A Brief Guide to the Great Equations (38 page)

We can bring the strange home, and bring it home with precision.

I have referred to the paths to these equations as journeys, but that metaphor can be misleading. It can mislead because it suggests smooth and steady progress toward a stable and predetermined destination, whereas the path to understanding that culminated in most of these equations was uneven and the travelers often wound up in a different place from where they thought they were heading. The metaphor also falsely suggests that the travelers were spectators taking in a vision of nature, rather than active participants in interactions with it who learned from their changing interactions and often changed their ideas in response.

But the metaphor does capture the way each step of the journey readjusted the perception of the travelers as new sights appeared and others disappeared, and as the overall landscape reorganized around new landmarks. What the travelers thought was important subtly changed as a new world slowly came into view. Such changes were not due to any specific development – to any single distinction, discovery, technique, or person – but to the journey itself. This is what philosophers mean by the historicity of human action. Each group of travelers inherits a landscape, a way of thinking, an accompanying set of dissatisfactions, and a direction to head in to resolve these dissatisfactions, and in the resulting journey the landscape is taken
up and transformed. At each step along the way the world seems to have a wild heterogeneity, possessing one order that does not seem inherent in the way the world appears to us – for the order we see in nature is due to our previous explorations and journeys – but to possess hints of another, inherent order that we might be able to see more clearly through inquiry. What Heaviside said of Maxwell’s work – ‘[I]t was only by changing its form of presentation that I was able to see it clearly’ – could be said by any of the individuals mentioned in this book. These individuals were discontent with what they saw, had an anticipatory vision of what might be, and the ability to organize an inquiry to seek it (philosophers call this process the hermeneutic circle). There will be no ‘final’ stopping point to the journey, for each new discovery – not to mention changing practical, instrumental, and theoretical contexts – works changes in the landscape. We will never stop being discontent, never stop anticipating, never stop organizing inquiry. Science could not happen in any other way, or it would be trivial or impossible.

Most of the time, however, we care more about the equations, and about the things they help us do, rather than about the journeys that led to them. We tend to pay attention to the part of the world directly beneath our gaze. This is understandable and there are good reasons for doing so. But we can also learn much from studying the journeys – paths from ignorance to knowledge – that the scientific community, and individual scientists, took to these equations.

First of all, we learn that the journeys are very different. Some journeys described in this book are short enough to take an individual just a few minutes. The Pythagorean theorem is an example; it allows someone with no mathematical training not only to grasp it but also to experience the thrill of discovery. Other journeys are extended: the journey to
F
=
ma
and
F
_g
=
Gm
₁
m
₂
/
r
²
justly can be said to have taken hundreds, even thousands, of years. Some journeys were taken in effect by a community of scientists constantly talking to one another, such as those to
E
=
mc
²
, to the second law of thermodynamics, and to the uncertainty principle.
Other journeys were traveled more or less solo, such as Einstein’s to his general equation for gravitation and Schrödinger’s to his wave equation, though such individuals in effect carried on conversations with colleagues even when working alone. There is, in short, no single road to discovery.

We also learn that equations are not simply scientific tools but have ‘social lives’, so to speak. We tend to view equations as inert and mute instruments, able to affect the world only when wielded by scientists and engineers. But equations are active and can exert an educational and even cultural force, instructing us about the world and occasionally reshaping the human perception of it. The Pythagorean theorem teaches each new generation of schoolchildren about the meaning of proof, while Newton’s law of gravitation taught certain political thinkers about the meaning of laws. The second law of thermodynamics helps keep in check humanity’s utopian visions of free energy, while Einstein’s
E
=
mc
²
and his general equation for gravitation reshaped the human understanding of space and time on a fundamental level. Schrödinger’s equation and Heisenberg’s uncertainty principle force us to rethink what being a ‘thing’ means.

Yet another thing we learn from these journeys is about the nature of scientific concepts. It is tempting to think that there is some pre-existing structure embedded in nature that we are only discovering and translating into mathematical language – that equations are descriptions rather than interpretations or creations. But how we translate depends on the journey we have already taken, on our dissatisfactions with it, and on how we responded to those dissatisfactions. We ‘fall up’, to adapt a phrase of George Steiner. It is thus misleading to picture science as proceeding solely by scientists producing new concepts, then testing and revising them. Two things are wrong with this picture. One is that the meaning of one concept depends on the meaning of all the others; a concept is one element of that fishbowl-like world that Newton discovered at the heart of the world we live in, and needs everything else in that fishbowl for its meaning. Testing one concept thinking you are testing it and not
everything else is like asking, Is New York to the right or left of Boston? without knowing where you are; without having the rest of the map. And we not only need the rest of the fishbowl, but the rest of our experience of the world as well. A scientific concept that we trust is really a concept plus that experience, and when our experience changes – new practices, new technologies – so does how the concept applies to the world. That’s why concepts never stay put, and always change or are being elaborated; a concept that tests right at one time can be inadequate at another. There is no right way to say something that does not include our experience with it. Concepts are thus not determinative but indicative; they ‘point’ to something based on our experience, in full awareness that what they point to is going to change with further inquiry. Philosophers call this ‘formal indication.’ Concepts are formal because we can evaluate them rigorously and test them as being adequate or inadequate; they belong to a closed system. Concepts are indicative because they point to and depend on other things for their adequacy – all our experiments and definitions and technology and open-ended connection to the world – and when these change so can the formal elements as well. In fact, we expect that it will. Historian of science Peter Galison has a wonderful description of this. It is the theorist’s experience, he writes, that:

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.

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.