In the black and white yin-yang symbol of Chinese philosophy, the yin represents disorder and the feminine, while the yang represents order and the masculine. In my married life the gender stereotype is reversed. My wife is order and I am chaos. She wants to plan what we are having for dinner every night for the coming week. I feel there isn’t much point thinking about it, “because what if we want to take the kids swimming one evening or meet up with friends?”
Our differing approaches have led to a few animated discussions (to say the least) and it was after one of these that I started to think about chaos theory: the mathematical theory behind the parable of the butterfly and the storm that gives rise to the so-called butterfly effect – the idea that the flap of the wing of a butterfly in the Amazon can set off the atmospheric events that lead to a storm off the coast of Texas weeks later. Perhaps, I thought, I can prove that planning is pointless. That disorder is inevitable and we might as well accept our lack of control.
Chaos theory was developed by Edward Lorenz, who published his breakthrough paper in 1963. Lorenz was an applied mathematician by training, but had moved to MIT to work on weather prediction. Lorenz’s colleagues had recently created artificial weather systems consisting of containers of water which, when heated and stirred, underwent unpredictable patterns of turbulence. Lorenz realised that he could create his own artificial weather system on a computer. By simulating the equations that modelled fluid dynamics, he could better understand the mechanisms behind turbulence, and thus the reason why the weather is so difficult to predict.
Lorenz reduced the weather to a set of 12 equations and simulated them on his state-of-the-art LGP-30 computer. The discovery of chaos came after Lorenz printed out a list of how the values of each of the variables in his weather simulation changed over time. When he reran the simulation again the next day – inputting the same starting values, exactly as they stood on the printout – he got a completely different printout. Something was wrong. When computers are given the same input, they should produce the same output.
Eventually, Lorenz realised that the difference occurred because the input to the code was specified to six decimal points, while the output on the printer was to three decimal points. He had used the printout of the first simulation to specify the initial values for the second simulation. But an input of a temperature of 14.956C gave a very different simulated weather forecast to an input of 14.956,181C centigrade. The fourth decimal point in the simulation is the butterfly of chaos: a small difference in starting conditions, the 0.000,181C difference between the two inputs, leads to a large difference in outcomes over time.
The butterfly parable can be misleading. It is not as if there is a single butterfly somewhere deep in the Amazon that, through a flap of its wings, causes a raging storm to hit Florida. A more accurate way of describing the butterfly effect is to say that in order to accurately predict storms in the North Atlantic two months in advance, we need to know about air disturbances everywhere on the planet, and that includes knowledge of whether that Amazonian butterfly flaps its wings or not. This insight is important to all aspects of our lives. A day at work will only turn out completely as planned if we can account for all the calls we might receive, all the people who might or might not arrive late for meetings, for the mood of everyone in the office and the mood of their partners and their partners’ parents. Creating perfection every day of the week is impossible, both at work and at home. We can’t account for everything when we look into the future. At this point I felt I had a failsafe argument to present to my wife over our next family dinner.
But then I started watching an oral history by Margaret Hamilton, who wrote the code for Lorenz’s weather simulations in 1961. Hamilton had an aversion to errors, a trait she had discovered in herself when she studied mathematics. While most of the other students in her class, all of them young men, carefully memorised every line of the proofs presented to them on the blackboard, Hamilton had noticed that Professor Florence Long, her teacher at Earlham College in Indiana, USA, never relied on memory. The professor derived each line of a proof from the previous one, as if seeing it for the first time, carefully illustrating how mathematical reasoning led inevitably to logical conclusions. Hamilton realised that, if each step follows on from another in a logical manner, then errors become impossible. It was this view that she had taken when approaching programming. She worked to eliminate all sources of error, so that when Lorenz ran her code, the only explanation for the unpredictable outcomes was chaos – and not a mistake on her part. The discovery of chaos was only possible because of Hamilton’s careful attention to detail.
Hamilton left Lorenz’s lab before he published his article on chaos theory to take a job at Homeland Security and then, in 1963, she applied for and was immediately offered a job as a computer programmer at Nasa. Six years later, it was software written by Hamilton and her team that was used on the Apollo 11 mission to the Moon in 1969. Her code was responsible for many of the spacecraft’s vital functions, including estimating its position and velocity, providing assistance with steering commands and helping the astronauts measure angles between planetary bodies. In space travel, butterfly uncertainties are generated in the high power thrust of the spaceship’s burners, the small errors in the spaceship’s positional measurements, or a miscalculation by an astronaut or by mission control. Hamilton wrote software that prioritised finding errors and eliminating them before they could generate chaos.
During the lunar descent, Hamilton’s software alerted Neil Armstrong and Buzz Aldrin to a problem with the hardware and that they needed to return the rendezvous radar to the correct position before landing. Six years of what Hamilton was the first to call software engineering made a difference when it mattered most. She showed it is possible to control the world around us in the short term, but to do so we need to act with extreme preparedness and precision.
The real lesson from chaos theory then is not that the future is beyond our control. The theory instead tells us that if something is important to us, we need to plan for it carefully. Family dinners are important to both me and my wife. We want to sit down as a family and talk about what has happened during the day. And good conversation is best enjoyed with a carefully planned meal. My wife was right. We need to think about details.
But chaos also imposes constraints. Careful planning for one aspect of our lives, such as family dinners, takes time from others. For my wife and me, this means that those things that matter less to both of us – that the grass is perfectly trimmed, that the car is clean or that the laundry is carefully folded – can be left (in good conscience) to the forces of disorder. This is not to say we never take care of these things. It is just that we accept that we might not always be on top of all of them.
Chaos theory is just one of many ways I have applied the mathematical thinking I use in my work to shape my approach to life. Sometimes I go too far. Like the time, when my wife was eight months pregnant, I did a statistical test on the outcome of several hundred games of Yahtzee we had played together during the pregnancy, to test if my tactical approach was better (it turned out it wasn’t and the test just made me a bit of a dick). But more often than not, thinking mathematically leads to a softer approach to my relationships with those close to me and the people I work with. I have used the theory of dynamical systems to study how to establish critical mass when trying to get people working together on projects. I have used entropy to think about how to best approach talking to students about how they feel their studies are progressing. And I have used complexity theory to help me negotiate social situations and even to think more deeply about who I am as an individual.
I am lucky, in that my wife is as tolerant of my mathematical ways of thinking as she is of the other forms of chaos I generate. She is a professor of mathematics education and often identifies the limitations in my theory, forcing me to rethink. We have, my Yahtzee mistake aside, found a way to combine our own different approaches to life and to share ideas together. We are, I think, a unique balance of yin and yang. We are at our own special place on the border between order and chaos. And there is no place I would rather be than here.
David Sumpter is Professor of Applied Mathematics at the University of Uppsala, Sweden. His book, Four Ways of Thinking (£18.99, Allen Lane) is published on 31 August. Buy a copy for £16.71 from guardianbookshop.com
