Mathematics and the Imagination

by Edward Kasner, James Newman

You don’t have to love math to enjoy a hand of cards, a night at the casino, or a puzzle. But your pleasure and prowess at games, gambling, and other numerically related pursuits can be heightened with this entertaining volume, in which the authors offer a fascinating view of some of the lesser-known and more imaginative aspects of mathematics.
A brief and breezy explanation of the new language of mathematics precedes a smorgasbord of such thought-provoking subjects as the googolplex (the largest definite number anyone has yet bothered to conceive of); assorted geometries — plane and fancy; famous puzzles that made mathematical history; and tantalizing paradoxes. Gamblers receive fair warning on the laws of chance; a look at rubber-sheet geometry twists circles into loops without sacrificing certain important properties; and an exploration of the mathematics of change and growth shows how calculus, among its other uses, helps trace the path of falling bombs.
Written with wit and clarity for the intelligent reader who has taken high school and perhaps college math, this volume deftly progresses from simple arithmetic to calculus and non-Euclidean geometry. It “lives up to its title in every way [and] might well have been merely terrifying, whereas it proves to be both charming and exciting." — Saturday Review of Literature.

Genres: Mathematics, Science, Nonfiction, Philosophy, Education, Stem

400 pages, Paperback

First published January 1, 1940

About the author

American mathematician

Reviews

[Leland · Rating 5 out of 5]

i picked up this book in olde city, philadelphia for $5. it is a hardback with aleph naught on the front: denotes cardinality.

the copyright is 1940, which is nice to have....
on the inside cover, a man named George wrote with a pencil to his friend:

"There's no royal road to geometry--" Euclid.

It's wonderful to think that in 1940, this book was in existence.... and stayed in the same city for over 60 years.

[Andrew Breslin · Rating 2 out of 5]

The cover of my edition bills this as a “modern classic." It was thus designated in 1967 (27 years after its original publication) and evidently spent the next half century doing what classics do: looking good on a bookshelf. There is remarkably little wear and tear on it for something that is a year older than I am. Certainly much less than I myself have suffered, but though I look terrible on a bookshelf, I have endured far more actual use.

It was probably something of a sensation in 1940, when there were no books about mathematics written for non-mathematicians and the notion of a book about math that the general public would read just for fun was comically absurd. But it hasn’t stood the test of time well. (The content that is. As noted, the actual book is in great shape.)

I’m not going to belabor my mediocre, lukewarm review but just wanted to mention two things Kasner got dead wrong:

1) In the second Chapter, Kasner estimates the “total possible number of moves in a game of chess” as 10^10^50. This figure is way off no matter what he was trying to express, which is not at all clear from his word choices.

What it looks like he is saying, based on the actual words he uses, is the highest number of individual moves that could be played in a single game. This has never been proven conclusively, but is somewhere in the neighborhood of about 5,000 moves.

If he is trying to express the total number of distinct games that are possible (which is what I think he meant), the first thing he should do is learn to use the language better, because that’s not what he said at all. The second thing he should do is read my blog post: “The number of different possible games of chess.” I don’t get very close to the actual answer, but there’s a pretty airtight upper boundary of about 10^17,000. The actual number is lower than this, and a lot lower than 10^10^50. But he can’t do either of the things I suggested, because he’s been dead for many years.

2) In that same chapter, a few pages later, he states:

“Since, for example, the class of men and the class of mathematicians are both finite, anyone realizing that some men are are not mathematicians would correctly conclude that the class of men is the larger of the two.”

Which, while accurate, is based on a flawed implied assumption that all mathematicians are men. There is, in fact, an entire subset of mathematicians who are not men. They are called “women.”

Anyhow, I’m getting rid of books I’ve read and don’t plan to reread and I’m pitching this modern classic. It’s a shame, because it looks good on a bookshelf.

[sigurd]

la mia è l'edizione Bompiani del 1948, ma non saprei come caricarla... avevo imparato a farlo su anobi, ma qui no. è questa: http://www.maremagnum.com/libri-antic...
chi si accontenta gode

a pagina 127 di questo libro, leggo: "tra le nostre più care condizioni, nessuna è più preziosa della concezione che possediamo sullo spazio e sul tempo, eppure nessuna è più difficile da spiegare. il pesce parlante delle favole di Grimm deve avere avuto una grande difficoltà a descrivere come si sta sempre bagnati, non avendo mai provato il piacere di essere asciutto. Noi abbiamo le stesse difficoltà nel parlare di spazio, non sapendo ciò che è , e nemmeno cosa sarebbe il non essere in esso..."
ehi, un attimo mi dico, acciderbolina, ma questo paragrafo mi ricorda qualcosa. ma certo, mi ricorda quel discorso di david poster wallace tenuto a dei laureandi, che fa parte di un suo libro famoso "questa è l'acqua", anche più che "matematica e immaginazione" di Kasner e Newman.
inizia così:
"Un saluto a tutti e le mie congratulazioni alla classe 2005 dei laureati del Kenyon college. Ci sono due giovani pesci che nuotano uno vicino all’altro e incontrano un pesce più anziano che, nuotando in direzione opposta, fa loro un cenno di saluto e poi dice “Buongiorno ragazzi. Com’è l’acqua?” I due giovani pesci continuano a nuotare per un po’, e poi uno dei due guarda l’altro e gli chiede “ma cosa diavolo è l’acqua?” È una caratteristica comune ai discorsi nelle cerimonie di consegna dei diplomi negli Stati Uniti di presentare delle storielle in forma di piccoli apologhi istruttivi. La storia è forse una delle migliori, tra le meno stupidamente convenzionali nel genere, ma se vi state preoccupando che io pensi di presentarmi qui come il vecchio pesce saggio, spiegando cosa sia l’acqua a voi giovani pesci, beh, vi prego, non fatelo. Non sono il vecchio pesce saggio. Il succo della storia dei pesci è solamente che spesso le più ovvie e importanti realtà sono quelle più difficili da vedere e di cui parlare".

la storiella "meno stupidamente convenzionale nel genere" è un racconto dei Grimm. e poiché gli americani sono del tutto privi di cultura letteraria (come me, che non ricordavo il racconto di Grimm), immagino che dfw abbia potuto trarre la sua storia da questo libro. matematica e immaginazione. sicuramente l'avrà sfogliato, il titolo si adatta alla sua facoltà intellettiva e ai suoi argomenti (scrisse anche un libro sulla matematica). si sarà imbattuto nella pagina 127 così come il piccolo Danny con il suo triciclo si imbatte nella camera 237 dell'overlook hotel. e avrà copiato la storiella, senza ritegno, senza citare la fonte.
SGAMATO!

[Roberto Rigolin F Lopes · Rating 5 out of 5]

We are in 1940, Kasner and Newman are compiling strange things you can see using mathematics (i.e. using a mathescope). If "imagination" triggered your curiosity, this book is far better than its title. Let us define "far" as one googleplex, which is enormous but finite. One googleplex of what? Up to you. Say fish steaks or kangaroos. It is just a matter of scale, you got infinity within 0 and 1. As you can see, this book may drive you mad.... A hell of a lot of wit and ingenuity within.

[Tonatiuh Dávila · Rating 2 out of 5]

Colección de curiosidades matemáticas, de fácil lectura y absolutamente dirigido a personas que no están familiarizados con las matemáticas o alguna ingeniería.

[Elazar · Rating 3 out of 5]

Not as much fun as I was hoping for...

[John Isles · Rating 4 out of 5]

Dated, but an enjoyable read. Some passages are obscure but most of the book can easily be understood by the general reader. I found only the account of calculus to be unduly labored.

[Mauricio · Rating 5 out of 5]

Es un libro escrito con un lenguaje muy accesible que introduce al lector al mundo de las matemáticas desde una óptica muy diferente a la tradicional utilizada en escuelas y colegios. Abarca conceptos de infinito, números trascendentales, geometría euclidiana, álgebra, juegos con matemáticas, espacios multidimencionales, geometrías no euclidianas, probabilidades, cálculo, y más. El libro no pretende ser exaustivo, si no que más bien es una introducción a las muchas ramas de las matemáticas. Totalmente recomendado para las personas que disfrutan las matemáticas pero no se han acercado a ellas formalmente.

[Vicent]

Pueden leerse fragmentos aquí:

http://books.google.es/books?id=zdBHM...

y aquí:

http://www.libraria.com.mx/fragmento/...

[Antony Monir · Rating 4 out of 5]

How should one read a book published almost a century ago on a topic that has advanced by leaps and bounds since? This is the principal question one should have when looking at this book because by modern standards, Mathematics and the Imagination is hardly the ideal popular mathematics book. Its terminology is dated, its language is awkward and the authors tend to use unfortunate racial terminology. To give this book its fair chance, I believe in reading it as a quasi-historical document, much like someone today would read Plato's dialogues. Just like these dialogues are flawed by modern philosophical standards, Mathematics and the Imagination has its issues. Yet, millions around the Earth still read Plato to this day, in hopes of understanding what made his works so important to history, and hopefully, to see how we can still get something out of a book so old. This is the attitude I took when reading Mathematics and the Imagination.

First, let's look at the content of this book. The authors cover a variety of topics from basic stuff like probability and calculus to more advanced topics such as set theory and topology. I believe that some chapters here are much stronger than others but I suspect this to be due to my bias in preference for some mathematical topics over others. I especially liked the topic on "rubber-sheet geometry", also known as topology, which I found to be particularly well written. This chapter explained the basic principles behind one of the most complex topics in mathematics in a way that felt intuitive. Other great chapters include the one on paradoxes (which is actually more like a chapter on curves of pursuit), the chapter on pi, i, and e, and the chapter on non-Euclidean geometry. These chapters show Mathematics and the Imagination at its absolute best, explaining the topics in a clear manner with relevant and well-drawn illustrations. Weaker chapters include the one on the calculus and probability, which are treated in a practically sterile manner. Of course, the hit-and-miss nature of these chapters is a bit exaggerated in this review as the weaker chapters are still enjoyable, if a bit inferior to the other chapters.

Next, it is important to note the influence of this book. Being published in 1940, I can confidently say that Mathematics and the Imagination is criminally unknown for its influence almost a century later. If you've ever used Google, you've been influenced by this book, although indirectly. For the name "Google" comes from the term "googol" which itself was coined by Kasner's nine year old nephew, and popularized by this book. Of course, the term googol is not the only thing we can credit this book with, though it is the most instantly recognizable. Instead, I would credit this book for popularizing books on mathematics written with the layperson in mind. This tradition continues to this day with authors such as Matt Parker. When discussing popularizations of technical topics such as physics and mathematics, there is always a fine line between a book that is too simple to be engaging and too complex to be understandable. I believe that Mathematics and the Imagination set the standard almost a century ago regarding how to strike that balance. While some sections focus too much on the history of mathematics rather than the actual topics at hand, I am glad to notice that this book doesn't shy away from more advanced topics and proofs. The chapter on paradoxes especially impressed me with the way it tackled curves of pursuit such as the tractrix. This is how I believe popularizations should be, covering topics that are interesting in a way that is intellectually engaging. The difference between academic mathematics and popular mathematics should be the choice of topics, not intellectual rigor.

To summarize, I think that while Mathematics and the Imagination is dated by modern standards, it is an essential book in the history of popular mathematics, and its influence should not be understated. However, I don't believe it is a book that everyone must read, not even those interested in mathematics as there are much better, more modern, books on the topic. Instead, if you want to read this book it should be for historical reference, though I do believe that you will get a lot out of it, much like I did. 4/5

[Alexa Daskalakis · Rating 5 out of 5]

Mathematics is not a discipline; it is the raw architecture of abstraction itself. Kasner and Newman do not merely present formulas or proofs—they unveil the cognitive machinery that makes structured thought possible. Mathematics and the Imagination is less about computation and more about how mathematical ideas emerge, evolve, and dictate the very way we categorize and understand reality.

Most people believe numbers describe things. They are wrong. Numbers construct the very space of possibility itself. From transfinite arithmetic to the structure of prime distributions, this book deconstructs the illusion that mathematics is a language we invented, instead revealing that it is a structure we discovered. The authors methodically dismantle the barriers between logic, philosophy, and intuition, demonstrating that mathematical abstraction is not an academic exercise but a constraint embedded in the nature of reason itself.

Take the concept of infinity. It is not merely a large number—it is a statement about the limits of formalization itself. The moment you engage with infinity mathematically, you encounter paradoxes that force a confrontation with the incompleteness of any system attempting to describe itself. Gödel’s incompleteness theorem lurks in the background of every discussion here, whether explicitly stated or not: no system of thought can fully capture its own foundations.

But what makes this book truly staggering is its ability to translate deep mathematical ideas into cognitive frameworks that extend far beyond mathematics itself. The discussions of curved space, non-Euclidean geometry, and the fourth dimension are not just intellectual curiosities; they are statements about the fundamental malleability of perception, the idea that space itself is not absolute but relational, dependent on axioms we take for granted.

The book does not just teach—it trains the mind to think in higher-order structures. It forces an awareness that all reasoning systems are contingent, all mathematical truths are embedded in deeper, more fundamental assumptions, and all human cognition operates within a framework that it can never fully escape.

If you read this correctly, you don’t just come away with knowledge. You come away with the realization that the very way you understand logic, abstraction, and structure itself is inherently incomplete. And that changes everything.

[Marta D'Agord · Rating 5 out of 5]

Kasner e Newman publicaram esse livro em Nova York em 1940. Foi publicado em espanhol na Biblioteca de Jorge Luís Borges com uma apresentação do mesmo. Como sabemos, Borges usa noções matemáticas em muitos contos e esse livro teria sido uma de suas fontes. A primeira edição brasileira é de 1968 pela Zahar. Havia lido alguns capítulos há alguns anos. Meu exemplar é da segunda edição de 1975, e acessei a vigésima edição, ainda da Simon & Schuster, de 1962. Desde 2001, a edição é pela Dover.

O estilo narra com leveza desafios históricos, tais como o seguinte: os gregos, para quem a Geometria era uma alegria e a Álgebra um mal necessário, rejeitaram os números negativos. Incapazes de enquadrá-los em sua Geometria, incapazes de representá-los em figuras, os gregos não os consideravam como números em absoluto(...) Não sendo tão conhecedores de Geometria, os chineses e hindus não tiveram nenhum escrúpulo com números que não podiam representar em figuras.

Esse estilo irreverente e anedótico fez escola, como encontramos em A janela de Euclides de Leonard Mlodinov (o original é de 2001) na contextualização do salto que os gregos realizariam a partir das práticas egípcias: Tales deu a Pitágoras a sugestão de Greeley, mas em vez de mandar o jovem para o oeste (o autor se refere à corrida do ouro na Califórnia), Tales recomendou o Egito.

O capítulo sobre as diversas geometrias é ainda atual. Escrito antes das fotos da vista da Terra pelos astronautas. Os autores não saberiam que vinte anos depois teríamos imagens da forma esférica de nosso planeta. Embora toda pessoa inteligente saiba que a superfície da Terra é curva e qualquer navegador use esse saber na sua prática, alguns até hoje pensam que nossas linhas retas são traçadas em um plano com curvatura zero. Quando os autores publicaram o livro, o último teorema de Fermat ainda não havia sido provado, o que somente aconteceu em 1995. Apesar disso, é um livro que apresenta os conceitos fundamentais da matemática sem subestimar o leitor leigo. Recomendo papel e lápis para ir praticando durante a leitura. Novas concepções aguardam quem confunde infinito com ilimitado ou não distingue o impossível matemático do impossível em Física.

[Marinho Lopes · Rating 4 out of 5]

Gostei bastante deste livro de divulgação de Matemática, embora por vezes não seja claro o público alvo: por um lado é talvez demasiado exigente para quem não tiver bons conhecimentos, por outro é demasiado simples e básico para quem os tiver (em grande parte do livro). É sem dúvida um problema com que provavelmente todos os autores de livros de divulgação de Matemática se deparam: para explorarem um pouco da beleza da Matemática vêm-se na necessidade de entrar em detalhes que poderão afastar grande parte do público. Este problema sentiu-se em particular no último capítulo sobre o Cálculo. Para mim foi interessante recordar muitas curiosidades diferentes, bem como observar a forma como a Matemática era entendida na altura em que o livro foi escrito. Também as referências à guerra e aos nazis são detalhes interessantes que nos dão um pouco do contexto histórico aquando da escrita do livro.

[James]

I actually got this book many, many years ago as a gift from Grandpa. I've read parts of it slowly over time. Recently I brought the book to work and finished the last chapters over my lunch breaks. The book covers an interesting set of topics - puzzles and paradoxes, googol and googolplex, inifinity, motion, pi, i, e, plane geometry and 'rubber sheet' geometry, topography, probability and chance, calculus... I was delighted to read much of it, and many of the topics were familiar enough to understand but things I had not necessarily learned in school. He has a lot of interesting examples of mathematics at work in the real world. Cool book!

[Arttu · Rating 3 out of 5]

For some time at the beginning of the book, I was worried it might turn out to be a dull experience for readers with a mathematical background. But many of the varied topics were surprisingly engaging, and I had a fun time explaining some of them to my fiancée (who graciously played along, preserving the purity of my joy.)

The book does a great job at discussing the mathematical concepts – such as different infinities, non-Euclidean geometry, paradoxes, probability – without the burden of notational rigour.

A fun old read; not a waste of time, but not a priority read either.

[Alice Wardle · Rating 3 out of 5]

Did not finish.

I expected it to be equally about the imagination than about complex mathematical theorems, but it was not. I read the first half, which was incredibly well-written and had parts that were funny and interesting. However, these parts became overshadowed by the mathematical explanation of things. It is probably a good book for someone who likes and understands mathematics a lot more than I do.

[Hessel Beijaard · Rating 5 out of 5]

Quite an enjoyable read. The authors - Kasner and Newman - understand mathematics as well as prosaic style and their pens orchestrate music from mathematical fact, scientific and philosophical insight, and, erudite as they are, from art, literature, and philosophy.

[Nilesh · Rating 5 out of 5]

It is quite a good book for getting introduction of many branches of maths like Non-Euclidean geometry,Topology,puzzles etc

[John · Rating 4 out of 5]

I haven't finished the book -- and I have the 1971 paperback edition with no ISBN, so that's an issue -- but what I read was extremely enjoyable.

[Leonardo]

Lo recomienda Walter Sosa Escudero

[Christopher Heckman · Rating 4 out of 5]

My copy was updated in 1989, but it didn't mention that the Four Color Theorem had been solved. (Fermat's Last Theorem hadn't been solved at that point.)

[Anthony Edridge · Rating 4 out of 5]

Even non-mathematicians can enjoy this book. The authors show not only how mathematics pervades our world and universe, but also our everyday experiences. Inclusion of historical characters and events bring life to this excellent work, often with humor and intriguing consequences.

Certainly this book fueled my imagination with the beauty of mathematics, and surprised me with all the many diverse applications in which mankind has found this topic beneficial and even necessary.

[Greggoss · Rating 5 out of 5]

Really good book but not exactly a quick read. It does a good job of putting enough detail without bogging the reader down with long proofs, huge numbers, confusing symbols or elaborate explanations when it refers to high level math and was a great book for anybody who is interested in math.

[Nivasilyev · Rating 5 out of 5]

wholly readable, inspirational, and enjoyable. The 4th dimension is explained in an easy to enjoy way. This book can be read at leisure in a few days as it has little information to verify or explore. The author encourages you to continue reading mathematics.

[Krollo · Rating 4 out of 5]

I only read an excerpt of this, but it was a great excerpt (New Names for Old). It was an interesting (if slightly dumbed down) take on the difference between mathematical language and colloquial langauge, as well as what we can do to make sure we don't get in a muddle.

[Brannon · Rating 5 out of 5]

This book has one of the most simple explanations of Calculus anywhere. And it's a fun book to read.

[Steve Cox · Rating 3 out of 5]

Never quite got into it.

[Maurizio Codogno · Rating 4 out of 5]

la mia recensione: http://xmau.com/notiziole/archives/00...

[Ken · Rating 5 out of 5]

The authors explain mathematics in 1940 terms in such detail that it boggles the mind, but they do it with a wonderful touch of humor.

[Tysseer Harak · Rating 5 out of 5]

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