Popular Maths Books

A big problem with doing an undergraduate degree in mathematics is that you do so much maths you get bored with it. This is made worse by the facts that learning maths is hard work and a lot of what you learn isn't all that interesting. You need it to do interesting stuff later on, in the same way as you need to learn how to spell to read interesting books, but sometimes learning maths is as tedious as learning to spell was when you were much younger.

To keep up your interest in maths I suggest you read some popular mathematics. Here are some suggestions. These are mostly fairly easy reading but will teach you important and interesting things about your discipline that we don't teach in lectures. Most of these are in the library - if they aren't, ask the library to order them.

This is a very incomplete list. Let me know if there's anything you'd like added.

The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, Keith J. Devlin.

At the start of the twentieth century the mathematician David Hilbert published a set of important problems (I think 23) that mathematicians should try to solve before the year 2000. At the start of this century the Clay Foundation in the US published a set of 7 problems, and offers $1,000,000 prizes for each. See http://www.claymath.org/millennium/. This book describes the problems in a form intended to be understandable by the non-specialist. I haven't seen the book, just a review that says Devlin fails to make the problems understandable (understandably) but that doesn't prevent it being very entertaining.

Uncle Petros and Goldbach's Conjecture, Apostolos Doxiadis.

Goldbach's Conjecture is a famous unsolved problem in number theory. It states that every even integer greater than 2 can be written as the sum of two prime numbers: 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, etc. It's been checked up to some huge number, and has inspired lots of partial results, incorrect proofs and new mathematical ideas. This is a novel about Uncle Petros who has gone mad trying to prove the conjecture and thinks he's done it. The author trained as a mathematician so the mathematical part is fairly good. An interesting feature is that a lot of real mathematicians appear as characters.

The Colossal Book of Mathematics, Martin Gardner.

For a long time Martin Gardner wrote a column called Mathematical Games in the Scientific American. He did this extraordinarily well and inspired many incipient mathematicians. This book constitutes 50 essays that originally appeared in there. He's written a lot of other excellent books, so if you can't get this one you should be able to find something else by him.

A Mathematician's Apology, G H Hardy.

Hardy was one of England's greatest twentieth century mathematicians. The Apology is an autobiography in which he explains what it's like to be a mathematician. It is much quoted by mathematicians and will tell you why the number 1729 is so important. Read the long introduction too.

Gödel, Escher, Bach, Douglas Hofstadter.

This was a cult book when it came out in 1979. Gödel was a mathematician/logician whose famous theorem states (roughly) that in any mathematical system there are statements that are true but unprovable. Escher was an artist who produced mathematical-looking and paradoxical pictures which you will have seen on t-shirts, posters etc. Bach is the composer. The book is a blend of mathematics, computer science, philosophy, musical theory and Lewis Carroll. It's very thick and few people have read it all the way through. However it's certainly worth looking at.

The Man Who Loved Only Numbers, Paul Hoffman.

The story of Paul Erdös. Erdös was an eccentric, brilliant and homeless mathematician who travelled the world giving lectures at universities and publishing a huge number of papers. For most of these he was a co-author with other mathematicians. It has become a tradition to say that your Erdös Number is 1 if you've published a joint paper with Erdös. If you haven't, but have published a joint paper with someone who has an Erdös Number of 1, then you get an Erdös number of 2, and so on. Lou Caccetta's Erdös Number is 1. Most of the other people in the Department have Erdös Numbers of 2. The American Mathematics Society generalises this idea at their MathSciNet site by providing a Collaboration Distance calculator.

The man who knew infinity, Robert Kanigel.

A biography of the Indian mathematician Srinivasa Ramanujan. Ramanujan was poor, brilliant, largely self-educated and dramatically discovered by Cambridge mathematicians Hardy and Littlewood who brought him to England. His health was ruined by English weather and food, and he returned to India to die.

A Beautiful Mind, Sylvia Nasar.

The book of 1998 that was made into a 2001 film of the same name, about Nobel Prize-winning schizophrenic genius John Nash. The film won the Oscar for Best Picture in 2002.

Notes on Fermat's Last Theorem, Alf van der Poorten, and Fermat's Enigma: the Epic Quest to Solve the World's Greatest Mathematical Problem, Simon Singh.

Everybody knows that 3² + 4² = 5² and many people know that 5² + 12² = 13². What about cubes instead of squares? It turns out that there are no positive integers a, b, and c such that aⁿ + bⁿ = cⁿ when n = 3 or when n is any larger integer. Pierre Fermat claimed to have a proof for this result around 1650 but said he didn't have enough room to write it in the margin of the book he was reading. Mathematicians then spent three and a half centuries trying to find proof, with Englishman Andrew Wiles eventually succeeding in 1994. Until then this was the most famous unsolved problem in mathematics (there's debate about the identity of the present front runner). There are a huge number of books written about the theorem and its proof. I mention these two because Singh's is very readable and because the author of the other much more technical work is a friend.

Pythagoras' Trousers, Margaret Wertheim.

More about physics than mathematics. It's an interesting history of modern physics with emphasis on the patriarchal nature of the discipline. The author sees physics as being similar to the Catholic Church in the ways it excludes women.

A New Kind of Science, Stephen Wolfram.

Wolfram is the founder of Wolfram Research which produces the Mathematica computer algebra system (an alternative to Maple). As a mathematician he worked on things called cellular automata, and the book argues that these are the answer to everything. The review I saw said the book was fascinating but Wolfram tends to give himself credit that should be shared with others.

Leaning towards Infinity, Sue Woolfe.

Here's a review from Kirkus Associates:

“Creating the feeling of a found document, prize-winning Australian writer Woolfe pieces together an intriguing and expansive novel of ideas–showing the ways in which love, motherhood, and mathematics wrap around the human soul. Three generations of Montrose women emerge from the narrative: Hypatia writes of her legendary mother Francis, a gifted and acclaimed mathematician; Francis, in turn, tells the story of her mother, the brilliant, breathtaking Juanita. Meanwhile, Hypatia frequently offers her own narrative in the form of disgruntled letters to Francis, or in the form of brief biographical commentaries on some of history's great mathematicians. The staggered segments of personal and historical chronology help shape the central story of Francis Montrose, who discovers for the world a whole new kind of number. Having devoted her life to building on the work of Juanita, Francis, a ridiculed amateur, is invited to a mathematics conference in Athens to present her incomprehensible conjectures, which are of “such fierce, austere beauty, you might think God is real.” What she is really hoping to give the world is a tribute to her beautiful, aloof mother. Juanita, raised in an Australian convent when her Spanish father was mysteriously assassinated and her mother took to gambling, is a savant, a secret mathematical genius who spends her later married life scribbling groundbreaking theories on scraps of paper. Trapped in a life of domesticity while dreaming of infinity, she pins her hopes on her beautiful son, but it is the plain and ignored Francis who inherits the gift of the abstract mind and becomes obsessed with becoming her mother. As the climax of the story, Hypatia tells of the renowned “missing days” when Francis completes her theory on a deserted Greek beach and finally slips out from under the domineering ghost of Juanita. A lovely novel, magical in its elevation of mathematics into a realm of divine beauty, charming in its depiction of the equally demanding sphere of motherhood.”
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