A Brief Guide to the Great Equations (15 page)
Euler’s equation, too, emblematized the way that its author had recast mathematics. Mathematics, like other sciences, does not develop along a predetermined track, but follows a historically contingent path in which each generation of scientists inherits assumptions, techniques, and concepts from the generation before, transforms them, and passes them on in turn. Thanks to this process, we perceive the field as structured in a particular way, as having a certain ontology, with different phenomena assigned to distinct domains. Every equation implicitly refers to this inherited structure. But Euler rearranged this ontology, reorganizing it so that analysis was at the centre, with geometry and algebra as neighbourhoods. Looking backward, mathematicians may take the latest organization as self-evident – which, no doubt, is why the mathematician Carl Friedrich Gauss is said to have remarked that anyone to whom
e
i
π
+ 1 = 0 is not obvious is not a mathematician. When you are fully literate, nothing comes as a surprise. But mathematicians are made not born; in infancy they are not yet mathematicians, and have to learn it – and in such learning often experience extensive transformations and reorganizations of mathematical knowledge that they have only partially acquired. The brief formula
e
i
π
+ 1 = 0 is the most succinct expression of this process.
14
There is yet one more, still deeper reason why this formula has become an icon. As Devlin once wrote of Euler’s equation, ‘Euler’s equation reaches down into the very depths of existence. It brings
together mental abstractions having their origins in very different aspects of our lives, reminding us once again that things that connect and bind together are ultimately more important, more valuable, and more beautiful than things that separate.’
Devlin’s remark suggests the chief reason why an equation such as Euler’s attracts value and interest beyond the particular scientific inquiries that gave birth to it. It serves as a clear and concise example of what an equation and formula does: it shows how what seemed to be disparate and even incompatible elements (rational, irrational, and imaginary numbers) are implicated in a unity, and does so concisely, with few moving parts, so to speak. It simultaneously simplifies, organizes, and unifies. It brings what equations do out into the open. It is an equation that shows what it is to be an Equation.
EQUATIONS AS ICONS
Journalist
: Do the Russians have anything like GISMO?
Scientist
(Rod Taylor): No, I’m sure they’d like it, though.
Journalist
: Can you give us the equation?
Scientist
: No, I’m sure they’d like that even better.
The Glass Bottom Boat
(1966)
Equations have a subtle influence on the fabric of our language and our thought far beyond science. Cloaking thoughts in mathematical dress seems to make them more authoritative, certain, precise, and eternal. Jokes, maxims, political bumper stickers, and uplifting self-help slogans are often revamped as equations: ‘Knowledge = Power’, ‘War = Killing People’, ‘Preparation + Patience = Success.’
Equations are written
about
humor, as the following diagnosis of the 2007 episode in which CBS Radio fired talk show host Don Imus for a racist remark: ‘White guy plus black slang equals comedy. But there’s where the equation breaks down. White guy plus black slang minus common sense equals tragedy.’
1
Or consider George Orwell’s famous equations from his novel
1984
, which, though obviously overtly false, point to a different kind of truth:
War = peace
Ignorance = strength
Freedom = slavery
While these have a superficial resemblance to equations of maths and science, they are really just metaphors in disguise. The ‘=’ sign in them does not mean the mathematical notion of ‘equal to’ or even equivalence. In mathematics, ‘=’ is quantitative, and means ‘is exactly the same as’, referring to the number of items in a set, or to a specific measurable amount. This is the foundation stone of the discipline. The way that knowledge is power, to take one example, is qualitative and much different, and must be addressed by broaching the philosophical complexities of the meanings of the seemingly self-evident words ‘sameness’, ‘equality’, and ‘is.’
Still, these fanciful equations are intriguing, for they exhibit the dangerous hope that other kinds of knowledge can be couched in equational terms, with neat packaging, balanced amounts, and simple units. Equations, that is, can seduce us into thinking that this is the way to think, and that other ways are inferior or even defective. A correspondent to a science magazine, after receiving an email from someone at a breakfast cereal company asking him to produce an equation for the best time to add the milk, made light of the way the public seems obsessed with finding equations for even the most trivial of actions. The letter unleashed comments from others – who had seen requests for equations for making sandwiches, parking cars, and ‘the perfect sitcom’ – warning that the practice had a dark side, not only because it was bad science, but because it encouraged irresponsible behaviour among scientists and mistaken views about the nature of science among the public.
2
Specific equations, too, can have a wide range of symbolic meanings. Take 2 + 2 = 4, the slightly elder sibling of 1 + 1 = 2. In fiction and reality, it has been used to symbolize
the superiority of the irrational over the rational, the rational over the irrational, and the divine over both the rational and the irrational.
3
In Dostoyevsky’s novel
Notes from Underground
, for instance, the narrator describes it as ‘insufferable’, as a ‘piece of insolence’, as sterile and rational, as something dead and beneath bare consciousness, which the narrator finds is ‘infinitely superior to two times two makes four.’ In George Orwell’s novel
1984
, on the other hand, Winston, the protagonist, uses 2 + 2 = 4 as a self-evident truth, the touchstone of sanity and rationality, available to thinking at any and every moment, the one shining light to grasp that objective reality, which assures and even guarantees for him that objective reality is there; for the Party, 2 + 2 = 4 is the final resistance that must be defeated in the way of the success of doublethink and the Party’s rule, the one outside standard that must be eradicated. Orwell, in turn, was only quoting a genuine slogan by the leaders of the Soviet Union, which used 2 + 2 = 5, written on billboards and in electric lights, as a symbol of optimism, of the ability of labour to triumph over nature, of the fact that ‘miracles could be worked through the sorcery of naked force.’
4
The correct equation – dry, rational, and stale – was false, it seems, because it did not capture human creative ability, while the incorrect one was true because it symbolized the way human creative ability can overcome limitations of nature. Meanwhile, the architect and inventor Buckminster Fuller liked to define synergy with the motto ‘1 + 1 = 4’, meaning that efficient and imaginative use of parts produces more than is possible with conventional methods. Finally, the eminent Oxford theologian Marilyn McCord Adams, arguing that ‘Human nature is not created to function independently, but in omnipresent partnership with its Maker’, speaks of the ‘self-effacing Spirit’ who ‘is ever the midwife of creative insight, subtly nudging, suggesting, directing, directing our attention until we leap to the discovery that 2 + 2 = 5.’
5
Novelists who have used equations in bizarre ways include Italo Calvino, whose book
Cosmicomics
features Einstein’s general relativity in one story. Another is Mark Leyner, whose book
Et Tu, Babe
includes a character who claims to have tattooed on his penis Max Planck’s energy formula
E
=
hv
, hence, something associated with radiation and power – and is humiliated to have to confess in front of a judge that it is actually
d
= 16
t
2
– Galileo’s law of falling bodies.
If equations have a dark side, it is that they can also tempt us to think that knowledge resides in the equation itself, rather than in the ongoing construction and renovation of the city of science (what Plato called more questioning). They can promote the erroneous view that science consists of a set of facts or beliefs to be memorized, rather than a quest for greater understanding to be achieved by moving beyond the facts or beliefs we already have.
The Scientific Equivalent of Shakespeare:
THE SECOND LAW OF THERMODYNAMICS
S
’ –
S
≥
0
DESCRIPTION
: The entropy of the world strives toward a maximum.
DISCOVERER
: An international cast of characters
DATE
: 1840s–1850s
A good many times I have been present at gatherings of people who, by the standards of the traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics. The response was cold: it was also negative. Yet I was asking something which is about the scientific equivalent of:
Have you read a work of Shakespeare’s
?
The Two Cultures
The first two laws of thermodynamics are easy to state. Rudolf Clausius, who formulated the second one and who coined the word ‘entropy’ as a name for a measure of disorder, expressed them this way: ‘The energy of the world is constant; the entropy of the world strives toward a maximum.’ A popular formulation in simple language is: ‘You can’t win. You can’t break even, either.’ Max Planck’s symbolic formulation of the second law, with
S
the entropy at an earlier time and
S
’ the entropy at a later time, is given above. This law is essential to the activities of the world around us. If you do not understand this, you can have little understanding of how the world works. This was surely C. P. Snow’s motivation in saying that asking if someone can describe the second law of thermodynamics is like asking, ‘Have you read a work of Shakespeare’s?’ It should be equally shameful for people who think themselves cultured to have to answer no to either question.
Ludwig Boltzmann (1844–1906)
Austrian physicist tormented by depression and mood swings. Uses statistical methods to show how agitations of swarms of tiny atoms give rise to bulk properties of matter. Famous for the Boltzmann equation,
S
=
k
log
W
. Deeply troubled by attacks on atomic theory, and thus his work, by prominent colleagues. In 1906, on vacation near Trieste, hangs himself while his wife and daughter are swimming. His equation is engraved on his tombstone in Vienna.
Lazare Carnot (1753–1823)
French military engineer specializing in the elimination of waste. Military duties interrupt his studies of inefficiency in water-powered machines. Does a jail spell when he seduces a woman betrothed to another man. Nicknamed ‘Organizer of Victory’ by French Revolutionaries thanks to his creative problem-solving for the cause. Fathers and homeschools two sons: Sadi and Hippolyte.
Sadi Carnot (1796–1832)
