Readers, I hope, will forgive me for a shameless bit of self-publicity about my latest book, The Great Equations: Breakthroughs in Science from Pythagoras to Heisenberg (Norton). But then the book is partly yours too, inspired as it was by the responses of Physics World readers to my request for suggestions of great equations (see “Critical Point: The greatest equations ever”). In the book, I chose to discuss not the most frequently mentioned equations, but those that seem to have engaged their discoverers in the most remarkable journeys.

The journey metaphor may seem misleading if taken to suggest smooth and steady progress to an already known destination. The scientific journeys I recount — which include those culminating in F=ma, and the equations of Maxwell and Schrödinger — were unpredicted, often protracted and erratic. The journey metaphor should also not imply that the travellers passively observed the changing scenery; in fact, the scientists interacted with their environment while altering it.

But the journey metaphor does capture one important aspect of the birth of these equations, which is how their originators’ ideas about what was important changed during the course of their research. Newton, Maxwell, Schrödinger and others each inherited a “landscape” or view of how knowledge about nature was organized. But during their research, new concepts — such as mass and force, entropy and displacement current, quanta and wave equations — appeared on the horizon, grew in importance and displaced others to assume positions as indispensable landmarks in the conceptual landscape.

For the ultimate destination of such scientists was not a particular location that they saw beforehand, but clarity. They were dissatisfied with what they had, perceived a vision of what might take its place, and were able to carry out the inquiry needed to realize it. At each step, they found the world to be somewhat discordant — not fully grasped — with hints of another, deeper order just over the horizon. This discordance is what makes newly realized equations seem, strangely, to be both discovered and invented.

Oliver Heaviside, who transformed Maxwell’s then-convoluted equations into their now-familiar versions, once remarked that “it was only by changing its form of presentation that I was able to see it [electromagnetism] clearly”. The sense of that remark — you transform to clarify — could have been said by any of the scientists mentioned in The Great Equations.

No royal road

Most of the time we are less interested in journeys than in where they take us. But we can learn much from them. One is just how varied such journeys are. Sometimes they are taken by scientists who talk and argue constantly with one another, as with the equations of thermodynamics and the uncertainty principle. Other journeys were undertaken by individuals working essentially by themselves, such as Einstein in his path to general relativity and Schrödinger to his wave equation, though such individuals in effect carried on conversations with colleagues even when working alone. There is no royal road to discovery.

Another thing we learn is that equations are not simply inert tools that work only in the hands of scientists and engineers. They can also exert an educational and even cultural force that shapes our view of the world. The Pythagorean theorem teaches us what proof means, the second law of thermodynamics keeps in check our dreams of free energy, Einstein’s equations changed our understanding of space and time, and the work of Schrödinger and Heisenberg forces us to rethink what being a “thing” means.

We also learn to appreciate how deeply affecting the scientific life can be. The scientists who took those journeys were never blasé, never disinterested. They were infused with curiosity, consternation, bafflement, frustration and wonder. And each scientist had what might be called a particular style. Some succeeded because they were only satisfied when they found what they were looking for, while others succeeded only because they were prepared to see something more than they expected.

Most of all, the journeys allow us to glimpse the mutability of nature and our role in it. The journeys teach us that nature could be otherwise — that it was otherwise for us until a moment ago, and for all we know it could change in the future. In such instances, we experience a transcendent moment in which a higher thought emerges in the middle of an existing one.

The critical point

The Great Equations ends by relating a conversation I had while writing the book, with an elderly physicist who expressed little comprehension and sympathy. To his workmanlike mind, the equations I mentioned seemed so obvious and logical that he could not picture not having known them, and he saw no value in making them more enigmatic. “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 him. At that moment, I finally realized exactly what I was trying to do. It was exactly the opposite — to undo that sense of obviousness and triviality, and to take readers back to the moment just before the equations were discovered, to appreciate how untrivial they are. Readers could, I hoped, thereby relive the wonder of the moment when the equations were first grasped — when they seemed simultaneously discovered and invented.

Scientists such as my physicist acquaintance tend to focus on the formal, discovered — what he meant by “trivial” — aspect of the birth of equations, whereas philosophers and historians tend to focus on the other aspect, having to do with their invention. It ought to be possible, I felt, to capture both aspects at once — which would, I thought, finally provide a more complete picture of the discovery process itself.