The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.

I am interested in mathematics only as a creative art.

G.H.Hardy  (from A Mathematician's Apology)
 
 
 
 

Mathematics, rightly viewed, possesses not only truth, but supreme beauty- a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of paintings or music,  yet sublimely pure and capable of a stern perfection such as only the greatest art can show.

Bertrand Russell
 
 

It is impossible to be a mathematician without being a poet in soul.

Sophia Kovalevskaya
 
 

A mathematician who is not also something of a poet will never be a complete mathematician.

Karl Weierstrass
 
 

One can find many quotations by famous mathematicians expressing similar sentiments. Here is a brief collection . Most people are exposed to mathematics in school, reading standard textbooks which may fail to convey any of the beauty or excitement of the subject,  and so the above quotations may seem quite baffling.  But there are many books written by mathematicians which attempt to convey this aspect of mathematics. The purpose of this page is to provide a guide to some of this literature, especially some of the classics which have influenced many young people to become mathematicians.   I will gradually add more and more books to the list at various levels of difficulty, all with the hope that readers who love poetry or music or art may be tempted to see what aesthetic pleasures mathematics has to offer.
 

1.   What is Mathematics?   An Elementary Approach to Ideas and Methods
             by Richard Courant and Herbert Robbins

This marvelous introduction to mathematics was first published in 1941. It is not a textbook, although it does have some exercises. A good knowledge of high school mathematics should be sufficient to understand the book, but it is not an easy book.  As Courant writes in his introduction, this book requires "a certain degree of intellectual maturity and a willingness to do some thinking on one's own. "  Although many of the topics discussed may be familiar from high school or college, this book goes more deeply into the underlying ideas and their roots. The reader will see that there are fascinating questions to ask about even the most elementary aspects of  mathematics.   Some of the topics discussed:  Prime Numbers,  Projective Geometry  and Non-Euclidean Geometry,  Algebraic and Transcendental Numbers, Unsolvability of the Three Construction Problems of Antiquity,   Minimal Surfaces (Soap Film Experiments), and Calculus.  The latest edition also has an additional chapter written by Ian Stewart which discusses the solution in recent years of the Four Color Problem and Fermat's Last Theorem.

It is natural for me to make this book the first on the list.  My interest in mathematics started in my high school geometry class.  The teacher told us about the impossibility of trisecting an angle by using just a straightedge and compass.  But I thought that I could do it.  There was a simple error in my method and, responding to my interest, my teacher lent me his copy of What is Mathematics? so that I could see the proof of the impossibility.  Later I bought my own copy and read the book from cover to cover.  Probably more than any other book that I read as a teenager, it was this book that convinced me to become a mathematician.
 

2.    The Enjoyment of Mathematics,   Selections from Mathematics for the Amateur
              by Hans Rademacher and Otto Toeplitz

This work was originally published in 1930 in German, and translated into English in 1957. Again,  a good knowledge of high school mathematics should allow a reader to understand this book. But it will not be easy reading.  However, the 28 chapters, each from just a few pages to ten or so, are rather independent of each other. So one could easily skip around.  Some of the topics covered:  Cantor's Theory of Sets,  The Five Platonic Solids,  Perfect Numbers,  A Property of the Number 30.   The latest edition is still available,  published by Princeton University Press.
 
 

 3.    Mathematics and the Imagination
                by Edward Kasner and James Newman.

This is another classic which was first published in 1940.  The latest edition was published in 1989, and seems to be out of print now. But one should easily be able to find a copy of this book in libraries.  It is a much more accessible book than the first two books suggested above. It is written in a rather chatty and humorous style, with many anecdotes about the history of mathematics and various philosophical digressions.  And the reader will learn some of the important ideas of modern mathematics along the way.   Here is a nice review  of this book.
 

4.     Poetry of the Universe
             by Robert Osserman

In the postlude to this delightful and very accessible book, the author writes:

        "I have tried in this book to trace a path through the mathematical landscape that leads to a view of one aspect of nature - the nature of the cosmos. By the cosmos, I mean the universe in its entirety, possessing an order, structure, and shape on its largest scale. That shape is not discernible or even describable without the language of mathematics. Studying mathematics in order to understand the laws of physics is not unlike learning enough of a foreign language to capture some of the special flavor and beauty of prose or poetry written in that language. In the process, one may well become fascinated by the language itself. And so it is with many parts of mathematics. Created in the first instance to provide deeper insights into the nature of the world around us, the language of mathematics develops its own structure and order, its own beauty and fascination. I hope that the dual nature of mathematics - its internal beauty and its power to reveal the hidden structure of the external world - will have become apparent in the course of this narrative."

Here is a brief review.
 
 

5.    Journey Through Genius: The Great Theorems of Mathematics
            by William Dunham

This book is a combination of history, biography, and mathematics. In his preface, the author writes:

            "For disciplines as diverse as literature, music, and art, there is a tradition of examining masterpieces - the "great novels," the "great symphonies," the "great paintings," - as the fittest and most illuminating objects of study. Books are written and courses are taught on precisely these topics in order to acquaint us with some of the creative milestones of the discipline and with the men and women who produced them.
          The present book offers an analogous approach to mathematics, where the creative unit is not the novel or the symphony, but the theorem. Consequently, this is not a typical math book in that it does not provide a step-by-step development of some branch of the subject. Nor does it stress the applicability of mathematics in determining planetary orbits, in understanding the world of computers, or, for that matter, in balancing your checkbook. Mathematics, of course, has been spectacularly successful in such applied undertakings. But it was not its worldly utility that led Euclid or Archimedes or Georg Cantor to devote so much of their energy and genius to mathematics.These individuals did not feel compelled to justify their work with utilitarian applications any more than Shakespeare had to apologize for writing love sonnets instead of cookbooks or van Gogh had to apologize for painting canvases instead of billboards."
 
The author begins with some of the "great theorems" of the ancient Greek mathematicians Hippocrates, Euclid, and Archimedes.  Later chapters are about Isaac Newton and the Bernoullis from the 17th century and Leonhard Euler from the 18th century.  The final chapter discusses the 19th century mathematician Georg Cantor and his ideas about infinity. But the readers will also learn about the surrounding culture, historical influences, anecdotes, etc.  For the most part, the book should be quite accessible. But Dunham does try to explain the proofs of the great theorems that he has chosen.  That is a valuable part of the book, but also the difficult part.  I would like to see a sequel to this book including more of  the "great theorems" of the 19th century and then going into the 20th century. 
 
 
 

6.     Fermat's Enigma: The Quest to Solve the World's Greatest Mathematical Problem
           by Simon Singh

Can there be unsolved mysteries concerning the familiar counting numbers 1, 2, 3, 4, . . . ?    The answer is yes, and such mysteries have occupied the minds of many mathematicians since antiquity.   One such mystery, which was solved in 1994 by Andrew Wiles, was  posed by the 17th century French mathematician Pierre de Fermat.  To explain it, consider the squares, the cubes, the  fourth powers, etc.:

SQUARES:   1,   4,   9,   16,   25,   36,   49,   64,   81,   100,   121,   144,  169,  196,   225,   256,   289,  .  .  .
(These are the numbers 1x1=1,   2x2=4,   3x3=9,   4x4=16,   5x5=25,   etc.)

CUBES:   1,   8 ,    27,   64,   125,   216,   343,   512,   729,  1000,  1331,  1728,   2197, . . .
(These are the numbers 1x1x1=1,   2x2x2=8,   3x3x3=27,   4x4x4=64,   etc.)

FOURTH POWERS:  1,   16,   81,   256,   625,   1296,   2401,   4096,   6561,   10000,   14641, . . .
(These are the numbers of the form  axaxaxa = a⁴,  where a = 1,  2,  3,  4, 5,   etc. )
 

Sometimes two squares have a sum which is another square. For example,  9+16=25,    25+144=169,   36+64=100,   64+225=289.   Ancient Greek mathematicians had a simple way to find all pairs of squares whose sum is another square.  This corresponds to the geometric problem of finding right triangles, each of whose sides have length equal to an integer.

Can two cubes have a sum which is another cube?  Can two fourth powers add up to another fourth power?  Can two fifth powers add up to another fifth power?   Can two sixth powers add up to another sixth power?    And so on: For any n > 2,  can two n-th powers have a sum which is also an n-th power?  (To be precise,  here n is supposed to be a positive integer greater than 2 and n-th powers are supposed to be n-th powers of positive integers.)   Fermat claimed that the answer to ALL of these questions is no,  and that he could prove it.   He never wrote down that proof (and probably was mistaken about it).  This problem (often referred to as "Fermat's Last Theorem")  captured the imagination of generations of mathematicians, and attempts to solve it stimulated the development of several important branches of mathematics.  Wiles solved the problem by proving a version of a very fundamental conjecture in number theory (called the Shimura-Taniyama Conjecture).   This settled the question because in the late 1980s it had been discovered that Fermat's Last Theorem can be deduced from a special case of the Shimura-Taniyama Conjecture.  (That in itself was a very astonishing discovery.)

Singh's book is a very popular and accessible account of the history of this problem, and sketches the extremely circuitous path to its eventual solution.   Here is a collection of  reviews and comments .    Singh also directed a Nova program called  The Proof.
 

7.     From Zero to Infinity, What Makes Numbers Interesting
           by Constance Reid

Fermat's Last Theorem, which is the subject of Simon Singh's book Fermat's Enigma, is a problem from one of the oldest branches of mathematics -  the Theory of Numbers. For readers who enjoyed that book and might be curious to learn more about this beautiful subject, there is an abundance of books to choose from.  They vary considerably in their expectations of the reader.   I have chosen just a few. The first (and least demanding) is this classic by Constance Reid, written in 1955 and now in its 4th edition.  The twelve chapters are appropiately named after numbers: Zero, One, Two, Three, . . . , Nine, Euler's Number e,  and Aleph-Zero.  The author delves into quite a variety of topics somehow connected with each of these numbers.  The author is not a mathematician (although her sister was a famous mathematical logician), but she has managed to explain quite a bit of mathematics in a very appealing and understandable way.

8.   An Introduction to the Theory of Numbers
             by G.H. Hardy and E.M.Wright

This is a wonderful, serious introduction to number theory. Actually, as the authors say in their preface, it is really a series of introductions to almost all sides of this many-sided branch of mathematics.  It is the most demanding book on this list. However, a good, thorough knowledge of high school mathematics and calculus together with a willingness to patiently study the proofs should be sufficient to understand virtually all of the chapters. This is a book to spend months with.   It was first published in 1938. I still have my treasured copy from 1962.
 
 

TO BE CONTINUED
 
 
 
 
 

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