Is God a Mathematician?

by Mario Livio

I had intended to send you my World as a New Year gift, and only two weeks ago I was quite determined to send you at least a part of it, if the whole work could not be copied in time. But I have to say that in the meantime I took the trouble to inquire in Leiden and Amsterdam whether Galileo’s World System was available, for I thought I had heard that it was published in Italy last year. I was told that it had indeed been published, but that all the copies had immediately been burnt at Rome, and that Galileo had been convicted and fined. I was so astonished at this that I almost decided to burn all my papers, or at least to let no one see them. For I could not imagine that he—an Italian, and as I understand, in the good graces of the Pope—could have been made a criminal for any other reason than that he tried, as he no doubt did, to establish that the Earth moves. I know that some Cardinals had already censured this view, but I thought I had heard it said that all the same it was taught publicly in Rome. I must admit that if the view is false, so too are the entire foundations of my philosophy [my emphasis], for it can be demonstrated from them quite clearly. And it is so closely interwoven in every part of my treatise that I could not remove it without rendering the whole work defective. But for all the world I did not want to publish a discourse in which a single word could be found that the Church would have disapproved of; so I preferred to suppress it rather than to publish it in a mutilated form.

  Descartes had indeed abandoned The World (the incomplete manuscript was eventually published in 1664), but he did incorporate most of the results in his Principles of Philosophy, which appeared in 1644. In this systematic discourse, Descartes presented his laws of nature and his theory of vortices. Two of his laws closely resemble Newton’s famous first and second laws of motion, but the others were in fact incorrect. The theory of vortices assumed that the Sun was at the center of a whirlpool created in the continuous cosmic matter. The planets were supposed to be swept around by this vortex like leaves in an eddy formed in the flow of a river. In turn, the planets were assumed to form their own secondary vortices that carried the satellites around. While Descartes’ theory of vortices was spectacularly wrong (as Newton ruthlessly pointed out later), it was still interesting, being the first serious attempt to formulate a theory of the universe as a whole that was based on the same laws that apply on the Earth’s surface. In other words, to Descartes there was no difference between terrestrial and celestial phenomena—the Earth was part of a universe that obeyed uniform physical laws. Unfortunately, Descartes ignored his own principles in constructing a detailed theory that was based neither on self-consistent mathematics nor on observations. Nevertheless, Descartes’ scenario, in which the Sun and the planets somehow disturb the smooth universal matter around them, contained some elements that much later became the cornerstone of Einstein’s theory of gravity. In Einstein’s theory of general relativity, gravity is not some mysterious force that acts across the vast distances of space. Rather, massive bodies such as the Sun warp the space in their vicinity, just as a bowling ball would cause a trampoline to sag. The planets then simply follow the shortest possible paths in this warped space.

  I have deliberately left out of this extremely brief description of Descartes’ ideas almost all of his seminal work in philosophy, because this would have taken us too far afield from the focus on the nature of mathematics (I shall return to some of his thoughts about God later in the chapter). I cannot refrain, however, from including the following amusing commentary that was written by the British mathematician Walter William Rouse Ball (1850–1925) in 1908:

  As to his [Descartes’] philosophical theories, it will be sufficient to say that he discussed the same problems which have been debated for the last two thousand years, and probably will be debated with equal zeal two thousand years hence. It is hardly necessary to say that the problems themselves are of importance and interest, but from the nature of the case no solution ever offered is capable either of rigid proof or disproof; all that can be effected is to make one explanation more probable than another, and whenever a philosopher like Descartes believes that he has at last finally settled a question it has been possible for his successors to point out the fallacy in his assumptions. I have read somewhere that philosophy has always been chiefly engaged with the inter-relations of God, Nature, and Man. The earliest philosophers were Greeks who occupied themselves mainly with the relations between God and Nature, and dealt with Man separately. The Christian Church was so absorbed in the relation of God to Man as entirely to neglect Nature. Finally, modern philosophers concern themselves chiefly with the relations between Man and Nature. Whether this is a correct historical generalization of the views which have been successively prevalent I do not care to discuss here, but the statement as to the scope of modern philosophy marks the limitations of Descartes’ writings.

  Descartes ended his book on geometry with the words: “I hope that posterity will judge me kindly, not only as to the things which I have explained, but also as to those which I have intentionally omitted so as to leave to others the pleasure of discovery” (figure 26). He could not have known that a man who was only eight years old the year Descartes died would take his ideas of mathematics as the heart of science one huge step forward. This unsurpassed genius had more opportunities to experience the “pleasure of discovery” than probably any other individual in the history of the human race.

  Figure 26

  And There Was Light

  The great eighteenth-century English poet Alexander Pope (1688–1744) was thirty-nine years old when Isaac Newton (1642–1727) died (figure 27 shows Newton’s tomb inside Westminster Abbey). In a well-known couplet, Pope attempted to encapsulate Newton’s achievements:

  Nature and Nature’s laws lay hid in night:

  God said, Let Newton be! And all was light.

  Almost a hundred years after Newton’s death, Lord Byron (1788–1824) added in his epic poem Don Juan the lines:

  And this is the sole mortal who could grapple,

  Since Adam, with a fall or with an apple.

  To the generations of scientists that succeeded him, Newton indeed was and remains a figure of legendary proportions, even if one disregards the myths. Newton’s famous quote “If I have seen further it is by standing on ye shoulders of Giants” is often presented as a model for the generosity and humility that scientists are expected to display about their greatest discoveries. Actually, Newton may have written that phrase as a subtly veiled sarcastic response to a letter from the person whom he regarded as his chief scientific nemesis, the prolific physicist and biologist Robert Hooke (1635–1703). Hooke had accused Newton on several occasions of stealing his own ideas, first on the theory of light, and later on gravity. On January 20, 1676, Hooke adopted a more conciliatory tone, and in a personal letter to Newton he declared: “Your Designes and myne [concerning the theory of light] I suppose aim both at the same thing which is the Discovery of truth and I suppose we can both endure to hear objections.” Newton decided to play the same game. In his reply to Hooke’s letter, dated February 5, 1676, he wrote: “What Des-Cartes [Descartes] did was a good step [referring to Descartes’ ideas on light]. You have added much several ways, & especially in taking the colours of thin plates into philosophical consideration. If I have seen further it is by standing on ye shoulders of Giants.” Since, far from being a giant, Hooke was quite short and afflicted with a severe stoop, Newton’s best-known quote might have simply meant that he felt he owed absolutely nothing to Hooke! The fact that Newton took every opportunity to insult Hooke, his statement that his own theory destroyed “all he [Hooke] has said,” and his refusal to take his own book on light, Opticks, to the press until after Hooke’s death, argue that this interpretation of the quote may not be too far-fetched. The feud between the two scientists reached an even higher peak when it came to the theory of gravity. When Newton heard that Hooke had claimed to be the originator of the law of gravity, he meticulously and vindictively erased every single reference to Hooke’s name from the last part of his book o
n the subject. To his friend the astronomer Edmond Halley (1656–1742), Newton wrote on June 20, 1686:

  Figure 27

  He [Hooke] should rather have excused himself by reason of his inability. For tis plain by his words he knew not how to go about it. Now is not this very fine? Mathematicians that find out, settle and do all the business must content themselves with being nothing but dry calculators and drudges and another that does nothing but pretend and grasp at all things must carry away all the invention as well of those that were to follow him as of those that went before.

  Newton makes it abundantly clear here why he thought that Hooke did not deserve any credit—he could not formulate his ideas in the language of mathematics. Indeed, the quality that made Newton’s theories truly stand out—the inherent characteristic that turned them into inevitable laws of nature—was precisely the fact that they were all expressed as crystal-clear, self-consistent mathematical relations. By comparison, Hooke’s theoretical ideas, as ingenious as they were in many cases, looked like nothing but a collection of hunches, conjectures, and speculations.

  Incidentally, the handwritten minutes of the Royal Society from 1661 to 1682, which were for a long time considered lost, suddenly surfaced in February 2006. The parchment, which contains more than 520 pages of script penned by Robert Hooke himself, was found in a house in Hampshire, England, where it is thought to have been stored in a cupboard for about fifty years. Minutes from December 1679 describe correspondence between Hooke and Newton in which they discussed an experiment to confirm the rotation of the Earth.

  Returning to Newton’s scientific masterstroke, Newton took Descartes’ conception—that the cosmos can be described by mathematics—and turned it into a working reality. In the preface to his monumental work The Mathematical Principles of Natural Philosophy (in Latin: Philosophiae Naturalis Principia Mathematica; commonly known as Principia), he declared:

  We offer this work as the mathematical principles of philosophy, for the whole burden of philosophy seems to consist in this—from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena: and to this end the general propositions in the first and second Books are directed. In the third Book we give an example of this in the explication of the System of the World; for by the propositions mathematically demonstrated in the former Books, in the third we derive from the celestial phenomena the force of gravity with which bodies tend to the Sun and the several planets. Then from these forces, by other propositions which are also mathematical, we deduce the motions of the planets, the comets, the moon, and the sea.

  When we realize that Newton truly accomplished in Principia everything he promised in the preface, the only possible reaction is: Wow! Newton’s innuendo of superiority to Descartes’ work was also unmistakable: He chose the title of his book to read Mathematical Principles, as opposed to Descartes’ Principles of Philosophy. Newton adopted the same mathematical reasoning and methodology even in his more experimentally based book on light, Opticks. He starts the book with: “My design in this book is not to explain the Properties of Light by Hypotheses, but to propose and prove them by Reason and Experiments: In order to which I shall premise the following definitions and Axioms.” He then proceeds as if this were a book on Euclidean geometry, with concise definitions and propositions. Then, in the book’s conclusion, Newton added for further emphasis: “As in Mathematicks, so in Natural Philosophy, the Investigation of difficult Things by the Method of Analysis, ought ever to precede the Method of Composition.”

  Newton’s feat with his mathematical tool kit was nothing short of miraculous. This genius, who by a historical coincidence was born in exactly the same year in which Galileo died, formulated the fundamental laws of mechanics, deciphered the laws describing planetary motion, erected the theoretical basis for the phenomena of light and color, and founded the study of differential and integral calculus. These achievements alone would have sufficed to earn Newton a place of honor in the gallery of the most prominent scientists. But it was his work on gravity that elevated him to the top place on the podium of the magicians—the one reserved for the greatest scientist ever to have lived. That work bridged the gap between the heavens and the Earth, fused the fields of astronomy and physics, and put the entire cosmos under one mathematical umbrella. How was that masterpiece—Principia—born?

  I Began to Think of Gravity Extending to the Orb of the Moon

  William Stukeley (1687–1765), an antiquary and physician who was Newton’s friend (in spite of the more than four decades in age separating them), eventually became the great scientist’s first biographer. In his Memoirs of Sir Isaac Newton’s Life we find an account of one of the most celebrated legends in the history of science:

  On 15 April 1726 I paid a visit to Sir Isaac at his lodgings in Orbels buildings in Kensington, dined with him and spent the whole day with him, alone…After dinner, the weather being warm, we went into the garden and drank thea, under the shade of some apple trees, only he and myself. Amidst other discourse, he told me he was just in the same situation, as when formerly [in 1666, when Newton returned home from Cambridge because of the plague], the notion of gravitation came into his mind. It was occasion’d by the fall of an apple, as he sat in contemplative mood. Why should that apple always descend perpendicularly to the ground, thought he to himself. Why should it not go sideways or upwards, but constantly to the earth’s centre? Assuredly, the reason is, that the earth draws it. There must be a drawing power in matter: and the sum of the drawing power in the matter of the earth must be in the earth’s centre, not in any side of earth. Therefore does this apple fall perpendicularly, or towards the centre. If matter thus draws matter, it must be in proportion of its quantity. Therefore the apple draws the earth, as well as the earth draws the apple. That there is a power, like that we here call gravity, which extends its self thro’ the universe…This was the birth of those amazing discoverys, whereby he built philosophy on a solid foundation, to the astonishment of all Europe.

  Irrespective of whether the mythical event with the apple actually occurred in 1666 or not, the legend sells Newton’s genius and unique depth of analytic thinking rather short. While there is no doubt that Newton had written his first manuscript on the theory of gravity before 1669, he did not need to physically see a falling apple to know that the Earth attracted objects near its surface. Nor could his incredible insight in the formulation of a universal law of gravitation stem from the mere sight of a falling apple. In fact, there are some indications that a few crucial concepts that Newton needed to be able to enunciate a universally acting gravitational force were only conceived as late as 1684–85. An idea of this magnitude is so rare in the annals of science that even someone with a phenomenal mind—such as Newton—had to arrive at it through a long series of intellectual steps.

  It may have all started in Newton’s youth, with his less-than-perfect encounter with Euclid’s massive treatise on geometry, The Elements. According to Newton’s own testimony, he first “read only the titles of the propositions,” since he found these so easy to understand that he “wondered how any body would amuse themselves to write any demonstrations of them.” The first proposition that actually made him pause and caused him to introduce a few construction lines in the book was the one stating that “in a right triangle the square of the hypothenuse is equal to the squares of the two other sides”—the Pythagorean theorem. Somewhat surprisingly perhaps, even though Newton did read a few books on mathematics while at Trinity College in Cambridge, he did not read many of the works that were already available at this time. Evidently he didn’t need to!

  The one book that turned out to be perhaps the most influential in guiding Newton’s mathematical and scientific thought was none other than Descartes’ La Géométrie. Newton read it in 1664 and re-read it several times, until “by degrees he made himself master of the whole.” The flexibility afforded by the notion of functions and their free variables appeared
to open an infinitude of possibilities for Newton. Not only did analytic geometry pave the way for Newton’s founding of calculus, with its associated exploration of functions, their tangents, and their curvatures, but Newton’s inner scientific spirit was truly set ablaze. Gone were the dull constructions with ruler and compass—they were replaced by arbitrary curves that could be represented by algebraic expressions. Then in 1665–66, a horrible plague hit London. When the weekly death toll reached the thousands, the colleges of Cambridge had to close down. Newton was forced to leave school and return to his home in the remote village of Woolsthorpe. There, in the tranquility of the countryside, Newton made his first attempt to prove that the force that held the Moon in its orbit around the Earth and the Earth’s gravity (the very force that caused apples to fall) were, in fact, one and the same. Newton described those early endeavors in a memorandum written around 1714:

  And the same year [1666] I began to think of gravity extending to the orb of the Moon, and having found out how to estimate the force with which [a] globe revolving within a sphere presses the surface of the sphere, from Kepler’s Rule of the periodical times of the Planets being in a sesquialternate proportion of their distances from the centres of their Orbs I deduced that the forces which keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centres about which they revolve: and thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth, and found them answer pretty nearly. All this was in the two plague years of 1665 and 1666, for in those years I was in the prime of my age for invention, and minded Mathematicks and Philosophy more than at any time since.

  Newton refers here to his important deduction (from Kepler’s laws of planetary motion) that the gravitational attraction of two spherical bodies varies inversely as the square of the distance between them. In other words, if the distance between the Earth and the Moon were to be tripled, the gravitational force that the Moon would experience would be nine times (three squared) smaller.