Mathematics: A Very Short Introduction

This short volume is an exciting and stimulating read! And it is also a rewarding experience for any non-mathematician, like myself, to be able to follow the carefully crafted arguments of one the world's leading mathematicians (see Tim Gowers' biography on wikipedia).There are three different ways of describing mathematics. The first is historical (or anthropological) and involves analyzing the development of the mathematics embedded in the various activities that are foundations of human civilization (accounting, navigation, architecture, astronomy, engineering, academic mathematics etc). The second is epistemological: that is, giving an account of mathematics as a branch of human knowledge and analyzing its distinguishing characteristics. Lastly, mathematics can be described by giving a tour of mathematics itself. This is the approach of the classic in this field, "What is Mathematics" by Richard Courant and, more recently, "Concepts of Modern Mathematics" by Ian Stewart. Timothy Gowers' "Mathematics A Very Short Introduction" is a combination of the latter two approaches. It is an extremely clear and accessible account of the analytic description of mathematics. At the same time the author manages to describe many of the key concepts of modern mathematics.The first chapter, "Models" presents mathematics as a way of modelling the world: kinematics, mechanics, fluid dynamics, computer networks etc. The implication is clear: the underlying power of mathematics lies in its ability to simplify and abstract features of reality. Throughout the book, the author refers to this as the "abstract method." This is the leitmotiv of the book. The author's intention is to demystify any Platonic notion of the autonomous existence of mathematical objects. He proposes the adage: "a mathematical object is what it does."The second chapter, "Numbers and Abstraction" illustrates the autonomy of the abstract method. The adage "a mathematical object is what it does" is illustrated by showing that all numbers are a consequence of applying a set of rules to counting numbers: - The commutative law for addition and multiplication - The associative law for addition and multiplication - The distributive law - The rules for multiplicative and additive inversesNegative numbers, rationals, irrational numbers and, ultimately, complex numbers are simply a consequence of applying these rules. Zero is similarly explained as the application of a rule: "From the abstract point of view, however, zero is very straightforward – it is just a new token introduced into our number system with the following special property: 0 is an additive identity: 0 + a = a for any number a. That is all you need to know about 0".The third chapter discusses the concept of mathematical proof. The author takes the classic proof by contradiction of the irrationality of root 2 as his starting example. However, instead of merely presenting the bare bones of the proof (as most authors do) he probes deeper by examining individual steps of the proof. This leads into a more general discussion of the nature of mathematical proof: - Why a mathematical proof is incontrovertible - Why many statements that seem intuitively obvious actually require proofs (and some of these can be extremely advanced) - And lastly, a pedagogical (or psychological?) observation: "if you do try to prove statements then you will understand them in a completely different and more interesting way."The fourth chapter, "Limits and Infinity" describes the underlying concepts of calculus. In keeping with the underlying analytic approach of the work, the concept of "infinity" is presented as a shorthand for a special kind of approximation: the limit. "Once again, we are regarding a statement that involves infinity as a convenient way of expressing a more complicated statement concerning approximations. Another word, which can be more suggestive, is 'limit'".Chapters 5 and 6 deal with the concepts of dimension and geometry respectively. They provide a really simple and clear introduction to the concepts of non-Euclidean geometry, and n-dimensional spaces (where n can even be a fractional number). With a pleasing symmetry the book returns to the thesis proposed in the first chapter: mathematics as a model of reality. The example given is Eddington's confirmation of the curvature of spacetime during the eclipse of 1919.Chapter 7, "Estimates and Approximations" presents an entirely different aspect of mathematics. Whereas the first chapters of the book explore the concept of mathematical certainty, this chapter reveals the other side: an inexact science concerned with approximations in a way that might seem alien to anyone who has only been exposed to the simplifications of high school math. These insights lead into a discussion of the Prime Number Theorem. As Timothy Gowers puts it: "Although the prime numbers are rigidly determined, they somehow feel like experimental data".Richard Courant wrote in the introduction to his classic work: "If the crystallized deductive form is the goal, intuition and construction are at least the driving forces". This truth is admirably conveyed in A Very Short Introduction. The chapters on dimension and geometry are bound to capture the imagination of the lay reader, even if these concepts follow ineluctably from axioms. If, as is likely to be the case, the Very Short Introduction has whetted your appetite, Timothy Gowers points readers to his website for additional materials.

El autor transmite una noción concisa de algunos temas de matemáticas avanzadas. No es necesario tener muchos conocimientos para poder leer este pequeño libro, lo cual verdaderamente vale la pena.

Tous les ouvrages de la collection vise à introduire le profane à un sujet complexe nécessitant beaucoup d'études en environ 120 pages afin d'ouvrir nous petits esprits sur des sujets qu'autrement nous aurions peiné à développer.C'est en anglais mais justement, ça permet de développer ses compétences linguistiques en lisant quelque chose d'utile.Celui si parle de mathématiques, j'ai tout compris et trouvé le sujet intéressant alors que mon niveau en algèbre ne doit probablement pas être mesure de réussir son bac.

Just love it!

The purpose of this book is not to teach you how to do math. (There are plenty of other books on the market than aim to do that.) Rather, its purpose is to help you get a better understanding of what mathematics is, how it works, why it works the way it does, and how mathematicians approach mathematical problems. The author, Timothy Gowers, is the Rouse Ball Professor of Mathematics at Cambridge University and is a recipient of the Fields Medal (the highest award given for achievement in mathematics scholarship, roughly equivalent to the Nobel Prize), so he definitely knows what he's talking about. Perhaps more importantly, he is able to communicate his ideas well – I wish all math professors were as clear and cogent as he is (that would have saved me lots of headaches back when I was an undergrad struggling through calculus). The approach he takes in this book is not at all what I had expected, but I have to admit that it works quite well. His main focus is on the abstract nature of mathematics: Sure, we can and do use math for practical applications, but at its heart mathematics is not about counting or measuring things in the "real world" around us; rather, it's about purely abstract concepts (numbers, lines, dimensions, etc.) that relate to each other according to a set of self-consistent rules. It doesn't matter to the mathematician whether there is anything out in the real world that corresponds to these abstract concepts – a mathematician, using nothing but the abstract rules of mathematics, can figure out the geometrical properties of, say, a 27-dimensional shape, regardless of whether or not there are actually that many spatial dimensions in our universe. Higher dimensions, imaginary numbers, infinities, even concepts that are more familiar to the average math student, such as irrational and negative numbers, make sense only in the abstract (in the real world you will never end up with -3 apples no matter how many apples you start with or how many you give away), and so that's how they must be approached. So, Gowers advises the math student (and also, more importantly, the math teacher) not to try to relate every mathematical concept to some real-world example, but to embrace the abstractness of mathematics and to treat it much in the same way that one might treat a game like chess. When we learn how to play a game, we don't imagine that it will have real-world applications; we learn it simply in order to play it. So, how do we learn how to play a game? Since all games have rules, we learn the game by learning its rules. Once we've learned those rules, playing the game is simply a matter of consistently applying those rules. Of course, playing the game *well* requires a degree of creativity and strategic thinking, but you can't play the game well until you have mastered the rules. Math, at least according to Gowers, is essentially the same: It's just a matter of learning a set of rules and applying them. You'll also need some creativity and strategic thinking to solve difficult problems, but you can't do anything without first mastering the rules. The rules are the essence of mathematics. At its heart, mathematics is not about counting or measuring real-world objects; it's about the application of self-consistent, abstract rules to abstract problems. (The fact that at least some aspects of mathematics do have practical, real-world applications is just a bonus.) So, I guess the central message of Gowers's book is that the key to learning (or teaching) math is to stop trying to relate every mathematical concept to something from the real world that you can easily visualize, and focus instead on learning (or teaching) the rules of the game and how to follow them. Before reading this book, if someone had asked me for advice on how to learn (or teach) math, I almost certainly would have said that it's best to try to relate mathematical concepts to real-world examples. After reading this book, I now understand why that might not have been such good advice after all.