*The Metaphysical Foundations of Modern Physical Science*

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### Chapter 1 Introduction (A)

### Chapter 1 Introduction (B)

### Chapter 2 Copernicus and Kepler (A)

### Chapter 2 Copernicus and Kepler (B)

* The Metaphysical Foundations of Modern Physical Science; A Historical and Critical Essay* (1924) was written by the American philosopher Edwin Arthur Burtt as his doctoral thesis. This work has had a significant influence upon the history and philosophy of science, as discussed by Floris Cohen in his

*The Scientific Revolution: A Historiographical Inquiry*and by Diane Davis Villemaire in her

*E.A. Burtt, Historian and Philosopher: A Study of the Author of The Metaphysical Foundations of Modern Physical Science*.

## Quotes[edit]

### Preface (1925)[edit]

- 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.
- Preface

### Chapter 1 Introduction (A) *Historical Problem Suggested by the Nature of Modern Thought*[edit]

- 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 visible universe itself was infinitely smaller than the realm of man. The medieval thinker never forgot that his philosophy was a religious philosophy**, with a firm persuasion of man's immortal destiny. The Unmoved Mover of Aristotle and the personal Father of the Christian had become one. There was an eternal Reason and Love, at once Creator and End of the whole cosmic scheme, with whom man as a reasoning and loving being was essentially akin. ...the religious experience to the medieval philosopher was the crowning scientific fact. Reason had become married to mystic inwardness and entrancement...

- 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.

### Chapter 1 Introduction (B) *The Metaphysical Foundations of Modern Science the Key to this Problem*[edit]

**Berkeley, Hume, Kant, Fichte, Hegel, James, Bergson all are united in one earnest attempt, the attempt to reinstate man with his high spiritual claims in a place of importance in the cosmic scheme.**The constant renewal of these attempts and their constant failure widely and thoroughly to convince men, reveals how powerful a grip the view they were attacking was winning over people's minds, and now, perhaps even more than in any previous generation, we find philosophers who are eager above all things to be intellectually honest, ready to give up the struggle as settled and surrender the field.

- That the scientific philosophy of the Greeks, with all its sublime passion for the very truth of things, arrived in its turn at an exalted philosophy of man, might be due to the circumstances insisted upon by some historians of thought, that
**the zenith of Greek metaphysics was attained quite consciously through the extension, to the physical realm, of concepts and methods already found helpful in dealing with personal and social situations.**...the misapplication being based in the last analysis on the unwarranted assumption that because man, while here, can know and use portions of his world, some ultimate and permanent difference is thereby made in that world.

**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.**

**The big puzzles of modern philosophers are all concerned with space and time.**Hume wonders how it is possible to know the future, Kant resolves by a*coup de force*the antinomies of space and time, Hegel invents a new logic in order to make the adventures of being a developing romance, James proclaims an empiricism of the 'flux,' Bergson bids us intuitively plunge into that stream of duration which is itself the essence of reality, and Alexander writes a metaphysical treatise on space, time, and deity.

- It might be that the reason for the failure of philosophy to assure man something more of that place in the universe which he once so confidently assumed is due to an inability to rethink a correct philosophy of man in the medium of this altered terminology.
**It might be that... modern philosophy had accepted uncritically certain important presuppositions, either in the form of meanings carried by these new terms or in the form of doctrines about man and his knowledge subtly insinuated with them**...

**During the last generation these ideas of science have been subjected to vigorous analysis and criticism by a group of keen thinkers**, who have asked themselves what modifications in the traditional conceptions would be demanded if we sought to overhaul them in the light of a broader and more consistently interpreted experience. At present**this critical investigation has culminated in a rather extensive transformation of the major concepts of scientific thinking**, furthered on the one hand by radical physical hypotheses of a gifted student of nature like Einstein, and on the other by the attempted reshaping of scientific methods and points of view by philosophers of science such as Whitehead, Broad, Cassirer.

- Just how did it come about that men began to think about the universe in terms of atoms of matter in space and time instead of the scholastic categories? Just when did teleological explanations, accounts in terms of use and the Good, become definitely abandoned in favour of the notion that true explanations, of man and his mind as well as of other things, must be in terms of their simplest parts? What was happening between the years 1500 and 1700 to accomplish this revolution? And then, what ultimate metaphysical implications were carried over into general philosophy in the course of the transformation? Who stated these implications in the form which gave them currency and conviction? How did they lead men to undertake such inquiries as that of modern epistemology? What effects did they have upon the intelligent modern man's ideas about his world?

- 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.

**As for pre-Newtonian science**, it is one and the same movement with pre-Newtonian philosophy, both in England and on the continent;**science was simply natural philosophy, and the influential figures of the period were both the greatest philosophers and the greatest scientists.**It is largely due to Newton himself that a real distinction came to be made between the two; philosophy came to take science, in the main, for granted, and another way to put our central theme is,*did not the problems to which philosophers now devoted themselves arise directly out of that uncritical acceptance?*

**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.**

**In his treatment of such ultimate concepts**, together with his doctrine of primary and secondary qualities, his notion of the nature of the physical universe and of its relation to human knowledge (in all of which he carried to a more influential position a movement already well advanced) in a word, in his decisive portrayal of the ultimate postulates of the new science and its successful method as they appeared to him,**Newton was constituting himself a philosopher rather than a scientist as we now distinguish them.**

**Imbedded directly and prominently in the**[Newton's]*Principia*... these metaphysical notions were carried wherever his**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.

- At the time of his death Leibniz was engaged in a heated debate on the nature of time and space with Newton's theological champion, Samuel Clarke. Berkeley's
*Commonplace Book*and*Principles*, still more his lesser works such as*The Analyst*,*A Defence of Free Thinking in Mathematics*, and*De Motu*, show clearly enough whom he conceived to be his deadly foe. Hume's*Enquiry Concerning Human Understanding*and*Enquiry Concerning the Principles of Morals*contain frequent references to Newton. The French Encyclopaedists and materialists of the middle of the eighteenth century felt themselves one and all to be more consistent Newtonians than Newton himself. In his early years Kant was an eager student of Newton, and his first works aim mainly at a synthesis of continental philosophy and Newtonian science. Hegel wrote an extended and trenchant criticism of Newton.

**The only way**to bring this issue to the bar of truth**is to plunge into the philosophy of early modern science, locating its key assumptions as they appear, and following them out to their classic formulation in the metaphysical paragraphs of Sir Isaac Newton.**The present is a brief historical study which aims to meet this need. ...At its close the reader will understand more clearly the nature of modern thinking and judge more accurately the validity of the contemporary scientific world-view.

### Chapter 2 Copernicus and Kepler (A) *The Problem of the New Astronomy*[edit]

- 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*.

**Both ancient and medieval observers had noted that in many respects nature appeared to be governed by the principle of simplicity**, and they had recorded the substance of their observations to this effect in the form of proverbial axioms which had become currently accepted bits of man's conception of the world. That falling bodies moved perpendicularly towards the earth, that light travelled in straight lines, that projectiles did not vary from the direction in which they were impelled, and countless other familiar facts of experience, had given rise to such common proverbs as:*'Natura semper agit per vias brevissimas'; 'natura nihil facit frustra'; 'natura neque redundat in superfluis neque deficit in necessariis'*[Nature always acts by the shortest path; nature does nothing in vain; nature never overflows into the unnecessary, nor is she deficient in what is necessary].**This notion, that nature performs her duties in the most commodious fashion, without extra labour, would have tended to decrease somewhat the repulsion which most minds must have felt at Copernicus; the cumbrous epicycles had been decreased in number, various irregularities in the Ptolemaic scheme were eliminated**...

**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 Renaissance had happened, namely the shifting of man's centre of interest in literature from the present to the golden age of antiquity. The Commercial Revolution had begun, with its long voyages and exciting discoveries of previously unknown continents and unstudied civilizations; the business leaders of Europe and the champions of colonialism were turning their attention from petty local fairs to the great untapped centres of trade in Asia and the Americas.
**The realm of man's previous acquaintance seemed suddenly small and meagre; men's thoughts were becoming accustomed to a widening horizon.**

**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.**

**Nicholas of Cusa... dared to teach that there is nothing at all without motion in the universe—the latter is infinite in all directions, possessing no centre—and that the earth travels its course in common with the other stars.**That**this widening of the intellectual horizon of the age, with the suggestion of new centres of interest, was a decisive factor in Copernicus' personal development**, the brief biographical sketch which he gives of himself in the*De Revolutionibus*strongly suggests.- Ref: Jasper Hawkins' English tr. Nicholas of Cusa's
*On Learned Ignorance*: Preface, Books I, II, III;*De Visione Dei*; [*De Coniecturis*]

- Ref: Jasper Hawkins' English tr. Nicholas of Cusa's

**The argument used**by Copernicus and other defenders of the new cosmography, like Gilbert of Colchester,**in answer to the objection that objects on its surface would be hurled off like projectiles if the earth were really in such a rapid motion the argument that rather would the supposed immense sphere of the fixed stars fly asunder implies that these men were already venturing to think of the heavenly bodies as homogeneous with the earth, to which the same principles of force and motion apply.**

- 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.

### Chapter 2 Copernicus and Kepler (B) *Metaphysical Bearings of the Pre-Copernican Progress in Mathematics*[edit]

**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.**

- In the later Middle Ages, there appeared a powerful revival of mathematical study, the same [geometrical] assumptions and methods were taken for granted... Roger Bacon eagerly adopted these assumptions and shared to the full this enthusiasm; two centuries after Bacon, the great and many-sided thinker, Leonardo da Vinci, stands out as the leader in this development. ...
**During the**next**century, that marked by the appearance of Copernicus' epoch-making book, this geometrical method**in mechanics and the other mathematico-physical sciences**was assumed by all important thinkers.**

- 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.- Note: an edition of
*Nova Scientia*was translated into English by Cyprian Lucar in 1588

- Note: an edition of

**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.

**The solution of quadratic and cubic equations in the sixteenth century was always sought by the geometrical method.**W. W. R. Ball gives an interesting example of this cumbrous mode of reaching such results in Cardanus' solution of the cubic equation:*x*^{3}+*qx*=*r*.**We can readily appreciate what a tremendous advance was in store for modern algebra when it finally succeeded in freeing itself from the shackles of spatiality.**- Footnote: W. W. R. Ball,
*A Short Account of the History of Mathematics*, p.224, ff.

- Footnote: W. W. R. Ball,

**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.
- Footnote: The boundary of the enclosing body on the side of the enclosed. Phys. IV. 4.
*τόποζ*is Aristotle's word.

- Footnote: The boundary of the enclosing body on the side of the enclosed. Phys. IV. 4.

**The sun and moon seem perfect circles, the stars but luminous points in pure space.**To be sure, they were held to be physical bodies of some sort, and so possessed more than geometrical characteristics, but there was no way of investigating such, hence**it must have been easy to raise no questions which would imply any difference between the realm of geometry and astronomical space. In fact, we know that, by many, astronomy was regarded as closer to the geometrical ideal of pure mathematics than arithmetic.**

- Typical lists of the mathematical sciences offered by Alfarabi and Roger Bacon place them in the order: geometry, astronomy, arithmetic, music. Of course, this was in part due to the higher dignity ascribed to the heavenly bodies and the fact that the main uses of arithmetic were commercial. But not wholly. Astronomy, more than arithmetic, was like geometry. It was nothing essentially but the geometry of the heavens;
**men readily felt**, therefore,**that whatever was true in geometry must be necessarily and fully true of astronomy.**

**If... astronomy is but a branch of geometry**, and if the transformation and reduction of algebraic equations is uniformly pursued by the geometrical method... indicating that such are still felt to be essentially geometrical problems,**shall we have to wait long for a thinker to appear who will raise the question**, why is not such reduction possible in astronomy?**If astronomy is mathematics it must partake of the relativity of mathematical values, the motions represented on our chart of the heavens must be purely relative, and it makes no difference as far as truth is concerned what point be taken as the point of reference for the whole spatial system.**

**Ptolemy**... against the champions of this or that cosmology of the heavens he had**dared to claim that it is legitimate to interpret the facts of astronomy by the simplest geometrical scheme which will 'save the phenomena,' no matter whose metaphysics might be upset. His conception of the physical structure of the earth, however, prevented him from carrying through**in earnest this principle of relativity, as his objections to the hypothesis that the earth moves amply show.- Footnote: For example, "if there were motion, it would be proportional to the great mass of the earth and would leave behind animals and objects thrown into the air."

- 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.

**To Copernicus' mind the question was not one of truth or falsity, not, does the earth move? He simply included the earth in the question which Ptolemy had asked with reference to the celestial bodies alone; what motions should we attribute to the earth in order to obtain the simplest and most harmonious geometry of the heavens**that will accord with the facts? ...ample proof of the continuity of his thought with the mathematical developments just recounted, and this is why he constantly appealed to mathematicians as those alone able to judge the new theory fairly.

- ...
**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.

- During the early Middle Ages the synthesis of Christian theology and Greek philosophy was accomplished with the latter in a predominantly Platonic, or rather Neo-Platonic cast. Now
**the Pythagorean element in Neo-Platonism**was very strong. All the important thinkers of the school**liked to express their favourite doctrines of emanation and evolution in terms of the number theory**, following Plato's suggestion in the Parmenides that**plurality unfolded itself from unity by a necessary mathematical process.**

**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.**

- Nicholas of Cusa found in the theory of numbers the essential element in the philosophy of Plato. The world is an infinite harmony, in which all things have their mathematical proportions. Hence "knowledge is always measurement," "number is the first model of things in the mind of the Creator"; in a word, all certain knowledge that is possible for man must be mathematical knowledge. The same strain appears strongly in Bruno, though in him even more than in Cusa the mystico-transcendental aspect of the number theory was apt to be uppermost.

- [Dominicus Maria de]
**Novara was Copernicus' friend and teacher**during the six years of his stay in Italy, and among the important facts which we know about him is this, that**he was a free critic of the Ptolemaic system of astronomy**, partly because of some observations which did not agree closely enough with deductions from it, but more especially**because he was thoroughly caught in this Platonic-Pythagorean current and felt that the whole cumbrous system violated the postulate that the astronomical universe is an orderly mathematical harmony.**- Footnote: Dorothy Stimson,
*The Gradual Acceptance of the Copernican Theory of the Universe*(1917) p. 25.

- Footnote: Dorothy Stimson,

- 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.**

**The orthodox Aristotelian school minimized the importance of mathematics.**Quantity was only one of the ten predicaments and not the most important. Mathematics was assigned an intermediate dignity between metaphysics and physics. Nature was fundamentally qualitative as well as quantitative; the key to the highest knowledge must, therefore, be logic rather than mathematics. With the mathematical sciences allotted this subordinate place in his philosophy,**it could not but appear ridiculous to an Aristotelian for any one to suggest seriously that his whole view of nature be set aside in the interest of a simpler and more harmonious geometrical astronomy.**

**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.**

**He**[Copernicus]**had himself become convinced that the whole universe was made of numbers, hence whatever was mathematically true was really or astronomically true.**Our earth was no exception it, too, was essentially geometrical in nature therefore the principle of relativity of mathematical values applied to man's domain just as to any other part of the astronomical realm.**The transformation to the new world-view, for him, was nothing but a mathematical reduction... of a complex geometrical labyrinth into a beautifully simple and harmonious system.**

- 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]."- Footnote: Joannis Kepleri,
*Astronomi Opera Omnia*, ed. Ch. Frisch. Vol.I, 112 ff. (1858)

- Footnote: Joannis Kepleri,

- 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.

**Kepler's achievements in mathematics would alone have been sufficient to win for him enduring fame**; he first enunciated clearly the principle of continuity in mathematics, treating the parabola as at once the limiting case of the ellipse and the hyperbola, and showing that parallel lines can be regarded as meeting at infinity; he introduced the word 'focus' into geometry; while in his*Stereometria Dolorum*, published 1615, he applied the conception to the solution of certain volumes and areas by the use of infinitesimals, thus preparing the way for Desargues, Cavalieri, Barrow, and the developed calculus of Newton and Leibniz.- Ref:
*Nova Stereometria Doliorum Vinariorum*[New Stereometry of Wine Barrels]

- Ref:

- 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 sun, according to Kepler, is God the Father, the sphere of the fixed stars is God the Son, the intervening ethereal medium, through which the power of the sun is communicated to impel the planets around their orbits, is the Holy Ghost.**To pronounce this allegorical trapping is not to suggest, of course, that Kepler's Christian theology is at all insincere ; it is rather that he had discovered an illuminating natural proof and interpretation of it, and the whole attitude, with its animism and allegorico-naturalistic approach, is quite typical of much thinking of the day. Kepler's contemporary Jacob Boehme, is the most characteristic representative of this type of philosophy.- Footnote:
*Joannis Kepleri Astronomi Opera Omnia*, ed. Christian Frisch (1858) Vol. I, p.11

- Footnote:

**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.

**It became the passion of his**[ Kepler's ]**life to penetrate and disclose**, for the "fuller knowledge of God through nature and the glorification of his profession,"**these deeper harmonies**, and the fact that he was not satisfied merely with mystical manipulation of numbers, or aesthetic contemplation of geometrical fancies, we owe to his long training in mathematics and astronomy, and in no small degree to the influence of the great**Tycho**, who**was the first competent mind in modern astronomy to feel ardently the passion for exact empirical facts.**

- 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.
- Footnote:
*Joannis Kepleri Astronomi Opera Omnia*, ed. Christian Frisch (1858) Vol. V, p.224, Cf. also Vol.I, p.143

- Footnote:

**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.

**Kepler's**enunciation of his third law, in the*Harmonices Mundi*, 1619, is imbedded in a laborious attempt to determine the music of the spheres according to precise laws, and express it in our form of music notation. ...They are decidedly of a piece with his**central aim**, namely to**establish more mathematical harmonies in the Copernican astronomy**quite irrespective of their fruitfulness for such further achievements as became the goal of later scientific labours.**They grow directly out of his whole philosophy of the aim and procedure of science, and the new metaphysical doctrines**which in rudimentary fashion he perceived to be**implied in the acceptance of Copernicanism**and in the adoption of such an aim.

**He**[Kepler]**has reached a new conception of**This notion of causality is substantially the*causality*, that is,*he thinks of the underlying mathematical harmony discoverable in the observed facts as the cause of the latter, the reason, as he usually puts it, why they are as they are*.**Aristotelian formal cause reinterpreted in terms of exact mathematics**; it also has obvious close relations with the rudimentary ideas of the early Pythagoreans.

**The exactness or rigour with which the causal harmony must be verified in phenomena is the new and important feature in Kepler.**Tycho had urged Kepler in a letter "to lay a solid foundation for his views by actual observation, and then by ascending from these to strive to reach the cause of things." Kepler,**however**, preferred to let Tycho gather the observations, for**he was antecedently convinced that genuine causes must always be in the nature of underlying mathematical harmonies.**...God created the world in accordance with the principle of perfect numbers, hence the mathematical harmonies in the mind of the creator furnish the cause "why the number, the size, and the motions of the orbits are as they are and not otherwise." Causality, to repeat, becomes reinterpreted in terms of mathematical simplicity and harmony.- Footnote:
*Joannis Kepleri Astronomi Opera Omnia*, ed. Christian Frisch (1858) Vol. I, p.10

- Footnote:

**A true hypothesis for Kepler must be a statement of the underlying mathematical harmony discoverable in the effects.**

- Kepler includes an interesting treatment of astronomical hypothesis in a letter written partly to refute Reimarus Ursus' position on the same subject.
**Kepler's thought is, that**...A true hypothesis is always a more inclusive conception, binding together facts which had hitherto been regarded as distinct; it reveals a mathematical order and harmony where before there had been unexplained diversity. And ...this more inclusive mathematical order is something discovered*of a number of variant hypotheses about the same facts, that one is true which shows why facts, which in the other hypotheses remain unrelated, are as they are, i.e., which demonstrates their orderly and rational mathematical connexion*.*in the facts*themselves.- Footnote:
*Joannis Kepleri Astronomi Opera Omnia*, ed. Christian Frisch (1858) Vol. I, p.238 ff.

- Footnote:

**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.**

- What further specific
**metaphysical doctrines**was Kepler led to adopt as a consequence of this notion of what constitutes the real world? For one thing, it**led him to appropriate in his own way the distinction between primary and secondary qualities**, which had been noted in the ancient world by the atomist and sceptical schools, and which was being revived in the sixteenth century in varied form by such miscellaneous thinkers as Vives, Sanchez, Montaigne, and Campanella.

**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.**

- In his [Kepler's] mathematical remains there is a brief criticism of Aristotle's treatment of the sciences, in which he declares that the fundamental difference between the Greek philosopher and himself was that the former traced things ultimately to qualitative, and hence irreducible distinctions, and was, therefore, led to give mathematics an intermediate place in dignity and reality between sensible things and the supreme theological or metaphysical ideas; whereas he had found means for discovering quantitative proportions between all things, and therefore gave mathematics the pre-eminence.

**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 apparently has no affiliations with the Democritan and Epicurean atomism, whose revival was destined to play an important part in post-Keplerian science.
**So far as his thought dwells upon the elementary particles of nature it is the geometrical atomism of the**that arouse his enthusiasm and interest. When he says that God made the world according to number he is thinking not about minute figured portions of space, but about*Timæus*and the ancient doctrine of the four elements that he inherited, but his interest is not in these; it is the mathematical relations revealed in the cosmos at large**these vaster numerical harmonies.**

**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.**

- Like other poor astronomers of the time, Kepler found in astrology a kind of service he could render which people without astronomical zeal were willing to pay for, a situation which he regarded as quite providential. But this does not at all mean that he did not thoroughly believe in astrology. Those who so maintain can hardly have read his essay
*De Funiamentis Aslrologiae Certioribus*...**there had been in the sixteenth century a powerful revival of interest and belief in astrology, and Kepler was prepared by his general philosophy of science to give it a comprehensive philosophical basis.**When the planets in their revolutions happen to fall in certain unusual relations, portentous consequences might very well ensue for human life—mighty vapours are perhaps projected from them, penetrate the animal spirits of men, stir their passions to an uncommon heat, with the result that wars and revolutions follow. ...the mathematical entities with which he is concerned are these larger astronomical harmonies rather than the elementary atoms.- Footnote:
*Joannis Kepleri Astronomi Opera Omnia*, ed. Christian Frisch (1858), Vol. I, p.417, ff. & 1477, ff.

- Footnote:

**We have... in Kepler's work, a second great event in the development of the metaphysics of modern science.**Aristotelianism had won out in the long preceding period of human thought because it seemed to make intelligible and rational the world of commonsense experience. Kepler early realized that the admission of validity to the Copernican world-scheme involved**a radically different cosmology**, a cosmology which could rest upon the revived Neo-Platonism for its general background, would find its historical justification in the remarkable developments in the sciences of mathematics and astronomy, and**which could lay bare a marvellous significance and a new beauty in the observed events of the natural cosmos by regarding them as exemplifications of simple, underlying numerical relations.**

**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.**

**He**[Kepler]**is entirely convinced on**For this side of his thought we have to thank his training in mathematics and in particular his association with that giant of careful star-observation, Tycho Brahe.*a priori*grounds that the universe is basically mathematical, and that all genuine knowledge must be mathematical, but he makes it plain that the laws of thought innate in us as a divine gift, cannot come to any knowledge of themselves; there must be the perceived motions which furnish the material for their exact exemplification.- Footnote:
*Joannis Kepleri Astronomi Opera Omnia*, ed. Christian Frisch (1858), Vol. V, p. 229

- Footnote:

- 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.**

### Chapter 3 Galileo (A) The Science of 'Local Motion'[edit]

**Were it not for his more stupendous achievements he**[Galileo],**like Kepler, would have won brilliant fame as a mathematician.**He invented a geometrical calculus for the reduction of complex to simple figures, and wrote an essay on continuous quantity. The latter was never published, but such was his mathematical name that Cavalieri did not publish his own treatise on the*Method of Indivisibles*as long as he hoped to see Galileo's essay printed.

- 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.

- As soon as he became competent in this new field [the mathematical study of mechanical motions] he eagerly embraced the Copernican system... and
**the Copernican attribution of motion to the earth gave him a powerful impetus to study more closely**, i.e., mathematically, such**motions of small parts of the earth**as occur in every-day experience, as we learn on the authority of his great English disciple, Hobbes.**Hence the birth of a new science, terrestrial dynamics**... - Footnote: Epistle Dedicatory to the "Elements of Philosophy Concerning Body",
*The English works of Thomas Hobbes of Malmesbury*, Molesworth edition (1839) Vol. I, p.viii. Note: Isaac Newton later to unified celestial and terrestrial dynamics.

**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.

### Chapter 3 Galileo (B) Nature as Mathematical Order—Galileo's Method[edit]

**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."- Footnote:
*Dialogues Concerning the Two Great Systems of the World*, Thomas Salusbury translation (1661) p. 99

- Footnote:

- "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."
- Footnote:
*Opere Complete di Galileo Galilei*, Firenze, 1842, ff.. Vol. IV, p.171 quoting Galileo's*The Assayer*(1623)

- Footnote:

**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*- Footnote: confer especially p.178

**It was this religious basis of his philosophy that made Galileo bold to declare that doubtful passages of scripture should be interpreted in the light of scientific discovery rather than the reverse. God has made the world an immutable mathematical system**, permitting by the mathematical method an absolute certainty of scientific knowledge. The disagreements of theologians about the meaning of scripture are ample testimony to the fact that here no such certainty is possible. Is it not obvious then which should determine the true meaning of the other?

- He [Galileo] quoted by way of orthodox support Tertullian's dictum that
**we know God first by nature, then by revelation.**

### Chapter 3 Galileo (C) The Subjectivity of Secondary Qualities[edit]

**Galileo makes the clear distinction between that in the world which is absolute, objective, immutable, and mathematical; and that which is relative, subjective, fluctuating, and sensible. The former is the realm of knowledge, divine and human; the latter is the realm of opinion and illusion.**The Copernican astronomy and the achievements of the two new sciences must break us of the natural assumption that sensed objects are the real or mathematical objects. They betray certain qualities, which, handled by mathematical rules, lead us to a knowledge of the true object, and these are the real or primary qualities, such as number, figure, magnitude, position, and motion, which cannot by any exertion of our powers be separated from bodies—qualities which also can be wholly expressed mathematically.**The reality of the universe is geometrical**; the only ultimate characteristics of nature are those in terms of which certain mathematical knowledge becomes possible.**All other qualities**, and these are often far more prominent to the senses,**are secondary, subordinate effects of the primary.**

**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.