Great Equations

This is a book by Robert P. Crease, subtitled The hunt for cosmic beauty in numbers, and it takes the reader from Pythagoras’s Theorem to quantum mechanics. In so doing, it pretty well covers the whole of Western physics – it's as much history as it is exposition – which makes it an ideal introduction for anyone with only a passing knowledge of physics and mathematics.

Crease takes us from Euclid to Newton, Euler, Boltzmann, Maxwell, Planck, Einstein, Schrodinger and Heisenberg. Even though he jumps from Euclid to Newton (chapters 1 to 2) he includes others who played a significant role: in particular, Plato, Aristotle, Copernicus, Galileo, Kepler and Descartes. It’s this historical context that gives the book a semblance of narrative, albeit an episodic one, and provides an appeal that may go beyond scientific nerds like myself. There is, in fact, very little mathematics in the book, yet he explains the physics behind the equations with eloquence and erudition. That’s quite an accolade, considering he covers the most seminal scientific discoveries and equations in all of Western history.

Only 2 of the ‘great equations’ are pure mathematics, the other 8 (4 of which are 20th Century) are all physics equations. The 2 exceptions are the well-known, and erroneously titled, Pythagoras’s Theorem (a² + b² = c²), and the lesser known, but no less iconic, Euler’s identity (e^iπ + 1 = 0). Euler’s identity is technically not really an equation because it contains no variables, and it’s derived from Euler’s equation: e^ix = cosx + isinx. But no book of ‘great equations’ could leave it out.

Pythagoras’s equation, as it relates to right angle triangles, was well known centuries before Pythagoras, and was discovered independently in various cultures, including India, China and Egypt. But even though the proof may well have been developed by Pythagoras or his school, it is Euclid’s proof that is best known. In fact, Euclid’s famous Elements, as Crease points out, is the first known work to provide mathematical proofs from stated axioms and became the standard by which mathematics has been mined ever since.

One of the historical and philosophical points that Crease makes is that, during the period from the Ancient Greeks to Newton, there were 2 recognised sources of knowledge and it was only during the renaissance that a conflict first arose, epitomised by Galileo’s famous clash with the Catholic Church.

What is not so well known is that Euclid’s Elements was the second most published book after the Bible following its initial typesetting in Venice in 1482. I find it most interesting that a mathematical volume should contest the Bible as a source of ‘truth’, during a period when Christianity was, politically, the most powerful force in Europe. Half a millennia on, this conflict still exists for some people, yet, for most of us, there is simply no contest, epistemologically.

Mathematics is a source of truth that no religious writings can match, because religious scriptures (of any persuasion) are completely open to diverse interpretations, dependent on the reader, whereas mathematical truths are both universal and epistemologically independent of the individual who discovers them.

Crease covers 2 of Newton’s equations: the second law of motion (F = ma) and the universal equation of gravitation (F_g = m₁m₂G/r²). Newton transformed the way we perceive abstract qualities like force, energy and gravity, which are, nevertheless, all tangible to our everyday experience. It was Newton’s discovery and consequential deployment of calculus (he called it fluxions) that opened up this world of physics from which we’ve never looked back. Despite the consequential discoveries of people like Maxwell, Planck, Einstein, Schrodinger and Heisenberg (all covered in this book) Newton’s equations are no less significant today than they were in his time, and no less relevant as humankind’s exploration of the solar system has demonstrated.

Euler’s identity is arguably of less significance to our everyday understanding of the universe (than Newton’s mathematical discoveries) yet no one who comes across it for the first time and appreciates its deep profundity can help but be gobsmacked by it. In one succinct formula it pulls together so many strands of mathematics: logarithms, trigonometry, calculus, power series and complex algebra. It’s all the more impressive when one realises it’s made up of 2 infinite series, that when combined gives the most unlikely relationship in mathematics between rational, irrational and imaginary numbers. The equation, as opposed to the better known identity (that is effectively a special case) is central to Schrodinger’s equation, developed a couple of centuries later.

Euler’s identity seems to encapsulate mathematical truth, which is why it has gained iconic status. As Richard Feynman wrote just months before his 15th birthday, when he first discovered it: ‘[it is] The most remarkable formula in math.’ Like the great Indian mathematician, Srinivasa Ramanujan, who also discovered it whilst still in high school, Feynman was disappointed to learn that Euler had made the connection a couple of centuries earlier. It’s not for nothing that it’s earned the title, God’s equation.

No book on great equations could leave out Einstein’s famous equation (E = mc²) that is a direct consequence of his special theory of relativity, and Crease provides a good exposition of how the theory developed and its logical consequence from a conceptual conflict between Maxwell’s equations and Newton’s mechanics. Crease also captures the other players like Fitzgerald, Lorentz and Poincare, which makes us realise that Einstein’s theories would have eventually evolved even without Einstein. But it was Einstein’s ambitious thought experiments that set him apart from his contemporaries and led him to the iconoclastic theories that history deservedly gives him credit for.

I’ve skipped over Maxwell’s equations and the second law of thermodynamics, yet in both cases, Crease points out that these discoveries transformed life as we know it. One was essential to the industrial revolution and the other to the communications industry that followed. He makes the salient point that few people appreciate the significance of great scientific discoveries and their impact on so-called civilisation the way they appreciate political changes and acts of war. He quotes Feynman, who once claimed that Maxwell’s equations would come to have a greater historical significance than the American Civil War, both being products of the 19th Century.

On my 50th birthday I was given a copy of Peter Watson’s A Terrible Beauty; A History of the People and Ideas that Shaped the Modern Mind. This ambitious book covers the entire 20th Century and was published to coincide with the dawn of the new Millennium. But, instead of covering the politics and wars that enveloped that century, Watson concentrated on the science and art, which he wrote about with equal erudition. It’s an extraordinary book and a great birthday present. I read it over a year, whilst travelling and working in North America, simultaneously writing my only published novel.

I mention Watson’s book because it encapsulates a point that Crease makes more than once: how the importance of scientific erudition often gets lost when scholars examine the history of the Western world. He makes this point specifically in regard to the 2 aforementioned 19th Century discoveries: Maxwell’s equations and the second law of thermodynamics.

Crease quotes Max Born in his introduction to Einstein’s equation of his General Theory of Relativity, who compared it to a work of art. I have to admit that was how I considered it when I first read about it in Einstein’s own words. I confess that I didn’t follow the physics and the mathematics at the time, yet I appreciated its significance and its beauty. Conceptually, Einstein realised that a falling body feels no force, which appears to contradict Newton’s formulation. He reformulated it so the motion of a falling body is a consequence of the geometry of space-time that is curved as a result of the existence of mass. This is an extraordinary intellectual achievement, especially when one realises that his equation maintains Newton’s inverse square law, thereby only disagreeing with Newton on relativistic grounds. Even the word genius sometimes seems inadequate when you apply it to Einstein; such was his vision, bravado and intellectual tenacity.

I wrote an exposition on Schrodinger’s equation back in May, and I’m proud to say it’s become my most popular post, though it’s strictly an introduction. One thing Crease does better than me is to explain the dichotomy between quantum mechanics and classical physics. In particular, he contends that the so-called collapse of the wave function is conceptually misleading. He argues that the wave function is a convenient mathematical device, like a plot device in a narrative (my analogy, not his) that no longer serves any purpose once a measurement is made. The wave function gives a probability that is confirmed statistically over many measurements, but determines nothing specific for a specific event.

The last great equation in his book is Heisenberg’s uncertainty principle expressed mathematically. By juxtaposing it with Schrodinger’s equation in the previous, second-to-last chapter, Crease demonstrates how the 2 antagonists used different mathematics to reach the same result. In other words, Schrodinger’s wave mechanics and Heisenberg’s matrices are mathematically equivalent (proven by Schrodinger), yet the different approaches led to arguments about what they meant conceptually and philosophically. Interestingly, Born played a key role in both interpretations: he realised that Schrodinger’s wave mechanics led to probabilities; and he realised that Heisenberg’s non-commutative algebra could be reformulated using matrices, and this led to the precursor of Heisenberg’s uncertainty principle with the conclusion pq does not equal qp, where p represents momentum and q represents position of a particle.

Crease’s account exemplifies the importance of ideas being challenged in their formative stages by people of comparable knowledge, and how the interaction between philosophy and science is a necessary factor in the advancement of scientific theories.

Like Schrodinger’s equations, Heisenberg’s uncertainty principle makes predictions that can be confirmed experimentally, yet the predictions can never be specific. Both equations, in different ways, highlight the inherent fuzziness that differentiates quantum mechanics from classical physics, whether it be Newtonian or relativistic. In fact, quantum mechanics and Einstein’s general theory of relativity have never been satisfactorily resolved, with the best contender being String Theory requiring 11 dimensions and predicting 10⁵⁰⁰ universes.

In his conclusion, Crease emphasises how the discovery process for theoreticians often involves a re-evaluation of what they set out to achieve. No where is this more apparent than in the early 20th century when physics underwent 2 revolutions, the epistemological and ontological consequences of which are still unresolved today.