A long historical and conceptual journey was required, therefore, to write even the simplest of equations. In 1910, Alfred North Whitehead and Bertrand Russell, two of the greatest mathematicians in history, published the
Principia Mathematica
, a famous, three-volume systematic textbook that derives the foundations of mathematics from the ground up in a purely logical way. When does the equation 1 + 1 = 2 first make its appearance? Well over halfway through volume one!
4
Thanks to this long journey, the word ‘equation’ eventually came to have a technical meaning as part of a specially constructed language – to refer to a statement that two measurable quantities, or sets of measurable quantities, are the same. (Strictly speaking, then, statements expressing inequalities are not equations.) In this codelike constructed language, indispensable to modern mathematics and science, symbols stand for sets of other things on which various operations (addition, subtraction, multiplication and division being the simplest) can be carried out.
5
Ever since this special technical language was developed, each individual equation has had two different types of discoveries. It was originally discovered by the first person to come across it – by the person or persons who introduced it into human culture. And it is rediscovered by each person who learns it since.
The journey to a particular equation has a different kind of setting than that of other historical turning points. The appearance of equations is not framed by bloody battlefields or by clashes of titanic political forces. Equations tend instead to emerge in quiet locations, such as studies and libraries, removed from distractions and encroachments. Maxwell wrote down his world-transforming equations in his study; Heisenberg began to piece together his on an isolated island. Such environments allow scientists to address their dissatisfactions, to explore the gnawing sense that the pieces at hand are not fitting together well and need some adjustment or the addition of something new. Scientists then can focus on some problem that often can be articulated with deceptive simplicity: What is the length of this side of a right-angled triangle? What is the strength of the force between celestial objects? How does electricity move? Can a given pair of seemingly contradictory theories be made to fit together?
Does this make sense
?
When the solution comes, it seems logical and even inevitable. This work is ‘universally received’, writes Roger Cotes, who
contributed a Preface to the second edition of Newton’s famous masterwork,
Mathematical Principles of Natural Philosophy
.
6
The discoverers often feel as if they’ve stumbled across something already there. Thus equations seem like treasures, spotted in the rough by some discerning individual, plucked and examined, placed in the grand storehouse of knowledge, passed on from generation to generation. This is so convenient a way to present scientific discovery, and so useful for textbooks, that it can be called the treasure-hunt picture of knowledge. It telescopes a difficult process and leaves us with an inventor, time, and place, and often a cause or purpose. An incident or moment, such as the fall of an apple, becomes a synecdoche that crystallizes the long discovery process. Generations of scholars then earn reputations criticizing the model and complicating this picture. The treasure-hunt picture is useful for everyone!
The treasure-hunt picture of the world, however useful, promotes the view that equations are essential features of the world, not created by human beings. And indeed, we are born into a world that already has equations that ‘we’ did not create. This is why equations sometimes appear to be not really of human origin, around long before we humans got here: On the eighth day, God created the equations, as the blueprint for His work. Or, as Galileo wrote, the Book of Nature is written in mathematical symbols.
But each and every equation had a human genesis. It was put together by a particular person at some specific place and time who felt a need – who was dissatisfied by what was at hand – and who wanted to make sense of things or sometimes merely wanted to make something that appeared hopelessly complicated easier to understand. Sometimes this creative process is buried in antiquity, as is true of the ‘Pythagorean’ theorem, whose principle was known long before Pythagoras. Sometimes the creative process is known in detail thanks to the correspondence, drafts, and notebooks of their inventors, as is the case with equations produced by Newton and Einstein. In each case, however, the equations cannot be said to be their work alone, for these scientists – even when working alone –
were involved in countless dialogues with other scientists in a shared process to make sense of nature.
When British scientist Oliver Heaviside rearranged Maxwell’s work into what is essentially their now-famous form – into the form that today is known as ‘Maxwell’s equations’ – he remarked that he was simply trying to understand Maxwell’s work more clearly. That motivation – sensing that one can express better something that one already knows, but vaguely – might be said of all inventors of equations.
After someone does come up with a new equation about some fundamental issue – when that person has answered his dissatisfaction – it changes both us and the world. Such equations thus do not simply instruct us how to calculate something, adding new tools to the same world, but do something ‘more’, as Harrison put it. In learning 1 + 1 = 2, his son did not merely input a new data point, but became transformed, possessing a new grip on the world. But along with this new grip comes new puzzles, and new dissatisfactions.
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Harrison’s description, finally, reminds us that equations can inspire wonder. Science is not a robotic activity in which we manoeuvre in or gaze at the world indifferently, but a form of life with a highly nuanced affective dimension. There is, of course, the celebratory, cork-popping joy that is a natural concomitant to a new discovery or achievement. But if that were the only emotion involved in science – the pleasure of making a discovery that ensures fame and fortune – it would be a sorry profession, for such moments are few and far between. Fortunately, the emotions of science are much more diverse, and thicker, than that. Doing science is accompanied by unfolding feelings at every moment – puzzlement, bafflement, curiosity, desire, the urge to find the answer, boredom that nothing is happening, frustration at getting nowhere, the thrill of being on the right track. Such feelings are always present, not deeply hidden, often overlooked, but easy to notice once we decide to pay attention to them.
When we understand an important equation for the first time, we glimpse deeper structures to the world than we suspected, in a way that reveals a deep connection between the way the world is and how we experience it. At such times, our reaction is not simply, ‘Yeah, that makes sense’, or even what is often called the ‘Aha!’ moment. This latter crude characterization goes hand in hand with the treasure-hunt picture of knowledge acquisition, for it simplifies and condenses the emotion of discovery into a single instant. The genuine emotion – wonder – is subtler, richer, and lengthier.
It is natural, though, even for scientists to stop wondering at equations, as they become more wrapped up in the world and their interests in it, and less attentive to the moments of disclosure in which its forms first appear. We lose wonder, indeed, at any instrument or object with which we grow too familiar. Equations can come to seem as just another set of tools that we find lying about in the world, or as onerous chores that we learn out of duty.
Pilots who learn too much about their craft, Mark Twain writes in
Life on the Mississippi
, often undergo a regrettable transformation. As they become increasingly skilled at reading the language of the river, they seem to grow correspondingly less able to appreciate its beauty and poetry. Features of the river – a floating log, a slanting mark on the water, a patch of choppy waves – that once aroused feelings of wonder and awe become increasingly appreciated only instrumentally, in terms of the use they have for piloting. Something similar is true of equations.
But great scientists are often still able to marvel at the breakthroughs of their predecessors. The physicist Frank Wilczek once wrote a series of articles on the simple equation expressing Newton’s second law of motion,
F
=
ma
, calling it ‘the soul of classical mechanics’, and exhibiting toward it the kind of appreciation that is appropriate to souls.
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The physicist and cosmologist Subrahmanyan Chandrasekhar wrote an entire book on Newton’s
Principia
, the book in which Newton proposed his second law of motion, comparing it to Michelangelo’s painting on the ceiling of the Sistine
Chapel. And a listener to Richard Feynman’s famous
Lectures on Physics
can detect throughout his unabashed and spontaneous wonder at the equations he is trying to teach his students. These three Nobel laureates each knew enough to maintain their wonder at the world and at the equations through which we know it.
This book aims to show that there is much more to equations than the simple tools they seem to be. Like other human artifacts, equations have social significance and exert cultural force. This book takes some great equations and provides brief accounts of who discovered them, what dissatisfactions lay behind their discovery, and what the equations say about the nature of our world.
DESCRIPTION
: The square of the length of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the lengths of the other two sides.
DISCOVERER
: Unknown
DATE
: Unknown
To this day, the theorem of Pythagoras remains the most important single theorem in the whole of mathematics.
The original journey to the Pythagorean theorem is forever shrouded in history. But we have countless stories of its rediscovery, both by people who taught it and by people who rediscovered it for themselves. These sometimes have been such powerful experiences as literally to have changed the lives and careers of those who have made them. The power and magic of the Pythagorean theorem arise from the fact that, while it is complex enough that its solution is not apparent at the outset, the proof process is condensed enough to constitute a single experience.
One person whose life it changed was the great political philosopher Thomas Hobbes (1588–1679). Until he was forty, Hobbes was
a talented scholar who showed little originality. He was well versed in the humanities but dissatisfied with his erudition. His principal achievement was an elegantly written if sometimes inaccurate translation of the ancient Greek historian Thucydides. He had little exposure to science, despite the exciting recent breakthroughs of Kepler, Galileo, and others, which were then revolutionizing the scholarly world.
One day, while passing through the library of an acquaintance, Hobbes saw a copy of Euclid’s
Elements
displayed on a table. This was not unusual: a gentleman who owned a handsome and expensive volume of an important work, such as a Bible, would not store it out of sight but would prominently exhibit it for the benefit of visitors, usually opened to a famous passage or psalm.
Euclid’s
Elements
was indeed like a Bible. It set out much of the mathematical wisdom of its time in axioms and postulates; scholars had been analyzing it ever since its appearance in about 300 bc; and its knowledge remained current. No other book at the time, except the Bible, had been as frequently copied or studied. The particular chapter and verse that Hobbes saw was Book I, Proposition 47, the Pythagorean theorem.
Hobbes took a look at the claim: The square described upon the hypotenuse of a right-angled triangle is equal to the sum of the squares described upon the other two sides. He was so astounded that he used a profanity that his acquaintance and first biographer John Aubrey refused to spell out: ‘“By G—”, Hobbes swore, “this is impossible!”’
1
Hobbes read on, intrigued. The demonstration referred him back to other propositions in the same book: Propositions 46, 14, 4, and 41. These referred to still others. Hobbes followed them and was soon convinced that the startling theorem was true.
2
‘This made him in love with geometry’, writes Aubrey, adding that Hobbes was a changed man. He started obsessively drawing geometrical figures and writing out calculations on his bedsheets and even on his thigh. He began to devote himself to mathematics, showed some talent – though his abilities remained modest – and
embroiled himself in controversies and hopeless mathematical crusades in a manner that still embarrasses his biographers and fans.
3
These episodes are not terribly interesting. What matters is that the theorem transformed him and his scholarship. As one commentator wrote of Hobbes’s initial encounter with the Pythagorean theorem, ‘everything he thought and wrote after that is modified by this happening.’
4
Hobbes began to chastise the moral and political philosophers of the day for their lack of rigor and for being unduly impressed by their predecessors. He compared them unfavorably with mathematicians, who proceeded slowly but surely from ‘low and humble principles’ that everyone understood and accepted. In books such as the
Leviathan
, Hobbes began to reconstruct political philosophy in a similar way, by first establishing clear definitions of terms, and then working out the implications in an orderly fashion. The Pythagorean theorem taught him a new way to reason, and to present the fruits of his reasoning persuasively, in ways that seemed necessary and universal.