Tag Archives: Maths

The Golden Ticket: P, NP, and the Search for the Impossible – Lance Fortnow ***

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There is good and bad news early on in this book about the P versus NP problem that haunts computing. The good news is that on the description I expected this to be a dull, heavy Screenshot_04_05_2013_11_02going book, and it’s not at all. Lance Fortnow makes what could be a fairly impenetrable and technical maths/computing issue light and accessible.

The bad news is that frustratingly he doesn’t actually tell you what P and NP mean for a long time, just gives rather sideways definitions of the problem along the lines of ‘P refers to the problems we can solve quickly using computers. NP refers to the problems to which we would like to find the best solution’, and also that he makes a couple of major errors early on, which make it difficult to be one hundred percent confident about the rest of the book.

The errors come in a section where he imagines a future where P=NP has been proved. This would mean you could write an algorithm to very efficiently match things and select from data. Fortnow suggests that our lives would be transformed. This is slightly cringe-making as fictional future histories often are, but the real problem is that he tells us that the algorithm would make it possible to do two things that I think just aren’t true.

First he says that from DNA you would be able to identify what a person looks like and their personality. Unfortunately, these are both strongly influenced by epigenetic/environmental issues. Anyone who knows adult identical twins (with the same basic DNA) will know that they can look quite different and certainly have very different personalities. And they will usually have been brought up in the same environment. Fortnow is forgetting one of the oldest essentials of computing – it doesn’t matter how good your algorithm is, GIGO – garbage in; garbage out.

The other, arguably worse error is that he says that it will be possible to have accurate weather forecasts going forward X days. This is so horribly wrong. He should have read my book Dice World. The reason you can’t predict the weather at all beyond about 10 days is nothing to do with the quality of the model/algorithm, it is because the system is chaotic. Firstly we just don’t know, and never can know, the initial conditions to enough decimal places not to deviate from the real world. When Lorenz first discovered chaos it was because he entered the starting values in his model to 4 decimal places rather than the 6 to which the model actually worked. It soon deviated from the previous run. We can’t measure things accurately enough. The other problem is that the weather system is so complex – hence the slightly misleading title of Lorenz’s famous paper Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas? – that we can’t possible take into account enough inputs to ever have so good a model as to go forwards that far. Sorry, Lance, it ain’t going to happen.

For the rest, the first half or so of the book goes along pretty well, gradually opening up the nature of P and NP, the problems that are of interest and the ‘hardest’ NP complete problems. I found the main example, used throughout, a hypothetical world called Frenemy where everyone is either a friend or enemy of everyone else confusing and not particularly useful, but Fortnow gets plenty of good stuff in. After that it’s as if he rather runs out of material and it gets a bit repetitious or has rather tangential chapters.

Overall, despite the flaws, a much better and more readable book than I thought it was going to be – but probably best for maths/computing buffs rather than the general popular science audience.

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Review by Brian Clegg

Game – Guesstimaster ****

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Some while ago on our old site I reviewed a game called The Art of Science, which was a science-based quiz. Although I thought the game was great, I have had a lot of trouble Screenshot_24_04_2013_16_20persuading a large enough group to play it – those who usually resort to any question but science in Trivial Pursuit (and that’s quite a lot of people) would struggle hugely. The trouble is that unless you are playing in an academic institution, the chances are there will be a proportion of people around the table who just aren’t interested in science.

Now we have another game from the same people where numbers are at the heart of things, but I think (I hope) that it will be more of a general interest. That’s because it (thankfully) isn’t a mathematical general knowledge quiz, but instead a quiz where the aim is to guess closest at the size of a number (how many hairs on a typical human head, for instance), with points for getting in the right order of magnitude and for being closest, plus an optional equivalent of the old Monopoly Chance cards.

Apart from the name, which like The Art of Science is a bit clumsy I think this game has great potential. I love the board which is a logarithmic spiral growing from 1 to vast numbers – it even has little hints along the way to the kind things that are on that scale.

Time will tell if it will be as difficult to get people to play Guesstimaster as its predecessor… but I hope not, because it’s a great idea. It’s not cheap – £39.59 including shipping – but it is something that is different and well worth a look.

Available from the Academic Board Games web shop.

Review by Brian Clegg

Dice World – Brian Clegg *****

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As human beings we are adept at seeing patterns. It’s how we understand the
world. But as Dice World makes plain, reality is all too often driven by
DiceWorldrandomness, without a pattern in sight. At an entertaining canter, Brian Clegg takes us through the way superstition turns correlation into causality; why economists are so bad at predicting real human responses; and how the power of statistics can reveal hidden truths that, if it weren’t for the logical walkthroughs, you just wouldn’t believe. The book starts by showing us how the world seemed an ordered place - briefly in-line with Newton’s clockwork universe – and then how the cracks began to show when it proved impossible to accurately predict the movement of just three bodies in space.

Chaos and randomness intertwine – chaos technically predictable but practically impossible to do so, while true randomness, the behaviour at the heart of quantum theory is totally unpredictable but often fits neat distributions. You’ll meet the smartest person in the world – and strange creatures like Schrödinger’s cat and
Maxwell’s demon; see why a window at night is a fiendishly complex quantum
device with randomness and probability at its heart; and find out what’s
going on with entropy, the end of the universe and free will. Oh and
discover how to get the best prediction of whether or not someone owns a
golden retriever.

In equal parts fascinating and mind-boggling this is a real revelation if
you have any interest in why things happen (and why they go wrong). We’re no
good at probability and we hate randomness. We rarely see either of them at
work – and yet they’re everywhere. Clegg has a gift for making this kind of
thing approachable and informative but still fun. With this book to hand
you’ve got your best chance of understanding just what’s going on in the
universe; and to have some laughs along the way. Not to mention discover how
to win a sports car rather than a goat. Which can’t be bad.

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Review by Peet Morris

Please note, this title is written by the editor of the Popular Science website. Our review is still an honest opinion – and we could hardly omit the book – but do want to make the connection clear.

Henri Poincaré – Jeremy Gray ***

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My first sight of this book filled me with a certain unease. It would be polite to call it chunky – in truth, at 542 pages plus appendices, it is obese. This initial feeling was not poincarehelped by a bizarre statement the author makes in the introduction. ‘This is a scientific biography of Henri Poincaré,’ he says. ‘It is confined entirely to his public life: his contributions to mathematics, to many branches of physics, technology, to philosophy and to public life. It presents him as a public figure in his intellectual and social world; it leaves the private man alone apart from a deliberately brief account of his childhood and education.’

No, no, no! This is a bizarre distortion of what a scientific biography should be. I am comfortable with keeping coverage of his childhood and education brief, as they are usually dull and not particularly illuminating. There are clear counter-examples, for example, with Newton’s formative years, which are absolutely crucial in understanding the scientist, but for many, these aspects are fairly irrelevant. But the point of a scientific biography, as opposed to a book about a person’s science pure and simple is that it puts the science into context – and that context must include the private life. Can you imagine a biography of Richard Feynman without his private life coming into it? This is a crazy viewpoint.

Even so I persevered, as I have always had Poincaré in my mind as one of those mathematicians beloved by other mathematicians but of little interest to the real world, so I wanted to find out more about the man (as much as Jeremy Gray would allow me) and his impact on science and technology. It was hard work. There’s an awful lot (some of it truly awful) about the subtleties of philosophy that gets in the way of much of the more interesting content. This is supposed to be a scientific biography, remember, not a philosophical one.

When there is a section that is more of interest (and the way the book is organized does not make it easy to find your way around), frankly it can verge on the unreadable. This is the worst kind of dry academic writing, combined with an approach to the science that is strongly mathematical in flavour and the author lacks any skill in actually explaining the science for anyone who doesn’t know the maths already.

There is always a danger in reading an academic tone and complaining that it’s not popular science because it was never intended to be. And this book is published by Princeton University Press. But I was told it was suitable for a general readership, and this is usually the case with scientific biographies. But I am afraid this is really only suitable for a very narrow audience with a purely academic interest in pure and applied mathematics and the philosophy behind it. Disappointing.

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Review by Brian Clegg

The Great Mathematical Problems – Ian Stewart *****

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As a science writer, whose only foray into maths has been to cover infinity – by far the sexiest and most intriguing mathematical topic – I am in awe of those who successfully popularize maths.

greatmathsBy comparison, science is easy. We all know from school that science can be dull, but if you go about it the right way, it is naturally fascinating, because it’s about how the universe we live in works. Admittedly maths has plenty of applications, but an awful lot of mathematics is about a universe we don’t live in. It can seem that many mathematicians spend their time doing the equivalent of arguing about the dietary habits of unicorns. Not really a proper job for a grown human being.

Probably the best of the current crop of popular maths writers is Ian Stewart. Certainly the most prolific – I don’t know how he finds the time for his day job. Stewart is decidedly variable in his books. Some of them are pure unicorn territory. I find myself turning page after page thinking ‘So what? I don’t care!’ But every now and then he gets it just right – and this is such an example.

Okay, there are occasional unicorn moments, where I had to skip through a page or two to avoid dropping off (when, for example, he gets altogether too excited about the prospect of constructing a regular 17 sided polygon using only a ruler and a pair of compasses), but they are rare indeed. Stewart takes on some of the greatest problems to face mathematicians through history – even the names are evocative, like Goldbach’s Conjecture and, of course, the Riemann Hypothesis. They sound like a Sherlock Holmes story. And Stewart makes them interesting. Which is truly wonderful.

In part the readability is because of a good smattering of stuff about the people – historical context is never more important than in popular maths – but he also pitches the mathematics itself at just the right level to keep our interest without going into mind-numbing detail, or being too summary. I am very wary of describing any book as a tour-de-force, but this one certainly comes close.

Even though Stewart does not keep things enthralling throughout – the dullest chapter is the one on Fermat’s Last Theorem, which I suspect is because Stewart focuses more on the maths here and less on the people, so excellently covered by Simon Singh – there is plenty in this book to keep the imagination alive. If you hate maths this is not going to make you a convert. But if, like me, you have a grudging admiration for maths but find a lot of it impenetrable or pointless, you should have a great time in Ian Stewart’s capable hands.

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Review by Brian Clegg

Thinking in Numbers – Daniel Tammet ****

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This collection of 25 essays by Daniel Tammet, probably best known for his feat of memorising vast quantities of digits of pi, is an enjoyable light way of getting an introduction to some of the reasons that maths is more than just a mechanism for doing science or adding up your shopping bills.

Some essay collections don’t work so well in book form, but these make excellent bite-sized nuggets, with Tammet ranging far and wide over a landscape that successfully pulls in poets, authors and playwrights as much as it does mathematicians. I loved, for instance, the parallels Tammet brings out between Tolstoy’s view of history and calculus.

Inevitably in such a collection there will be some pieces that appeal less to an individual reader. I was less interested in the more autobiographical essays, but I am sure they would appeal to others. If I’m being picky I’d also say Tammet is occasionally a little loose factually. So, for instance, he says the odds of him being in a particular location is 1 in 2 – he’s either there or he’s not. That’s a very strange way of defining odds, which usually means the probability of something: and clearly there isn’t a 1 in 2 chance of him being (say) in my kitchen.

Overall, though, a very enjoyable and informative read.

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Review by Brian Clegg

The Logician and the Engineer – Paul J. Nahin ***

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For its target readership this is an excellent book – and I have to say as someone outside that market I really enjoyed some parts – but the fact remains it is aimed at a pretty narrow segment. There’s even a little section at the front of the book that effectively says ‘read this to see if you can cope with the rest.’

The bits I found particularly appealing were a few introductory logic problems (though I’m not sure I agreed with all  the conclusions) and the pocket biographies of mathematician George Boole and information engineer Claude Shannon. However, while technically qualified to deal with the other parts of the book, in truth I couldn’t be bothered – it was too much like hard work.

For bits of it I would have to wade through far too much grunt maths, and for other bits would have had to think far too hard about electronic circuits and the logic circuits beloved of basement dwellers on computer science courses. (Or was it just my university that confined the computer scientists to the basement?)

I think the author makes the mistake that many academics make when trying to write for a broader audience: they carry through too much of the textbook, and find that the aspects that often encourage people to remember things in that context (often because they involve repetitious grunt work) actually prevent popular science readers from getting the message. It’s a shame, because the subjects are interesting, but unless you are the kind of person who designs logic circuits for fun, this is probably not the book you’d want to see.

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Review by Brian Clegg

Thinking Statistically – Uri Bram ****

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This is a delightful little book (just three chapters) introducing three of the fundamental aspects of statistics that can get us confused: selection bias, edogeneity (effectively missing external factors which are influencing the outcome) and the use of Bayesian statistics, an approach that is very powerful but makes it easy to go astray.

I wouldn’t quite describe this as a popular science book – there are probably rather too many equations – but it is excellent both as providing a bit of understanding for those making use of statistical methods (it’s all too easy to just crank the handle without understanding what you are doing and thereby come up with the wrong results) and as  an introduction for the general reader who isn’t put off by a little bit of jargon and equations in what is, nonetheless, a very readable little book.

Thinking Statistically is short enough to read in a couple of hours, and I think it’s a credit to the author that I thought ‘Oh, really, I wanted more!’ when I got to the end. Uri Bram’s aim is to get the reader taking a more statistical viewpoint. Not necessarily wheeling out the statistical big guns every time you make a decision, but at least being aware of the statistical processes you are undergoing mentally, often unconsciously.

If you would like to know a bit more about statistics, but find the whole business a bit baffling, this is a good place to start.

You may wonder what the cover has to do with statistics. So did I. The simple answer is nothing.

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Review by Brian Clegg

Poor Robin’s Prophecies – Benjamin Wardhaugh ***

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This is an unusual one. It’s reminiscent of that quote on Wagner’s music. Not Woody Allen’s (I can’t listen to that much Wagner. I start getting the urge to conquer Poland.) or Oscar Wilde’s (I like Wagner’s music better than anybody’s. It is so loud that one can talk the whole time without other people hearing what one says.) but Rossini’s - Mr. Wagner has beautiful moments but awful quarters of an hour.

That probably makes the book sound worse than it is (unless you like Wagner). The concept is brilliant. It is looking back at a seventeenth/eighteenth century phenomenon and using it as a hook on which to hang an assessment of the everyday approach to maths in England in that time. The phenomenon in question is Poor Robin’s Prophesies, a long running almanac. In general almanacs were annual publications that threw in what the authors thought of as lots of useful information, from saints’ days to tide tables, with a good dollop of astrology to add zest. Poor Robin was initially primarily a satirical attack on the other almanacs, including saints days like Robin Hood and the day Jane fell off the hen-roost.

Author Benjamin Wardhaugh is at his best when looking at the almanacs and their quirky view on life in those interesting times. Where the book falls down a little is the lengthy sections on how the basics of maths were taught back then, including lengthy commentary on some maths notebooks of the period. I am interested in maths, but these parts left me cold.

There is no doubt there are some real delights here, primarily in the bits that have little to do with science or maths and everything to do with the culture of the period. And it should be of interest to any historian of mathematics. But it’s not a book for everyone.

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Review by Brian Clegg

The Signal and the Noise – Nate Silver ****

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It was really interesting coming to this book soon after reading The Black Swan, as in some ways they cover similar ground – but take a very different approach. I ought to say straight away that this book is too long at a wrist-busting 534 pages, but on the whole it is much better than its rival. Where Black Swan is written in a highly self-indulgent fashion, telling us far too much about the author and really only containing one significant piece of information, Signal and Noise has much more content. (Strangely, the biggest omission is properly covering Taleb’s black swan concept.)

What we’re dealing with is a book about forecasting, randomness, probability and chance. You will find plenty about all the interesting stuff – weather forecasting, the stock market, climate change, political forecasts and more, and with the exception of one chapter which I will come back to in a moment it is very readable and well-written (though inevitably takes a long time to get through). It has one of the best explanations of Bayes’ theorem I’ve ever seen in a popular science book, and (properly to my mind) makes significant use of Bayesian statistics.

What’s not to like? Well, frankly, if you aren’t American, you might find it more than a trifle parochial. There is a huge section on baseball and predicting baseball results that is unlikely to mean anything to the vast majority of the world’s readers. I’m afraid I had to skip chunks of that. And there’s a bizarre chapter about terrorism. I have two problems with this. One is the fawning approach to Donald Rumsfeld. Nate Silver seems so thrilled Rumsfeld gives him an interview that he treats his every word as sheer gold. Unfortunately, he seems to miss that for much of the world, Rumsfeld is hardly highly regarded (that parochialism again).

There is also a moment where Silver falls for one of the traps he points out that it’s easy to succumb to in analyzing data. On one subject he cherry picks information to present the picture he wants. He contrasts the distribution of deaths in terrorist attacks in the US and Israel, pointing out that where the US numbers follow a rough power law, deaths in Israel tail off before 100 people killed in an incident, which he puts down to their approach to security. What he fails to point out is that this is also true of pretty well every European country, none of which have Israeli-style security.

I also couldn’t help point out one of the funniest typos I have ever seen. He quotes physicist Richard Rood as saying ‘At NASA, I finally realised that the definition of rocket science is using relatively simple psychics to solve complex problems.’ Love it Bring on the simple psychics.

Overall, despite a few issues it was a good read with a lot of meat on probability and forecasting and a good introduction to the basics of Bayesian statistics thrown in. Recommended.

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Review by Brian Clegg