You try adding a minus sign to a term – but cannot because the theory then violates parity; you try adding a term with more particles in it – forbidden because the theory now is nonrenormalizable and so demands an infinite number of parameters; you try leaving a particle out of the theory – now the law has uninterpretable probabilities; you subtract a different term – all your particles vanish into the vacuum; you split a term in two – now charge is not conserved; and you still have to satisfy conservation laws of angular momentum, linear momentum, energy, lepton number, and baryon number. Such constraints do not all issue axiomatically from a single, governing theory. Rather, they are the sum total of a myriad of interpenetrating commitments of practice. Some, such as the conservation of energy, are over a century old. Others, such as the conservation of parity, survived for a very long time before being discarded. And yet others, such as the demand for naturalness – that all free parameters arise in ratios on the order of unity – have their origin in more recent memory. Some are taken by the research community to present nearly insuperable barriers to violation, while others merely flash a yellow cautionary light on being pushed aside. But taken together, the superposition of such constraints makes some phenomena virtually impossible to posit, and others (such as black holes) almost impossible to avoid.1
Yet another thing we learn from these journeys is that science is a deeply affective process. Those who do not realize this, or who think that scientific experience involves a dry conceptual part plus a separate emotional part, do not understand science or human creativity. It is possible – and indeed useful for some purposes – to divide up the scientific process into a conceptual part and an affective part, but this is an artificial model, something that comes afterward. Studying these journeys allows us to bore underneath the levels of abstraction that conceal how science truly works. We encountered the role of dissatisfaction, for instance, in many of these journeys, and also saw episodes of curiosity, consternation, bafflement, and wonder. We saw the difference between expectation and alertness – between scientists who expected something and could only take notice when that expectation was fulfilled, and scientists who were alert in the sense that they were prepared to hear something more than what they expected. We encountered affects not only in what motivates discovery, but also in the scientists’ response to it. The affective response to a discovery is not simply ‘OK, I get it now, this belongs here and that over there, I had it wrong and I get it now’, but something much more nuanced and powerful. Nor is the affective response limited to discovery. As Leon Lederman wrote, to pin one’s hopes on making a discovery that will bring fame and fortune ‘is not a life.’ He continued, ‘The fun and excitement must be daily – in the challenge of creating an instrument and seeing it work, the joy of communicating to colleagues and students, the pleasure of learning something new in lectures, corridors, and journals.’2
But our wonder is not only at what we have learned, but at something still more profound. In certain moments of wonder, we glimpse the connection between ourselves and nature; we glimpse the mutability of nature and our role in it. We experience that nature could be otherwise – more, that it was otherwise until a moment ago, and for all we know it could change in the future. In such moments, we experience an Emersonian moment of a higher thought in the middle of the existing one, a more profound feeling befalling us that we experience at once as new and old, surprising and familiar, there and not there before us, uncanny and domestic.
Shortly before finishing this book, I found myself struggling to describe the project to an eminent, elderly physicist who expressed little sympathy for books about science accessible to nonscientists. No magic for him! The equations, when fully grasped, seem so obvious, or so complete, or so logical that, once grasped, we cannot imagine not having known them. He approached science the way he thought a purely professional workman should, and urged me to do likewise. ‘Such equations’, he told me, ‘would not be wonderful if people realized how trivial they are. You should help them do so.’
I could have hugged the old man. He helped me put my finger on what I was trying to do. Which was just the opposite – to show how equations are not trivial, to recover the dissatisfactions that led us to seek them, and to restore the wonder to the moment when we first grasped them. The wonder at the moment when they arrived, seemingly simultaneously discovered and invented, when they seemed more concise statements of what we already know, something (like the Pythagorian theorem) so secure that it seems that it was already ‘in us’ and simply remembered. Scientists of the sort as my elderly physicist acquaintance tend to be focused on the formal (what he meant by ‘trivial’) part, whereas philosophers and other scholars in the humanities tend to focus on the other part. It ought to be possible to have both parts at once: the sciences and the humanities together, anosognosia cured, Twain’s young and old pilots viewing the water, the slave boy with his eye on the diagram and Socrates with his on human life, the formal part and the meaningful, affective part put back together in the originary unity from which they sprang. If we can, we will recover the wonder of Richard Harrison’s child discovering 1 + 1 = 2, and view equations as the key ‘not only to what was wonderful in the outside world, but what was wonderful in him and all of us.’ Such a moment would be a fully human response to the world.
NOTES
INTRODUCTION
1 Modern astrologers seem strangely untroubled by, and even ignorant of, the fact that constellations do not come in neat packages – and that the sun and planets pass by them, and sometimes by entirely different constellations, at different times than confidently asserted by the dates given in newspaper horoscopes. Someone ought to file a malpractice claim.
2 See I. Bernard Cohen, The Triumph of Numbers: How Counting Shaped Modern Life (New York: W. W. Norton, 2005).
3 In response to this need to use something to stand for numbers or other things – symbols – the ancient Egyptians, Babylonians, and Greeks developed different ways of symbolizing numbers and quantities. Much ancient mathematics then consisted of solving specific cases and inviting the reader to generalize. For example, the famous Rhind papyrus, an Egyptian manuscript from about 1650 bc, contains what amount to rudimentary equations, based on examples, for figuring out the areas of triangles, rectangles, circles, and the volumes of prisms and cylinders. The papyrus also demonstrated solutions for practical problems, such as how to determine equalities between loaves of bread of different consistencies and different amounts of barley. It even discussed exemplary problems that are not practical but conceptually interesting, such as the following: ‘There are seven houses; in each house there are seven cats; each cat kills seven mice; each mouse has eaten seven grains of barley; each grain would have produced seven ‘hekat.’ What is the sum of all the enumerated things?’ Many different versions of this problem have cropped up ever since, such as the Mother Goose rhyme ‘The Man from St. Ives’ – who had seven wives, each of whom had seven sacks, each of which contained seven cats, each of which had seven kittens. In their equations, the Egyptians used symbols that consisted of hieroglyphs looking like pairs of legs that seem to be walking in the direction the book is written for addition, or in the opposite direction for subtraction.
4 A. N. Whitehead and B. Russell, Principia Mathematica, vol. 1 (Cambridge: Cambridge University Press, 1957), p. 362: ‘From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2.’
5 Today, equations are classified in several different ways. One is by their degree, or the character of its biggest exponent. In linear equations, so-called because they describe lines (examples include 4x + 3y = 11 and y = 2x + 1), the unknown numbers x or y are not raised to any power and are said to be of the first degree. When the unknown is squared, the equation is called quadratic; when cubed, it is a cubic equation; after that it is an equation of the fourth, fifth, sixth degrees, and so on. And when the solution to an equation is not a number but a function – when it is said
to contain ‘derivatives’ – it is called a differential equation.
6 In Isaac Newton, The Principia: Mathematical Principles of Natural Philosophy, trans. I. B. Cohen and Anne Whitman (Berkeley: University of California Press, 1999), p. 391.
7 This has led some people to compare equations and poems. Both involve special uses of language often above the heads of untrained readers that seek to express truths concisely and with precision, and that allow us to understand otherwise inaccessible things, changing our experience in the process. Equations ‘state truths with a unique precision, convey volumes of information in rather brief terms, and often are difficult for the uninitiated to comprehend’, writes Michael Guillen in his book Five Equations that Changed the World: The Power and Poetry of Mathematics. Guillen adds, ‘And just as conventional poetry helps us to see deep within ourselves, mathematical poetry helps us to see far beyond ourselves.’ Graham Farmelo, the editor of It Must Be Beautiful: Great Equations of Modern Science, likewise compared equations and poems. Both, he noted, are composed of abstractions with which we address the world, even though many individual terms do not have specific referents. While ‘poetry is the most concise and highly charged form of language’, Farmelo writes, equations are ‘the most succinct form of understanding of the aspect of physical reality they describe.’
Many other differences exist, of course, between poems and equations. Equations can seem fearful. For they are not only beyond our understanding, but refer to powers beyond our control – which can make us feel helpless and resentful. Poems generally refer directly to the intuitive human experience of the surrounding world, invoke more than inform, and do so in a way that can have an impact on that intuitive experience. Equations, by contrast, do not refer to direct human experience, but to specially defined quantities – such as acceleration, energy, force, mass, the speed of light, to name a few – that are measured in laboratories. They cannot be plucked or dug up from anywhere, ordered from a catalogue, or held in your hand like apples and balls. And equations have a special structure that poems lack – they state that one group of these quantities is equal to (or greater to or less than, in their looser sense) another.
Such quantities – what equations refer to – are not always easy to identify. Consider the old saw about the Army captain seeking to hire a lieutenant, who posed to each of the three candidates the same question: ‘How much is 1 + 1?’
Candidate 1 answered, ‘Two, of course.’
Candidate 2 answered, ‘Well, that all depends on what 1 represents. It might be a vector, in which case its value could be anything from 0 to +2.’
And Candidate 3 answered, ‘How much would you like it to be?’
The predictable punch line, of course, is that the job goes to Candidate 3. Part of the joke – Candidate 2’s contribution – relies on an equivocation: between a number and the magnitude of a vector. But the buildup shows that the ties between equations and the world are not as simple as it appears. Still, specifying the relations among specially defined quantities allows equations to transform our encounters with the world in several ways – by pointing out new things, by giving us more power, and by reorganizing the way we see. Poems don’t do it that way.
8 Frank Wilczek, ‘Whence the Force of F = ma? I: Culture Shock’, Physics Today, October 2004, pp. 11–12; ‘Whence the Force of F = ma? II: Rationalizations’, Physics Today, December 2004, pp. 10–11; ‘Whence the Force of F = ma? III: Cultural Diversity’, Physics Today, July 2005, pp. 10–11.
1 ‘The Basis of Civilization’: The Pythagorean Theorem
1 John Aubrey, Brief Lives, ed. Richard Barber (Great Britain: Boydell Press, 1982), p. 152.
2 This story might seem too apocryphal to be true, a retrospective ‘Eureka!’ moment, but most biographers believe it. Often we realize only later the significance of a moment whose meaning we are only dimly aware of at the time. Hobbes’s recent biographer A. P. Martinich (Hobbes: A Biography, pp. 84–85) argues forcefully for the truth of the story. Martinich adds, ‘The importance of geometry on Hobbes’s philosophy can hardly be exaggerated… What came to impress Hobbes was not so much the axioms, theorems, and proofs of geometry itself, but the method of connecting one thing with another on a foundation that could not be doubted. It was the method, not the substance, of geometry that staggered him.’
3 He never became a true professional, though, and fell into traps of the sort that enthusiastic amateurs often do. These included pursuing impossible problems like trying to square the circle, trisect an angle, and double a cube, each of which Hobbes erroneously thought he had achieved.
4 Leo Strauss, The Political Philosophy of Hobbes: Its Basis and Its Genesis (Chicago: University of Chicago Press, 1959), p. 29.
5 Reid McInvale, ‘Circumambulation and Euclid’s 47th Proposition’, http://www.io.com/~janebm/summa.html (accessed April 11, 2008). See also James Anderson, The Constitutions of the Free-Masons (1723): ‘[T]he Greater Pythagoras, prov’d the Author of the 47th Proposition of Euclid’s first Book, which, if duly observ’d, is the Foundation of all Masonry, sacred, civil, and military…’ Little Masonic Library [rev. ed.], vol. 1 (Richmond, VA: Macoy, 1977), pp. 203–4.
6 O. Neugebauer and A. Sachs, ‘Mathematical Cuneiform Texts’, in American Oriental Series, vol. 29 (New Haven: American Oriental Society, 1945), p. 38; Eleanor Robson, ‘Neither Sherlock Holmes nor Babylon: A Reassessment of Plimpton 322’, Historia Mathematica 28 (2001), pp. 167–206.
7 The Baudhāyana, for instance, says that ‘the diagonal of an oblong produces by itself both the areas which the two sides of the oblong produce separately’ [quoted in David Smith, History of Mathematics, vol. 1 (New York: Dover, 1958), p. 98], but simply declares this as a fact without further justification. ‘[W]e must remember’, writes one scholar, ‘that they were interested in geometrical truths only as far as they were of practical use, and that they accordingly gave to them the most practical expression’ [G. Thibaut, The Śulvasūtras (Calcutta: Papatist Mission Press, 1875), p. 232].
8 Christopher Cullen, Astronomy and Mathematics in Ancient China: The Zhou Bi Suan Jing (Cambridge: Cambridge University Press, 1996), p. xi. But as Cullen observes, ‘the process is more verbal than computational’, and ‘to illustrate it by a carefully labelled Euclidean diagram when none is referred to in the text is perhaps only a way of misleading oneself’ about what the author is up to, for ‘nothing worthy of being called computation is involved’ (p. 80). The height of the sun, by the way, is 80,000 li, or about 40,000 kilometers or 24,000 miles. A later Chinese text called the Ziu Zhang Suan Shu (Nine Chapters on the Mathematical Art, from about ad 250), has the rule somewhat more explicitly treated. The Ziu Zhang’s concerns are mainly practical; its first chapter is ‘field measurement’, while later chapters are on canals, taxation, and other matters. Its ninth and final chapter is ‘Kou ku’, or ‘base and altitude’, kou or ‘leg’ meaning the short side of a right triangle, ku or ‘thigh’ the long side (hsien meant the hypotenuse or line strung between two points). The chapter contained twenty-four problems on properties of right triangles. But ‘proof is not their preoccupation’, says historian G.E.R. Lloyd. ‘[T]heir style of mathematical reasoning has more to do with exploring analogies and common structures (in groups of problems, procedures, formulae) than with demonstration as such – a style that itself remains close to that favoured in other genres, including poetry, also remarkable for its interest in correlations, complementarities, parallelisms. The contrast, here, with the Greek opposition of proof and persuasion – fueled by the quest for incontrovertibility – could hardly be more striking’ [G.E.R. Lloyd, Demystifying Mentalities (Cambridge: Cambridge University Press, 1990), pp. 121–22]. ‘And that is the main point’, Cullen observes after noting Lloyd’s comments: ‘even when an ancient Chinese mathematician gives a proof, it is not very important to him in comparison with his real aim of explaining the use of the methods he is expounding to solve specific problems.’ He adds (p. 89), ‘Why should it
be otherwise?’
9 The hypotenuse rule was well known to Greek authors who lived a century or so after Pythagoras, and none of these authors attributes it to Pythagoras. Aristotle – who is good about attributing credit where it’s due – also knew the proof, but says nothing of any tie to Pythagoras. The idea of a proof began to emerge in the fifth century bc, and culminated in the fourth with Plato’s discussion of the distinction between persuasion and demonstration, with Aristotle’s discussion of the nature of proof, and finally with Euclid’s Elements, a book that presents mathematical knowledge entirely in the form of proofs. There remains a major difference between the early writings exhibiting knowledge of mathematical rules in obtaining practical results, and the later Greek idea of formal proof. ‘Practice is one thing, having the explicit concept another’, writes Lloyd. ‘[T]o give a formal proof of a theorem or proposition requires at the very least that the procedure used be exact and of general validity, establishing by way of a general, deductive justification the truth of the theorem or proposition concerned’ (Lloyd, Demystifying Mentalities, pp. 73, 74). This, Lloyd continues, was first defined, as far as we know, not just in Greece but anywhere, by Aristotle. Though some individuals make claims for earlier discoveries of the Pythagorean theorem, in Mesopotamia, India, and China, for instance, ‘[I]n the key texts we find no distinction observed between exact procedures and approximate ones. Both are used apparently indiscriminately, and that suggests that their authors were not concerned with proving their results at all, but merely with the concrete problems of altar construction’ (p. 75). It is true that the first proof of the hypotenuse formula is traditionally ascribed to Pythagoras (ca. 569–475 bc), by authors who lived about half a millennium later, around the time of the birth of Christ. But this attribution may well be, as Lloyd remarks, the result of the tendency of ‘the late Greek commentators to make overoptimistic attributions of sophisticated ideas to the heroic founders of Greek philosophy’ (p. 80). The culprit seems to be a certain Apollodorus, about whom nothing is known except his remark that Pythagoras sacrificed oxen upon discovering a ‘famous theorem.’ Apollodorus’s remark was then relied on by many other authors – who include Plutarch, Athenaeus, Diogenes Laertius, Porphyry, and Vitruvius. Some authors embellished the story, while others express skepticism about the sacrifice, given that the Pythagoreans had strictures against rituals in which blood was shed. ‘What is both uncontroversial and of first rate importance for the subsequent development of Greek science’, Lloyd concludes (p. 87), ‘is the role that Euclid’s Elements itself had as providing the model for the systematic demonstration of a body of knowledge. Thereafter proof more geometrico became all the rage, and not just in geometry, but also for example in optics, in parts of music theory, in statics and hydrostatics, in parts of theoretical astronomy, and not just in the would-be exact sciences, but in some of the life sciences as well.’
A Brief Guide to the Great Equations Page 27