Continuity and Infinitesimals

7/25/2019 Continuity and Infinitesimals

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Stanford Encyclopediaof Philosophy

Continuity and Infinitesimals

First published Wed Jul 27, 2005; substantive revision Fri Sep 6,

2013

The usual meaning of the word continuousis unbroken oruninterrupted: thus a continuous entitya continuumhas nogaps. We commonly suppose that space and time are continuous,and certain philosophers have maintained that all natural processesoccur continuously: witness, for example, Leibniz's famousapothegm natura non facit saltusnature makes no jump. Inmathematics the word is used in the same general sense, but has had

to be furnished with increasingly precise definitions. So, forinstance, in the later 18th century continuity of a function was takento mean that infinitesimal changes in the value of the argumentinduced infinitesimal changes in the value of the function. With theabandonment of infinitesimals in the 19th century this definitioncame to be replaced by one employing the more precise concept oflimit.

Traditionally, an infinitesimalquantityis one which, while notnecessarily coinciding with zero, is in some sense smaller than anyfinite quantity. For engineers, an infinitesimal is a quantity so smallthat its square and all higher powers can be neglected. In the theoryof limits the term infinitesimal is sometimes applied to anysequence whose limit is zero. An infinitesimal magnitudemay beregarded as what remains after a continuum has been subjected to an

exhaustive analysis, in other words, as a continuum viewed in thesmall. It is in this sense that continuous curves have sometimesbeen held to be composed of infinitesimal straight lines.


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Infinitesimals have a long and colourful history. They make an earlyappearance in the mathematics of the Greek atomist philosopherDemocritus (c. 450 B.C.E.), only to be banished by themathematician Eudoxus (c. 350 B.C.E.) in what was to becomeofficial Euclidean mathematics. Taking the somewhat obscure

form of indivisibles, they reappear in the mathematics of the latemiddle 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.In recent years, however, the concept of infinitesimal has beenrefounded on a rigorous basis.

1. Introduction: The Continuous, the Discrete, and theInfinitesimal2. The Continuum and the Infinitesimal in the Ancient Period3. The Continuum and the Infinitesimal in the Medieval,Renaissance, and Early Modern Periods4. The Continuum and the Infinitesimal in the 17th and 18thCenturies5. The Continuum and the Infinitesimal in the 19th Century6. Critical Reactions to Arithmetization7. Nonstandard Analysis8. The Constructive Real Line and the IntuitionisticContinuum9. Smooth Infinitesimal AnalysisBibliographyAcademic Tools

Other Internet ResourcesRelated Entries

1. Introduction: The Continuous, the

Discrete, and the Infinitesimal

We are all familiar with the idea of continuity. To be continuous[1]


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is to constitute an unbroken or uninterrupted whole, like the oceanor the sky.A continuous entitya continuumhas no gaps.

Opposed to continuity is discreteness: to be discrete[2]is to beseparated, like the scattered pebbles on a beach or the leaves on atree. Continuity connotes unity; discreteness, plurality.

While it is the fundamental nature of a continuum to be undivided, itis nevertheless generally (although not invariably) held that anycontinuum admits of repeated or successive divisionwithout limit.This means that the process of dividing it into ever smaller parts willnever terminate in an indivisibleor an atomthat is, a part which,lacking proper parts itself, cannot be further divided. In a word,

continua are divisible without limitor infinitely divisible.The unityof a continuum thus conceals a potentially infinite plurality. Inantiquity this claim met with the objection that, were one to carryout completelyif only in imaginationthe process of dividing anextended magnitude, such as a continuous line, then the magnitudewould be reduced to a multitude of atomsin this case,extensionless pointsor even, possibly, to nothing at all. But then,

it was held, no matter how many such points there may beeven ifinfinitely manythey cannot be reassembled to form the originalmagnitude, for surely a sum of extensionless elements still lacks

extension[3]. Moreover, if indeed (as seems unavoidable) infinitelymany points remain after the division, then, following Zeno, themagnitude may be taken to be a (finite) motion, leading to theseemingly absurd conclusion that infinitely many points can be

touched in a finite time.

Such difficulties attended the birth, in the 5thcentury B.C.E., of theschool of atomism.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 had in actualityto beconceived as being compounded from such atoms. In attackinginfinite divisibility the atomists were at the same time mounting a


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claim that the continuous is ultimately reducible to the discrete,whether it be at the physical, theoretical, or perceptual level.

The eventual triumph of the atomic theory in physics and chemistry

in the 19thcentury paved the way for the idea of atomism, as

applying to matter, at least, to become widely familiar: it might wellbe said, to adapt Sir William Harcourt's famous observation inrespect of the socialists of his day, We are all atomists now.Nevertheless, only a minority of philosophers of the past espousedatomism at a metaphysical level, a fact which may explain why theanalogous doctrine upholding continuity lacks a familiar name: thatwhich is unconsciously acknowledged requires no name. Peirce

coined the term synechism(from Greek syneche,continuous) forhis own philosophya philosophy permeated by the idea of

continuity in its sense of being connected[4]. In this article I shallappropriate Peirce's term and use it in a sense shorn of its Peirceanovertones, simply as a contrary to atomism. I shall also use the termdivisionism for the more specific doctrine that continua areinfinitely divisible.

Closely associated with the concept of a continuum is that of

infinitesimal.[5]An infinitesimal magnitudehas been somewhathazily conceived as a continuum viewed in the small, an ultimatepart of a continuum. In something like the same sense as a discreteentity is made up of its individual units, its indivisibles, so, it wasmaintained, a continuum is composed of infinitesimal

magnitudes, its ultimate parts. (It is in this sense, for example, thatmathematicians of the 17thcentury held that continuous curves arecomposed of infinitesimal straight lines.) Now the coherence ofa continuum entails that each of its (connected) parts is also acontinuum, and, accordingly, divisible. Since points are indivisible,it follows that no point can be part of a continuum. Infinitesimalmagnitudes, as parts of continua, cannot, of necessity, be points:

they are, in a word, nonpunctiform.

Magnitudes are normally taken as being extensivequantities, like


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mass or volume, which are defined over extended regions of space.By contrast, infinitesimal magnitudes have been construed asintensivemagnitudes resembling locally defined intensive quantitiessuch as temperature or density. The effect of distributing orintegrating an intensive quantity over such an intensive magnitude

is to convert the former into an infinitesimal extensive quantity: thustemperature is transformed into infinitesimal heat and density intoinfinitesimal mass. When the continuum is the trace of a motion, theassociated infinitesimal/intensive magnitudes have been identifiedaspotentialmagnitudesentities which, while not possessing truemagnitude themselves, possess a tendencyto generate magnitudethrough motion, so manifesting becoming as opposed to being.

An infinitesimal numberis one which, while not coinciding withzero, is in some sense smaller than any finite number. This sense hasoften been taken to be the failure to satisfy the Principle ofArchimedes, which amounts to saying that an infinitesimal numberis one that, no matter how many times it is added to itself, the resultremains less than any finite number. In the engineer's practicaltreatment of the differential calculus, an infinitesimal is a number sosmall that its square and all higher powers can be neglected. In thetheory of limits the term infinitesimal is sometimes applied to anysequence whose limit is zero.

The concept of an indivisibleis closely allied to, but to bedistinguished from, that of an infinitesimal. An indivisible is, bydefinition, something that cannot be divided, which is usually

understood 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 thesehave in common the feature of being unextended; extended entitiessuch as lines, surfaces, and volumes prove a much richer source ofindivisibles. Indeed, if the process of dividing such entities were

to terminate, as the atomists maintained, it would necessarily issuein indivisibles of a qualitatively different nature. In the case of astraight line, such indivisibles would, plausibly, be points; in the


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case of a circle, straight lines; and in the case of a cylinder dividedby sections parallel to its base, circles. In each case the indivisible inquestion is infinitesimal in the sense ofpossessing one fewer

dimension than the figure from which it is generated. In the 16thand

17

th

centuries indivisibles in this sense were used in the calculationof areas and volumes of curvilinear figures, a surface or volumebeing thought of as a collection, or sum, of linear, or planarindivisibles respectively.

The concept of infinitesimal was beset by controversy from itsbeginnings. The idea makes an early appearance in the mathematicsof the Greek atomist philosopher Democritus c. 450 B.C.E., only to

be banished c. 350 B.C.E. by Eudoxus in what was to becomeofficial 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 andtimelets, infinitesimals played an essential role in Barrow's

method for finding tangents by calculation, which appears in hisLectiones Geometricaeof 1670. As evanescent quantitiesinfinitesimals were instrumental (although later abandoned) inNewton's development of the calculus, and, as inassignablequantities, in Leibniz's. The Marquis de l'Hpital, who in 1696published the first treatise on the differential calculus (entitledAnalyse des Infiniments Petits pour l'Intelligence des Lignes

Courbes), invokes the concept in postulating that a curved line maybe regarded as being made up of infinitely small straight linesegments, and that one can take as equal two quantities differingby an infinitely small quantity.

However useful it may have been in practice, the concept ofinfinitesimal could scarcely withstand logical scrutiny. Derided by

Berkeley in the 18thcentury as ghosts of departed quantities, in

the 19thcentury execrated by Cantor as cholera-bacilli infecting

mathematics, and in the 20throundly condemned by Bertrand


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Russell as unnecessary, erroneous, and self-contradictory, theseuseful, but logically dubious entities were believed to have beenfinally supplanted in the foundations of analysis by the limit concept

which took rigorous and final form in the latter half of the 19th

century. By the beginning of the 20

th

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 asa heuristic device for the derivation of correct results in theapplication of the calculus to physical problems. Differential

geometers of the stature of Lie and Cartan relied on their use in theformulation of concepts which would later be put on a rigorousfooting. And, in a technical sense, they lived on in the algebraists'investigations of nonarchimedean fields.

A new phase in the long contest between the continuous and thediscrete has opened in the past few decades with the refounding of

the concept of infinitesimal on a solid basis. This has been achievedin two essentially different ways, the one providing a rigorousformulation of the idea of infinitesimal number, the other ofinfinitesimal magnitude.

First, in the nineteen sixties Abraham Robinson, using methods ofmathematical logic, created nonstandard analysis,an extension ofmathematical analysis embracing both infinitely large andinfinitesimal numbers in which the usual laws of the arithmetic ofreal numbers continue to hold, an idea which, in essence, goes backto Leibniz. Here by an infinitely large number is meant one whichexceeds every 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 every

statement of ordinary analysis involving limits has a succinct andhighly intuitive translation into the language of infinitesimals.


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The second development in the refounding of the concept ofinfinitesimal took place in the nineteen seventies with theemergence of synthetic differential geometry,also known as smoothinfinitesimal analysis. Based on the ideas of the Americanmathematician F. W. Lawvere, and employing the methods of

category theory, smooth infinitesimal analysis provides an image ofthe world in which the continuous is an autonomous notion, notexplicable in terms of the discrete. It provides a rigorous frameworkfor mathematical analysis in which every function between spaces issmooth (i.e., differentiable arbitrarily many times, and so inparticular continuous) and in which the use of limits in defining thebasic notions of the calculus is replaced by nilpotent infinitesimals,

that is, of quantities so small (but not actually zero) that some powermost usefully, the squarevanishes. Smooth infinitesimalanalysis embodies a concept of intensive magnitude in the form ofinfinitesimal tangent vectorsto curves. A tangent vector to a curveat a pointpon it is a short straight line segment lpassing throughthe point and pointing along the curve. In fact we may take lactually to be an infinitesimalpartof the curve. Curves in smooth

infinitesimal analysis are locally straight and accordingly may beconceived as being composed of infinitesimal straight lines in del'Hpital's sense, or as being generated by an infinitesimal tangentvector.

The development of nonstandard and smooth infinitesimal analysishas breathed new life into the concept of infinitesimal, andespecially in connection with smooth infinitesimal analysis

supplied novel insights into the nature of the continuum.

2. The Continuum and the Infinitesimal in

the Ancient Period

The opposition between Continuity and Discreteness played a

significant role in ancient Greek philosophy. This probably derivedfrom the still more fundamental question concerning the One andthe Many, an antithesis lying at the heart of early Greek thought (see


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Stokes [1971]). The Greek debate over the continuous and thediscrete seems to have been ignited by the efforts of Eleaticphilosophers such as Parmenides (c. 515 B.C.E.), and Zeno (c. 460

B.C.E.) to establish their doctrine of absolute monism[6]. They wereconcerned to show that the divisibility of Being into parts leads tocontradiction, so forcing the conclusion that the apparently diverse

world is a static, changeless unity.[7]In his Way of TruthParmenides asserts that Being is homogeneousand continuous.However in asserting the continuity of Being Parmenides is likelyno more than underscoring its essential unity. Parmenides seems tobe claiming that Being is more than merely continuousthat it is, infact, a single whole, indeed an indivisiblewhole. The single

Parmenidean existent is a continuum without parts, at once acontinuum and an atom. If Parmenides was a synechist, his absolutemonism precluded his being at the same time a divisionist.

In support of Parmenides' doctrine of changelessness Zenoformulated his famous paradoxes of motion. (see entry on Zeno'sparadoxes) TheDichotomyandAchillesparadoxes both rest

explicitly on the limitless divisibility of space and time.

The doctrine ofAtomism,[8]which seems to have arisen as anattempt at escaping the Eleatic dilemma, was first and foremost aphysical theory. It was mounted by Leucippus (fl. 440 B.C.E.) andDemocritus (b. 460457 B.C.E.) who maintained that matter wasnot divisible without limit, but composed of indivisible, solid,

homogeneous, spatially extended corpuscles, all below the level ofvisibility.

Atomism was challenged by Aristotle (384322 B.C.E.), who wasthe first to undertake the systematic analysis of continuity anddiscreteness. A thoroughgoing synechist, he maintained thatphysical reality is a continuous plenum, and that the structure of acontinuum, common to space, time and motion, is not reducible toanything else. His answer to the Eleatic problem was thatcontinuous magnitudes are potentially divisible to infinity, in thesense that they may be divided anywhere, though they cannot be


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divided everywhereat the same time.

Aristotle identifies continuity and discreteness as attributes applying

to the category of Quantity[9]. As examples of continuous quantities,or continua, he offers lines, planes, solids (i.e., solid bodies),

extensions, movement, time and space; among discrete quantities he

includes number[10]and speech[11]. He also lays down definitionsof a number of terms, including continuity. In effect, Aristotledefines continuity as a relationbetween entities rather than as anattributeappertaining to a single entity; that is to say, he does notprovide an explicit definition of the concept of continuum. Heobserves that a single continuous whole can be brought into

existence by gluing together two things which have been broughtinto contact, which suggests that the continuity of a whole shouldderive from the way itspartsjoin up. Accordingly for Aristotlequantities such as lines and planes, space and time are continuous byvirtue of the fact that their constituent parts join together at somecommon boundary. By contrast noconstituent parts of a discretequantity can possess a common boundary.

One of the central theses Aristotle is at pains to defend is theirreducibility of the continuum to discretenessthat a continuumcannot be composed of indivisibles or atoms, parts which cannotthemselves be further divided.

Aristotle sometimes recognizes infinite divisibilitythe property ofbeing divisible into parts which can themselves be further divided,the process never terminating in an indivisibleas a consequence ofcontinuity as he characterizes the notion. But on occasion he takesthe property of infinite divisibility as definingcontinuity. It is thisdefinition of continuity that figures in Aristotle's demonstration ofwhat has come to be known as the isomorphismthesis, which assertsthat either magnitude, time and motion are all continuous, or theyare all discrete.

The question of whether magnitude is perpetually divisible intosmaller units, or divisible only down to some atomic magnitude


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leads to the dilemma of divisibility(see Miller [1982]), a difficultythat Aristotle necessarily had to face in connection with his analysisof the continuum. In the dilemma's first, or nihilistichorn, it isargued that, were magnitude everywhere divisible, the process ofcarrying out this division completely would reduce a magnitude to

extensionless points, or perhaps even to nothingness. The second, oratomistic,horn starts from the assumption that magnitude is noteverywhere divisible and leads to the equally unpalatable conclusion(for Aristotle, at least) that indivisible magnitudes must exist.

As a thoroughgoing materialist, Epicurus[12](341271 B.C.E.)could not accept the notion of potentiality on which Aristotle's

theory of continuity rested, and so was propelled towards atomismin both its conceptual and physical senses. Like Leucippus andDemocritus, Epicurus felt it necessary to postulate the existence ofphysical atoms, but to avoid Aristotle's strictures he proposed thatthese should not be themselves conceptually indivisible, but shouldcontainconceptually indivisible parts. Aristotle had shown that acontinuous magnitude could not be composed ofpoints,that is,

indivisible units lacking extension, but he had not shown that anindivisible unit must necessarily lack extension. Epicurus metAristotle's argument that a continuum could not be composed ofsuch indivisibles by taking indivisibles to be partless units ofmagnitude possessing extension.

In opposition to the atomists, the Stoic philosophers Zeno of Cition(fl. 250 B.C.E.) and Chrysippus (280206 B.C.E.) upheld the

Aristotelian position that space, time, matter and motion are allcontinuous (see Sambursky [1963], [1971]; White [1992]). And,like Aristotle, they explicitly rejected any possible existence of voidwithin the cosmos. The cosmos is pervaded by a continuousinvisible substance which they calledpneuma(Greek: breath).This pneumawhich was regarded as a kind of synthesis of air andfire, two of the four basic elements, the others being earth and water

was conceived as being an elastic medium through whichimpulses are transmitted by wave motion. All physical occurrenceswere viewed as being linked through tensile forces in the pneuma,


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and matter itself was held to derive its qualities form the bindingproperties of the pneuma it contains.


the Medieval, Renaissance, and EarlyModern 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 Master

in Book VI of the Physics,that continua cannot be composed ofindivisibles. On the other hand, the avowed infinitude of the Deityof scholastic theology, which ran counter to Aristotle's thesis thatthe infinite existed only in a potential sense, emboldened certain ofthe Schoolmen to speculate that the actual infinite might be foundeven outside the Godhead, for instance in the assemblage of pointson a continuous line. A few scholars of the time, for example Henryof Harclay (c. 12751317) and Nicholas of Autrecourt (c. 130069)chose to follow Epicurus in upholding atomism reasonable andattempted to circumvent Aristotle's counterarguments (see Pyle[1997]).

This incipient atomism met with a determined synechist rebuttal,initiated by John Duns Scotus (c. 12661308). In his analysis of theproblem of whether an angel can move from place to place with a

continuous motion he offers a pair of purely geometrical argumentsagainst the composition of a continuum out of indivisibles. One ofthese arguments is that if the diagonal and the side of a square wereboth composed of points, then not only would the two becommensurable in violation of Book X of Euclid, they would evenbe equal. In the other, two unequal circles are constructed about acommon centre, and from the supposition that the larger circle is

composed of points, part of an angle is shown to be equal to thewhole, in violation of Euclid's axiom V.


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William of Ockham (c. 12801349) brought a considerable degree

of dialectical subtlety[13]to his analysis of continuity; it has been

the subject of much scholarly dispute[14]. For Ockham the principaldifficulty presented by the continuous is the infinite divisibility of

space, and in general, that of any continuum. The treatment ofcontinuity in the first book of his Quodlibetof 13227 rests on theidea that between any two points on a line there is a thirdperhapsthe first explicit formulation of the property of densityand on thedistinction between a continuumwhose parts form a unity from acontiguumof juxtaposed things. Ockham recognizes that it followsfrom the property of density that on arbitrarily small stretches of aline infinitely many points must lie, but resists the conclusion that

lines, or indeed any continuum, consists of points. Concerned,rather, to determine the sense in which the line may be said toconsist or to be made up of anything., Ockham claims that no partof the line is indivisible, nor is any part of a continuum indivisible.While Ockham does not assert that a line is actually composed ofpoints, he had the insight, startling in its prescience, that a punctateand yet continuous line becomes a possibility when conceived as a

dense array of points, rather than as an assemblage of points incontiguous succession.

The most ambitious and systematic attempt at refuting atomism in

the 14thcentury was mounted by Thomas Bradwardine (c. 1290 1349). The purpose of his Tractatus de Continuo(c. 1330) was toprove that the opinion which maintains continua to be composed of

indivisibles is false. This was to be achieved by setting forth anumber of first principles concerning the continuumakin to theaxioms and postulates of Euclid'sElementsand thendemonstrating that the further assumption that a continuum iscomposed of indivisibles leads to absurdities (see Murdoch [1957]).

The views on the continuum of Nicolaus Cusanus (140164), achampion of the actual infinite, are of considerable interest. In hisDe Mente Idiotaeof 1450, he asserts that any continuum, be itgeometric, perceptual, or physical, is divisible in two senses, the one


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Johann Kepler (15711630) made abundant use of infinitesimals inhis calculations. In hisNova Stereometriaof 1615, a work actuallywritten as an aid in calculating the volumes of wine casks, heregards curves as being infinilateral polygons, and solid bodies asbeing made up of infinitesimal cones or infinitesimally thin discs

(see Baron [1987], pp. 108116; Boyer [1969], pp. 106110). Suchuses are in keeping with Kepler's customary use of infinitesimals ofthe same dimension as the figures they constitute; but he also usedindivisibles on occasion. He spoke, for example, of a cone as beingcomposed of circles, and in hisAstronomia Novaof 1609, the workin which he states his famous laws of planetary motion, he takes thearea of an ellipse to be the sum of the radii drawn from the focus.

It seems to have been Kepler who first introduced the idea, whichwas later to become a reigning principle in geometry, of continuouschange of a mathematical object, in this case, of a geometric figure.In hisAstronomiae pars Opticaof 1604 Kepler notes that all theconic sections are continuously derivable from one another boththrough focal motion and by variation of the angle with the cone ofthe cutting plane.

Galileo Galilei (15641642) advocated a form of mathematicalatomism in which the influence of both the Democritean atomistsand the Aristotelian scholastics can be discerned. This emergeswhen one turns to the First Day of Galileo'sDialogues ConcerningTwo New Sciences(1638). Salviati, Galileo's spokesman, maintains,contrary to Bradwardine and the Aristotelians, that continuous

magnitude is made up of indivisibles, indeed an infinite number ofthem. Salviati/Galileo recognizes that this infinity of indivisibleswill never be produced by successive subdivision, but claims tohave a method for generating it all at once, thereby removing it fromthe realm of the potential into actual realization: this method forseparating and resolving, at a single stroke, the whole of infinityturns out simply to the act of bending a straight line into a circle.

Here Galileo finds an ingenious metaphysical application of theidea of regarding the circle as an infinilateral polygon. When thestraight line has been bent into a circle Galileo seems to take it that


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that the line has thereby been rendered into indivisible parts, that is,points. But if one considers that these parts are the sides of theinfinilateral polygon, they are better characterized not as indivisiblepoints, but rather as unbendable straight lines, each at once part of

and tangent to the circle[15]. Galileo does not mention thispossibility, but nevertheless it does not seem fanciful to detect thegerm here of the idea of considering a curve as a an assemblage of

infinitesimal unbendable straight lines.[16]

It was Galileo's pupil and colleague Bonaventura Cavalieri (15981647) who refined the use of indivisibles into a reliablemathematical tool (see Boyer [1959]); indeed the method of

indivisibles remains associated with his name to the present day.Cavalieri nowhere explains precisely what he understands by theword indivisible, but it is apparent that he conceived of a surfaceas composed of a multitude of equispaced parallel lines and of avolume as composed of equispaced parallel planes, these beingtermed the indivisibles of the surface and the volume respectively.While Cavalieri recognized that these multitudes of indivisibles

must be unboundedly large, indeed was prepared to regard them asbeing actually infinite, he avoided following Galileo intoensnarement in the coils of infinity by grasping that, for the methodof indivisibles to work, the precise number of indivisiblesinvolved did not matter. Indeed, the essence of Cavalieri's methodwas the establishing of a correspondence between the indivisibles oftwo similar configurations, and in the cases Cavalieri considers it

is evident that the correspondence is suggested on solely geometricgrounds, rendering it quite independent of number. The verystatement of Cavalieri's principle embodies this idea: if plane figuresare included between a pair of parallel lines, and if their interceptson any line parallel to the including lines are in a fixed ratio, thenthe areas of the figures are in the same ratio. (An analogousprinciple holds for solids.) Cavalieri's method is in essence that ofreduction of dimension: solids are reduced to planes withcomparable areas and planes to lines with comparable lengths.While this method suffices for the computation of areas or volumes,it cannot be applied to rectify curves, since the reduction in this case


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would be to points, and no meaning can be attached to the ratio oftwo points. For rectification a curve has, it was later realized, to beregarded as the sum, not of indivisibles, that is, points, but rather ofinfinitesimal straight lines, its microsegments.

Ren Descartes (15961650) employed infinitesimalist techniques,including Cavalieri's method of indivisibles, in his mathematicalwork. But he avoided the use of infinitesimals in the determinationof tangents to curves, instead developing purely algebraic methodsfor the purpose. Some of his sharpest criticism was directed at thosemathematicians, such as Fermat, who used infinitesimals in theconstruction of tangents.

As a philosopher Descartes may be broadly characterized as asynechist. His philosophical system rests on two fundamentalprinciples: the celebrated Cartesian dualismthe division betweenmind and matterand the less familiar identification of matter andspatial extension. In theMeditationsDescartes distinguishes mindand matter on the grounds that the corporeal, being spatiallyextended, is divisible, while the mental is partless. The identificationof matter and spatial extension has the consequence that matter iscontinuous and divisible without limit. Since extension is the soleessential property of matter and, conversely, matter alwaysaccompanies extension, matter must be ubiquitous. Descartes' spaceis accordingly, as it was for the Stoics, a plenum pervaded by acontinuous medium.

The concept of infinitesimal had arisen with problems of ageometric character and infinitesimals were originally conceived asbelonging solely to the realm of continuous magnitude as opposedto that of discrete number. But from the algebra and analytic

geometry of the 16thand 17thcenturies there issued the concept ofinfinitesimal number. The idea first appears in the work of Pierre deFermat (see Boyer [1959]) (160165) on the determination of

maximum and minimum (extreme) values, published in 1638.

Fermat's treatment of maxima and minima contains the germ of the


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fertile technique of infinitesimal variation, that is, theinvestigation of the behaviour of a function by subjecting itsvariables to small changes. Fermat applied this method indetermining tangents to curves and centres of gravity.


the 17th and 18th Centuries

Isaac Barrow[17](163077) was one of the first mathematicians tograsp the reciprocal relation between the problem of quadrature andthat of finding tangents to curvesin modern parlance, betweenintegration and differentiation. In hisLectiones Geometricaeof1670, Barrow observes, in essence, that if the quadrature of a curvey=f(x) is known, with the area up toxgiven by F(x), then thesubtangent to the curvey= F(x) is measured by the ratio of itsordinate to the ordinate of the original curve.

Barrow, a thoroughgoing synechist, regarded the conflict between

divisionism and atomism as a live issue, and presented a number ofarguments against mathematical atomism, the strongest of which isthat atomism contradicts many of the basic propositions ofEuclidean geometry.

Barrow conceived of continuous magnitudes as being generated bymotions, and so necessarily dependent on time, a view that seems to

have had a strong influence on the thinking of his illustrious pupilIsaac Newton[18](16421727). Newton's meditations during theplague year 166566 issued in the invention of what he called theCalculus of Fluxions, the principles and methods of which werepresented in three tracts published many years after they were

written[19]:De analysi per aequationes numero terminoruminfinitas; Methodus fluxionum et serierum infinitarum;andDe

quadraturacurvarum. Newton's approach to the calculus rests, evenmore firmly than did Barrow's, on the conception of continua asbeing generated by motion.


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But Newton's exploitation of the kinematic conception went muchdeeper than had Barrow's. InDe Analysi,for example, Newtonintroduces a notation for the momentary increment (moment)evidently meant to represent a moment or instant of timeof theabscissa or the area of a curve, with the abscissa itself representing

time. This momenteffectively the same as the infinitesimalquantities previously introduced by Fermat and BarrowNewtondenotes by oin the case of the abscissa, and by ovin the case of thearea. From the fact that Newton uses the letter vfor the ordinate, itmay be inferred that Newton is thinking of the curve as being agraph of velocity against time. By considering the moving line, orordinate, as the moment of the area Newton established the

generality 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, that is, onthe method of indivisibles. Newton's explicit treatment ofintegration as inverse differentiation was the key to the integral

calculus.

In theMethodus fluxionumNewton makes explicit his conception ofvariable quantities as generated by motions, and introduces hischaracteristic notation. He calls the quantity generated by a motion afluent, and its rate of generation afluxion. The fluxion of a fluentxis denoted by , and its moment, or infinitely small incrementaccruing in an infinitely short time o, by o. The problem of

determining a tangent to a curve is transformed into the problem offinding the relationship between the fluxions and when presentedwith an equation representing the relationship between the fluentsxandz. (A quadrature is the inverse problem, that of determining thefluents when the fluxions are given.) Thus, for example, in the case

of the fluentz=xn, Newton first forms + o= ( + o)n, expands

the right-hand side using the binomial theorem, subtractsz=xn,divides through by o, neglects all terms still containing o, and so

obtains = nxn1 .

x

x

x z

z z x x

z x


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Newton later became discontented with the undeniable presence ofinfinitesimals in his calculus, and dissatisfied with the dubiousprocedure of neglecting them. In the preface to theDe quadraturacurvarumhe remarks that there is no necessity to introduce into themethod of fluxions any argument about infinitely small quantities.

In their place he proposes to employ what he calls the method ofprime and ultimate ratio. This method, in many respects ananticipation of the limit concept, receives a number of allusions inNewton's celebrated Principia mathematica philosophiae naturalisof 1687.

Newton developed three approaches for his calculus, all of which he

regarded as leading to equivalent results, but which varied in theirdegree of rigour. 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, theinfinitesimal quantities forming it may be replaced by any suitablefinite magnitudessuch as velocities or fluxionshaving the sameratio. This is the method of fluxions. Recognizing that this methoditself required a foundation, Newton supplied it with one in the formof the doctrine of prime and ultimate ratios, a kinematic form of thetheory of limits.

The philosopher-mathematician G. W. F. Leibniz[20](16461716)was greatly preoccupied with the problem of the composition of thecontinuumthe labyrinth of the continuum, as he called it.

Indeed we have it on his own testimony that his philosophicalsystemmonadismgrew from his struggle with the problem of

just how, or whether, a continuum can be built from indivisibleelements. Leibniz asked himself: if we grant that each real entity iseither a simple unity or a multiplicity, and that a multiplicity isnecessarily an aggregation of unities, then under what head should ageometric continuum such as a line be classified? Now a line is

extended and Leibniz held that extension is a form of repetition, so,a line, being divisible into parts, cannot be a (true) unity. It is then amultiplicity, and accordingly an aggregation of unities. But of what


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sort of unities? Seemingly, the only candidates for geometric unitiesare points, but points are no more than extremities of the extended,and in any case, as Leibniz knew, solid arguments going back toAristotle establish that no continuum can be constituted from points.It follows that a continuum is neither a unity nor an aggregation of

unities. Leibniz concluded that continua are not realentitiesat all;as wholes preceding their parts they have instead a purely idealcharacter. In this way he freed the continuum from the requirementthat, as something intelligible, it must itself be simple or acompound of simples.

Leibniz held that space and time, as continua, are ideal, and

anything real, in particular matter, is discrete, compounded ofsimple unit substances he termed monads.

Among the best known of Leibniz's doctrines is the PrincipleorLaw of Continuity. In a somewhat nebulous form this principle hadbeen employed on occasion by a number of Leibniz's predecessors,including Cusanus and Kepler, but it was Leibniz who gave to theprinciple a clarity of formulation which had previously beenlacking and perhaps for this reason regarded it as his owndiscovery (Boyer 1959, p. 217). In a letter to Bayle of 1687,Leibniz gave the following formulation of the principle: in anysupposed transition, ending in any terminus, it is permissible toinstitute a general reasoning in which the final terminus may beincluded. This would seem to indicate that Leibniz consideredtransitions of any kind as continuous. Certainly he held this to be

the case in geometry and for natural processes, where it appears asthe principleNatura non facit saltus. According to Leibniz, it is theLaw of Continuity that allows geometry and the evolving methodsof the infinitesimal calculus to be applicable in physics. ThePrinciple of Continuity also furnished the chief grounds forLeibniz's rejection of material atomism.

The Principle of Continuity also played an important underlyingrole in Leibniz's mathematical work, especially in his developmentof the infinitesimal calculus. Leibniz's essaysNova Methodusof


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1684 andDe Geometri Reconditaof 1686 may be said to representthe official births of the differential and integral calculi,respectively. His approach to the calculus, in which the use ofinfinitesimals, plays a central role, has combinatorial roots,traceable to his early work on derived sequences of numbers. Given

a curve determined by correlated variablesx,y, he wrote dxand dyfor infinitesimal differences, or differentials,between the valuesxandy: and dy/dxfor the ratio of the two, which he then took torepresent the slope of the curve at the corresponding point. Thissuggestive, if highly formal procedure led Leibniz to evolve rulesfor calculating with differentials, which was achieved byappropriate modification of the 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 ais constant, then

da = 0

d(ax) =a dx

d(x+yz) =dx+ dy dz

d(xy) =xdy+ydx

d(x/y) = [xdy+ydx]/y2

d(xp) =pxp1dx, also for fractionalp

But behind the formal beauty of these rulesan early manifestationof what was later to flower into differential algebrathe presence ofinfinitesimals makes itself felt, since Leibniz's definition of tangentemploys both infinitely small distances and the conception of acurve as an infinilateral polygon.

Leibniz conceived of differentials dx, dyas variables ranging over


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differences. This enabled him to take the important step of regardingthe symbol das an operatoracting on variables, so paving the wayfor the iteratedapplicationof d, leading to the higher differentials

ddx= d2x, d3x= dd2x, and in general dn+1x= ddnx. Leibnizsupposed that the first-order differentials dx, dy,. wereincomparably smaller than, or infinitesimal with respect to, thefinite quantities x, y,, and, in general, that an analogous relation

obtained between the (n+1)th-order differentials dn+1xand the nth-

order differentials dnx. He also assumed that the nthpower (dx)nof a

first-order differential was of the same order of magnitude as an nth-

order differential dnx, in the sense that the quotient dnx/(dx)nis a

finite quantity.

For Leibniz the incomparable smallness of infinitesimals derivedfrom their 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

they were governed by the same laws as the latter.

Leibniz's attitude toward infinitesimals and differentials seems tohave 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 consistent forhim to maintain, as he did, that infinitesimal quantities themselvesare no less idealsimply useful fictions, introduced to 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 ofthese was Johann Bernoulli (16671748). A letter of his to Leibnizwritten in 1698 contains the forthright assertion that inasmuch as

the number of terms in nature is infinite, the infinitesimal exists ipsofacto. One of his arguments for the existence of actualinfinitesimals begins with the positing of the infinite sequence 1/2,


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1/3, 1/4,. If there are ten terms, one tenth exists; if a hundred, thena hundredth exists, etc.; and so if, as postulated, the number of termsis infinite, then the infinitesimal exists.

Leibniz's calculus gained a wide audience through the publication in

1696, by Guillaume de L'Hpital (16611704), of the firstexpository book on the subject, theAnalyse des Infiniments PetitsPour L'Intelligence des Lignes Courbes. This is based on twodefinitions:

1. Variable quantities are those that continually increase ordecrease; and constant or standing quantities are those that

continue the same while others vary.2. The infinitely small part whereby a variable quantity iscontinually increased or decreased is called the differential ofthat quantity.

And two postulates:

1. Grant that two quantities, whose difference is an infinitely

small quantity, may be taken (or used) indifferently for eachother: or (what is the same thing) that a quantity, which isincreased or decreased only by an infinitely small quantity,may be considered as remaining the same.

2. Grant that a curve line may be considered as the assemblageof an infinite number of infinitely small right lines: or (what isthe same thing) as a polygon with an infinite number of sides,

each of an infinitely small length, which determine thecurvature of the line by the angles they make with each other.

Following Leibniz, L'Hpital writes dxfor the differential of avariable quantityx. A typical application of these definitions andpostulates is the determination of the differential of a productxy:

d(xy) = (x+ dx)(y+dy) xy=ydx+xdy+ dxdy=ydx+xdy.

Here the last step is justified by Postulate I, since dxdyis infinitely


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small in comparison to ydx+ xdy.

Leibniz's calculus of differentials, resting as it did on somewhatinsecure foundations, soon attracted criticism. The attack mounted

by the Dutch physician Bernard Nieuwentijdt[21](16541718) in

works of 1694-6 is of particular interest, since Nieuwentijdt offeredhis own account of infinitesimals which conflicts with that ofLeibniz and has striking features of its own. Nieuwentijdt postulatesa domain of quantities, or numbers, subject to a ordering relation ofgreater or less. This domain includes the ordinary finite quantities,but it is also presumed to contain infinitesimal and infinite quantitiesa quantity being infinitesimal, or infinite, when it is smaller, or,

respectively, greater, than any arbitrarily given finite quantity. Thewhole domain is governed by a version of the Archimedeanprinciple to the effect that zero is the only quantity incapable ofbeing multiplied sufficiently many times to equal any givenquantity. Infinitesimal quantities may be characterized as quotientsb/mof a finite quantity bby an infinite quantity m. In contrast withLeibniz's differentials, Nieuwentijdt's infinitesimals have the

property that the product of any pair of them vanishes; in particulareach infinitesimal is nilsquare in that its square and all higherpowers are zero. This fact enables Nieuwentijdt to show that, forany curve given by an algebraic equation, the hypotenuse of thedifferential triangle generated by an infinitesimal abscissalincrement ecoincides with the segment of the curve between xand x+ e. That is, a curve truly isan infinilateral polygon.

The major differences between Nieuwentijdt's and Leibniz's calculiof infinitesimals are summed up in the following table:

Leibniz Nieuwentijdt

Infinitesimals arevariables

Infinitesimals areconstants

Higher-order Higher-order


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infinitesimals exist infinitesimals do notexist

Products ofinfinitesimals are not

absolute zeros

Products ofinfinitesimals are

absolute zeros

Infinitesimals can beneglected wheninfinitely small withrespect to otherquantities

(First-order)infinitesimals can neverbe neglected

In responding to Nieuwentijdt's assertion that squares and higherpowers of infinitesimals vanish, Leibniz objected that it is ratherstrange to posit that a segment dxis different from zero and at thesame time that the area of a square with side dxis equal to zero(Mancosu 1996, 161). Yet this oddity may be regarded as aconsequence apparently unremarked by Leibniz himself of

one of his own key principles, namely that curves may beconsidered as infinilateral polygons. Consider, for instance, the

curve y= x2. Given that the curve is an infinilateral polygon, theinfinitesimal straight stretch of the curve between the abscissae 0and dxmust coincide with the tangent to the curve at the origin inthis case, the axis of abscissae between these two points. But

then the point (dx, dx2) must lie on the axis of abscissae, which

means that dx2= 0.

Now Leibniz could retort that that this argument depends cruciallyon the assumption that the portion of the curve between abscissae 0and dxis indeed straight. If this be denied, then of course it does not

follow that dx2= 0. But if one grants, as Leibniz does, that that thereis an infinitesimal straight stretch of the curve (a side, that is, of an

infinilateral polygon coinciding with the curve) between abscissae 0and e, say, which does not reduce to a single point then ecannot be


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equated to 0 and yet the above argument shows that e2= 0. Itfollows that, if curves are infinilateral polygons, then the lengthsof the sides of these latter must be nilsquare infinitesimals.Accordingly, to do full justice to Leibniz's (as well asNieuwentijdt's) conception, twosorts of infinitesimals are required:first, differentials obeying= as laid down by Leibniz the samealgebraic laws as finite quantities; and second the (necessarilysmaller) nilsquare infinitesimals which measure the lengths of thesides of infinilateral polygons. It may be said that Leibnizrecognized the need for the first, but not the seccond type ofinfinitesimal and Nieuwentijdt, vice-versa. It is of interest to notethat Leibnizian infinitesimals (differentials) are realized in

nonstandard 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 algebraic

rules as finite quantities forced Leibniz and the defenders of hisdifferential calculus into treating infinitesimals, in the presence offinite quantities, as ifthey were zeros, so that, for example,x+ dxistreated as if it were the same asx. This was justified on the groundsthat differentials are to be taken as variable, not fixed quantities,decreasing continually until reaching zero. Considered only in themoment of their evanescence, they were accordingly neithersomething nor absolute zeros.

Thus differentials (or infinitesimals) dxwere ascribed variously thefour following properties:

1. dx 02. neitherdx= 0 nordx 0

3. dx2= 0

4. dx0

where stands for indistinguishable from, and 0 stands for


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becomes vanishingly small. Of these properties only the last, inwhich a differential is considered to be a variable quantity tending

to 0, survived the 19thcentury refounding of the calculus in terms of

the limit concept[22].

The leading practitioner of the calculus, indeed the leading

mathematician of the 18thcentury, was Leonhard Euler[23](170783). Philosophically Euler was a thoroughgoing synechist. RejectingLeibnizian monadism, he favoured the Cartesian doctrine that theuniverse 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:that differentials must be zeros, and dy/dxthe quotient 0/0. Since forany number , 0 = 0, Euler maintained that the quotient 0/0

could represent any number whatsoever[24]. For Euler quaformalistthe calculus was essentially a procedure for determining the value of

the expression 0/0 in the manifold situations it arises as the ratio ofevanescent increments.

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 concreteelements of continua, treating them not as atoms or monads in the

strict senseas parts of a continuum they must of necessity bedivisiblebut as being of sufficient minuteness to preserve theirrectilinear shapeunder infinitesimal flow, yet allowing their volumeto undergo infinitesimal change. This idea was to becomefundamental in continuum mechanics.

While Euler treated infinitesimals as formal zeros, that is, as fixedquantities, his contemporary Jean le Rond d'Alembert (171783)took a different view of the matter. Following Newton's lead, heconceived of infinitesimals or differentials in terms of the limit


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concept, which he formulated by the assertion that one varyingquantity is the limit of another if the second can approach the othermore closely than by any given quantity. D'Alembert firmly rejectedthe idea of infinitesimals as fixed quantities, and saw the idea oflimit as supplying the methodological root of the differential

calculus. For d'Alembert the language of infinitesimals ordifferentials was just a convenient shorthand for avoiding thecumbrousness of expression required by the use of the limit concept.

Infinitesimals, differentials, evanescent quantities and the like

coursed through the veins of the calculus throughout the 18th

century. Although nebulouseven logically suspectthese

concepts provided,faute de mieux,the tools for deriving the greatwealth of results the calculus had made possible. And while, with

the notable exception of Euler, many 18thcentury mathematicianswere ill-at-ease with the infinitesimal, they would not risk killingthe goose laying such a wealth of golden mathematical eggs.Accordingly they refrained, in the main, from destructive criticismof the ideas underlying the calculus. Philosophers, however, were

not fettered by such constraints.

The philosopher George Berkeley (16851753), noted both for hissubjective idealist doctrine of esse est percipiand his denial ofgeneral ideas, was a persistent critic of the presuppositionsunderlying the mathematical practice of his day (see 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 mathematicalconcepts by abstraction, that is, by the mental suppression ofextraneous features of perceived objects so as to focus on properties

singled 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


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percepts to serve as signs for all percepts of a similar form.

At first Berkeley poured scorn on those who adhere to the conceptof infinitesimal. maintaining that the use of infinitesimals inderiving mathematical results is illusory, and is in fact eliminable.

But later he came to adopt a more tolerant attitude towardsinfinitesimals, regarding them as useful fictions in somewhat thesame way as did Leibniz.

In The Analystof 1734 Berkeley launched his most sustained andsophisticated critique of infinitesimals and the whole metaphysics of

the calculus. Addressed To an Infidel Mathematician[25], the tract

was written with the avowed purpose of defending theology againstthe scepticism shared by many of the mathematicians and scientistsof the day. Berkeley's defense of religion amounts to the claim thatthe reasoning of mathematicians in respect of the calculus is no lessflawed than that of theologians in respect of the mysteries of thedivine.

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 asevanescent quantities and prime and ultimate ratios Newton has infact violated the law of noncontradiction by first subjecting aquantity to an increment and then setting the increment to 0, that is,denying that an increment had ever been present. As for fluxionsand evanescent increments themselves, Berkeley has this to say:

And what are these fluxions? The velocities ofevanescent increments? And what are these sameevanescent increments? They are neither finitequantities nor quantities infinitely small, nor yetnothing. May we not call them the ghosts of departedquantities?

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


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The opposition between continuity and discreteness plays asignificant role in the philosophical thought of Immanuel Kant(17241804). His mature philosophy, transcendental idealism, restson the division of reality into two realms. The first, thephenomenalrealm, consists of appearances or objects of possible experience,

configured by the forms of sensibility and the epistemic categories.The second, the noumenalrealm, consists of entities of theunderstanding to which no objects of experience can evercorrespond, that is, things-in-themselves.

Regarded as magnitudes, appearances are spatiotemporally extendedand continuous, 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 continuous extensivemagnitudes, but as objects of sensation or perception they are,according to Kant, intensivemagnitudes. By an intensive magnitudeKant means a magnitude possessing a degreeand so capable ofbeing apprehended by the senses: for example brightness ortemperature. Intensive magnitudes are entirely free of the intuitionsof space or time, and can only be presented as unities. But, likeextensive magnitudes, they are continuous. Moreover, appearancesare always presented to the senses as intensive magnitudes.

In the Critique of Pure Reason(1781) Kant brings a new subtlety(and, it must be said, tortuousity) to the analysis of the opposition

between continuity and discreteness. This may be seen in the secondof the celebrated Antinomies in that work, which concerns thequestion of the mereological composition of matter, or extendedsubstance. Is it (a) discrete, that is, consists of simple or indivisibleparts, or (b) continuous, that is, contains parts within parts adinfinitum? Although (a), which Kant calls the Thesisand (b) theAntithesiswould seem to contradict one another, Kant offers proofs

of both assertions. The resulting contradiction may be resolved, heasserts, by observing that while the antinomy relates to the divisionof appearances, the arguments for (a) and (b) implicitly treat matter


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or substance as things-in-themselves. Kant concludes that bothThesis and Antithesis presuppose an inadmissible condition andaccordingly both fall to the ground, inasmuch as the condition,under which alone either of them can be maintained, 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 the sequenceof divisions is finite; the Antithesis, that it is infinite. These cannotbe both be true of the completed(or at least completable) sequenceof divisions which would result from taking matter or substance as a

thing-in-itself.[26]Now since the truth of both assertions has beenshown to follow from that assumption, itmust be false, that is,matter and extended substance are appearances only. And forappearances, Kant maintains, divisions into parts are notcompletable in experience, with the result that such divisions can beconsidered, in a startling phrase, neither finite nor infinite. Itfollows that, for appearances, both Thesis and Antithesis arefalse.

Later in the CritiqueKant enlarges on the issue of divisibility,asserting that, while each part generated by a sequence of divisionsof an intuited whole is given with the whole, the sequence'sincompletability prevents itfrom forming a whole; a fortiorinosuch sequence can be claimed to be actually infinite.

5. The Continuum and the Infinitesimal inthe 19th Century

The rapid development of mathematical analysis in the 18thcenturyhad not concealed the fact that its underlying concepts not onlylacked rigorous definition, but were even (e.g., in the case of

differentials and infinitesimals) of doubtful logical character. Thelack of precision in the notion of continuous functionstill vaguelyunderstood as one which could be represented by a formula and


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whose associated curve could be smoothly drawnhad led todoubts concerning the validity of a number of procedures in whichthat concept figured. For example it was often assumed that everycontinuous function could be expressed as an infinite series by

means of Taylor's theorem. Early in the 19thcentury this and otherassumptions began to be questioned, thereby initiating an inquiryinto what was meant by a function in general and by a continuousfunction in particular.

A pioneer in the matter of clarifying the concept of continuousfunction was the Bohemian priest, philosopher and mathematicianBernard Bolzano (17811848). In hisRein analytischer Beweisof

1817 he defines a (real-valued) functionfto be continuous at a pointxif the differencef(x+ ) f(x) can be made smaller than anypreselected quantity once we are permitted to take was small as weplease. This is essentially the same as the definition of continuity interms of the limit concept given a little later by Cauchy. Bolzanoalso formulated a definition of the derivative of a function free ofthe notion of infinitesimal (see Bolzano [1950]). Bolzano repudiated

Euler's treatment of differentials as formal zeros in expressions suchas dy/dx, suggesting instead that in determining the derivative of afunction, increments x, y, , befinally setto zero. For Bolzanodifferentials have the status of ideal elements, purely formalentities such as points and lines at infinity in projective geometry, or(as Bolzano himself mentions) imaginary numbers, whose use willnever lead to false assertions concerning real quantities.

Although Bolzano anticipated the form that the rigorous formulationof the concepts of the calculus would assume, his work was largelyignored in his lifetime. The cornerstone for the rigorousdevelopment of the calculus was supplied by the ideasessentiallysimilar to Bolzano'sof the great French mathematician Augustin-Louis Cauchy (17891857). In Cauchy's work, as in Bolzano's, acentral role is played by a purely arithmetical concept of limit freed

of all geometric and temporal intuition. Cauchy also formulates thecondition for a sequence of real numbers to converge to a limit, and

states his familiar criterion for convergence[27], namely, that a


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sequence is convergent if and only if s

n+r sncan be made

less in absolute value than any preassigned quantity for all randsufficiently large n. Cauchy proves that this is necessary forconvergence, but as to sufficiency of the condition merely remarkswhen the various conditions are fulfilled, the convergence of the

series is assured. In making this latter assertion he is implicitlyappealing to geometric intuition, since he makes no attempt todefine real numbers, observing only that irrational numbers are to beregarded as the limits of sequences of rational numbers.

Cauchy chose to characterize the continuity of functions in terms ofa rigorized notion of infinitesimal, which he defines in the Cours

d'analyseas a variable quantity [whose value] decreasesindefinitely in such a way as to converge to the limit 0. Here is hisdefinition of continuity. Cauchy's definition of continuity of f(x) inthe neighbourhood of a value aamounts to the condition, in modernnotation, that lim

xaf(x) = f(a). Cauchy defines the derivative f(x)

of a function f(x) in a manner essentially identical to that ofBolzano.

The work of Cauchy (as well as that of Bolzano) represents a crucialstage in the renunciation by mathematiciansadumbrated in thework of d'Alembertof (fixed) infinitesimals and the intuitive ideasof continuity and motion. Certain mathematicians of the day, suchas Poisson and Cournot, who regarded the limit concept as no morethan a circuitous substitute for the use of infinitesimally smallmagnitudeswhich in any case (they claimed) had a real existencefelt that Cauchy's reforms had been carried too far. But traces ofthe traditional ideas did in fact remain in Cauchy's formulations, asevidenced by his use of such expressions as variable quantities,infinitesimal quantities, approach indefinitely, as little as one

wishes and the like[28].

Meanwhile the German mathematician Karl Weierstrass (181597)

was completing the banishment of spatiotemporal intuition, and theinfinitesimal, from the foundations of analysis. To instill completelogical rigour Weierstrass proposed to establish mathematical


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essentially indemonstrable; he ascribes to it, rather, the status of anaxiom by which we attribute to the line its continuity, by which wethink continuity into the line. It is not, Dedekind stresses, necessaryfor spaceto be continuous in this sense, for many of its propertieswould remain the same even if it were discontinuous.

The filling-up of gaps in the rational numbers through the creationof new point-individuals is the key idea underlying Dedekind'sconstruction of the domain of real numbers. He first defines a cuttobe a partition (A1, A2) of the rational numbers such that every

member ofA1is less than every member ofA2. After noting that

each rational number corresponds, in an evident way, to a cut, he

observes that infinitely many cuts fail to be engendered by rationalnumbers. The discontinuity or incompleteness of the domain ofrational numbers consists 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 amental act, and remains quite distinct from its associated cut.

Dedekind goes on to show how the domain of cuts, and thereby theassociated domain of real numbers, can be ordered in such a way asto possess the property of continuity, viz. if the system of all real

numbers divides into two classes 1, 2such that every number a1of the class 1is less than every number a2of the class 2, then

there exists one and only one number by which this separation isproduced.

The most visionary arithmetizer of all was Georg Cantor[32]

(18451918). Cantor's analysis of the continuum in terms of infinitepoint sets led to his theory of transfinite numbers and to the eventualfreeing of the concept of set from its geometric origins as acollection 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 on


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the rational numbers. Following Cauchy, he calls a sequence a1, a2,

, an, of rational numbers afundamental sequenceif there exists

an integerNsuch that, for any positive rational , there exists anintegerNsuch that |a

n+m a

n| < for all mand all n>N. Any

sequence satisfying this condition is said to have a definitelimit b. Dedekind had taken irrational numbers to be mentalobjects associated with cuts, so, analogously, Cantor regards thesedefinite limits, as nothing more thanformal symbolsassociated withfundamental sequences. The domainBof such symbols may beconsidered an enlargement of the domainAof rational numbers.After imposing an arithmetical structure on the domainB, Cantor isemboldened to refer to its elements as (real) numbers. Nevertheless,he still insists that these numbers have no existence except asrepresentatives of fundamental sequences. Cantor then shows thateach point on the line corresponds to a definite element ofB.Conversely, each element ofBshould determine a definite point onthe line. Realizing that the intuitive nature of the linear continuumprecludes a rigorous proof of this property, Cantor simply assumesit as an axiom, just as Dedekind had done in regard to his principle

of continuity.

For Cantor, who began as a number-theorist, and throughout hiscareer cleaved to the discrete, it was numbers, rather than geometricpoints, that possessed objective significance. Indeed theisomorphism between the discrete numerical domainBand thelinear continuum was regarded by Cantor essentially as a device for

facilitating the manipulation 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 infinite inlength, 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 of points on a linear continuumwith a domain of numbers enabled the sizesof point sets to becompared in a definite way, using the well-grounded idea of one-


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one correspondencebetween sets of numbers.

Cantor's investigations into the properties of subsets of the linearcontinuum are presented in six masterly papers published during187984, ber unendliche lineare Punktmannigfaltigkeiten(On

infinite, linear point manifolds). Remarkable in their richness ofideas, these papers provide the first accounts of Cantor'srevolutionary theory of infinite sets and its application to theclassification of subsets of the linear continuum. In the fifth of these

papers, the Grundlagenof 1883,[33]are to be found some ofCantor's most searching observations on the nature of thecontinuum.

Cantor begins his examination of the continuum with a tartsummary of the controversies that have traditionally surrounded thenotion, remarking that the continuum has until recently beenregarded as an essentially unanalyzable concept. It is Cantor'sconcern to develop the concept of the continuum as soberly andbriefly as possible, and only with regard to the mathematicaltheory

of sets. This opens the way, he believes, to the formulation of anexact concept of the continuum. Cantor points out that the idea ofthe continuum has heretofore merely been presupposed bymathematicians concerned with the analysis of continuous functionsand the like, and has not been subjected to any more thoroughinspection.

Repudiating any use of spatial or temporal intuition in an exact

determination 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 , calling each

such an arithmetical pointof Gn.

The distance between two such

points is given by


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Cantor defines an arithmeticalpoint-set in Gnto be any aggregate

of points of the points of the space Gnthat is given in a lawlike

way.

After remarking that he has previously shown that all spaces Gn

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 either

the power of the set of natural numbers or that of (0,1),[34]Cantorturns to the definition of the general concept of a continuum withinGn.. For this he employs the concept of derivativeor derived setof a

point set introduced in a paper of 1872 on trigonometric series.

Cantor had defined the derived set of a point set Pto be the set oflimit pointsof P,where a limit point of Pis a point of Pwithinfinitely many points of Parbitrarily close to it. A point set is

calledperfectif it coincides with its derived set[35]. Cantor observesthat this condition does not suffice to characterize a continuum,since perfect sets can be constructed in the linear continuum whichare dense in no interval, however small: as an example of such a set

he offers the set[36]

consisting of all real numbers in (0,1) whoseternary expansion does not contain a 1.

Accordingly an additional condition is needed to define acontinuum. Cantor supplies this by introducing the concept of aconnectedset. A point set Tis connected in Cantor's sense if for anypair of its points t, tand any arbitrarily small number there is a

finite sequence of points t1

, t2

,, tnof Tfor which the distances

[tt1], [t1t2], [t2t3], , [tnt], are all less than . Cantor now defines a

continuum to be a perfect connected point set.

Cantor has advanced beyond his predecessors in formulating what isin essence a topologicaldefinition of continuum, one that, while still

dependent on metric notions, does not involve an order relation[37].

It is interesting to compare Cantor's definition with the definition ofcontinuum in modern general topology. In a well-known textbook(see Hocking and Young [1961]) on the subject we find a continuum


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defined as a compact connected subset of a topological space. Nowwithin any boundedregion of Euclidean space it can be shown thatCantor's continua coincide with continua in the sense of the moderndefinition. While Cantor lacked the definition of compactness, hisrequirement that continua be complete (which led to his rejecting

as continua such noncompact sets as open intervals or discs) is notfar away from the idea.

Throughout Cantor's mathematical career he maintained anunwavering, even dogmatic opposition to infinitesimals, attackingthe efforts of mathematicians such as du Bois-Reymond and

Veronese[38]to formulate rigorous theories of actual infinitesimals.

As far as Cantor was concerned, the infinitesimal was beyond therealm of the possible; infiinitesimals were no more than castles inthe air, or rather just nonsense, to be classed with circular squaresand square circles. His abhorrence of infinitesimals went so deepas to move him to outright vilification, branding them as Cholera-bacilli of mathematics. Cantor's rejection of infinitesimalsstemmed from his conviction that his own theory of transfinite

ordinal and cardinal numbers exhausted the realm of the numerable,so that no further generalization of the concept of number, inparticular any which embraced infinitesimals, was admissible.

6. Critical Reactions to Arithmetization

Despite the great success of Weierstrass, Dedekind and Cantor in

constructing the continuum from arithmetical materials, a number of

thinkers of the late 19thand early 20thcenturies remained opposed,in varying degrees, to the idea of explicating the continuum conceptentirely in discrete terms. These include the philosophers Brentanoand Peirce and the mathematicians Poincar, Brouwer and Weyl.

In his later years the Austrian philosopher Franz Brentano (1838

1917) became preoccupied with the nature of the continuous (seeBrentano [1988]). In its fundamentals Brentano's account of thecontinuous is akin to Aristotle's. Brentano regards continuity as


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something given in perception, primordial in nature, rather than amathematical construction. He held that the idea of the continuous isabstracted from sensible intuition. Brentano suggests that thecontinuous is brought to appearance by sensible intuition in threephases. First, sensation presents us with objects having parts that

coincide. From such objects the concept of boundaryis abstracted inturn, and then one grasps that these objects actually containcoincident boundaries. Finally one sees that this is all that isrequired in order to have grasped the concept of a continuum.

For Brentano the essential feature of a continuum is its inherentcapacity to engender boundaries, and the fact that such boundaries

can be grasped as coincident. Boundaries themselves possess aquality which Brentano callsplerosis(fullness). Plerosis is themeasure of the number of directions in which the given boundaryactually bounds. Thus, for example, within a temporal continuumthe endpoint of a past episode or the starting point of a future onebounds in a single direction, while the point marking the end of oneepisode and the beginning of another may be said to bound doubly.In the case of a spatial continuum there are numerous additionalpossibilities: here a boundary may bound in all the directions ofwhich it is capable of bounding, or it may bound in only some ofthese directions. In the former case, the boundary is said to exist infull plerosis; in the latter, inpartial plerosis. Brentano believed thatthe concept of plerosis enabled sense to be made of the idea that aboundary possesses parts, even when the boundary lacksdimensions altogether, as in the case of a point. Thus, while the

present or now is, according to Brentano, temporally unextendedand exists only as a boundary between past and future, it stillpossesses two 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, likeAristotle he held that existence in the strict sense meansexistence now, it necessarily followed that existing things exist

only as boundaries of what has existed or of what will exist, or both.

Brentano took a somewhat dim view of the efforts of


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mathematicians to construct the continuum from numbers. Hisattitude varied from rejecting such attempts as inadequate to

according them the status of fictions[39]. This is not surprisinggiven his Aristotelian inclination to take mathematical and physicaltheories to be genuine descriptions of empirical phenomena rather

than idealizations: in his view, if such theories were to be taken asliteral descriptions of experience, they would amount to nothingbetter than misrepresentations.

Brentano's analysis of the continuum centred on itsphenomenological and qualitative aspects, which are by their verynature incapable of reduction to the discrete. Brentano's rejection of

the mathematicians' attempts to construct it in discrete terms is thushardly surprising.

The American philosopher-mathematician Charles Sanders Peirce's

(18391914) view of the continuum[40]was, in a sense,intermediate between that of Brentano and the arithmetizers. LikeBrentano, he held that the cohesiveness of a continuum rules out the

possibility of it being a mere collection of discrete individuals, orpoints, in the usual sense. And even before Brouwer, Peirce seemsto have been aware that a faithful account of the continuum willinvolve questioning the law of excluded middle. Peirce also heldthat any continuum harbours an unboundedly largecollection ofpointsin his colourful terminology, a supermultitudinouscollectionwhat we would today call aproper class. Peircemaintained that if enough points were to be crowded together bycarrying insertion of new points between old to its ultimate limitthey wouldthrough a logicaltransformation of quantity intoqualitylose their individual identity and become fused into a truecontinuum.

Peirce's conception of the number continuum is also notable for thepresence in it of an abundance of infinitesimals, Peirce championed

the retention of the infinitesimal concept in the foundations of thecalculus, both because of what he saw as the efficiency ofinfinitesimal methods, and because he regarded infinitesimals as


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constituting the glue causing points on a continuous line to losetheir individual identity.

The idea of continuity played a central role in the thought of thegreat French mathematician Henri Poincar[41](18541912). While

accepting the arithmetic definition of the continuum, he questionsthe fact that (as with Dedekind and Cantor's formulations) the(irrational) numbers so produced are mere symbols, detached fromtheir origins in intuition. Unlike Cantor, Poincar accepted theinfinitesimal, even if he did not regard all of the concept'smanifestations as useful.

The Dutch mathematician L. E. J. Brouwer (18811966) is bestknown as the founder of the philosophy of (neo)intuitionism(seeBrouwer [1975]; van Dalen [1998]). Brouwer's highly idealist viewson mathematics bore some resemblance to Kant's. For Brouwer,mathematical concepts are admissible only if they are adequatelygrounded in intuition, mathematical theories are significant only ifthey concern entities which are constructed out of something givenimmediately in intuition, and mathematical demonstration is a formof construction in intuition. While admitting that the emergence ofnoneuclidean geometry had discredited Kant's view of space,Brouwer held, in opposition to the logicists (whom he calledformalists) that arithmetic, and so all mathematics, must derivefrom temporal intuition.

Initially Brouwer held without qualification that the continuum is

not constructible 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 fluidity wasachieved by admitting as points, not only fully defined discretenumbers such as 2, , e, and the likewhich have, so to speak,already achieved beingbut also numbers which are in a

perpetual state of becoming in that there the entries in their decimal(or dyadic) expansions are the result of free acts of choice by asubject operating throughout an indefinitely extended time. The


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resulting choice sequencescannot be conceived as finished,completed objects: at any moment only an initial segment is known.In this way Brouwer obtained the mathematical continuum in a waycompatible with his belief in the primordial intuition of timethatis, as an unfinished, indeed unfinishable entity in a perpetual state of

growth, a medium of free development. In this conception, themathematical continuum is indeed constructed, not, however, byinitially shattering, as did Cantor and Dedekind, an intuitivecontinuum into isolated points, but rather by assembling it from acomplex of continually changing overlapping parts.

The mathematical continuum as conceived by Brouwer displays a

number of features that seem bizarre to the classical eye. Forexample, in the Brouwerian continuum the usual law ofcomparability, namely that for any real numbers a, beither a< bora= bor a> b, fails. Even more fundamental is the failure of the lawof excluded middle in the form that for any real numbers a, b, eithera= bor a b. The failure of these seemingly unquestionableprinciples in turn vitiates the proofs of a number of basic results ofclassical analysis, for example the Bolzano-Weierstrass theorem, aswell as the theorems of monotone convergence, intermediate value,

least upper bound, and maximum value for continuous functions[42].

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 ofintuition than does its classical counterpart. Far from being bizarre,

the failure of the law of excluded middle for points in theintuitionistic continuum may be seen as fitting in well with thecharacter of the intuitive continuum.

In 1924 Brouwer showed that every function defined on a closedinterval of his continuum is uniformly continuous. As aconsequence the intuitionistic continuum is indecomposable,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 of the


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continuum has in recent years become the subject of intensivemathematical investigation.

Hermann Weyl (18851955), one of most versatile mathematicians

of the 20thcentury, was preoccupied with the nature of the

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Continuity and Infinitesimals