For a certain kind of public intellectual, it is a badge of honour to confess mathematical illiteracy. Anybody who confesses alphabetical illiteracy instead would be considered an object of pity labouring under a crippling disability. In fact, about the same number of people (3-6 per cent of the population) suffer from dyscalculia as from dyslexia, while possessing otherwise normal intelligence. But dyscalculia is much less studied because claims of innumeracy are more common and socially acceptable.
If you think about it, this is a strange reflection on our cultural values. Mathematical ability is more fundamental to the human condition than writing. Tribes in the Amazon and New Guinea live perfectly happily without any concept of writing. But they all possess numerical systems, even if it is as rudimentary as “one”, “two” and “many”.
When Bellos (who studied maths and philosophy) became a journalist, he was puzzled by innumeracy among his colleagues. (Scarcely the only fault of the profession!) So, he started investigating, not maths but the history, sociology and neuro-scientific research that had gone into understanding how the brain abstracts concepts of space, time and quantity.
Anthropological (and zoological) research suggests that the ability to compare quantities is crucial. How many fruit on a tree? How many enemies in the jungle? The earliest evidence of counting is a 35,000-year-old baboon fibula marked off as a tally stick.
To assuage his curiosity about how human beings think mathematically, Bellos embarked on a journey through “numberland”. He examined the ways in which different cultures handled maths problems and the development of the edifice of modern maths. That led to interactions with an array of maths researchers, gameplayers, social scientists, neuro-scientists and zoologists investigating animal intelligence.
None of the maths in “numberland” is terribly complicated. Anybody who passed their secondary school should be able to follow it. It is more about the evolution of mathematical thought and method, presented in a quirky and entertaining manner with many anecdotes thrown in.
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The basic material is standard. There are no rigorous proofs on offer. Nor does Bellos venture into heavily philosophical terrain like Hofstadter. He starts with counting, and describes how different cultures count. He follows through with the development of different methods of math notations, the invention of the zero, different number bases, etc.
Then he moves onto Pythagoras, the eternally fascinating Pi and other transcendentals, algebraic equations, infinite series, golden ratios, etc. The later chapters that centre on probability theory, combinatorials, set theory, statistical distributions and topology might mildly stretch somebody who has forgotten (or bunked) the requisite high-school and undergrad classes.
The presentation is unfailingly entertaining. In every instance, the author has managed to compile an interesting combination of trivia, historical details and unusual playful insights and lemmas. For example, Lincolnshire shepherds used a base 20 number system and chimpanzees actually outgunned humans at high-speed numerical versions of Kim’s game. An ape that can count will register numbers flashed on a computer screen much quicker than humans do.
There’s a chapter entirely devoted to recreational maths. He explains the underlying concepts of sudoku, for instance. In addition, each chapter contains many interesting asides and divergences. There’s a sidestep into numerology and a section on the art of lightning-fast calculations via computational shortcuts. Hindutva fanatics will be offended by his very sceptical take on vedic “science” as opposed to vedic maths but the explanation of the slokas of vedic maths and their tricks of cross-multiplication are very well-presented.
Some of the material will be completely new to average, even mathematically aware readers. Bellos went to Japan to study both the abacus and origami (now at the cutting-edge of topological investigations). He comes up with some unorthodox origami theorems and with a very lucid explanation of how abacus usage and mental abacus training work.
There is an awesome description of a nine-year mental abacus champion multitasking as she plays the Japanese equivalent of antakshari, while adding 30 consecutive numbers in her head. There is also an introduction to Cornell topologist Daina Taimina’s experimental topological crochet patterns of shapes that nobody had ever tried to physically map.
Rather than head into metaphysics, the author focuses on cataloguing the findings and speculations of neuro-science in studying maths thinking. Somebody counting by abacus (or mentally using abacus-methods) is employing entirely different parts of the brain from somebody counting conventionally. Children, Amazon tribesmen and synaesthetics also seem to “see” the number line differently from adults who are trained to count.
There are interesting philological speculations. Does chanting tables musically or using a language where the counting system is consistent and regular (“ten-one” rather than “eleven”) lead to quicker learning? Does it help mnemonically if, as in Sanskrit, each number is associated with attributes (3=fire, 4=wind, etc)?
Bellos has an easy, approachable writing style as well as excellent teaching skills. That’s an unusual and felicitous combination in a maths geek. He’s not the late Martin Gardner and he’s not trying to be. This is not a book of puzzles though it can be treated as such.
It’s an excellent introduction to the world of numbers and to our enduring relationship with numbers. Coupled up to the superb efforts of Apostolos Doxiadis (Logicomix, Uncle Petros and Goldbach’s Conjecture) and Simon Singh (Fermat’s Last Theorem, The Code Book), it marks a welcome new trend in popularising mathematics.
ALEX’S ADVENTURES IN NUMBERLAND
Alex Bellos
Bloomsbury
433 pages; £18.99
