Mathematics: A Very Short Introduction by Timothy Gowers, on Learning to Think Abstractly
Although mathematics is constantly present in our daily lives, the subject itself still meets some resistance. To make it more accessible and understandable, Timothy Gowers, a Fields Medal winner, uses different philosophical approaches to show the most important differences between advanced mathematics and the mathematics we learn at school.
Author: Timothy Gowers, Fields Medal 1998
Title: Mathematics: A Very Short Introduction (2002)
Format: English paperback (Oxford University Press), 160 pages
Scope: A short, non-technical introduction to how mathematicians think
Amazon: www.amazon.es/dp/0192853619 (ASIN 0192853619)
Learning to Think Abstractly
In the preface, Gowers explains that the main aim of the book is to help people learn to think abstractly, because when they do, many philosophical difficulties simply disappear. That means understanding more clearly concepts such as infinity, curved space, and rational and irrational numbers.
Models, Proofs and Estimates
Along the way he explains what a mathematical model is, why proofs matter, how mathematicians use approximations and estimates, and why high-dimensional geometry is less mysterious than it sounds. The examples are well chosen and the tone is friendly, never condescending.
What Is Inside
The chapters move from models, where Gowers shows how mathematicians simplify a real situation into something they can reason about, to numbers and abstraction, proofs, limits and infinity, dimension, geometry and estimates. Along the way he gives some classic arguments in full, such as the proof that the square root of two cannot be written as a fraction, so the reader experiences mathematical reasoning rather than just hearing about it. The final chapter answers frequently asked questions about mathematicians themselves, including whether they really do their best work young.
The Abstract Method
Gowers’s central idea is that a mathematical object is defined by what you can do with it, not by what it "really is". Chess offers his favourite analogy: the king is not a piece of wood but whatever obeys the rules for moving a king. Once you think of numbers, infinity or a four-dimensional space in that way, many puzzles that trouble beginners, such as what the square root of minus one "is", lose their mystery. It is a liberating idea for anyone who was put off mathematics at school.
The Author
Sir Timothy Gowers is a Cambridge mathematician who received the Fields Medal in 1998 for work linking functional analysis and combinatorics. He also edited the large Princeton Companion to Mathematics (2008), and he has been influential beyond research: he launched the online Polymath project, in which many mathematicians collaborate openly on a problem, and helped start a boycott of high journal prices in 2012.
Where It Fits
The book belongs to Oxford University Press’s Very Short Introductions series, and it is one of the most admired titles in it. Readers who want more can move on to the Princeton Companion or to popular books such as Ian Stewart’s.
Who Should Read It
Sixth-formers and first-year students wondering what university mathematics is really like, teachers looking for ways to explain why proof matters, and adults who were told at school that they were "no good at maths". Gowers asks for no more than school algebra, yet by the end the reader has a clear idea of how professional mathematicians think. The book is short enough to read in a weekend and rich enough to reread, and the reviewer’s verdict, that it changes the way you look at the subject, is shared by many readers. It is also a good antidote to the idea that mathematics is mostly calculation: Gowers shows that it is above all about ideas and the arguments that connect them.
The People Who Do It
The book ends with a chapter about the people who do mathematics, raising philosophical and sociological questions whose answers help unravel some of the mysteries of the world of numbers. It is a small book that changes the way you look at the subject, and a very good gift for a student about to choose a degree.
As an Amazon Associate, this site earns from qualifying purchases.
Part of the reading recommendations featured on books.sapiensverbum.com.
