Stewart is a mathematician and prolific author, having written over 40 books on all aspects of maths, as well as publishing guides to the maths used in Terry Pratchett’s Discworld books, authoring half a dozen textbooks for students and co-authoring a couple of science fiction novels.
He writes in a marvellously clear style, but more importantly, he is interesting: he sees the world in an interesting way and manages to convey the wonder and strangeness and powerful insights which seeing the world in terms of patterns and shapes, numbers and maths, gives you.
He wants to help us see the world as a mathematician sees it, full of clues and information which can lead us to deeper and deeper appreciation of the patterns and harmonies all around us.
1. The Natural Order
Thus he begins the book by describing just some of nature’s patterns: the regular movements of the stars in the night sky; the sixfold symmetry of snowflakes; the stripes of tigers and zebras; the recurring patterns of sand dunes; rainbows; the spiral of a snail’s shell; why nearly all flowers have petals arranged in one of the following numbers 5, 8, 13, 21, 34, 55, 89; the regular patterns or ‘rhythms’ made by animals scuttling, walking, flying and swimming.
2. What Mathematics is For
Mathematics is brilliant at helping us to solve puzzles. it is a more or less systematic way of digging out the rules and structures that lie behind some observed pattern or regularity, and then using those rules and structures to explain what’s going on. (p.16)
Kepler discovers the planets move in ellipses. The nature of acceleration, ‘not a fundamental quality, a rate of change’. Newton and Leibniz invent calculus to help us work out rates of change.
Two of the main things that maths are for are 1. providing the tools which let scientists understand what nature is doing 2. providing new theoretical questions for mathematicians to explore further. Applied and pure mathematics.
He mentions one of the oddities, paradoxes or thought-provoking things that comes up in many science books which is the eerie way that good mathematics, whatever its source, eventually turns out to be useful, to be applicable to the real world, to explain some aspect of nature. Many philosophers have wondered why. Is there a deep congruence between the human mind and the structure of the universe? Did God make the universe mathematically and implant an understanding of maths in us? Is the universe made of maths?
Stewart’s answer is simple and elegant: he thinks that nature exploits every pattern that there is, which is why we keep discovering patterns everywhere. We humans express these patterns in numbers, but it isn’t the numbers nature uses – it’s the patterns and shapes and possibilities which the numbers express, or define.
Mendel noticing the numerical relationships with which characteristics of peas are expressed when they are crossbred. The double helix structure of DNA. The computer simulation of the evolution of the eye from an initial mutation providing for skin cells sensitive to light, published by Daniel Nilsson and Susanne Pelger in 1994.
Resonance = the relationship between periodically moving bodies in which their cycles lock together so that they take up the same relative positions at regular intervals. The cycle time is the period of the system. The individual bodies have different periods. The moon’s rotational period is the same as its revolution around the earth, so there is a 1:1 resonance of its orbital and rotational period.
Mathematics doesn’t just analyse, it can predict, predict how all kinds of systems will work, from the aerodynamics which keep planes flying to the amount of fertiliser required to increase crop yield to the complicated calculations which keep communications satellites in orbit round the earth and therefore sustain the internet and mobile phone networks.
Time lags: the gap between a new mathematical idea being developed and its practical implementation can be a century or more: it was 17th century interest in the vibration of a violin string which led, three hundred years later, to the invention of radio, radar and TV.
3. What Mathematics is About
The word ‘number’ does not have any immutable, God-given meaning. (p.42)
Numbers are the most prominent part of mathematics and everyone is taught arithmetic at school, but numbers are just one type of object that mathematics is interested in.
The invention of numbers. Fractions. Some time in the Dark Ages the invention of 0. The invention of negative numbers, then of square roots. Irrational numbers. ‘Real’ numbers.
Whole numbers 1, 2, 3… are known as the natural numbers. If you include negative whole number, the series is known as integers. Positive and negative numbers are known as rational numbers. Then there are real numbers and complex numbers. Five systems in total.
But maths is also about operations such as addition, subtraction, multiplication and division. And functions, also known as transformations, rules for transforming one mathematical object into another. Many of these processes can be thought of as things which help to create data structures.
Maths is like a landscape with similar proofs and theories clustered together to create peaks and troughs.
4. The Constants of Change
Newton’s basic insight was that changes in nature can be described by mathematical processes. Stewart explains how detailed consideration of what happens to a cannonball fired out of a cannon helps us towards Newton’s fundamental law, that force = mass x acceleration.
Newton invented calculus to help work out solutions to moving bodies. Its two basic operations – integration and differentiation – mean that, given one element – force, mass or acceleration – you can work out the other two. Differentiation is the technique for finding rates of change; integration is the technique for ‘undoing’ the effect of differentiation.
Calculating rates of change is a crucial aspect of maths, engineering, cosmology and many other areas.
5. From Violins to Videos
A fascinating historical recap of how initial investigations into the way a violin string vibrates gave rise to formulae and equations which turned out to be useful in mapping electricity and magnetism, which turned out to be aspects of the same fundamental force, understanding which underpinned the invention of radio, radar, TV etc, taking in contributions from Michael Faraday, James Clerk Maxwell, Heinrich Hertz and Giulielmo Marconi.
Stewart makes the point that mathematical theory tends to start with the simple and immediate and grow ever-more complicated. This is because you have to start somewhere.
6. Broken Symmetry
A symmetry of an object or system is any transformation that leaves it invariant. (p.87)
There are many types of symmetry. The most important ones are reflections, rotations and translations.
7. The Rhythm of Life
The nature of oscillation and Hopf bifurcation (if a simplified system wobbles, then so must the complex system it is derived from) leads into a discussion of how animals, specifically animals with legs, move, which is by staggered or syncopated oscillations, oscillations of muscles triggered by neural circuits in the brain.
This is a subject Stewart has written about elsewhere and is something of an expert on. The seven types of quadrupedal gait are: the trot, pace, bound, walk, rotary gallop, transverse gallop, and canter.
8. Do Dice Play God?
Stewart’s take on chaos theory.
Chaotic behaviour obeys deterministic laws, but is so irregular that to the untrained eye it looks pretty much random. Chaos is not complicated, patternless behaviour; it is much more subtle. Chaos is apparently complicated, apparently patternless behaviour that actually has a simple, deterministic explanation. (p.130)
19th century scientists thought that, if you knew the starting conditions, and then the rules governing any system, you could completely predict the outcomes. In the 1970s and 80s it became increasingly unclear that this was wrong. It is impossible because you can never define the starting conditions with complete certainty.
Thus all real world behaviours are subject to ‘sensitivity to initial conditions’. From minuscule divergences at the start point, cataclysmic differences may eventually emerge.
He then explains the concept of phase space developed by Henri Poincaré: this is an imaginary mathematical space that represents all possible motions in a given dynamic system. The phase space is the 3-D place in which you plot the behaviour in order to create the phase portrait. Instead of having to define a formula and worrying about identifying every number of the behaviour, the general shape can be determined.
Much use of phase portraits has shown that dynamic systems tend to have set shapes which emerge and which systems move towards. These are called attractors.
9. Drops, Dynamics and Daisies
The book ends by drawing a kind of philosophical conclusion.
Chaos theory has all sorts of implications but the one he closes on is this: the world is not chaotic; if anything, it is boringly predictable. And at the level of basic physics and maths, the laws which seem to underpin it are also schematic and simple. And yet what we are only really beginning to appreciate is how complicated things are in the middle.
It is as if nature can only get from simple laws (like Newton’s incredibly simple law of thermodynamics) to fairly simple outcomes (the orbit of the planets) via almost incomprehensibly complex processes. To end, Stewart gives us three examples of the way apparently ‘simple’ phenomena in nature derive from stupefying complexity:
- what exactly happens when a drop of water falls off a tap
- computer modelling of the growth of fox and rabbit populations
- why petals on flowers are arranged in numbers derived from the Fibonacci sequence
In all three cases the underlying principles seem to be resolvable into laws and functions – and we see water dropping off taps or flowerheads all the time – and yet the intermediate steps are bogglingly complex.
Coda: Morphomatics
He ends the book with an epilogue speculating, hoping and wishing for a new kind of mathematics which incorporates chaos theory and the other elements he’s discussed – a theory and study of form, which takes everything we already know about mathematics and seeks to work out how the almost incomprehensible complexity we are discovering in nature gives rise to all the ‘simple’ patterns which we see around us.
Related links
Reviews of other science books
Cosmology
- The Perfect Theory by Pedro G. Ferreira (2014)
- The Origin Of The Universe: To the Edge of Space and Time by John D. Barrow (1994)
- The Last Three Minutes: Conjectures about the Ultimate Fate of the Universe by Paul Davies (1994)
- The Black Cloud by Fred Hoyle (1957)
Environment / human impact
- The Sixth Extinction: An Unnatural History by Elizabeth Kolbert (2014)
- Guns, Germs and Steel by Jared Diamond (1997)
- The Sixth Extinction by Richard Leakey and Roger Lewin (1995)
Genetics
- The Double Helix by James Watson (1968)
Maths
- Nature’s Numbers: Discovering Order and Pattern in the Universe by Ian Stewart (1995)
- Innumeracy: Mathematical Illiteracy and Its Consequences by John Allen Paulos (1988)
- A Mathematician Reads the Newspaper: Making Sense of the Numbers in the Headlines by John Allen Paulos (1995)
Particle physics
Psychology
- Irrationality: The Enemy Within by Stuart Sutherland (1992)
