Posts Tagged ‘Galileo Galilei’

Burtt: Foundations of Modern Science

April 25, 2012

The Metaphysical Foundations of Modern Science
E.A. Burtt
(Dover, 2003) [1932]
352 p.

That man is the product of causes which had no prevision of the end they were achieving; that his origin, his growth, his hopes and fears, his loves and his beliefs, are but the outcome of accidental collocations of atoms; that no fire, no heroism, no intensity of thought and feeling, can preserve an individual life beyond the grave; that all the labours of the ages, all the devotion, all the inspiration, all the noonday brightness of human genius, are destined to extinction in the vast death of the solar system, and that the whole temple of Man’s achievement must inevitably be buried beneath the debris of a universe in ruins — all these things, if not quite beyond dispute, are yet so nearly certain, that no philosophy which rejects them can hope to stand. Only within the scaffolding of these truths, only on the firm foundation of unyielding despair, can the soul’s habitation henceforth be safely built.

— Bertrand Russell, Mysticism and Logic

I begin with this quotation because it gives us a vivid portrait of the predicament into which the metaphysics of modern science has led us. We have arrived at a picture of the world, and an understanding of our own place within it, which is, in a great many respects, hostile not only to the conception of human nature that reigned prior to the modern period, but, one is tempted to say, to even the most basic notion of man as a rational and moral creature. This situation, which I in certain moods can see only as an impasse, has come about in part because we have adopted a particular view of the natural world. It is the burden of E.A. Burtt’s classic book on the philosophy of science to outline this view, and to describe the historical circumstances in which it developed.

It developed out of something, and it is worth trying to sketch the basic contours of what preceded it. For late medieval man, nature was qualitative and inherently intelligible. Things has natures which were in principle knowable, and the whole natural order, though not itself intelligent, was nonetheless teeming with teleological relations. The texture of the world was thick: objects presented themselves to the understanding as unities, rich with colour and sound, and the beauty they conveyed to the mind was a modest but real intimation of a deeper, more permanent order. If man was considered to be, in some sense, above nature, this did not prevent his being at home in the world, for it was a world in which the human experience of will and desire, or the love of beauty, or the longing for knowledge was perfectly intelligible.

The birth of modern science did away with this view of things, perhaps with good intentions, sometimes with good reasons, and unquestionably with great success. Eventually it bequeathed us a world in which we appear as aliens, a world devoid of purposes, stripped of meaning, colourless and silent, comprised solely of bodies moving in space and time in a manner described by mathematical relations. We see the world as a massive machine, functioning according to fixed principles, best understood by examining its basic parts, and wholly governed by temporal (or, in Aristotelian terms, efficient) causation. Paradoxically, given the concomitant massive increase in our capacity to manipulate the natural world to serve our ends, the very framework whereby the world might be intelligible to us has been dismantled; we are reduced to speculation and inference based on neural signals produced by particles impinging on our sensory organs. The realm of qualities, purposes, and meaning, which can scarcely be entirely dispensed with, but which can find no place in the world so conceived, has been confined to scattered, and increasingly mysterious, things called ‘minds’. And now, with the turning of the wheel, the attempt is made to close the circle: to absorb even minds, hitherto the shelter for all those aspects of reality not compatible with the mechanistic, mathematical framework, into the framework itself. Our situation is, to say the very, very least, dramatic.

A thorough rehearsal of the historical development of the modern view would be a book-length project — indeed, it would be this very book — but I can sketch the main trajectory. Generally speaking, there are two important streams of thought to consider: the mathematical and the empirical. Both had roots in the medieval period. Though largely independent as they developed, they both informed the thought of Isaac Newton, who formulated an influential fusion of the two.

The revival of interest in Pythagorean thought was an important factor. Pythagoras had famously claimed that the world was made of “number”, and though the meaning of this claim was perhaps somewhat mysterious, it exerted a certain fascination. Late medieval astronomers showed a particular interest, and for intelligible reasons. It is easy to see, for example, how the sciences of astronomy and geometry, a physical science and a mathematical one, were considered closely related. In fact, Burtt argues that in the minds of at least some astronomers, astronomy just was geometry: astronomers studied the geometry of the heavens. To such men, it was natural, and even tempting, to believe that what was true in geometry was also true, in some sense, in the heavens. Thus when Copernicus proposed his heliocentric theory of the cosmos, the fact that it was mathematically simpler than the prevailing Ptolomeic model was interesting, and suggested to some, if not in Copernicus’ generation then certainly in the succeeding ones, that its mathematical simplicity was itself providing physical insight into the actual structure of the cosmos.

Johannes Kepler made a more radical claim: he argued that the mathematical order discernible in nature was itself the cause of the observed facts about the world. The real world was, in his mind, just the mathematical harmony discoverable in it. The strangeness of this idea ought to impress us: it was not that the world exhibited certain regularities such that aspects of it could be modelled using mathematical concepts exhibiting those same regularities — what we might call an instrumental use of mathematics — but rather that a mathematical description penetrated to the core of being, yielding a foundational understanding of the natural world. This essentialist view of mathematics was to prove very influential. An epistemological consequence followed: genuine knowledge of the world amounted to knowledge of its mathematical structure; mathematics provided not just a description of the natural world, but an explanation of it.

Kepler’s ideas influenced Galileo, who also believed that mathematical order implied necessity in nature. Galileo’s special contributions were, first, to explicitly abandon final causality as a principle of explanation in the physical sciences, and, second, to clarify the distinction, still hazy for Kepler, between the emerging concepts of primary and secondary qualities. The idea that final causality should be given up in favour of efficient causality had medieval precendent (in the thought of John Buridan, for instance), but until Galileo’s time it had not gained much traction. No doubt the waning influence of Aristotle was part of the reason why the time was ripe, and it is likely that the appeal of mathematical physics was another factor: it is more difficult (though not obviously impossible) for final causes to be given a mathematical formalism. To those seeking to construct a mathematical description of nature, therefore, and especially to those who believed that nature was intrinsically mathematical, final causes could have no appeal and provide no insight. The interesting question for these men was no longer ‘why’, but only ‘how’. The world so conceived was mechanical in substance: it consisted of bodies moving in space and time according to fixed mathematical relations. (Indeed, space and time now began to acquire status as fundamental metaphysical notions, which they certainly had not had in Aristotelian thought.) It is crucial to notice, in this context, that it was the method, inspired by a particular view of the natural world, that disposed with final causes, rather than, say, a particular discovery about the world.

The distinction between primary and secondary qualities was motivated — and, arguably, created — by the adoption of the mathematical concept of nature as well. Primary qualities are those features of an object that truly inhere in it, which cannot be separated from it. Secondary qualities, on the other hand, though we commonly ascribe them to objects, do not truly belong to them. For an Aristotelian, for instance, the redness of a red ball may be accidental, but it is still truly a property of the red ball that it is red, whereas for the early moderns like Galileo the ball only seems red, but it is not actually so; its redness is a secondary quality ascribed to the ball on the basis of certain peculiarities of the human senses; its redness exists only in the mind. The distinction between primary and secondary qualities arose for early modern scientists because they were committed to a mathematical view of nature, yet certain features of the natural world were not amenable to mathematical treatment. Those aspects of the world which could be treated mathematically — size, shape, position, motion, magnitude — were called “primary” and were considered real properties of objects, whereas those aspects which resisted mathematical treatment — colour, sound, smell, not to mention more intangible qualities like beauty or goodness — were called “secondary” and were relocated from objects to minds. Thus, on this view, objects in the external world possess only primary qualities, and second qualities are confined to mental life. Indeed, “man is hardly more than a bundle of secondary qualities”. Burtt comments on this state of affairs:

Observe that the stage is fully set for the Cartesian dualism on the one side the primary, the mathematical realm; on the other the realm of man. And the premium of importance and value as well as of independent existence all goes with the former. Man begins to appear for the first time in the history of thought as an irrelevant spectator and insignificant effect of the great mathematical system which is the substance of reality.

The mention of Descartes is natural enough at this juncture, but before continuing that line of clear and distinct thought it is worthwhile to pause a moment to reflect on the motives and the evidence for the mechanistic, mathematical view of the world. If Burtt is correct, this conception of the world is by no means a discovery of the sciences, but rather a methodological stipulation. What evidence is there for it? The question is more difficult to answer than one might expect. The incredible success that the sciences have enjoyed in describing a vast range of physical phenomena strongly suggests that there is something right about the general view, for under its guidance we seem to have gained real insight into the physical world. Moreover, we know that the atomic hypothesis is broadly correct: there really are particles moving around in space and time. But this is not really contested; the question is not whether this view is correct, so far as it goes, but whether it provides an exhaustive description. Is there nothing more to the world than these particles? The fact that the investigations of the sciences have never discovered anything which could not be fit into the mathematical framework, while sometimes cited as evidence for the truth of the framework, is nothing of the sort. Methodological limitations are being conflated with ontological ones. Is it, after all, a coincidence that the world as conceived by the mathematical physicist answers so perfectly to his needs?

Returning to Descartes, it is clear that his division of the world into res extensa and res cogitans was a natural development of the distinction between primary and secondary qualities: primary qualities belonged to the former and secondary qualities to the latter. Descartes, too, was convinced from an early age that mathematics was the key to genuine knowledge; his entire philosophical project was constructed on that assumption. Even more than some of the other early modern natural philosophers, Descartes was attracted by the idea that nature was not just mathematical, but geometrical. He resisted the idea that motion could be reduced to mathematical formulae only by attributing to bodies non-geometric qualities (such as mass); his famous vortex theory was a remarkable, though unsuccessful, attempt to produce a geometric theory of gravity. With Descartes the idea that nature is purely mathematical becomes tautological, for he defined the world external to the mind as consisting only of extended objects possessing primary qualities, with everything else pushed into the subjective realm of mind. In consequence, the mental realm was, for him, not a possible object of scientific study, for it consisted precisely of those qualities, attributes, and powers which eluded scientific methods.

Not everyone, however, was content with a sharp distinction between the physical and mental. Hobbes attacked Cartesian dualism, and made an attempt to subsume everything, including mind, into the res extensa. He was not successful, but his following has waxed greatly in the meantime. The question of whether that project can possibly succeed is an exceedingly interesting one that can, however, not deter us now. Instead, I simply note that, whether on the Cartesian or the Hobbesian side, many of the basic concepts were shared: efficient causality, mathematical description, bodies in motion, reductionism, and mechanism. The formulation of the metaphysics of modern science was substantially complete.

We have yet, however, to take account of the second principal stream of thought that informed the Newtonian synthesis: the empirical tradition. The principal figure here is Robert Boyle. Empiricists were, in general, less radical than their counterparts in the mathematical tradition. They resisted the push to reductionism, making productive use of concepts such as heat, weight, hardness, brittleness, etc. which could not obviously be ascribed to individual atoms. Boyle had moderate views: he valued qualitative descriptions, maintained the reality of secondary qualities, and was willing to entertain the existence of final causes. He also took a modest view of human knowledge, being suspicious of grand explanatory systems and thinking it often necessary to be satisfied with probable explanations rather than certainties. Paradoxically, it was he who began to point out certain skeptical consequences of the ideas propounded by those intent on obtaining genuine and certain (that is, mathematical) knowledge: if the picture of the world as conceived by Galileo and Descartes was correct, if the soul knows the world only through the effects of bodies impinging upon the senses, and if the world is not intrinsically ordered toward intelligibility, skeptical consequences follow. I will return to this point below. We should also note, however, that despite some differences, Boyle also accepted many of the new assumptions of natural philosophy. His view of man was largely Cartesian: “engines endowed with wills”.

In Isaac Newton these two traditions found a common advocate and were, to a large degree, integrated with one another. Newton’s basic method was, first, to work from observation and experiment to principles (in keeping with the empirical tradition), and then from principles to other phenomena (as in the mathematical tradition). Experiments were always involved at both the beginning and the end of an investigation, and the physical principles were always expressed mathematically. His synthesis has proved remarkably robust. Burtt notes, “Newton enjoys the remarkable distinction of having become an authority paralleled only by Aristotle to an age characterized through and through by rebellion against authority”. Though some of his scientific ideas have been superseded, his basic approach to scientific studies and the metaphysical system within which it was expressed remain dominant today.

Naturally, the emergence of the modern metaphysics of nature had an effect on theology. The relationship of God and the world has always been an important theological question, and it could not but be touched by a revolution in our views of nature. The repercussions within theological circles were sometimes comical — or would have been, had so much not been at stake. Henry More, for instance, gave this list of attributes: “one, simple, immobile, eternal, perfect, independent, existing by itself, subsisting through itself, incorruptible, necessary, immense, uncreated, uncircumscribed, incomprehensible, omnipresent, incorporeal, permeating and embracing all things, essential being, actual being, pure actuality” — as attributes of space! Space, he argued, was “divine presence”; even God, being real, was thought to be a res estensa! Malebranche too said something similar. Robert Boyle, as before, was more moderate in his views, but was nonetheless clearly under the influence of the mechanical worldview. He stressed, very wisely, that God was known naturally and normally through the world’s regularity, not through irregularities (that is, miracles); in his view, God maintained the “general concourse” of the universe as an harmonious whole. His view of God tended toward the Deist; he described God, using a phrase that was to have an unfortunate legacy, as the artificer of “a rare clock”. This general view he bequeathed also to Newton, who made a hash of it: he thought of God as providentially intervening in the world to “repair” it when necessary. For instance, he believed that God needed to intervene to keep the stars (which would tend to collapse together under the influence of universal gravitation) apart from one another. Burtt dryly notes that “to stake the present existence and activity of God on imperfections in the cosmic engine was to court rapid disaster for theology”. As time passed, under pressure from thinkers like Hume and Kant, the need for (and the knowability of) this God became more doubtful. The general story is familiar enough, but it is worth contrasting the God so conceived with the conception of God that was compatible with medieval metaphysics: in the medieval view, God had no purpose, but was the ultimate object of purpose, the final end of everything; natural processes were thus themselves examples of his providential action. In the modern view, he was demoted to custodial duties, his actions confined to the service of a greater end: the order and mathematical harmony of the universe.

God, however, has not been the only victim of skepticism in the light of modern metaphysics. I noted earlier the paradox that a view born principally of a desire for genuine and sure knowledge of the natural world should itself produce skepticism about that same knowledge, yet it is quite true. A universe consisting merely of atoms moving in space inclines one more or less strongly toward nominalism — that is, to the view that the world is not inherently intelligible, our concepts being merely conventions that do not correspond to real things. Moreover, the ascent of atheism itself intensified skepticism, for if the world is not underwritten by an intelligence, what reason have we to suppose it can be grasped by our intellects? “It was by no means an accident,” writes Burtt, “that Hume and Kant, the first pair who really banished God from metaphysical philosophy, likewise destroyed by a sceptical critique the current overweening faith in the metaphysical competence of reason. They perceived that the Newtonian world without God must be a world in which the reach and certainty of knowledge is decidedly and closely limited, if indeed the very existence of knowledge at all is possible.” And, in a kind of reductio ad absurdam of the mechanistic metaphysics, the effort to extend it into the mental realm results, as it apparently must according to the terms available, in the obliteration of specifically mental life itself and those things belonging to it, such as the very concept of knowledge. It is the ultimate apotheosis of skepticism. But that is a topic for another time.

At the end of this long analysis, I suppose the question hanging in the air is: if not this, then what? How should I know? I am as beholden to the modern assumptions as much as anyone — and, as a physicist, I am perhaps beholden more than most. Yet I can see the problems clearly enough, and I can see, too, that the positive arguments in favour of the currently dominant view are surprisingly weak. It seems likely to me that we are guilty of allowing our method to dictate our ontology, which is a clear fallacy.

Yet it is far from clear how best to respond to the situation. One possible step would be to reappraise the rejection of final causality. The sciences have in any case never been entirely consistent in rejecting them: biologists in particular find it hard to resist making teleological claims when they discuss their subject, and there may be resources within physics as well for a restoration of final causes (I am thinking of teleological interpretations of the action principle in both classical and quantum mechanics). It is sometimes thought that final causes, having to do with goal-directedness and purpose, require the existence of a presiding or immanent intelligence or will, which requirement seems to imply either personification of nature or theism, but actually this is not true; Aristotelian final causes imply neither. Second, we may reconsider our commitment to reductionism: even if it is true (as it is) that the world is comprised of particles in motion, is it really true that an understanding of the properties of those particles is, in principle, sufficient to understand everything else? Are the physical properties of ink molecules on a sheet of paper really enough to account for the meaning the written word conveys? It seems obvious that a bridge is out somewhere. A richer metaphysics could provide room, once again, for serious and honest engagement with non-mathematical aspects of reality. But I am a feeble philosopher, and such things are far beyond my competence.

In the meantime we are left with a view which, though having been wonderfully successful in certain respects, ultimately has no place in it for you and me: rational beings who think about things from a first-person perspective and act in the world out of our own freedom. As such, the battle is joined.

Grant: The Foundations of Modern Science in the Middle Ages

June 29, 2011

The Foundations of Modern Science in the Middle Ages
Their Religious, Institutional, and Intellectual Contexts
Edward Grant
(Cambridge, 1996)

261 p.

It is not easy to think of an aspect of medieval culture that enjoys popular acclaim today (unless we count the paternalistic medieval code of chivalry or the militaristic medieval ideal of knightly valour — do you see what I mean?), though it is a favourite pastime to tabulate all the many ways in which medieval society was thoroughly awful. Perhaps no aspect of medieval culture is more eagerly and easily derided than its science: impotent, absurd, hide-bound, airy, and wrong. It is a cunning strategy from our point of view, of course, for what better grounds on which to criticize others than those on which we stand most confidently? The bad reputation of medieval science dates from the seventeenth century, and is perhaps exemplified most famously in Galileo’s character (in his Dialogue) Simplicio, a dim-witted Aristotelian whose flounderings serve as a foil for the boldness and intelligence of the new scientists — and, in particular, of Galileo himself.

Fair enough, I suppose. The old order, which certainly merited criticism, was also open to caricature. Today, however, from a distance of several hundred years, we can look back at the dispute and try to discriminate the just criticisms from the unjust, and perhaps to trace certain threads of the fabric of the new science back into the Middle Ages, supplying the appreciation that was withheld or neglected at the time.

This is very much the objective of Edward Grant’s fine book, a study of the debt which modern science owes to medieval Europe. Grant is a leading scholar in medieval science, and has published numerous books on various aspects of the subject; this book feels like a condensation and summation of a lifetime of learning. Its conclusions, as is fitting, are balanced: there were important elements of the new science, he argues, that were genuinely new and which were not anticipated by medieval thinkers, but, at the same time, there were real and significant respects in which the medieval period made the birth of the new science possible.

To the extent that the new science was stimulated by encounters with Greek science, the Middle Ages deserves credit for having made the Greek texts available in a language that Europeans could understand. When knowledge of Greek was largely lost in the later Roman Empire, the scientific texts were preserved by Eastern Christians — whether Orthodox or Nestorian — and were then translated into Arabic in the aftermath of the Islamic conquests. Through the eleventh and twelfth centuries these texts found their way back to Europe, and from Arabic to Latin, through a dedicated and far-sighted translation effort, centred on the Iberian peninsula. They were eagerly taken up for study in the universities, as is well known.

The universities themselves deserve comment. We are sometimes inclined to take universities for granted, but it is well to consider what a rare and, in some respects, peculiar institution a university is. Apart from distant and singular Greek models like the Academy and Lyceum, there was really no precedent for the medieval university. Medieval scholars were self-consciously aware that the institution was not intended to serve the practical needs of society. There was a strong emphasis, which will come as no surprise to those familiar with medieval ideas about the liberal arts and the relative merits of the contemplative and active life, that the university was grounded in a love of learning and an appreciation for the intrinsic value of knowledge. The university enjoyed a wide liberty for free inquiry; interventions by civil and ecclesiastical authorities were remarkably rare. Its curriculum was structured around the trivium (grammar, rhetoric, logic) and the quadrivium (arithmetic, geometry, astronomy, music). It is significant, for our present considerations, that the quadrivium had a significant mathematical component, for we all know how important mathematics was, and is, to the natural sciences.

A third important factor that prepared the ground for science was the emergence, within the universities, of men well-trained in both natural philosophy and theology. Theology was the highest field of study, the ‘queen of the sciences’, and one could gain entry to a program of theological study only after having obtained a thorough grounding in more elementary subjects, including philosophy. The fact that theologians had, as a matter of course, also studied natural philosophy meant that there was no artificial bifurcation between the two fields of study, and certainly no motive for hostility; rather, the theologian-natural philosophers were able to relate the two fields of study to one another with relative ease. The interest they took in natural philosophy was often, naturally enough, from a theological vantage point — as a means to better understand and interpret Scripture, for instance — but that does not alter the essential point, which is that there was a group of elite scholars in Europe who understood and valued natural philosophy.

Those, then, are several historical and contextual factors that, Grant argues, created a climate in which interest in scientific questions could flourish. But were there any specific medieval intellectual contributions to the sciences themselves? Grant argues that medieval science was divisible into two parts: natural philosophy, concerned with the principles of nature at a fairly general level, and the exact sciences, such as optics or statics, in which specific scientific questions were addressed. The medieval contribution was principally to the former; scholars of the period mastered the ancient methods of the exact sciences, but did not add substantially to them.

In the light of that fact — which might reasonably be considered a failure — it is worthwhile to pause briefly to consider several of the most common criticisms of medieval science. One, of course, is that the medieval period did little to advance the exact sciences. Grant, as was just said, does not contest the charge, but argues that some allowance must be made for the difficult conditions under which medieval scholars laboured: it was a period in which, owing to the relatively small scholarly community and the mutable media available for recording and transmitting knowledge, ‘knowledge was as likely to vanish as to be preserved’. Consequently, ‘an enormous effort would have been required just to maintain the status quo’. We are, I think, sometimes too apt to forget that fact. Another common criticism of medieval science is that it was unfruitful because it was not experimental; this, again, is true to a large extent, but is the merit of an experimental approach so very evident at the outset? Grant argues that within the Aristotelian intellectual tradition which dominated the high medieval period experimental science actually seemed to be a superfluous, if not actually obstructive, endeavour. Aristotle’s physics implied that the nature of a thing would be manifest most clearly under natural conditions; to study something under ‘unnatural’, laboratory conditions, in which objects were manipulated, constrained, or otherwise tampered with, would therefore not have taught one about their true nature. Overcoming this objection required overcoming Aristotelian physics (which was, in some sense, the whole adventure of medieval science). Instead, medieval scholars preferred to argue deductively from principles. Finally, it is sometimes said that medieval scholars wasted their time with pseudo-sciences like alchemy, astrology, and divination; but this is false: these subjects were not part of the medieval curriculum. Interest in them belongs principally to the early modern period, contemporaneous with Bacon, Descartes, and Galileo.

Returning, then, to the question of whether medieval scholars made any significant concrete contributions to scientific progress, we should look to conceptual advances more than to experimental results. And we find, perhaps surprisingly, a quite wonderful litany of contributions that are so basic to us as to be almost invisible. Medieval natural philosophers argued over and clarified ideas about causality, necessity, contingency, and degrees of certitude. They studied, under Aristotle’s guidance, types of causes, and some (like John Buridan) actually anticipated the early modern philosophers by rejecting, for better or worse, final causes in nature. They developed conceptual frameworks for discussing infinities and infinitesimals, and mathematical treatments of qualities. They developed a language for describing kinematics, and proposed precise definitions for concepts such as uniform motion, uniform acceleration, and instantaneous velocity. They distinguished intensive and extensive qualities. They proposed principles of simplicity and economy of explanation. Crucially, they adopted the concept of ‘the common course of nature’, which granted to the natural world an integrity and consistency that made it intelligible and fit for scientific scrutiny, not immune from divine intervention but nonetheless having an orderly structure of its own.

This is not to claim that their ideas about these matters were all correct; in many cases they were not. It is to claim — and it seems to me a significant point — that many of the breakthroughs in early modern science did not occur because new questions were asked, but because new answers were given to old questions. The conversation was in many respects already happening; the questions were thought worthy of study; the shelves were stacked with proposals and counter-proposals.

Among the more interesting points Grant makes about the conceptual developments in medieval natural philosophy concerns the impact of the famous, or infamous, Condemnation of 1277. This was one of the infrequent ecclesiastical interventions into the intellectual life of the university: the Bishop of Paris issued a condemnation of some 219 philosophical and theological propositions. In his book on scholasticism, Josef Pieper reflected ruefully on the chilling influence these condemnations had on the dialogue between theology and philosophy, and between theological authority and philosophical inquiry. Grant has a more positive appraisal because of the unforeseen positive impact the condemnations had on natural philosophy. The burden of several of the condemnations, in particular, had been to insist that God’s power can be limited by nothing save logical contradiction. This was an idea fraught with peril for natural philosophy, for it might have had the effect of dissolving the order of nature into radically contingent and potentially capricious ‘happenings’, its character dependent from moment to moment on the changeability of God’s omnipotent will. (Something very like this seems to have afflicted Islamic natural philosophy, and also eventually certain Christian thinkers like William of Ockham.)

Instead, however, the doctrine of God’s omnipotence had a milder and more fruitful effect: it began to loosen the stranglehold that Aristotelian physics had on the medieval imagination. The Aristotelian view of nature, however correct it was as a description of our world, was not the only way God might have structured the world he created. This thought inspired medieval natural philosophers to speculate about possible worlds that might differ in one respect or another from ours, worlds in which one or another of the principles of Aristotelian physics did not apply. They found that certain of these ‘natural impossibilities’ were logically defensible — that vacua might exist, for instance, either within or beyond our cosmos, or that other worlds might exist. The arguments Aristotle had offered in defence of his positions were therefore subjected to critique and found wanting in certain respects. This process was immensely important for the development of the sciences, for modern science could not have emerged until people took seriously the idea that Aristotelian science might be wrong.

In closing, I would like to examine a few specific technical developments of medieval science that seem particularly closely related to developments usually associated with early modern science. In particular I will briefly examine some medieval arguments about the earth’s axial rotation, about motion and kinematics, and about the concepts of inertia and momentum.

Certain medieval natural philosophers entertained the thought that the earth might rotate axially once each day. It seemed a more elegant and economical way of explaining the observed daily rotation of the celestial sphere. The principal objection to the idea, of course, is that we seem to be stationary; natural philosophers considered whether that was a sound objection. John Buridan posed the problem as one of relative motion, and he argued that if the earth was rotating an arrow shot directly upward would fall to the ground in a different spot, since the ground would have rotated some distance while it was aloft. (In other words, he did not have the concept of inertia.) Nicole Oresme, however, who was one of the greatest natural philosophers of the later Middle Ages, pointed out that if the earth was rotating then evidently the atmosphere was also rotating with it (else we would always feel a wind from the same direction), and so the arrow aloft would be carried by the air and fall to earth exactly where it was launched. This is not quite a correct explanation, but it is probably about as good as one can do without the concept of inertia. Oresme, in fact, went systematically through all of the objections to the earth’s axial rotation and found them all wanting; he therefore concluded that there was no good reason why the earth should not rotate. He had, however, no positive case to make for its actual rotation. It is interesting to note that several of his arguments reappear in the writings of Copernicus.

I have already mentioned above that medieval natural philosophers made several important conceptual contributions to kinematics. They were motivated to do so by a quite general interest in the augmentation and diminution of qualities — the increase of grace in the soul, for instance, or the reddening of leaves in the autumn, or the acceleration of a moving body. A mathematics to describe this variation in quantifiable qualities was developed, and the concepts of uniform motion and uniformly accelerated motion were articulated, principally by a group of men at Oxford’s Merton College. (They are collectively called ‘the Oxford calculators’.) Perhaps their finest achievement was a derivation of the mean speed theorem; they gave the theorem verbally, not algebraically. Nicole Oresme later gave a geometric proof that was in all essentials identical to the geometric proof given by Galileo, for whom the mean speed theorem was foundational to the new science of motion. It is possible that Galileo learned the theorem from medieval treatises, which circulated widely in Italy, but this has not actually been demonstrated.

Some interesting modifications of Aristotle were made on the topic of the dynamics of motion. Aristotle had argued that the velocity of an object was proportional to its weight and inversely proportional to the resistance to its motion (v ~ W ~ 1/R); this led, however, to infinite velocities when the resistance was zero, which was one of the reasons Aristotle offered for the impossibility of a vacuum. Thomas Bradwardine, at Oxford, suggested instead that velocity was proportional to the applied force and inversely proportional to a combination of weight and resistance (v ~ F/(W + R)). This was wrong — it is acceleration that is proportional to force — but it was interesting because it behaved nicely in a vacuum (R = 0). This allowed Bradwardine to think about the possibility of motion in a vacuum, and to propose the idea that a medium was a retarding factor imposed upon a more basic, if hypothetical, vacuum case. This was a powerful idea that was to be of central importance to Galileo and Newton. Bradwardine also argued for a somewhat different set of ideas, in which motion did not depend on weight or size, but on an intensive quality called ‘internal resistance’. This is rather similar to Galileo’s use of ‘specific weight’ in the same context in an early manuscript (De motu, c.1590), which was later supplanted by the more universal (and correct) claim that motion is independent of the constitution of an object (Two New Sciences, 1638). Again, despite the suggestive parallels, no connection between Bradwardine and Galileo has been demonstrated on this matter.

Finally, we can look briefly at early ideas related to inertia and momentum. John Buridan argued that the motion of an object which results from an impressed force was determined by the object’s ‘impetus’, which he defined as a combination of its speed and its ‘quantity of matter’. I do not know what he meant by ‘quantity of matter’, but if he meant something like ‘mass’ then his impetus would be what we call momentum. In modern physics the impressed force is equal to the time rate of change of momentum, so Buridan was not too far off. He conceived of impetus as something that would be preserved unless diminished by an outside force, which is again suggestive of the modern concept of conservation of momentum. For Buridan, however, impetus was a cause of motion, rather than simply a quantity of motion, and in this he differed sharply from the modern view. Also, despite his idea that impetus would be preserved in the absence of outside forces, he did not conceive of rest and uniform motion as comparable states, nor of the possibility of infinite uniform motion — which was, in any case, an impossibility in an Aristotelian cosmos.

In each of these examples we see something that I find quite fascinating: challenges to the Aristotelian framework and the proposal of creative ideas that bore a certain family resemblance to the ideas that became basic to physics during the sixteenth and seventeenth centuries. This supports, I think, the claim made earlier that the birth of modern science was in important respects a consequence of new answers being given to old questions. The brilliance and power of those new answers is beyond dispute, but the probity and intelligence of the questions ought also to have our respect. Grant makes a strong case for the claim that modern science could not have developed without the preparatory work — cultural, institutional, literary, and conceptual — of medieval scholars, and, as such, it seems long past time for their contributions to be acknowledged.