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



We are all familiar with the idea of *continuity*. To be
 continuous[1]
 is to constitute an unbroken or uninterrupted whole, like the ocean
or the sky. A continuous entity--a *continuum*--has
no "gaps". Opposed to continuity is *discreteness*:
to be
 discrete[2]
 is to be separated, like the scattered pebbles on a beach or the
leaves on a tree. Continuity connotes unity; discreteness,
plurality.



While it is the fundamental nature of a continuum to be
*undivided*, it is nevertheless generally (although not
invariably) held that any continuum admits of repeated or successive
*division without limit*. This means that the process of
dividing it into ever smaller parts will never terminate in an
*indivisible* or an *atom*--that is, a part which,
lacking proper parts itself, cannot be further divided. In a word,
continua are *divisible without limit* or *infinitely
divisible*. The unity of a continuum thus conceals a potentially
infinite plurality. In antiquity this claim met with the objection
that, were one to carry out completely--if only in
imagination--the process of dividing an extended magnitude, such
as a continuous line, then the magnitude would be reduced to a
multitude of atoms--in this case, extensionless points--or
even, possibly, to nothing at all. But then, it was held, no matter
how many such points there may be--even if infinitely
many--they cannot be "reassembled" to form the
original magnitude, for surely a sum of extensionless elements still
lacks
 extension.[3]
 Moreover, if indeed (as seems unavoidable) infinitely many points
remain after the division, then, following Zeno, the magnitude may be
taken to be a (finite) motion, leading to the seemingly absurd
conclusion that infinitely many points can be "touched" in
a finite time.



Such difficulties attended the birth, in the fifth century BCE, of the
school of *atomism*. The founders of this school, Leucippus and
Democritus, claimed that matter, and, more generally, extension, is
not infinitely divisible. Not only would the successive division of
matter ultimately terminate in atoms, that is, in discrete particles
incapable of being further divided, but matter had *in
actuality* to be conceived as being compounded from such atoms. In
attacking infinite divisibility the atomists were at the same time
mounting a 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 nineteenth century paved the way for the idea of
"atomism", as applying to matter, at least, to become
widely familiar: it might well be said, to adapt Sir William
Harcourt's famous observation in respect of the socialists of
his day, "We are all atomists now". Nevertheless, only a
minority of philosophers of the past espoused atomism at a
metaphysical level, a fact which may explain why the analogous
doctrine upholding continuity lacks a familiar name: that which is
unconsciously acknowledged requires no name. Peirce coined the term
*synechism* (from Greek *syneche*,
"continuous") for his own philosophy--a philosophy
permeated by the idea of continuity in its sense of "being
 connected".[4]
 In this article I shall appropriate Peirce's term and use it in
a sense shorn of its Peircean overtones, simply as a contrary to
atomism. I shall also use the term "divisionism" for the
more specific doctrine that continua are infinitely divisible.



Closely associated with the concept of a continuum is that of
 *infinitesimal*.[5]
 An *infinitesimal magnitude* has been somewhat hazily
conceived as a continuum "viewed in the small", an
"ultimate part" of a continuum. In something like the same
sense 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 ultimate
parts. (It is in this sense, for example, that mathematicians of the
seventeenth 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 of
a continuum. Infinitesimal magnitudes, as parts of continua, cannot,
of necessity, be points: they are, in a word,
*nonpunctiform*.



Magnitudes are normally taken as being *extensive* quantities,
like mass or volume, which are defined over extended regions of space.
By contrast, infinitesimal magnitudes have been construed as
*intensive* magnitudes resembling locally defined intensive
quantities such as temperature or density. The effect of
"distributing" or "integrating" an intensive
quantity over such an intensive magnitude is to convert the former
into an infinitesimal extensive quantity: thus temperature is
transformed into infinitesimal heat and density into infinitesimal
mass. When the continuum is the trace of a motion, the associated
infinitesimal/intensive magnitudes have been identified as
*potential* magnitudes--entities which, while not
possessing true magnitude themselves, possess a *tendency* to
generate magnitude through motion, so manifesting
"becoming" as opposed to "being".



An infinitesimal *number* is one which, while not coinciding
with zero, is in some sense smaller than any finite number. This sense
has often been taken to be the failure to satisfy the *Principle of
Archimedes*, which amounts to saying that an infinitesimal number
is one that, no matter how many times it is added to itself, the
result remains less than any finite number. In the engineer's
practical treatment of the differential calculus, an infinitesimal is
a number so small that its square and all higher powers can be
neglected. In the theory of limits the term
"infinitesimal" is sometimes applied to any sequence whose
limit is zero.



The concept of an *indivisible* is closely allied to, but to be
distinguished from, that of an infinitesimal. An indivisible is, by
definition, something that cannot be divided, which is usually
understood to mean that it has no proper parts. Now a partless, or
indivisible entity does not necessarily have to be infinitesimal:
souls, individual consciousnesses, and Leibnizian monads all
supposedly lack parts but are surely not infinitesimal. But these have
in common the feature of being unextended; extended entities such as
lines, surfaces, and volumes prove a much richer source of
"indivisibles". Indeed, if the process of dividing such
entities were to terminate, as the atomists maintained, it would
necessarily issue in indivisibles of a qualitatively different nature.
In the case of a straight line, such indivisibles would, plausibly, be
points; in the case of a circle, straight lines; and in the case of a
cylinder divided by sections parallel to its base, circles. In each
case the indivisible in question is infinitesimal in the sense of
*possessing one fewer dimension than the figure from which it is
generated*. In the sixteenth and seventeenth centuries
indivisibles in this sense were used in the calculation of areas and
volumes of curvilinear figures, a surface or volume being thought of
as a collection, or sum, of linear, or planar indivisibles
respectively.



The concept of infinitesimal was beset by controversy from its
beginnings. The idea makes an early appearance in the mathematics of
the Greek atomist philosopher Democritus c. 450 BCE, only to be
banished c. 350 BCE by Eudoxus in what was to become official
"Euclidean" mathematics. We have noted their reappearance
as indivisibles in the sixteenth and seventeenth centuries: in this
form they were systematically employed by Kepler, Galileo's
student Cavalieri, the Bernoulli clan, and a number of other
mathematicians. In the guise of the beguilingly named
"linelets" and "timelets", infinitesimals
played an essential role in Barrow's "method for finding
tangents by calculation" (1670 [1916: 119]), which appears in
his *Lectiones Geometricae* of 1670. As "evanescent
quantities" infinitesimals were instrumental (although later
abandoned) in Newton's development of the calculus, and, as
"inassignable quantities", in Leibniz's. The Marquis
de L'Hopital, who in 1696 published the first treatise on
the differential calculus (entitled *Analyse des Infiniments Petits
pour l'Intelligence des Lignes Courbes*), invokes the
concept in postulating that "a curved line may be regarded as
being made up of infinitely small straight line segments" (1696:
3 [Postulate II]), and that "one can take as equal two
quantities differing by an infinitely small quantity" (1696: 2
[Postulate 1]).



However useful it may have been in practice, the concept of
infinitesimal could scarcely withstand logical scrutiny. Derided by
Berkeley in the eighteenth century as "Ghosts of departed
Quantities" (1734: 59), in the nineteenth century execrated by
Cantor as "cholera-bacilli" infecting mathematics (1893
[1965: 505], translated by Fisher 1981: 116), and in the twentieth
roundly condemned by Bertrand Russell as "unnecessary,
erroneous, and self-contradictory" (1903: 345), these useful,
but logically dubious entities were believed to have been finally
supplanted in the foundations of analysis by the limit concept which
took rigorous and final form in the latter half of the nineteenth
century. By the beginning of the twentieth century, the concept of
infinitesimal had become, in analysis at least, a virtual
"unconcept".



Nevertheless the proscription of infinitesimals did not succeed in
extirpating them; they were, rather, driven further underground.
Physicists and engineers, for example, never abandoned their use as a
heuristic device for the derivation of correct results in the
application of the calculus to physical problems. Differential
geometers of the stature of Sophus Lie and Elie Cartan relied on
their use in the formulation of concepts which would later be put on a
"rigorous" footing. And, in a technical sense, they lived
on in the algebraists' investigations of non-Archimedean
fields.



A new phase in the long contest between the continuous and the
discrete has opened in the past few decades with the refounding of the
concept of infinitesimal on a solid basis. This has been achieved in
two essentially different ways, the one providing a rigorous
formulation of the idea of infinitesimal *number*, the other of
infinitesimal *magnitude*.



First, in the 1960s Abraham Robinson, using methods of mathematical
logic, created *nonstandard analysis*, an extension of
mathematical analysis embracing both "infinitely large"
and infinitesimal numbers in which the usual laws of the arithmetic of
real numbers continue to hold, an idea which, in essence, goes back to
Leibniz. Here by an infinitely large number is meant one which exceeds
every positive integer; the reciprocal of any one of these is
infinitesimal in the sense that, while being nonzero, it is smaller
than every positive fraction \(1/n\). Much of the usefulness of
nonstandard analysis stems from the fact that within it every
statement of ordinary analysis involving limits has a succinct and
highly intuitive translation into the language of infinitesimals.



The second development in the refounding of the concept of
infinitesimal took place in the 1970s with the emergence of
*synthetic differential geometry*, also known as *smooth
infinitesimal analysis*. Based on the ideas of the American
mathematician F. W. Lawvere, and employing the methods of category
theory, smooth infinitesimal analysis provides an image of the world
in which the continuous is an autonomous notion, not explicable in
terms of the discrete. It provides a rigorous framework for
mathematical analysis in which every function between spaces is smooth
(i.e., differentiable arbitrarily many times, and so in particular
continuous) and in which the use of limits in defining the basic
notions of the calculus is replaced by *nilpotent
infinitesimals*, that is, of quantities so small (but not actually
zero) that some power--most usefully, the square--vanishes.
Smooth infinitesimal analysis embodies a concept of intensive
magnitude in the form of *infinitesimal tangent vectors* to
curves. A tangent vector to a curve at a point \(p\) on it is a short
straight line segment \(l\) passing through the point and pointing
along the curve. In fact we may take \(l\) actually to be an
infinitesimal *part* of the curve. Curves in smooth
infinitesimal analysis are "locally straight" and
accordingly may be conceived as being "composed of"
infinitesimal straight lines in de L'Hopital's
sense, or as being "generated" by an infinitesimal tangent
vector.



The development of nonstandard and smooth infinitesimal analysis has
breathed new life into the concept of infinitesimal,
and--especially 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 derived
from the still more fundamental question concerning the One and the
Many, an antithesis lying at the heart of early Greek thought (Stokes
1971). The Greek debate over the continuous and the discrete seems to
have been ignited by the efforts of Eleatic philosophers such as
Parmenides (c. 515 BCE), and Zeno (c. 460 BCE) to establish their
doctrine of absolute
 monism.[6]
 They were concerned to show that the divisibility of Being into parts
leads to contradiction, so forcing the conclusion that the apparently
diverse world is a static, changeless
 unity.[7]
 In his *Way of Truth* Parmenides asserts that Being is
*homogeneous* and *continuous*. However in asserting the
continuity of Being Parmenides is likely no more than underscoring its
essential unity. Parmenides seems to be claiming that Being is more
than merely continuous--that it is, in fact, a single whole,
indeed an *indivisible* whole. The single Parmenidean existent
is a continuum without parts, at once a continuum and an atom. If
Parmenides was a synechist, his absolute monism precluded his being at
the same time a divisionist.



In support of Parmenides' doctrine of changelessness Zeno
formulated his famous paradoxes of motion. (see entry on
 Zeno's paradoxes)
 The *Dichotomy* and *Achilles* paradoxes both rest
explicitly on the limitless divisibility of space and time.



The doctrine of
 *Atomism*,[8]
 which seems to have arisen as an attempt at escaping the Eleatic
dilemma, was first and foremost a physical theory. It was mounted by
Leucippus (fl. 440 BCE) and Democritus (b. 460-457 BCE) who
maintained that matter was not divisible without limit, but composed
of indivisible, solid, homogeneous, spatially extended corpuscles, all
below the level of visibility.



Atomism was challenged by Aristotle (384-322 BCE), who was the
first to undertake the systematic analysis of continuity and
discreteness. A thoroughgoing synechist, he maintained that physical
reality 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 are
potentially divisible to infinity, in the sense that they may be
divided *anywhere*, though they cannot be divided
*everywhere* at 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 definitions of a number of terms, including
continuity. In effect, Aristotle defines continuity as a
*relation* between entities rather than as an
*attribute* appertaining to a single entity; that is to say, he
does not provide an explicit definition of the concept of
*continuum*. He observes that a single continuous whole can be
brought into existence by "gluing together" two things
which have been brought into contact, which suggests that the
continuity of a whole should derive from the way its *parts*
join up (see *Physics* V, 3). Accordingly for Aristotle
quantities such as lines and planes, space and time are continuous by
virtue of the fact that their constituent parts "join together
at some common boundary" (*Catergories*, VI [MOMM]). By
contrast *no* constituent parts of a discrete quantity can
possess a common boundary.



One of the central theses Aristotle is at pains to defend is the
irreducibility of the continuum to discreteness--that a continuum
cannot be "composed" of indivisibles or atoms, parts which
cannot themselves be further divided.



Aristotle sometimes recognizes *infinite
divisibility*--the property of being divisible into parts
which can themselves be further divided, the process never terminating
in an indivisible--as a consequence of continuity as he
characterizes the notion. But on occasion he takes the property of
infinite divisibility as *defining* continuity. It is this
definition of continuity that figures in Aristotle's
demonstration of what has come to be known as the *isomorphism
thesis*, which asserts that either magnitude, time and motion are
all continuous, or they are all discrete.



The question of whether magnitude is perpetually divisible into
smaller units, or divisible only down to some atomic magnitude leads
to the *dilemma of divisibility* (Miller 1982), a difficulty
that Aristotle necessarily had to face in connection with his analysis
of the continuum. In the dilemma's first, or *nihilistic*
horn, it is argued that, were magnitude everywhere divisible, the
process of carrying out this division completely would reduce a
magnitude to extensionless points, or perhaps even to nothingness. The
second, or *atomistic*, horn starts from the assumption that
magnitude is not everywhere divisible and leads to the equally
unpalatable conclusion (for Aristotle, at least) that indivisible
magnitudes must exist.



As a thoroughgoing materialist,
 Epicurus[12]
 (341-271 BCE) could not accept the notion of potentiality on
which Aristotle's theory of continuity rested, and so was
propelled towards atomism in both its conceptual and physical senses.
Like Leucippus and Democritus, Epicurus felt it necessary to postulate
the existence of physical atoms, but to avoid Aristotle's
strictures he proposed that these should not be themselves
conceptually indivisible, but should *contain* conceptually
indivisible parts. Aristotle had shown that a continuous magnitude
could not be composed of *points*, that is, indivisible units
lacking extension, but he had not shown that an indivisible unit must
necessarily lack extension. Epicurus met Aristotle's argument
that a continuum could not be composed of such indivisibles by taking
indivisibles to be partless units of magnitude possessing
extension.



In opposition to the atomists, the Stoic philosophers Zeno of Cition
(fl. 250 BCE) and Chrysippus (280-206 BCE) upheld the
Aristotelian position that space, time, matter and motion are all
continuous (Sambursky 1954 [1956], 1959; White 1992). And, like
Aristotle, they explicitly rejected any possible existence of void
within the cosmos. The cosmos is pervaded by a continuous invisible
substance which they called *pneuma* (Greek:
"breath"). This pneuma--which was regarded as a kind
of synthesis of air and fire, two of the four basic elements, the
others being earth and water--was conceived as being an elastic
medium through which impulses are transmitted by wave motion. All
physical occurrences were viewed as being linked through tensile
forces in the pneuma, and matter itself was held to derive its
qualities from the "binding" properties of the pneuma it
contains.


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



The scholastic philosophers of Medieval Europe, in thrall to the
massive authority of Aristotle, mostly subscribed in one form or
another to the thesis, argued with great effectiveness by the Master
in Book VI of the *Physics*, that continua cannot be composed
of indivisibles. On the other hand, the avowed infinitude of the Deity
of scholastic theology, which ran counter to Aristotle's thesis
that the infinite existed only in a potential sense, emboldened
certain of the Schoolmen to speculate that the actual infinite might
be found even outside the Godhead, for instance in the assemblage of
points on a continuous line. A few scholars of the time, for example
Henry of Harclay (c. 1275-1317) and Nicholas of Autrecourt (c.
1300-69) chose to follow Epicurus in upholding atomism
reasonable and attempted to circumvent Aristotle's
counterarguments (Pyle 1997).



This incipient atomism met with a determined synechist rebuttal,
initiated by John Duns Scotus (c. 1266-1308). In his analysis of
the problem of "whether an angel can move from place to place
with a continuous motion" (*Opus Oxoniense*, see Grant 1974, SS52)
he offers a pair of purely
geometrical arguments against the composition of a continuum out of
indivisibles. One of these arguments is that if the diagonal and the
side of a square were both composed of points, then not only would the
two be commensurable in violation of Book X of Euclid, they would even
be equal. In the other, two unequal circles are constructed about a
common center, and from the supposition that the larger circle is
composed of points, part of an angle is shown to be equal to the
whole, in violation of Euclid's axiom V.



William of Ockham (c. 1280-1349) 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 principal difficulty presented by the continuous is
the infinite divisibility of space, and in general, that of any
continuum. The treatment of continuity in the first book of his
*Quodlibet* of 1322-7 rests on the idea that between any
two points on a line there is a third--perhaps the first explicit
formulation of the property of *density*--and on the
distinction between a *continuum* "whose parts form a
unity" (QQ, 510) from a *contiguum* of juxtaposed
things. Ockham recognizes that it follows from the property of density
that on arbitrarily small stretches of a line 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 to consist or to be
made up of anything" (QQ, 507), Ockham claims that "no
part of the line is indivisible, nor is any part of a continuum
indivisible" (QQ, 507). While Ockham does not assert that a line
is actually "composed" of points, he had the insight,
startling in its prescience, that a punctate and yet continuous line
becomes 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 the
fourteenth century was mounted by Thomas Bradwardine (c. 1290 -
1349). The purpose of his *Tractatus de Continuo* (c. 1330) was
to "prove that the opinion which maintains continua to be
composed of indivisibles is false" (Murdoch 1957: 54). This was
to be achieved by setting forth a number of "first
principles" concerning the continuum--akin to the axioms
and postulates of Euclid's *Elements*--and then
demonstrating that the further assumption that a continuum is composed
of indivisibles leads to absurdities (Murdoch 1957).



The views on the continuum of Nicolaus Cusanus (1401-64), a
champion of the actual infinite, are of considerable interest. In his
*De Mente Idiotae* of 1450, he asserts that any continuum, be
it geometric, perceptual, or physical, is divisible in two senses, the
one ideal, the other actual. Ideal division "progresses to
infinity"; actual division terminates in atoms after finitely
many steps (see Stones 1928: 447).



Cusanus's realist conception of the actual infinite is reflected
in his quadrature of the circle (Boyer 1939 [1959: 91]). He took the
circle to be an *infinilateral* regular polygon, that is, a
regular polygon with an infinite number of (infinitesimally short)
sides. By dividing it up into a correspondingly infinite number of
triangles, its area, as for any regular polygon, can be computed as
half the product of the apothem (in this case identical with the
radius of the circle), and the perimeter. The idea of considering a
curve as an infinilateral polygon was employed by a number of later
thinkers, for instance, Kepler, Galileo and Leibniz.



The early modern period saw the spread of knowledge in Europe of
ancient geometry, particularly that of Archimedes, and a loosening of
the Aristotelian grip on thinking. In regard to the problem of the
continuum, the focus shifted away from metaphysics to technique, from
the problem of "*what* indivisibles were, or
*whether* they composed magnitudes" to "the new
marvels one could *accomplish* with them" (Murdoch 1957:
325) through the emerging calculus and mathematical analysis. Indeed,
tracing the development of the continuum concept during this period is
tantamount to charting the rise of the calculus. Traditionally,
geometry is the branch of mathematics concerned with the continuous
and arithmetic (or algebra) with the discrete. The infinitesimal
calculus that took form in the sixteenth and seventeenth centuries,
which had as its primary subject matter *continuous variation*,
may be seen as a kind of synthesis of the continuous and the discrete,
with infinitesimals bridging the gap between the two. The widespread
use of indivisibles and infinitesimals in the analysis of continuous
variation by the mathematicians of the time testifies to the
affirmation of a kind of mathematical atomism which, while logically
questionable, made possible the spectacular mathematical advances with
which the calculus is associated. It was thus to be the infinitesimal,
rather than the infinite, that served as the mathematical stepping
stone between the continuous and the discrete.



Johann Kepler (1571-1630) made abundant use of infinitesimals in
his calculations. In his *Nova Stereometria* of 1615, a work
actually written as an aid in calculating the volumes of wine casks,
he regards curves as being infinilateral polygons, and solid bodies as
being 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's
customary use of infinitesimals of the same dimension as the figures
they constitute; but he also used indivisibles on occasion. He spoke,
for example, of a cone as being composed of circles, and in
his *Astronomia Nova* of 1609, the work in which he states his
famous laws of planetary motion, he takes the area 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, which was
later to become a reigning principle in geometry, of *continuous
change of a mathematical object*, in this case, of a geometric
figure. In his *Astronomiae pars Optica* of 1604 Kepler notes
that all the conic sections are continuously derivable from one
another both through focal motion and by variation of the angle with
the cone of the cutting plane.



Galileo Galilei (1564-1642) advocated a form of mathematical
atomism in which the influence of both the Democritean atomists and
the Aristotelian scholastics can be discerned. This emerges when one
turns to the First Day of Galileo's *Dialogues Concerning Two
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 of them. Salviati/Galileo recognizes that this infinity of
indivisibles will never be produced by successive subdivision, but
claims to have a method for generating it all at once, thereby
removing it from the realm of the potential into actual realization:
this "method for separating and resolving the whole of infinity
at a single stroke" (1638 [NE: 92-93; 1914: 48]) turns out
simply to the act of bending a straight line into a circle. Here
Galileo finds an ingenious "metaphysical" application of
the idea of regarding the circle as an infinilateral polygon. When the
straight line has been bent into a circle Galileo seems to take it
that that the line has thereby been rendered into indivisible parts,
that is, points. But if one considers that these parts are the sides
of the infinilateral polygon, they are better characterized not as
indivisible points, but rather as unbendable straight lines, each at
once part of and tangent to the
 circle.[15]
 Galileo does not mention this possibility, but nevertheless it does
not seem fanciful to detect the germ 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
(1598-1647) who refined the use of indivisibles into a reliable
mathematical tool (Boyer 1939 [1959]); indeed the "method of
indivisibles" remains associated with his name to the present
day. Cavalieri nowhere explains precisely what he understands by the
word "indivisible", but it is apparent that he conceived
of a surface as composed of a multitude of equispaced parallel lines
and of a volume as composed of equispaced parallel planes, these being
termed 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 as being actually infinite, he avoided following Galileo into
ensnarement in the coils of infinity by grasping that, for the
"method of indivisibles" to work, the precise
"number" of indivisibles involved *did not matter*.
Indeed, the essence of Cavalieri's method was the establishing
of a correspondence between the indivisibles of two
"similar" configurations, and in the cases Cavalieri
considers it is evident that the correspondence is suggested on solely
geometric grounds, rendering it quite independent of number. The very
statement of Cavalieri's principle embodies this idea: if plane
figures are included between a pair of parallel lines, and if their
intercepts on any line parallel to the including lines are in a fixed
ratio, then the areas of the figures are in the same ratio. (An
analogous principle holds for solids.) Cavalieri's method is in
essence that of reduction of dimension: solids are reduced to planes
with comparable 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
would be to points, and no meaning can be attached to the
"ratio" of two points. For rectification a curve has, it
was later realized, to be regarded as the sum, not of indivisibles,
that is, points, but rather of infinitesimal straight lines, its
microsegments.



Rene Descartes (1596-1650) employed infinitesimalist
techniques, including Cavalieri's method of indivisibles, in his
mathematical work. But he avoided the use of infinitesimals in the
determination of tangents to curves, instead developing purely
algebraic methods for the purpose. Some of his sharpest criticism was
directed at those mathematicians, such as Fermat, who used
infinitesimals in the construction of tangents.



As a philosopher Descartes may be broadly characterized as a
synechist. His philosophical system rests on two fundamental
principles: the celebrated Cartesian dualism--the division
between mind and matter--and the less familiar identification of
matter and spatial extension. In the *Meditations* Descartes
distinguishes mind and matter on the grounds that the corporeal, being
spatially extended, is divisible, while the mental is partless. The
identification of matter and spatial extension has the consequence
that matter is continuous and divisible without limit. Since extension
is the sole essential property of matter and, conversely, matter
always accompanies extension, matter must be ubiquitous.
Descartes' space is accordingly, as it was for the Stoics, a
plenum pervaded by a continuous medium.



The concept of infinitesimal had arisen with problems of a geometric
character and infinitesimals were originally conceived as belonging
solely to the realm of continuous magnitude as opposed to that of
discrete number. But from the algebra and analytic geometry of the
sixteenth and seventeenth centuries there issued the concept of
*infinitesimal number*. The idea first appears in the work of
Pierre de Fermat (1601-65) on the determination of maximum and
minimum (extreme) values, published in 1638 (Boyer 1939 [1959: 155]).



Fermat's treatment of maxima and minima contains the germ of the
fertile technique of "infinitesimal variation", that is,
the investigation of the behavior of a function by subjecting its
variables to small changes. Fermat applied this method in determining
tangents 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 the
reciprocal relation between the problem of quadrature and that of
finding tangents to curves--in modern parlance, between
integration and differentiation. In his *Lectiones Geometricae*
of 1670, Barrow observes, in essence, that if the quadrature of a
curve \(y = f(x)\) is known, with the area up to \(x\) given by
\(F(x)\), then the subtangent to the curve \(y = F(x)\) is measured by
the ratio of its ordinate 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 of
arguments against mathematical atomism, the strongest of which is that
atomism contradicts many of the basic propositions of Euclidean
geometry.



Barrow conceived of continuous magnitudes as being generated by
motions, and so necessarily dependent on time, a view that seems to
have had a strong influence on the thinking of his illustrious pupil
Isaac
 Newton[18]
 (1642-1727). Newton's meditations during the plague year
1665-66 issued in the invention of what he called the
"Calculus of Fluxions", the principles and methods of
which were presented in three tracts published many years after they
were written: *De analysi per aequationes numero terminorum
infinitas*; *Methodus fluxionum et serierum infinitarum*;
and *De quadratura curvarum*. Newton's approach to the
calculus rests, even more firmly than did Barrow's, on the
conception of continua as being generated by motion.



But Newton's exploitation of the kinematic conception went much
deeper than had Barrow's. In *De Analysi*, for example,
Newton introduces a notation for the "momentary increment"
(*moment*)--evidently meant to represent a moment or
instant of time--of the abscissa or the area of a curve, with the
abscissa itself representing time. This
"moment"--effectively the same as the infinitesimal
quantities previously introduced by Fermat and Barrow--Newton
denotes by \(o\) in the case of the abscissa, and by \(ov\) in the
case of the area. From the fact that Newton uses the letter \(v\) for
the ordinate, it may be inferred that Newton is thinking of the curve
as being a graph of velocity against time. By considering the moving
line, or ordinate, as the moment of the area Newton established the
generality of and reciprocal relationship between the operations of
differentiation and integration, a fact that Barrow had grasped but
had not put to systematic use. Before Newton, quadrature or
integration had rested ultimately "on some process through which
elemental triangles or rectangles were added together" (Baron
1969 [1987: 268]), that is, on the method of indivisibles.
Newton's explicit treatment of integration as inverse
differentiation was the key to the integral calculus.



In the *Methodus fluxionum* Newton makes explicit his
conception of variable quantities as generated by motions, and
introduces his characteristic notation. He calls the quantity
generated by a motion a *fluent*, and its rate of generation a
*fluxion*. The fluxion of a fluent \(x\) is denoted by
\(\dot{x}\), and its moment, or "infinitely small increment
accruing in an infinitely short time \(o\)", by \(\dot{x}
o\). The problem of determining a tangent to a curve is transformed
into the problem of finding the relationship between the fluxions
\(\dot{x}\) and \(\dot{z}\) when presented with an equation
representing the relationship between the fluents \(x\) and \(z\). (A
quadrature is the inverse problem, that of determining the fluents
when the fluxions are given.) Thus, for example, in the case of the
fluent \(z = x^n\), Newton first forms \(\dot{z} + \dot{z} o =
(\dot{x} + \dot{x} o)^n\), expands the right-hand side using the
binomial 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 of
infinitesimals in his calculus, and dissatisfied with the dubious
procedure of "neglecting" them. In the preface to
the *De quadratura curvarum* he remarks that there is no
necessity to introduce into the method of fluxions any argument about
infinitely small quantities. In their place he proposes to employ what
he calls the method of *prime and ultimate ratio*. This method,
in many respects an anticipation of the limit concept, receives a
number of allusions in Newton's celebrated *Principia
mathematica philosophiae naturalis* of 1687.



Newton developed three approaches for his calculus, all of which he
regarded as leading to equivalent results, but which varied in their
degree of rigor. The first employed infinitesimal quantities which,
while not finite, are at the same time not exactly zero. Finding that
these eluded precise formulation, Newton focussed instead on their
ratio, which is in general a finite number. If this ratio is known,
the infinitesimal quantities forming it may be replaced by any
suitable finite magnitudes--such as velocities or
fluxions--having the same ratio. This is the method of fluxions.
Recognizing that this method itself required a foundation, Newton
supplied it with one in the form of the doctrine of prime and ultimate
ratios, 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 the
composition of the continuum--the "labyrinth of the
continuum", as he called it. Indeed we have it on his own
testimony that his philosophical
system--*monadism*--grew from his struggle with the
problem of just how, or whether, a continuum can be built from
indivisible elements. Leibniz asked himself: if we grant that each
real entity is either a simple unity or a multiplicity, and that a
multiplicity is necessarily an aggregation of unities, then under what
head should a geometric 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 a multiplicity, and accordingly an aggregation of
unities. But of what sort of unities? Seemingly, the only candidates
for geometric unities are points, but points are no more than
extremities of the extended, and in any case, as Leibniz knew, solid
arguments going back to Aristotle 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 real entities* at all; as "wholes preceding
their parts" they have instead a purely ideal character. In this
way he freed the continuum from the requirement that, as something
intelligible, it must itself be simple or a compound of simples.



Leibniz held that space and time, as continua, are ideal, and anything
real, in particular matter, is discrete, compounded of simple unit
substances he termed *monads*.



Among the best known of Leibniz's doctrines is the
*Principle* or *Law of Continuity*. In a somewhat
nebulous form this principle had been employed on occasion by a number
of Leibniz's predecessors, including Cusanus and Kepler, but it
was Leibniz who gave to the principle 



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



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



> 
> 
> in any supposed transition, ending in any terminus, it is permissible
> to institute a general reasoning in which the final terminus may be
> included. (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 held
this to be the case in geometry and for natural processes, where it
appears as the principle *Natura non facit saltus*. According
to Leibniz, it is the Law of Continuity that allows geometry and the
evolving methods of the infinitesimal calculus to be applicable in
physics. The Principle of Continuity also furnished the chief grounds
for Leibniz's rejection of material atomism.



The Principle of Continuity also played an important underlying role
in Leibniz's mathematical work, especially in his development of
the infinitesimal calculus. Leibniz's essays *Nova
Methodus* of 1684 and *De Geometria Recondita* of 1686 may
be said to represent the official births of the differential and
integral calculi, respectively. His approach to the calculus, in which
the use of infinitesimals, plays a central role, has combinatorial
roots, 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, or
*differentials*, between the values \(x\) and \(y\): and
\(\Dy/\Dx\) for the ratio of the two, which he then took to represent
the slope of the curve at the corresponding point. This suggestive, if
highly formal procedure led Leibniz to evolve rules for calculating
with differentials, which was achieved by appropriate modification of
the rules of calculation for ordinary numbers.



Although the use of infinitesimals was instrumental in Leibniz's
approach to the calculus, in 1684 he introduced the concept of
differential without mentioning infinitely small quantities, almost
certainly in order to avoid foundational difficulties. He states
without 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 early
manifestation of what was later to flower into differential
algebra--the presence of infinitesimals makes itself felt, since
Leibniz's definition of tangent employs both infinitely small
distances and the conception of a curve as an infinilateral
polygon.



Leibniz conceived of differentials \(\Dx , \Dy\) as variables ranging
over differences. This enabled him to take the important step of
regarding the symbol \(\D\) as an *operator* acting on
variables, so paving the way for the *iterated 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 infinitesimal
with respect to, the finite quantities \(x\), \(y\),..., and, in
general, 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 assumed
that the \(n^{\textrm{th}}\) power \((\Dx )^n\) of a first-order
differential was of the same order of magnitude as an
\(n^{\textrm{th}}\)-order differential \(\D^{n} x\), in the sense that
the quotient \(\D^{n} x/(\Dx )^n\) is a finite quantity.



For Leibniz the incomparable smallness of infinitesimals derived from
their failure to satisfy Archimedes' principle; and quantities
differing only by an infinitesimal were to be considered equal. But
while infinitesimals were conceived by Leibniz to be incomparably
smaller 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
to have been that they furnished the elements from which to fashion a
formal grammar, an algebra, of the continuous. Since he regarded
continua as purely ideal entities, it was then perfectly consistent
for him to maintain, as he did, that infinitesimal quantities
themselves are no less ideal--simply 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 his
followers did not hesitate to do so. Among the most prominent of these
was Johann Bernoulli (1667-1748). A letter of his to Leibniz
written in 1698 contains the forthright assertion that "inasmuch
as the number of terms in nature is infinite, the infinitesimal exists
*ipso facto*" (Boyer 1939 [1959: 239], quoting Leibniz,
*Mathematische Schriften*, III (Part 2), 555). One of his
arguments for the existence of actual infinitesimals begins with the
positing of the infinite sequence 1/2, 1/3, 1/4,.... If there are
ten terms, one tenth exists; if a hundred, then a hundredth exists,
etc.; and so if, as postulated, the number of terms is infinite, then
the infinitesimal exists.



Leibniz's calculus gained a wide audience through the
publication in 1696, by Guillaume de L'Hopital
(1661-1704), of the first expository book on the subject, the
*Analyse des Infiniments Petits Pour L'Intelligence des
Lignes Courbes*. This is based on two definitions:


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



And two postulates:


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



Following Leibniz, L'Hopital writes \(\Dx\) for the
differential of a variable quantity \(x\). A typical application of
these definitions and postulates is the determination of the
differential of a product \(xy\):

 

\[
\begin{align}
\D (xy) & = (x + \Dx )(y +\Dy ) - xy \\
& = y \Dx + x \Dy + \Dx \Dy \\
& = y \Dx + x \Dy .\\
\end{align}
\]


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



Leibniz's calculus of differentials, resting as it did on
somewhat insecure foundations, soon attracted criticism. The attack
mounted 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 which
conflicts with that of Leibniz and has striking features of its own
(see Mancosu 1996: 158-160). Nieuwentijdt postulates a domain
of quantities, or numbers, subject to a ordering relation of greater
or less. This domain includes the ordinary finite quantities, but it
is also presumed to contain infinitesimal and infinite
quantities--a quantity being infinitesimal, or infinite, when it
is smaller, or, respectively, greater, than any arbitrarily given
finite quantity. The whole domain is governed by a version of the
Archimedean principle to the effect that zero is the only quantity
incapable of being multiplied sufficiently many times to equal any
given quantity. Infinitesimal quantities may be characterized as
quotients \(b/m\) of a finite quantity \(b\) by an infinite quantity
\(m\). In contrast with Leibniz's differentials,
Nieuwentijdt's infinitesimals have the property that the product
of any pair of them vanishes; in particular each infinitesimal is
*nilsquare*, 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 the
differential triangle generated by an infinitesimal abscissal
increment \(e\) coincides with the segment of the curve between \(x\)
and \(x + e\). That is, a curve truly *is* an infinilateral
polygon.



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




| **Leibniz** | **Nieuwentijdt** |
| --- | --- |
| Infinitesimals are variables | Infinitesimals are constants |
| Higher-order infinitesimals exist | Higher-order infinitesimals do not exist |
| Products of infinitesimals are not absolute zeros | Products of infinitesimals are absolute zeros |
| Infinitesimals can be neglected when infinitely small with
respect to other quantities | (First-order) infinitesimals can never be neglected |



In responding to Nieuwentijdt's assertion that squares and
higher powers of infinitesimals vanish, Leibniz objected that it is
rather strange to posit that a segment \(\Dx\) is different from zero
and at the same time that the area of a square with side \(\Dx\) is
equal to zero (Mancosu 1996: 161). Yet this oddity may be regarded as
a consequence--apparently unremarked by Leibniz himself--of
one of his own key principles, namely that curves may be considered as
infinilateral polygons. Consider, for instance, the curve \(y = x^2\).
Given that the curve is an infinilateral polygon, the infinitesimal
straight stretch of the curve between the abscissae 0 and \(\Dx\) must
coincide with the tangent to the curve at the origin--in this
case, the axis of abscissae--between these two points. But then
the point \((\Dx , \Dx ^2)\) must lie on the axis of abscissae, which
means that \(\Dx ^2 = 0\).



Now Leibniz could retort that that this argument depends crucially on
the assumption that the portion of the curve between abscissae 0 and
\(\Dx\) is indeed straight. If this be denied, then of course it does
not follow that \(\Dx ^2 = 0\). But if one grants, as Leibniz does,
that that there is an infinitesimal straight stretch of the curve (a
side, that is, of an infinilateral polygon coinciding with the curve)
between abscissae 0 and \(e\), say, which does not reduce to a single
point then \(e\) cannot be equated to 0 and yet the above argument
shows that \(e^2 = 0\). It follows that, if curves are infinilateral
polygons, then the "lengths" of the sides of these latter
must be nilsquare infinitesimals. Accordingly, to do full justice to
Leibniz's (as well as Nieuwentijdt's) conception,
*two* sorts of infinitesimals are required: first,
"differentials" obeying--as laid down by
Leibniz--the same algebraic laws as finite quantities; and second
the (necessarily smaller) nilsquare infinitesimals which measure the
lengths of the sides of infinilateral polygons. It may be said that
Leibniz recognized the need for the first, but not the second type of
infinitesimal and Nieuwentijdt, vice-versa. It is of interest to note
that Leibnizian infinitesimals (differentials) are realized in
nonstandard analysis, and nilsquare infinitesimals in smooth
infinitesimal analysis (for both types of analysis see below). In fact
it has been shown to be possible to combine the two approaches, so
creating an analytic framework realizing both Leibniz's and
Nieuwentijdt's conceptions of infinitesimal.



The insistence that infinitesimals obey precisely the same algebraic
rules as finite quantities forced Leibniz and the defenders of his
differential calculus into treating infinitesimals, in the presence of
finite quantities, *as if* they were zeros, so that, for
example, \(x + \Dx\) is treated as if it were the same as \(x\). This
was justified on the grounds that differentials are to be taken as
variable, not fixed quantities, decreasing continually until reaching
zero. Considered only in the "moment of their
evanescence", they were accordingly neither something nor
absolute zeros.



Thus differentials (or infinitesimals) \(\Dx\) were ascribed variously
the 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 "indistinguishable
from", and "\(\rightarrow 0\)" stands for
"becomes vanishingly small". Of these properties only the
last, in which a differential is considered to be a variable quantity
tending to 0, survived the nineteenth century refounding of the
calculus in terms of the limit
 concept.[21]



The leading practitioner of the calculus, indeed the leading
mathematician of the eighteenth century, was Leonhard
 Euler[22]
 (1707-83). Philosophically Euler was a thoroughgoing synechist.
Rejecting Leibnizian monadism, he favored the Cartesian doctrine that
the universe is filled with a continuous ethereal fluid and upheld the
wave theory of light over the corpuscular theory propounded by
Newton.



Euler rejected the concept of infinitesimal in its sense as a quantity
less than any assignable magnitude and yet unequal to 0, arguing: that
differentials must be zeros, and \(\Dy/\Dx\) the quotient \(0/0\).
Since for any number \(\alpha\), \(\alpha \cdot 0 = 0\), Euler
maintained that the quotient \(0/0\) could represent any number
 whatsoever.[23]
 For Euler *qua* formalist the calculus was essentially a
procedure for determining the value of the expression \(0/0\) in the
manifold situations it arises as the ratio of evanescent
increments.



But in the mathematical analysis of natural phenomena, Euler, along
with a number of his contemporaries, did employ what amount to
infinitesimals in the form of minute, but more or less concrete
"elements" of continua, treating them not as atoms or
monads in the strict sense--as parts of a continuum they must of
necessity be divisible--but as being of sufficient minuteness to
preserve their rectilinear *shape* under infinitesimal flow,
yet allowing their *volume* to undergo infinitesimal change.
This idea was to become fundamental in continuum mechanics.



While Euler treated infinitesimals as formal zeros, that is, as fixed
quantities, his contemporary Jean le Rond d'Alembert
(1717-83) took a different view of the matter. Following
Newton's lead, he conceived of infinitesimals or differentials
in terms of the limit concept, which he formulated by the assertion
that one varying quantity is the limit of another if the second can
approach the other more closely than by any given quantity.
D'Alembert firmly rejected the idea of infinitesimals as fixed
quantities, and saw the idea of limit as supplying the methodological
root of the differential calculus. For d'Alembert the language
of infinitesimals or differentials was just a convenient shorthand for
avoiding the cumbrousness 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 eighteenth
century. Although nebulous--even logically suspect--these
concepts provided, *faute de mieux*, the tools for deriving the
great wealth of results the calculus had made possible. And while,
with the notable exception of Euler, many eighteenth century
mathematicians were ill-at-ease with the infinitesimal, they would not
risk killing the goose laying such a wealth of golden mathematical
eggs. Accordingly they refrained, in the main, from destructive
criticism of the ideas underlying the calculus. Philosophers, however,
were not fettered by such constraints.



The philosopher George Berkeley (1685-1753), noted both for his
subjective idealist doctrine of *esse est percipi* and his
denial of general ideas, was a persistent critic of the
presuppositions underlying the mathematical practice of his day
(Jesseph 1993). His most celebrated broadsides were directed at the
calculus, but in fact his conflict with the mathematicians went
deeper. For his denial of the existence of abstract ideas of any kind
went in direct opposition with the abstractionist account of
mathematical concepts held by the majority of mathematicians and
philosophers of the day. The central tenet of this doctrine, which
goes back to Aristotle, is that the mind creates mathematical concepts
by *abstraction*, that is, by the mental suppression of
extraneous features of perceived objects so as to focus on properties
singled out for attention. Berkeley rejected this, asserting that
mathematics as a science is ultimately concerned with objects of
sense, its admitted generality stemming from the capacity of percepts
to serve as signs for all percepts of a similar form.



At first Berkeley poured scorn on those who adhere to the concept of
infinitesimal. maintaining that the use of infinitesimals in deriving
mathematical results is illusory, and is in fact eliminable. But later
he came to adopt a more tolerant attitude towards infinitesimals,
regarding them as useful fictions in somewhat the same way as did
Leibniz.



In *The Analyst* of 1734 Berkeley launched his most sustained
and sophisticated critique of infinitesimals and the whole metaphysics
of the calculus. Addressed *To an Infidel
 Mathematician*,[24]
 the tract was written with the avowed purpose of defending theology
against the scepticism shared by many of the mathematicians and
scientists of the day. Berkeley's defense of religion amounts to
the claim that the reasoning of mathematicians in respect of the
calculus is no less flawed than that of theologians in respect of the
mysteries of the divine.



Berkeley's arguments are directed chiefly against the Newtonian
fluxional calculus. Typical of his objections is that in attempting to
avoid infinitesimals by the employment of such devices as evanescent
quantities and prime and ultimate ratios Newton has in fact violated
the law of noncontradiction by first subjecting a quantity to an
increment and then setting the increment to 0, that is, denying that
an increment had ever been present. As for fluxions and evanescent
increments 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 finite
> quantities nor quantities infinitely small, nor yet nothing. May we
> not call them the ghosts of departed quantities? (1734: 59)
> 
> 
> 



Nor did the Leibnizian method of differentials escape Berkeley's
strictures.



The opposition between continuity and discreteness plays a significant
role in the philosophical thought of Immanuel Kant (1724-1804).
His mature philosophy, *transcendental idealism*, rests on the
division of reality into two realms. The first, the
*phenomenal* realm, consists of appearances or objects of
possible experience, configured by the forms of sensibility and the
epistemic categories. The second, the *noumenal* realm,
consists of "entities of the understanding to which no objects
of experience can ever correspond" (Korner 1955: 94), that
is, things-in-themselves.



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



As objects of knowledge, appearances are continuous *extensive*
magnitudes, but as objects of sensation or perception they are,
according to Kant, *intensive* magnitudes. By an intensive
magnitude Kant means a magnitude possessing a *degree* and so
capable of being apprehended by the senses: for example brightness or
temperature. Intensive magnitudes are entirely free of the intuitions
of space or time, and "can only be presented as
unities". But, like extensive magnitudes, they are
continuous. Moreover, appearances are 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 second of the celebrated Antinomies in that work, which concerns
the question of the mereological composition of matter, or extended
substance. Is it (a) discrete, that is, consists of simple or
indivisible parts, or (b) continuous, that is, contains parts within
parts *ad infinitum*? Although (a), which Kant calls the
*Thesis* and (b) the *Antithesis* would seem to
contradict one another, Kant offers proofs of both assertions. The
resulting contradiction may be resolved, he asserts, by observing that
while the antinomy "relates to the division of
appearances", the arguments for (a) and (b) implicitly treat
matter or substance as things-in-themselves. Kant concludes that both
Thesis and Antithesis "presuppose an inadmissible
condition" and accordingly "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 of
matter as a thing-in-itself, which in turn leads to the mistake of
taking the division of matter into parts to subsist independently of
the act of dividing. In that case, the Thesis implies that the
sequence of divisions is finite; the Antithesis, that it is infinite.
These cannot both be true of the *completed* (or at least
completable) sequence of divisions which would result from taking
matter or substance as a
 thing-in-itself.[25]
 Now since the truth of both assertions has been shown to follow from
that assumption, *it* must be false, that is, matter and
extended substance are appearances only. And for appearances, Kant
maintains, divisions into parts are not completable in experience,
with the result that such divisions can be considered, in a startling
phrase, "neither finite nor infinite". It follows that,
for appearances, both Thesis and Antithesis are *false*.



Later in the *Critique* Kant enlarges on the issue of
divisibility, asserting that, while each part generated by a sequence
of divisions of an intuited whole is given with the whole, the
sequence's incompletability prevents *it* from forming a
whole; *a fortiori* no such sequence can be claimed to be
actually infinite.


## 5. The Continuum and the Infinitesimal in the Nineteenth Century



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



A pioneer in the matter of clarifying the concept of continuous
function was the Bohemian priest, philosopher and mathematician
Bernard Bolzano (1781-1848). In his *Rein analytischer
Beweis* of 1817 he defines a (real-valued) function \(f\) to be
continuous at a point \(x\) if the difference \(f(x + \omega) - f(x)\)
can be made smaller than any preselected quantity once we are
permitted to take \(w\) as small as we please. This is essentially the
same as the definition of continuity in terms of the limit concept
given a little later by Cauchy. Bolzano also formulated a definition
of the derivative of a function free of the notion of infinitesimal
(Bolzano 1851 [1950]). Bolzano repudiated Euler's treatment of
differentials as formal zeros in expressions such as \(\Dy/\Dx\),
suggesting instead that in determining the derivative of a function,
increments \(\Delta x,\) \(\Delta y,\) ..., be *finally
set* to zero. For Bolzano differentials have the status of
*ideal elements*, purely formal entities such as points and
lines at infinity in projective geometry, or (as Bolzano himself
mentions) imaginary numbers, whose use will never lead to false
assertions concerning *real* quantities.



Although Bolzano anticipated the form that the rigorous formulation of
the concepts of the calculus would assume, his work was largely
ignored in his lifetime. The cornerstone for the rigorous development
of the calculus was supplied by the ideas--essentially similar to
Bolzano's--of the great French mathematician Augustin-Louis
Cauchy (1789-1857). In Cauchy's work, as in
Bolzano's, a central role is played by a purely arithmetical
concept of limit freed of all geometric and temporal intuition. Cauchy
also formulates the condition for a sequence of real numbers to
converge to a limit, and states his familiar criterion for
 convergence,[26]
 namely, that a sequence \(\langle s\_n\rangle\) is convergent if and
only if \(s\_{n+r} - s\_n\) can be made less in absolute value than any
preassigned quantity for all \(r\) and sufficiently large \(n\).
Cauchy proves that this is necessary for convergence, but as to
sufficiency of the condition merely remarks "when the various
conditions are fulfilled, the convergence of the series is
assured" (Kline 1972: 951). In making this latter assertion he
is implicitly appealing to geometric intuition, since he makes no
attempt to define real numbers, observing only that irrational numbers
are to be regarded as the limits of sequences of rational numbers.



Cauchy chose to characterize the continuity of functions in terms of a
rigorized notion of infinitesimal, which he defines in the *Cours
d'analyse* as "a variable quantity [whose value]
decreases indefinitely in such a way as to converge to the limit
0" (Kline 1972: 951). Here is his definition of
continuity. Cauchy's definition of continuity of \(f(x)\) in the
neighborhood of a value \(a\) amounts to the condition, in modern
notation, that \(\lim\_{x\rightarrow a}f(x) = f(a)\). Cauchy defines
the derivative \(f '(x)\) of a function \(f(x)\) in a manner
essentially identical to that of Bolzano.



The work of Cauchy (as well as that of Bolzano) represents a crucial
stage in the renunciation by mathematicians--adumbrated in the
work of d'Alembert--of (fixed) infinitesimals and the
intuitive ideas of continuity and motion. Certain mathematicians of
the day, such as Poisson and Cournot, who regarded the limit concept
as no more than a circuitous substitute for the use of infinitesimally
small magnitudes--which in any case (they claimed) had a real
existence--felt that Cauchy's reforms had been carried too
far. But traces of the traditional ideas did in fact remain in
Cauchy's formulations, as evidenced by his use of such
expressions as "variable quantities", "infinitesimal
quantities", "approach indefinitely", "as
little 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 the
infinitesimal, from the foundations of analysis. To instill complete
logical rigor Weierstrass proposed to establish mathematical analysis
on 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 mathematical
atomism. In pursuit of this goal Weierstrass had first to formulate a
rigorous "arithmetical" definition of real number. He did
this by defining a (positive) real number to be a countable set of
positive rational numbers for which the sum of any finite subset
always remains below some preassigned bound, and then specifying the
conditions under which two such "real numbers" are to be
considered equal, or strictly less than one another.



Weierstrass was concerned to purge the foundations of analysis of all
traces of the intuition of continuous motion--in a word, to
replace the variable by the static. For Weierstrass a variable \(x\)
was simply a symbol designating an arbitrary member of a given set of
numbers, 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 also
formulated the familiar \((\varepsilon , \delta)\) definition of
continuous
 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 problem
of formulating rigorous definitions of continuity and the real numbers
was mounted by Richard Dedekind (1831-1916). Dedekind focussed
attention on the question: exactly what is it that distinguishes a
continuous domain from a discontinuous one? He seems to have been the
first to recognize that the property of density, possessed by the
ordered set of rational numbers, is insufficient to guarantee
continuity. In *Continuity and Irrational Numbers* (1872) he
remarks that when the rational numbers are associated to points on a
straight 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 straight
line's "absence of gaps, completeness, continuity"
(1872: section 3 [1999: 771]). Dedekind regards this principle as
being essentially indemonstrable; he ascribes to it, rather, the
status of an axiom "by which we attribute to the line its
continuity, by which we think continuity into the line" (1872
[1999: 771-772]. It is not, Dedekind stresses, necessary for
*space* to be continuous in this sense, for "many of its
properties 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]) is
the key idea underlying Dedekind's construction of the domain of
real numbers. He first defines a *cut* 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 each
rational number corresponds, in an evident way, to a cut, he observes
that infinitely many cuts fail to be engendered by rational numbers.
The discontinuity or incompleteness of the domain of rational numbers
consists precisely in this latter fact.



It is to be noted that Dedekind does not identify irrational numbers
with cuts; rather, each irrational number is newly
"created" by a mental act, and remains quite distinct from
its associated cut. Dedekind goes on to show how the domain of cuts,
and thereby the associated domain of real numbers, can be ordered in
such a way as to possess the property of *continuity*, 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 is
> produced. (1872: section 5 [1999: 776])
> 
> 
> 



The most visionary "arithmetizer" of all was Georg
 Cantor[31]
 (1845-1918). Cantor's analysis of the continuum in terms
of infinite point sets led to his theory of transfinite numbers and to
the eventual freeing of the concept of set from its geometric origins
as a collection of points, so paving the way for the emergence of the
concept of general abstract set central to today's mathematics.
Like Weierstrass and Dedekind, Cantor aimed to formulate an adequate
definition of the real numbers which avoided the presupposition of
their prior existence, and he follows them in basing his definition on
the rational numbers. Following Cauchy, he calls a sequence \(a\_1,\)
\(a\_2 ,\)..., \(a\_n,\)... of rational numbers a
*fundamental sequence* if there exists an integer \(N\) such
that, for any positive rational \(\varepsilon\), there exists an
integer \(N\) 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 a
*definite limit b*. Dedekind had taken irrational numbers to be
*mental objects* associated with cuts, so, analogously, Cantor
regards these definite limits, as nothing more than *formal
symbols* associated with fundamental sequences (Dauben 1979:
38). The domain \(B\) of such symbols may be considered an enlargement
of the domain \(A\) of rational numbers. After imposing an
arithmetical structure on the domain \(B\), Cantor is emboldened to
refer to its elements as (real) *numbers*. Nevertheless, he
still insists that these "numbers" have no existence
except as representatives of fundamental sequences. Cantor then shows
that each point on the line corresponds to a definite element of
\(B\). Conversely, each element of \(B\) should determine a definite
point on the line. Realizing that the intuitive nature of the linear
continuum precludes a rigorous proof of this property, Cantor simply
assumes it 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 his career
cleaved to the discrete, it was numbers, rather than geometric points,
that possessed objective significance. Indeed the isomorphism between
the discrete numerical domain \(B\) and the linear continuum was
regarded by Cantor essentially as a device for facilitating the
manipulation of numbers.



Cantor's arithmetization of the continuum had the following
important consequence. It had long been recognized that the sets of
points of any pair of line segments, even if one of them is infinite
in length, can be placed in one-one correspondence. This fact was
taken to show that such sets of points have no well-defined
"size". But Cantor's identification of the set of
points on a linear continuum with a domain of numbers enabled the
*sizes* of point sets to be compared in a definite way, using
the well-grounded idea of *one-one correspondence* between sets
of numbers.



Cantor's investigations into the properties of subsets of the
linear continuum are presented in six masterly papers published during
1879-84, *Uber unendliche lineare
Punktmannichfaltigkeiten* ("On infinite, linear point
manifolds"). Remarkable in their richness of ideas, these papers
provide the first accounts of Cantor's revolutionary theory of
infinite sets and its application to the classification of subsets of
the linear continuum. In the fifth of these papers, the
*Grundlagen* of 1883, are to be found some of Cantor's
most searching observations on the nature of the continuum.



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



> 
> 
> develop the concept of the continuum as soberly and briefly as
> possible, and only with regard to the *mathematical* theory of
> sets. (1883 [1999: SS10, para. 2, p. 903]) 
> 
> 
> 



This opens the way, he believes, to the formulation of an exact
concept of the continuum. Cantor points out that the idea of the
continuum has heretofore merely been presupposed by mathematicians
concerned with the analysis of continuous functions and the like, and
has "not been subjected to any more thorough inspection"
(1883 [1999: SS10, para. 3, p. 904]).



Repudiating any use of spatial or temporal intuition in an exact
determination of the continuum, Cantor undertakes its precise
arithmetical definition. Making reference to the definition of real
number he has already provided (i.e., in terms of fundamental
sequences), he introduces the \(n\)-dimensional arithmetical space
\(G\_n\) as the set of all \(n\)-tuples of real numbers \(\langle
x\_1,x\_2 ,\ldots ,x\_n\rangle\), calling each such an *arithmetical
point* of \(G\_{n.}\) The distance between two such points is given
by

 

\[
\sqrt{(x'\_1 - x\_1)^2 + (x'\_2 - x\_2)^2 + \ldots (x'\_n - x\_n)^2}
\]


Cantor defines an *arithmetical* point-set in \(G\_n\) to be any
"aggregate of points of the points of the space \(G\_n\) that is
given in a lawlike way" (1883 [1999: SS10, 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 either
the power of the set of natural numbers or that of
 (0,1),[32]
 Cantor turns to the definition of the general concept of a continuum
within \(G\_{n}.\) For this he employs the concept of
*derivative* or *derived set* of a point set introduced
in a paper of 1872 on trigonometric series. Cantor had defined the derived
set of a point set \(P\) to be the set of *limit points* of
\(P,\) where a limit point of \(P\) is a point of \(P\) with
infinitely many points of \(P\) arbitrarily close to it. A point set
is called *perfect* if it coincides with its derived
 set.[33]
 Cantor observes that this condition does not suffice to characterize
a continuum, since perfect sets can be constructed in the linear
continuum which are dense in no interval, however small: as an example
of such a set he offers the
 set[34]
 consisting of all real numbers in (0,1) whose ternary expansion does
not contain a "1".



Accordingly an additional condition is needed to define a continuum.
Cantor supplies this by introducing the concept of a
*connected* set. A point set \(T\) is connected in
Cantor's sense if for any pair of its points \(t, t'\) and any
arbitrarily small number \(\varepsilon\) there is a finite sequence of
points \(t\_1,\) \(t\_2,\)..., \(t\_n\) of \(T\) for which the
distances \([tt\_1],\) \([t\_1 t\_2],\) \([t\_2 t\_3],\) ..., \([t\_n
t'],\) are all less than \(\varepsilon\). Cantor now defines a
continuum to be a perfect connected point set.



Cantor has advanced beyond his predecessors in formulating what is in
essence a *topological* 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 the
definition of continuum in modern general topology. In a well-known
textbook (Hocking & Young 1961: 43) on the subject we find
a *continuum* defined as a compact connected subset of a
topological space. Now within any *bounded* region of Euclidean
space it can be shown that Cantor's continua coincide with
continua in the sense of the modern definition. While Cantor lacked
the definition of compactness, his requirement that continua be
"complete" (which led to his rejecting as continua such
noncompact sets as open intervals or discs) is not far away from the
idea.



Throughout Cantor's mathematical career he maintained an
unwavering, even dogmatic opposition to infinitesimals, attacking the
efforts of mathematicians such as du Bois-Reymond and
Veronese[36]
to formulate rigorous theories of actual infinitesimals. As far as
Cantor was concerned, the infinitesimal was beyond the realm of the
possible; infinitesimals were no more than "castles in the air,
or rather just nonsense" (1893 [1965: 506], translated by Fisher
1981: 118), to be classed "with circular squares and square
circles" (1893 [1965: 507], translated by Fisher 1981: 118). His
abhorrence of infinitesimals went so deep as to move him to outright
vilification, branding them as "Cholera-bacilli of
mathematics" (1893 [1965: 505], translated by Fisher 1981:
116). Cantor's rejection of infinitesimals stemmed 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, in particular 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 nineteenth and early twentieth centuries remained
opposed, in varying degrees, to the idea of explicating the continuum
concept entirely in discrete terms. These include the philosophers
Brentano and Peirce and the mathematicians Poincare, Brouwer
and 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 account
of the continuous is akin to Aristotle's. Brentano regards
continuity as something given in perception, primordial in nature,
rather than a mathematical construction. He held that the idea of the
continuous is abstracted from sensible intuition. Brentano suggests
that the continuous is brought to appearance by sensible intuition in
three phases. First, sensation presents us with objects having parts
that coincide. From such objects the concept of *boundary* is
abstracted in turn, and then one grasps that these objects actually
*contain* coincident boundaries. Finally one sees that this is
all that is required in order to have grasped the concept of a
continuum.



For Brentano the essential feature of a continuum is its inherent
capacity to engender boundaries, and the fact that such boundaries can
be grasped as coincident. Boundaries themselves possess a quality
which Brentano calls *plerosis* ("fullness").
Plerosis is the measure of the number of directions in which the given
boundary actually bounds. Thus, for example, within a temporal
continuum the endpoint of a past episode or the starting point of a
future one bounds in a single direction, while the point marking the
end of one episode and the beginning of another may be said to bound
doubly. In the case of a spatial continuum there are numerous
additional possibilities: here a boundary may bound in all the
directions of which it is capable of bounding, or it may bound in only
some of these directions. In the former case, the boundary is said to
exist in *full plerosis*; in the latter, in *partial
plerosis*. Brentano believed that the concept of plerosis enabled
sense to be made of the idea that a boundary possesses
"parts", even when the boundary lacks dimensions
altogether, as in the case of a point. Thus, while the present or
"now" is, according to Brentano, temporally unextended and
exists only as a boundary between past and future, it still possesses
two "parts" or aspects: it is both the end of the past and
the beginning of the future. It is worth mentioning that for Brentano
it was not just the "now" that existed only as a boundary;
since, like Aristotle he held that "existence" in the
strict sense means "existence *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 mathematicians to
construct the continuum from numbers. His attitude varied from
rejecting such attempts as inadequate to according them the status of
 "fictions".[37]
 This is not surprising given his Aristotelian inclination to take
mathematical and physical theories to be genuine descriptions of
empirical phenomena rather than idealizations: in his view, if such
theories were to be taken as literal descriptions of experience, they
would amount to nothing better than
"misrepresentations".



Brentano's analysis of the continuum centered on its
phenomenological and qualitative aspects, which are by their very
nature incapable of reduction to the discrete. Brentano's
rejection of the mathematicians' attempts to construct it in
discrete 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 the
arithmetizers. Like Brentano, he held that the cohesiveness of a
continuum rules out the possibility of it being a mere collection of
discrete individuals, or points, in the usual sense. And even before
Brouwer, Peirce seems to have been aware that a faithful account of
the continuum will involve questioning the law of excluded middle.
Peirce also held that any continuum harbors an *unboundedly
large* collection of points--in his colorful terminology, a
*supermultitudinous* collection--what we would today call
a *proper class*. Peirce maintained that if
enough points were to be crowded together by carrying insertion of new
points between old to its ultimate limit they would--through
a *logical* "transformation of quantity into
quality"--lose their individual identity and become fused
into a true continuum.



Peirce's conception of the number continuum is also notable for
the presence in it of an abundance of *infinitesimals*, Peirce
championed the retention of the infinitesimal concept in the
foundations of the calculus, both because of what he saw as the
efficiency of infinitesimal methods, and because he regarded
infinitesimals as constituting the "glue" causing points
on a continuous line to lose their individual identity.



The idea of continuity played a central role in the thought of the
great French mathematician Henri
 Poincare[39]
 (1854-1912). While accepting the arithmetic definition of the
continuum, he questions the fact that (as with Dedekind and
Cantor's formulations) the (irrational) numbers so produced are
mere symbols, detached from their origins in intuition. Unlike Cantor,
Poincare accepted the infinitesimal, even if he did not regard
all of the concept's manifestations as useful.



The Dutch mathematician L. E. J. Brouwer (1881-1966) is best
known as the founder of the philosophy of (neo)*intuitionism*
(Brouwer [1975]; van Dalen 1998). Brouwer's highly idealist
views on mathematics bore some resemblance to Kant's. For
Brouwer, mathematical concepts are admissible only if they are
adequately grounded in intuition, mathematical theories are
significant only if they concern entities which are constructed out of
something given immediately in intuition, and mathematical
demonstration is a form of construction in intuition. While admitting
that the emergence of noneuclidean geometry had discredited
Kant's view of space, Brouwer held, in opposition to the
logicists (whom he called "formalists") that arithmetic,
and so all mathematics, must derive from temporal intuition.



Initially Brouwer held without qualification that the continuum is not
constructible from discrete points, but was later to modify this
doctrine. In his mature thought, he radically transformed the concept
of point, endowing points with sufficient fluidity to enable them to
serve as generators of a "true" continuum. This fluidity
was achieved by admitting as "points", not only fully
defined discrete numbers such as \(\sqrt{2},\) \(\pi,\) \(e,\) and the
like--which have, so to speak, already achieved
"being"--but 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 a subject operating throughout an indefinitely extended time. The
resulting *choice sequences* cannot 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 way
compatible with his belief in the primordial intuition of
time--that is, as an unfinished, indeed unfinishable entity in a
perpetual state of growth, a "medium of free development".
In this conception, the mathematical continuum is indeed
"constructed", not, however, by initially shattering, as
did Cantor and Dedekind, an intuitive continuum into isolated points,
but rather by assembling it from a complex of continually changing
overlapping parts.



The mathematical continuum as conceived by Brouwer displays a number
of features that seem bizarre to the classical eye. For example, in
the Brouwerian continuum the usual law of comparability, namely that
for 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 of
excluded middle in the form that for any real numbers \(a, b\), either
\(a = b\) or \(a \ne b\). The failure of these seemingly
unquestionable principles in turn vitiates the proofs of a number of
basic results of classical analysis, for example the
Bolzano-Weierstrass theorem, as well as the theorems of monotone
convergence, intermediate value, least upper bound, and maximum value
for continuous
 functions.[40]



While the Brouwerian continuum may possess a number of negative
features from the standpoint of the classical mathematician, it has
the merit of corresponding more closely to the continuum of intuition
than does its classical counterpart. Far from being bizarre, the
failure of the law of excluded middle for points in the intuitionistic
continuum may be seen as fitting in well with the character of the
intuitive continuum.



In 1924, Brouwer showed that every function defined on a closed
interval of his continuum is uniformly continuous. As a consequence
the intuitionistic continuum is *indecomposable*, that is,
cannot be split into two disjoint parts in any way whatsoever. In
contrast with a discrete entity, the indecomposable Brouwerian
continuum cannot be composed of its parts. Brouwer's vision of
the continuum has in recent years become the subject of intensive
mathematical investigation.



Hermann Weyl (1885-1955), one of most versatile mathematicians
of the twentieth century, was preoccupied with the nature of the
continuum (Bell 2000). In his *Das Kontinuum* of 1918 he
attempts to provide the continuum with an exact mathematical
formulation free of the set-theoretic assumptions he had come to
regard as objectionable. As he saw it, there is an unbridgeable gap
between intuitively given continua (e.g., those of space, time and
motion) on the one hand, and the discrete exact concepts of
mathematics (e.g., that of real number) on the other. For Weyl the
presence of this split meant that the construction of the mathematical
continuum could not simply be "read off" from intuition.
Rather, he believed that the mathematical continuum must be treated
and, in the end, justified in the same way as a physical theory.
However much he may have wished it, in *Das Kontinuum* Weyl did
not aim to provide a mathematical formulation of the continuum as it
is presented to intuition, which, as the quotations above show, he
regarded as an impossibility (at that time at least). Rather, his goal
was first to achieve *consistency* by putting the
*arithmetical* notion of real number on a firm logical basis,
and then to show that the resulting theory is *reasonable* by
employing it as the foundation for a plausible account of continuous
process in the objective physical world.



Later Weyl came to repudiate atomistic theories of the continuum,
including that of his own *Das Kontinuum*. He accordingly
welcomed Brouwer's construction of the continuum by means of
sequences generated by free acts of choice, thus identifying it as a
"medium of free Becoming" which "does not dissolve
into a set of real numbers as finished entities" (Weyl 1921,
50). Weyl felt that Brouwer, through his doctrine of intuitionism, had
come closer than anyone else to bridging that "unbridgeable
chasm" between the intuitive and mathematical continua. In
particular, he found compelling the fact that the Brouwerian continuum
is not the union of two disjoint nonempty parts--that it is
indecomposable. "A genuine continuum", Weyl says,
"cannot be divided into separate fragments". In later
publications he expresses this more colorfully by quoting Anaxagoras
to the effect that a continuum "defies the chopping off of its
parts 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 largely
abandoned. And so the situation remained for a number of years. The
first signs of a revival of the infinitesimal approach to analysis
surfaced in 1958 with a paper by C. Schmieden and D. Laugwitz. But the
major breakthrough came in 1960 when it occurred to the mathematical
logician Abraham Robinson (1918-1974) that 



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



This insight led to the creation of *nonstandard
 analysis*,[41]
 which Robinson regarded as realizing Leibniz's conception of
infinitesimals and infinities as ideal numbers possessing the same
properties as ordinary real numbers.



After Robinson's initial insight, a number of ways of presenting
nonstandard analysis were developed. Here is a sketch of one of
them.



Starting with the classical real line \(\Re\), a set-theoretic
universe--the *standard universe*--is first
constructed over it: here by such a universe is meant a set \(U\)
containing \(\Re\) which is closed under the usual set-theoretic
operations of union, power set, Cartesian products and subsets. Now
write \(\fU\) for the structure \((U, \in)\), where \(\in\) is the
usual membership relation on \(U\): associated with this is the
extension \(\cL(U)\) of the first-order language of set theory to
include 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 a *nonstandard universe*, satisfying the following key
principle:




**Saturation Principle**. Let \(\Phi\) be a collection of
\(\cL(U)\)-formulas with exactly one free variable. If \(\Phi\) is
*finitely satisfiable* in \(\fU\), that is, if for any finite
subset \(\Phi '\) of \(\Phi\) there is an element of \(U\) which
satisfies all the formulas of \(\Phi '\) in \(\fU\), then there is an
element of \(^\*U\) which satisfies all the formulas of \(\Phi\) in
\(^\*\fU\).




The saturation property expresses the intuitive idea that the
nonstandard 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 not
necessarily be an element of \(U\) satisfying *all* the members
of \(\cP\). The saturation of \(^\*\fU\) guarantees the existence of an
element of \(^\*U\) which satisfies, in \(^\*\fU\), all the members of
\(\cP\). For example, suppose the set \(\bbN\) of natural numbers is a
member of \(U\); for each \(n \in \bbN\) let \(P\_n (x)\) be the
property \(x \in \bbN \amp n \lt x\). Then clearly, while each finite
subcollection of the collection \(\cP = \{P\_n : n \in \bbN\}\) is
satisfiable in \(\fU\), the whole collection is not. An element of
\(^\*U\) satisfying \(\cP\) in \(^\*\fU\) will then be an "natural
number" greater than every member of \(\bbN\), that is, an
*infinite* number.



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




**Transfer Principle**. If \(\sigma\) is any sentence of
\(\cL(U)\), then \(\sigma\) holds in \(\fU\) if and only if it holds
in \(^\*\fU\).




The transfer principle may be seen as a version of Leibniz's
continuity principle: it asserts that all first-order properties are
preserved in the passage to or "transfer" from the
standard to the nonstandard universe.



The members of \(U\) are called *standard sets*, or
*standard objects*; those in \(^\*U - U\) *nonstandard
sets* or *nonstandard objects*: \(^\*U\) thus consists of
both standard and nonstandard objects. The members of \(^\*U\) will also
be 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
\(\hat{A}\)--the *inflate* of \(A\)--consisting of
all 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 regarded
as the same set \(A\) viewed from a nonstandard vantage point. While
clearly \(A \subseteq \hat{A}\), \(\hat{A}\) may contain
"nonstandard" elements not in \(A\). It can in fact be
shown that *infinite* standard sets always get
"inflated" in this way. Using the transfer principle, any
function \(f\) between standard sets automatically extends to a
function--also written \(f\)--between their inflates.



If \(\fA = (A, R)\) is a mathematical structure, we may
consider the structure \(\hat{\fA} = (\hat{A},\hat{R})\). From the
transfer principle it follows that \(\fA\) and \(\hat{\fA}\) have
precisely 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 real
number may be identified with a set of natural numbers. \(\Re\) may be
regarded as an ordered field, and the same is therefore true of its
inflate \(\hat{\Re}\), since the latter has precisely the same
first-order properties as \(\Re\). \(\hat{\Re}\) is called the
*hyperreal line*, and its members *hyperreals*. A
standard hyperreal is then just a real, to which we shall refer for
emphasis as a *standard real*. Since \(\Re\) is infinite,
nonstandard hyperreals must exist. The saturation principle implies
that there must be an *infinite* (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 the
sense of exceeding 0 and yet being smaller than 1/\(n+1\) for every
\(n \in \bbN\). In general, we call a hyperreal *a
infinitesimal* if its absolute value \(\lvert a\rvert\) is less
than 1/\(n+1\) for every \(n \in \bbN\). In that case the set \(I\) of
infinitesimals contains not just 0 but a substantial number (in fact,
infinitely many) other elements. Clearly \(I\) is an additive subgroup
of \(\Re\), that is, if \(a, b \in I\), then \(a - b \in I\).



The members of the inflate \(\hat{\bbN}\) of \(\bbN\) are called
*hypernatural numbers*. As for the hyperreals, it can be shown
that \(\hat{\bbN}\) also contains nonstandard elements which must
exceed every member of \(\bbN\); these are called *infinite*
hypernatural numbers.



For hyperreals \(a, b\) we define \(a \approx b\) and say that \(a\)
and \(b\) are *infinitesimally close* if \(a - b \in I\). This
is an equivalence relation on the hyperreal line: for each hyperreal
\(a\) we write \(\mu(a)\) for the equivalence class of \(a\) under
this relation and call it the *monad* of \(a\). The monad of a
hyperreal \(a\) thus consists of all the hyperreals that are
infinitesimally close to \(a\): it may be thought of as a small cloud
centered at \(a\). Note also that \(\mu\)(0) \(= I\).



A hyperreal \(a\) is *finite* if it is not infinite; this means
that \(\lvert a\rvert \lt n\) for some \(n \in \bbN\) . It is not
difficult to show that finiteness is equivalent to the condition of
*near-standardness*; here a hyperreal \(a\) is
*near-standard* if \(a \approx r\) for some standard real
\(r\).



Much of the usefulness of nonstandard analysis stems from the fact
that statements of classical analysis involving limits or the
\((\varepsilon , \delta)\) criterion admit succinct, intuitive
translations into statements involving infinitesimals or infinite
numbers, in turn enabling comparatively straightforward proofs to be
given of classical theorems. Here are some examples of such
 translations:[43]


* Let \(\langle s\_n\rangle\) be a standard infinite sequence of real
numbers and let \(s\) be a standard real number. Then \(s\) is the
limit of \(\langle s\_n\rangle\) within \(\Re\), \(\lim\_{n\rightarrow
\infty} s\_n = s\) in the classical sense, if and only if \(s\_n \approx
s\) *for all* infinite subscripts \(n\).
* A standard sequence \(\langle s\_n\rangle\) converges if and only
if \(s\_n \approx s\_m\) for all infinite \(n\) and \(m\).
(Cauchy's criterion for convergence.)



Now suppose that \(f\) is a real-valued function defined on some open
interval \((a, b)\). We have remarked above that \(f\) automatically
extends 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 is
necessary 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 monad
of \(x\_0\) 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 nonstandard
formulations.


## 8. The Constructive Real Line and the Intuitionistic Continuum



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



In constructive mathematics a problem is counted as solved only if an
explicit solution can, in principle at least, be produced. Thus, for
example, "There is an \(x\) such that \(P(x)\)" means
that, in principle at least, we can explicitly produce an \(x\) such
that \(P(x)\). This fact led to the questioning of certain principles
of classical logic, in particular, the law of excluded middle, and the
creation of a new logic, intuitionistic logic (see entry on
 intuitionistic logic).
 It also led to the introduction of a sharpened definition of real
numbers--the constructive real numbers. A constructive real
number is a sequence of rationals \((r\_n) = r\_1, r\_2,\ldots\) such
that, 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 may
be regarded as a real number by identifying it with the real number
\((\alpha , \alpha ,\ldots)\). The set \(R\) of all constructive real
numbers is the constructive real line.



Now of course, for any "given" real number there are a
variety 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 that
two real numbers are equal is to say that they are equivalent in this
sense.



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 corresponding *apartness 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 called *real numbers*. A real number
\(x\) is *positive* if \(x \gt 0\) and *negative* if
\(-x \gt 0\). The relation \(\ge\) (*greater than or equal to*)
is defined by

 

\[x \ge y \Longleftrightarrow \forall z(y \gt z \Rightarrow x \gt z).\]


The relations \(\lt\) and \(\le\) are defined in the usual way; \(x\)
is *nonnegative* if \(0 \le x\). Two real numbers are
*equal* 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 the
usual subsets of \(R\); for instance \(N^+\) is identified with the
set of elements of \(R\) of the form \(1 + 1 + \ldots + 1\).



These relations and operations are subject to the following three
groups of axioms, which, taken together, form the system
**CA** of axioms for *constructive analysis*, or
the *constructive 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 notions *bounded above, bounded
below, and bounded* are defined as in classical mathematics, and
the *least upper bound*, if it exists, of a
 nonempty[44]
 set \(S\) of real numbers is the unique real number \(b\) such
that


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



**Special Properties of \(\gt\).**



* *Archimedean axiom*. For each \(x \in R\) such that \(x
\ge 0\) there exists \(n \in N\) such that \(x \lt n\).
* *The least upper bound principle*. Let \(S\) be a nonempty
subset of \(R\) that is bounded above relative to the relation
\(\ge\), such that for all real numbers \(a, b\) with \(a \lt b\),
either \(b\) is an upper bound for \(S\) or else there exists \(s \in
S\) with \(s \gt a\). Then \(S\) has a least upper bound.




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


* \(\neg(x \gt x)\)
* \(x \ge x\)
* \(x \gt y \wland y \gt z \Longrightarrow x \gt z\)
* \(\neg(x \gt y \wland y \ge x)\)
* \((x \gt y \ge z) \Longrightarrow x \gt z\)
* \(\neg(x \gt y) \Longleftrightarrow y \ge x\)
* \(\neg \neg(x \ge y) \Longleftrightarrow \neg \neg(y \gt x)\)
* \((x \ge y \ge z) \Longrightarrow x \ge z\)
* \((x \ge y \wland y \ge x) \Longrightarrow x = y\)
* \(\neg(x \gt y \wland x = y)\)
* \(x \ge 0 \Longleftrightarrow \forall \varepsilon \gt 0(x \lt
\varepsilon)\)
* \(x + y \gt 0 \Longrightarrow (x \gt 0 \wlor y \gt 0)\)
* \(x \gt 0 \Longrightarrow -x \lt 0\)
* \((x \gt y \wland z \lt 0) \Longrightarrow yz \gt xz\)
* \(x \apart 0 \Longrightarrow x^2 \gt 0\)
* \(1 \gt 0\)
* \(x^2 \gt 0\)
* \(0 \lt x \lt 1 \Longrightarrow x \gt x^2\)
* \(x^2 \gt 0 \Longrightarrow x \apart 0\)
* \(n \in N^{+} \Longrightarrow n^{-1} \gt 0\)
* if \(x \gt 0\) and \(y \ge 0\), then \(\exists n\in Z(nx \gt
y)\)
* \(x \gt 0 \Longrightarrow x^{-1} \gt 0\)
* \(xy \gt 0 \Longrightarrow(x \ne 0 \wlor y \ne 0)\)
* 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 of
**CA**. Are there any other models, that is, models not
isomorphic 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 is
possible for the Dedekind and Cantor reals to fail to be isomorphic,
despite the fact that they are both models of **CA**.



In constructive analysis, a real number is an infinite (convergent)
sequence of rational numbers generated by an effective rule, so that
the constructive real line is essentially just a restriction of its
classical counterpart. Brouwerian intuitionism takes a more liberal
view of the matter, resulting in a considerable enrichment of the
arithmetical continuum over the version offered by strict
constructivism. As conceived by intuitionism, the arithmetical
continuum admits as real numbers not only infinite sequences
determined in advance by an effective rule for computing their terms,
but also ones in whose generation free selection plays a part. The
latter are called *(free) choice sequences*. Without loss of
generality we may and shall assume that the entries in choice
sequences are natural numbers.



While constructive analysis does not formally contradict classical
analysis, and may in fact be regarded as a subtheory of the latter, a
number of intuitionistically plausible principles have been proposed
for the theory of choice sequences which render intuitionistic
analysis divergent from its classical counterpart. One such principle
is *Brouwer'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 for
which \(Q(\alpha , n)\) holds, then \(n\) can already be determined on
the 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. Another
such principle is *Bar Induction*, a certain form of induction
for well-founded sets of finite
 sequences.[46]
 Brouwer used Bar Induction and the Continuity Principle in proving
his Continuity Theorem that every real-valued function defined on a
closed interval is uniformly continuous, from which, as has already
been observed, it follows that the intuitionistic continuum is
indecomposable.



Brouwer gave the intuitionistic conception of mathematics an
explicitly subjective twist by introducing the *creative
subject*. The creative subject was conceived as a kind of
idealized mathematician for whom time is divided into discrete
sequential stages, during each of which he may test various
propositions, attempt to construct proofs, and so on. In particular,
it can always be determined whether or not at stage \(n\) the creative
subject has a proof of a particular mathematical proposition \(p\).
While the theory of the creative subject remains controversial, its
purely mathematical consequences can be obtained by a simple postulate
which 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 the
creative subject has a proof of \(p\) at stage \(n\); \(a\_n = 0\)
otherwise. Now if the construction of these sequences is the only use
made of the creative subject, then references to the latter may be
avoided by postulating the principle known as *Kripke's
Scheme*



> 
> 
> 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 remarkable
consequences 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, assuming
the Continuity Principle and Kripke's Scheme, it remains
indecomposable even if one pricks it with a pin. The intuitionistic
continuum has, as it were, a syrupy nature, so that one cannot simply
take away one point. If in addition Bar Induction is assumed, then,
still more surprisingly, indecomposability is maintained even when all
the rational points are removed from the continuum.



Finally, it has been shown that a natural notion of infinitesimal can
be developed within intuitionistic mathematics (Vesley 1981), the idea
being that an infinitesimal should be a "very small" real
number in the sense of not being known to be
distinguishable--that is, strictly greater than or less
than--zero.


## 9. Smooth Infinitesimal Analysis



A major development in the refounding of the concept of infinitesimal
took place in the 1970s with the emergence of *synthetic
differential geometry*, also known as *smooth infinitesimal
analysis*
 (SIA).[47]
 Based on the ideas of the American mathematician F. W. Lawvere, and
employing the methods of category theory, smooth infinitesimal
analysis provides an image of the world in which the continuous is an
autonomous notion, not explicable in terms of the discrete. It
provides a rigorous framework for mathematical analysis in which every
function between spaces is smooth (i.e., differentiable arbitrarily
many times, and so in particular continuous) and in which the use of
limits in defining the basic notions of the calculus is replaced by
*nilpotent infinitesimals*, that is, of quantities so small
(but not actually zero) that some power--most usefully, the
square--vanishes. Since in SIA all functions are continuous, it
embodies in a striking way Leibniz's principle of continuity
*Natura non facit saltus*.



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

 

\[\tag{1}
\delta y = f '(x)\delta x + A(\delta x)^2,\] 


where \(f '(x)\) is the derivative of \(f(x)\) and \(A\) is a quantity
whose value depends on both \(x\) and \(\delta x\). Now if it were
possible to take \(\delta x\) so *small* (but not demonstrably
identical with 0) that \((\delta x)^2 = 0\) then (1) would assume the
simple form

 

\[\tag{2}
f(x + \delta x) - f(x) = \delta y = f '(x) \delta x.\]


We shall call a quantity having the property that its square is zero a
*nilsquare infinitesimal* or simply a *microquantity*.
In SIA "enough" microquantities are present to ensure that
equation (2) holds *nontrivially* for *arbitrary*
functions \(f: \bR \rightarrow \bR\). (Of course (2) holds trivially
in standard mathematical analysis because there 0 is the sole
microquantity in this sense.) The meaning of the term
"nontrivial" here may be explicated in following way. If
we replace \(\delta x\) by the letter \(\varepsilon\) standing for an
arbitrary microquantity, (2) assumes the form




\[\tag{3}
f(x + \varepsilon) - f(x) = \varepsilon f '(x).\]



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

 

\[f(x + \varepsilon) - f(x) = \varepsilon D\]


holds for all microquantities \(\varepsilon\).



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



\[\tag{4}
f(\varepsilon) = f(0) + \varepsilon D,\]


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

 

\[\Delta = \{x: x \in \bR \land x^2 = 0\}.\]


Then it is postulated that, for any \(f: \Delta \rightarrow \bR\),
there is a *unique \(D \in \bR\)* such that equation (4) holds
for all \(\varepsilon\). This says that the graph of \(f\) is a
straight line passing through \((0, f\)(0)) with slope \(\Delta\).
Thus any function on \(\Delta\) is what mathematicians term
*affine*, and so this postulate is naturally termed the
*principle* of *microaffineness*. It means that
\(\Delta\) *cannot be bent or broken*: it is subject only to
*translations and rotations*--and yet is not (as it would
have to be in ordinary analysis) identical with a point. \(\Delta\)
may be thought of as an entity possessing position and attitude, but
lacking true extension.



Now consider the space \(\Delta^{\Delta}\) of maps from \(\Delta\) to
itself. It follows from the microaffineness principle that the
subspace \((\Delta^{\Delta})\_0\) of \(\Delta^{\Delta}\) consisting of
maps 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 \(\Delta\) by
evaluation: for \(f \in \Delta^{\Delta},\) \(f \cdot \varepsilon = f
(\varepsilon)\). Its subspace \((\Delta^{\Delta})\_0\) is a submonoid
naturally identified as the space of *ratios of
microquantities*. 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 ratios
of microquantities. This was essentially the view of Euler, who
regarded (real) numbers as representing the possible results of
calculating the ratio 0/0. For this reason Lawvere has suggested that
\(\bR\) be called the space of *Euler reals*.



If we think of a function \(y = f(x)\) as defining a curve, then, for
any \(a\), the image under \(f\) of the "microinterval"
\(\Delta + a\) obtained by translating \(\Delta\) to \(a\) is straight
and coincides with the tangent to the curve at \(x = a\). In this
sense each curve is "infinitesimally straight".



From the principle of microaffineness we deduce the important
*principle of microcancellation*, viz.

 

\[
\text{If } \varepsilon a = \varepsilon b \text{ for all } \varepsilon, \text{ then } a = b.
\]


For the premise asserts that the graph of the function \(g: \Delta
\rightarrow \bR\) defined by \(g(\varepsilon) = a \varepsilon\) has
both slope \(a\) and slope \(b\): the uniqueness condition in the
principle of microaffineness then gives \(a = b\). The principle of
microcancellation supplies the exact sense in which there are
"enough" infinitesimals in smooth infinitesimal
analysis.



From the principle of microaffineness it also follows that *all
functions on \(\bR\) are continuous*, that is, *send
neighboring 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 \(x, y\), note
that \(y = x + \varepsilon\) with \(\varepsilon\) in \(\Delta\), so
that

 

\[
\begin{align}
f(y) - f(x) & = f(x + \varepsilon) - f(x)\\
& = \varepsilon f '(x).\\
\end{align}
\]


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



In fact, since equation
 (3)
 holds for any \(f\), it also holds for its derivative \(f '\); it
follows that functions in smooth infinitesimal analysis are
differentiable arbitrarily many times, thereby justifying the use of
the term "smooth".



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

 

\[(fg)' = f'g + fg'.\]


To do this we compute

 

\[
\begin{align}
(fg)(x + \varepsilon) & = (fg)(x) + \varepsilon(fg)'(x) \\
& = f(x)g(x) + \varepsilon(fg)'(x),\\
(fg)(x + \varepsilon) & = f(x + \varepsilon)g(x + \varepsilon) \\
& = [f(x) + f'(x)]\cdot[g(x) + g'(x)]\\
& = f(x)g(x) + \varepsilon(f'g + fg') + \varepsilon^2 f'g'\\
& = f(x)g(x) + \varepsilon(f'g + fg'),
\end{align}
\]


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



A *stationary 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 it
follows from microcancellation that \(f'(a) = 0\). This is
*Fermat's rule*.



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



> 
> *Constancy Principle*. If every point in an interval \(J\) is a
> stationary point of \(f: J \rightarrow \bR\) (that is, if \(f'\) is
> identically 0), then \(f\) is constant.
> 



Put succinctly, "universal local constancy implies global
constancy". It follows from this that two functions with
identical derivatives differ by at most a constant.



In ordinary analysis the continuum \(\bR\) is connected in the sense
that it cannot be split into two non empty subsets neither of which
contains a limit point of the other. In smooth infinitesimal analysis
it has the vastly stronger property of *indecomposability*: it
cannot be split *in any way whatsoever* into two disjoint
nonempty 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 \(f\) is
constant. For we have

 

\[(f(x) = 0 \text{ or } f(x) = 1) \amp(f(x + \varepsilon) = 0 \text{ or } f(x + \varepsilon) = 1).\]


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 \(f\) is
continuous. 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 are
*incompatible with the law of excluded middle of classical
logic*. This incompatibility can be demonstrated in two ways, one
informal and the other rigorous. First the informal argument. Consider
the 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 of
excluded middle held, each real number would then be either equal or
unequal to 0, so that the function \(f\) would be defined on the whole
of \(\bR\). But, considered as a function with domain \(\bR,\) \(f\)
is clearly discontinuous. Since, as we know, in smooth infinitesimal
analysis every function on \(\bR\) is continuous, \(f\) cannot have
domain \(\bR\)
 there.[50]
 So the law of excluded middle fails in smooth infinitesimal analysis.
To put it succinctly, *universal continuity implies the failure of
the law of excluded middle*.



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



(\*)
 *for all infinitesimal \(\varepsilon\), not \(\varepsilon \ne
0\).* 


Now suppose that the law of excluded middle were to hold. Then we
would have, for any \(\varepsilon\), either \(\varepsilon = 0\) or
\(\varepsilon \ne 0\). But (\*) allows us to eliminate the second
alternative, 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 \(1 = 0\). So
again the law of excluded middle must fail.



The "internal" logic of smooth infinitesimal analysis is
accordingly not full classical logic. It is, instead,
*intuitionistic* logic, that is, the logic derived from the
constructive interpretation of mathematical assertions. In our brief
sketch we did not notice this "change of logic" because,
like much of elementary mathematics, the topics we discussed are
naturally treated by constructive means such as direct
computation.



What are the *algebraic* and *order structures* on
\(\bR\) in SIA? As far as the former is concerned, there is little
difference from the classical situation: in SIA \(\bR\) is equipped
with the usual addition and multiplication operations under which it
is a field. In particular, \(\bR\) satisfies the condition that each
\(x \ne 0\) has a multiplicative inverse. Notice, however, that since
in SIA no microquantity (apart from 0 itself) is provably \(\ne 0\),
microquantities are not required to have multiplicative inverses (a
requirement which would lead to inconsistency). From a strictly
algebraic standpoint, \(\bR\) in SIA differs from its classical
counterpart only in being required to satisfy the principle of
infinitesimal 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 the
trichotomy law

 

\[x \lt y \wlor y \lt x \wlor x = y,\]


and accordingly \(\lt\) must be a *partial*, rather than a
*total* ordering. Since microquantities do not have
multiplicative inverses, and \(\bR\) is a field, any microquantity
\(\varepsilon\) must satisfy

 

\[\neg \varepsilon \lt 0 \wland \neg \varepsilon \gt 0.\]


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

 

\[\varepsilon \le 0 \wland \varepsilon \ge 0.\]


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

 

\[ J = \{x \in \bR : x \le 0 \wland x \ge 0\}\]


of elements neither less nor greater than 0. These three may be
thought of as the infinitesimal neighborhoods of 0 defined
*algebraically, logically, and order-theoretically*,
respectively.



In certain models of SIA the system of *natural numbers*
possesses some subtle and intriguing features which make it possible
to introduce another type of infinitesimal--the so-called
*invertible* infinitesimals--resembling those of
nonstandard analysis, whose presence engenders yet another
infinitesimal neighborhood of 0 properly containing all those
introduced above.



In SIA the set \(\bN \) of natural numbers can be defined to be the
smallest subset of \(\bR\) which contains 0 and is closed under the
operation of adding 1. In some models of SIA, \(\bR\) satisfies the
*Archimedean principle* that every real number is majorized by
a natural number. However, models of SIA have been constructed
(Moerdijk & Reyes 1991) in which \(\bR\) is not Archimedean in
this sense. In these models it is more natural to consider, in place
of \(\bN \), the set \(\bNs\) of *smooth natural numbers*
defined by

 

\[\bN^\* = \{x \in \bR: 0 \le x \wland \sin \pi x = 0\}.\]


\(\bNs\) 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 property *provided that in
the definition \(\bN \) is replaced by \(\bNs\)*. In these models,
then, \(\bN \) is a proper subset of \(\bNs\): the members of \(\bNs -
\bN \) may be considered *nonstandard integers*. Multiplicative
inverses of nonstandard integers are infinitesimals, but, being
themselves invertible, they are of a different type from the ones we
have considered so far. It is quite easy to show that they, as well as
the infinitesimals in \(J\) (and so also those in \(\Delta\) and
\(I)\) are all contained in the set--a further infinitesimal
neighborhood of 0--

 

\[K = \{x \in \bR: \forall n \in \bN (-\tfrac{1}{n} \lt x \lt \tfrac{1}{n})\}\]


of *infinitely small* elements of \(\bR\). The members of the
set

 

\[\In = \{x \in K: x \ne 0\}\]


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



Finally, a brief word on the *models* of SIA. These are the
so-called smooth toposes, categories (see entry on
 category theory)
 of a certain kind in which all the usual mathematical operations can
be performed but whose internal logic is intuitionistic and in which
every map between spaces is smooth, that is, differentiable without
limit. It is this "universal smoothness" that makes the
presence of infinitesimal objects such as \(\Delta\) possible. The
construction of smooth toposes (Moerdijk & Reyes 1991) guarantees
the consistency of SIA with intuitionistic logic. This is so despite
the evident fact that SIA is not consistent with classical logic.