Reading list

Mathematics: Reading List

(WITH REFERENCE TO Maths reading list – Cambridge )

Author	Title	Notes	Available in RGS Library?
Abbott, Edwin A.	Flatland: A Romance of Many Dimensions		Yes
Acheson, David	1089 and All That		Yes
Alcock, Lara	How to Study for a Mathematics Degree
Averbach, Bonnie and Chein, Orin	Problem Solving Through Recreational Mathematics	One can never have enough maths puzzles! This is another great collection, from easier ones to some that will leave you stumped, through quite enough variety to please all tastes, and to give an introduction to all the main areas of mathematics while you have fun making your way through it. Lots of practice problems, and hints and solutions to most puzzles.
Bellos, Alex	Alex’s Adventures in Numberland		Yes
Berlinski, David	Infinte Ascent – A Short History of Mathematics
Bewerdorff, Jörg	Luck, Logic and White Lies	Learn about (some of) the maths behind risk, uncertainty and gambling. Debunk some popular myths, and improve your chances of winning at games! Very readable, clear and practical. A good introduction.
Bollobas, B (ed.)	Littlewood’s Miscellany	This collection, first published in 1953, contains some wonderful insights into the development and lifestyle of a great mathematician as well as numerous anecdotes, mathematical (Lion and Man is excellent) and not-so-mathematical. The latest edition contains several worthwhile additions, including a splendid lecture entitled `The Mathematician’s Art of Work’, (as well as various items of interest mainly to those who believe that Trinity Great Court is the centre of the Universe). Thoroughly recommended.
Bondi, C (ed.)	New Applications of Mathematics	Twelve chapters by different authors, starting with functions and ending with supercomputers. There is material here which many readers will already understand, but treated from a novel point of view, and plenty of less familiar but still very understandable material.	Yes
Burton, David	The History of Mathematics
Cheng, Eugenia	Cakes, Custard and Category Theory: Easy Recipes for Understanding Complex Maths	Entertaining, innovative, and packed with infectious enthusiasm and unexpectedly mathematical recipes. The baking metaphors may seem a little forced at times, but on the whole work well.
Clegg, Brian	A Brief History of Infinity
Cook, Mariana	Mathematics: An Outer View of an Inner World	Another, more modern, biographical approach. This book gives a compelling and immediate introduction to some of the most amazing mathematicians of our time, not just through a glimpse of their brilliant mathematical work, but also of their experience as fathers, daughters, husbands, wives… Each portrait is personal and in the voice of the mathematicians themselves. You will find out what inspired them to pursue maths, and no doubt be inspired yourself to participate in the joy of mathematical discovery.
Courant, Robbins and Stewart	What is Mathematics?	A new edition, revised by Ian Stewart, of a classic. It has chapters on numbers (including 1), logic, cubics, duality, soap-films, etc. The subtitle (An elementary approach to ideas and methods) is rather optimistic: challenging would be a more appropriate adjective, though interesting or instructive would do equally well. Stewart has resisted the temptation to tamper: he has simply updated where appropriate – for example, he discusses the solution to the four-colour problem and the proof of Fermat’s Last Theorem.	Yes
Davis, P.J. and Hersh R.	The Mathematical Experience	This gives a tremendous foretaste of the excitement of discovering mathematics. A classic.	Yes
Derbyshire, John	Unknown Quantity – A Real and Imaginary History of Algebra
Devlin, Keith	Mathematics: The New Golden Age		Yes
Devlin, Keith	The Millennium Problems		Yes
Devlin, Keith	The Unfinished Game
du Sautoy, Marcus	Finding Moonshine: A Mathematician’s Journey Through Symmetry	This book has had exceptionally good reviews. The title is self-explanatory. The book starts with a romp through the history and winds up with some very modern ideas. You even have the opportunity to discover a group for yourself and have it named after you.
du Sautoy, Marcus	The Music of the Primes	This is a wide-ranging historical survey of a large chunk of mathematics with the Riemann Hypothesis acting as a thread tying everything together. The Riemann Hypothesis is one of the big unsolved problems in mathematics { in fact, it is one of the Clay Institute million dollar problems { though unlike Fermat’s last theorem it is unlikely ever to be the subject of pub conversation. Du Sautoy’s book is bang up to date, and attractively written. Some of the maths is tough but the history and storytelling paint a convincing (and appealing) picture of the world of professional mathematics.	Yes
du Sautoy, Marcus	What We Cannot Know: Explorations at the Edge of Knowledge
Dudley, Underwood	Is Mathematics Inevitable? A Miscellany
Dunham, William	Journey Through Genius
Elwes, Richard	MATHS 1001
Elwes, Richard	Maths in 100 Key Breakthroughs
Feynman, R.P.	Surely You’re Joking, Mr Feynman	Autobiographical anecdotes from one of the greatest theoretical physicists of the last century, which became an immediate best-seller. You learn about physics, about life and (most puzzling of all) about Feynman. Very amusing and entertaining.	Yes
Gardner, Martin	The Colossal Book of Mathematics	Over 700 pages of Gardner for under 20 pounds is an astonishing bargain. You will be hooked by the very first topic in the book if you haven’t seen it before (and probably even if you have): a Diophantine problem involving a monkey and some coconuts \| can’t say more without writing a spoiler. At the beginning, about 60 other books by Martin Gardner are listed, none of which will disappoint.	Yes
Gibilisco, S	Reaching for Infinity	A short and comfortable, though mathematical, read about different sorts of infinity. It has theorems, too, which are good for you. An example:א 0 + א 1 = א 1. This probably needs a bit of explanation. Loosely speaking: א 0 (pronounced `aleph’ zero) is the number of integers (which is the same as the number of rational numbers) and א 1 is the next biggest infinity. There is another infinity, c = , which is the number of real numbers. The continuum hypothesis says that c = א 1, but it was not realised until 1963 that this cannot be proved or disproved.
Gleick, James	Chaos: Making a New Science	Sometimes, at interview, candidates are asked whether they have read any good mathematics books recently. There was a time when nine out of ten candidates who expressed a view named this one. Before that, it was Douglas Hofstadter’s Gödel, Escher, Bach (Penguin, 1980). Surely they couldn’t all have been wrong?	Yes
Gleick, James	The Information
Goldstein, Rebecca	Incompleteness – The Proof and Paradox of Kurt Gödel
Gowers, Tim	Mathematics: A Very Short Introduction
Gowers, Timothy	Mathematics: A Very Short Introduction	Gowers is a Fields Medallist (the Fields medal is the mathematical equivalent of the Nobel prize), so it is not at all surprising that what he writes is worth reading. What is surprising is the ease and charm of his writing. He touches lightly many areas of mathematics, some that will be familiar (Pythagoras) and some that may not be (manifolds) and has something illuminating to say about all of them. The book is small and thin: it will fit in your pocket. You should get it.	Yes
Gray, Jeremy	Hilbert’s Challenge
Gura, Ein-Ya and Maschler, Michael M.	Insights into Game Theory: An Alternative Mathematical Experience	This book arose from Ein-Ya Gura’s PhD dissertation. It provides an introduction to the field of Game Theory – the mathematical analysis of competitive strategies – for an audience without a background in higher mathematics. Although the book avoids formal mathematical notation, rigorous proofs are given of some of the major results of the field. And you can also use the many exercises provided to help consolidate the material.
Hall, N. (ed.)	The New Scientist Guide to Chaos	This comprises a series of articles on various aspects of chaotic systems together with some really amazing photographs of computer-generated landscapes. Chaos is what happens when the behaviour of a system gets too complicated to predict; the most familiar example is the weather, which apparently cannot be forecast accurately more than five days ahead. The articles here delve into many diverse systems in which chaos can occur and include a piece by the guru (Mandelbrot) and one about the mysterious new constant of nature discovered by Feigenbaum associated with the timescale over which dynamical systems change in character.
Hardy, G.H.	A Mathematician’s Apology	Hardy was one of the best mathematicians of the first part of this century. Always an achiever (his New Year resolutions one year included proving the Riemann hypothesis, making 211 not out in the fourth test at the Oval, finding an argument for the non-existence of God which would convince the general public, and murdering Mussolini), he led the renaissance in mathematical analysis in England. Graham Greene knew of no writing (except perhaps Henry James’s Introductory Essays) which conveys so clearly and with such an absence of fuss the excitement of the creative artist. There is an introduction by C.P. Snow.
Hellman, Hal	Great Feuds in Mathematics
Hodges, Andrew	Alan Turing: The Enigma	A great biography of Alan Turing, a pioneer of modern computing. The title has a double meaning: the man was an enigma, committing suicide in 1954 by eating a poisoned apple, and the German code that he was instrumental in cracking was generated by the Enigma machine. The book is largely nonmathematical, but there are no holds barred when it comes to describing his major achievement, now called a Turing machine, with which he demonstrated that a famous conjecture by Hilbert is false.
Hodges, Andrew	Alan Turing: The Enigma
Hodgkin, Luke	A History of Mathematics – From Mesopotamia to Modernity
Hoffman, P.	Archimedes’ Revenge	This is not a difficult read, but it covers some very interesting topics: for example, why democracy is mathematically unsound, Turing machines and travelling salesmen. Remarkably, there is no chapter on chaos.	Yes
Hoffman, Paul	The Man Who Loved Only Numbers	An excellent biography of Paul Erdös, one of the most prolific mathematicians of all time. Erdös wrote over 1500 papers (about 10 times the normal number for a mathematician) and collaborated with 485 other mathematicians. He had no home; he just descended on colleagues with whom he wanted to work, bringing with him all his belongings in a suitcase. Apart from details of Erdös’s life, there is plenty of discussion of the kind of problems (mainly number theory) that he worked on.	Yes
Hofstadter, Douglas	Gödel, Escher, Bach: an Eternal Golden Braid		Yes
Hollingdale, S	Makers of Mathematics	There are not many books on the history of mathematics which are pitched at a suitable level. Hollingdale gives a biographical approach which is both readable and mathematical. You might also try E.T. Bell Men of Mathematics (Touchstone Books, Simon and Schuster, 1986). Historians of mathematics have a lot to say about this (very little of it complimentary) but it is full of good stories which have inspired generations of mathematicians.	Yes
Houston, Kevin	How to Think like a Mathematician	This sounds like the sort of book that elderly people think that young people should read. However, there is lots of good mathematics in it (including many interesting exercises) as well as lots of good advice. How can you resist a book the first words of which (relating to the need for accurate expression) are: Question: How many months have 28 days? Mathematician’s answer: All of them.
Kanigel, Robert	The Man Who Knew Infinity	The life of Ramanujan, the self-taught mathematical prodigy from a village near Madras. He sent Hardy samples of his work from India, which included rediscoveries of theorems already well known in the West and other results which completely baffled Hardy. Some of his estimates for the number of ways a large integer can be expressed as the sum of integers are extraordinarily accurate, but seem to have been plucked out of thin air.	Yes
Kline, Morris	Mathematics for the Non-Mathematician
Körner, T.W	Calculus for the Ambitious	You can and should supplement your sixth-form calculus with Körner’s latest offering. You will find here some familiar ideas seen from unfamiliar angles and almost certainly much that is unfamiliar; multivariable calculus for example (when functions depend on more than one variable). This excerpt from introduction gives you a flavour of the style: When leaving a party, Brahms is reported to have said `If there is anyone here whom I have not offended tonight, I beg their pardon.’ If any logician, historian of mathematics, numerical analyst, physicist, teacher of pedagogy or any other sort of expert picks up this book to see how I have treated their subject, I can only repeat Brahms apology.
Körner, T.W	The Pleasures of Counting	A brilliant book. There is something here for anyone interested in mathematics and even the most erudite professional mathematicians will learn something new. Some of the chapters involve very little technical mathematics (the discussion of cholera outbreaks which begins the book, for example) while others require the techniques of a first or second year undergraduate course. However, you can skip through the technical bits and still have an idea what is going on. You will enjoy the account of Braess’s paradox (a mathematical demonstration of the result, which we all know to be correct, that building more roads can increase journey times), the explanation of why we should all be called Smith, and the account of the Enigma code{breaking. These are just a few of the topics Körner explains with enviable clarity and humour.
Kovalevskaya, Sonya	A Russian Childhood	Sonya Kovalevskaya was the first woman in modern times to hold a lectureship at a European university: in 1889 she was appointed a professor at the University of Stockholm, in spite of the fact that she was a woman (with an unconventional private life), a foreigner, a socialist and a practitioner of the new Weierstrassian theory of analysis. Her memories of childhood are non-mathematical but fascinating. She discovered in her nursery the theory of infinitesimals: times being hard, the walls had been papered with pages of mathematical notes.
Lauwerier, H	Fractals, Images of Chaos	Poincaré recurrence, Julia sets, Mandelbrot, snowflakes, the coastline of Norway, nice pictures; in fact, just what you would expect to find. But this has quite a bit of mathematics in it and also a number of programs in basic so that you can build your own fractals. It is written with the energy of a true enthusiast.
Liebeck, Martin	A Concise Introduction to Pure Mathematics	This is really excellent. Liebeck provides a simple, nicely explained, appetizer to a wide variety of topics (such as number systems, complex numbers, prime factorisation, number theory, infinities) that would be found in any first year course. His approach is rigorous but he stops before the reader can get too bogged down in detail. There are worked examples (e.g. `Between any two real numbers there is an irrational’) and exercises, which have the same light touch as the text.
Maor, Eli	To Infinity and Beyond	Not much hard mathematics here, but lots of interesting mathematical ideas (prime numbers, irrationals, the continuum hypothesis, Olber’s paradox (why is the sky dark at night?) and the expanding universe to name but a few), fascinating history and lavish illustrations. The same author has also written a whole book about one number (e The Story of a Number), also published by Princeton (1994, 1998).
McLeish, J	Number	The development of the theory of numbers, from Babylon to Babbage, written with humour and erudition. Hugely enjoyable.
Odifreddi, Piergiorgio	The Mathematical Century: The 30 Greatest Problems of the Last 100 Years
Parker, Matt	Things to Make and Do in the Fourth Dimension
Paulos, J.A.	Beyond Numeracy	Bite-sized essays on fractals, game-theory, countability, convergence and much more. It is a sequel to his equally entertaining, but less technical, Numeracy.	Yes
Pesic, Peter	Abel’s Proof
Pickover, Clifford A.	The Mαth βook	The subtitle is `From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics’. Each left hand page has a largely non-mathematical description of one of the great results in mathematics and each right hand page has a relevant illustration. There is just enough mathematical detail to allow you to understand the result and pursue it (if you fancy it), via google. The book is beautifully produced. The illustration for the page on Russell and Whitehead’s Principia Mathematica, said here to be the 23rd most important non-fiction book of the 20th century, is the proposition occurring several hundred pages into the book, that 1 + 1 = 2.	Yes
Polya, George	How to Solve It		Yes
Polya, George and Kilpatrick, Jeremy	The Stanford Mathematics Problem Book: With Hints and Solutions
Seife, Charles	Zero: The Biography of a Dangerous Idea
Sewell, Michael (ed.)	Mathematics Masterclasses: Stretching the Imagination
Siklos, Stephen	Advanced Problems in Mathematics: Preparing for University	This is a combined and much improved version by Stephen Siklos of his two previous booklets on STEP problems: Advanced Problems in Core Mathematics (2003) and Advanced Problems in Mathematics (1996). It is recommended as preparation for any undergraduate mathematics course, even for students who do not plan to take the Sixth Term Examination Paper (STEP- the examination normally used as a basis for conditional offers to Cambridge). It contains a selection of STEP{like problems complete with discussion and solutions. The problems are different from most A-level questions, being much longer (`multi-step’ is the current terminology) and sometimes covering material from apparently unconnected areas of mathematics. They are more like the sort of problems that you encounter in a university mathematics course, although they are based on the syllabuses of school mathematics. Working through one or both of these booklets would be an excellent way of getting your mathematics up to speed again after the summer break. The book is free to read online at http://www.openbookpublishers.com or can be downloaded as a pdf for a small fee, or bought as a paperback.	Yes
Singh, Simon	Fermat’s Last Theorem	You must read this story of Andrew Wiles’s proof of Fermat’s Last Theorem, including all sorts of mathematical ideas and anecdotes; there is no better introduction to the world of research mathematics. You must also see the associated BBC Horizon documentary if you get the chance. Singh’s later The Code Book (Fourth Estate) is not so interesting mathematically, but is still a very good read.	Yes
Singh, Simon	The Code Book		Yes
Smullyan, Raymond S.	Logical Labyrinths	This book is a fun and engaging collection of logical puzzles, combined with a rigorous mathematical introduction to logic. The carefully graded and entertaining progression leads you to the more formal logical reasoning through a journey that is always challenging enough but manageable and rewarding. Enjoy!
Stewart, Ian	17 Equations That Changed the World		Yes
Stewart, Ian	Flatterland: Like Flatland Only More So
Stewart, Ian	From Here to Infinity	This is a revised version of Problems in Mathematics (1987); revised of necessity, as the author says, because some of the problems now have solutions \| an indication of the speed at which the frontiers of mathematics are receding. Topics discussed include solving the quintic, colouring, knots, infinitesimals, computability and chaos. In the preface, it is guaranteed that the very least you will get from the book is the understanding that mathematical research is not just a matter of inventing new numbers; what you will in fact get is an idea of what real mathematics is.
Stewart, Ian	Game, Set and Math	Stewart is one of the best current writers of mathematics (recreational or otherwise). This collection (which includes a calculation which shows why you need only be marginally the better player to win a tennis match \| whence the title) was originally written in French: some of the puns seem to have suffered in translation, but the joie de vivre shines through. You might also like Stewart’s book on Chaos, Does God Play Dice? (Penguin, 1990). Excellent writing again but, unlike the chaos books mentioned below, no colour pictures. The title is a quotation from Einstein, who believed (probably incorrectly) that the answer was no; he thought that theories of physics should be deterministic, unlike quantum mechanics which is probabilistic.	Yes
Stewart, Ian	Letters to a Young Mathematician
Stewart, Ian	Professor Stewart’s Cabinet of Mathematical Curiosities		Yes
Stillwell, John	Mathematics and Its History
Surowiecki, James	The Wisdom of Crowds
Tao, Terence	Solving Mathematical Problems	Tao is another Fields Medalist. He subtitles this little book `a personal perspective’ and there is probably no one better qualified to give a personal perspective on problem solving: at 13, he was the youngest ever (by some margin) gold medal winner in International Mathematical Olympiad. There are easy problems (as well as hard problems) and good insights throughout. The problems are mainly geometric and algebraic, including number theory (no calculus).
Wells, David	The Penguin Dictionary of Curious and Interesting Numbers	A brilliant idea. The numbers are listed in order of magnitude with historical and mathematical information. Look up 1729 to see why it is `among the most famous of all numbers’. Look up to discover that this is the density of closely-packed identical spheres in what is believed by many mathematicians (though it was at that time an unproven hypothesis) and is known by all physicists and greengrocers to be the optimal packing. Look up Graham’s number (the last one in the book), which is inconceivably big: even written as a tower of powers it would take up far more ink than could be made from all the atoms in the universe. It is an upper bound for a quantity in Ramsey theory whose actual value is believed to be about 6. A book for the bathroom to be dipped into at leisure. You might also like Wells’s The Penguin Dictionary of Curious and Interesting Geometry (Penguin, 1991) which is another book for the bathroom. It is not just obscure theorems about triangles and circles (though there are plenty of them); far-reaching results such as the hairy ball theorem (you can’t brush the hair at everywhere) and fixed point theorems are also discussed.

Mathematics and Philosophy: Reading List

Author	Title	Available in RGS Library?
Ayer, A.J.	The Central Questions in Philosophy	Yes
Blackburn, Simon	Think
Glover, Jonathan	Causing Death and Saving Lives	Yes
Hollis, Martin	Invitation to Philosophy
Moore, A.W.	The Infinite
Nagel, Thomas	What Does It All Mean?	Yes
Plato	One or more of Protagoras, Meno or Phaedo
Russell, Bertrand	The Problems of Philosophy	Yes
Strawson, P.F.	Introduction to Logical Theory