Mathematics: A Very Short Introduction (Very Short Introductions) 1st Edition by Timothy Gowers (PDF)

Description

The aim of this book is to explain, carefully but not technically, the differences between advanced, research-level mathematics, and the sort of mathematics we learn at school. The most fundamental differences are philosophical, and readers of this book will emerge with a clearer understanding of paradoxical-sounding concepts such as infinity, curved space, and imaginary numbers. The first few chapters are about general aspects of mathematical thought. These arefollowed by discussions of more specific topics, and the book closes with a chapter answering common sociological questions about the mathematical community (such as “Is it true that mathematicians burn out at the age of 25?”)ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐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.

⭐I’ve been feeling lacking in my mathematics understanding lately. Every time I encounter something mathematically as of late I feel a small fear creep over me; fear of the unknown. I used this to get back in touch with my mathematical side after a few years off from the discipline; very refreshing.He touches on many different aspects of the field of mathematics. He provides a lucid intro to a bunch of fields within mathematics and touches on how to approach them.It was a bit too short. It was an introduction after all. He does seem pretty jazzed about prime numbers which is cool(I think prime numbers are pretty cool myself). I am curious if he has a background in cryptography though. It also made me think that certain fields of mathematics are pretty popular right now(such as theoretical computer science) and if that’s unfairly pulling attention away from the other important sub fields of math.I was surprised and enjoyed by the FAQ at the end. He touched on the fact that very few women pursue mathematics(the same problem exists in engineering). It made me wonder if this has a recursive effect in that a lack of a balanced gender population in a particular field ends up driving more of the majority gender away from it(maybe compounded by the socioeconomic effects of the surrounding environment); that is, a lack of females actually causes less men to get interested in mathematics(and vice versa).Either way great intro to such a sublime field of study. 5

⭐An introduction to mathematics could be just that; elementary arithmetic and geometry, or it could be an outline history, or finally, it could introduce the philosophical aspects of the subject. Gowers does none of those, although he does touch on the history and philosophy of mathematics. This is really an introduction to higher mathematics, for readers who have reached what in Britain is GCSE standard, roughly eleventh grade in the US.Philosophically, Gowers is a pragmatist. To him, problematic concepts like infinity and irrational numbers have meaning in as much as they are useful, and are true in as much as they give true results. As a European, Gowers credits Wittgenstein with these ideas. An American author would have credited William James. Gowers sidesteps rather than resolves philosophical problems, thus giving reassurance to mathematicians and irritation to philosophers.The book is a random selection of topics rather than a continuous narrative, but succeeds because each topic is fascinating and the writing is clear throughout.Under “Further Reading”, Gowers includes his own website address, where you can find sections that did not make it into the book. What a good idea! The site is as full of good stuff as the book, and gives links to further sites that will give you as much mathematics as you will ever want.

⭐There was always a sort of discontent in my mind that I couldn’t grasp why it is not possible to divide by 0. As it turns out, there might be no meaning in a physical sense, but these phenomenon can be explained from the system of axioms. The rules are constructed in such a way, so that complications of philosophical character are avoided. There might be other sets of axioms and therefore other types of mathematics.A clear explanation of how we can operate in higher dimensions; even though we can’t imagine what it means to have 64 dimensions, we can still calculate similarly defined concepts as in 2 and 3 dimensions, such as lines, vertices, areas and so on. It, frankly, astonished me that we can discuss something what we can’t imagine, based on strict logic (‘abstract method’).I hold a belief, that a real professional in his field can explain any concept to a layman. That is what Timothy showed in his book- well structured, logically consistent presentation of the essence of mathematics. I had similar feelings after reading “A short introduction to Accounting”.

⭐Absolutely great. Let me explain. Firstly about the level of the book. It is elementary but not basic. There are some concepts and especially some proofs that require some focus to grasp. The amount of equations is minimal, but still some maturity is required to fully appreciate the contents. I would say that the best audience of the book is a beginning undergrad in a stem program, a motivated and mature high school students with a keen interest on maths or a layman with some maturity of mathematical concepts. I was searching for books to recommend to my students and after finishing that i would highly recommend it. What the book does, is to introduce us to the world of mathematics, what it isall about and how mathematicians approach different ideas and the way Timothy does that is by explaining an array of different ideas from model building, basic arithmetic, formal proofs, complex numbers, number theory, coordinate systems, infinity, higher dimensions, non Eucledian geometries, approximations etc. In the final chapter there are some comments on common misconceptions about mathematics. There are so many gems in every chapter and the way they are connected makes you understand why mathematicians treat them in that way. From the geometric(!) proof of the irrationality of φ, to ”visualizing” objects in higher dimensions and being able to grasp some of their properties, to how the curvature of different geometries affects the shortest path. The reason of 5 stars is all of the above plus finally understanding why airplanes tend to move towards the poles (had to finish my bachelor in physics without ever anyone touching that concept…) plus the 2 most basic ideas of the book, that the meaning of a mathematical object is the rules it obeys and how it is related to other objects and how to extend familiar ideas to unfamiliar territory. These two concepts and how the are used in what Timothy calls the ”abstract method” are the spine of the book and worth buying just for those two only. True gem.

⭐For me this is a far superior insight into the nature of mathematics and its place in the world than Hardy’s ‘A Mathematician’s Apology’. Gowers is sensitive and at the same time guided by common sense. He cuts to the chase and says it how it is. And you get the sense that he passionately loves mathematics so much that he can describe it plainly and dispassionately just how it is: and still make it magical.

⭐Having read a few of the Very Short Introduction (VSI) series, I wanted to revisit some of the joy of university life by returning to mathematics, the subject which I studied as an undergraduate. With the title as it is, one might wonder what sort of level as it’s pitched at. Here, one could be lulled into a false sense of security by mistaking it for “Arithmetic: A Very Short Introduction”. Do not expect this to be “a very simple introduction”. To anyone who has studied maths at university, this will be a very simple book. To anyone studying maths at A-level, they should find it a little challenging in places, but it should provide good food for thought, building on some familiar principles. The author says that it should be OK for anyone with a good GCSE grade, though I would express a little scepticism at that sentiment.That said, I do think it’s a marvellous little book. One of the first things that Gowers discusses is the cumulative nature of maths. i.e. some things can be very simply stated, but only in terms of other things which need to be well-defined and understood. The danger then is how far back do you regress to be able to find ground which is widely understood?Gowers deals with this brilliantly by having his opening chapter on mathematical modelling. In so doing, he grounds mathematics in the real, the physical, the tangible, instead of diving off into the realms of pure mathematics straight away. Though I must admit, the appeal to me comes about primarily because he enunciates the way I have thought of maths for most of the last 3 decades.From here, he starts to ask some more fundamental questions about the nature of numbers, including complex numbers (but not quaternions) and some “proper” algebra though he cunningly avoids the use of terms such group, ring or field whilst ensuring the reader is familiar with their rules by means of definitions followed-up with examples. He also touches on some rules regarding logarithms which perplex some people, but are dealt with very well.He then goes on to probably the most important idea in maths: proof. Though touching on a little philosophy, he tries to skirt around it and give a robust exposition of what a mathematician means when (s)he talks about proof and how it differs from the more lackadaisical use of the word in everyday (and even some other areas of scientific) usage.Though any book on serious mathematics probably ought to contain a good amount on calculus, Gowers avoids this quite ostentatiously. Rather, he lays the groundwork for an understanding of it with a chapter on limits and infinity. In so doing, one might think he’s dodged a potential bullet of losing the interest of readers, though I think that anyone who hasn’t done calculus but who has understood this chapter will be well-placed to start studying calculus.Moving on, we start to get a bit more geometrical. The first of these two chapters looks solely at the idea of dimensionality. One might instinctively have an idea of what a dimension is or how to count them. However, maths is rather more refined than such instinctive generalities and Gowers gives us some examples that any student would find in a 1st year linear algebra course. If anything, this is the acid test for those considering doing maths at university; if this is incomprehensible to you, then it’s best to turn away. But if, on the other hand, you can see there’s something there that can be grasped, if you don’t quite get it exactly at this stage, then you’re standing in good stead. The second of the geometrical chapters looks at much more fundamental geometry, looking at the classic issue of Euclid’s 5th postulate and the consequences of abandoning it.The last main chapter is on one of my least favourite topics: estimates and approximations. This was a topic I did to death in my 2nd year at uni, not getting on with the lecturer (an angry Scot called Alan) and having a nightmare on the exam, barely scraping a 2:1 in it. Gowers doesn’t hide from the fact that this is not neat maths, as most of the rest of the book is, though he spares the reader from an insight into the gory detail of numerical analysis (or num anal, as we disparagingly called it). If there’s any downside to this wonderful introduction, it is this chapter. Not because it is badly written or poorly explained, it isn’t. But merely because it lacks the sexy panache that the rest of the topics have.This is redeemed somewhat at the very end when he puts in some frequently asked questions, along with his answers. If anything, these give a far better insight into the working life of the mathematician than anything found in the rest of the book. He addresses some myths and common (mis)conceptions, giving an honest assessment on some issues where he needn’t have done so.Overall, it’s a great book. If you hated maths at school, then this isn’t the book to get you back into it and loving it. But for anyone with a keen interest in science and how it’s done, then this is a gem. If you know any A-level students considering doing maths at uni, give them this book. Hey, they are students, so £8 for a little book is a lot of money. Go on!

⭐I have already a growing number of “very short introduction..” on my bookshelf and quite enjoy reading them.I think its a very good series except for the text font and layout which I am not a big fan. I find them hard to read sometimes but they do have the advantage of being pocket size so handy.This one on Mathematics brought be back to my young age at school. I would have been so pleased to have a math teacher such as Mr Gower.He has a talent to explain complex things simply. Some worked examples may be a bit derouting for some of us who dont use maths every day but you honnestly do not have to read everything. I read this book with a relaxed attitude, trying to enjoy more than to learn. The book is also loaded with diagrams which helps you further to understand some key concepts.What I found fascinating was that some maths conjectures are still not resolved to this date. Finally, the last sections on “FAQ” is very useful and instructive.7/10

⭐I only have O-level maths from fifty years ago, so most of this book was new to me. I can’t claim to understand much of it, but at least I now know what people mean when they talk about things like imaginary and complex numbers and multi-dimensional space. Professor Gowers has a gift for explaining complex ideas in simple terms : for example, working with numbers is like playing chess without the pieces, or that it is possible to conceptualise multi-dimensional and we don’t have to worry about visualising it.

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