol{u}\) for each element \(u\) of \(U\).Now, using the well-known compactness theorem for first-order logic,\(\fU\) is extended to a new structure \(^*\fU = (^*U\), \(^*{\in})\),called anonstandard universe, satisfying the following keyprinciple:

The saturation property expresses the intuitive idea that thenonstandard universe is very rich in comparison to the standard one.Indeed, while there may exist, for each finite subcollection \(\cF\)of a given collection of properties \(\cP\), an element of \(U\)satisfying the members of \(\cF\) in \(\fU\), there may notnecessarily be an element of \(U\) satisfyingall the membersof \(\cP\). The saturation of \(^*\fU\) guarantees the existence of anelement of \(^*U\) which satisfies, in \(^*\fU\), all the members of\(\cP\). For example, suppose the set \(\bbN\) of natural numbers is amember of \(U\); for each \(n \in \bbN\) let \(P_n (x)\) be theproperty \(x \in \bbN \amp n \lt x\). Then clearly, while each finitesubcollection of the collection \(\cP = \{P_n : n \in \bbN\}\) issatisfiable in \(\fU\), the whole collection is not. An element of\(^*U\) satisfying \(\cP\) in \(^*\fU\) will then be an “naturalnumber” greater than every member of \(\bbN\), that is, aninfinite number.

From the saturation property it follows that \(^*\fU\) satisfies theimportant

The transfer principle may be seen as a version of Leibniz’scontinuity principle: it asserts that all first-order properties arepreserved in the passage to or “transfer” from thestandard to the nonstandard universe.

The members of \(U\) are calledstandard sets, orstandard objects; those in \(^*U - U\)nonstandardsets ornonstandard objects: \(^*U\) thus consists ofboth standard and nonstandard objects. The members of \(^*U\) will alsobe referred to as *-sets or *-objects Since \(U\subseteq ^*U\), under this convention every set (object) is also a*-set (object) The \(^*\)-members of a *-set \(A\) are the*-objects \(x\) for which \(x \mathbin{^*{\in}} A\).

If \(A\) is a standard set, we may consider the collection\(\hat{A}\)—theinflate of \(A\)—consisting ofall the *-members of \(A\): this is not necessarily a set nor even a*-set. The inflate \(\hat{A}\) of a standard set \(A\) may be regardedas the same set \(A\) viewed from a nonstandard vantage point. Whileclearly \(A \subseteq \hat{A}\), \(\hat{A}\) may contain“nonstandard” elements not in \(A\). It can in fact beshown thatinfinite standard sets always get“inflated” in this way. Using the transfer principle, anyfunction \(f\) between standard sets automatically extends to afunction—also written \(f\)—between their inflates.

If \(\fA = (A, R)\) is a mathematical structure, we mayconsider the structure \(\hat{\fA} = (\hat{A},\hat{R})\). From thetransfer principle it follows that \(\fA\) and \(\hat{\fA}\) haveprecisely the same first-order properties.

Now suppose that the set \(\bbN\) of natural numbers is a member of\(U\). Then so is the set \(\Re\) of real numbers, since each realnumber may be identified with a set of natural numbers. \(\Re\) may beregarded as an ordered field, and the same is therefore true of itsinflate \(\hat{\Re}\), since the latter has precisely the samefirst-order properties as \(\Re\). \(\hat{\Re}\) is called thehyperreal line, and its membershyperreals. Astandard hyperreal is then just a real, to which we shall refer foremphasis as astandard real. Since \(\Re\) is infinite,nonstandard hyperreals must exist. The saturation principle impliesthat there must be aninfinite (nonstandard) hyperreal,^[42] that is, a hyperreal \(a\) such that \(a \gt n\) for every \(n \in\bbN\). In that case its reciprocal 1/\(a\) is infinitesimal in thesense of exceeding 0 and yet being smaller than 1/\(n+1\) for every\(n \in \bbN\). In general, we call a hyperrealainfinitesimal if its absolute value \(\lvert a\rvert\) is lessthan 1/\(n+1\) for every \(n \in \bbN\). In that case the set \(I\) ofinfinitesimals contains not just 0 but a substantial number (in fact,infinitely many) other elements. Clearly \(I\) is an additive subgroupof \(\Re\), that is, if \(a, b \in I\), then \(a - b \in I\).

The members of the inflate \(\hat{\bbN}\) of \(\bbN\) are calledhypernatural numbers. As for the hyperreals, it can be shownthat \(\hat{\bbN}\) also contains nonstandard elements which mustexceed every member of \(\bbN\); these are calledinfinitehypernatural numbers.

For hyperreals \(a, b\) we define \(a \approx b\) and say that \(a\)and \(b\) areinfinitesimally close if \(a - b \in I\). Thisis an equivalence relation on the hyperreal line: for each hyperreal\(a\) we write \(\mu(a)\) for the equivalence class of \(a\) underthis relation and call it themonad of \(a\). The monad of ahyperreal \(a\) thus consists of all the hyperreals that areinfinitesimally close to \(a\): it may be thought of as a small cloudcentered at \(a\). Note also that \(\mu\)(0) \(= I\).

A hyperreal \(a\) isfinite if it is not infinite; this meansthat \(\lvert a\rvert \lt n\) for some \(n \in \bbN\) . It is notdifficult to show that finiteness is equivalent to the condition ofnear-standardness; here a hyperreal \(a\) isnear-standard if \(a \approx r\) for some standard real\(r\).

Much of the usefulness of nonstandard analysis stems from the factthat statements of classical analysis involving limits or the\((\varepsilon , \delta)\) criterion admit succinct, intuitivetranslations into statements involving infinitesimals or infinitenumbers, in turn enabling comparatively straightforward proofs to begiven of classical theorems. Here are some examples of such translations:^[43]

• Let \(\langle s_n\rangle\) be a standard infinite sequence of realnumbers and let \(s\) be a standard real number. Then \(s\) is thelimit of \(\langle s_n\rangle\) within \(\Re\), \(\lim_{n\rightarrow\infty} s_n = s\) in the classical sense, if and only if \(s_n \approxs\)for all infinite subscripts \(n\).
• A standard sequence \(\langle s_n\rangle\) converges if and onlyif \(s_n \approx s_m\) for all infinite \(n\) and \(m\).(Cauchy’s criterion for convergence.)

Now suppose that \(f\) is a real-valued function defined on some openinterval \((a, b)\). We have remarked above that \(f\) automaticallyextends to a function—also written \(f\)—on\(\widehat{(a,b)}\).

• In order that the standard real number \(c\) be the limit of\(f(x)\) as \(x\) approaches \(x_0\), \(\lim_{x\rightarrow x_0}f(x) =c\), with \(x_0\) a standard real number in \((a, b)\), it isnecessary and sufficient that \(f(x) \approx f(x_0)\) for all \(x\approx x_0\).
• The function \(f\) is continuous at a standard real number \(x_0\)in \((a, b)\) if and only if \(f(x) \approx f(x_0)\) for all \(x\approx x_0\). (This is equivalent to saying that \(f\) maps the monadof \(x_0\) into the monad of \(f(x_0\).)
• In order that the standard number \(c\) be the derivative of \(f\)at \(x_0\) it is necessary and sufficient that \[ \frac{f(x) - f(x_0)}{x - x_0} \approx c \] for all \(x\ne x_0\) in the monad of \(x_0\).

Many other branches of mathematics admit neat and fruitful nonstandardformulations.

8. The Constructive Real Line and the Intuitionistic Continuum

The original motivation for the development of constructivemathematics was to put the idea of mathematical existence on aconstructive or computable basis. While there are a number ofvarieties of constructive mathematics (Bridges & Richman 1987),here we shall focus on Bishop’s constructive analysis (Bishop& Bridges 1985; Bridges 1994, 1999; and Bridges & Richman1987) and Brouwer’s intuitionistic analysis (Dummett 1977).

In constructive mathematics a problem is counted as solved only if anexplicit solution can, in principle at least, be produced. Thus, forexample, “There is an \(x\) such that \(P(x)\)” meansthat, in principle at least, we can explicitly produce an \(x\) suchthat \(P(x)\). This fact led to the questioning of certain principlesof classical logic, in particular, the law of excluded middle, and thecreation of a new logic, intuitionistic logic (see entry onintuitionistic logic). It also led to the introduction of a sharpened definition of realnumbers—the constructive real numbers. A constructive realnumber is a sequence of rationals \((r_n) = r_1, r_2,\ldots\) suchthat, for any \(k\), a number \(n\) can be computed in such a way that\(\lvert r_{n+p} - r_{n}\rvert \le 1/k\). Each rational number a maybe regarded as a real number by identifying it with the real number\((\alpha , \alpha ,\ldots)\). The set \(R\) of all constructive realnumbers is the constructive real line.

Now of course, for any “given” real number there are avariety of ways of giving explicit approximating sequences for it.Thus it is necessary to define an equivalence relation,“equality on the reals”. The correct definition here is:\(r =_{\Re} s\) iff for any \(k\), a number \(n\) can be found so that\(\lvert r_{n+p} - s_{n+p}\rvert \le 1/k\), for all \(p\). To say thattwo real numbers are equal is to say that they are equivalent in thissense.

The real number line can be furnished with an axiomatic description.We begin by assuming the existence of a set \(R\) with

• a binary relation \(\gt\) (greater than)
• a correspondingapartness relation \(\apart\) defined by\(x \apart y \Leftrightarrow x \gt y\) or \(y \gt x\)
• a unary operation \(x \mapsto -x\)
• binary operations \((x, y) \mapsto (x + y\) (addition)and \((x, y) \mapsto xy\) (multiplication)
• distinguished elements 0 (zero) and 1 (one) with\(0 \ne 1\)
• a unary operation \(x \mapsto x^{-1}\) on the set of elements\(\ne 0\).

The elements of \(R\) are calledreal numbers. A real number\(x\) ispositive if \(x \gt 0\) andnegative if\(-x \gt 0\). The relation \(\ge\) (greater than or equal to)is defined by

The relations \(\lt\) and \(\le\) are defined in the usual way; \(x\)isnonnegative if \(0 \le x\). Two real numbers areequal if \(x \ge y\) and \(y \ge x\), in which case we write\(x = y\).

The sets \(N\) of natural numbers, \(N^+\) of positive integers, \(Z\)of integers and \(Q\) of rational numbers are identified with theusual subsets of \(R\); for instance \(N^+\) is identified with theset of elements of \(R\) of the form \(1 + 1 + \ldots + 1\).

These relations and operations are subject to the following threegroups of axioms, which, taken together, form the systemCA of axioms forconstructive analysis, ortheconstructive real numbers (Bridges 1999).

Field Axioms

• \(x + y = y + x\)
• \((x + y) + z = x + (y + z)\)
• \(0 + x = x\)
• \(x + (-x) = 0 xy = yx\)
• \((xy)z = x(yz)\)
• \(1 x = x\)
• \(xx^{-1} = 1\) if \(x \apart 0\)
• \(x(y + z) = xy + xz\)

Order Axioms

• \(\neg(x \gt y \wland y \gt x)\)
• \(x \gt y \Longrightarrow \forall z(x \gt z \wlor z \gt y)\)
• \(\neg(x \apart y) \Longrightarrow x = y\)
• \(x \gt y \Longrightarrow \forall z(x + z \gt y + z)\)
• \((x \gt 0 \wland y \gt 0) \Longrightarrow xy \gt 0\).

The last two axioms introduce special properties of \(\gt\) and\(\ge\). In the second of these the notionsbounded above, boundedbelow, and bounded are defined as in classical mathematics, andtheleast upper bound, if it exists, of a nonempty^[44] set \(S\) of real numbers is the unique real number \(b\) suchthat

• \(b\) is an upper bound for \(S\), and
• for each \(c \lt b\) there exists \(s \in S\) with \(s \gtc\).

Special Properties of \(\gt\).

The following basic properties of \(\gt\) and \(\ge\) can then beestablished.

• \(\neg(x \gt x)\)
• \(x \ge x\)
• \(x \gt y \wland y \gt z \Longrightarrow x \gt z\)
• \(\neg(x \gt y \wland y \ge x)\)
• \((x \gt y \ge z) \Longrightarrow x \gt z\)
• \(\neg(x \gt y) \Longleftrightarrow y \ge x\)
• \(\neg \neg(x \ge y) \Longleftrightarrow \neg \neg(y \gt x)\)
• \((x \ge y \ge z) \Longrightarrow x \ge z\)
• \((x \ge y \wland y \ge x) \Longrightarrow x = y\)
• \(\neg(x \gt y \wland x = y)\)
• \(x \ge 0 \Longleftrightarrow \forall \varepsilon \gt 0(x \lt\varepsilon)\)
• \(x + y \gt 0 \Longrightarrow (x \gt 0 \wlor y \gt 0)\)
• \(x \gt 0 \Longrightarrow -x \lt 0\)
• \((x \gt y \wland z \lt 0) \Longrightarrow yz \gt xz\)
• \(x \apart 0 \Longrightarrow x^2 \gt 0\)
• \(1 \gt 0\)
• \(x^2 \gt 0\)
• \(0 \lt x \lt 1 \Longrightarrow x \gt x^2\)
• \(x^2 \gt 0 \Longrightarrow x \apart 0\)
• \(n \in N^{+} \Longrightarrow n^{-1} \gt 0\)
• if \(x \gt 0\) and \(y \ge 0\), then \(\exists n\in Z(nx \gty)\)
• \(x \gt 0 \Longrightarrow x^{-1} \gt 0\)
• \(xy \gt 0 \Longrightarrow(x \ne 0 \wlor y \ne 0)\)
• if \(a \lt b\), then \(\exists r\in Q(a \lt r \lt b)\)

The constructive real line \(R\) as introduced above is a model ofCA. Are there any other models, that is, models notisomorphic to \(R\). If classical logic is assumed,CA is a categorical theory and so the answer is no.But this is not the case within intuitionistic logic, for there it ispossible for the Dedekind and Cantor reals to fail to be isomorphic,despite the fact that they are both models ofCA.

In constructive analysis, a real number is an infinite (convergent)sequence of rational numbers generated by an effective rule, so thatthe constructive real line is essentially just a restriction of itsclassical counterpart. Brouwerian intuitionism takes a more liberalview of the matter, resulting in a considerable enrichment of thearithmetical continuum over the version offered by strictconstructivism. As conceived by intuitionism, the arithmeticalcontinuum admits as real numbers not only infinite sequencesdetermined in advance by an effective rule for computing their terms,but also ones in whose generation free selection plays a part. Thelatter are called(free) choice sequences. Without loss ofgenerality we may and shall assume that the entries in choicesequences are natural numbers.

While constructive analysis does not formally contradict classicalanalysis, and may in fact be regarded as a subtheory of the latter, anumber of intuitionistically plausible principles have been proposedfor the theory of choice sequences which render intuitionisticanalysis divergent from its classical counterpart. One such principleisBrouwer’s Continuity Principle: given a relation\(Q(\alpha , n)\) between choice sequences \(\alpha\) and numbers\(n\), if for each \(\alpha\) a number \(n\) may be determined forwhich \(Q(\alpha , n)\) holds, then \(n\) can already be determined onthe basis of the knowledge of a finite number of terms of \(\alpha\).^[45] From this one can prove a weak version of the Continuity Theorem,namely, that every function from \(R\) to \(R\) is continuous. Anothersuch principle isBar Induction, a certain form of inductionfor well-founded sets of finite sequences.^[46] Brouwer used Bar Induction and the Continuity Principle in provinghis Continuity Theorem that every real-valued function defined on aclosed interval is uniformly continuous, from which, as has alreadybeen observed, it follows that the intuitionistic continuum isindecomposable.

Brouwer gave the intuitionistic conception of mathematics anexplicitly subjective twist by introducing thecreativesubject. The creative subject was conceived as a kind ofidealized mathematician for whom time is divided into discretesequential stages, during each of which he may test variouspropositions, attempt to construct proofs, and so on. In particular,it can always be determined whether or not at stage \(n\) the creativesubject has a proof of a particular mathematical proposition \(p\).While the theory of the creative subject remains controversial, itspurely mathematical consequences can be obtained by a simple postulatewhich is entirely free of subjective and temporal elements.

The creative subject allows us to define, for a given proposition\(p\), a binary sequence \(\langle a_n\rangle\) by \(a_n = 1\) if thecreative subject has a proof of \(p\) at stage \(n\); \(a_n = 0\)otherwise. Now if the construction of these sequences is the only usemade of the creative subject, then references to the latter may beavoided by postulating the principle known asKripke’sScheme

For each proposition \(p\) there exists an increasing binary sequence\(\langle a_n\rangle\) such that \(p\) holds if and only if \(a_n =1\) for some \(n\).

Taken together, these principles have been shown to have remarkableconsequences for the indecomposability of subsets of the continuum.Not only is the intuitionistic continuum indecomposable (that is,cannot be partitioned into two nonempty disjoint parts), but, assumingthe Continuity Principle and Kripke’s Scheme, it remainsindecomposable even if one pricks it with a pin. The intuitionisticcontinuum has, as it were, a syrupy nature, so that one cannot simplytake away one point. If in addition Bar Induction is assumed, then,still more surprisingly, indecomposability is maintained even when allthe rational points are removed from the continuum.

Finally, it has been shown that a natural notion of infinitesimal canbe developed within intuitionistic mathematics (Vesley 1981), the ideabeing that an infinitesimal should be a “very small” realnumber in the sense of not being known to bedistinguishable—that is, strictly greater than or lessthan—zero.

9. Smooth Infinitesimal Analysis

A major development in the refounding of the concept of infinitesimaltook place in the 1970s with the emergence ofsyntheticdifferential geometry, also known assmooth infinitesimalanalysis (SIA).^[47] Based on the ideas of the American mathematician F. W. Lawvere, andemploying the methods of category theory, smooth infinitesimalanalysis provides an image of the world in which the continuous is anautonomous notion, not explicable in terms of the discrete. Itprovides a rigorous framework for mathematical analysis in which everyfunction between spaces is smooth (i.e., differentiable arbitrarilymany times, and so in particular continuous) and in which the use oflimits in defining the basic notions of the calculus is replaced bynilpotent infinitesimals, that is, of quantities so small(but not actually zero) that some power—most usefully, thesquare—vanishes. Since in SIA all functions are continuous, itembodies in a striking way Leibniz’s principle of continuityNatura non facit saltus.

In what follows, we use bold \(\bR\) to distinguish the real line inSIA from its counterparts in classical and constructive analysis. Inthe usual development of the calculus, for any differentiable function\(f\) on the real line \(\bR, y = f(x)\), it follows fromTaylor’s theorem that the increment \(\delta y = f(x + \delta x)- f(x)\) in \(y\) attendant upon an increment \(\delta x\) in \(x\) isdetermined by an equation of the form

where \(f '(x)\) is the derivative of \(f(x)\) and \(A\) is a quantitywhose value depends on both \(x\) and \(\delta x\). Now if it werepossible to take \(\delta x\) sosmall (but not demonstrablyidentical with 0) that \((\delta x)^2 = 0\) then (1) would assume thesimple form

We shall call a quantity having the property that its square is zero anilsquare infinitesimal or simply amicroquantity.In SIA “enough” microquantities are present to ensure thatequation (2) holdsnontrivially forarbitraryfunctions \(f: \bR \rightarrow \bR\). (Of course (2) holds triviallyin standard mathematical analysis because there 0 is the solemicroquantity in this sense.) The meaning of the term“nontrivial” here may be explicated in following way. Ifwe replace \(\delta x\) by the letter \(\varepsilon\) standing for anarbitrary microquantity, (2) assumes the form

Ideally, we want the validity of this equation to be independent of\(\varepsilon\) , that is, given \(x\), for it to hold forall microquantities \(\varepsilon\). In that case thederivative \(f '(x)\) may bedefined as the unique quantity\(D\) such that the equation

holds for all microquantities \(\varepsilon\).

Setting \(x = 0\) in this equation, we get in particular

for all \(\varepsilon\).It is equation (4)that is takenas axiomatic in smooth infinitesimal analysis. Let us write\(\Delta\) for the set of microquantities, that is,

Then it is postulated that, for any \(f: \Delta \rightarrow \bR\),there is aunique \(D \in \bR\) such that equation (4) holdsfor all \(\varepsilon\). This says that the graph of \(f\) is astraight line passing through \((0, f\)(0)) with slope \(\Delta\).Thus any function on \(\Delta\) is what mathematicians termaffine, and so this postulate is naturally termed theprinciple ofmicroaffineness. It means that\(\Delta\)cannot be bent or broken: it is subject only totranslations and rotations—and yet is not (as it wouldhave to be in ordinary analysis) identical with a point. \(\Delta\)may be thought of as an entity possessing position and attitude, butlacking true extension.

Now consider the space \(\Delta^{\Delta}\) of maps from \(\Delta\) toitself. It follows from the microaffineness principle that thesubspace \((\Delta^{\Delta})_0\) of \(\Delta^{\Delta}\) consisting ofmaps vanishing at 0 is isomorphic to \(\bR\).^[48] The space \(\Delta^{\Delta}\) is a monoid^[49] under composition which may be regarded as acting on \(\Delta\) byevaluation: for \(f \in \Delta^{\Delta},\) \(f \cdot \varepsilon = f(\varepsilon)\). Its subspace \((\Delta^{\Delta})_0\) is a submonoidnaturally identified as the space ofratios ofmicroquantities. The isomorphism between \((\Delta^{\Delta})_0\)and \(\bR\) noted above is easily seen to be an isomorphism of monoids(where \(\bR\) is considered a monoid under its usual multiplication).It follows that \(\bR\) itself may be regarded as the space of ratiosof microquantities. This was essentially the view of Euler, whoregarded (real) numbers as representing the possible results ofcalculating the ratio 0/0. For this reason Lawvere has suggested that\(\bR\) be called the space ofEuler reals.

If we think of a function \(y = f(x)\) as defining a curve, then, forany \(a\), the image under \(f\) of the “microinterval”\(\Delta + a\) obtained by translating \(\Delta\) to \(a\) is straightand coincides with the tangent to the curve at \(x = a\). In thissense each curve is “infinitesimally straight”.

From the principle of microaffineness we deduce the importantprinciple of microcancellation, viz.

For the premise asserts that the graph of the function \(g: \Delta\rightarrow \bR\) defined by \(g(\varepsilon) = a \varepsilon\) hasboth slope \(a\) and slope \(b\): the uniqueness condition in theprinciple of microaffineness then gives \(a = b\). The principle ofmicrocancellation supplies the exact sense in which there are“enough” infinitesimals in smooth infinitesimalanalysis.

From the principle of microaffineness it also follows thatallfunctions on \(\bR\) are continuous, that is,sendneighboring points to neighboring points. Here two points \(x,y\) on \(\bR\) are said to be neighbors if \(x - y\) is in \(\Delta\),that is, if \(x\) and \(y\) differ by a microquantity. To see this,given \(f: \bR \rightarrow \bR\) and neighboring points \(x, y\), notethat \(y = x + \varepsilon\) with \(\varepsilon\) in \(\Delta\), sothat

But clearly any multiple of a microquantity is also a microquantity,so \(\varepsilon f '(x)\) is a microquantity, and the resultfollows.

In fact, since equation(3) holds for any \(f\), it also holds for its derivative \(f '\); itfollows that functions in smooth infinitesimal analysis aredifferentiable arbitrarily many times, thereby justifying the use ofthe term “smooth”.

Let us derive a basic law of the differential calculus, theproduct rule:

To do this we compute

since \(\varepsilon^2 = 0\). Therefore \(\varepsilon(fg)' =\varepsilon(f'g + fg')\), and the result follows bymicrocancellation.

Astationary point \(a\) in \(\bR\) of a function \(f\):\(\bR \rightarrow \bR\) is defined to be one in whose vicinity“infinitesimal variations” fail to change the value of\(f\), that is, such that \(f(a + \varepsilon) = f(a)\) for all\(\varepsilon\). This means that \(f(a) + \varepsilon f '(a)= f(a)\),so that \(\varepsilon f'(a) = 0\) for all \(\varepsilon\), whence itfollows from microcancellation that \(f'(a) = 0\). This isFermat’s rule.

An important postulate concerning stationary points that we adopt insmooth infinitesimal analysis is the

Constancy Principle. If every point in an interval \(J\) is astationary point of \(f: J \rightarrow \bR\) (that is, if \(f'\) isidentically 0), then \(f\) is constant.

Put succinctly, “universal local constancy implies globalconstancy”. It follows from this that two functions withidentical derivatives differ by at most a constant.

In ordinary analysis the continuum \(\bR\) is connected in the sensethat it cannot be split into two non empty subsets neither of whichcontains a limit point of the other. In smooth infinitesimal analysisit has the vastly stronger property ofindecomposability: itcannot be splitin any way whatsoever into two disjointnonempty subsets. For suppose \(\bR = U \cup V\) with \(U \cap V =\varnothing\). Define \(f: \bR \rightarrow \{0, 1\}\) by \(f(x) = 1\)if \(x \in U, f(x) = 0\) if \(x \in V\). We claim that \(f\) isconstant. For we have

This gives four possibilities:

1. \(f(x) = 0 \wamp f(x + \varepsilon) = 0\)
2. \(f(x) = 0 \wamp f(x + \varepsilon) = 1\)
3. \(f(x) = 1 \wamp f(x + \varepsilon) = 0\)
4. \(f(x) = 1 \wamp f(x + \varepsilon) = 1\)

Possibilities (ii) and (iii) may be ruled out because \(f\) iscontinuous. This leaves (i) and (iv), in either of which \(f(x) = f(x+ \varepsilon)\). So \(f\) is locally, and hence globally, constant,that is, constantly 1 or 0. In the first case \(V = \varnothing\) ,and in the second \(U = \varnothing\).

We observe that the postulates of smooth infinitesimal analysis areincompatible with the law of excluded middle of classicallogic. This incompatibility can be demonstrated in two ways, oneinformal and the other rigorous. First the informal argument. Considerthe function \(f\) defined for real numbers \(x\) by \(f(x) = 1\) if\(x = 0\) and \(f(x) = 0\) whenever \(x \ne 0\). If the law ofexcluded middle held, each real number would then be either equal orunequal to 0, so that the function \(f\) would be defined on the wholeof \(\bR\). But, considered as a function with domain \(\bR,\) \(f\)is clearly discontinuous. Since, as we know, in smooth infinitesimalanalysis every function on \(\bR\) is continuous, \(f\) cannot havedomain \(\bR\) there.^[50] So the law of excluded middle fails in smooth infinitesimal analysis.To put it succinctly,universal continuity implies the failure ofthe law of excluded middle.

Here now is the rigorous argument. We show that the failure of the lawof excluded middle can be derived from the principle of infinitesimalcancellation. To begin with, if \(x \ne 0\), then \(x^2 \ne 0\), sothat, if \(x^2 = 0\), then necessarily not \(x \ne 0\). This meansthat

(*)

for all infinitesimal \(\varepsilon\), not \(\varepsilon \ne0\).

Now suppose that the law of excluded middle were to hold. Then wewould have, for any \(\varepsilon\), either \(\varepsilon = 0\) or\(\varepsilon \ne 0\). But (*) allows us to eliminate the secondalternative, and we infer that, for all \(\varepsilon\), \(\varepsilon= 0\). This may be written

for all \(\varepsilon\), \(\varepsilon \cdot 1 = \varepsilon\cdot 0\),

from which we derive by microcancellation the falsehood \(1 = 0\). Soagain the law of excluded middle must fail.

The “internal” logic of smooth infinitesimal analysis isaccordingly not full classical logic. It is, instead,intuitionistic logic, that is, the logic derived from theconstructive interpretation of mathematical assertions. In our briefsketch we did not notice this “change of logic” because,like much of elementary mathematics, the topics we discussed arenaturally treated by constructive means such as directcomputation.

What are thealgebraic andorder structures on\(\bR\) in SIA? As far as the former is concerned, there is littledifference from the classical situation: in SIA \(\bR\) is equippedwith the usual addition and multiplication operations under which itis a field. In particular, \(\bR\) satisfies the condition that each\(x \ne 0\) has a multiplicative inverse. Notice, however, that sincein SIA no microquantity (apart from 0 itself) is provably \(\ne 0\),microquantities are not required to have multiplicative inverses (arequirement which would lead to inconsistency). From a strictlyalgebraic standpoint, \(\bR\) in SIA differs from its classicalcounterpart only in being required to satisfy the principle ofinfinitesimal cancellation.

The situation is different, however, as regards the order structure of\(\bR\) in SIA. Because of the failure of the law of excluded middle,the order relation \(\lt\) on \(\bR\) in SIA cannot satisfy thetrichotomy law

and accordingly \(\lt\) must be apartial, rather than atotal ordering. Since microquantities do not havemultiplicative inverses, and \(\bR\) is a field, any microquantity\(\varepsilon\) must satisfy

Accordingly, if we define the relation \(\lt\) by \(x \lt y\) iff\(\neg(y \lt x)\), then, for any microquantity \(\varepsilon\) wehave

Using these ideas we can identify threeinfinitesimalneighborhoods of 0 on \(\bR\) in SIA, each of which is includedin its successor. First, the set \(\Delta\) of microquantities itself,next, the set \(I = \{x \in \bR : \neg x \ne 0\}\) of elementsindistinguishable from 0; finally, the set

of elements neither less nor greater than 0. These three may bethought of as the infinitesimal neighborhoods of 0 definedalgebraically, logically, and order-theoretically,respectively.

In certain models of SIA the system ofnatural numberspossesses some subtle and intriguing features which make it possibleto introduce another type of infinitesimal—the so-calledinvertible infinitesimals—resembling those ofnonstandard analysis, whose presence engenders yet anotherinfinitesimal neighborhood of 0 properly containing all thoseintroduced above.

In SIA the set \(\bN \) of natural numbers can be defined to be thesmallest subset of \(\bR\) which contains 0 and is closed under theoperation of adding 1. In some models of SIA, \(\bR\) satisfies theArchimedean principle that every real number is majorized bya natural number. However, models of SIA have been constructed(Moerdijk & Reyes 1991) in which \(\bR\) is not Archimedean inthis sense. In these models it is more natural to consider, in placeof \(\bN \), the set \(\bNs\) ofsmooth natural numbersdefined by

\(\bNs\) is the set of points of intersection of the smooth curve \(y= \sin \pi x\) with the positive \(x\)-axis. In these models \(\bR\)can be shown to possess the Archimedean propertyprovided that inthe definition \(\bN \) is replaced by \(\bNs\). In these models,then, \(\bN \) is a proper subset of \(\bNs\): the members of \(\bNs -\bN \) may be considerednonstandard integers. Multiplicativeinverses of nonstandard integers are infinitesimals, but, beingthemselves invertible, they are of a different type from the ones wehave considered so far. It is quite easy to show that they, as well asthe infinitesimals in \(J\) (and so also those in \(\Delta\) and\(I)\) are all contained in the set—a further infinitesimalneighborhood of 0—

ofinfinitely small elements of \(\bR\). The members of theset

of invertible elements of \(K\) are naturally identified asinvertible infinitesimals. Being obtained as inverses of“infinitely large” reals (i.e., reals \(r\) satisfying\(\forall n\in \bN (n \lt r) \lor \forall n\in \bN (r \lt -n)\)) themembers of \(\In\) are the counterparts in SIA of the infinitesimalsof nonstandard analysis.

Finally, a brief word on themodels of SIA. These are theso-called smooth toposes, categories (see entry oncategory theory) of a certain kind in which all the usual mathematical operations canbe performed but whose internal logic is intuitionistic and in whichevery map between spaces is smooth, that is, differentiable withoutlimit. It is this “universal smoothness” that makes thepresence of infinitesimal objects such as \(\Delta\) possible. Theconstruction of smooth toposes (Moerdijk & Reyes 1991) guaranteesthe consistency of SIA with intuitionistic logic. This is so despitethe evident fact that SIA is not consistent with classical logic.

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Acknowledgments

For a comprehensive account of the evolution of the concepts ofcontinuity and the infinitesimal, see Bell (2005), on which thepresent article is based.