Mathematics: The New Golden Age

by Keith Devlin

This lively and comprehensive new survey sets out to show just how mistaken people who think they can't enjoy mathematics really are. Dr. Devlin's television appearances and popular GUARDIAN column have revealed his remarkable gift for making difficult topics beautifully accessible. In this book he offers a glimpse of the extraordinary vistas and bizarre universes opened up by contemporary mathematics.

Genres: Mathematics, Science, Nonfiction, Physics

304 pages, Paperback

First published January 1, 1988

About the author

Dr. Keith Devlin is a co-founder and Executive Director of the university's H-STAR institute, a Consulting Professor in the Department of Mathematics, a co-founder of the Stanford Media X research network, and a Senior Researcher at CSLI. He is a World Economic Forum Fellow and a Fellow of the American Association for the Advancement of Science. His current research is focused on the use of different media to teach and communicate mathematics to diverse audiences. He also works on the design of information/reasoning systems for intelligence analysis. Other research interests include: theory of information, models of reasoning, applications of mathematical techniques in the study of communication, and mathematical cognition. He has written 26 books and over 80 published research articles. Recipient of the Pythagoras Prize, the Peano Prize, the Carl Sagan Award, and the Joint Policy Board for Mathematics Communications Award. He is "the Math Guy" on National Public Radio.

Reviews

[WarpDrive · Rating 4 out of 5]

This book represents a compendium of some of the major development in modern higher mathematics in areas such as number theory, topology, group theory.
This book is really good at conveying the fascinating and sometime totally unexpected inter-connectedness between apparently unrelated fields in mathematics.
Moreover, the author's love for mathematics, as something not just extremely useful to progress, but inherently beautiful and fun, is visible and contagious.

What I like about this book is also that it goes into the math to a much more significant extent than most popular science books, yet managing to keep things at a level sufficiently easy to understand. The target audience is the competent maths amateur, interested in the relatively recent (say after WWII) developments in higher mathematics.
The only issue I have is that the author in some cases gets so infuriatingly and tantalizingly close to the actual details of a major result, and then it just closes with the statement "the full result can only be understood by specialists". The resulting frustration is compounded by the fact that the bibliographic references are very limited and unsatisfactory. But this is a minor fault.

Let's get into the actual contents of this book:
- after an initial, introductory chapter about prime numbers and number theory, there is a pretty nice chapter (still at pretty introductory level, unfortunately) about axiomatic set theory and Cantor's famous "continuum problem". To be honest, I was expecting more detail here: for example, when talking about axiomatic set theories such as Zermelo–Fraenkel, the axiom of choice is not discussed, and the axioms of the ZF theory itself are not discussed either. Real pity, as we are dealing with the foundational basis of the whole of mathematics here.
- There is then a section about complex numbers. While complex numbers are explained in a reasonable way, there is no complex analysis, and the quaternions are touched only very briefly. This is a real pity and a missed opportunity, as complex analysis is one of the most beautiful as well as useful branches of Mathematics.
- This is followed by a really nice chapter titled "beauty from chaos", which explains in a really beautiful and approachable way the concepts of fractional dimensions, chaotic dynamics, and the amazingly beauty of fractals. The Menger Sponge (D=2.7268), an amazing object which has zero volume enclosed by an infinite surface area, is also presented. Fractals are not just beautiful, but also useful, and we can see examples of fractals in in the realms of engineering, electronics, chemistry, medicine, even urban planning and public policy.
- We have then another very enjoyable chapter about the critically important (to science in general, not just maths) concept of symmetry (and symmetry groups). The author even manages to explain at conceptual level, and in a very approachable way, the classification theorem of finite simple groups. The "monster" (sporadic simple group consisting of complex matrices of order 196,833) is also explained. Galois groups and the solution to the general quintic equation are also explained quite well.
- There is then a very nice chapter about the famous Hilbert's Tenth Problem (solution to Diophantine Equations). The degree-25 prime-generating polynomial discovered in 1977 is also explained.
- This is followed by couple of very nice chapters about some fundamental concepts and results in topology, where the author manages to beautifully explain, in a clear and precise manner, fundamental concepts such as the topological invariants represented by the Euler characteristic and "orientability". A beautiful introduction to the theory of manifolds (including concepts such as associated differentiation structures) is also a real enjoyment to read. Mind-blowing facts such as the peculiar richness and complexity of the 4-dimensional space as opposed to all other dimensional situations, are also presented beautifully. By the way, the four-color problem is also nicely treated.
- The final two chapters are nothing of particular brilliance, but decently good anyway; they explain Fermat's last theorem and some introductory concepts about the efficiency of algorithms.

Overall, a pretty good introductory overview of some selected recent developments in modern maths, targeted at the competent amateur, with some pretty good chapters.

4-star.

[Robert · Rating 3 out of 5]

Nice general historical survey.

[Pirata · Rating 3 out of 5]

This book sketches the beauty and wonders of mathematics.

[Timothy Chklovski]

This is a book I read mostly "just for fun", as well as to understand more about how interest in mathematics can be developed in hound students.
Although I have some grasp and appreciation of elementary mathematics (having done the obligatory 10,000 hours to be on us team for the international math olympiad, and continued with a few more math courses to get a degree in math from mit), I was thoroughly impressed with just how powerful the modern mathematical accomplishments are. It's like someone who has done some bouldering learning about ascent of K2 -- the current achievements are vast in scope and depth. The author does a fantastic job of brining those to life, helping you appreciate 5000 page proofs classifying simple groups, providing glimmers of approach to solving Fermat's last theorem, and giving one a flavor of how rich number theory is -- the exposition about e ^ pi sqrt(163) still surprises me (there is no reason that I understand that a number as small as that should get you within 12 decimal places of an integer in that formula -- see Wikipedia entry in 163 for a bit more).

Incidentally, the recent growth in both population, wealth, and communication have combined to produce a crop of mathematicians who may well be as good as the greats of earlier centuries, who had to work alone, and split their time between pure math and other endeavors, as eg in Laplace's case.

It's true that reading this book provides just a glimpse over the fence -- you will not be able to understand any of the big proofs in their completeness after this, just as reading an interview with Kasparov will allow you to fully appreciate his chess games. But, just as parkour or beatbox videos, simply reading this book is apt to make the reader feel that we live in a world where exceptional people can pull off jaw-dropping, awe-inspiring stunts.
Highly recommended as a dose of optimism about human potential.

[Maurizio Codogno · Rating 4 out of 5]

L'anno scorso Bollati Boringhieri ha molto opportunamente ripubblicato la seconda edizione di questo saggio di Keith Devlin, il cui titolo inglese che fa più o meno "Matematica: la nuova età dell'oro" è molto più incisivo. Il testo racconta alcune scoperte matematiche avvenute tra il 1960 e il 1995. Occhei, "racconta" è un termine un po' fuorviante: purtroppo la matematica che si fa oggi è troppo astratta per poter essere spiegata anche a chi ha una cultura universitaria di base, figuriamoci a lettori acculturati sì ma non certo specialisti del campo. Tanto per dare un'idea, nell'introduzione Devlin spiega che ha dovuto chiedere consulenze ad alcuni suoi colleghi.... Quello che però si può fare, e che Devlin ha fatto egregiamente, è dare un'idea di come si è arrivati ai problemi trattati nel libro, dare una spiegazione dell'enunciato del problema - cosa che non è sempre così facile come nel caso dell'Ultimo Teorema di Fermat - e infine raccontare quali sono state le linee di attacco che hanno infine portato alla soluzione, Il risultato finale è un bell'affresco che permette al lettore di comprendere come l'avanzamento della matematica non sia un'azione continua e lineare ma proceda in un certo senso come le altre scienze, per tentativi ed errori, e nuove idee prese da rami a prima vista scorrelati. La traduzione di Annarosa Giannetti, Agnese Manassero e Laura Servidei è generalmente chiara anche per chi non ha una formazione matematica specifica; peccato che nella riedizione non siano stati corretti alcuni refusi.

[Unni Krishnan · Rating 2 out of 5]

This had a set of decent descriptions on the important 'open' (some are no more open) problems of mathematics. The premise of the book was (at least it appeared so to me) that the problems and proofs would be described in a pop-science (ok, pop-maths) way to an average-skilled mathematics enthusiast. I'm well aware that this is an impossible task. So, no wonder I ended up with having a little bit more knowledge on the peripheral aspects of the problems and their proofs.
However the initial chapters were easily comprehensible. But when it comes to advanced topology and number theory, things started to go out of hand (from me). As mentioned somewhere in the book, the open problems seem to be deceptively simple and very inviting to be examined only later to be realized how complex they are, in fact.
So, the book is quite OK if you are looking for the thrill of examining some tough problems without being insistent to understand them in detail. (In my opinion, if you want to appreciate them for all the beauty, then you need to go through the proper proof explained in the reference books)

[Brent · Rating 5 out of 5]

This is a great read for people with even the tiniest interest in mathematics. It's precise and clearly written.

[Frenje]

note to self: KIV, by the Stanford prof who just won the Carl Sagan Prize for popularizing science

[K P · Rating 5 out of 5]

10/10

[Massimo Redaelli · Rating 3 out of 5]

Ben scritto, meno dettagliato e ampio dell'equivalente scritto da Odifreddi (Matematica del '900), è un'ottima introduzione non tecnica ma seria.

[Monaco Obbediente · Rating 5 out of 5]

Un libro che riesce a descrivere perfettamente la bellezza della matematica.