#Êàáóë Î.Ê.
Theories of continuity and infinitesimals: Four philosophers of the nineteenth
century
KABUL ORALBAY KURMANBAYULY
KAZAKHISTAN
Theories of continuity and
infinitesimals: Four philosophers of the nineteenth century
Òåîðèè
ïðååìñòâåííîñòè è èíôèíèòåçèìàëåé: ×åòûðå ôèëîñîôîâ äåâÿòíàäöàòîãî âåêà
Ñàáàқòàñòûқ òåîðèÿñû ìåí èíôèíèòåçèìàëèÿ: Îí
òîғûçûíøû ғàñûðäûң òөðò ôèëîñîôû
The concept of
continuity recurs in many different philosophical contexts. Aristotle and Kant
believed it to be an essential feature of space and time. Medieval scholars
believed it to be the key to unlock the mysteries of motion and change.
Bertrand Russell believed that, while everyone talked about continuity, no one
quite knew what it was they were talking about.
The subject of this
article is mathematical continuity in particular. By mathematical continuity, I
mean continuity as it applies to or is found in mathematical systems such as
sets of numbers. Mathematical continuity is a relatively recent concern. The
need to address whether numerical systems are continuous came about with the
creation of calculus, specifically, of limit theory.
The article focuses on
four mathematicians/philosophers from the late nineteenth and early twentieth
centuries who were concerned with mathematical continuity. Richard Dedekind and
Georg Cantor, in the 1870s and 1880s, developed the concept of a
'point-continuum;' i.e. a continuum composed of discrete entities, such as a
collection of numbers arranged on a straight line. Paul du Bois-Reymond, in
1882, and Charles S. Peirce, especially in his post-1906 essays, criticized
this compositional point-continuum. Du Bois-Reymond believed infinitesimals
were necessary for continuity; Peirce believed no compositional continuum could
ever satisfy our intuitions.
My ultimate conclusions
are that (1) the concept of the mathematical point-continuum does suffer from
philosophical difficulties, (2) the concept of the infinitesimal is neither as
philosophically problematic nor as mathematically useless as is often charged,
but that (3) infinitesimals by themselves cannot solve the problems raised by a
compositional view of continuity.
Àííîòàöèÿ
Ïîíÿòèå íåïðåðûâíîñòè
ïîâòîðÿåòñÿ â ðàçëè÷íûõ ôèëîñîôñêèõ êîíòåêñòàõ. Àðèñòîòåëü è Êàíò ñ÷èòàëè åå
âàæíîé îñîáåííîñòüþ ïðîñòðàíñòâà è âðåìåíè. Ñðåäíåâåêîâûå ó÷åíûå ñ÷èòàëè, ÷òî
ýòî êëþ÷, ÷òîáû îòïåðåòü òàéíû äâèæåíèÿ è èçìåíåíèÿ. Áåðòðàí Ðàññåë ñ÷èòàåò,
÷òî, õîòÿ âñå ãîâîðèëè î ïðååìñòâåííîñòè, íèêòî òî÷íî íå çíàë, ÷òî èìåííî îíè
ãîâîðèëè.
Ïðåäìåòîì äàííîé ñòàòüè
ÿâëÿåòñÿ ìàòåìàòè÷åñêîå ïðååìñòâåííîñòü â ÷àñòíîñòè. Ïî ìàòåìàòè÷åñêîé
íåïðåðûâíîñòè, ÿ èìåþ â âèäó ïðååìñòâåííîñòü, êàê îíà êàñàåòñÿ èëè íàõîäèòñÿ â
ìàòåìàòè÷åñêèå ñèñòåìû, òàêèå êàê íàáîðû ÷èñåë. Íóæíî ðàçîáðàòüñÿ,
äåéñòâèòåëüíî ëè ÷èñëîâûå ñèñòåìû íåïðåðûâíîãî ïîÿâèëàñü ñ ñîçäàíèÿ
ìàòåìàòè÷åñêîãî àíàëèçà, â ÷àñòíîñòè, òåîðèè îãðàíè÷åíèé.
Ñòàòüÿ ôîêóñèðóåòñÿ íà ÷åòûðåõ ìàòåìàòèêîâ/ôèëîñîôîâ êîíöà
äåâÿòíàäöàòîãî è íà÷àëà äâàäöàòîãî âåêîâ, êîòîðûå áûëè ñâÿçàíû ñ ìàòåìàòè÷åñêîé
íåïðåðûâíîñòè. Ðè÷àðä Äåäåêèíä è Ãåîðã Êàíòîð â 1870-õ è 1880-õ ãã. ðàçðàáîòàíà
êîíöåïöèÿ 'òî÷êà-êîíòèíóóì;' ò. å. êîíòèíóóì ñîñòîèò èç äèñêðåòíûõ åäèíèö,
òàêèõ êàê íàáîð ÷èñåë, ðàñïîëîæåííûõ íà ïðÿìîé ëèíèè. Ïîëü Äþáóà-Ðåéìîí, â 1882
ãîäó, è ×àðëüç Ñ. Ïèðñ, îñîáåííî â ñâîåì ïîñòå-1906 ýññå, êðèòèêóåò ýòó
êîìïîçèöèþ òî÷êè-êîíòèíóóìà. Äþáóà-Ðåéìîí ñ÷èòàë áåñêîíå÷íî ìàëûå íåîáõîäèìû
äëÿ íåïðåðûâíîñòè; Ïèðñ ñ÷èòàë, íèêàêîé êîìïîçèöèîííîé êîíòèíóóì ìîãëè óäîâëåòâîðèòü íàøè èíòóèöèè.
Ìîè îêîí÷àòåëüíûå
âûâîäû, ÷òî (1) Ïîíÿòèå ìàòåìàòè÷åñêîé òî÷êè-êîíòèíèóìà, íå ñòðàäàåò ôèëîñîôñêèå
òðóäíîñòè, (2) Ïîíÿòèå áåñêîíå÷íî ìàëîé íè êàê ôèëîñîôñêè ïðîáëåìàòè÷íûì íè êàê
ìàòåìàòè÷åñêè áåñïîëåçíî, êàê ýòî ÷àñòî âçèìàåòñÿ, íî ÷òî (3) áåñêîíå÷íî ìàëûå
ñàìè ïî ñåáå íå ìîãóò ðåøèòü ïðîáëåìû, ïîäíÿòûå êîìïîçèöèîííûé çðåíèÿ
ïðååìñòâåííîñòè.
Àңäàòïà
Ñàáàқòàñòûқ òүñ³í³ã³ òүðë³ ôèëîñîôèÿëûқ
êîíòåêñòåðäå үçä³êñ³ç қàéòàëàíàäû. Àðèñòîòåëü ìåí Êàíòòûң îíû
êåң³ñò³ê æәíå óàқûòòûң ìàңûçäû åðåêøåë³ã³ äåï
ñàíàäû. Îðòàғàñûðëûқ ғàëûìäàð áұë қîçғàëûñ
ïåí өçãåð³ñò³ң құïèÿ құëïûí àøàòûí ê³ëò äåï
ïàéûìäàäû. Ðàññåë Áåðòðàííûң îéûíøà áàðëûғû ñàáàқòàñòûқ
òóðàëû àéòқàíûìåí, åøê³ì өç³í³ң íå æàéûíäà
ñөéëåãåíäåð³í á³ëìåéä³.
Îñû ìàқàëàíûң ìәí³
ìàòåìàòèêàëûқ ñàáàқòàñòûқ áîëûï òàáûëàäû.
Ìàòåìàòèêàëûқ үçä³êñ³çä³ê äåï ìàòåìàòèêàëûқ
ñàáàқòàñòûқ àéòàìûí, îғàí ìàòåìàòèêàëûқ
æүéåëåðìåí ñàíäàð æèûíòûғû ñèÿқòû òүñ³í³êòåð æàòàäû.
Ñàíäûқ æүéå үçä³êñ³çä³ã³ øûí ìәí³íäå
ìàòåìàòèêàëûқ òàëäàó, àòàï àéòқàíäà, øåêòåó òåîðèÿñûíûң
құðûëóûìåí ïàéäà áîëäûìà ñîíû қàðàñòûðó êåðåê.
Ìàқàëà îí òîғûçûíøû
ғàñûðäûң ñîңû ìåí æèûðìàñûíøû ғàñûðäûң
áàñûíäàғû ìàòåìàòèêàëûқ үçä³êñ³çä³êïåí áàéëàíûñòû òөðò
ìàòåìàòèê/ôèëîñîôòàðäûң åңáåêòåð³ìåí òұéûқòàëàäû.
Ðè÷àðä Äåäåêèíä æәíå Ãåîðã Êàíòîð 1870-1880 ææ. 'êîíòèíóóì-íүêòåñ³'
òұæûðûìäàìàñûí әç³ðëåãåí, êîíòèíóóì ñàíäàð æèûíòûғû ñåê³ëä³
ñûçûқòûқ æүéåäå îðíàëàñқàí äèñêðåòò³ á³ðë³ê. Ïîëü
Äþáóà-Ðåéìîí, 1882 æûëû, ×àðëüç Ñ. Ïèðñ, әñ³ðåñå өç³í³ң
ïîñò-1906 ýññåñ³íäå ñûí îñû êîíòèíóóì íүêòåñ³ êîìïîçèöèÿñûí ñûíғà àëäû.
Äþáóà-Ðåéìîí øåêñ³ç øàғûíäûқ үçä³êñ³çä³ê үø³í
қàæåòò³; Ïèðñ åøқàíäàé äà êîìïîçèöèÿëûқ êîíòèíóóì
á³çä³ң òүéñ³ã³ì³çä³ қàíàғàòòàíäûðà àëìàғàí áîëàð
åä³ äåï ñàíàäû.
Ìåí³ң ñîңғû қîðûòûíäûëàðûì, áұë (1)
Ìàòåìàòèêàëûқ êîíòèíóóì-íүêòåñ³ ұғûìû
ôèëîñîôèÿëûқ қèûíäûқòàðäàí çàðäàï øåêïåéä³, (2) Øåêñ³ç
ê³ø³ë³ê òүñ³í³ã³ äå ôèëîñîôèÿëûқ ïðîáëåìàëû, ìàòåìàòèêàëûқ
ïàéäàñûç äåï æè³ ëûíғàíûìåí, á³ðàқ áұë (3) øåêñ³ç ê³ø³ë³ê
êөòåð³ëãåí êîìïîçèöèÿëûқ ñàáàқòàñòûқ ìәñåëåëåð³
өçä³ã³íåí øåøå àëìàéäû.
The usual meaning of the
word continuous is “unbroken” or “uninterrupted”: thus a continuous
entity—a continuum—has no “gaps.” We commonly suppose that space and time
are continuous, and certain philosophers have maintained that all natural
processes occur continuously: witness, for example, Leibniz's famous
apothegm natura non facitsaltus—“nature makes no jump.” In mathematics the
word is used in the same general sense, but has had to be furnished with
increasingly precise definitions. So, for instance, in the later 18th century
continuity of a function was taken to mean that infinitesimal changes in the
value of the argument induced infinitesimal changes in the value of the
function. With the abandonment of infinitesimals in the 19th century this
definition came to be replaced by one employing the more precise concept
of limit.
Traditionally,
an infinitesimal quantity is one which, while not necessarily
coinciding with zero, is in some sense smaller than any finite quantity. For
engineers, an infinitesimal is a quantity 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.
An infinitesimal magnitude may be regarded as what remains after a
continuum has been subjected to an exhaustive analysis, in other words, as a
continuum “viewed in the small.” It is in this sense that continuous curves
have sometimes been held to be “composed” of infinitesimal straight lines.
Infinitesimals have a
long and colourful history. They make an early appearance in the mathematics of
the Greek atomist philosopher Democritus (c. 450 B.C.E.), only to be banished
by the mathematician Eudoxus (c. 350 B.C.E.) in what was to become official
“Euclidean” mathematics. Taking the somewhat obscure form of “indivisibles,”
they reappear in the mathematics of the late middle ages and later played an
important role in the development of the calculus. Their doubtful logical
status led in the nineteenth century to their abandonment and replacement by
the limit concept. In recent years, however, the concept of infinitesimal has
been refounded on a rigorous basis.
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
anindivisible 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 5th century B.C.E., 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 19th 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
17th 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 consciousness’s, 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
16th and 17th 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 B.C.E., only to be banished c. 350 B.C.E. 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”, which appears in
his LectionesGeometricae 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'Hôpital, who in 1696 published the first treatise on the
differential calculus (entitled Analyse des InfinimentsPetits pour
l'Intelligence des LignesCourbes), invokes the concept in postulating that “a
curved line may be regarded as being made up of infinitely small straight line
segments,” and that “one can take as equal two quantities differing by an
infinitely small quantity.”
However useful it may
have been in practice, the concept of infinitesimal could scarcely withstand
logical scrutiny. Derided by Berkeley in the 18th century as “ghosts of
departed quantities”, in the 19th century execrated by Cantor as
“cholera-bacilli” infecting mathematics, and in the 20th roundly condemned
by Bertrand Russell as “unnecessary, erroneous, and self-contradictory”, 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 19th century. By the
beginning of the 20th 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 Lie and 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
nonarchimedean 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 nineteen
sixties Abraham Robinson, using methods of mathematical logic, creatednonstandard
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 nineteen
seventies 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 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'Hôpital'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.
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 (see 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 B.C.E.), and Zeno (c. 460 B.C.E.) 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 divisionism.
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 B.C.E.)
and Democritus (b. 460–457 B.C.E.) 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 B.C.E.), 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”. 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”. 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 (see
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
B.C.E.) 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 B.C.E.) and Chrysippus
(280–206 B.C.E.) upheld the Aristotelian position that space, time, matter and
motion are all continuous (see Sambursky [1963], [1971]; 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 form the “binding” properties of the pneuma it
contains.
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 (see 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” 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 centre, 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” 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.”, Ockham claims that
“no part of the line is indivisible, nor is any part of a continuum
indivisible.” 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 14th 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.” 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
(see Murdoch [1957]).
The views on the
continuum of NicolausCusanus (1401–64), a champion of the actual infinite, are
of considerable interest. In his De MenteIdiotae 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.
Cusanus's realist
conception of the actual infinite is reflected in his quadrature of the circle
(see Boyer [1959], p. 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” (see Murdoch [1957], p.
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 16th and
17th 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 (see Baron [1987], pp. 108–116; Boyer [1969], pp.
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, at a single
stroke, the whole of infinity” 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 (see Boyer [1959]); indeed the “method of
indivisibles” remains associated with his name to the present day. Cavalieri
nowhere explains precisely what he understands by 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.
René 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 MeditationsDescartes 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 16th and 17th centuries there
issued the concept of infinitesimal number. The idea first appears in the
work of Pierre de Fermat (see Boyer [1959]) (1601–65) on the determination of
maximum and minimum (extreme) values, published in 1638.
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 centres of gravity.
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 LectionesGeometricae 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[19] : De
analysi per aequation esnumero 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 ova 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”, 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 afluent, and its rate of
generation a fluxion. The fluxion of a fluent x is denoted
by x·, and its moment, or “infinitely small increment accruing in an infinitely
short time o”, by x·o. The problem of determining a tangent to a
curve is transformed into the problem of finding the relationship between the
fluxions x· and 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 = xn, Newton first forms z· + z·o =
(x· + x·o)n, expands the right-hand side using the binomial theorem,
subtracts z = xn, divides through by o, neglects all terms
still containing o, and so
obtains z· = nxn−1 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 quadraturacurvarum 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 rigour. The first employed
infinitesimal quantities which, while not finite, are at the same time not
exactly zero. Finding that these eluded precise formulation, Newton focused
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[20] (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
1959, p. 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.” 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 facitsaltus. 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 Methodusof 1684 and De GeometriRecondita 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:
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 ddx = d2x, d3x = dd2x, and in
general dn+1x = ddnx. 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)th-order
differentials dn+1x and the nth-order differentials dnx. He
also assumed that the nth power (dx)n of a first-order
differential was of the same order of magnitude as an nth-order
differential dnx, in the sense that the quotient dnx/(dx)nis a finite
quantity.
For Leibniz the
incomparable smallness of infinitesimals 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.” 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'Hôpital (1661–1704), of the first expository book on the subject,
the Analyse des InfinimentsPetits Pour L'Intelligence des LignesCourbes.
This is based on two definitions:
Variable quantities are
those that continually increase or decrease; and constant or standing
quantities are those that continue the same while others vary.
The infinitely small
part whereby a variable quantity is continually increased or decreased is
called the differential of that quantity.
And two postulates:
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.
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.
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 = x2. 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, dx2) must lie on the axis of abscissae, which means
that dx2 = 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 dx2 = 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 e2 = 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 seccond 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 asx. 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.
The leading practitioner
of the calculus, indeed the leading mathematician of the 18th century, was
Leonhard Euler[23] (1707–83).
Philosophically Euler was a thoroughgoing synechist. Rejecting
Leibnizianmonadism, he favoured the Cartesian doctrine that the universe is
filled with a continuous ethereal fluid and upheld the wave theory of light
over the corpuscular theory propounded by Newton.
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 α, α · 0 = 0, Euler maintained that the quotient 0/0 could represent any
number whatsoever[24].
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 Rondd'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 18th 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 18th 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
ofesseestpercipi and his denial of general ideas, was a persistent critic
of the presuppositions underlying the mathematical practice of his day (see
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[25],
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?
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”, 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 be 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.[26]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.
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
Δ possible. The construction of smooth toposes (see Moerdijk and 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.
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