What makes a great equation? Robert P Crease seeks your candidates and criteria
In It Must Be Beautiful: Great Equations of Modern Science, Graham Farmelo assembled and edited essays on 11 great equations of the 20th century. Six are from physics: E = hf, E = mc2, Einstein’s general-relativity equation, the Schrödinger wave equation, the Dirac equation and the Yang-Mills equation. The other five include the Drake equation on the likelihood of us forming radio contact with extraterrestrial life, and Shannon’s equations on information transmission.
Farmelo shuns defining greatness in equations, but compares them with poems. Both are composed of abstractions with which we address the world, even though many individual terms do not refer to anything specific. While “poetry is the most concise and highly charged form of language”, he says, equations are “the most succinct form of understanding of the aspect of physical reality they describe”. We sense greatness in equations as well as poems, even though we do not have an objective measure for it.
Wardrobe numbers and overcoats
Whether a particular equation is “great” obviously has something to do with the properties of the equation itself, such as simplicity and symmetry. This would seem to favour three-letter equations like F = ma, E = hf and E = mc2.
But clearly other criteria come into play as well. For example, we have to consider the relationship of the equation with the world – otherwise it would just remain hieroglyphs. Uniquely, equations do not refer directly to things but to quantities measured from special situations staged in the laboratory. Force, energy, time or acceleration do not lie around like ordinary objects; measurements of these quantities have to be “read off” from events that have been specially conceived, prepared and systematized. The relationship between a measurement and what it measures is thus not like that of a word to an object, but – as Einstein once remarked – more like that of “wardrobe number to overcoat”. Equations, as it were, link the wardrobe numbers to one another – what links wardrobe numbers to overcoats is laboratory preparation and measurement.
To use an analogy of the science philosopher Patrick Heelan, a laboratory is like a garden where special kinds of things are grown in an environment that is isolated (although never completely) from the life outside. The special things that emerge within the laboratory walls are thus artifacts – like greenhouse orchids – which may exist only momentarily, but their properties help us to understand and explain that wider and wilder external life. The laboratory creates the conditions under which special things appear that show themselves as structures of the world. Neither the world inside nor outside the laboratory is a static environment, however; both are mediated by technology and continually changing, which allows new things, concepts and interests.
Whether an equation is great, it seems to me, has something to do not only with the properties of the equation, but also with the scope and depth of the phenomena to which it refers. This would favour equations dealing with fundamental things such as space, time, fields and energy. Great equations can seem to be “wiser even than their discoverers” about such fundamental things, as Hertz said of Maxwell’s equations, for “we get more out of them than was originally put into them”. This is why, Hertz felt, that “mathematical formulae have an independent existence of their own”.
Inside and outside science, furthermore, equations can acquire what one might call a cultural “flesh”. That is, equations are more than bare and abstract scientific tools but can develop a lore, history and meaning of their own. This can happen to even the
most elementary of equations. In US high schools, students are often reminded of Ohm’s law using the phrase “Rhode Island equals Vermont”. During the Second World War, faced with the urgency to churn out radio operators speedily, one radar school outside Chicago taught it in three versions – V = IR, I = V/R and R = V/I – because it was faster to teach these equations than it was to teach trainees how to manipulate equations.
Equations can exemplify epistemological and moral lessons about science, and appear to be signposts toward entry into nature’s deeper mysteries. The convoluted story of Schrödinger’s equation, and its competition with Heisenberg’s matrix mechanics, is sometimes used to instruct physics students on the different ways of practising science.
In popular culture, meanwhile, E = mc2 has come to stand for science – even human knowledge – itself. It is a staple of popular cartoons and images of science, and even turned up in the recent movie School of Rock, in which a washed-out rock musician is stuck teaching junior-high-school kids.
The French intellectual Roland Barthes observed that while photographs of Einstein often show him next to a blackboard covered with impenetrable symbols and equations, cartoons often portray him, chalk in hand, next to a clean blackboard on which he has written down this particular formula as if it had just come to him. Barthes observed that this equation restores the image of “knowledge reduced to a formula…science entirely contained in a few letters”. It has become a Gnostic image symbolizing knowledge at once ultimate and esoteric: “The unity of nature, the ideal possibility of a fundamental reduction of the world, the unfastening power of the word, the age-old struggle between a secret and an utterance, the idea that total knowledge can only be discovered all at once, like a lock that opens after a thousand unsuccessful attempts.”
The critical point
Equations were not always as prominent in, nor as identified with, scientific methodology as they are now. At the beginning of modern science, laws such as Galileo’s law that we now express in the form of equations (d = 1/2at2) were expressed through proportionalities (d α t2). One may well wonder about the role played by the increasing centrality of equations in the advance of science.
The equations in Farmelo’s book are all from the 20th century. Which equations would be on the list if it were expanded to include the greatest equations of all time? I invite you to send me your candidates, the reasons why they deserve to be on the list, and what value, if any, you find in discussing their greatness. I shall report on the results in a future column.
• What is your shortlist of the greatest equations in science, and what makes them great? Send your thoughts to Robert P Crease at the address or e-mail given below, or by fax to +1 631 632 7522.