An intensive study of the classic English thinkers early taught me that no one could hope to appreciate the motives underlying their work till he had mastered the philosophy of the one Englishman whose authority and influence in modern times has rivalled that of Aristotle over the late medieval epoch—Sir Isaac Newton.
How curious, after all, is the way in which we moderns think about our world ! And it is all so novel, too. The cosmology underlying our mental processes is but three centuries old—a mere infant in the history of thought—and yet we cling to it with the same embarrassed zeal with which a young father fondles his new-born baby.
Philosophers never succeed in getting quite outside the ideas of their time so as to look at them objectively-this would, indeed, be too much to expect. ...But philosophers do succeed in glimpsing some of the problems involved in the metaphysical notions of their day and take harmless pleasure in speculating at them in more or less futile fashion.
The central place of epistemology in modern philosophy is no accident... Knowledge was not a problem for the ruling philosophy of the Middle Ages ; that the whole world which man's mind seeks to understand is intelligible to it was explicitly taken for granted. That people subsequently came to consider knowledge a problem implies that they had been led to accept certain different beliefs about the nature of man and about the things which he tries to understand.
It may very well be that the truly constructive ideas of modern philosophy are not cosmological ideas at all, but such ethico-social concepts as 'progress,' control,' and the like.
In the last analysis it is the ultimate picture which an age forms of the nature of its world that is its most fundamental possession. It is the final controlling factor in all thinking whatever.
All of us tend easily to be caught in the point of view of our age and to accept unquestioningly its main presuppositions.
For the dominant trend in medieval thought, man occupied a more significant and determinative place in the universe than the realm of physical nature, while for the main current of modern thought, nature holds a more independent, more determinative, and more permanent place than man.
For the Middle Ages man was in every sense the centre of the universe. The whole world of nature was believed to be teleologically subordinate to him and his eternal destiny. Toward this conviction the two great movements which had become united in the medieval synthesis, Greek philosophy and Judeo-Christian theology, had irresistibly led.
This view underlay medieval physics. The entire world of nature was held not only to exist for man's sake, but to be likewise immediately present and fully intelligible to his mind. Hence the categories in terms of which it was interpreted were not those of time, space, mass, energy, and the like ; but substance, essence, matter, form, quality, quantity... the facts and relations observed in man's unaided sense-experience of the world and the main uses which he made it serve.
Man was believed to be active in his acquisition of knowledge—nature passive. When he observed a distant object, something proceeded from his eye to that object rather than from the object to his eye. And, of course, that which was real about objects was that which could be immediately perceived about them by human senses.
What more natural than to hold that these regular, shining lights were made to circle round man's dwelling-place, existed in short for his enjoyment, instruction, and use? The whole universe was a small, finite place, and it was man's place. He occupied the centre ; his good was the controlling end of the natural creation.
The world of nature existed that it might be known and enjoyed by man. Man in turn existed that he might "know God and enjoy him forever."
The whole natural world in its present form was but a moment in a great divine drama...
What a contrast between the audacious philosophy of Dante reposeful, contemplative, infinitely confident and this view! To Russell, man is but the chance and temporary product of a blind and purposeless nature, an irrelevant spectator of her doings, almost an alien intruder on her domain.
Just as it was thoroughly natural for medieval thinkers to view nature as subservient to mans knowledge, purpose, and destiny ; so now it has become natural to view her as existing and operating in her own self-contained independence, and so far as mans ultimate relation to her is clear at all, to consider his knowledge and purpose somehow produced by her, and his destiny wholly dependent on her.
It is difficult for the modern mind, accustomed to think so largely in terms of space and time, to realize how unimportant these entities were for scholastic science. Spatial and temporal relations were accidental, not essential characteristics. Instead of spatial connexions of things, men were seeking their logical connexions ; instead of the onward march of time, men thought of the eternal passage of potentiality into actuality.
Professor Cassirer himself has done work on modern epistemology which will long remain a monumental achievement in its field. But a much more radical historical analysis needs to be made.
We must grasp the essential contrast between the whole modern world-view and that of previous thought, and use that clearly conceived contrast as a guiding clue to pick out for criticism and evaluation, in the light of their historical development, every one of our significant modern presuppositions.
The whole magnificent movement of modern science is essentially of a piece ; the later biological and sociological branches took over their basic postulates from the earlier victorious mechanics, especially the all-important postulate that valid explanations must always be in terms of small, elementary units in regularly changing relations. To this has likewise been added, in all but the rarest cases, the postulate that ultimate causality is to be found in the motion of the physical atoms.
A student of the history of physical science will assign to Newton a further importance which the average man can hardly appreciate. ...the separation ...of positive scientific inquiries from questions of ultimate causation.
The history of mathematics and mechanics for a hundred years subsequent to Newton appears primarily as a period devoted to the assimilation of his work and the application of his laws to more varied types of phenomena. So far as objects were masses, moving in space and time under the impress of forces as he had defined them, their behaviour was now, as a result of his labours, fully explicable in terms of exact mathematics.
Newton... not only found a precise mathematical use for concepts like force, mass, inertia ; he gave new meanings to the old terms space, time, and motion, which had hitherto been unimportant but were now becoming the fundamental categories of men's thinking.
Imbedded directly and prominently in the Principia... these metaphysical notions were carried wherever his [Newton's] scientific influence penetrated, and borrowed a possibly unjustified certainty from the clear demonstrability of the gravitational theorems to which they are appended as Scholia.
The fact that his [Newton's] treatment of these great themes... was covered over by this cloak of positivism, may have become itself a danger.
The old set of categories, involving, as it appeared, the now discredited medieval physics, was no longer an alternative to any competent thinker. In these circumstances it is easy to understand how modern philosophy might have been led into certain puzzles which were due to the unchallenged presence of these new categories and presuppositions.
Post-Newtonian philosophers... were philosophizing quite definitely in the light of his [Newton's] achievements, and with his metaphysics especially in mind. ...Of course, these men do not accept Newton as gospel truth they all criticize some of his conceptions, especially force and space—but none of them subjects the whole system of categories which had come to its clearest expression in the great Principia to a critical analysis.
There were no known celestial phenomena which were not accounted for by the Ptolemaic method with as great accuracy as could be expected without more modern instruments. Predictions of astronomical events were made which varied no more from the actual occurrence than did predictions made by a Copernican.
The motions of the heavenly bodies could be charted according to Ptolemy just as correctly as according to Copernicus.
To the senses it must have appeared incontestable that the earth was a solid, immovable substance, while the light ether and the bits of starry flame at its not too distant limit floated easily around it day by day.
The earth is to the senses the massive, stable thing; the heavens are by comparison, as revealed in every passing breeze and every flickering fire, the tenuous, the unresisting, the mobile thing.
There had been built up on the basis of... supposedly unshakeable testimony of the senses a natural philosophy of the universe which furnished a fairly complete and satisfactory background for man's thinking.
The assertion that a body projected vertically in the air must fall considerably to the west of its starting-point if the Copernican theory be correct, had to wait for its refutation till Galileo laid the foundations of modern dynamics.
The objection that according to Copernicus the fixed stars ought to reveal an annual parallax, due to the 186,000,000-mile difference in the position of the earth every six months, were not answered till Bessel's discovery of such a parallax in 1838.
In Copernicus' day the non-appearance to the senses of any stellar parallax implied, if his theory be sound, the necessity of attributing to the fixed stars a distance so immense that it would have been dismissed by all but a few as ridiculously incredible.
Even had there been no religious scruples whatever against the Copernican astronomy, sensible men all over Europe, especially the most empirically minded, would have pronounced it a wild appeal to accept the premature fruits of an uncontrolled imagination, in preference to the solid inductions, built up gradually through the ages, of men's confirmed sense experience.
Contemporary empiricists, had they lived in the sixteenth century, would have been first to scoff out of court the new philosophy of the universe.
Copernicus... could plead only that his conception threw the facts of astronomy into a simpler and more harmonious mathematical order.
That such a tremendous shift in the point of reference could be legitimate was a suggestion quite beyond the grasp of people trained for centuries to think in terms of a homocentric philosophy and a geocentric physics.
The unprecedented religious upheaval of the times had contributed powerfully to loosen men's thinking. Rome had been taken for granted as the religious centre of the world for well over a thousand years ; now there appeared a number of distinct centres of religious life besides Rome.
There was a renouncement... of man's former centres of interest and a fixation on something new.
In this ferment of strange and radical ideas, widely disseminated by the recent invention of printing, it was not so difficult for Copernicus to consider seriously for himself and suggest persuasively to others that a still greater shift than any of these must now be made, a shift of the centre of reference in astronomy from the earth to the sun.
London and Paris had become like Rome; in the absence of evidence to the contrary it is to be conceived that the distant celestial bodies are like the earth.
It is a commonplace to mathematicians, that save for the last two centuries, during which higher algebra has to a considerable extent freed men's minds from dependence on spatial representations in their mathematical thinking, geometry has always been the mathematical science par excellence.
As Kepler remarks, the certainty possible in exact mathematical reasoning is allied at every step with visible extended images, hence many who are quite incompetent in abstract thinking readily master the geometrical method.
In ancient times... arithmetic developed in close dependence on geometry.
Whenever Plato (as in the Meno) turns to mathematics for an illustration of some pet contention... the proposition used is always one that can be presented geometrically.
The famous Pythagorean doctrine that the world is made of numbers is apt to appear quite unintelligible to moderns till it is recognized that what they meant was geometrical units, i.e., the sort of geometrical atomism that was taken over later by Plato in his Timaus. They meant that the ultimate elements of the cosmos were limited portions of space.
The Nova Scienza of Tartaglia, published in 1537, applies this method [of applying geometry] to certain problems of falling bodies and the maximum range of a projectile, while Stevinus (1548-1620) uses a definite scheme for the representation of forces, motions, and times by geometrical lines.
When in the fifteenth and sixteenth centuries a more extended use began to be made of algebraic symbols, mathematicians were able only very gradually to detach their thinking from continued dependence on geometrical representation.
The popular objects of mathematical inquiry in these [15th and 16th] centuries dealt primarily with the theory of equations, and in particular with methods for the reduction and solution of quadratic and cubic equations.
Pacioli... (died about 1510), was mainly interested in using the growing algebraic knowledge to investigate the properties of geometrical figures.
Throughout the ancient and medieval period to the time of Galileo, astronomy was considered a branch of mathematics, i.e., of geometry. It was the geometry of the heavens.
Our current conception of mathematics as an ideal science, of geometry in particular as dealing with an ideal space, rather than the actual space in which the universe is set, was a notion quite unformulated before Hobbes, and not taken seriously till the middle of the eighteenth century, though it was dimly felt after by a few Aristotelian opponents of Copernicus.
The space of geometry appears to have been the space of the real universe to all ancient and medieval thinkers who give any clear clue to their notion of the matter. In the case of the Pythagoreans and Platonists the identity of the two was an important metaphysical doctrine; in the case of other schools the same assumption seems to have been made, only its bearings were not thought out along cosmological lines.
When some, like Aristotle, defined space in a quite different manner, it is noticeable that the definition is still such that the needs of geometers are fully met.
Copernicus was the first astronomer to carry it [mathematical relativity of spatial system's reference] through in earnest, with full appreciation of its revolutionary implications.
The particular event which led Copernicus to consider a new point of reference in astronomy was his discovery that the ancients had disagreed about the matter. Ptolemy's system had not been the only theory advanced.
...for the sixty years that elapsed before Copernicus' theory was confirmed in more empirical fashion, practically all those who ventured to stand with him were accomplished mathematicians, whose thinking was thoroughly in line with the mathematical advances of the day.
To follow Ptolemy in ancient times meant merely to reject the cumbrous crystalline spheres. To follow Copernicus was a far more radical step, it meant to reject the whole prevailing conception of the universe.
For many minds of the age at least, there was an alternative background besides Aristotelianism, in terms of which their metaphysical thinking might go on, and which was more favourable to this astonishing mathematical movement.
The interest in mathematics evidenced by such freethinkers as Roger Bacon, Leonardo, Nicholas of Cusa, Bruno, and others, together with their insistence on its importance, was in large part supported by the existence and pervading influence of this Pythagorean stream.
This somewhat submerged but still pervasive Platonism... regarded a universal mathematics of nature as legitimate (though, to be sure, just how this was to be applied was not yet solved); the universe is fundamentally geometrical; its ultimate constituents are nothing but limited portions of space; as a whole it presents a simple, beautiful, geometrical harmony.
Copernicus could take the step because... he had definitely placed himself in this dissenting Platonic movement. ...It was no accident that he became familiar with the remains of the early Pythagoreans, who almost alone among the ancients had ventured to suggest a non-geocentric astronomy.
Now during the half-century after Copernicus, no one was bold enough to champion his theory save a few eminent mathematicians like Rheticus and a few incorrigible intellectual radicals like Bruno. In the late eighties and early nineties, however, certain corollaries of Copernicus' work were seized upon by the youthful Kepler, then in his student days...
Copernicus had himself noted the greater importance and dignity which seemed to be attributed to the sun in the new world-scheme, and had been eager to find mystical as well as scientific justification for it.
Copernicus had formed a rudimentary conception of scientific hypothesis, accommodated to his new astronomical method. A true hypothesis is one which binds together rationally (i.e., for him mathematically) things which had before been held distinct; it reveals the reason, in terms of that which unites them, why they are as they are.
With him [ Kepler ] nature's simplicity and unity was a commonplace. "Natura simplicitatem amat [Nature loves simplicity]." "Amat ilia unitatem [She loves unity]." "Numquam in ipsa quicquam otiosum aut superfluum exstitit [Never existing in actual rest or superfluity]." "Natura semper quod potest per faciliora, non agit per ambages difficiles [Nature always travels the more facile, not via the difficult circuitous routes]."
Kepler's profound attainments in the science of mathematics could not but lead him to feel with full force all those considerations which had influenced the mind of his predecessor [Copernicus].
His teacher of mathematics and astronomy at Tubingen, Mastlin, who had been strongly attracted by the greater order and harmony attainable in the Copernican scheme, was an adherent of the new astronomy at heart, though he had so far expressed himself only with the greatest caution.
The Neo-Platonic background, which furnished the metaphysical justification for much of this mathematical development (at least as regards its bearing on astronomy) awoke Kepler's full conviction and sympathy. Especially did the aesthetic satisfactions gained by this conception of the universe as a simple, mathematical harmony, appeal vigorously to his artistic nature.
Founder of exact modern science though he was, Kepler combined with his exact methods and indeed found his motivation for them in certain long discredited superstitions, including what it is not unfair to describe as sunworship.
The connexion between Kepler, the sun-worshipper, and Kepler, the seeker of exact mathematical knowledge of astronomical nature, is very close.
It was very fortunate for Kepler that he was just plunging into such profound labours at the time when Tycho Brahe, the greatest giant of observational astronomy since Hipparchus, was completing his life-work of compiling a vastly more extensive and incomparably more precise set of data than had been in the possession of any of his predecessors.
Kepler joined with his speculative superstitions an eagerness to find precise formulae confirmed in the data; it was the observed world about which he was philosophizing, hence "without proper experiments I conclude nothing," hence also his refusal to neglect variations between his deductions and the observations which would not have troubled the ancients.
Kepler's thinking was genuinely empirical in the modern sense of the term.
These [ Kepler's three laws ] were not especially important to Kepler's own mind, being only three out of scores of interesting mathematical relations which, as he pointed out, were established between the observed motions if the Copernican hypothesis be true.
A true hypothesis for Kepler must be a statement of the underlying mathematical harmony discoverable in the effects.
Such a mathematico-aesthetic conception of causality and hypothesis already implies a new metaphysical picture of the world... Such hypotheses as these, Kepler maintained, are precisely what give us the true picture of the real world, and the world thus revealed is a bigger and far more beautiful realm than man's reason had ever before entered.
For Kepler... the real qualities are those caught up in this mathematical harmony underlying the world of the senses, and which, therefore, have a causal relation to the latter. The real world is a world of quantitative characteristics only; its differences are differences of number alone.
Kepler's position led to an important doctrine of knowledge. Not only is it true that we can discover mathematical relations in all objects presented to the senses; all certain knowledge must be knowledge of their quantitative characteristics, perfect knowledge is always mathematical.
We have in Kepler the position clearly stated that the real world is the mathematical harmony discoverable in things. The changeable, surface qualities which do not fit into this underlying harmony are on a lower level of reality; they do not so truly exist. All this is thoroughly Pythagorean and Neo-Platonic in cast, it is the realm of the Platonic ideas suddenly found identical with the realm of geometrical relationships.
Kepler... quotes with approval the famous saying of Plato, that God ever geometrizes; he created the world in accordance with numerical harmonies, and that is why he made the human mind such that it can only know by quantity.
The task involved revising ... the traditional ideas of causality, hypothesis, reality, and knowledge; hence Kepler offers us the fundamentals of a metaphysic based in outline upon the early Pythagorean speculations, but carefully accommodated to the new ideal and method.
The acquisition of further empirical facts in astronomy by Galileo and his successors showed that the astronomical and physical universe was enough like what Copernicus and Kepler had dared to believe, for them to become established as fathers of the outstanding movement of human thought in modern times, instead of being consigned to oblivion as a pair of wild-minded apriorists.
His [Kepler's] outlook and method were as fully dominated by an aesthetic as by a purely theoretic interest, and the whole of his work was overlaid and confused by crude inherited superstitions which the most enlightened people of his time had already discarded.
At the youthful age of twenty-five he was appointed professor of mathematics at the University of Pisa, largely because of the fame won by some papers on the hydrostatic balance, the properties of the cycloid, and the centre of gravity in solids.
The famous event in the Cathedral of Pisa, when he observed that the swings of the great hanging lamp were apparently isochronous, had just preceded, and in part inspired, his [Galileo's] first interest in mathematics, hence the mathematical study of mechanical motions became quite naturally the focus of his work.
Others before him [Galileo] had asked why heavy bodies fall; now, the homogeneity of the earth with the heavenly bodies having suggested that terrestrial motion is a proper subject for exact mathematical study, we have the further question raised: how do they fall? with the expectation that the answer will be given in mathematical terms.
It was this reduction of terrestrial motions to terms of exact mathematics which, fully as much as the significant astronomical discoveries that empirically confirmed Copernicanism, measured his [Galileo's] import to those of his contemporaries who were fitted to appreciate this stupendous advance in human knowledge.
Galileo's practical mechanical inventions are themselves sufficiently remarkable. In his early years he invented a pulsimeter, operating by means of a small pendulum, and also a contrivance for measuring time by the uniform flow of water. Later he became the inventor of the first crude thermometer, and in the last year of his life sketched out complete plans for a pendulum clock. His achievements in the early development of the telescope are known to all students.
Our expectation that the reduction of the motions of bodies to exact mathematics must carry large metaphysical bearings to Galileo's mind will not be disappointed.
Nature presents herself to Galileo, even more than to Kepler, as a simple, orderly system, whose every proceeding is thoroughly regular and inexorably necessary. " Nature . . . doth not that by many things, which may be done by few."
Philosophy is written in that great book which ever lies before our eyes—I mean the universe—but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles, and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.
Galileo is continually astonished at the marvellous manner in which natural happenings follow the principles of geometry, and his favorite answer to the objection that mathematical demonstrations are abstract and possess no necessary applicability to the physical world, is to proceed to further geometrical demonstrations, in the hope that they will become their own proof to all unprejudiced minds.
Galileo could hardly have become the doughty figure in the overthrow of Aristotelianism that he appeared to his contemporaries had it not been for his popularly verifiable discoveries, which showed clearly to men's senses that some of Aristotle's statements were false.
As with Kepler, so with Galileo, this mathematical explanation of nature must be in exact terms; it is no vague Pythagorean mysticism that the founder of dynamics has in mind.
Viewed as a whole, Galileo's method then can be analysed into three steps, intuition or resolution, demonstration, and experiment. ...That Galileo actually followed these three steps in all of his important discoveries in dynamics is easily ascertainable from his frank biographical paragraphs, especially in the Dialogues Concerning Two New Sciences
He [Galileo] quoted by way of orthodox support Tertullian's dictum that we know God first by nature, then by revelation.
Of the utmost moment was Galileo's further assertion that these secondary qualities are subiective. ...Galileo fell definitely in line with the Platonic identification of the realm of changing opinion with the realm of sense experience, and became the heir to all the influences emanating from the ancient atomists which had been recently revived in the epistemology of such thinkers as Vives and Campanella.