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Dictionary of the History of Ideas | ||

#### MATHEMATICS IN

CULTURAL HISTORY

The history of mathematics is but a thin ribbon across

the fabric of general history, that is, of the cultural

history of events, insights, and ideas. Yet there are

major problems of general history that are meaning-

fully refracted in the history of mathematics, and one

such problem, on which we shall concentrate, is the

problem of explaining the decline of ancient civili-

zation in the West. On the other hand, it may happen

that mathematics is materially involved in a problem

in general history, and yet cannot contribute to its

illumination; one such problem, on which we intend

to comment, is the problem of the rise and spread of

tions in which the judgment of mathematics on the

eminence of an era is different from that of other fields

of knowledge, even of physics proper, and in several

brief preliminary sections we shall rapidly review a

few problems of this kind from periods beginning with

and following upon the Renaissance.

But before beginning our reviews we wish to state

that the problems to be encountered will be formulated

in broad, summary, and even simplistic terms. This is

to be expected. As a mode of rational cognition mathe-

matics appears rather early and is quite central, but

as a mode of intellectual activity it is rather primitive.

For this reason mathematics is effective because of its

strength rather than its delicacy, even when involved

in sensibilities. Therefore, when participating in the

analysis of problems from general history, the history

of mathematics is at its best when the problems are

stated in large, manifest, even crude fashion rather than

in a localized, esoteric, and delicate manner.

*The Renaissance.* There is a much-studied problem

whether and in what sense there indeed was a Renais-

sance—in the fifteenth and sixteenth centuries, say—as

institutionalized by Jacob Burckhardt; that is, whether

there is a marked-off era which interposes itself be-

tween medieval and modern times. (For a history of

the problems see W. K. Ferguson, K. H. Dannenfeldt,

T. Helton, also E. Panofsky.) George Sarton, the leading

historian of science, once made the statement (which

he later greatly modified) that “from the scientific point

of view, the Renaissance was *not* a renaissance” (quoted

in Dannenfeldt, p. 115). Of course, nobody would deny

that there was a Copernicus in the sixteenth century

or that the introduction of printing created a great

spurt in many compartments of knowledge, scientific

or other; but the question, which Sarton's statement

was intended to answer, is whether during the era of

the Renaissance, disciplines like physics, chemistry,

biology, geology, economics, history, philosophy, etc.,

were developed in a manner which sets these two

centuries off from both the Middle Ages and the seven-

teenth century.

Now, with regard to mathematics no such doubts

need, or even can arise. There was indeed a mathe-

matics of the Renaissance that was original and dis-

tinctive in its drives and characteristics. Firstly, there

was a school, mainly, but not exclusively, represented

by Germans (Peurbach, Regiomontanus, and others),

that sharply advanced the use of symbols and notations

in arithmetic and algebra. Secondly, and strikingly, an

Italian school sharply advanced the cause of the alge-

bra of polynomial equations when it solved equations

of the third and fourth degree in terms of radicals

(Scipione del Ferro, Ludovico Ferrari, Nicolo Tartag

lia, Geronimo Cardano). Thirdly, a French school,

mainly represented by François Viète, achieved a syn-

thesis of these two developments. And finally, an “in-

ternational” school laid the foundation for the eventual

rise of analysis by introducing and studying two special

classes of functions, trigonometric functions and loga-

rithms. Regiomontanus, Rheticus, Johann Werner, and

later Viète gradually made trigonometry an inde-

pendent part of mathematics, and Henry Briggs and

John Napier (and perhaps also Jost Bürger) created the

logarithmic (and hence also exponential) functions.

After that, for over three centuries trigonometric and

logarithmic functions were the stepping-stones leading

to the realm of analysis, for students of mathematics

on all levels.

There is a near-consensus among social historians

that the rise of arithmetic and algebra during the

Renaissance was motivated “preponderantly by the rise

of commerce in the later Middle Ages; so that the

challenges to which the rise of algebra was the response

were predominantly the very unlofty and utilitarian

demands of counting houses of bankers and merchants

in Lombardy, Northern Europe and the Levant”

(Bochner, *Role*..., p. 38). On this explanation, the

socioeconomic needs that were thus satisfied were not

those of the “industrialist” but those of the merchant,

in keeping with the fact that in the late Middle Ages,

and soon after, the general economy was dominated

not by the producer but by the trader (ibid.).

*The Seventeenth Century.* There was one thing that

the mathematics of the Renaissance era did not

achieve. It did not continue creatively the mathematics

of Euclid, Archimedes, and Apollonius, even if it did

begin to translate and study their works. The exploita-

tion in depth of Greek mathematics was achieved only

in the seventeenth century, in the era of the Scientific

Revolution. It is this development that created the

infinitesimal calculus, and thus molded the world image

which we have inherited today. For instance, Johannes

Kepler anticipated the infinitesimal calculus in two

ways; by his approach to Archimedean calculations of

volumes, and also by his manner of reasoning which

led him to conclude that planetary orbits are not circles

but ellipses; and without his intimate knowledge of

Apollonius this conclusion would have hardly come

about. Next, René Descartes' avowed aim in his *La
Géométrie* was to solve or re-solve geometrical prob-

lems of the great Greek commentator Pappus (third

century A.D.) who was groping for topics and attitudes

in geometry that were beyond his pale. Now, Des-

cartes' technical equipment was the operational appa-

ratus of Viète, and it was this fusion of Viète with

Pappus that created our coordinate and algebraic ge-

ometry. Next, Isaac Barrow, the teacher of Isaac

(theory of proportion in lieu of our theory of positive

real numbers) for the eventual unfolding of analysis.

And, finally, Newton himself, by an extraordinary

tour-de-force, pressed his forward-directed

*Principia*

(

*Principles of Natural Philosophy*) into the obstructive

mold of backward-directed but powerful Archimedism.

That this impressment was even harder on Newton's

readers than on Newton is attested to by a memorable

statement (in the nineteenth century) of William

Whewell:

The ponderous instrument of synthesis [meaning Archimed-

ism], so effective in his hands, has never since been grasped

by one who could use it for such purposes; and we gaze

at it with admiring curiosity, as on some gigantic implement

of war, which stands idle among the memorials of ancient

days, and makes us wonder what manner of man he was

who could wield as a weapon what we can hardly lift as

a burden

(Whewell, I, 408).

To sum it up, it was not the Renaissance proper of

the fifteenth and sixteenth centuries that saw the

“rebirth” of antiquity as much as it was the succeeding

Scientific Revolution of the seventeenth century. What

the Renaissance did was to contribute a prerequisite

algebraic symbolism, and this was something that, from

our retrospect, antiquity might have additionally pro-

duced out of itself but did not.

*The Age of Enlightenment.* The merits and achieve-

ments of this age are subject to most diverse and diver-

gent interpretations. For instance, Carl Becker, in a

small but unforgettable book, *The Heavenly City of
the Eighteenth-century Philosophers* (1932), proposed

that, on the whole, the intellectual attitudes of the age

were medieval rather than modern, Voltaire or no

Voltaire. Notwithstanding assertions by some historians

of science, the advancement in physical science was

rather circumscribed. For instance, the eighteenth

century did not achieve much in the theory of elec-

tricity, at any rate not before Henry Cavendish and

C. A. Coulomb, who were active towards the end of

the century after the sheen of the Enlightenment had

already been dimmed. Also, optics virtually stood still

for over a century. After the lustrous works of Newton,

Christiaan Huygens, and others in the seventeenth

century almost nothing memorable happened in optics

till the early part of the nineteenth century when

Thomas Young, Étienne-Louis Malus, Augustin Fresnel,

William Hyde Wollaston, and Joseph von Fraunhofer

began to crowd the field. Furthermore, Immanuel

Kant's famous

*Critiques*came toward the end of the

eighteenth century, after the “true” Enlightenment had

begun to fade; and Kant's predecessors in metaphysics

earlier in the century had been not a whit better than

Kant had depicted them.

But the eminence of mathematics in the era of

Enlightenment is clear-cut. It was a very great century.

Mathematicians like G. W. Leibniz, Jacob and John

Bernoulli, A. C. Clairaut, L. Euler, J. le R. D'Alembert,

P. L. M. de Maupertuis, J. L. Lagrange, and P. S. de

Laplace made it as distinctive a century as any since

Pythagoras among the Greeks, or even since the age

of Hammurabi in Mesopotamia. Also the age had one

feature that made it simply unique. It fused mathe-

matics and mechanics in a manner and to a degree

that were unparalleled in any other era, before or after.

Also, in mathematics, far from being a “medieval” age

as in Carl Becker's conception of the eighteenth cen-

tury, it was a very “modern” age. Monumental as

Newton's *Principia* (1686) may have been, it is Lag-

range's *Mécanique analytique* (1786) that became the

basic textbook of our later physical theory. Lagrange's

treatise is old-fashioned, but *readable,* Newton's treatise

is “immortal,” but antiquarian, and very difficult.

It is not easy to follow in depth the growth of this

mathematics in relation to other developments of the

era. Socioeconomic motivations do not account for its

high level, and there are no explanations from general

philosophy that are convincing. Immanuel Kant, for

instance, shows no familiarity at all with the advanced

mathematics of his times.

*The Nineteenth Century.* In the history of knowl-

edge, even more complex than the era of Enlighten-

ment is an era following it. It is, schematically, the

half-century centered in the year 1800, that is the

half-century 1776-1825, which has been called the Age

of Eclosion by Bochner. During this era there was a

great and sudden outburst of knowledge in all academic

disciplines; and in historical studies of various kinds,

there emerged a “critical” approach to the evaluation

of source material, which is sometimes called “higher

criticism.” In mathematics there was no such sudden

increase of activity, but an analogue to the “higher

criticism” did begin to manifest itself. A critical ap-

proach to so-called foundations of mathematics began

to spread, and even novel patterns of insight were

emerging. Thus, C. F. Gauss in his *Disquisitiones arith-
meticae,* which appeared in 1801, explicitly formulated

the statement that any integer can be represented as

a product of prime numbers, and uniquely so. Implic-

itly the statement is already contained, between the

lines, in Book 7 of Euclid's

*Elements.*But explicitly

the statement is a kind of “existence” and “uniqueness”

theorem, for which even number theorists like Fermat,

Wallis, Euler, and Lagrange were “not ready” yet.

Afterwards, in the course of the nineteenth century,

mathematics filled all those “gaps” in Greek mathe-

matics, which the Greeks themselves could not or

would not close. Thus, the nineteenth century finally

the role of axioms in general, as well as in geometry;

and it finally constructed a “Euclidean” space as a

background space for (Greek) mathematical figures and

astronomical orbits. In a peculiar logical sense, Greek

mathematics was “completed” only around 1900, but

by the developments which brought this about, it was

also rendered “antiquarian” by them. On the other

hand, “humanistic” Greek works like those of Homer

and Aeschylus, Herodotus and Thucydides, and Plato

and Aristotle were not “completed” in a similar sense,

but they have, on the other hand, not been rendered

“antiquarian,” as shelves full of reissues of these works

in paperback can attest.

This completes our preliminary observations, and we

now turn to two topics from antiquity for a somewhat

less summary analysis.

*Phonetic Writing.* We are taking it for granted that

within the history of civilization there is a correlation,

an important one, between the rise of organized

knowledge and the emergence of adequate writing, the

degree of adequacy being measured by the degree of

phonetic articulation. With a suitable definition of

“phonetic,” we may say that Chinese writing is a fully

developed phonetic system, and has been so since its

appearance about the middle of the second millennium

B.C. (Gelb, p. 85). Now, mathematics is the oldest

organized knowledge there is, or nearly so, and our

problem is the task of assessing the role of mathematics

in this correlation between knowledge and writing.

On hard evidence, the presence of organized mathe-

matics is first attested around 1800 B.C. Now, our first

intricacy is the fact that this first evidence for organ-

ized mathematics appears in two separate areas simul-

taneously, in Mesopotamia and in Egypt (Neugebauer,

Chs. 1-3). This simultaneity of appearance cannot be

readily explained by invoking a “cultural diffusion” or

only a “stimulus diffusion” because by intent and con-

tent these two mathematics are very different from

each other (Neugebauer, loc. cit.; van der Waerden,

Chs. 1-3). Both these systems of mathematics are

documented by writing that is highly organized (see

Gelb, pp. 168ff., for Egypt, and Kramer, p. 306, for

Mesopotamia), but of seemingly different provenance.

In both geographical areas the mathematics in use

appears to be a full part of organized knowledge in

general. Also, from our retrospect, the Mesopotamian

mathematics even gives the impression of having been

in the very vanguard of organized knowledge, although

to an “average Mesopotamian” of the era, the code

of Hammurabi may have been more important, and,

above all, much more familiar.

At this point it might be expected that if writing

and knowledge reach a certain stage of organization

conjointly, then organized mathematics would do so

too. But this would be a very hasty expectation, as can

be seen from the development in an area that geo-

graphically and intellectually was very proximate to

both Mesopotamia and Egypt, namely, the land of the

Bible, where writing of various modes was organizing

itself in the latter half of the second millennium B.C.,

and in this area it was even advancing towards its

ultimate stage, namely, to the stage of becoming en-

tirely alphabetical (Gelb, pp. 134-53). At the same

time, juridical, ethical, and sacerdotal knowledge was

organizing itself too, and much of it became ultimately

knowledge for the ages. Yet, the history of mathematics

knows absolutely nothing about an indigenous mathe-

matics also springing up in this area at the same time.

This is our most serious intricacy.

A different problem arises—and it is a research task

rather than a conundrum—when we take into account

Greek achievements, from the first millennium B.C. It

is a fact that the Greeks made writing fully alphabeti-

cal. They thus created “a writing which expresses the

single sounds of a language” (Gelb, p. 197), and their

script was undoubtedly more advanced than Babylon-

ian script from around 1800 B.C. Similarly, the mathe-

matics which the Greeks began shaping within their

own thought patterns in the sixth century B.C. was,

even from the first, undoubtedly more advanced than

the Babylonian mathematics from around 1800 B.C.

Now, the research task arising is to determine whether

these two advances are commensurate in extent. Greek

mathematics drew, as heavily as it could, on all the

accumulated mathematics that preceded it. Never-

theless, it is immediately clear that the Greek intellec-

tual innovation in organized mathematics—as also in

organized knowledge of any kind—was far greater than

the parallel advance in the phonetic quality of writing.

An analysis in depth might counteract this impression.

But such an analysis could not be an easy one, because

it would also have to account for the fact that modern

mathematics is immeasurably superior to the Greek

creation, although “from the Greek period up to the

present nothing has happened in the inner structural

development of writing” (Gelb, p. 184).

Finally, we wish to observe that both for organized

mathematics and organized writing it is equally diffi-

cult to decide whether in China either came into being

independently of the West, or in direct dependence

on the West, or by a combination of the two possi-

bilities at different times. For organized mathematics

the problem of its geographic propagation in Asia was

already known, more or less, in the nineteenth century,

perhaps under the impact of the corresponding prob-

lem for language as it is spoken. In the twentieth

century the problem for mathematics has received less

early or organized, probably because for mathematics

the problem has been less able to exploit achievements

in archeology than those in writing. For instance, for

writing there is a balanced account of the problem in

Gelb, Ch. VIII, but there seems to be no analogous

account for mathematics.

*Dissolution of Ancient Civilization.* By “ancient

civilization” we mean, as usual, the large conglom-

eration of component civilizations of so-called antiq-

uity that severally came into being in and around the

Mediterranean Littoral. In the last centuries B.C. and

first centuries A.D., the conglomeration merged into one

compound civilization, creating “one civilized world,”

the self-styled *oikouméne.* The passing-away of this

civilization constitutes one of history's greatest prob-

lems. The central part of this problem is the seemingly

“formal” question of determining when ancient civili-

zation terminated, if indeed it did “terminate”; or,

rather, since the termination was probably a process

of some duration, when the process of termination

began and when it ended.

It is widely accepted that the process ended towards

the close of the fifth century A.D., as symbolized by

the fact that in 476 A.D. “the last claimant to the

Roman throne in the West was deposed” (Kagan, p.

viii). An almost lone dissenter among historians, but

a leading one, is Henri Pirenne, who maintains that

ancient civilization was brought to an end—and then

rather violently so—only in the seventh century A.D.,

by the widespread militancy of Islam. Ancient civili-

zation was then not actually “killed,” but rather pushed

away from its Mediterranean habitat into more north-

ern parts of Europe, where it became isolated and

immobilized for centuries (Havighurst, 1958).

Another kind of dissent, a qualified one, is caused

inevitably by the fact that there was an Eastern Roman

Empire which lasted very long, until 1453 A.D. Most

Byzantinists of today find that this empire was very

viable in 476 A.D. and beyond, until the onset of the

Crusades at any rate. They do not deny that there was

a ruin of the Western Empire in the fifth century, but

most of them do not allow that this ruin came about,

to a meaningful degree, by decay from within. They

argue that the causes for decay from within would have

been present also in the Eastern half of the Empire,

and should have had the same destructive effect at the

time (Jones, Ch. 24; J. B. Bury, quoted in Kagan, pp.

7-10), so that, in their view, only military and other

external causes remain. A very outspoken champion

of this view is Lynn White, Jr., and he is severely

critical of the details, attitudes, and emphases in

Edward Gibbon's key work, *Decline and Fall of the
Roman Empire* (White, pp. 291-311).

With regard to the beginning of the process of the

decline of ancient civilization widely spaced dates have

been proposed, explicitly or by implication. F. W.

Walbank finds “... the germs of the illness of antiquity

already present in the Athens of the fifth century B.C.”

(Kagan, p. viii). This is a very early date indeed. Less

extreme is the finding of M. I. Rostovtzeff that “decline

began as early as the second century B.C.” (Kagan, p.

2). Finally “according to Gibbon, the Roman Empire

reached its zenith in the age of the Antonines [second

century A.D.] after which the decline set in” (White,

p. 25). Thus, by implication, in the view of Gibbon

the decline set in around 200 A.D., after the era of the

“Five Good Emperors” (90-180 A.D.).

We proceed now to review the cogency of the above

dates in the light of the history of mathematics.

The history of mathematics fully corroborates the

familiar textbook assertion that around 500 A.D. a

large-scale decline occurred. Mathematics as an intel-

lectual activity—as an academic discipline, so to

speak—was suddenly lost from sight, as if swallowed

up by a wave of a flood, or buried by a sandstorm.

In the Latinized West there had been a mathematics

bearing the telltale mark of a Greek heritage even

when dealing with non-Greek or extra-Greek topics

of the mathematical corpus. It was this kind of mathe-

matics that suddenly disappeared. Also in the West,

this kind of mathematics came fully to surface only

after seven centuries or so, in the famous *Liber abaci*

of Leonardo of Pisa (Fibonacci) around 1200 A.D. This

is not to say that in the intervening centuries mathe-

matics was unknown in the West. A mathematics of

a kind was of course included in school curricula. There

were some translations from the Arabic, especially

during the so-called twelfth-century Renaissance.

There was also a certain pursuit of “utilitarian” math-

ematics, even during the so-called Carolingian Renais-

sance (Smith, I, 175-220). But, before the work of

Fibonacci, this pursuit did not evince a quest for the

kind of originality, if only on a modest scale, with

which mathematics had been imbued since the sixth

century B.C., when the Greeks had begun to weave

mathematics into the texture of their rationality.

This decay of Greek mathematics did not spare the

Eastern Roman empire, which, by an official reckoning,

lasted from 529 to 1453, from Justinian to the fall of

Constantinople. As far as mathematics is concerned this

Empire might have never been. There is an encyclo-

pedic treatise which lists the extant Byzantine writings,

including works on mathematics and cognate subjects

(Krumbacher, pp. 620-26). The mathematical works

of the collection bespeak sterility and stagnation, and

only in the art of warfare were there some elements

of originality (Taton, p. 446).

Finally, we note that, in the judgment of a leading

student of chronological innovations (Ginzel, III,

178ff.), at the beginning of the sixth century A.D. the

Roman Canonist Dionysius Exiguus founded our

present-day system of designating years by A.D. and

B.C. This event, even if conceived very modestly by

its author, was, from our retrospect, mathematically

tinged, and it took place in Rome, after its “official”

fall, even if not long after it. Thus, this may be viewed

as a mathematical corroboration of the fact—which

was stressed, from very different approaches, by Henri

Pirenne, Lynn White, and probably others—that the

political fall of Rome in 476 A.D. was not, instantly,

also a social and intellectual disintegration.

This achievement of the Roman Canonist, modest

as it may appear to be when viewed in isolation, cannot

be overestimated as a determinant of history. The

Greeks never quite succeeded in introducing a com-

parable dating of years. Mathematically this would

have amounted to introducing a coordinate system on

the time axis, and the Greeks never achieved this.

Furthermore, it is remarkable that the “Christian” era

of Dionysius not only became a “common” era but

has turned out to be the most durable era ever. The

French Revolution of 1793, the Russian Revolution of

1917, the Italian Revolution of 1922, and the German

Revolution of 1933, each attempted to introduce a new

era beginning with itself, but none really succeeded

or even made the attempt in earnest. Our “common”

era, however “Christian” by origin, has become a

standard institution that cannot be tampered with.

We now turn to the question of mathematical evi-

dence for the beginning of the decline of ancient civi-

lization. Firstly, mathematics clearly concurs with the

assertion of Rostovtzeff that a general decay began in

the second century B.C. Secondly, mathematics can

offer no tangible corroboration of the fact, known from

general history, that life in the Greco-Roman world

was much bleaker in the third century A.D. than in

the preceding one. Thirdly, and finally, mathematics

can even corroborate the thesis of Walbank that germs

of some of the illnesses of antiquity can already be

found in the Athens of the fifth century B.C.; namely,

by a tour-de-force, we may elicit from the nature of

Greek mathematics some peculiar comment of the

thesis, which can be interpreted as a corroboration of

it, in a sense.

Greek mathematics built on a considerable body of

mathematics that had preceded it, but it was never-

theless a singular achievement of ancient civilization

as a whole, and a hallmark of its Hellenic aspect in

particular. Naively or boldly, the Greeks made a fresh

start. They were inspired to begin from a new begin-

ning and they succeeded. They erected an edifice of

mathematics that was a veritable “system” in our

present-day sense of the word. The bricks and stones

in the edifice may have been Egyptian, Babylonian,

or other, but the structure was Greek. This Greek

mathematics attained its intellectual acme in the

achievements of Archimedes. Isaac Newton even com-

posed his *Principia* in an Archimedean *mise-en-scène,*

but he acknowledged no indebtedness to an Egyptian

calculus of fractions, or even a Babylonian calculus of

quadratic equations, even if he knew anything about

them at all.

From what is known, this Greek mathematics

showed the first signs of being itself around 600 B.C.

It then grew and kept unfolding for about four centur-

ies, that is, till about 200 B.C., and the last of these

four centuries, that is the era from 300 B.C. to 200 B.C.,

was a culminating one. In fact, around 300 B.C. Euclid

composed his *Elements,* Archimedes flourished around

250 B.C., and around 200 B.C., Apollonius produced his

monumental *Conics.*

But after that, in the second century B.C., unexpect-

edly and inexplicably, as if on a signal, the development

of this mathematics came almost to a halt. After 200

B.C. it began to level off, to loose its impetus, and then

to falter, bringing to the fore only such works as those

of a Nicomedes, Dioclos, and Hypsicles. The phenom-

enon was no passing setback but a permanent recession,

the beginning of a decline. It was, for mathematics,

the beginning of the end in the conception of Ros-

tovtzeff, even if the true end itself, that is the final

extinction of Greek mathematics in its own phase,

came only considerably later, around 500 A.D., that is,

around the time of the fall of Rome.

It is noteworthy though, that the second century B.C.,

as if to almost show that it was not entirely down and

out, produced the astronomer Hipparchus, famed dis-

coverer of the precession of the equinoxes; and that

the second century A.D., as if to lay claim to being

indeed a “good” century, brought forth his great “suc-

cessor” Claudius Ptolemy, author of the majestic

*Almagest,* and of a *Geography.* It must be stated how-

ever, emphatically, that in “basic” mathematics

Ptolemy was in no wise farther along than Archimedes,

even if the *Almagest,* as an astronomical text, was a

live text still for Copernicus in the first half of the

sixteenth century and began to become antiquarian

only half a century later in consequence of the mathe-

matically articulated innovations of Kepler.

*Limitations of Greek Mathematics.* The reasons that

have been variously adduced for the dissolution of

ancient civilization—the overextension of the *oikou-
méne* so that “the stupendous fabric yielded to the

pressure of its own weight”; inadequacy of industriali-

zation and too much involvement with slave labor;

gradual absorption of the educated classes by the

masses”; “the pitiful poverty of Western Rome”; etc.

(Kagan, pp. xi and xii)—may all help to account for

the

*ultimate*extinction of Greek mathematics, around

500 A.D., after a gradual decline of long duration. One

may even add the view of J. B. Bury that “the gradual

collapse... was the consequence of

*a series of con-*

tigent events.No general cause can be assigned that

tigent events.

made it inevitable” (ibid.).

But, in the case of mathematics there is one peculiar

fact which no such reasons from general history can

really account for. It is the fact that in the second

century B.C., much before the ultimate extinction, the

decline of mathematics from the heights which it had

attained in the preceding century was seemingly too

large, too brusque, and too unmotivated by internal

developments to be satisfactorily explained by general

reasons of this kind. By standard criteria of advance-

ment, mathematics in the third century B.C. was in a

state of upward development, and it suggests itself that

the rather sudden break in the development after 200

B.C. may have been due, at least in part, to some

particular reasons applying to mathematics only. This

is indeed our suggestion, and we shall attempt to for-

mulate it.

In the third century B.C., Greek mathematics was

not only very good, but it also reached a climax. By

this we mean that it reached a level of development

that was maximal relative to the intellectual base,

mathematical and philosophical, on which it had been

erected and on which it rested. Therefore, mathematics

could have continued to develop in the second century

B.C. and later only if the overall intellectual base on

which it rested could also have been broadened in the

process. But of this kind of broadening of the total

intellectual setting of mathematics, Greek civilization

in the second century B.C. was no longer capable. The

general intellectual basis for Greek mathematics, which

in a sense never broadened or deepened, was laid in

the sixth and fifth centuries B.C., especially the latter,

and in this peculiarly conceived sense it can be said

that, as far as mathematics is concerned the decline

of Greek civilization reaches back even into the fifth

century B.C. (Walbank).

In order to demonstrate that the mathematics of

Archimedes and Apollonius was overripe relative to

its intellectual basis we shall compare the conceptual

setting in Archimedes and Apollonius with the corre-

sponding setting in Newton's *Principia* (1686), even if

Newton's work came nineteen centuries later. A com-

parison of the works of Archimedes and Apollonius

(and Pappus) with *La Géométrie* (1637) of Descartes,

which was published half a century before the *Prin*

*cipia,* would not serve our present purpose, because

Descartes does not retain the setting of antiquity. On

the contrary, he radically changed the technical setting

by a full recourse to the apparatus of algebraic sym-

bolism as made ready for him by Viète. Not so Newton.

He was most expert in the handling of this apparatus,

and on occasions he employed it more penetratingly

than Descartes and others; but, for reasons best known

to himself he elected to cast the *Principia* in a mold

of Archimedean technicalities, outwardly, that is. In

the *Principia* there are hardly any analytical formulas;

but there are circumlocutions and verbalized formulae

which, at times, seem to be as condensed and stero-

typed as in Archimedes. This makes for hard reading

nowadays, but it makes it easy to isolate differences

of approach and setting. The differences are enormous,

and we list the following ones.

Newton prominently introduced an underlying

overall space, his absolute space, as a background space

for both mathematics and mechanics. The Greeks

achieved nothing like it. They certainly did not intro-

duce a space for mechanics and mathematics jointly.

They did introduce a “place” for events in nature

which perhaps served as a space of mechanics, but they

most certainly did not ever introduce a space of math-

ematics, or any kind of space of perception, physical,

logical, or ontological. In mathematics, they had “loci”

for individual figures when constructed, but not a space

for such figures before being constructed. In short, the

Greeks did not have any kind of space in the sense

of Descartes, or Newton, or John Locke.

Newton expressly introduced in his mechanics a

*translational momentum* (quantity of motion), defining

it, for a mass particle moving on a straight line, as

the product *m · v* in which the factor *m* is the constant

amount of mass of the particle and *v* is its instantaneous

velocity. Archimedes, in his theory of the lever, ought

to have introduced the conception of a rotational mo-

mentum, defining it as the product *l · p* in which the

factor *l* is the length of an arm of the lever and *p* is

the weight suspended from this arm. But Archimedes

did not introduce such a concept, nor did Greek math-

ematical thought ever conceptualize a product like

*l · p;* and mechanics went on marking time for almost

2000 years.

Even more significantly, Newton had the concept

of a function constantly in his thinking, however cov-

ertly. Altogether since the seventeenth century the

concept of a function kept on occurring in many facets

and contexts, in mathematics as well as in other areas

of cognition. Greek cognition, however, never had the

notion of function, anywhere. Even the absence of

products like *l · p* from Greek thinking was part of the

absence of functions, inasmuch as in mathematics of

*l · p*for variable values of

*l*and

*p,*

is a function on the set of pairs (

*l,p*). More centrally,

in cognition today the most important component of

the concept of function is the notion of

*relation,*how-

ever elusive it may be, to define or even describe what

a relation is. Aristotle, the creator of the academic

discipline of logic, did not anticipate the importance

of

*relation*(which he terms

*pros ti*), nor did the Stoic

logicians after him. But in modern developments, the

creation of an algebra or logic by the American

philosopher-logician Charles Sanders Peirce was his

most outstanding logical achievement.

Operationally, functions occur in Newton's *Principia*

in the following way. If a mass particle moves on the

*x*-axis and *t* denotes the time variable, then Newton

covertly assumes that there is a function *x*(*t*) which is

the instantaneous distance of the particle from a fixed

origin. He forms the derivative *dx/dt* for variable *t,*

which is a new function *v = v*(*t*). It is the instantaneous

velocity of the motion. He multiplies this by the con-

stant value *m* of the mass, thus introducing the instan-

taneous quantity of motion *m · v*(*t*), which is again a

function in the variable *t.* Newton then crowns these

covert assumptions with the hypothesis, which is ap-

parently due to himself, that the force *F* which brings

about the motion is, at every instant, equal to the rate

of change of the quantity of motion, *F = d*(*mv*)/*dt.*

This hypothesis, coupled with Newton's specific law

of gravitation, created our exact science of today. The

Greeks did not conceive of any part of this entire

context of assumptions and hypothesis, not because

they were unable to form a derivative of a function,

but because they did not have in their thinking the

concept of a function that is a prerequisite to forming

the various derivatives involved. By maturity of insight,

Archimedes was better equipped than Newton to carry

out the limit process that is involved in the formation

of a derivative, if only the concept of function and

the entire prerequisite setting had been given to him.

The Greek lack of familiarity with the concept of

function does not manifest itself only in mathematical

mechanics, which, to the Greeks was a relatively eso-

teric topic, but also in the entire area of geometry,

which, by a common conception, was a stronghold of

Greek rationality. There is a purely geometrical con-

text, common to Archimedes and Newton, in which

Newton does, and Archimedes does not have functions

in his thinking. Namely, Newton views the tangent to

a curve at a point of the curve as the limiting position

of a secant through the fixed point and a variable point

of the curve, so that, in effect, he performs the opera-

tion of differentiation on “hidden” coordinate func-

tions. Greek mathematics, however, never broke

through to this all-important view, but persisted in the

view, known from Euclid, that a tangent to a curve

is a straight line which in its entire extent coincides

with the curve at one point only. Archimedes tries to

adhere to this Euclidean definition even in his essay

on (Archimedean) spirals, in spite of the complication,

of which he is apparently aware, that any straight line

in the plane of the spiral intersects it in more than

one point. Without putting it into words, Archimedes

overcomes the complication by a simple adjustment,

but he does not advance towards the modern concep-

tion of a tangent as in Newton.

*Epilogue.* In modern mathematics the Greek limita-

tions which we have adduced were overcome mainly

by conceptual innovations, namely by the creation of

abstractions, and of escalations of abstractions, which

do not conform with the cognitive texture of Greek

classical philosophy and general knowledge. There is

an all-pervading difference between modern mathe-

matical abstractions and, say, Platonic ideas; reductions

of the one to the other, as frequently attempted in

philosophy of mathematics, are forced and unconvinc-

ing. There are analogies and parallels between the two,

but not assimilations and subordinations. The Greeks

could form the (Platonic) idea of a “general” triangle,

quadrangle, pentagon, and even of a “general” poly-

gon, but the conception of a background space for

Euclid's geometry was somehow no longer such an idea

and eluded them. A Platonic idea, even in its most

“idealistic” form, was still somehow object-bound,

which a background space for mathematics no longer

is. Nor could the Greeks form the “idea” of a rotational

momentum, for it simply is no longer an “idea,” and

cannot be pressed into the mold of one. It is, quanti-

tatively, a product *l · p* in which *l* and *p* represent

“ideally” heterogenous objects, but are nevertheless

measured by the same kind of positive real number;

and real numbers themselves are already abstractions

pressing beyond the confines of mere “ideas.” Such

fusion of several abstractions into one was more than

the Greek could cope with; and the formation of

(Newton's) translational momentum, as presented skel-

etally above, was even farther beyond their intellectual

horizon.

If it is granted that Greek mathematics has been thus

circumscribed, it becomes a major task of the history

of ideas—and not only of the history of mathe-

matics—to determine by what stages of medieval de-

velopment, gradual or spontaneous—the inherited

mathematics was eventually made receptive to sym-

bolic and conceptual innovations during the Renais-

sance and after.

This task is inseparable from the task of determining

the originality and effectiveness of medieval Arabic

knowledge, mathematical and other, and its durable

of setting will it be possible to comprehend the course

of mathematics in its conceptual and cultural aspects.

## *BIBLIOGRAPHY*

Carl Becker, *The Heavenly City of the Eighteenth-Century
Philosophers* (New Haven, 1932). Salomon Bochner,

*Eclosion*

and Synthesis, Perspectives on the History of Knowledge

and Synthesis, Perspectives on the History of Knowledge

(New York, 1969); idem,

*The Role of Mathematics in the*

Rise of Science(Princeton, 1966). Carl Boyer,

Rise of Science

*History of*

Mathematics(New York, 1968). Karl H. Dannenfeldt, ed.,

Mathematics

*The Renaissance; Medieval or Modern?*(Boston, 1959).

Wallace K. Ferguson,

*The Renaissance in Historical*

Thought: Five Centuries of Interpretation(Boston, 1948).

Thought: Five Centuries of Interpretation

I. J. Gelb,

*A Study of Writing*(Chicago, 1952). F. K. Ginzel,

*Handbuch der mathematischen und technischen Chronolo-*

gie,3 vols. (Leipzig, 1914). Alfred F. Havighurst, ed.,

gie,

*The*

Pirenne Thesis: Analysis, Criticism, and Revision(Boston,

Pirenne Thesis: Analysis, Criticism, and Revision

1958). Tinsley Helton, ed.,

*The Renaissance: A Reconsidera-*

tion of the Theories and Interpretations of the Age(Madison,

tion of the Theories and Interpretations of the Age

1961). Donald Kagan, ed.,

*Decline and Fall of the Roman*

Empire: Why Did It Collapse?(Boston, 1962). Samuel Noah

Empire: Why Did It Collapse?

Kramer,

*The Sumerians: Their History, Culture, and Char-*

acter(Chicago, 1963). Karl Krumbacher,

acter

*Geschichte der*

byzantinischen Litteratur von Justinian bis zum Ende des

oströmischen Reiches:(

byzantinischen Litteratur von Justinian bis zum Ende des

oströmischen Reiches:

*527-1453*), 2nd ed. (Munich, 1897).

Otto Neugebauer,

*The Exact Sciences in Antiquity,*2nd ed.

(Providence, 1957). Isaac Newton,

*Mathematical Principles*

of Natural Philosophy,trans. A. Motte (Berkeley, 1946; rev.

of Natural Philosophy,

F. Cajori, 1962). Erwin Panofsky,

*Renaissance and Renas-*

cences in Western Art(Stockholm, 1960). David Eugene

cences in Western Art

Smith,

*History of Mathematics,*2 vols. (Boston, 1923), esp.

Vol. I. René Taton, ed.,

*A History of Science,*trans. A. F.

Pomerans, 4 vols. (New York, 1963), Vol. I,

*Ancient and*

Medieval Science.B. L. van der Waerden,

Medieval Science.

*Science Awaken-*

ing(Groningen, 1954). William Whewell,

ing

*History of the*

Inductive Sciences from the Earliest to the Present Time,3

Inductive Sciences from the Earliest to the Present Time,

vols. (London, 1837), 3rd ed. (New York, 1869), esp. Vol.

I. Lynn White, Jr.,

*The Transformation of the Roman World:*

Gibbon's Problem after Two Centuries(Berkeley and Los

Gibbon's Problem after Two Centuries

Angeles, 1966).

SALOMON BOCHNER

[See also Infinity; Mathematical Rigor; Newton on Method;Relativity; Renaissance Humanism; Space; Symmetry.]

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