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 4
x
+ 3
y
= 11 and
y
= 2
x
+ 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
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.’
10
To name one, Otto Neugebauer, who first deciphered the
Pythagorean triplets of Plimpton 322, cited old Babylonian tablets as ‘sufficient proof that the ‘Pythagorean’ theorem was known more than a thousand years before Pythagoras.’ Otto Neugebauer,
The Exact Sciences in Antiquity
(Providence: Brown University Press, 1993), p. 36.
11
Francis M. Cornford,
Before and After Socrates
(Cambridge: Cambridge University Press, 1972), pp. 72–73.
12
American Mathematical Monthly
1, no. 1 (January 1894), p. 1.
13
Elisha S. Loomis,
The Pythagorean Proposition: Its Proofs Analysed and Classified
. Publ. by The Masters and Wardens Association of the 22nd Masonic District of the Most Worshipful Grand Lodge of Free and Accepted Masons of Ohio, 1927; and
The Pythagorean Proposition: Its Demonstrations Analysed and Classified
(Ann Arbor, MI: Edwards Brothers, 1940). He ended the first book thus: ‘FINAL THOUGHT: Is it an all-embracing truth? The generalization of the Pythagorean Theorem so as to conform to and include the data of geometries other than that of Euclid, as was done by Riemann in 1854, and later, 1915, by Einstein in formulating and positing the general theory of relativity, seems to show that the truth implied in this theorem is destined to become the fundamental factor in harmonizing past, present and future theories relative to the underlying laws of our universe.’