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The purpose of this chapter is to describe, in broad outlines, the contributions to the theory of space and time which were made in the period from Aristotle to Newton. These two figures stand at the opposite endpoints of a period which extends for over 2000 years. The influence of the Aristotelian point of view for the intervening period can hardly be overestimated. The significance of a thinker can be measured both by his followers and his critics. The followers of Aristotle in the middle ages and early renaissance were numerous and far outnumbered his detractors. But the true measure of the importance of his thought, or what passed for his thought, is that, in addition, he was the prime target of all the major critics during the period. Newton, on the other end, signaled the birth of a new age in physical theory, the development of a mathematical physics which was the forerunner of our contemporary physical theories. Even where Newton has been supplanted or corrected the methods he developed and the ideal that his physical theory presented pervade contemporary science.
There is no way we can hope to do justice to the complexities and intricacies of medieval and renaissance contributions to the discussion of the nature of space and time. At best, we can try to reduce the complex lines of development to some manageable proportions. Part of the problem is that much primary scholarship, e.g., production of critical editions of primary texts and their analysis, remains to be done. For this reason, a complete history of the development of theories of space and time for this period cannot yet be written. Our aim is the much more modest one of suggesting what were some of the major lines of influence.
A portion of the period in question includes a period of decline of Greek science and the subsequent low point in learning and intellectual activity in Europe which is known as the Dark Ages. Thus, section II of this chapter contains a brief description of that decline and the subsequent rise of Christianity and its influence on the development of science. While Europe was in the so-called Dark Ages, Greek learning and ideals were being preserved in the Arabic speaking communities of the Islamic empire and in the Byzantine Empire. The role of Byzantine and Arabic scholars in preserving Greek thought and keeping the light of learning more or less bright is a crucial one in the subsequent "rediscovery" of the learning of antiquity in the West. Here much scholarly work needs to be done, but the subsequent history of the development of theories of space and time cannot be properly understood without some mention of the efforts of these transmitters and preservers, one or two of whom were original scholars in their own right.
Section III then goes on to briefly describe the rediscovery of learning in the West which can be conveniently punctuated by two events: The 12th century renaissance due to the discovery and translation of Arabic editions of Aristotle as well as commentaries by Arabic writers, and the 15th century renaissance in Italy which was characterized by the recovery of and rekindling of interest in Plato's work. As a matter of fact, there was a more or less constant filtering of texts from the Arabs and the Byzantines to the West from the 12th century on, and so to characterize the rebirth of intellectual activity in Europe as due solely to these two events is somewhat misleading. Nevertheless, these two renaissances serve to mark off especially high peaks of activity and the present work is not the proper place to examine all the nuances of the transmission of Greek science to the West.
In Section IV, we turn to an examination of the development of the concept of space from Aristotle to Newton. Aristotle's definition of place was the focal point of continual criticism and we begin by examining some of the main lines of criticism that were leveled against it. As a substitute, most critics suggested some version of Aristotle's rejected "interval" view of place. The major lines of thought are discussed in this chapter.
The conflict between faith and reason meant that theological considerations entered in to the discussion and the major points raised by these concerns are outlined. Finally, a brief survey of some of the more important scientific considerations relevant to the concept of space are considered. The net result was a shift in consensus from the Aristotelian position to a view which, with modifications to be discussed more fully in the next chapter, was adopted by Newton as one of the cornerstones of his new natural philosophy based on mathematical principles. In section V, a similar analysis is provided for the evolution of the concept of time during the period in question.
One final cautionary word. Since we know how the story turns out, there is an unfortunate tendency to read the historical developments as if they were leading up to the concepts adopted by Newton as the foundation of his physical theory. Each intermediary thinker then appears as a bricklayer contributing one or more bricks to a building under the ultimate design of Newton. The selected nature of the material of this chapter, chosen with such hindsight in mind, is likely to reinforce this tendency. However, such inclinations should be strongly resisted. We repeat that we cannot hope to do justice to the complexities and intricacies of medieval and renaissance speculations about the nature of space and time in the scope of a single chapter. It should always be borne in mind that contributions which we have singled out as significant steps in a historical procession cannot be completely understood except in the light of their own historical context. Unfortunately, this book is not the place to develop these contexts in any detail. The net result is an inevitable, if regrettable, distortion of the historical record.
II. The Decline of Greek Science
A. The Hellenistic Period.
Aristotle died in 324 B.C. His legacy was a comprehensive cosmological system which, however, he did not pretend to be either complete or final. Many details, e.g., the exact motions of the planets and the details of projectile motion, were either left for other investigators or included in Aristotle's scheme in only a very sketchy sense. Furthermore, Aristotle's system was not the only available scientific world view in ancient times. There was, of course, the Platonic system, the Atomistic point of view, and, with the development of Hellenistic culture, a number of alternative systems based on Stoic and Epicurean principles were developed. The emergence of the Aristotelian system as the embodiment of the scientific world view of Western man prior to the scientific revolution in the 17th century was a long and complex process.
The century following the death of Aristotle was the highpoint in the development of Greek mathematical thought. The center of intellectual activity had shifted from Athens to Alexandria. Euclid (c. 300-c. 260 B.C.) codified what was known about geometry into his Elements, which became the basis for all standard texts on the subject until the 20th century. The most important figure of the time was Archimedes of Syracuse (287-212 B.C.) who did important work on conic sections and virtually single handedly invented the science of statics. After this period there was a general decline of Greek civilization in the ancient world. The details are somewhat controversial but the main points can be highlighted fairly simply. First, there was an increase in the power of Rome and the Romanization of the ancient world. Whereas the Greeks were speculative thinkers, the Romans tended to be practical men interested in building roads and buildings, and creating an empire. They were not terribly original thinkers. Thus, they failed to significantly contribute to or continue the Greek scientific methods and traditions. Ptolemy (85-165 A.D.), who constructed the basic astronomical system which prevailed until Copernicus (1453), and Galen (129-199 A.D.), whose views on medicine were accepted as authoritative until the 16th century, are generally considered to be the last two great original thinkers of the ancient world.
B. The Conflict with Christianity.
After 200 A.D. the Roman empire began to fall apart, creating great economic and political strains and, in the process, contributing further to the decline of Greek civilization and intellectual effort. In 330 A.D., the Roman empire split into parts, an Eastern part and a Western part. The Eastern part became the Byzantine empire, and through the efforts of Byzantine scholars and the contact of the Arabs with the Byzantine empire, Greek learning was preserved to be rediscovered by the West (i.e., Europe) in the 12th century A.D. In the West, Rome was sacked by the Huns in 410 A.D. and a general period of decline set in.
Two significant intermediary figures stand out: The first is Boethius (c. 510 A.D.). Boethius saw the coming dark ages and was trying to preserve the wisdom of the ancient thinkers before the final darkness fell. Unfortunately, he got involved in politics and was strangled before he could translate all of Aristotle. His handbooks, however, provided the major source (for the West) of knowledge about the ancient world until the rediscovery of the primary Greek works in the 12th century. The other figure is Cassiodorus (c. 540 A.D.). Cassiodorus was one of the founders of the monastery system and the practice of having monks copy texts. It was the monasteries which eventually provided one of the seeds of the rebirth of intellectual activity in the West.
There were, of course, some original thinkers after 200 A.D. (for example, John Philoponus (c. 600 A.D.), whose criticisms of Aristotle were highly influential in shaping subsequent Islamic, Judaic and Christian thought), but they were few and far between. The general trend after 200 A.D. was to preserve rather than increase knowledge. Thus, much effort was expended in the production of handbooks and compendia (collected selections in i.e., handbooks which themselves were derived from earlier handbooks). The net effect was the haphazard preservation of some of the results of Greek learning, but the loss of the spirit, method and aim of Greek science. The major philosophical systems of Aristotle and Plato, for example, survived only in fragmentary and garbled form.
Another factor contributing to the decline of ancient science in the West was the rise of Christianity. As the Roman Empire declined in power, the church gained in power. In 312 A.D., the Roman Church became, in effect, the state religion and this increased its influence enormously. The reaction of the early church fathers to science and Greek learning was mixed. Paul and Tertullian, for example, were opposed to Greek learning in general. Saint Augustine, on the other hand, was sympathetic to the scientific learning of the Greeks but argued that faith and Christian doctrine was to take precedence over any truths that science or reason could provide. There is nothing in many of the Christian doctrines themselves, e.g., belief in immortality, promoting the soul over the body, belief in miracles, etc. which was new. Greek science had co-existed with religions that held similar doctrines. The growing political and social influence of the Christian Church, however, proved inimical to the development of natural science. First, all the church fathers agreed that scientific knowledge, was, at best, subservient to and not as important as divine revelation. The general idea was that if God reveals the truth to you, why waste time puttering around in a laboratory trying to discover the secrets of this world when you should be praying for salvation in the next? Second, as the Church became the ladder to success and power, potential scientists were siphoned off into the clergy.
The conflict between faith (Revelation) and reason, as it emerged in the 13th century and ultimately led to the confrontation between Galileo and the church, dates back to the early days of Christianity. The attitude of the early church leaders to science and reason, as we just mentioned, was mixed. On the one hand, there were those who held that Revelation was the only source of true knowledge and that a good Christian need know nothing more. The defenders of this view were wont to point to the scriptures, e.g., to Paul 2.8 Col. "Take care that no one comes to take you captive by the empty deceptions of philosophy, based on a man-made tradition of teaching concerning the elements of the material world, and not based on Christ." The most outspoken defender of this view was Tertullian (c. 200 A.D.): "What indeed has Athens to do with Jerusalem? . . . With our faith, we desire no further belief. For this is our primary faith, that there is nothing which we ought to believe besides."
On the other hand, there were those writers, notably Saint Augustine (354-430 A.D.), who, while relying on the primacy of faith still found room for reason and philosophy. The basic idea behind this position was that the truth as determined by philosophy and reasons was incomplete and incoherent. One could not, as Plato had thought, reach the divine by reason alone, one had to start from revelation, but, in the light of revelation, the rational truths of philosophy and science would fall into place. Thus, Augustine says, "Understanding is the reward of faith. Therefore, seek not to understand in order to believe, but believe in order to understand." These sentiments were echoed in the 12th century by St. Anselm (1033-1109 A.D.): Credo ut intelligiam (I believe in order that I may understand). Augustine then went on to construct a theology based on his reading of Plato and the so called neo-Platonists, who emphasized the more mystical elements of Plato's philosophy. This new-Platonist
view became the dominant Christian philosophy until the revival of Aristotle in the 12th century A.D.
Because of the decline of learning in the West, and the haphazard translation in copying of ancient texts during the period from 400-1000 A.D. in Europe, Aristotle was known in the West primarily as a logician, not as a scientist. The scientific world view that survived as the epitome of ancient wisdom was Plato's. This can be credited to an influential translation of part of the Timaeus by Chalcidius in the 4th century A.D., and, in part, indirectly through the influence of St. Augustine.
C. The Arabic Connection
An important link in the transmission of ancient Greek philosophy and science to the West was provided by Arab intermediaries. After Constantine (330 A.D.) established Constantinople as the Eastern capital of the Roman Empire and declared Christianity to be the state religion, there were a number of conflicts among different Christian sects as to the nature of Christ. One group, the so called Nestorian Christians were expelled from the Empire and set up schools in Syria and Persia. These Nestorian Christians preserved much of the Hellenistic science and philosophy and translated it from Greek in Syriac. While intellectual activity declined in the Western Roman Empire, these Nestorian Christians and other groups in the Eastern Roman Empire, the Byzantine empire, kept the spark of Greek thought alive. In the seventh century A.D., Mohammed had a revelation and Islam was born. In the course of the next 100 years, the word of Islam was spread by Arab conquests from Syria to Egypt. In 850, for reasons that are not completely understood the Caliphs in Baghdad who ruled the Islamic empire subsidized a massive translation project in an attempt to recover the, by then almost lost, Greek culture. The translators were, for the most part, Nestorian Christians who translated from their Syriac copies into Arabic.* The discovery of Aristotle and Plato had an enormous influence on Islamic thought and was immediately pressed into service to provide a rational foundation for Islamic faith. It must be noted, however, that the Arabs did not have access to the complete works of Plato and Aristotle. What they had was the product of 1,000 years of Hellenistic influence and a long period of low level intellectual activity. The assimilation of Greek thought produced a flurry of intellectual activity for approximately 300 years. The first philosophers, at Baghdad, tended to be Platonists. The later Islamic philosophers, centered in Spain, tended to be more Aristotelian in outlook. The most eminent of these latter was Averroes (1126-1198) whose influence on subsequent medieval thought in Europe was very great. He was known to later Scholastic thinkers as "The Commentator," as Aristotle himself became known as "The Philosopher."
The Arabic commentators were not merely transmitters of Greek knowledge to the West, they also contributed original work of their own in line with the particular interests of their times and society. The full impact and significance of these writers is not yet known due to the long neglect of their writings. The scholarly tradition in Islam after the 13th century gradually died out due, * There were other lines of transmission of Greek thought to Arabic as well. For asurvey, see O'Leary (1949). in part, to the political and social instability that prevailed during the next few hundred years. The area of Islamic philosophy is a fertile one for scholarship. Much work still needs to be done.*
For our immediate purposes, the importance of the Arabic commentators lies in the fact that they picked up on the criticisms that Philoponus and others leveledagainst Aristotle's views on space and time and transmitted them to the Christian thinkers in the West. D. The Impact of Greek Thought on the Middle Ages.
We have already had occasion to remark on the fact that philosophy, and the basic idea that man, using his powers of reason, is capable of comprehending the universe, is a peculiarly Greek invention. When the religious traditions of Islam, Judaism and Christianity came into contact with the products of Greek science and philosophy each went through a similar set of stages. On the one hand, Greek science and philosophy in general, and Aristotelianism, in particular, seemed like just the thing to provide each faith with a rational foundation. On the other hand, Aristotelian philosophy was incompatible with many of the doctrines and tenets of the religious views. Thus, a tension developed, in each tradition, between defenders of faith over reason and those, on the other hand, who sought to reconcile Athens with Jerusalem, or Mecca, as the case might be. In the 12th and 13th century A.D., the interaction between Greek thought and the religious traditions reached a peak. Three mediators appeared on the scene, virtually at once, each striving to reconcile Greek philosophy (Aristotelianism for the most part) with his respective faith: Averroes (11281196) for Islam; Maimonides (1135-1204) for Judaism and Saint Thomas Aquinas * For a survey of the period in question see Walzer (1962). A more comprehensive treatment which is somewhat dated is de Boer (1967). (1224-1274 A.D.) for Christianity. Their efforts had an enormous influence on the subsequent development of philosophy and science, in general, and views about space and time, in particular.
III. The Renaissance in the West
The recovery of Greek learning in the West was punctuated by two Renaissances. The first occurred in the 12th century when the Aristotelianism preserved by the Arabs was translated from Arabic into Latin. The second occurred in the 15th century when the collapse of the Byzantine Empire flooded Italy with refugee Greek scholars who brought with them the Platonic heritage and the work of the great Greek mathematicians of the Alexandrean period.*
For 300 years, from approximately 600 A.D. to 900 A.D., learning in the West came to a virtual standstill. What was left of the Western Roman Empire was under constant seige, in the south by Islamic imperialism and, from the north, first, by invading Germanic tribes and then by the Vikings. In the 7th century, the monastery system was established and the collection and copying of texts and the rudimentary education of monks was conducted therein. The monks were taught to write and to read Latin. In the 10th century, cathedral schools were established at the seats of the bishops. Education was gradually expanded to train secular leaders for a world which was rapidly becoming more complex due to increased political stability and, in an age of expansion, the development of commerce and the growth of cities.
The state of learning in the 11th century in the West is illustrated by
*Actually Greek material was flowing from Byzantium to the West continually from the 12th century on through contacts in Sicily and Constantinople. The 15th century, however, saw a rapid explosion of interest in ancient Greece and is, therefore, singled out as more significant. a correspondence between Ragimbold of Cologne and Radolf of Liege, two of the foremost mathematicians of the day, on mathematics. The name of Euclid is unknown to them, as is the Pythagorean theorem. They puzzle over a simple theorem in Boethius's Geometricum, a collection of odd mathematical facts and theorems from the ancients. The theorem in question is: The sum of the interior angles of a triangle is 180 degrees. The questions they asked included: What is an interior angle? What is an obtuse angle? What is an acute angle? On the questions of the ratio of the diagonal to the side of a square,
they wonder what the ratio of d/s is. They hazard the guess that d/s equals 7/5 or 17/12. That the ratio is not a rational number does not occur to them.
Later, Franco of Liege (c. 1083), the foremost mathematician of his day, wrote a famous book on squaring the circle, i.e., constructing a square which has the same area as a given circle. Of course, this problem has no solution, something that the Greeks knew quite well. Again, in Boethius, Franco found the formula d2/4 equals 22/7, where "d" is the diameter of a circle. Taking this as an exact ratio (!), he wonders "How did the ancients discover this?" The best he can come up with is: by cutting up a parchment circle and subtly pasting it back together again (see Dijksterhuis, 1961, 99-108).
The 12th century renaissance.
In 1085, Toledo, Spain, an outpost of Western Islam, fell into Christian hands. In 1134 A.D., the systematic translation of the entire Greek corpus as preserved by the Arabs and discovered in Spain, began. The obvious superiority of learning in these texts to what was then current was soon recognized. The general attitude of the 12th century thinkers was that just as the truth of Christianity was revealed in the Scriptures, so the truths of natural science were all to be found in the ancient texts; it was merely a matter of recovering what had been known already.
The idea that there are authorities in matter of religion was already an established principle from the time of Augustine. The ultimate authority was the Scriptures, but the mantle was gradually extended to cover the early church fathers, including later Augustine and also, ultimately, the consensus of the contemporary church. The extension of the idea of authority to the ancient writers on natural philosophy was also made. That medieval philosophical method known as Scholasticism was the result of the attempt to reconcile conflicts among authorities -- both conflicts amongst religious authorities, and also conflicts between authorities in religion and authorities in natural philosophy. This reliance on authority, which was contrary to the empirical attitude of Aristotle himself, hampered the development of modern scientific methodology. But, the fact of the matter is that, however obvious it may seem to us that the method of observation and test is the right method for doing science, it is certainly not self evidently the right method. The 12th century was still under the general sway of Plato's faith in the power of human reason to discover absolute truths. It was widely held that the road to scientific knowledge was that of discovering what the ancients had said and using our power of reason to understand what it was that they had said. Thus, the subsequent transformation of the Aristotelian system into an absolutely authoritative one was not something that Aristotle himself ever intended or would have wished, but, rather, was a reflection of the thinking of medieval Christianity.
The recovery of the Aristotelian corpus posed both a challenge to Christianity and an opportunity for Christianity. The challenge that Aristotelianism presented was that there was here a systematic philosophy which seemed to establish, through the use of reason, truths which were in conflict with the Christian faith. Thus, Aristotle's philosophy entailed the following conclusions: (l) The natural world is eternal; (2) There is no resurrection or personal immortality; (3) A denial of the existence of divine providence. In addition, the Islamic sources from which the Aristotelian philosophy was recovered, were coupled with the extreme rationalism of philosophers such as Avicenna and Averroes. Their position seemed to be that where reason conflicted with revelation, reason must be followed. A school of Latin Averroeists developed and followed this line.
The opportunity that Aristotelianism presented was that there seemed to be here a systematic philosophy which could be used to defend Christianity against other religions which seem to have a broader philosophical base, e.g. Islam and Judaism.* A series of churchmen, Alexander of Hales, Albertus Magnus and Thomas Aquinas defused the Averroeistic elements in Aristotle because they saw the systematic Aristotelian world view as capable of providing a unified
* Recall that a similar attempt to reconcile Aristotelianism with Islam was being carried out in Islamic Spain by Averroes and with Judaism in Egypt by Maimonides at roughly the same time. framework within which a rational defense of Christianity was possible. Aquinas (1224-1274 A.D.) worked out the details. He tried to strike a balance between reason and faith. He used a method, developed by Abelard (1079-1140 A.D.) called Sic et Non (Yes and No), to show that conflicts between authorities could be resolved when it was seen that the conflicts were not real but only apparent. In contrast to the early church fathers, Aquinas supported natural science because he felt that although false scientific views could promote false ideas about God, and hence, draw men away from God, true scientific views would promote true ideas about God and thereby draw men to God. By using Aristotle's systematic philosophy, in which everything is connected into a tight coherent picture Aquinas welded natural philosophy and Christian theology together. While this device appeared to make it possible to reconcile reason, on the one hand, and faith, on the other, it ultimately contributed to the undermining of church authority and the conflict between church and science in the 15th and 16th centuries. The problem was that the resulting package was so interlocked that all the pieces seemed essential. But, there are various aspects of the Aristotelian picture of the physical world which are rather easily undermined, e.g., the geocentrism, the denial of a vacuum, and the doctrine of the four elements. When, in fact, these aspects of the Aristotelian physical picture began to be undermined, the associated Christian doctrines which were now wedded to this cosmological picture, were also apparently undermined as well.
The condemnation of 1277.
The marriage of Aristotelianism with Christianity was not all smooth sailing. The Latin Averroeists, centered at the University of Paris, came under increasing attack from the Church authorities. Finally, in 1277 the Bishop of Paris, Stephen Tempier, condemned 219 propositions with the threat of excommunication for anyone who taught, defended or even listened to any of the propositions in question. The condemned propositions which are directly related to our concerns included:
66. That God could not move the heaven in a straight line, the reason being that He would then leave a vacuum. (49)*
80. That the reasoning of the Philosopher [i.e., Aristotle] proving that the motion of the heaven is eternal is not sophistic, and that it is surprising that profound men do not perceive this.
85. That the world is eternal as regards all the species included in it, and that time, motion, agent and receiver are eternal, because the world comes from the infinite power of God and it is impossible that there be something new in the effect without there being something new in the cause. (87)
86. That eternity and time have no existence in reality but only in the mind. (200)
88. That time is infinite as regards both extremes, for although it is impossible for an infinity to be passed through when one of its parts had to be passed through, it is not impossible for an infinity to be passed through when none of its parts have to be passed through. (205)
*Lerner and Mahdi (1972), pp. 335-354, contains English translations of all 219 propositions. The numbers to the left are those from the Lerner and Mahdi text. The numbers in parentheses on the right are the original ordering.
89. That it is impossible to refute the arguments of the Philosopher [again: Aristotle] concerning the eternity of the world unless we say that the will of the first being embraces incompatibles. (89)
92. That with all the heavenly bodies coming back to the same point after a period of thirty-six thousand years, the same effects as now exist will reappear. (6) [cf. the ancient doctrine of eternal return]
190. That he who generates the world in its totality posits a vacuum, because place necessarily precedes that which is generated in it; and so before the generation of the world there would have been a place with nothing in it, which is a vacuum. (201)
The condemnation had a profound effect in the subsequent development of Western philosophy. The spread of Latin Averroeism was checked and the spread of Thomistic Aristotelianism was slowed. The development of Christian Aristotelianism was not, however, completely stopped and gradually Aristotelianism, in particular the Thomistic version came to dominate the schools. Thomas was cannonized in 1323, and in 1324, the condemnation of specifically Thomistic theses on the original 1277 list was revoked.
The exact influence of the condemnation on later medieval thinking about space and time is difficult to assess, but that it did contribute to the undermining of the Aristotelian position on these issues is clear (cf., e.g., Grant, 1969, p. 50 f.; Ariotti, 1972,; Grant, 1964, 271).
The 15th century renaissance.
While Aristotelianism came to dominate the Schools in the later middle ages, there still were pockets of Platonist influence in Paris and London. But the "geometrical" approach of the Timaeus and its neo-Platonist twists was overwhelmed by the comprehensive and interlocking nature of the Aristotelian system. Then, in the 15th century, the situation radically changed. The decisive event was the collapse of the Byzantine Empire and the fall of Constantinople to the Turks in 1453. Whereas the Western Roman Empire had ceased to exist as an integral political unit after the 6th century A.D., the Eastern Roman Empire had survived as an intact unit from 330 A.D. to 1453 A.D. During that time, Greek culture and science had been preserved and taught.
When Constantinople fell, the Greek scholars fled to the West. They brought with them not only the body of Platonic work, and Greek poetry and literature which had been ignored by the Arabs, but also the works of Euclid and Archimedes and the Greek atomists. The first Platonic academy was established in Italy in 1462 and the subsequent development of natural philosophy was strongly infused with a geometrical approach characteristic of Platonism, and especially, Alexandrean science. The scientific revolution which has shaped our modern vision of science and nature, and which can be dated from the publication of Copernicus's book "On the revolution of heavenly Bodies" (De Revolutionibus orbium caelestium) in 1543, owes much to this 15th century influx of Platonism and Greek mathematical thought.
Let us turn from a consideration of these social, political, and religious issues to a consideration of some of the scientific work that had to be done in order to effect the transition from Aristotle to Newton. As has already been suggested, the Aristotelain world view was never the monolithic, unchanging conception that later apologists for the scientific revolution sometimes claimed it to be. Indeed, even the traditional picture of Scholastic philosophy as slavish imitators of Aristotle, constantly deferring to Authority in order to settle disputes, is very misleading. The medieval philosophers were not the slavish followers of Authority that they have sometimes been portrayed to be--(especially by Enlightenment philosophers concerned to 'overthrow' Scholastic thought). For one thing, there were different authorities: religious, philosophical, and physical. Within each group, e.g., of the religious authorities, there were conflicts and disagreements. The task that the Scholastic philosophers set themselves to perform was to try to reconcile these authorities or discover which of them should be accepted. In order to do this, they employed reason and argumentation, which got very sbutle and complex as all the ramifications of a given problem were worked out. What they did not, in general, do, was propose experiments to test the physical implications of the competing views they juggled. This lack of an experimental attitude is one of the characteristics that distinguishes the medieval from the modern tradition. Modern science, as we know it, developed outside the traditional schools and Universities devoted to the promulgation of Aristotelian doctrine.
As a matter of fact, the details of the Aristotelian system were continually being subjected to scrutiny and reevaluation. The power of the Aristotelian model lay not in the accuracy of all its details but in the comprehensiveness of the global picture it presented. Thus, in order to be overthrown, certain fundamental ideas which were inherent and central to the Aristotelian picture had to be modified. In accordance with the theme of this book we focus on the evolution of the concepts of space and time from their Aristotelian formulation to the form in which they were incorporated by Isaac Newton in the Mathematical Principles of Natural Philosophy [Philosophiae naturalis principia mathematica] in 1687, the book which formed the basis of the development of classical physics.
IV The Development of the Concept of Space From Aristotle to Newton
A. The Closed World of Antiquity.
Except for the Greek atomists, who did not exert an appreciable influenceon the development of physical thought in the middle ages, the prevailing Greek view was that the cosmos was finite. The Stoics, who flourished in the Hellenistic period, did propose that the finite cosmos was embedded at the center (!) of an infinite space which was filled with pneuma, somewhat akin to a cohesive ether. But the Stoics never developed an ongoing program in physical science and by the time of the fall of Rome, their works were lost and known only through paraphrases (cf. Sambursky, 1962).
The Aristotelian system was the subject of a steady but intermittent interest, first by the Byzantine scholars and later by the Arabic commentators before being rediscovered in the West. For the most part, these scholars were preservers of the Aristotelian tradition and not themselves innovators. A notable exception is John Philoponus who worked at Alexandria in the early 6th century. His lucid and telling criticisms of Aristotelian positions were highly influential in the subsequent development of Aristotelian commentary. We will examine some of Philoponus's fundamental criticisms of Aristotle in the upcoming sections. First, let us summarize the basic Aristotelian position on space as it was understood by the medieval commentators.
Three basic aspects of Aristotle's view on space were the subject of much critical commentary.
1. Aristotle's characterization of place.
2. Aristotle's rejection of the void.
3. Aristotle's view that space was finite.
Aristotle's characterization of place.
Recall that Aristotle defined the place of an object as the surface which contains the object. For an ordinary 3-dimensional object, its surface is 2-dimensional. Thus, Aristotle does not identify the place of a thing with the (3-dimensional) volume which it occupies, but rather with the (2-dimensional) surface which surrounds it. In fact, Aristotle rejects the suggestion that there is an independently existing 3-dimensional space, which is occupied by a body. This is the view that the place of an object is the interval stretching between the extremities of a thing (P3) which Aristotle explicitly rejected (cf. p. 5-11).
In addition to Aristotle's explicit arguments against the interval view which we considered in Chapter 5, an implicit consideration was Aristotle's opinion that 3-dimensionality entails corporeality, that is, that if something has 3-dimensions it must be a body. If this is accepted, then if the place of an object were a 3-dimensional (bodily) interval, a penetration of bodies would exist, which Aristotle felt led to conceptual difficulties (cf. 5-21). From the time of John Philoponus onwards, this view came under increasing attack. Those who opted to reject it leaned towards Aristotle's rejected interval view. But, there were difficulties with the interval view. Not the least of which was that the interval view seemed to entail that there must be voids in nature.
Aristotle's rejection of the void
Consideration of Aristotle's concept of place naturally led the commentators to look at Aristotle's arguments against the possibility of voids. Those, like Philoponus, who rejected Aristotle's concept of place in favor of the interval
~ Actually, earlier in the Physics 20915-7, and in the Categories 5a 6-14, Aristotle does seem to characterize space as 3-dimensional. Medieval thinkers generally ignored this remark and took the 2-dimension view of the later Physics as Aristotle's true position (Grant, 1974, p. 138). Compare Patrizi's remarks on this contradiction in Brickman, 1944, p. 229. view were constrained to somehow answer or evade Aristotle's view on the void (cf. Grant, 1978; Wolfson, 1929). Since Aristotle's rejection of the void was connected with his theory of motion, those who favored the existence of voids were faced with the problem of coming up with an alternative account of motion (cf. Grant,1964).
Aristotle's view that space was finite.
The finiteness of space follows from Aristotle's characterization of places as be;ng all inside the finite Heavenly Sphere. Attacks on the Aristotelian views of place and the void naturally led to speculation about whether the spatial extent of the universe might be finite. Questions were raised about Aristotle's contention that the cosmos itself is not in a place, although all places are in it (cf. 6-22 ff.). If, then, the cosmos was in a place, perhaps there could be places outside the cosmos. These would be void places, since every material thing is inside the cosmos. But voids are qualitatively homogeneous and, as such, there would be no reason not to expect them to continue without limit. The universe has become infinite.
A number of different arguments were raised concerning these points in the years intervening between Aristotle and Newton. We turn to a consideration of some of them. B. Difficulties with Aristotle's concept of place.
Within the limits of a brief survey it would be impossible to catalogue all the objections raised against Aristotle's concept of place. In what follows we single out for discussion some of the characteristic arguments which were influential in undermining Aristotle's 2-dimensional view.
We begin with an argument alleged to be of great antiquity which was reechoed throughout the centuries from the time of Aristotle to the scientific revolution. Our source is Simplicius, the 6th century A.D. commentator from Athens. He attributes the argument to Archytas of Tarentun (428-347 B.C.), a contemporary of Plato, and also to Stoic critics of Aristotle. The issue is whether or not there can be places outside the heavenly sphere. The following dilemma, as reported by Simplicius, suggests there must be: . . . let it be assumed that someone standing motionless at the extremity [of the world] extends his hand upward. Now if his hand does extend, they [that is, the Stoics] take it that there is something [i.e., some place] beyond the sky to which the hand extends. But, if the arm could not be extended, then something will exist outside that prevents the extension of the hand; but if he then stands at the extremity of this [obstacle that prevents the extension of the hand] and extends his hand, the same question as before [is asked], since something could be shown to exist beyond that being. The place of the sublunar world. This objection was raised by Philoponus among others (Jammer, 1960, p. 53). The problem is to determine the place of the sublunar world. According to Aristotle, the place of the sublunar world is the concave surface of the heavenly that sphere. A problem with this answer is that the heavenly spere is constantly moving, and for Aristotle the place of a thing is something which is unmoving. When the immediate container of a thing is in motion, as in the case of a ship in a flowing river (cf. 5-14), Aristotle's move is to define the place of the ship as the first unmoved container (in the case of the ship, this would be the
* This is taken from Grant (1969), p. 41. Compare Henry, 1979, p. 564, fr. 94. A similar argument is found in Lucretius Book I. river bank). However, this strategy won't work for the sublunar world since the heavenly sphere is the largest container there is.
The moving outer sphere.
Another objection results from just this fact that the heavenly sphere is in constant motion. According to Aristotle, the heavenly sphere is not in a place. But Aristotle also argues that local motion just is change of place. How then can the heavenly sphere both be moving (i.e., changing its place) and also not be in a place at all (Jammer, 1960, 53 f.)?
These difficulties inclined Philoponus to adopt the rejected Aristotelian interval concept of place. We will return to these arguments for the interval view in the next section.
The whole is the sum of its parts.
Finally, we examine an argument that appears in the late medieval Jewish commentator Hasdai Crescas (c. 1340-1411). In a commentary on Maimonides written in 1400, Crescas advances some further arguments against Aristotle's concept of place. A number of these arguments raise difficulties with the attempt to reconcile Aristotle's definition of place and his theory of natural places (cf. Wolfsen, 1929, 195-199); Jammer, 1960, 74-76). We pass over these, however, and turn to an argument which points out a difficulty, on Aristotle's view, with the relation between the place of a whole and the places of its parts. The problem is this. Consider a cube whose sides are 2 meters long.
Figure 6-2a The place of this object, according to Aristotle, is its enclosing surface, which is a quantity of surface area equal to 24 square meters (6 faces times 4 square meters per face). Now divide the cube into 8 equal cubes whose sides are 1 meter each.
Figure 6-2b surface of the respective smaller cube. The total place of all the parts is the total surface area of the cubes. Each cube has a surface area of 6 square meters. Since there are 8 cubes, the total surface area of all the smaller cubes is equal to 48 square meters. Thus, the place of the parts is greater than the place of the whole. (Wolfson, 1929, 199, the argument occurs implicitly in Crescas and explicitly in his student Joseph Albo (1380-1444), cf. Wolfson, 1929, 457; Jammer, 1960, 76).
Albo points out that this contradicts a claim by Euclid to the effect that "equal bodies occupy equal places" (quoted by Jammer, 1960, 76). The interesting thing about this criticism is that it implicitly introduces a further adequacy condition on an appropriate concept of space (cf. the discussion on 5-10). Unlike the other criticisms which we have considered and which point out basic inconsistencies internal to Aristotle's view, this objection imports/further consideration into the discussion. We ultimately accept this argument as telling against Aristotle because the extended concept of space that it certifies proves to have useful consequences for our study of nature.
Although Crescas wrote in Hebrew, his work was known to the later Renaissance figures who laid the foundation for the concept of space incorporated by Newton into his physical theory.
This brief survey barely scratches the surface of the material relating to objections to Aristotle's concept of place. An exhaustive treatment of all the criticisms and their interrelations does not exist. Nevertheless, one basic point emerges with clarity. Despite the longevity of Aristotelian physical theory, cogent criticisms against a central concept of that system, the concept of place, were being formulated from the very beginning. Reflection on these criticisms led a small set of commentators to try to develop the rejected interval view in its stead. We turn now to a brief consideration of these efforts.
C. The Development of the Idea of Space as an Interval.
The authority and integrity of the Aristotelian system was such that, despite the inconsistencies in Aristotle's view of place, his 2-dimensional view generally prevailed until the 16th century (Grant, 1964, 138). Nevertheless there were a number of supporters of the interval view from late antiquity.
Damascus, a 6th century contemporary of Philoponus, explicitly maintained the 3-dimensionality of space (Sambursky, 1962, 5).
Philoponus, in addition to criticizing Aristotle's view put forward a positive doctrine of his own. He defended the three dimensional nature of space and the idea that the place of a body was the empty interval occupied by the body. The three dimensional interval he held to be incorporeal, i.e., not a body, thereby explicitly rejecting the association of 3-dimensionality with corporeality in Aristotle (Sambursky, 1962, 5). Philoponus was well aware that a 3-dimensional interval view of place committed one to a void. He dealt with this in two ways. First, he argued that although space was fundamentally void by nature, in fact, there were no actual voids in nature since as soon as one body left its place, another body moved in to occupy it. Thus, Philoponus agreed with Aristotle that the cosmos was, in fact, a plenum, although he disagreed with Aristotle about the, in principle, possibility of a void. Recall that Aristotle's principles of motion are incompatible with the existence, even in principle, of a void. Thus, Philoponus had to produce an alternative account of motion to go along with his acceptance of the possibility of voids. This he proceeded to do.
* For a discussion of his unique view of space see Jammer, 1960, 56 f.
** Unfortunately, the bulk of Philoponus's extant commentaries on Aristotle are not in English. However, there are extended selections in Cohen and Drabkin (1958, pp. 217-223). Compare the discussion in Clagett (1963, 207-216).
The infinite velocity argument against the void.
As an example of Philoponus's reasoning, consider his objection to one of Aristotle's arguments against a void. Aristotle argued that no voids could exist because if they did, objects falling through them would travel infinitely fast, which is impossible. Aristotle reasoned as follows. He had argued that the velocity of an object (V) was directly proportional to its weight (W) and inversely proportional to the resistance of the medium (R), i.e., V = R . Since a void would offer no resistance to a falling object, the velocity of all objects in a void, regardless of their weight would be V = WO , or infinite.
Philoponus rejected this argument. He accepted Aristotle's contention that the speed of a falling object was, in large part, determined by its weight, but he rejected Aristotle's formula V = R . Instead, Philoponus said, each body has a limiting velocity which is its velocity through the void. This limiting velocity is such that heavier bodies will fall faster than lighter ones through a void (Cohen and Drabkin, 1958, 217). The effect of a resisting medium is to increase the time it takes a body to fall a given distance. Consider two objects of unequal weights Wl and W2 such that Wl is greater than W2. Let each fall through a void for a distance D. The limiting velocity of Wl (Vl) will be greater than the limiting velocity of W2 (V2). Since distance (D), time (T), and velocity (V) are related as V = T ~ we obtain
Thus, on Aristotle's view, if Wl = 2W2 then l2 = 2ll . Philoponus objects: But this is completely erroneous, and our view may be corroborated by actual observation more effectively than by any sort of verbal argument. For if you let fall from the same height two weights of which one is many times as heavy as the other, you will see that the ratio of the times required for the motion does not depend on the ratio of the weights, but that the difference in time is a very small one. And so, if the difference in the weights is not considerable, that is, if one is, let us say, double the other, there will be no difference, in time, though the difference in weight is by no means negligible, with one body weighing twice as much as the other (Cohen and Drabkin, 220).
One remarkable feature of this argument is that the underlined section describes an experiment which Galilei allegedly performed (for the first time) by dropping two weights from the leaning tower of Pisa in his attempt to confute the Aristotelians in the 17th century. In fact, Galilei probably did not perform the experiment, and whether the idea originated with Philoponus or whether he was merely reporting an existing anti-Aristotelian tradition is not clear (cf.Grant, 1964).
* Since V= A similar argument, undoubtedly influenced by Philoponus through Arabic intermediaries, appears in Crescas (Wolfson, 1929, pp. 183, 403 ff.). But, the prejedice against voids was not so easily overcome and until it could be the idea of space as an interval had to await further developments before it was accepted by a majority of thinkers.
The identification of space and light.
Another line of thought which contributed to the development of the 3dimensional interval view of space can be found in Proclus (410-485 A.D.). Proclus was a neo-Platonic head of the Athenian Academy. He held that space was a body, despite the Aristotelian arguments against this view, albeit a very subtle one (Sambursky, 1962, 8). Summing up all the arguments, space is thus an immovable, indivisible, immaterial body. Thus it must obviously be the most immaterial of all bodies . . . Of all these light is the simplest body (among the elements fire is the most incorporeal, but light is the essence of Fire), and therefore it is manifest that space is light, the purest among all bodies. Let us now imagine two spheres, one made of light and the other of a manifold of bodies, equal in volume. One sphere is placed round the centre of the world, and the other is immersed in the first sphere. The whole universe will then seem to be in its place and to rotate in the immovable light. It will not be translated from its place, and in this respect it will resemble space, but each of its parts will be in rotary motion, and in that respect the universe will be less than space. (from Simplicius, quoted by Sambursky, 1962, 8). Sambursky goes on to characterize Proclus's view as follows: Space, thus identified by Proclus with light, is elevated to the highest rank of reality accorded to light in neoPlatonic teachings, but his picture of the luminous sphere representing space with all the attributes given to it, and carrying the material sphere of the universe, suggests a specific notion which, as Simplicius reminds us, already appears in Porphyry's [232-c. 301 A.D.] writings. It is the idea that the soul is clad in a luminous garment called 'the luminous vehicle of the soul', an expression that is also found in several places in Proclus's commentary on the Timaeus. In his theory of space Proclus transfers this picture of an immaterial apparel of the soul, made of light, to the world soul, and makes space, the luminous sphere, the vehicle of the material world (Sambursky, 1962, 8-9).
This should suffice to give the general flavor of typical neo-Platonic lines of thought. In later centuries it became aligned with mystical and religious themes, and, as such, exerted a strong influence on Renaissance and post-Renaissance thought. Proclus's theory identifying space and light influenced early scientists such as Robert Grossteste (1175-1253) and Witelo (c. 1220) who were instrumental in the early development of the study of optics in the late Middle Ages in the West. Proclus was also read by later Renaissance figures such as Fransiscus Patrizi (1529-1597) who laid the metaphysical foundations for Newton's concept of Absolute space (see Jammer, 1960, 34 F,; Henry, 1979, 556-7; for further discussion of Patrizi, see section F, pp. 6-46 ff. below).
The significance of this approach for the development of the Newtonian view of space was that it provided another line of argument designed to establish the independent existence of space as a 3-dimensional arena in which the cosmos was situated. This, of course, reflects the strong influence of the Timaeus in this tradition
We have seen, in this brief survey, that resistance to the Aristotelian conception of place and advocacy of the 3-dimensional interval view has a long history. Despite this, however, it must be reemphasized that for most of the period in question (i.e. from late antiquity (500 A.D.) to the 15th century Renaissance),Aristotle's view generally prevailed. We turn now to a consideration of theological influences on the transition from the Aristotelian to the Newtonian view of space. D. God and space in the transition from Aristotle to Newton.
Starting with the Old Testament, there has been a continual association of God with Space. The effect, during the period under discussion, was to introduce religious and theological considerations which bore on the shaping of the concept of space as it emerged in the scientific revolution in the 17th century. Insofar as space was identified as an attribute or aspect of God, it assumed an absolute character as something which would exist even if no objects existed. As an absolute existent, possibly devoid of all objects, space would be, by nature, a void. Finally, insofar as God is infinite in power, it seems an unnecessary restriction on his power to maintain that space is finite and not infinite. Thus, three of the characteristics of Newton space, that it is infinite, that it is a void and that it is absolute (existing whether or not physical objects exist) are all features which theological considerations served to suggest and reinforce. We turn to a brief discussion of some of the details of the theological concerns which led to this view of space.
Place as a name for God.
As Copenhaver points out, in the Judaic tradition, a hesitancy to pronounce the name of God led rabbinical writers to coin a number of other descriptive designations, e.g., "the Glary of the World" to refer to God. A number of these designations involved references to place or related concepts (Copenhaver, 1980, 490). As early as the 1st century A.D., place (MAKOM in Hebrew) was an accepted designation for God. The idea was developed in later Judaic literature and became a cornerstone of the cabalistic tradition, a version of Jewish mysticism with neo-Platonic elements. After the fall of Constantinople in 1453, cabalistic learning was introduced into Christianity by a leading figure of the 15th century Renaissance, Pico della Mirandola (1463-1494). It spread northward and influenced the views of Pierre Gassendi (1592-1655), Henry More (1614-1687) and Newton himself (for further details see Jammer, 1960, 26-34 and Copenhaver, 1980).
The result was to suggest that space was an attribute or property of God. As such, space was conceived as something absolute and divine.
The condemnation of 1277.
We have already had occasion to mention the condemnation of 219 propositions in 1277 and its general effect on the subsequent development of medieval thought.
The 49th proposition, in particular, exerted a significant influence on the subsequent acceptance of the reality of an extra mundane void (cf. 6-16 above). The proposition in question is 66 on the list in Lerner and Mahdi. (For further details on the arguments over the void in the middle ages see Grant, 1964, 1969, 1974).
The condemnation of the 49th proposition rejects the view that God could not move the entire universe in a straight line and is a direct attack on the Aristotelian cosmology. In supporting the thesis that God could so move the cosmos, the way is paved for (l) accepting the reality of places outside the finite cosmos, places to which God would move the Cosmos, and (2) accepting the reality of an extra-cosmic void. Thomas Bradwardine (1290-1349) and Nicole Oresme (d. 1382) both cited the condemnation of proposition 49 in their arguments for the reality of an extramundane void (Grant, 1969, 51).
Once the reality of an extramundane void became palatable, it is a small step to the claim that such a void must be infinite. First, as a void with no features except the potentiality to be the place for the physical world and the (possible) actual place of the divine presence, there is no reason for the void to terminate at one point rather than another. Hence, it must be infinite. Secondly, once the idea that God could do things which are not allowed in Aristotelian physics took hold (partially as a result of the condemnation of 1277 of propositions which asserted otherwise), to deny that the void was infinite seemed like an unjustified limitation of divine power.
The history of the development of the idea of an infinite void in medieval
Christian thought is a long and complicated one. For details we must refer the 1981). reader to more specialized sources (Grant, 1964, 1969, 1974,/ Two points perhaps should be made at this point. First, the medieval discussions on the void usually distinguished between three kinds of voids and arguments for one kind did not commit to the existence of other kinds of voids. There were (l) extramundane voids (2) large voids between the spheres inside the cosmos and (3) small voids existing between individual particles (Grant, 1969). One way to reconcile Aristotle with Christianity was to admit the possibility of "supernatural" voids outside the cosmos and accept the Aristotelian physics for denying voids inside the cosmos. Thus, while Aristotle was the authority on things natural, God was free to act in supernatural ways outside the cosmos (Grant, 1969, 59). Secondly, the early medieval writers on the extramundane void, such as Bradwardine and Oresme, thought of the infinite extramundane void as a dimensionless expanse (Grant, 1969). By the 17th century, the extracosmic void had become a 3-dimensional space which was fit to serve as the framework of classical mechanics. How did this occur? Grant conjectures It is evident that . . . [in the 17th century . . . the position developed in the Middle Ages was widely adopted--that God in virtue of this infinite power and immensity must necessarily occupy and fill an infinite void space. But whereas in the Middle Ages God's nondimensionality was deemed paramount [on the grounds that ascribing 3 dimensions to God was a limitation on his nature] so that He was held to occupy an imaginary nondimensional extramundane infinite void, in the seventeenth century he is thought to fill a three-dimensional void and thereby has become a physically extended three-dimensional incorporeal being. How did this happen? If I may be allowed to conjecture and speculate--for there is little that would at present pass for solid evidence in tracing this transition--it seems to me bound up with the Aristotelian cosmos and its collapse in the seventeenth century (Grant, 1969, 59).
The close association of God with 3-dimensional infinite space was later to provide one of the focal points for Leibniz's criticisms of the Newtonian world view (cf. Chapter 7 below, and Alexander, 1956).
Henry More (1614-1687)
The trend towards identifying space as an essential attribute of God finds its pre-Newtonian culmination in the writings of Henry More. More was an English neo-Platonist whose thought was influenced by the revival of Platonism in the 15th century, by the resurgence of atomism and the new scientific philosophy being promoted by Galileo Galilei (1564-1642) and Rene Descartes (1596-1650), as well as by the Jewish mystical cabalistic tradition. More accepted the view that space was an infinite extended something. The question was, what kind of thing? Descartes had urged that space and matter were one and the same. Thus, for Descartes space was a material something. More rejected this view, and argued that space was an infinitely extended spiritual thing.
According to More, the material universe, which is created, is finite and embedded in infinite space. More identified Space with Spirit with God. In order to support this identification, he listed 20 properties which are associated with God and which he also claims to be true of Space. These properties include: being One, being Simple, being Immobile, being Eternal, being Complete, being Independent, Existing in Itself, Subsisting by Itself, being Incorruptible, being Necessary, being Immense, being Uncreated, being Uncircumscribed, being Omnipresent, being Incorporeal, being All-penetrating and being All-embracing (see Koyre, 1957, 148). The extent to which Newton followed More in identifying space with God or a property of God is a matter of some dispute. That the characteristics of Newtonian absolute space match the properties listed from More above, there can be no doubt. Whether this entails that space itself was divine for Newton is another matter (cf. McGuire, 1978).
* Koyre, 1957, marshalls a strong argument for the influence of More on Newton. For a somewhat more reserved appraisal of More's influence on Newton, see McGuire, 1978. This completes our brief discussion of some of the theological considerations which played a role in the development of the Newtonian concept of space. Needless to say, perhaps, this survey is quite incomplete and a number of thinkers who made significant contributions have not been mentioned at all. To do them, and this whole period, justice would require another book in itself.
E. The Impact of Science on the Development of the Concept of Space.
The 17th century in Europe saw the rise of a new methodology for studying nature in what has been labelled the Scientific Revolution. This development did not spring up out of thin air but was the result of the convergence of a number of trends that had been developing since the explosion of intellectual activity in the West in the 12th century. Among the factors contributing to the scientific revolution we may list (l) the rediscovery of Greek atomistic theories in the 15th century; (2) the Copernican revolution in the 16th century; (3) the growth of interest in mathematics fostered by Latin translations of Archimedes and Apollonius; (4) the development of an experimental method; and (5) the invention of movable type in 1448 which made multiple copies of texts readily available.
The scientific revolution contributed to the transformation of the concept of space. We discuss three key factors.
The Copernican Revolution.
From the time of Aristotle to the publication of De Revolutionibus by Nicholas Copernicus (1473-1543) in 1543, the predominant view of the universe was the Aristotelian earth-centered universe. Aristotle's crude astronomy had been replaced by that of Ptolemy after the 1st century A.D., but the essential earth-centered cosmos was the heart of Ptolemy's system as well. In order to get a better agreement with empirical observations (in accord with the ancient slogan "Save the Appearances") Ptolemy introduced a number of complications but ultimately preserved the traditional aim of trying to reduce the motions of the heavenly bodies to a set of circular motions (see Kuhn, 1957, for further details). The history of astronomy from Ptolemy to Copernicus was basically the history of a series of corrections and emendations to the basic Ptolemaic scheme.
Copernicus's book revolutionized man's ideas about the heavens and the place of the earth in the universe. In De Revolutionibus, he developed an astronomical system which initiated what has come to be known as the Copernican Revolution. The central idea of the Copernican Revolution is that the sun, not the Earth, is at rest in the center of the universe. The nocturnal motion of the heavenly bodies was to be explained not by having the heavens revolve around the earth but rather having the earth rotate on its axis.
Since Copernicus held that the sun was at the center of the universe, he was still committed to the idea that the universe was finite in extent (since an infinite universe has no "center"). But, although he was committed to the idea of a finite universe, the move to an infinite universe was not far off. Aristotle had argued against the infinity of the heavens on the grounds that they revolved around the Earth once every 24 hours, which an infinite body could not do. Suppose, for example, that some star which is at an infinite distance from the earth revolves around the earth once every 24 hours. Since the star is an infinite distance away, then the circle around which it revolves must have an infinite circumference. Thus, the star could only revolve around the earth in a finite time if it traveled at an infinite speed. But, the idea that something could travel at an infinite speed was held by Aristotle to be impossible. However, on the Copernican view, it is the earth which rotates on its axis not the heavens which revolve. Since the earth is a finite sphere, there is no problem in having it rotate completely once every 24 hours. Therefore, this particular line of argument against the infinity of space is undermined, in principle, by the Copernican move. It was left to Thomas Digges (1576), an English follower of Copernicus, to draw the requisite conclusion from the Copernican system. Digges argued explicitly for the view that the expanse of stars in the heavens is infinite.
Although Copernicus argued that it was the sun and not the earth which was the center of the universe, there are other aspects of the Aristotelian system which he did not reject. In particlar, he still accepted the Aristotelian view that the motion of heavenly bodies was naturally in a circle. This idea, that there are two kinds of natural motions, straight line motions which are characteristic of earthly phenomena, and circular motions which are characteristics of heavenly phenomena had to be abandoned before the split between the sublunar realm and the heavenly realm, which is a central feature of Aristotelian cosmology, could be undermined. This final step in the abandonment of Aristotle's view occurred only with the publication of Newton's Principia in 1687.
Johannes Kepler (1571-1630) helped undermine another central dogma of the Aristotelian position when he showed that the orbit of the planet Mars was best represented by an ellipse not a circle. The use of ellipses to represent planatary orbits at one stroke produced an enormous simplification in the representation of the movements of the heavenly bodies. Up until the time of Kepler, even in the Copernican system, the orbits of planets had to be represented by complicated systems of interlocking circles some of which were rotating off center with respect to others. Kepler argued that a single ellipse could replace the many circular motions that had hither to been necessary to represent the motion of each planet. Not only was the representation simpler, but Kepler's view contributed to undermining the Aristotelian view that circular motion was the natural motion of heavenly bodies. Kepler showed that the orbits of the planets could be represented and codified in three laws of planetary motion which are: ~
Kepler's First Law: The orbits of the planets are ellipses. The sun occupies one focus of this ellipse.
Kepler's Second Law: The radius vector from the sun to the planet sweeps over equal areas in equal times.
Kepler's Third Law: T /R = K for all planets, where T = the period of the planet (i.e. the time it takes to complete one orbital revolution) and R = the average radius of the planetary orbit.
Figure of ellipse goes here A1=A2
t2 - tl = t3 - t4
The first two laws were published by Kepler in 1609. The third law was published ten years later in 1619.
Although Kepler accepted the basic heliocentric system of Copernicus and added, through his laws, important modifications of his own, he did not accept the implicit infinitism of the Copernican position. Kepler was working with data which were pre-telescopic. The data had been collected by Tycho Brahe, the foremost observational astronomer of all times up to the invention (or: perfection) of the telescope as an astronomical instrument by Galilei in 1612. Thus, the number of stars which could be observed was, although large, no where near as many as can be seen with even the simplist of telescopes. Kepler, in fact, rejected the idea that the universe was infinite in extent on the basis of this pre-telescopic data. His argument was essentially as follows. First, there can be no visible stars at infinite distance. If a star in the diameter S were at an infinite distance, S would have to be infinite since S ~ ~r, where S is the subtended arc, for small angle ~: a result of elementary geometry. But this is impossible.
Second, there could be no invisible stars at an infinite distance either, since, they too would have to be infinitely large. Kepler reasoned as follows. Suppose there were such an invisible star at an infinite distance from the earth. The star would then lie on the circumference of a circle centered on the earth which is infinite in extent. Let the star's diameter be designated by d. The diameter d is some proper fraction of the circumference of the circle on which it exists. But the circumference of the circle on which it sits is infinite, therefore, d must be some proper fraction of infinity. But no finite d is a proper fraction of infinity. Therefore, d itself must be infinite as well. Reflective readers will recognize in this argument an echoing of arguments which already were raised with respect to infinity by Zeno.*
Even after Galilei perfected the telescope as an instrument for astronomical observation, revealing that, among other things, the Milky Way, which hitherto had appeared to be merely a wispy cloud in the sky, was in fact composed of millions upon millions of stars, Kepler did not change his mind. Telescopic evidence in and of itself is inadequate to reject the "invisible star at an infinite distance" argument. The effect of the Copernican revolution was, thus, not to establish that space is infinite but, rather, to remove serious obstacles to this view by undermining the Aristotelian cosmology.
Experimental evidence for the void
One set of technical developments relevant to our story was the development of efficient pumps and the development of metallurgical techniques. This combination allowed for the production of artificial vacuua's epitomized by the famous demonstration by Otto von Guericke in Magdeburg in 1654. By pumping
* In fact, the assumption that d is some proper fraction of an infinite circle centered on the earth is false. But, a proper understanding of infinity is needed to appreciate this fact. Such was not forthcoming until the l9th century. the air from a globe made of two hemispheres, von Guericke produced a vacuum which held the hemispheres together even against the tug of two teams of horses pulling in opposite directions. The ability to produce such artificial vacuua helped to undermine what doubts remained about the possibility of voids within the sublunar sphere. It should be noted, however, that von Guericke's experiments did not convince everyone of the reality of voids. One influential dissenter was Rene Descartes, of whom we shall say more presently.
The mathematization of nature.
Other than the development of the experimental method, the single most significant methodological innovation of the scientific revolution was the conviction that natural phenomena and processes could be understood mathematically. The Greek influences at work here were Archimedes (for Leonardo da Vinci (14521519) and Galilei) and the Pythagorean tradition (for the more mystical minded such as Kepler). Galilei captured this spirit in the following remark: Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and to read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth. (Drake, 1957, 237-8).
One of the consequences of this attitude was the development of a mathematical analysis of motion. Galilei is the culmination of a long medieval tradition building towards the use of functional relationships to describe motion (see Dijksterhuis, 1961 for details). The mathematical analysis of motion
means that space and time must also be conceived mathematically (for local motion, recall, is just changes of place with respect to time). The net effect was to reinforce the view that space was a quantity (i.e. something measurable) which existed independently of the objects which moved in it. Space becomes a geometrical structure.
Rene Descartes (1596-1650) was another pioneer in the mathematical analysis of nature. It was Descartes who first conceived of the idea of representing spatial magnitudes and positions on diagrams with coordinate axes which became known as Cartesian coordinate systems. Descartes, thus, contributed to the geometrization of space.
In addition, Descartes is significant because he saw himself (and was perceived by his contemporaries) as having produced a comprehensive synthesis of science, religion and philosophy which rivaled Aristotle's (Dijksterhuis, 1961, 408). Descartes also formulated a physical theory that was to compete with Newton's for over a hundred years. Finally, he formulated a research program which, with some modifications, was adopted by the Newtonians. Descartes' plan was to produce a completely mechanical explanation of natural phenomena, that is, an explanation of all natural phenomena in terms of matter and motion. In this respect, Descartes' program is reminiscent of that of the ancient atomists whose slogan was "Nothing exists but atoms and the void." In virtue of Descartes' peculiar theory of matter, which he identified with space, the Cartesian program can be characterized as "everything is to be explained in terms of geometry and motion."
Spatial magnitudes were called by Descartes "extensions." By a thought experiment, Descartes identified extension (or space) with matter. The argument goes as follows. Descartes held that the essential nature of something was whatever could not be thought away from the thing in question. For example, grass is green. Is being green part of the essence of being grass? The Cartesian test is to try to imagine grass not being green. If it is possible to imagine grass not being green then being green is not part of the essence of being grass. Since it is possible to imagine grass which is not green, then greenness is not part of the essence of grass. Now consider any material object. According to Descartes, all the properties of that object can be "thought away" except for the fact that the object is extended, i.e., occupies a volume. Thus, Descartes concludes that the essence of matter is extension or space. Note that it follows, on Descartes' view, that there can be no voids, i.e., extensions with no matter in them. This explains why Descartes did not accept von Guericke's experiments as proving that vacuua could exist. They can't exist, according to Descartes, just as "square circles" can't exist no matter what experimental evidence someone might adduce to show that they did.
One might wonder, if space and matter are the same, what Descartes took to be the difference between individual objects like tables and chairs, or planets and people, which we ordinarily think of as material objects, and the (more or less) empty spaces between them. The key, according to Descartes, is motion. When God created the universe he put various parts of space into relative motion with respect to one another. What we perceive as "solid" objects are merely pieces of space in suitable relative motion to one another. The details of the motions Descartes left for his successors to work out. As an aid, he formulated 7 laws of motion which could be used as the basis for explaining all motions. Unfortunately for Descartes' original program, his laws turned out to be inconsistent with one another and in conflict with experimental fact (see Blackwell, 1966, for a discussion).
Despite the failure of Descartes' original program, his mechanical philosophy became the paradigm of research in the 18th and l9th centuries with the slight modification that matter was assumed as an irreducible factor which in conjunction with motion sufficed to explain everything.
Descartes' views on matter and motion were attacked by many of his contemporaries and near contemporaries such as More, Isaac Barrow (1630-77) and Newton himself. Isaac Barrow, in particular, developed a concept of space as a capacity for containing objects which is similar to that adopted by Newton. As Newton was a student at Cambridge while Barrow was teaching there, and, in fact, succeeded Barrow as Professor of Mathematics, it is reasonable to conclude that Barrow's work influenced Newton. F. Metaphysical Considerations.
In addition to the theological and scientific arguments against the Aristotelian conception of space, there were metaphysical arguments as well. Complete emancipation from the Aristotelian position required a rejection of the metaphysics implicit in that view. The metaphysical position underlying Scholastic Aristotelianism came to be known as the doctrine of substance and accident. The aim of metaphysics, classically understood, is to articulate the most fundamental categories of reality. The substance/accident doctrine held that everything real was either a substance or an accident. By a substance was meant some thing in which properties, attributes or, as they came to be known, accidents could inhere. An accident was then some property which characterized a substance. Consider, for example, a wooden chair. The chair is a substance; being wooden is an accident or property of the chair. The basic difference between substances and accidents was, roughly, that substances could exist in and of themselves, but accidents could only exist if they were attached to some substance. The Cheshire
* For a discussion of Barrow's view of space and his possible influence on Newton see Strong, 1970. For Barrow's views on Time see below p. 6-68. cat in Alice in Wonderland captured to a T the intuition underlying this metaphysical position. You recall that after Alice has met the Cheshire cat and had a pleasant conversation with it, the cat begins to fade away until only its smile remains. What makes this so funny is that we all feel that a smile is not the kind of thing that can exist on its own. The Cheshire cat may be smiling or it may not be, but the smile can't be sometimes "catted" (') and sometimes not. This intuition is reflected in the substance/accident view by our categorizing the Cheshire cat as a substance and its smile (when it has one) as an accident.
That the substance/accident view implicitly underlies Aristotle's view of place (or space) is revealed by the difficulty he had in accepting the interval view of place. To adopt the interval view of place was to accept the idea that the essence of space is its 3-dimensionality. Being 3-dimensional, however, is or seems to be a property or attribute. As a property or attribute it must be a property or attribute of some substance. When this is connected with the general Greek tendency to think of substances as material or corporeal, we are led to conclude that the space of a thing is a 3-dimensional body of some kind different from the body whose place it is. With this insight in mind you might go back and reread the Aristotelian arguments against the void, especially those which rest on the "penetration" arguments (cf. 5-21 ff: above).
For those, therefore, who wanted to adopt the 3-dimensional view of space, a metaphysical hurdle had to be overcome. Various strategems were employed to try and reconcile the 3-dimensional view of space with the basic substance/ accident metaphysics. In order to avoid the "penetration" arguments, Plotinus and others distinguished between corporeal and incorporeal substances. Space was held to be an incorporeal substance and, thus, penetrable by corporeal objects. More identified space with a divine or spiritual substance. Descartes,
as we have just seen, took the heroic course of identifying space with body, i.e., accepting the view that space was a corporeal substance. His view avoids the Aristotelian penetrability objections by identifying a body with its underlying space leaving only one body not two, and hence, no penetration problem.
By the 17th century, however, the need for an alternative metaphysics was becoming evident to many thinkers. Space (and time as well, see p. 6-69 below) as they functioned in the new physics being developed did not fit easily into either category.
Interestingly enough, the Condemnation of 1277 seems to have played a role in undermining the traditional substance/accident view (Grant, 1974, 142f.). Among the condemned propositions were the following:
196. That to make an accident exist without a subject has the nature of an impossiblity implying contradiction. (140)*
197. That God cannot make an accident exist without a subject or make more than one dimension exist simultaneously. (141)
198. That an accident existing without a subject is not an accident except in an equivocal sense, and that it is impossible for quantity or dimension to exist by itself, for this would make it [i.e., an accident, into] a substance. (139)
The condemning theologians were not, of course, concerned with the substance/ accident doctrine as applied to space or time. As Lerner and Mahdi note, the condemned propositions
* Lerner and Mahdi, 1972, 354. Again the numbers on the left reflect the order of presentation in the Lerner and Mahdi selection, while the numbers on the right in parentheses indicate the order in which they occurred in the original condemnation. are directed implicitly against the theological doctrine of the Eucharist, which states that the accidents of bread and wine are left without any subject of inherence once the substance of the bread and wine has been converted into the body and blood of Christ (Lerner and Mahdi, 1972, 354). Nevertheless, the effect was to affirm, against Aristotelian metaphysics, God's power to create self-subsisting accidents. If the accidents of the bread and wine could subsist without an underlying substance why not the 3-dimensionality of space?
The credit for first raising an explicit attack on the substance/accident distinction as it relates to space must go to the Renaissance scholar Francescus Patrizi (1529-1597) who, in 1577 became the first Professor of Platonic philosophy at the University of Ferrara (Henry, 1979, 552). It is fitting, perhaps, that a Platonist should play such a major role in undermining the metaphysics implicit in the Aristotelian view of place. Patrizi was influenced by such advocates of 3-dimensional space as Philoponus and Crescas, as well as the general revival of Platonism in Italy in the latter half of the 15th century and the particular influence of Bernardino Telesio (1509-1588) (see Jammer, 1960, 83 f.). Telesio had argued that space was a quantity (i.e. an "accident") which, however, existed in its own right. As such, he held it to be incorporeal, causally inert, homogeneous and immobile (Jammer, 1960, 84). It was left to Patrizi to draw out the metaphysical implications of this view. After affirming that space is an infinitely existing void expanse which exists prior to the existence of bodies ("For if bodies came into being and no locus [place] were provided for them, where would they be located?"), he goes on to ask: "What then, finally, is this Space which existed before the world, after which came the world, which contains the world, and extends beyond it? (Brickman, 1944, 23 ff.). Patrizzi's answer and the spirit of his position can best be appreciated by letting him speak for himself: To be sure, this Space . . . if it is something, . . . will be either a substance or an accident. If it is a substance, it is either some incorporeal thing or a body. If it is an accident, it is either a quantity, a quality, or some other such thing. Now I maintain that space per se in itself , since it is prior to the world and outside it, is . . . a different thing from the world. For the world is a body, while space is not a body at all. It is therefore not embraced by any of the categories, and is prior to and outside them. our emphasis What then is it? Space . . . is substantial extension . . . subsisting per se, inhering in nothing else. It is not an Aristotelian quantity. And if it be a quantity of some sort , it is not that of the Aristotelian categories, . . . Nor can it be called an accident, for it is not the attribute of any substance.
Is it then a substance? If substance is id quod per se substat [that which subsists in itself] . . . or . . . is quae aliis substat [that which underlies other things] . . . or . . . is quae nulla aliarum rerum eget ad esse [that which requires no other thing in order to be] . . . or . . . is primum omnium entium [prior to all things] . . . then . . . Space is above all a substance . . .
For all these reasons, therefore, it is very clear that Space is above all a substance, but not the "substance" of the Aristotelian categories . . . It is a different sort
of substance outside the table of categories. What then is it, a body or an incorporeal substance? Neither, but a mean between the two . . . it is an incorporeal body and a corporeal non-body Brickman, 1944, 240-1.
This line of reasoning with its paradoxical conclusions shows Patrizi trying to struggle free of the Aristotelian system of thought and finally having to settle for a paradoxical formulation: Space is a "corporeal incorporeal and incorporeal corporeal." This expression betrays Patrizi's neoPlatonic heritage: Patrizi uses the same term to apply to light, and notes the similarity between the two although he does not go as far as Proclus who identified Space with light.
Despite the cumbersome and obscure terminology it is clear that Patrizi is making a fundamental break with Aristotelian ways of thinking. Space is something which must be an absolute existent, a set of loci (places) within which objects exist. It exists independently of objects and functions as an infinitely extended Platonic receptacle.
The significance of Patrizi for later thinkers lies in his influence on the views of such men as Pierre Gassendi (1592-1655) (see Henry, 1979; McGuire, 1978; Jammer, 1960). Gassendi was the foremost exponent of atomism in France in the early 17th century. As such, he accepted the reality of the void as postulated by the later Greek atomists and adopted Patrizi's position that space was something which defied categorization in Aristotle's terms. Gassendi, in turn, directly or indirectly influenced Newton. Newton certainly read Walter Charleton's 1654 epitome of Gassendi's philosophy and in it Charleton drops Patrizi's terminology and talks merely about "spatial dimensions" (Henry, 1979, 572; Westfall, 1980, 89). This completes our brief survey of some of the complex lines of thought leading from the Aristotelian concept of place to the concept of space as an autonomous, absolute, receptacle upon which Newton constructed his classical physics. Before we turn to a discussion of the development of the concept of time, a summary of the basic trends may prove helpful. F. Converging Themes.
We may distinguish four lines of argument leading from Aristotle's concept of space to that of Newton: Theological, Physical, Mathematical and Metaphysical.
The connection of God with place or space goes back to the Old Testament and early rabbinical writers. The condemnation of 1277 rejected the views that God could not make an infinite space or create a void and thus lent support for the tendency to identify space as an absolute structure devoid of qualities but serving as a receptacle for all objects. More took this suggestion to its logical conclusion by identifying space with God's immensity. The net result of these speculations was to lend credence to the view that space was a something in its own right capable of existing prior to and independently of objects.
The Copernican revolution and Kepler's laws of planetary motion served to undermine the physical basis of Aristotle's cosmology and so paved the way for thinking of space as a physically infinite expanse. The experimental results relating to the production of artificial vacuua were instrumental in reinforcing the view that space as such was a void, causally inert, arena within which objects existed.
The 15th century Renaissance coincided with a revival of interest in Greek mathematics. This revival and the subsequent efforts of Galilei and Descartes
to produce mathematical analyses of local motion, directly or indirectly, reinforced the view that Space was a quantity in and of itself. Physical space became for the first time, a mathematical structure. In the course of the subsequent 300 years, an analysis of that structure yielded a number of surprizes (see Chapters 9 ff. below).
Finally, the shift from the Aristotelian to the Newtonian view of space was characterized by a fundamental metaphysical shift. Space was conceived of as an entity which did not fit easily into the Aristotelian system of categories. The emerging concept of Space is quite Platonic. It is conceived of as an independently existing mathematical structure which serves as the background or matrix for physical objects and processes. Unlike Plato's receptacle, however, the mathematical space of the 17th century was infinite.
The result of these converging themes was to set the stage for the revolution in physical theory which was initiated by Newton. Newton's physical theory, the first viable mathematical physics, became the paradigm in terms of which all subsequent scientific theories were judged until the 20th century when another revolution in our idea about physics and space and time was initiated by Einstein. The fact that the Newtonian ideal of science reigned supreme for less than 200 years whereas the Aristotelian ideal reigned supreme for almost 2,000 years does not detract from Newton. It is, rather, a testament to the power of the mathematical mode of analysis, perfected by Galilei, Descartes and Newton, to uncover the secrets of the physical world around us. V. The Development of the Concept of Time from Aristotle to Newton. A. Aristotle's Relational Theory of Time.
Aristotle, recall, characterizes time as the measure or number of a motion. In urging that although there are many motions in the world, there is only one time, he strongly suggests that time is the measure of the (uniform) motion of the sphere.
Time, thus, for Aristotle is an accident or property of motion. If the motion of the sphere, (and all other motions as well) were to cease, then time would cease as well. Time is like the smile of the Cheshire cat: no cat, no smile; similarly, no motion, no time. This view has been called, appropriately enough, the "celestial reductionism of time" (Ariotti, 1969). Time has been reduced to an aspect of the motion of the heavenly sphere.
The idea that time could exist in and of itself was not prevalent in the thinking of Greek antiquity. Recall that Plato had characterized time as the moving image of eternity and, like Aristotle, explicitly connected the passage of time with the motion of the sphere.
According to Sambursky, the only representative in Greek antiquity of the view that time could exist in and of itself was Strato of Lampsascus (c. 275 B.C.). Strato, working at Alexandria, held that "Time is a quantity wherein all changes and actions are contained" (Sambursky, 1960, 10-11). This view, which is, in essence, that of Newton, did not prevail until the 17th century.
Finally, note that Aristotle's concern is primarily with time as a physical measure. Thus, Aristotle's theory is a physical theory of time. But, time has other aspects as well. It is intricately connected with our sense of consciousness and, thus, has a fundamentally psychological aspect as well. Except for the one brief remark at Physics 223a23 where Aristotle wonders if time would exist were there no souls to measure the motions of the heavenly sphere, he does not delve into this aspect of temporality. The earliest extended critics of Aristotle whose writings we still possess, however, focussed on these aspects as central. We turn now to a brief discussion of their views. B. Plotinus' Metaphysical Theory of Time.
Plotinus (c. 204-270 A.D.) was born in Egypt, educated in Alexandria and was a center of a group of students of philosophy in Rome from 244 to 270 A.D. He was an admirer of Plato's philosophy and developed it along spiritual and mystical lines. His views were instrumental in shaping the Platonic heritage as it was passed down through late antiquity and the middle ages.* He wrote a number of works including a collection of essays outlining his views on diverse topics called the Enneads. One book of the third Ennead, "On Eternity and Time," is devoted to his theory of time and contains the most ancient extensive criticism of Aristotle's theory of time that we still possess. After classifying theories of time according to whether they (l) identify time with motion, or (2) identify time with some moving object, or (3) identify time with some attribute of motion, and having rejected all the alternatives, except for Aristotle's, he focusses in on it. He criticizes Aristotle for failing to provide a standard of uniform motion. Aristotle says that the motion of the sphere is uniform but since it and time measure each other, how does Aristotle know that the motion is uniform (cf. p. 5-47 above). Secondly, Aristotle tells us that time is the number of a motion but he doesn't tell us what kind of number it is. Without telling us this, how are we, Plotinus asks, to distinguish
ordinary numbers which are not time from those numbers which are time? (Plotinus, 1967, III, ch. 9, / 331.)
Thirdly, and most fundamentally, Plotinus argues that Aristotle's characterization tells us what time measures but not what time is
(Plotinus, 1967, III, ch. 9,/ In general, Plotinus objects to Aristotle's
physical quantitative approach to time as being too narrow and failing to reveal the essential nature of time.
* For a general survey of the philosophy of Plotinus, see Rist, 1967. In developing his positive theory of the nature of time, Plotinus draws on Plato's account in the Timaeus. With Plato, he sees time as a reflection, in this world, of Eternity. He characterizes Eternity as "infinite life that is ever complete" (Callahan, 1948, 93). The basic idea is that Eternity is the timeless existence characteristic of Being. Following Plato's cosmology, Plotinus sees Eternity and the World of Forms as existing timelessly and unchanging (logically prior) to the existence of the world. The World Soul contemplates the Forms in their completeness. In this picture of tranquility, there arises a certain restlessness in the World Soul. This sets the World Soul in motion by some stimulus. The World Soul creates the Universe in imitation of the intelligible Being which exists eternally. Time then, Plotinus says, is the life of the soul as Eternity was the life of Being.
Thus, for Plotinus, as for Aristotle, time is connected with motion. The difference is that for Aristotle, time is a measure of physical motion, whereas for Plotinus, time is the very life of the moving (thinking) soul. Which soul? The World Soul and incidentally all the individual souls as well. Does this mean that eachperSOn has his own time? Yes and no. Yes, each person does have his own time but, because the nature of all souls is, according to Plotinus, the same, there is only one time for the entire universe.
This view is not particularly conducive to physical research. It succeeds in telling us what time really is (perhaps) but does not suggest any way in which that knowledge might be useful in understanding the physical world. Plotinus' view is, in this sense, typically Platonic. Nevertheless, Plotinus' theory influenced a number of thinkers, St. Augustine among them, who in turn shaped the views of time held by many in the middle ages. C. Augustine's Psychological Theory of Time.
Augustine was born in North Africa in 354 A.D. He was made Bishop of Hippo (North Africa) in 395, and he died in 430. Augustine was the most eminent of the early Church fathers. During the period 397-401 he composed the Confessions, an account of his early dissolute youth and consequent conversion to Christianity. Book 11 of the Confessions contains an account of the nature of time which begins with his formulation of the Aristotelian puzzle about the existence of time which we cited earlier (5-31 above). Augustine was influenced by Plato and Plotinus but was constrained to temper their views by his Christian faith.
Augustine accepted, with modifications, the connection between time and the soul suggested by Plotinus. Whereas for Plotinus, time, as the life (movement) of the World Soul produces (physical) motion (since the restless motion of the World Soul is what 'creates' the physical world of change), for Augustine time is an activity of soul completely independent of physical motion. Augustine accepts Aristotle's view that time is a measure of motion, adding, however, that the time which is a measure of motion is (essentially) an activity of the soul (see Callahan, 1948, 166 f.). For Augustine, the activity of the soul, rather than the motion of the sphere, serves as an absolute standard for temporal duration (Callahan, 1948, 170 ff.).
As for the problem of temporal measurement, Augustine solves it in a radical way. The problem, recall, is that the past and the future don't seem to exist in the present. If they don't exist then how can they be measured? When we use a ruler to measure a spatial length, the length is "all there" so to speak, throughout the process of measuring. When we use a clock to measure time, however, the ticks marking off the seconds, once ticked, are gone. Once gone, how can we effect the measurement of any temporal interval since the only moment that does exist is the present? It is as if every time one tried to measure a spatial interval, only one of the endpoints could be determined. Augustine's solution is radical and simple. He accepts that only the present moment as experienced by an observer is real. The past only exists as a memory. The future only exists as an expectation. In determining the lengths of temporal durations, these memories and expectations exist in the present along with the present moment. What the soul does, in effect, is measure the intervals as they are remembered (or for future intervals, as they are anticipated). Thus, physically, only the present exists. The rest of time is a function of psychological processes involving memory traces and expectations. Augustine answers Aristotle's question about whether time exists if no souls exist to number the motion of the sphere in the negative. No measuring souls, no time.
Time was created when God created the heaven and earth. Thus, Augustine rejects questions such as: "What was God doing before He made the heavens and the earth" as meaningless (Augustine, Confessions, XI, chapter 14). God, for Augustine, is Eternal, where this does not mean that God exists in time forever, but rather that God is "outside" time. D. God and Time in the Transition from Aristotle to Newton.
Just as in the case of the revolution of the concept of space, so religious concerns have helped toshape the evolution of our concept of time. The fundamental problem with Aristotle's view of time, from the perspective of Judaism, Islam and Christianity, is the doctrine that time is eternal. Insofar as time is the measure of motion this implies that motion is eternal which in turn implies that the world is eternal. This, however, directly contradicts the Genesis account of the creation of the world. Three options were available to meet this challenge. Either one could deny that time was eternal (i.e., infinitely extended into the past), as did Augustine, or one could deny the Aristotelian connection between time and motion, as did Crescas, or one could adopt a compromise position intermediate between the two, as did Aquinas (cf. Ariotti, 1972, 103). Aquinas on time.
Aquinas accepts the Aristotelian identification of time as the measure of physical motion. In his commentary on Aristotle's physics, he argues that
. . . among all . . . circular motion, the first motion which revolves the whole firmament in daily motion is the most uniform and regular. Hence that circular motion, as first and more simple and more regular is the measure of all motions. Moreover, it is necessary that a regular motion be the measure or number of the others. For every measure ought to be most certain, and this is found in things which are uniformly related.
Therefore, from this we can conclude that, if the first circular motion measures all motion, and if motions insofar as they are measured by some motion are measured by time, then it is necessary to say that time is the measure of the first circular motion . . . (Aquinas, 1963, 256 f.)
On the eternity of time, he parts company with Aristotle. The Christian faith holds that only God has always existed. Time, with the world, was created, and, thus, is not infinite. However, A~uinas goes on to say, there is a duration before time, namely the eternity of God, which has no extension of either before or after, as does time, but is a simultaneous whole. This does not have the same nature (ratio) as time, jsut as divine magnitude is not the same as corporeal magnitude (Aquinas, 1963, 487). In the Summa Theologica, Aquinas says: God is before the world in duration yet before does not mean a priority in time, but of eternity, or perhaps, if you like, an endlessness of imaginary time [our emphasis] (Ariotti, 1972, 103). The compromise that Aquinas effects by this last remaFk is to distinguish between real (physical) time,which is finite, and imaginary time, associated with the eternal enduring of God, which is, if you will, infinite (cf. Maimonides, 1956, 171). A similar device was used by medieval writers to argue for the infinite extent of space. A distinction was drawn between real physical space, which was finite, and imaginary space, which extended to infinity, and was associated with God's immensity (see Grant, 1981, for further discussion). In both cases, the distinction eventually collapsed and imaginary time (space) became indentified with a duration (extension) that existed independently of motion (objects).
One can detect the hint of Plotinian and Augustinian views about Eternity in Aquinas' account. The crucial difference is that Aquinas, perhaps inadvertantly, suggests a closer relation between time and eternity than Plotinus or Augustine would allow through his suggestion that God can be thought to endure throughout imaginary time.
The condemnation of 1277.
As in the case of space, the condemnation of 1277 had an effect on the development of theories about the nature of time (Hutton, 1977, 348; Ariotti, 1872, 92 f.). The eternalness of time was one of the condemned propositions (number 87 on the original list, see p. 6-16 above) as well as the doctrine that "eternity and time have no existence in reality but only in the mind" (number 200 on the original list. See p. 6-16 above).
Crescas' critique of Aristotle.
Crescas explicitly rejects the Aristotelian view (as understood by Maimonides) that "Time is an accident that is consequent on motion and is conjoined with it" (Wolfson, 1929, 283). Crescas argues that time is equally a measure of rest as it is of motion. Thus, he is not rejecting the Aristotelian view that time is an accident, rather he is rejecting the view that time is an accident of motion. In fact, he explicitly rejects the suggestion that time is an (Aristotelian) substance (Wolfson, 1929, 289). What, then, is time an accident of? The correct definition, he suggests, is that time "is the measure of the duration [our emphasis] of motion or of rest between two instants" (Wolfson, 1929, 289). Duration, thus characterized, is something independent of either rest or motion. Wolfson suggests that it is modelled on Plotinus' concept of Eternity. Duration is a temporal eternity, as it were (Wolfson, 1929, 633-664, esp. 654 f .).
On the basis of this line of reasoning, Crescas accepts the infinity of time, although he takes the world to be created in accordance with the account in Genesis (Wolfson, 1929, 291). Crescas' views were, in turn, available to the Italian renaissance through the work of Pico della Mirandola (1463-1494) and, Wolfson argues, Giordano Bruno (1548-1600) as well (Wolfson, 1929, 36. For the relationship between these thinkers, see Hutton, 1977, esp. 362) .
In addition to arguing that duration is eternal, Crescas also holds that temporal order, i.e. the "before-after" relation exists prior to the creation of the world (Wolfson, 1929, 291) . Crescas' student Joseph Albo (1380-1444), adopts a slightly different position which seems to have had a number of adherents during the period. As Wolfson reports, Albo distinguished two kinds of time, (l) absolute time, which is "unmeasured duration" and which existed prior to the world and will exist af ter the world ceases to exist and (2) measured time, which is the time we measure by the motion of the heavens (Wolfson, 1929, 656 f.). For Albo, only measured time is "true" time; absolute time is, in a sense, only potentially time (Wolfson, 1929, 658; cf. Aquinas and Maimonides on "imaginary" time). A similar distinction was argued for by Christian philosophers such as Duns Scotus (1270-1308) and Nicholas of Cusa (1401-1464) (see Ariotti, 1973b, 146 f.). Cusa identified duration with God, as did Nicole Oresme (d. 1382) and later More (Ariotti, 1973a, 143). More, in turn, influenced Isaac Barrow (16301677) and Newton. E. The Impact of Science on the Development of the Concept of Time.
In this section we examine some of the scientific innovations which shaped the development of the concept of time. As in the case of space, the Copernican revolution and its aftermath had a significant impact on changing prevalent ideas about time. In addition, we single out for consideration the "mathematization of time" effected by Galilei and Barrow, and relevant technological considerations which resulted in the need for and development of more accurate clocks.
The Copernican revolution.
Ariotti argues that the Copernican revolution aided in undermining the Aristotelian connection between time and motion, and by so doing set the stage for the subsequent development of the concept of absolute physical time as it appears in the work of Barrow and Newton (Ariotti, 1973a, 37). Copernicus, himself, retained the Aristotelian concept of time, but his heliocentric system does not fit in easily with this conception.
The empirical kernel of the Aristotelian view is the supposed uniform motion of the heavenly sphere. In the Copernican system, where the earth revolves around the sun, the uniformity of motion disappears from the system.
* The irregularities in the motions of the planets were noticed by Greek astronomers and great pains were taken to try to accommodate them to models using uniform motions (cf. Vlastos, 1975).
N4 ~ N3 L2~/Nl
\ --I / vernal equinox
The north pole rotates through
NlN4Nl, in a direction opposite
to the revolution of the earth
around the sun. One complete
rOtation takes 26,000 years- Figure 6-6
The two years do not coincide because the equinox slowly shifts to the west as the sun moves to the east. In fact, even successive tropical years are not congruent because the shift is irregular. Successive sidereal years, it turns out, are not congruent either because of a number of factors including variations in the earth's orbital motion and shifts in the background of fixed stars. Even so, determining sidereal years observationally would have ~een very difficult in Copernicus's time. A similar problem occurs when one considers a "day." Copernicus distinguishes between two days: (l) a rotational day, defined by one complete rotation of the earth on its axis and (2) the solar day, defined by the time between two successive sunrises.
The solar days are not constant because of the eccentricity of the earth's orbit around the sun. This
( t4 - t3~(t2 - tl~ 1
means that the orbital speed of the earth around the sun varies from point to point. The net effect is that the time between successive sunrises varies from day to day. Copernicus took the rotational day to be constant, but because of the earth's motion around the sun, was left without an operational means of determining it (Ariotti, 1973a, 37 ff.).
The net effect is that there are a lot of circular motions in the Copernican system, but no empirical grounds for singling out any to be the measure of uniform motion which time is supposed to be. The situation gets worse with Kepler's realization that the orbital paths of the planets is irregular (cf. the equal areas law) and, hence, the motions of the planets cannot serve as a standard of uniform time as well (Ariotti, 1973a, 41 ff.).
* In fact, the rotational day is not a constant since the tides act as a dragging force inducing variation. But, this was not understood until Newton's day. These empirical considerations do not imply that time has an absolute character independent of motion. What they do, at best, is undermine the dominant theory of the age which identifies physical time as a measure of physical motion. Insofar as an absolute conception is lurking in the (theological) wings, the effect, at best, can be to reinforce inclinations to move toward an absolute conception of time.
It is hard for us today, when time is money and young children wear the latest sophisticated watches capable of measuring time to incredible accuracy, to appreciate ages where time and its accurate measurement were not of great concern. Yet for most of human history people have survived with little regard or need for accurate measures of time. In the middle ages, one impetus towards the development of reliable timepieces was the concern of Christian monks to determine the proper time for prayers. The major impetus towards accurate timekeeping in the late medieval and renaissance period was due, however, to the commercial expansion of Europe. As European nations embarked on the age of discovery and exploration they ventured across the open seas in the search for new sources of raw materials and new markets. The need for reliable methods of navigation became crucial. Determining latitude at sea is a relatively easy task. One need only measure the height of the noon sun or aknown star over the horizon in order to determine latitude. Determinations of longitude are something
Figure 6-8 else. In order to use the stars to determine longitude one must have an accurate measure of time. Given that the procedure is simple. One can determine, for example, that on a given day a certain star will appear at a given position in the sky at a particular place (say, Greenwich) at various times (at that place) during the night. In order to determine location at some other place, one notes the sky as it appears locally and determines the time at Greenwich which corresponds to the observed configuration of the stars in the sky. By comparing local time (on a clock which was synchronized with a clock in Greenwich) with the Greenwich time corresponding to the observed configuration, one can determine how far one is east or west ofGreenwich. Each hour of difference corresponds to 15 . The catch is that one must have an accurate clock that, once synchronized with the Greenwich clock, will keep regular time. Constructing such a clock was a ma~or research problem from the 16th to the 18th centuries (see Howse, 1980).
Creenwich: prire reridian
Figure 6-9 In 1583, Galilei noticed that the swings of an undamped pendulum were isochronous, that is, each successive swing took the same time to complete. The principle of the Grandfather clock was born. Unfortunately, a pendulum clock is of little use at sea, since the rolling of a ship interferes with the regular motion of the pendulum bob. The first commercially successful clock useful for navigational purposes was constructed in 1727 (Howse, 1980, 67ff.).
The search for reliable and accurate measures of time does not affect the question of whether time is connected essentially with motion or not. Even Galilei's pendulum clock measures time by a motion, namely, the periodic motion of the pendulum bob. What the search for reliable clocks does is foster the mathematization of time, that is, it encourages the view that time is a quantity which can be described mathematically. In this area as well, Galilei was a pioneer.
The mathematization of time.
Although Galilei does not provide an explicit discussion of the nature of time, he is credited with being the first to conceive of time as a mathematical coordinate which can be utilized in the mathematical analysis of motion (Sambursky, 1960, 99; Ariotti, 1973, 150). By carefully distinguishing between velocity and acceleration (change invelocity) Galilei, for the first time was able to provide a functional relationship between the time, speed and distance travelled by a falling body. The idea that physical quantities and processes could be so represented was a major innovation. One of the reasons for the failure of the
* Actually, a pendulum clock is affected by gravity. Since the pull of gravity varies from point to point on the earth, a pendulum clock which keeps accurate time at one latitude will not keep accurate time at another. Galilei, however, did not realize this. Greeks to develop an adequate mathematical analysis of motion, it has been suggested, was their failure to think in terms of functional relationships between quantities (see Sambursky, 1962, 11; and Dijksterhuis, 1961, p. 53. See also Owen in Salmon, 1975).
The idea that time is a mathematical coordinate is conducive to the view that time is a framework within which processes and events occur. By conceptualizing time as such a framework, its connection with those"contained" processes and events is weakened. It begins to take on the character of an analogue to Plato's receptacle. This view is found fully articulated in Isaac Barrow, the first Lucasian professor of mathematics at Cambridge while Newton was a student there (Newton succeeded Barrow as the second Lucasian professor in 1669).
Barrow urges that Time has many analogies with a line, either straight or circular ['], and therefore may be conveniently represented by it; for time has length alone, is similar in all its parts, and be looked upon as constituted from a simple addition of successive instants or as from a continuous flow of one instant; . . . (Child, 1904, 37)
Barrow has evidently not thought through all the implications of his analogy. As we have remarked before, time has two aspects, length (or duration) and order. Barrow in this remark ignores aspects of temporal order, since to represent time as a circular line, in effect, denies that any unique before/ after relation can be defined with respect to its points. Nevertheless, the fundamental point is that time, represented by a line, can be used as an independent variable to which the motions of bodies can be functionally related.
Barrow argues that time, in itself, is "not an acutal existence, but a certain capacity or possibility of existence; just as space denotes a capacity for intervening length" (Child, 1904, 35). Time is declared to be independent of motion, although Barrow allows that motions are used to measure time. Time, he says, "pursues the even tenor of its way . . . [and] . . . [w]e evidently must regard Time as passing with a steady flow . . . " (Child, 1904, 35-6). How do we know that time does pursue the "even tenor of its way?" By comparing it to some "handy steady motion" such as the motion of the Sun. The motion of the Sun, in turn, is deemed to be steady by comparison with some handy instrument of our own making "designed to be moved uniformly by successive repetitions of its own peculiar motion" (Child, 1904, 37). In fact, Barrow concludes that the "first and original measures of Time" are not the motions of the heavenly bodies, but rather motions which we observe our instruments to undergo, e.g., the periodic swings of a pendulum (Child, 1904, 57). By "first and original" Barrow must mean logically or conceptually first, since, historically, the "even tenor" of time seems clearly to have been suggested by the regular motion of the sun and stars. The circularity in the reasoning seems to escape Barrow, and unlike Newton (as we shall see) and some of the earlier adherents to the doctrine of absolute time, such as Albo, Barrow seems to have accepted the idea that there are exact measuresof the "even tenor" of the flow of absolute Time.
The extent of Barrow's influence on Newton cannot be determined precisely although the similarity of ideas and approaches suggest there was some influence. Newton certainly read Barrow's work but, if he borrowed anything, he transformed it into a system which Barrow did not dream of. F. Metaphysical Considerations.
Again, as in the case of space, metaphysical considerations played a role in shaping the development of the concept of time.
In the 15th century renaissance, a number of thinkers, influenced by the revival of Platonism, advanced considerations which suggested that time was not, as Aristotle had it, an accident of motion, but something more absolute and self-subsistent. Among them were Bernardino Telesio (1509-1588), Giordano Bruno (1548-1600) and Francescus Patrizi (see Hutton, 1977, for an extended discussion of their contributions and possible interrelations of their thought).
Telesio argues that 'Time in no way depends on motion, but . . . it exists by itself, and what characteristics it has, it has of itself, and more from motion' (Quoted by Hutton, 1977, 354). In addition, Telesio rejects Aristotle's epistemological argument for the dependence of time on motion (Hutton, 1977, 355). Recall that Aristotle had argued, in effect, that
Pl: Awareness of time depends upon awareness of motion. Therefore
C : The existence of time is dependent upon the existence of motion (cf. p. 5-35 above).
Telesio rejects this argument as invalid. From the fact that one is aware of the passage of time only insofar as one is aware of some motion, it does not follow that time would not pass, if no motion existed. Barrow repeats this argument in his geometrical lectures (Child, 1904, 36).
Bruno, possibly influenced by Crescas (cf. 6-61 above), argues that time and duration are one and the same. Duration, in turn, is, like space, a unity comTnon to all objects it contains (Hutton, 1977, 356). With Aristotle, however, he holds that time is eternal; eternity is, he says, perpetual time (Hutton, 1977, 358) .
Patrizi, who argues against the substance/accident metaphysics as inapplicable to space, apparently does not take the same tack in his critique of Aristotle's concept of time (Hutton, 1977, 338 f.). This task is left to Gassendi, who was, recall, familiar with Patrizi's work. Gassendi argues It is commonly held that every being is either substance or accident, and that every substance is corporeal or incorporeal, and every accident (since it belongs to a substance . . . ) is either corporeal or incorporeal; and that the first characteristic of all bodies is quantity . . . it is commonly held that Place and Time are corporeal accidents with the consequence that if there were no bodies upon which they could depend neither Place nor Time would exist. However, our opinion is that even if there were no bodies there would be both a constant Place and a flowing Time; whence it is contended that Place and Time do not depend upon bodies and are not corporeal accidents. But neither on that account are they incorporeal accidents . . . they are, rather, certain incorporeal entities which differ in genus [kind] from what are usually termed substances and accidents. Whence it comes about that Being in its most general sense is not adequately divided into Substance and Accident, but Place and Time must be added as the two other members of this division. That is to say, every being is either a substance or an accident or a place in which all substances and accidents are or a time in and by which all substances and accidents perdure. This is a fact because there is no substance, there is no accident which is not somewhere or in some place or other, which does not exist sometimes or at some particular time, so much so that even if this or that substance or accident would perish, nonetheless Place would continue to be and Time to flow. Whence it is that place and time are to be regarded as true Things and as entirely real . . . Even though . . . substance and accident did not exist, nevertheless ~Place and Time] would exist; neither do they ri.e., Place and Time] depend upon the intellect as Chimeras do, for whether the Intellect thinks or not, place perdures and time flows along. (Quoted by Ariotti, 1973, 160)
Here we have a clear statement of an object-independent, mind-independent concept of space and time, a concept that was to become the foundation of the Newtonian system. G. Converging Themes.
As with space, we may distinguish four lines of argument leading from Aristotle's concept of time to that of Newton: Theological, Physical, Mathematical and Metaphysical.
Two major theological themes emerged in the Middle Ages as a response to Aristotle's theory of time. First, Aristotle's claim that time, and, hence, physical motion were infinite had to be reconciled with the religious conviction common to Islam, Judaism and Christianity that the world was created a finite number of years ago. Secondly, the sense in which God endures even when the world does not needed to be clarified. The neo-Platonic distinction between timeless eternity and time played a significant role in the reconciliation of relipion with Aristotle. Some thinkers identified God's duration with eternity and qualified it as imaginative or hypothetical time. This contributed to the breakdown of the sharp distinction between the two as envisaged by Plotinus, e.g., and lent credence to the anti-Aristotelian view that the essence of time was some kind of everlasting duration independent of the existence or nonexistence of correlated physical motions.
The condemnation of 1277, which rejected the doctrine that time is eternal certainly played some role in the subsequent development of concepts about time. Exactly what role it played, however, cannot be determined before further research into the period is done.
The major input from physical theory to the transformation of the concept of time was, undoubtedly, the Copernican revolution. The extended effect of the revolution was to shatter the fundamental structure of the Aristotelian cosmology and the centrality of the role of circular motions in accounting for the motions of the heavenly bodies. That the earth rotated on its axis and, the implication that the heavens could be infinite meant that no particular motion of the heavens could be singled out as primary or fundamental. Recall that Aristotle's case for the unity of time and its uniform rate rested on an appeal to the circular motions and their supposed uniform character. Without these motions to serve as primary any motions can be used to measure time. When these motions are not congruent, one is forced either to accept that there are many times or to accept that the essence of time was distinct from motion. We know of no pre-Newtonian figure who accepted the former alternative.
The mathematization of time effected by Galilei and developed by Barrow was a fundamental achievement. To a certain extent it must be admitted that Aristotle's concept of time is mathematical too. After all, he does see that time is the number of a motion. But, the Aristotelian perspective suffers from two grave defects, at least form the point of view of the furtherence of physical theory. First, there is no clear distinction in Aristotle between different kinds of motion, e.g., accelerated vs. non-accelerated motions. There is, in fact, no clear specification of what motion is, nothing that compares to the cleanness of the formula: velocity = distance/time (cf. Owen, in Salmon). This, in fact, is the second point. Neither Aristotle nor any other Greek developed the concept of a functional relationship between physical variables. Without such a concept it is hard to see that time, as a physical quantity, functions essentially as an independent variable which can be functionally related to other dependent variables such as velocity, distance covered, acceleration or the like. Galilei's fundamental contribution to the development of the theory of time was to establish the fruitfulness of this approach. In effect, time becomes a mathematical quantity which serves as the common denominator in terms of which other variables can be related to one another and accurate mathematical representations of physical phenomena become possible. As such, time is something which functions as a framework existing independently of the processes whose progress is measured in terms of it. It, like space, transcends the Aristotelian categorical option: substance or accident.
The basic metaphysical input into the oppostion to Aristotle's view of time is ultimately the neo-Platonic distinction between eternity and time. As it became transformed over the centuries it led to the development of an idea of duration which was independent of motion. It was left to Gassendi to articulate the culmination of this line of attack: namely, that time like space was neither an Aristotelian substance nor an Aristotelian accident but rather a new kind of Being. V. Conclusion
Thus ends our Cook's Tour through Medieval and Renaissance conceptions of space and time. Our purpose in this chapter has not been to provide a systematic explication or even a full survey of medieval themes, but only to suggest how some medieval lines of attack on the Aristotelian model of space and time paved the way for the development of the Newtonian model.
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