“Measuring the world” – Daniel Kehlmann

What a fantastically funny read! “Die Vermessung der Welt” is two biographies in one, lovingly weaving together the lives of Carl Friedrich Gauß (Gauss) and Alexander von Humboldt, both of whom lived and changed German science in the late 18th and early 19th century.

Both men, as good scientists, had an obsession with knowledge. Humboldt, tiny and possibly homosexual Prussian aristocrat with a famous older brother and a thirst for adventure, without much talent for human interaction, tirelessly and with little regard for his fellow travellers, measured most of South America, in the process accidentally tasting human flesh, forming strange attachments to a stray dog, mapping the canal connecting the Amazon and Orinoco rivers and stealing human corpses. Gauß, from a poor family and reliant on patronage, impatient with the slow wit of everyone else he met, at least partially aware of the social faux-pas he committed, turned his mind to several big mathematical problems at a time and completed his masterpiece, the Disquisitiones Arithmeticae, in his early 20s, and was known as the prince of mathematicians.

Kehlmann’s genius lies in the exclusive use of indirect speech between all of his characters, which creates a comical distance between reality and these extraordinary protagonists although I cannot vouch for the English translation as I read this book in German.

This book has to be taken with a pinch of salt, however, as some liberties appear to have been taken with facts and many anecdotes are so droll they must have been invented by the author. But the alternating (and then joining) chapters of Humboldt versus Gauß are so hilarious and the characters Kehlmann shapes so infuriatingly strange and German that I personally wouldn’t care if it was pure fiction.

“Small World” (or “Nexus”) – Mark Buchanan

It was a mistake to read both Barabási’s “Linked” and this book by science journalist Mark Buchanan. At least in such a short space of time, but I’m tempted to say you’ll really ever only need one or the other. Published within a few months of each other in 2002, not only do these two books cover exactly the same topic (small world, scale-free networks in everday life) but they use the same anecdotes (lame Bill Clinton jokes, Internet hacker stories, the excitement of having an Erdös number – mine is 4!) and even some of the same chapter titles. I can only imagine how mortifying it must have been to find out about each other.

As this review must necessarily be a comparison to the book I read first, let me just say they both have slightly different focal points. Although both authors are physicists by training, Buchanan gives a little more detail about the biological networks found, like the neural connections of C.elegans, while Barabási concentrates more on his own playground – the internet and WWW.

Overall, Buchanan’s style surprisingly is a little less readable than Barabási’s, or maybe I was just not as tolerant the second time round. There is quite an abstract chapter about rivers and the modelling of their meandering course to the sea and Small World is longer without saying much more than Linked.

To be fair, I would still recommend this book as an introduction to network theory. I just felt a little cheated.

“Linked” – Albert-László Barabási

Having some vague notions about connectedness and network topologies I had been looking to learn more about this buzz area. If you are in the same boat, “Linked” is an excellent place to start as it offers nicely sized chunks of information on scale-free networks, power law distributions and the small world phenomenon.

The subtitle “How everything is connected to everything else and what it means for business, science and everyday life”, while being too long to include in the title of this post, gives an idea of what this book is all about.

For example, the omnipresent Six Degrees of Separation is explained and de-mystified (did you know it was first documented in a Hungarian short story and that Kevin Bacon is not actually the most connected actor in Hollywood?) to show that there is a short path between any two nodes (proteins, actors, websites, friends) in many networks, even though that number is not necessarily six.

Barabási opens each chapter with a human-interest anecdote; this and the fact that the only equations he uses are demoted to footnotes make for fluent reading and an accessible style. You will read about the 9/11 attacks, the Clinton-Lewinsky scandal, networks of cellular metabolism, the Internet and the World Wide Web (perhaps, like me, you will finally understand what the difference between the latter two is).

“When will a thinking machine, orders of magnitude faster than a human brain, emerge spontaneously from billions of interconnected modules?” Apart from a tiny sub-chapter about how our wired up planet might give rise to a self-aware super-computer, Barabási’s portrait of networks is all positive and instills in the reader the excitement that made him study them in so much depth. I couldn’t recommend it more.

“The Maths Gene”, Keith Devlin

This book had been on my shelf for months, but I’d been looking forward to the promised explanation of “why everyone has it, but most people don’t use it”. Devlin is quick to point out that the maths gene is of course a metaphor for an innate ability and he then provides lots of evidence for our inherent number sense and mathematical mind.

Cool baby experiments (hooray!) illustrate his points by showing that in a human’s first year of life, “number is apparently a more important ‘invariable’ than color, shape or appearance”: A baby’s surprise, measured by amount of time spent looking at a presented object, is much greater when two objects turn into one than when, say, a ball turns into a rattle.

Another memorable example of how deeply anchored our sense of arithmetic is invokes the mental number line that most people “see” when picturing numbers. Although for most of us this is subconscious, it is nice to find out that I am one of 14 % of people who are aware of doing it. I felt personally addressed again when Devlin mentions how people who become fluent in a second language keep doing arithmetic in their native tongue because it is remembered by sound patterns.

One of his most important messages is the distinction between arithmetic and higher mathematics. While many people (proudly) claim to be “terrible at maths”, the times tables have little to do with mathematics. Maths is instead the science of patterns; patterns like fractals, for example, self-similar figures like broccoli or the Koch snowflake, made of ever-smaller triangles stuck onto the middle of each side of an equilateral triangle. This is fascinating stuff and it’s great to have mathematics pointed out in everyday phenomena: Animal fur, flower shapes and even wallpaper all get fair mention. Elsewhere, Devlin offers a slightly cheesy mathematical soap opera and a fake missing person story to make the reader take note. Maybe he keeps his examples deliberately down-to-earth to fight against the stereotypes of the head-in-the-clouds mathematician, but I don’t care – it works.

A lot of effort is spent on linking maths to language, because Devlin’s main argument is that mathematical and linguistic ability are two parts of the same coin. This is a nice concept but given that I was slightly bored with The Language Instinct, there were too many language syntax trees in this book for me.

Devlin advances his own theory of why all humans have the capability for abstract thought, which he equates to mathematical ability, and it goes something like this: during evolution, the brains of hominids grew and changed structure, which allowed for off-line thought, which was a great advantage and led to maths and language. It takes several chapters to arrive at this point and I’m in no real position to summarise but I think those are the bones of the theory.

As weak Homo sapiens became able to recognise and act on more and more intricate (abstract) patterns, he didn’t have to spend as much time and effort on just surviving and hence we can spend our time today reading, painting, travelling and carrying out other useless activities.

“I do not believe that a basic mathematical ability is any more unusual than an ability to talk”, Devlin says, but people are put off by the notation and, often, teaching methods. “It is a great pity that for so many years our teaching methods have obscured one of humankind’s greatest conceptual inventions”.

This book is really quite the declaration of love to higher mathematics and should be of considerable interest to non-mathematicians. Hopefully you’ll come out of reading “The Maths Gene” convinced that mathematics, rather than being all about numbers and equations, is the discovery of fundamental facts in an abstract world invented by the human brain.