Short works

Books : reviews

Ian Stewart.
Galois Theory.
Chapman and Hall. 1973

Ian Stewart, David O. Tall.
The Foundations of Mathematics.
OUP. 1977

This is a book for readers in transition from ‘school mathematics’ to the fully-fledged type of thinking used by professional mathematicians. It should prove useful for first-year students in universities, polytechnics, and colleges, to sixth-formers contemplating further study, and anyone else interested in the critical change in mathematical thinking from intuition to rigour. It develops the more formal approach as a natural outgrowth of the pattern of underlying ideas, building on a school mathematics background to develop the viewpoint of an advanced practising mathematician. The topics covered include: the nature of mathematical thinking; a review of the intuitive development of familiar number systems; sets, relations, functions; an introduction to logic as used by practising mathematicians, methods of proof (including how a mathematical proof is written); development of axiomatic number systems from natural numbers and proof by induction to the construction of the real and complex numbers; the real numbers as a complete ordered field; cardinal numbers; foundations in retrospect.

Ian Stewart, John Jaworski.
Seven Years of Manifold, 1968-1980.
Shiva. 1981

(read but not reviewed)

Mathematical essays, spoofs, jokes, and cartoons

Ian Stewart, David O. Tall.
Complex Analysis: the hitch-hiker's guide to the plane.
CUP. 1983

Ian Stewart.
The Problems of Mathematics.
OUP. 1987

(read but not reviewed)

Ian Stewart.
Does God Play Dice?: the new mathematics of chaos.
Penguin. 1989

rating : 2.5 : great stuff

Ian Stewart.
Another Fine Math You've Got Me Into....
Freeman. 1992

rating : 3.5 : worth reading

Welcome to the mathematical wonderland of Ian Stewart. On your journey you will also confront a number of brain-teasing puzzles and games and gain a fresh understanding of some significant mathematical ideas. You will also meet a host of curious characters and endure dozens of jokes and puns. Call these stories fractured mathematical fairy tales&mdasg;engaging, challenging, and loads of fun!

Featuring: Tile and Error; Cogwheels of the Mind; A Vine Math You’ve Got Me Into; Maxdoch Murwell, Market Manipulator; The Group-Theorist of Notre Dame; Sofa, So Good; and more…

A selection of 16 "mathematical games"columns from Pour la Science, the French edition of Scientific American. (The columns are in English here.)

Ian Stewart, Martin Golubitsky.
Fearful Symmetry: is God a geometer?.
Penguin. 1992

rating : 3 : worth reading

Jack Cohen, Ian Stewart.
The Collapse of Chaos: simple laws in a complex world.
Viking. 1994

rating : 3 : worth reading
review : 1 February 1996

The first half of the book is a good overview of 'conventional' reductionist science, taking everything apart and looking at the content. It provides the background for the novel second half.

The second half puts everything back together, showing how complex systems can have emergent properties that appear at the 'whole' level, but be invisible on the 'part' level. Reductionism isn't enough, the context is crucial. This second half is necessarily less polished, because the authors are trying to pull concepts together in a new way. But it's a fun read, and very thought-provoking.

p421. Feedback between spaces with different geographies tends to produce new types of behaviour that are seriously different from anything that you find in either system alone.

Ian Stewart.
Nature's Numbers: discovering order and pattern in the universe.
Phoenix. 1995

rating : 4 : passes the time
review : 17 November 2007

This is a brief overview of the way mathematics permeates many aspects of the natural world. It is somewhat like an executive summary of the later, deeper, better, Life's Other Secret. Read that instead.

Ian Stewart, Jack Cohen.
Figments of Reality: the origins of the curious mind.
CUP. 1997

rating : 2.5 : great stuff
review : 22 February 1998

This extremely well-written book lucidly weaves together themes of self-organising complexity, co-evolution, cultural capital, and an explanation style that takes into account external 'complicity' as well as internal 'reductionist' ideas. It develops some of the authors' earlier ideas, in particular in The Collapse of Chaos. Stewart and Cohen take the view that in order to explain (human) intelligence, we need to understand its co-evolution with culture.

Evolution is explained using the mathematical idea of phase spaces, here the 'phase spaces of the possible'. Evolution progresses in directions constrained by its phase space, much as the behaviour of a dynamical system evolves in accordance with its phase space. The crucial ingredient, though, is that the phase space is determined by the existing system, and as the system evolves, so does the phase space. So the rules of the game change as evolution progresses. Some evolutionary changes make small 'private' changes to the phase space; the more interesting ones make qualitative, 'public' changes, opening up whole new regions of possibility not available before. For example, early bacteria produced a toxic by-product, oxygen; the new oxygen-rich atmosphere allowed whole new kinds of organisms to evolve. But the most interesting change in the rules of the game from our human perspective was the ability to pass on 'cultural capital' to the next generation, so that each new generation does not have to start from scratch: yolk in eggs, a nest as a protected environment, learning survival tricks from the troupe, and, eventually, being able to tap in to all of the 'extelligence' of human culture.

An interesting analogy with a 'snooker break' (if you don't know what that is, just read the book -- it's worth it!) is made, about how each generation needs to set up the right state for the next generation, not only for it to survive, but for it to be able to set up the right circumstances for its next generation, and so on. But as well as this stability, the rules of the game are always changing, so the system must also be flexible enough to keep evolving, or it will eventually fail to survive.

There is so much more here than I can summarise in this review. Hardly a page goes by without an interesting new idea or insight. We get the idea of human cultures providing a 'make a human kit', used to transmit the ideas needed to maintain the 'snooker break' of the culture itself. We get a reasoned explanation of the fallacy of Searle's Chinese Room argument, and a discussion of why Roger Penrose's approach to explaining intelligence is mistaken. We get the ideas of Dennett's 'multiple drafts' theory of emergent intelligence, as a 'pandemonium' of sub-intelligences, taken further, in an interesting chapter in free will, which also addresses the problem of genetic determinacy (with a wonderful little scene about a mythical gene for autoincarceration, which made me laugh out loud -- much to the confusion of the people around me!). And we get an optimistic speculation on the future of human evolution -- now cultural rather than physical -- and how our thinking is becoming more complex, rather than linear and simplex, as we discover and take account of the subtle ways seemingly unrelated events can interact. Read it yourself, and find out how much more good stuff is here.

Ian Stewart.
Life's Other Secret: the new mathematics of the living world.
Penguin. 1998

rating : 2.5 : great stuff
review : 26 November 2007

Ian Stewart, Jack Cohen.
Aspect. 2000

rating : 3.5 : worth reading
review : 17 June 2003

Not only has extraterrestrial life been discovered, but it seems to be at war with us. That's the first conclusion when someone rearranges the moons of Jupiter to deflect a comet straight at the Earth. But maybe it's just a misunderstanding? A group of scientists is sent out Jupiter to try to make contact, and explain we'd really rather they sent the comet somewhere else. But the aliens turn out to be hard to contact, and the scientists need a bit of help from a maverick archaeologist and her animal-loving nephew.

This is in some sense the fictional version of the authors' Evolving the Alien, explaining just how alien aliens will be. They certainly succeed on the physiological front -- we get giant gasbags floating in the Jovian atmosphere, with a really weird genetics, and more. But I think they are less successful on the psychological front -- the aliens seem just extreme versions of certain human types.

The plot is rather uneven, and takes a while to get going. Once it does get going, it races along, with some nice twists and cliffhangers. There are zillions of coincidences -- but they are the kind I can almost forgive -- they are mostly used to explain parts of the plot (a sort of docudramatic infodump), and to reduce the cast of characters to a manageable level -- rather than to drive the plot off in unnatural directions.

But this is really a book of fun ideas. We get the physiology of the Jovian and other aliens, the explanation for ice at the lunar poles, the neo-Zen Buddhists, the mechanism to get to Jupiter in a hurry, the very cheeky explanation of the microwave background radiation, and lots of great little details. These ideas, and the whole plot-line, overcome the sometimes uneven, and occasionally clumsy, storytelling.

Ian Stewart.
The Beauty of Numbers in Nature: mathematical patterns and principles from the natural world.
Ivy Press. 2001

Ever wondered why spirals are found everywhere, from whirlpools and galaxies, to shells and DNA? Do you know how zebras form their stripes, or why each snowflake is so intricate and unique? The beauty of mathematics is all around us, from the very smallest scales of molecules and bacteria right up to the scale of our entire universe – the largest structure of all. In this fascinating and accessible illustrated guide, Ian Stewart explains the numbers and mathematical concepts that underpin everything we can observe in nature. With over 150 full-colour photographs and diagrams, this book explores humankind’s desire to understand the beauty and mind-boggling complexity of the natural world in all its many varied forms.

Ian Stewart.
Flatterland: like Flatland, only more so.
Pan. 2001

Jack Cohen, Ian Stewart.
Evolving the Alien: the science of extraterrestrial life == What Does a Martian Look Like?.
Ebury Press. 2002

rating : 3.5 : worth reading
review : 30 October 2002

For many years now Jack Cohen has been giving a lecture on "The Possibility of Life on Other Planets", or POLOOP for short. This is the book of that lecture, and it stresses just how alien alien life will be.

First Contact could happen any time, and is likely within, say, another 100,000 years. Don't hold your breath, though. The aliens will have evolved somewhere incredibly unpleasant to us, will resemble nothing we have yet conceived, will be sexually bizarre or have unrecognisable reproductive tricks, will behave in incomprehensible ways, will use technology we don't understand, and will communicate in a manner that we've never thought of, and which we can't even recognise as communication. They will be here for reasons that make no sense to us, and they probably won't find humans the least bit interesting. We will find it almost impossible to understand them.

The main thrust of the book is that the astrobiologists have got it all wrong. They have too narrow an outlook, assuming that life has to be a 'carbon-copy' (lovely pun!) of 'life as we know it' on Earth. And yet the more we find out about the life on Earth, the more bizarre are the specimens discovered.

the possibilities on our own planet are known to be much broader than the astrobiologists permit themselves to envisage anywhere.

The authors prefer the term "xenoscience", as it allows a wider range of possibilities to be explored. Much of what we see on Earth is "parochial" -- an accident that happened once during evolution here, but would very likely not happen if evolution were run again -- five fingered limbs, elbows and knees, nose above mouth, and so on. The most convincing evidence against little green (or Grey) men isn't that they are little and green, but that they are men. The universals are more interesting -- eyes, flight, fur, photosynthesis, and sex seem to be some such universals, having evolved several times on earth. But it is a little difficult to work out what truly is universal, given the small sample space we have available. And even though "absence of evidence isn't evidence of absence", the authors are careful to point out that

Lack of evidence against is not evidence for.

Even so, they have nothing against more exotic life based on processes other than DNA, on other than carbon, or on other than chemistry (magnetic vortices in the sun, nuclear reactions on neutron stars, ...). The only requirement is for some process that supports sufficient complexity, and, as we are discovering, that's nearly anything. And life changes the world in which it lives.

it genuinely would not surprise us to find ... that the universe is much older than the twelve billion years or so that physicists currently estimate, from present-day data attributed to the Big Bang. The data might be misleading, because 'living' processes could have been parasitic on, for example, the inflation phase of the universe. Or the remnants of such 'lifeforms' have led us to assume that there had to have been such an inflation phase.

So the universe is probably teeming with life. But looking for extra-terrestrial life with something like the SETI programme is probably doomed to failure. Aliens probably won't use radio, any more than they would use smoke signals. An even if they do use radio, it will be efficiently encoded and compressed.

Lachmann's theorem: 'any sufficiently advanced communication is indistinguishable from noise'.

Not all the book is about xenoscience. A lot of it is about science in general. We get digs at TV programmes:

This programme was known at the BBC Natural History Unit as 'Making it up as you go along With Dinosaurs'.

We get tongue-in-cheek commentary on certain Big Science projects:

... the human genome project was excellent basic science, because the first thing it turned up [that humans have about 34,000 genes instead of the expected 100,000] was strong evidence that its entire theoretical basis was wrong.

We get little interesting details about biology.

Human ears have more nerve fibres going out to the ear from the brain ... than they have nerves leading from the cochlea to the brain.

We get a calm and thoughtful tirade against pseudo-science of all stripes.

science is the best defence against believing what we want to.

Yes, science is a social construct ... But it is a construct that has been constrained at eery step by external reality. If the world's scientists all got together and decided that elephants are weightless and rise into the air if they are not held down by ropes, it would still be foolish to stand under a cliff when a herd of elephants was leaping off the edge.
     In science, there has to be a reality check. That check is carried out by beings who perceive reality through imperfect and biased senses, so it cannot be perfect -- but it has to survive some very stringent scrutiny. Far more stringent than any scrutiny of religious beliefs, artistic merit, or a trial in a court of law.

The problem with this kind of credulity is not that there are prosaic explanations. It is that so many different supernatural explanations are possible.
     ... the real problem is that if we are convinced that Geller really does have the power to bend spoons using his mind alone, then we don't know whether we must change our minds about metallurgy, about what 'mind' is, or about the relationship of mind-mechanism to metals.
     ... We could start a scientific programme to find out -- but then, the first thing we would do is repeat the experiment, trying to find the trickery that is overwhelmingly likely to be there. Only when we had excluded that would we start the more difficult process of rewriting our entire theory of the universe.
     Either we think scientifically -- or at least rationally and critically -- about this kind of event, or we end up unable to think sensibly about anything. You choose.

And we get lots of links to science fiction, one place where aliens get portrayed every day, for better (mainly books) or for worse (mainly TV: men in rubber suits). The text is dotted with little summaries of interesting fictional aliens, and comments on how SF writers have more imagination than astrobiologists.

But the dinosaurs met their end because of a meteorite that just happened to occupy the same bit of space as the Earth. Some SF authors have suggested, not very seriously, that this was Gaia's way of getting rid of the dinosaurs -- who clearly could not protect the Gaia system from big meteorites -- and replacing them with another set of big terrestrial vertebrates, one of whom might be able to.

One of the things the authors talk about is the "hidden book" behind the better SF tales: the book with all the science worked out to back up what gets foregrounded in the story. Jack Cohen himself has famously helped write several of these hidden books, albeit sometimes retrospectively! For example, the rationalisation of Star Trek's Tribbles is beautiful.

The content is certainly deep, interesting, and novel. So why "only" a 3.5 rating? Well, it's simply that I am left feeling that several layers of polish have been left off, and a few punchlines omitted. Some of the arguments could be better organised, and there is some repetition. If you've ever been to one of Jack's talks, this has a bit of their hop-scotch flavour: an interesting factoid here, a confusing allusion there, certainly all woven together to be very entertaining, and here with rather more depth and detail (but fewer cartoons!). But I would have like a little more in the book, some more of the "hidden book" that is behind this one. Don't get me wrong: I've been waiting for the book of POLOOP for ever, and maybe that has raised my expectations too high. This is a good book; I'm not sorry to have bought it in hardback. Yet the cover provocatively asks, "What does an alien look like?", and I still don't know.

Ian Stewart, Jack Cohen.
Aspect. 2004

Ian Stewart.
Math Hysteria: fun and games with mathematics.
OUP. 2004

Ian Stewart.
Professor Stewart's Cabinet of Mathematical Curiosities.
Profile. 2008

Ian Stewart.
Seventeen Equations that Changed the World.
Profile. 2012

Two sets of numbers and symbols separated by an equals sign can change the world. And not only that: they can reveal the laws of nature and unlock the secrets of the universe.

Professor Ian Stewart shows how seventeen extraordinary equations have opened new areas of experience, from electronics and global communications to radar, lasers, space craft and the atom bomb. His narrative spans two and half millennia, from the ancient Greeks to the seductive formulation that led the world’s banking systems to near-oblivion. He shows how insights from the past can inform work centuries later: Pythagoras’s theorem inspired Einstein and, yet more curiously, the stumbling calculations of a sixteenth-century gambler led to an impossible number that became a key to the baffling world of quantum physics.

Ian Stewart, David O. Tall.
Complex Analysis: the hitch-hiker's guide to the plane: 2nd edn.
CUP. 2018


Terry Pratchett, Ian Stewart, Jack Cohen.
The Science of Discworld.
Ebury Press. 1999

rating : 3.5 : worth reading
review : 30 November 2000

Despite its title, this isn't a book about the science underlying Terry Pratchett's stories of Discworld -- that world runs on magic, after all -- it's about the science that governs Roundworld, our world.

It is structured as alternating chapters. The first of each pair tells the story of the Wizards at the Unseen University, and how a magical accident results in the creation of a pocket universe -- ours. With the unwilling help of Rincewind, they observe this puzzling universe as it evolves from the Big Bang to beyond our current time, and they try to understand in their own magical terms. The second of each pair of chapters then explains what the Wizards are observing from a scientific point of view.

The explanations build on ideas from Stewart and Cohen's earlier books, The Collapse of Chaos and Figments of Reality, including those of extelligence, and Ant Country. The ideas cover an enormous breadth, and because of this, there is necessarily some lack of depth, so it helps to have read the previous books to get some of this missing background.

It could also do with a "further reading" list. For example, they refer in passing to Deep Time -- that evolution operates over incomprehensibly vast periods of time, and so doesn't fit well with the kind of stories we tell about it -- and I would have liked a pointer to somewhere that explores this in more detail. There is rather a lot of this feeling that certain sentences have a lot packed into them, but with no pointer to somewhere to help the unpacking.

Lack of depth notwithstanding, I like this book. There are some nice points made about the way our high tech is beginning to look like magic from the outside, and some good discussions about the way "lies-to-children" -- over-simplified explanations -- are a necessary stepping-stone on the way to deeper explanations. [I was a bit taken aback to read on page 152: "In April 1969, Neil Armstrong stepped down onto the surface of the moon". This isn't a "lie-to-children", this is a simple factual mistake.]

In particular, I like the way the alternating fact and fiction chapters allow the separation of our current best understanding of the facts from the wildly speculative illustrative examples. The fictional details about evolution on Earth -- the rise of intelligence in certain crabs, lizards, and what not, only for them to be wiped out by the next ice age or meteor strike -- makes the point that our current civilisation is just as vulnerable to whatever the universe chooses to throw at us, but that the planet as a whole is very resilient.

So, some fascinating deep ideas lurking just round the corners. Maybe their next book will explore them further?

Terry Pratchett, Ian Stewart, Jack Cohen.
The Globe.
Ebury Press. 2002

rating : 3.5 : worth reading
review : 10 October 2003

The first Science of Discworld gave us two parallel stories: the story of evolution on Earth, culminating with humanity escaping to the stars just before it was wiped out by yet another natural cataclysm, coupled with the story of the actual science behind it all. In this second volume we have the same format, with the (fictional) story being of how the humans got themselves that far, and the (scientific) story of how culture develops, and (nicely recursive here) the importance of stories within that culture.

The "plot" is this: evil misery-eating elves come to prey on the humans, by making them superstitious and thus fearful. The wizards try to stop this, but their first attempts make things even worse, until eventually they discover Shakespeare. The fun is seeing how (fiction) and why (science) they eventually succeed.

Like most Stewart and Cohen journeys, there is a lot packed into a small space, not always unpacked (or often, unpackable only if you've read previous travels), yet all fascinating nevertheless. We get excursions on chaos and complexity; on extelligence, cultural privilege, and the importance of stories; side-swipes at science, art, and religion; and loads more.

Some of the material has been covered before, but is well worth covering again and differently, because it is so import and difficult to intuit. The parts on DNA and complexity fall in this category.

p188. Whether something is a message depends upon context, too: sender and receiver must agree upon a protocol for turning meanings into symbols and back again. Without this protocol a semaphore is just a few bits of wood that flap about. Tree branches are bits of wood that flap about, too, but no one ever tries to decode the message being transmitted by the tree. ...
In biological development the protocol that gives meaning to the DNA message is the laws of physics and chemistry. ... An organism's complexity is not determined by the number of bases in its DNA sequence, but by the complexity of the actions initiated by those bases within the context of biological development. That is, by the meaning of the DNA 'message' when it is received by a finely-tuned, up-and-running biochemical machine.

p305. Chaos is just one of the practical reasons why it's generally impossible to predict the future (and get it right). Here we'll examine a rather different one: complexity. Chaos afflicts the prediction method, but complexity afflicts the rules. Chaos occurs because it is impossible to say in practice what the state of the system is, exactly. In a complex system, it may be impossible to say what the range of possible states of the system is, even approximately. Chaos throws a spanner in the works of the scientific prediction machine, but complexity turns that machine into a small cube of crumpled scrap metal.

Most of the work is taken up with examining the importance of stories in the development and continuation of culture. It would be interesting to speculate on how some of these ideas might be tested.

p327. Our minds make stories, and stories make our minds. ... Stories map out the phase space of existence. ...
Some stories are Worlds of If, a way for minds to try out hypothetical choices and imagine their consequences. Word-play in the Nest of the Mind. And some of those stories have such a compelling logic that narrative imperative takes over, and they transmute into plans. A plan is a story together with the intention of making it come true.

p342. in an article defending fairy stories [G. K. Chesterton] disputed the suggestion that stories tell children that there are monsters. Children already know there are monsters, he said. Fairy stories tell them that monsters can be killed.

And towards the end, we get to see how science can help us make better stories, stories that are better suited to reality, rather than just how we would like things to be. Science stops us, and our stories, from fooling ourselves.

p344. The ideal of the scientific method is often not realised. Its usual statement is an oversimplification in any case, but the basic worldview captures the essence. Think critically about what you are told. Do not accept the word of authority unthinkingly. Science is not a belief system: no belief system instructs you to question the system itself. Science does. (There are many scientists, however, who treat it as a belief system. Be wary of them.)

So, a fun read, and lots of food for thought (even if that food is served up as a splendid hors d'oeuvre course of masses of delightful toothfuls, rather than a good solid main course of meat and potatoes (although potatoes do play an important role...)).

Terry Pratchett, Ian Stewart, Jack Cohen.
Darwin's Watch.
Ebury Press. 2005

rating : 3.5 : worth reading
review : 30 December 2005

In this third outing of The Science of Discworld, the authors have their usual fun, this time with evolution. The format is the same -- chapters describing how the Wizards of Discworld are trying to put right problems in their accidental creation -- our Roundworld, alternating with chapters describing the science behind it all. This time something strange has happened on Roundworld -- it seems that suddenly Darwin never wrote his Origin of Species -- and it put science back so that the people never escaped snowball earth 500 years later (don't ask -- just read the books!)

Evolution, Paley's watchmaker, and the history of Origin of Species gets an outing here -- as does a supporting cast of infinite numbers, cosmology, and complexity. It makes a fun read -- but it's probably not the best place to start, if only to understand many of the jokes about the Wizards attempts to change Roundworld.

Terry Pratchett, Ian Stewart, Jack Cohen.
Judgement Day.
Ebury Press. 2013