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What Is Mathematics? An Elementary Approach to Ideas and Methods
For more than two thousand years a familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but does not lead to real understanding or to greater intellectual independence. This new edition of Richard Courant's and Herbert Robbins's classic work seeks to address this problem. Its goal is to put the meaning back into mathematics.
Written for beginners and scholars, for students and teachers, for philosophers and engineers, What is Mathematics? Second Edition is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world. Covering everything from natural numbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving. With chapters largely independent of one another and sections that lead upward from basic to more advanced discussions, readers can easily pick and choose areas of particular interest without impairing their understanding of subsequent parts.
Brought up to date with a new chapter by Ian Stewart, What is Mathematics? Second Edition offers new insights into recent mathematical developments and describes proofs of the Four-Color Theorem and Fermat's Last Theorem, problems that were still open when Courant and Robbins wrote this masterpiece, but ones that have since been solved.
Formal mathematics is like spelling and grammar - a matter of the correct application of local rules. Meaningful mathematics is like journalism - it tells an interesting story. But unlike some journalism, the story has to be true. The best mathematics is like literature - it brings a story to life before your eyes and involves you in it, intellectually and emotionally. What is Mathematics is like a fine piece of literature - it opens a window onto the world of mathematics for anyone interested to view.
Written for beginners and scholars, for students and teachers, for philosophers and engineers, What is Mathematics? Second Edition is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world. Covering everything from natural numbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving. With chapters largely independent of one another and sections that lead upward from basic to more advanced discussions, readers can easily pick and choose areas of particular interest without impairing their understanding of subsequent parts.
Brought up to date with a new chapter by Ian Stewart, What is Mathematics? Second Edition offers new insights into recent mathematical developments and describes proofs of the Four-Color Theorem and Fermat's Last Theorem, problems that were still open when Courant and Robbins wrote this masterpiece, but ones that have since been solved.
Formal mathematics is like spelling and grammar - a matter of the correct application of local rules. Meaningful mathematics is like journalism - it tells an interesting story. But unlike some journalism, the story has to be true. The best mathematics is like literature - it brings a story to life before your eyes and involves you in it, intellectually and emotionally. What is Mathematics is like a fine piece of literature - it opens a window onto the world of mathematics for anyone interested to view.
592 pages, Paperback
First published January 1, 1941
628 people are currently reading
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About the author
Richard Courant
136 books59 followersRichard Courant (January 8, 1888 – January 27, 1972) was a German American mathematician. He is best known by the general public for the book What is Mathematics?, co-written with Herbert Robbins.
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Displaying 1 - 30 of 67 reviews
December 4, 2013
This is the ideal book to read during the last years at school, or early in an undergraduate degree. School maths is often about understanding mathematical concepts intuitively via rote learning, repetition and visualization. This book teaches you how mathematical knowledge is arrived at by looking carefully at a few simple ideas and building upon them.
This book is also ideal reading for anybody who wants to improve their general mathematical literacy. Perhaps many people who don't know much about mathematics think that it is about arithmetic or calculation - it is easy to get that impression when school mathematics centers mostly around solving uninteresting problems, which often involve a lot of calculation.
This book shows that mathematics is about understanding the properties and behaviour of mathematical objects. What are numbers? Why do they behave the way they do when we carry out operations on them? Why can some shapes be constructed with ruler and compass, but some not?
The book requires patience and time to read. Don't read it if you are in a hurry.
This book is also ideal reading for anybody who wants to improve their general mathematical literacy. Perhaps many people who don't know much about mathematics think that it is about arithmetic or calculation - it is easy to get that impression when school mathematics centers mostly around solving uninteresting problems, which often involve a lot of calculation.
This book shows that mathematics is about understanding the properties and behaviour of mathematical objects. What are numbers? Why do they behave the way they do when we carry out operations on them? Why can some shapes be constructed with ruler and compass, but some not?
The book requires patience and time to read. Don't read it if you are in a hurry.
January 22, 2012
"The difficulty is that it is easy to prove a problem is easy, but hard to prove that it is hard!"
I'm probably not unique in that I made it through a lot of math without ever really understanding what I was doing, just flipping equations around until an answer poured out. I had a trigonometry professor who at least pushed understanding what you were doing. It was in my first semester of the calculus that math started to make sense. "What Is Mathematics?" was a clear inspiration for some of the lectures in that course. This book ties it all up with a nice bow. The long chapter on maxima and minima -- went on long, I was ready to put the book down and then it made this brilliant transition to the calculus.
I'm sure the more math you've had, the more you will get out of this book -- some sections may be rough if you haven't had some trig and the calculus and I was a little lost more than once anyway.
I'm probably not unique in that I made it through a lot of math without ever really understanding what I was doing, just flipping equations around until an answer poured out. I had a trigonometry professor who at least pushed understanding what you were doing. It was in my first semester of the calculus that math started to make sense. "What Is Mathematics?" was a clear inspiration for some of the lectures in that course. This book ties it all up with a nice bow. The long chapter on maxima and minima -- went on long, I was ready to put the book down and then it made this brilliant transition to the calculus.
I'm sure the more math you've had, the more you will get out of this book -- some sections may be rough if you haven't had some trig and the calculus and I was a little lost more than once anyway.
June 9, 2019
در دوران خستگی و دلزدگی از کنکور و سوالات مسخرهٔ مشتق و انتگرالش، دوستداشتیترین کتابم بود. در سیستمی که دو سال آخر مدرسه رو کلا از هر معنایی تهی کرده، کمکی بود برام تا سرکی بکشم به مبانی ریاضیات و با مفاهیم جالبی آشنا بشم و مطمئن بشم که ریاضی، به همین مسخرگی و آبدوخیاریای که معلمهای مدرسه و ریاضی عمومی دانشگاه درسش میدن نیست!
خیلی متاسفم که این کتاب اینقدر کم قدر دید و اینقدر افراد کمی خوندنش، ولی همینکه میبینم به چاپ هفتم رسیده، خودش مایهٔ دلگرمیه. هنوز دیگرانی هم هستن که چنین شوقی داشته باشن
خیلی متاسفم که این کتاب اینقدر کم قدر دید و اینقدر افراد کمی خوندنش، ولی همینکه میبینم به چاپ هفتم رسیده، خودش مایهٔ دلگرمیه. هنوز دیگرانی هم هستن که چنین شوقی داشته باشن
Currently reading
July 11, 2010Einstein put it best when he said: "[this book is:] a lucid representation of the fundamental concepts and methods of the whole field of mathematics." It really is.
April 14, 2017
This is a book which changed my life. I originally read it when I was 17, and it was the book which confirmed to me that I wanted to study mathematics at university. Re-reading it now, it is a clear and inspiring look at a wide range of interesting mathematics, presenting an elementary view of how it is viewed from a university (rather than a high school) perspective. Some parts are a little dated - even when I read it in the 1980s, the amount of geometrical expertise expected was much more than was present in the O and A level syllabus. The equivalent today would probably include a lot more about logic, as the basis for computer science. But I still found myself getting interested all over again in parts of the subject I haven't thought about for years.
June 21, 2010
Wow, that was absolutely beautiful! Thanks for the heads-up, Alex! I really, really wish I'd have had this book back as a senior or junior in high school; I'd have been able to plan out my math classes early on much better (why on earth did I take Dynamics and Bifurcations?!?! and how did I avoid taking a serious vector spaces/manifold calculus class? would have been nice for upper-level physics! oh well!). I think this will be my default high school graduation present to anyone with an inkling of mathematical talent from now on (rather than the more intimidating Road to Reality, which likely just frightens people).
As they said in Mathematical Review, "A work of extraordinary perfection."
As they said in Mathematical Review, "A work of extraordinary perfection."
December 20, 2021
این کتاب را سال ۸۹ خریدم. ترم اول دانشگاه که با نجفی که اون موقع توی شورای شهر بود کلاس داشتیم. اون بهمون معرفی کرد یا یادم نیست ازش مطلب می آورد.
رابطه من با ریاضی رابطه ای عجیبه؛ بخشی از هویتم از خوب بودن ریاضیاتم در دوران مدرسه شکل گرفته است. اعتماد به نفس، تشویق هایی که مثل چی ازشون لذت می بردم و... تجربه تدریس هم از همین رابطه شروع شد. سوم ابتدایی بودم و خانه مان میدان خراسان. دبستان جهان آرا می رفتم. وسط سال تحصیلی منتقل شده بودم و برای اولین بار سر کلاس یک مرد نشستم. جانباز بود و یک پاشم را می کشید. سخت گیر و خاص بود. نمی دانم چقدر وقتی برای جدول ضرب یاد گرفتن می بردمان نمازخانه مدرسه در زیرزمین و کاغذ و مقوا و قیچی می داد تا جدول ها را بچسبانیم بهم روی علاقه ام به ریاضی تاثیر گذاشت. ولی وقتی یکی از بچه ها را با اجازه ناهید، مادرم می فرستاد خانه تا درس بدهمش و این قصه نقل خانواده و اطرافیان می شد و آن ها لوسم می کردند، اصلا یحتمل از همانجاها با درس دادن میانه ام خوب شد و حسابی مدیون ریاضیات شدم.
ریاضیات دوستی بود که من ازش حسابی استفاده می کردم. و شاید خیلی نبوده ها را جبران می کرد. چه وقتی که شادی ناشی از حل مساله لذت وافری بهم می داد، چه وقتی راهنمایی روز "مدرسه در دست بچه ها" من معلم ریاضی می شدم و می رفتم سر کلاس ها. یادم نرفته که آقای محمدی معلم ریاضی سوم راهنمایی بهم گفت تو درس جدید را بده، و من درس جدید را از مبتکران خواندم و درسش دادم و بچه ها به من یا شاید خود آقای محمدی گفتند بهتر از ایشان درس دادم.
صحنه پیش روی برای من سه یا چهار مرتبه ای بود که همان دوران و بلبشوی آزمونهای سمپاد و بقیه مدارس، چند بار سوالاتی مطرح شد در کلاس که معلم در حلش می ماند و دست در موهای لختش می کرد و من زودتر از او حلش می کردم.
آقای عزیزی اما جنس دیگری داشت. بزرگی و منشش، کتش و جلیقه اش و بند ساعتش. شیوه صحبت کردنش، وقارش و اینکه همیشه می گفت مثلثات درس زندگیه، زندگی من را ساخت. بماند که یادش رفت که من را ببرد سر کلاس های مدرسه دخترانه فرزانگان. گفته بود به بچه ها که اگر در المپیاد قبول بشید می برمتان آنجا سر کلاسهایم بنشینید. قبول شدم با اینکه دومی بودن و با سومی ها رقابت می کردیم. هم ریاضی هم کامپیوتر که آن هم اعم سوالاتش "ترکیبیات" ریاضی بود.
چگونه بدون شمارش بشماریم ترجمه بتول جذبی را می خواندیم و یک کتاب زرد برای ترکیبیات.
بگذریم دانشگاه اما دیگر راضیم نمی کرد. می خواستم بفهمم چرا دو و دو چهار می شود و دقیقا همین بدیهیات حسابی برایم مطرح شده بودند. نمی دانم چرا ولی آن دوران اول دانشگاه درگیری های فلسفی ام خیلی زیاد شده بود. این کتاب در فهم برخی از بدیهیات ریاضی کمکم کرد. چیزهایی که حسابی روی مخم بودند.
وقتی افتادم زندان بعد چند ماه این کتاب هم یکی از کتابهایی بود که به ناهید گفتم تا برایم پست کند.
الان زندانم و گرچه فکر می کردم این رفاقت با ریاضیات و البته استفاده های من ازش تمام شده. اما نشده. چند ماه پیش برادر کسی اینجا امتحان ریاضی داشت و می دانست من تحصیل کرده ام ازم پرسید که چیزی یادمه از ریاضی عمومی دانشگاه گفتم آره کارم تدریس بوده یک دورانی. برگه را شب بعد آورد نمی نویسم چطور!
تا صبح همه اش را حل کردم و باز لذت بردم هم از حل و بیشتر از تعریفای اون پسر توی بند
این کتاب معرکه است. کامل نخوانده امش ولی از کتابهایی است که از خوندنش خیلی راضیم. خلاهای مهمی را پر می کند. حتی تا حدی خلا فلسفه علم که برای همه کسانی که با درس و مشق ارتباط دارند وجود دارد.
رابطه من با ریاضی رابطه ای عجیبه؛ بخشی از هویتم از خوب بودن ریاضیاتم در دوران مدرسه شکل گرفته است. اعتماد به نفس، تشویق هایی که مثل چی ازشون لذت می بردم و... تجربه تدریس هم از همین رابطه شروع شد. سوم ابتدایی بودم و خانه مان میدان خراسان. دبستان جهان آرا می رفتم. وسط سال تحصیلی منتقل شده بودم و برای اولین بار سر کلاس یک مرد نشستم. جانباز بود و یک پاشم را می کشید. سخت گیر و خاص بود. نمی دانم چقدر وقتی برای جدول ضرب یاد گرفتن می بردمان نمازخانه مدرسه در زیرزمین و کاغذ و مقوا و قیچی می داد تا جدول ها را بچسبانیم بهم روی علاقه ام به ریاضی تاثیر گذاشت. ولی وقتی یکی از بچه ها را با اجازه ناهید، مادرم می فرستاد خانه تا درس بدهمش و این قصه نقل خانواده و اطرافیان می شد و آن ها لوسم می کردند، اصلا یحتمل از همانجاها با درس دادن میانه ام خوب شد و حسابی مدیون ریاضیات شدم.
ریاضیات دوستی بود که من ازش حسابی استفاده می کردم. و شاید خیلی نبوده ها را جبران می کرد. چه وقتی که شادی ناشی از حل مساله لذت وافری بهم می داد، چه وقتی راهنمایی روز "مدرسه در دست بچه ها" من معلم ریاضی می شدم و می رفتم سر کلاس ها. یادم نرفته که آقای محمدی معلم ریاضی سوم راهنمایی بهم گفت تو درس جدید را بده، و من درس جدید را از مبتکران خواندم و درسش دادم و بچه ها به من یا شاید خود آقای محمدی گفتند بهتر از ایشان درس دادم.
صحنه پیش روی برای من سه یا چهار مرتبه ای بود که همان دوران و بلبشوی آزمونهای سمپاد و بقیه مدارس، چند بار سوالاتی مطرح شد در کلاس که معلم در حلش می ماند و دست در موهای لختش می کرد و من زودتر از او حلش می کردم.
آقای عزیزی اما جنس دیگری داشت. بزرگی و منشش، کتش و جلیقه اش و بند ساعتش. شیوه صحبت کردنش، وقارش و اینکه همیشه می گفت مثلثات درس زندگیه، زندگی من را ساخت. بماند که یادش رفت که من را ببرد سر کلاس های مدرسه دخترانه فرزانگان. گفته بود به بچه ها که اگر در المپیاد قبول بشید می برمتان آنجا سر کلاسهایم بنشینید. قبول شدم با اینکه دومی بودن و با سومی ها رقابت می کردیم. هم ریاضی هم کامپیوتر که آن هم اعم سوالاتش "ترکیبیات" ریاضی بود.
چگونه بدون شمارش بشماریم ترجمه بتول جذبی را می خواندیم و یک کتاب زرد برای ترکیبیات.
بگذریم دانشگاه اما دیگر راضیم نمی کرد. می خواستم بفهمم چرا دو و دو چهار می شود و دقیقا همین بدیهیات حسابی برایم مطرح شده بودند. نمی دانم چرا ولی آن دوران اول دانشگاه درگیری های فلسفی ام خیلی زیاد شده بود. این کتاب در فهم برخی از بدیهیات ریاضی کمکم کرد. چیزهایی که حسابی روی مخم بودند.
وقتی افتادم زندان بعد چند ماه این کتاب هم یکی از کتابهایی بود که به ناهید گفتم تا برایم پست کند.
الان زندانم و گرچه فکر می کردم این رفاقت با ریاضیات و البته استفاده های من ازش تمام شده. اما نشده. چند ماه پیش برادر کسی اینجا امتحان ریاضی داشت و می دانست من تحصیل کرده ام ازم پرسید که چیزی یادمه از ریاضی عمومی دانشگاه گفتم آره کارم تدریس بوده یک دورانی. برگه را شب بعد آورد نمی نویسم چطور!
تا صبح همه اش را حل کردم و باز لذت بردم هم از حل و بیشتر از تعریفای اون پسر توی بند
این کتاب معرکه است. کامل نخوانده امش ولی از کتابهایی است که از خوندنش خیلی راضیم. خلاهای مهمی را پر می کند. حتی تا حدی خلا فلسفه علم که برای همه کسانی که با درس و مشق ارتباط دارند وجود دارد.
April 16, 2015
In another life, many many moons ago, I received this book as a gift from an English math professor I met on a plane. It was not a completely haphazard encounter: we were both on our way back from the Hong Kong IMO. I was a contestant (not a really successful one), and he was an elderly member of the committee... I did not realize who he was during our pleasant conversation; I only learned that from our team leader after we landed in Frankfurt and parted way with the British contingent. In a couple of weeks, this book arrived in the mail :D
It was my second "real math" book. When I say *real* math, I mean that it was not a problem book or a textbook, but it explored the history and reasoning behind all the ideas I held so dear to my heart and mind. The first one was about Platonic solids ... This one is about Mathematics as a whole ... I will always cherish the afternoons spent going over this book, figuring out both Math and English. I cannot even remember most of the math covered consciously -- but I remember the feeling... For some reason I do Math for a living now, so it must be that some of the ideas in Courant's book stuck.
If you are in high-school, there are probably books which would be more modern in exposition -- you should probably look into one of those. If you are a bit more "advanced" math-lover, you will appreciate the patina in the text and the weight behind the name "Courant".
It was my second "real math" book. When I say *real* math, I mean that it was not a problem book or a textbook, but it explored the history and reasoning behind all the ideas I held so dear to my heart and mind. The first one was about Platonic solids ... This one is about Mathematics as a whole ... I will always cherish the afternoons spent going over this book, figuring out both Math and English. I cannot even remember most of the math covered consciously -- but I remember the feeling... For some reason I do Math for a living now, so it must be that some of the ideas in Courant's book stuck.
If you are in high-school, there are probably books which would be more modern in exposition -- you should probably look into one of those. If you are a bit more "advanced" math-lover, you will appreciate the patina in the text and the weight behind the name "Courant".
August 13, 2009
От тази книга тръгна моят интерес към математиката. Показва красотата на точните науки, водейки читателя за ръка по много интересна пътечка.
May 7, 2016
This is a magnificent book for laymen who want to know about mathematics. This is a sensational book for amateurs who wish to widen and deepen understanding of mathematics. This is an indispensable book for educators who need to find wonderful examples to inspire children and students. Everyone does not want to miss this.
I especially love the geometry part of this book, which contains chapter 3, 4 & 7.
Chapter 3 introduces the ruler and compass construction. This chapter firstly discusses the classical geometry of Des Cartes, namely how to construct the finite sums, differences, products, quotients and square roots of given line section. Then explains the construtable problems. Finally proves the impossibilities of the three ancient famous classical construction problems, squaring the circle, doubling the cube and angle trisection, which cost mathematicians hundreds of years to finally figure out. Readers will be stunned by the connection between algebraic fields and classical geometry, which is an important subject named group theory in modern mathematics.
Chapter 4 talks about geometries beyond Euclidean. Readers' recognition about what the geometry is will be totally subverted by finding out that shape, length and area of a figure might not be intrinsic. Put in another word, one might never distinguish a square from a rectangle and never measure a unique value of its side length or area. Actually, these concepts are undefined in some geometries. Felix Klein defines geometry in Erlangen program as a subject studying the invariant properties of space under group transformations. Projective geometry, studying the invariant properties of space under projection, is one of the most famous and intriguing geometries. (Actually, projective geometry directly leads to the definition from Klein.) In projective geometry, dual roles of line and point make it extremely interesting. Readers will enjoy it from homogeneous coordinate and conic curves. Moreover, authors give a very brief introduction to Riemannian and pseudo Riemannian geometry. The most general, abstract geometry, which contain all other geometries, named topology, is the main topic of Chapter 5, which is surely interesting and readable but I will not talk about it.
Chapter 7 shows some famous extreme values problems in geometry. If those two chapters relate geometry to algebra, then this chapter relates it to analysis. Readers will see how differential calculus and method of variation play the significant role in such problems. Both of which have strong geometric intuitions and applications. The method of variation has been applied to solving problems in optics, classical mechanics, and now relativity and quantum mechanics. Soap film problem is such a very interesting and even artistic example because readers can easily conduct experiments and find out how beautiful the mathematics is. The mean curvature is natural to describe in minimum surface, and has great implications in geometric analysis, a fresh and active study subject.
I especially love the geometry part of this book, which contains chapter 3, 4 & 7.
Chapter 3 introduces the ruler and compass construction. This chapter firstly discusses the classical geometry of Des Cartes, namely how to construct the finite sums, differences, products, quotients and square roots of given line section. Then explains the construtable problems. Finally proves the impossibilities of the three ancient famous classical construction problems, squaring the circle, doubling the cube and angle trisection, which cost mathematicians hundreds of years to finally figure out. Readers will be stunned by the connection between algebraic fields and classical geometry, which is an important subject named group theory in modern mathematics.
Chapter 4 talks about geometries beyond Euclidean. Readers' recognition about what the geometry is will be totally subverted by finding out that shape, length and area of a figure might not be intrinsic. Put in another word, one might never distinguish a square from a rectangle and never measure a unique value of its side length or area. Actually, these concepts are undefined in some geometries. Felix Klein defines geometry in Erlangen program as a subject studying the invariant properties of space under group transformations. Projective geometry, studying the invariant properties of space under projection, is one of the most famous and intriguing geometries. (Actually, projective geometry directly leads to the definition from Klein.) In projective geometry, dual roles of line and point make it extremely interesting. Readers will enjoy it from homogeneous coordinate and conic curves. Moreover, authors give a very brief introduction to Riemannian and pseudo Riemannian geometry. The most general, abstract geometry, which contain all other geometries, named topology, is the main topic of Chapter 5, which is surely interesting and readable but I will not talk about it.
Chapter 7 shows some famous extreme values problems in geometry. If those two chapters relate geometry to algebra, then this chapter relates it to analysis. Readers will see how differential calculus and method of variation play the significant role in such problems. Both of which have strong geometric intuitions and applications. The method of variation has been applied to solving problems in optics, classical mechanics, and now relativity and quantum mechanics. Soap film problem is such a very interesting and even artistic example because readers can easily conduct experiments and find out how beautiful the mathematics is. The mean curvature is natural to describe in minimum surface, and has great implications in geometric analysis, a fresh and active study subject.
February 13, 2017
I am strongly disappointed with this book. I think it only delayed my progress in mathematics. And here's why: written supposedly for non-mathematicians it contains so many abstract concepts and strict reasoning that it's virtually impossible to apprehend it being a person with a non-technical background (put aside that I work as a software developer for almost 5 years now). For me, the book is more like Wikipedia than like a guide for newcomers.
My point is that for a novice, a tour into something new should contain hooks tying new ideas to the person's previous experience. And this book doesn't provide such approach at all.
My point is that for a novice, a tour into something new should contain hooks tying new ideas to the person's previous experience. And this book doesn't provide such approach at all.
December 7, 2012
I wanted to like this book. I really did. But it's just too old. It will never be a classic; it's just outdated.
There are references to ongoing problems to solve Hilbert's 10th problem, but no acknowledgement of Alan Turing's contributions (probably because his work was still classified or unknown at the time of publication). Godel's "recent work on incompleteness" is a footnote to the summary of Russell's "great work", which made me snicker.
The notation is pretty bad as well. Many of the fonts are difficult to read, and some of the shorthand is less than modern.
This may have been a good introduction/refresher at some point in time, but there have to be better books out there now. For one, your time would be better spent reading Heath's translation of Euclid. Some day I hope to make time for The Princeton Companion to Mathematics. I'll see how it compares.
There are references to ongoing problems to solve Hilbert's 10th problem, but no acknowledgement of Alan Turing's contributions (probably because his work was still classified or unknown at the time of publication). Godel's "recent work on incompleteness" is a footnote to the summary of Russell's "great work", which made me snicker.
The notation is pretty bad as well. Many of the fonts are difficult to read, and some of the shorthand is less than modern.
This may have been a good introduction/refresher at some point in time, but there have to be better books out there now. For one, your time would be better spent reading Heath's translation of Euclid. Some day I hope to make time for The Princeton Companion to Mathematics. I'll see how it compares.
April 6, 2021
A very good philosophically inclined text offering lots of inticing tidbits about lots of topics. Lots of context on varied topics, including differencial calculus, topology, numbers, geoometry, etc. Definitely worth a read for anyone willing to either brush up on stuff or review or just have some cerebral fun.
December 23, 2008
This is one of the best high school level extensions in mathematics. Originally called -
Mathematics for Lori - it was written for his daughter.
Mathematics for Lori - it was written for his daughter.
July 1, 2019
Q. what is this book about?
As a first approximation, here are the(abridged) contents of the book(more on all of these later):
Chapter 1 : natural numbers(intuitive "definition" of integers, representations of integers in non-decimal bases.
mathematical induction and proofs of geometric and arithmetical series based on the principle of induction.
Elementary number theory(congruence, divisibility etc.)
Chapter 2: number systems(definitions of Q and R(not formal definitions, but reasonably well-explained), cardinality of reals. complex numbers and operations with them, algebraic and transcendental numbers.
Chapter 3 : geometrical constructions(limits of ruler and compass constructions, proofs of the impossibility of two of the three famous Greek construction problems(trisecting and angle and all that)
Chapter 4: mainly projective geometry, remarks on non-euclidean geometries, discussion of the axiomatic method.
Chapter 5: Basics of Topology.
Chapter 6: functions and limits
Chapter 7: Maximum and minimum problems
Chapter 8 : Differential and integral calculus
-----------------------------------------
Q. how rigorous is the book?
A. varies chapter by chapter. Generally, the proofs are provided, but they are not absolutely rigorous. Sometimes, the proofs are merely hinted at, this can be seen particularly on the chapter on Topology.
--------------------------
Q. how difficult is the book?
A. Again, varies chapter by chapter. But this is mainly a consequence of the changes in mathematics education since the book was published. I think that the chapters on geometry, especially chapter 4, were quite difficult, but I think that this is partly because of the lack of proper education in geometry in current school curricula. Some other chapters, by contrast, were easy. Indeed, I only skimmed chapters 6 and 8 because I felt I was already familiar with most of the content.
--------------------------
The title might be misleading. One might think that this is a book that talks about mathematics, without any actual mathematics being done. This is not the case. This can mostly be thought of as an ordinary textbook on elementary mathematics. Now, by this I don't mean that the contents are easy or shallow, they are not. You could say that the book is a response to the question :
" What happens if you start thinking more deeply about certain areas of high-school mathematics?"
There are however, some small philosophical and historical sketches, but this does not make it a philosophy of mathematics book.
My main problem with the book was the over-emphasis on classical geometry. Although it is a beautiful subject, and I learnt many things that I'm certain I would not have known about in a standard undergraduate curriculum(say, projective geometry), It is far-removed from most of the modern undergraduate mathematics curriculum, and seems to me to be almost a dead subject.
This is not, by itself, a problem. But I was reading this book mostly to prepare for university mathematics, and I Don't think that it did a very good job in that.
However, in some ways, This was a type of book I had always wished to have read. I always had the suspicion that my knowledge of elementary mathematics was woefully incomplete, and there are certainly many deep theorems and beautiful results to be found there. Indeed, there are, and you can find them in this book.
As a first approximation, here are the(abridged) contents of the book(more on all of these later):
Chapter 1 : natural numbers(intuitive "definition" of integers, representations of integers in non-decimal bases.
mathematical induction and proofs of geometric and arithmetical series based on the principle of induction.
Elementary number theory(congruence, divisibility etc.)
Chapter 2: number systems(definitions of Q and R(not formal definitions, but reasonably well-explained), cardinality of reals. complex numbers and operations with them, algebraic and transcendental numbers.
Chapter 3 : geometrical constructions(limits of ruler and compass constructions, proofs of the impossibility of two of the three famous Greek construction problems(trisecting and angle and all that)
Chapter 4: mainly projective geometry, remarks on non-euclidean geometries, discussion of the axiomatic method.
Chapter 5: Basics of Topology.
Chapter 6: functions and limits
Chapter 7: Maximum and minimum problems
Chapter 8 : Differential and integral calculus
-----------------------------------------
Q. how rigorous is the book?
A. varies chapter by chapter. Generally, the proofs are provided, but they are not absolutely rigorous. Sometimes, the proofs are merely hinted at, this can be seen particularly on the chapter on Topology.
--------------------------
Q. how difficult is the book?
A. Again, varies chapter by chapter. But this is mainly a consequence of the changes in mathematics education since the book was published. I think that the chapters on geometry, especially chapter 4, were quite difficult, but I think that this is partly because of the lack of proper education in geometry in current school curricula. Some other chapters, by contrast, were easy. Indeed, I only skimmed chapters 6 and 8 because I felt I was already familiar with most of the content.
--------------------------
The title might be misleading. One might think that this is a book that talks about mathematics, without any actual mathematics being done. This is not the case. This can mostly be thought of as an ordinary textbook on elementary mathematics. Now, by this I don't mean that the contents are easy or shallow, they are not. You could say that the book is a response to the question :
" What happens if you start thinking more deeply about certain areas of high-school mathematics?"
There are however, some small philosophical and historical sketches, but this does not make it a philosophy of mathematics book.
My main problem with the book was the over-emphasis on classical geometry. Although it is a beautiful subject, and I learnt many things that I'm certain I would not have known about in a standard undergraduate curriculum(say, projective geometry), It is far-removed from most of the modern undergraduate mathematics curriculum, and seems to me to be almost a dead subject.
This is not, by itself, a problem. But I was reading this book mostly to prepare for university mathematics, and I Don't think that it did a very good job in that.
However, in some ways, This was a type of book I had always wished to have read. I always had the suspicion that my knowledge of elementary mathematics was woefully incomplete, and there are certainly many deep theorems and beautiful results to be found there. Indeed, there are, and you can find them in this book.
June 21, 2019
So I returned the book after page 65.
The premise is promising: bridging high school and college mathematics by going through all major areas of mathematics from scratch and showing everything in proof with exercise problems. While one can learn a lot of the actual rationale behind the mathematical principles we study with rote learning (like why we get a positive number from multiplying two negative numbers), the explanation ,for this reader, is often too technical and requires multiple reading to follow its logic....and this reader here is already doing more than quite well in high school calculus. Yet, there are times when this reader spent the whole evening to get through 2 pages... The lack of any tip for the exercise here is not helping either. As a book to bring expertise knowledge to the public, it's hard to see why the author decided to convey his content in this dry, technical tone.
The author did suggest readers to skip all the harder parts and enjoy the journey but, argh, just so frustrating.
The premise is promising: bridging high school and college mathematics by going through all major areas of mathematics from scratch and showing everything in proof with exercise problems. While one can learn a lot of the actual rationale behind the mathematical principles we study with rote learning (like why we get a positive number from multiplying two negative numbers), the explanation ,for this reader, is often too technical and requires multiple reading to follow its logic....and this reader here is already doing more than quite well in high school calculus. Yet, there are times when this reader spent the whole evening to get through 2 pages... The lack of any tip for the exercise here is not helping either. As a book to bring expertise knowledge to the public, it's hard to see why the author decided to convey his content in this dry, technical tone.
The author did suggest readers to skip all the harder parts and enjoy the journey but, argh, just so frustrating.
October 22, 2016
This book has risen to the top to be one of my favorite math books. The other is David Foster Wallace's book on Infinity. What makes them fun are these are prose books on math. The rigor is there but you do not need to work through the actual with a pencil and paper. Instead the author walks you through the logic.
You will find this book very difficult at times. But that's the point. It's fun to be challenged.
You don't need any knowledge of advanced math to read this book. What makes it fun is that the author walks you through the most famous proofs in all of mathematics simplifying them to simple equations that you can solve in your head. Some of these are difficult and you will need to pull out paper and pencil. But there is a lot of joy in understanding, for example, how the proof of Fermat's Last Theorem works or what is an algebraic expression.
You will find this book very difficult at times. But that's the point. It's fun to be challenged.
You don't need any knowledge of advanced math to read this book. What makes it fun is that the author walks you through the most famous proofs in all of mathematics simplifying them to simple equations that you can solve in your head. Some of these are difficult and you will need to pull out paper and pencil. But there is a lot of joy in understanding, for example, how the proof of Fermat's Last Theorem works or what is an algebraic expression.
October 24, 2016
Passed a lot of math courses but failed to make any sense out of them ? If your answer is yes, this book is for you.
As Albert Einstein said this book is a lucid representation of fundamental concepts and methods of whole field of mathematics.
But, it is not a casual mathematics history book which you can flip through at leisure. It is a serious read involving actual mathematics and should be treated as a reference book and should be refrenced from time to time, specially if you are in high school or college.
As Albert Einstein said this book is a lucid representation of fundamental concepts and methods of whole field of mathematics.
But, it is not a casual mathematics history book which you can flip through at leisure. It is a serious read involving actual mathematics and should be treated as a reference book and should be refrenced from time to time, specially if you are in high school or college.
December 22, 2012
One of my favorite books on mathematics. Also, one of Einstein's favorites.
July 24, 2018
This is a good book to read in your first years in the maths degree, even for prospective mathematics students who wants an advance of how university mathematics are. I think this is a high school level extension, but a very strong extension. Besides, Courant and Robbins give their opinion about the necessity of the mathematical concepts and ideas, just as their location in Mathematics, understanding these as an interconnected discipline.
The edition I read is a revision by Ian Stewart, who added some commentaries and wrote extensions to several of the chapters in the light of recent progress (well, recent in 1996 when this edition was published).
The chapters are almost entirely independent of one another, each one about a different mathematical topic, like natural numbers, ruler and compass contructions, projective geometry, topology, calculus or the number system. Often the beginning is easier to understand, and then the path leads gradually upward. A reader who wants only a general vision may be content with a selection of material, avoiding the more detailed discussions.
It's a book not to read in a day. It requires time and patience. You will need paper and pencil near you. And to really enjoying it you have to have certain previous knowledge in issues like trigonometry or calculus. If not, you can be a little lost sometimes... but there is always the chance to jump the darkest paragraphs and continue later.
The book includes many exercises proposed to the advanced reader, some are easier and some a bit more difficult. The authors come out with them between discussiones, and Ian Stewarts add a bunch of them at the end of the book.
The edition I read is a revision by Ian Stewart, who added some commentaries and wrote extensions to several of the chapters in the light of recent progress (well, recent in 1996 when this edition was published).
The chapters are almost entirely independent of one another, each one about a different mathematical topic, like natural numbers, ruler and compass contructions, projective geometry, topology, calculus or the number system. Often the beginning is easier to understand, and then the path leads gradually upward. A reader who wants only a general vision may be content with a selection of material, avoiding the more detailed discussions.
It's a book not to read in a day. It requires time and patience. You will need paper and pencil near you. And to really enjoying it you have to have certain previous knowledge in issues like trigonometry or calculus. If not, you can be a little lost sometimes... but there is always the chance to jump the darkest paragraphs and continue later.
The book includes many exercises proposed to the advanced reader, some are easier and some a bit more difficult. The authors come out with them between discussiones, and Ian Stewarts add a bunch of them at the end of the book.
April 26, 2020
lots of interesting things to ponder in here, especially for an PhD dropout. I most enjoyed the problems I haven't worked rigorously before- see e.g. knot theory, soap bubbles, and Steiners problems. what keeps me from giving this book a five is it switching in between:
-material I find pretty basic- such as the introduction of calculus, and prim d
-material which is hazy and I've mostly covered before, such as complex numbers or topology
-material which is fascinating, and I want to try and follow the proofs, but they aren't too 100% rigor- such as the Prime Number Theorem
-material which blew me away (the railway rod problem) but then in the appendix of recent developments we are told the solution is wrong
i was given this book a long time ago. if I had the bandwidth to read something like this in highschool, if I could have handled it, it would have been eye opening. now, it gives me more of a feeling of my past life, which parts of me miss, but is also long gone.
-material I find pretty basic- such as the introduction of calculus, and prim d
-material which is hazy and I've mostly covered before, such as complex numbers or topology
-material which is fascinating, and I want to try and follow the proofs, but they aren't too 100% rigor- such as the Prime Number Theorem
-material which blew me away (the railway rod problem) but then in the appendix of recent developments we are told the solution is wrong
i was given this book a long time ago. if I had the bandwidth to read something like this in highschool, if I could have handled it, it would have been eye opening. now, it gives me more of a feeling of my past life, which parts of me miss, but is also long gone.
December 10, 2025
Was ist Mathematik? – Eine lange Geschichte darüber, warum gute Regeln manchmal die falschen sind
Richard Courant und Herbert Robbins haben mit „Was ist Mathematik?“ einen Titel gewählt, der klingt wie eine Einladung – und sich bei näherer Lektüre als elegante Zumutung entpuppt. Wer erwartet, hier eine freundliche Einführung in die Welt der Zahlen zu erhalten, bekommt stattdessen eine intellektuelle Langstrecke serviert: eine ebenso lehrreiche wie leicht grausame Geschichte darüber, was passiert, wenn eine Hochkultur über Jahrhunderte hinweg an den falschen Werkzeugen festhält. Das Buch ist weniger eine Definition der Mathematik als ihre Autobiographie im Modus der Selbstbeschränkung.
Im Zentrum steht ein bemerkenswert simples Regelwerk, das die griechische Mathematik über Jahrtausende dominierte: erlaubt waren genau zwei Instrumente – ein Zirkel und ein Lineal, wohlgemerkt ohne Skala. Diese bewusste Askese, gedacht als Ausdruck geometrischer Reinheit, entwickelte sich rückblickend zu einem der folgenreichsten Experimente der Geistesgeschichte. Mit diesen beiden Werkzeugen stellte sich die Antike drei Aufgaben, die harmlos klingen, sich jedoch als unüberwindbare intellektuelle Sperren erwiesen: die Dreiteilung (Trisektion) eines beliebigen Winkels, die Verdoppelung des Würfels und die Quadratur des Kreises, also die Konstruktion eines Quadrats mit derselben Fläche wie ein gegebener Kreis.
Courant und Robbins führen diese Probleme nicht als Kuriositäten vor, sondern als Schlüssel zum Verständnis mathematischer Erkenntnis. Generation um Generation versuchte sich an diesen Aufgaben, verfeinerte Konstruktionen, erfand Hilfslinien, verwarf Ansätze und begann erneut. Das Ergebnis war nicht Fortschritt im Sinne einer Lösung, sondern eine jahrhundertelange Kultur der eleganten Verzweiflung. Besonders die Quadratur des Kreises entwickelte sich zu einem Monument des Scheiterns: Der Widerstand der Zahl π gegen jede geometrische Zähmung beschäftigte antike Geometer ebenso wie mittelalterliche Tüftler und moderne Hobby-Mathematiker. Das Buch verdichtet diese endlosen Bemühungen mit trockenem Humor zu einem Lehrstück darüber, wie produktiv Irrwege sein können – und wie hartnäckig Menschen an ihnen festhalten.
Doch diese Tragödie beginnt tiefer. Lange bevor die klassischen Konstruktionsprobleme ihre volle Wirkung entfalteten, stießen die Griechen auf ein Problem, das ihr Weltbild erschütterte: die Existenz inkommensurabler Größen. Die Entdeckung, dass etwa √2 sich nicht als Verhältnis ganzer Zahlen ausdrücken lässt, war keine technische Randnotiz, sondern ein philosophischer Schock. Plötzlich geriet die Vorstellung ins Wanken, die Welt lasse sich vollständig durch harmonische Zahlenverhältnisse erfassen. Fragen nach Stetigkeit, Bewegung und Unendlichkeit drängten sich auf – und wurden zunächst nicht gelöst, sondern umgangen. Courant und Robbins zeigen eindrücklich, wie diese frühe Begegnung mit dem Unmessbaren paradoxerweise nicht zur Öffnung, sondern zur Verkrampfung führte: Statt den Zahlbegriff zu erweitern, klammerte man sich umso fester an die Geometrie.
In dieser Perspektive erscheint die griechische Mathematik nicht als Vorstufe moderner Algebra, sondern als bewusst alternative Tradition, die ihre eigenen Möglichkeiten systematisch begrenzte. Sie wusste oft erstaunlich genau, welche Bauanleitung sie eigentlich brauchte – verweigerte sich aber dem Werkzeug, das sie hätte umsetzen können. Das Buch liest sich hier wie ein Lehrstück über Prinzipientreue als Erkenntnishindernis.
Die große Wendung setzt erst viel später ein, und Courant und Robbins inszenieren sie mit sichtbarem Vergnügen. Die Erlösung kommt nicht aus der Geometrie selbst, sondern von außen: aus der Algebra und der Zahlentheorie des 19. Jahrhunderts. Mit dem Aufkommen eines erweiterten Zahlbegriffs, der irrationalen, algebraischen und schließlich transzendenten Zahlen einen präzisen Platz zuweist, kippt die Perspektive vollständig. Was jahrtausendelang als ungelöst galt, erweist sich nun als prinzipiell unlösbar. Der entscheidende Schlag fällt 1882, als Ferdinand von Lindemann die Transzendenz von π beweist – und damit endgültig klarmacht, dass die Quadratur des Kreises mit Zirkel und Lineal nie möglich war. Nicht, weil die Versuche schlecht waren, sondern weil die Aufgabe selbst außerhalb des erlaubten Zahlenkosmos lag.
Hier liegt die eigentliche Pointe des Buches: Die antiken Mathematiker scheiterten nicht an mangelnder Intelligenz, sondern an den Grenzen ihres Regelwerks. Courant und Robbins machen deutlich, dass mathematische Größe nicht nur im Lösen von Problemen besteht, sondern auch im Beweisen ihrer Unlösbarkeit. Erst dieser negative Beweis beendet die jahrhundertelangen Spiele endgültig – und verleiht dem Scheitern im Nachhinein Würde und Sinn.
„Was ist Mathematik?“ ist weit mehr als ein populäres Einführungswerk. Es ist eine intellektuelle Archäologie der Selbstbeschränkung, eine stille Tragödie über 2000 Jahre mathematischer Askese und zugleich eine Feier des Moments, in dem sich das Denken neue Werkzeuge erlaubt. Wer dieses Buch liest, versteht nicht nur, warum die Antike so lange ohne Algebra auskam, sondern auch, warum manchmal erst ein radikaler Perspektivwechsel nötig ist, um Fortschritt zu ermöglichen.
Als Fazit bleibt ein tröstlicher Gedanke: Wer jemals an der Quadratur des Kreises verzweifelt ist, war nicht zu dumm – er spielte lediglich nach Regeln, die das Gewinnen unmöglich machten. Mathematik, so zeigt dieses Buch, ist nicht nur die Kunst des Rechnens, sondern auch die Fähigkeit, rechtzeitig die eigenen Werkzeuge zu hinterfragen.
Richard Courant und Herbert Robbins haben mit „Was ist Mathematik?“ einen Titel gewählt, der klingt wie eine Einladung – und sich bei näherer Lektüre als elegante Zumutung entpuppt. Wer erwartet, hier eine freundliche Einführung in die Welt der Zahlen zu erhalten, bekommt stattdessen eine intellektuelle Langstrecke serviert: eine ebenso lehrreiche wie leicht grausame Geschichte darüber, was passiert, wenn eine Hochkultur über Jahrhunderte hinweg an den falschen Werkzeugen festhält. Das Buch ist weniger eine Definition der Mathematik als ihre Autobiographie im Modus der Selbstbeschränkung.
Im Zentrum steht ein bemerkenswert simples Regelwerk, das die griechische Mathematik über Jahrtausende dominierte: erlaubt waren genau zwei Instrumente – ein Zirkel und ein Lineal, wohlgemerkt ohne Skala. Diese bewusste Askese, gedacht als Ausdruck geometrischer Reinheit, entwickelte sich rückblickend zu einem der folgenreichsten Experimente der Geistesgeschichte. Mit diesen beiden Werkzeugen stellte sich die Antike drei Aufgaben, die harmlos klingen, sich jedoch als unüberwindbare intellektuelle Sperren erwiesen: die Dreiteilung (Trisektion) eines beliebigen Winkels, die Verdoppelung des Würfels und die Quadratur des Kreises, also die Konstruktion eines Quadrats mit derselben Fläche wie ein gegebener Kreis.
Courant und Robbins führen diese Probleme nicht als Kuriositäten vor, sondern als Schlüssel zum Verständnis mathematischer Erkenntnis. Generation um Generation versuchte sich an diesen Aufgaben, verfeinerte Konstruktionen, erfand Hilfslinien, verwarf Ansätze und begann erneut. Das Ergebnis war nicht Fortschritt im Sinne einer Lösung, sondern eine jahrhundertelange Kultur der eleganten Verzweiflung. Besonders die Quadratur des Kreises entwickelte sich zu einem Monument des Scheiterns: Der Widerstand der Zahl π gegen jede geometrische Zähmung beschäftigte antike Geometer ebenso wie mittelalterliche Tüftler und moderne Hobby-Mathematiker. Das Buch verdichtet diese endlosen Bemühungen mit trockenem Humor zu einem Lehrstück darüber, wie produktiv Irrwege sein können – und wie hartnäckig Menschen an ihnen festhalten.
Doch diese Tragödie beginnt tiefer. Lange bevor die klassischen Konstruktionsprobleme ihre volle Wirkung entfalteten, stießen die Griechen auf ein Problem, das ihr Weltbild erschütterte: die Existenz inkommensurabler Größen. Die Entdeckung, dass etwa √2 sich nicht als Verhältnis ganzer Zahlen ausdrücken lässt, war keine technische Randnotiz, sondern ein philosophischer Schock. Plötzlich geriet die Vorstellung ins Wanken, die Welt lasse sich vollständig durch harmonische Zahlenverhältnisse erfassen. Fragen nach Stetigkeit, Bewegung und Unendlichkeit drängten sich auf – und wurden zunächst nicht gelöst, sondern umgangen. Courant und Robbins zeigen eindrücklich, wie diese frühe Begegnung mit dem Unmessbaren paradoxerweise nicht zur Öffnung, sondern zur Verkrampfung führte: Statt den Zahlbegriff zu erweitern, klammerte man sich umso fester an die Geometrie.
In dieser Perspektive erscheint die griechische Mathematik nicht als Vorstufe moderner Algebra, sondern als bewusst alternative Tradition, die ihre eigenen Möglichkeiten systematisch begrenzte. Sie wusste oft erstaunlich genau, welche Bauanleitung sie eigentlich brauchte – verweigerte sich aber dem Werkzeug, das sie hätte umsetzen können. Das Buch liest sich hier wie ein Lehrstück über Prinzipientreue als Erkenntnishindernis.
Die große Wendung setzt erst viel später ein, und Courant und Robbins inszenieren sie mit sichtbarem Vergnügen. Die Erlösung kommt nicht aus der Geometrie selbst, sondern von außen: aus der Algebra und der Zahlentheorie des 19. Jahrhunderts. Mit dem Aufkommen eines erweiterten Zahlbegriffs, der irrationalen, algebraischen und schließlich transzendenten Zahlen einen präzisen Platz zuweist, kippt die Perspektive vollständig. Was jahrtausendelang als ungelöst galt, erweist sich nun als prinzipiell unlösbar. Der entscheidende Schlag fällt 1882, als Ferdinand von Lindemann die Transzendenz von π beweist – und damit endgültig klarmacht, dass die Quadratur des Kreises mit Zirkel und Lineal nie möglich war. Nicht, weil die Versuche schlecht waren, sondern weil die Aufgabe selbst außerhalb des erlaubten Zahlenkosmos lag.
Hier liegt die eigentliche Pointe des Buches: Die antiken Mathematiker scheiterten nicht an mangelnder Intelligenz, sondern an den Grenzen ihres Regelwerks. Courant und Robbins machen deutlich, dass mathematische Größe nicht nur im Lösen von Problemen besteht, sondern auch im Beweisen ihrer Unlösbarkeit. Erst dieser negative Beweis beendet die jahrhundertelangen Spiele endgültig – und verleiht dem Scheitern im Nachhinein Würde und Sinn.
„Was ist Mathematik?“ ist weit mehr als ein populäres Einführungswerk. Es ist eine intellektuelle Archäologie der Selbstbeschränkung, eine stille Tragödie über 2000 Jahre mathematischer Askese und zugleich eine Feier des Moments, in dem sich das Denken neue Werkzeuge erlaubt. Wer dieses Buch liest, versteht nicht nur, warum die Antike so lange ohne Algebra auskam, sondern auch, warum manchmal erst ein radikaler Perspektivwechsel nötig ist, um Fortschritt zu ermöglichen.
Als Fazit bleibt ein tröstlicher Gedanke: Wer jemals an der Quadratur des Kreises verzweifelt ist, war nicht zu dumm – er spielte lediglich nach Regeln, die das Gewinnen unmöglich machten. Mathematik, so zeigt dieses Buch, ist nicht nur die Kunst des Rechnens, sondern auch die Fähigkeit, rechtzeitig die eigenen Werkzeuge zu hinterfragen.
March 3, 2016
Skimmed through and it is quite useful, very similar to one of Felix Klein's books in terms of style. A little too mathematics for my purposes which is more towards physics, some insights are good nonetheless.
January 3, 2023
I liked it at first, but jumping from topic to topic started leaving me exhausted after a while. I guess I like a more focused math book.
September 4, 2021
el review compiado de otra persona en goodreads pero tiene casi todo lo que yo diria
Q. what is this book about?
As a first approximation, here are the(abridged) contents of the book(more on all of these later):
Chapter 1 : natural numbers(intuitive "definition" of integers, representations of integers in non-decimal bases.
mathematical induction and proofs of geometric and arithmetical series based on the principle of induction.
Elementary number theory(congruence, divisibility etc.)
Chapter 2: number systems(definitions of Q and R(not formal definitions, but reasonably well-explained), cardinality of reals. complex numbers and operations with them, algebraic and transcendental numbers.
Chapter 3 : geometrical constructions(limits of ruler and compass constructions, proofs of the impossibility of two of the three famous Greek construction problems(trisecting and angle and all that)
Chapter 4: mainly projective geometry, remarks on non-euclidean geometries, discussion of the axiomatic method.
Chapter 5: Basics of Topology.
Chapter 6: functions and limits
Chapter 7: Maximum and minimum problems
Chapter 8 : Differential and integral calculus
-----------------------------------------
Q. how rigorous is the book?
A. varies chapter by chapter. Generally, the proofs are provided, but they are not absolutely rigorous. Sometimes, the proofs are merely hinted at, this can be seen particularly on the chapter on Topology.
--------------------------
Q. how difficult is the book?
A. Again, varies chapter by chapter. But this is mainly a consequence of the changes in mathematics education since the book was published. I think that the chapters on geometry, especially chapter 4, were quite difficult, but I think that this is partly because of the lack of proper education in geometry in current school curricula. Some other chapters, by contrast, were easy. Indeed, I only skimmed chapters 6 and 8 because I felt I was already familiar with most of the content.
--------------------------
The title might be misleading. One might think that this is a book that talks about mathematics, without any actual mathematics being done. This is not the case. This can mostly be thought of as an ordinary textbook on elementary mathematics. Now, by this I don't mean that the contents are easy or shallow, they are not. You could say that the book is a response to the question :
" What happens if you start thinking more deeply about certain areas of high-school mathematics?"
There are however, some small philosophical and historical sketches, but this does not make it a philosophy of mathematics book.
My main problem with the book was the over-emphasis on classical geometry. Although it is a beautiful subject, and I learnt many things that I'm certain I would not have known about in a standard undergraduate curriculum(say, projective geometry), It is far-removed from most of the modern undergraduate mathematics curriculum, and seems to me to be almost a dead subject.
This is not, by itself, a problem. But I was reading this book mostly to prepare for university mathematics, and I Don't think that it did a very good job in that.
However, in some ways, This was a type of book I had always wished to have read. I always had the suspicion that my knowledge of elementary mathematics was woefully incomplete, and there are certainly many deep theorems and beautiful results to be found there. Indeed, there are, and you can find them in this book
Q. what is this book about?
As a first approximation, here are the(abridged) contents of the book(more on all of these later):
Chapter 1 : natural numbers(intuitive "definition" of integers, representations of integers in non-decimal bases.
mathematical induction and proofs of geometric and arithmetical series based on the principle of induction.
Elementary number theory(congruence, divisibility etc.)
Chapter 2: number systems(definitions of Q and R(not formal definitions, but reasonably well-explained), cardinality of reals. complex numbers and operations with them, algebraic and transcendental numbers.
Chapter 3 : geometrical constructions(limits of ruler and compass constructions, proofs of the impossibility of two of the three famous Greek construction problems(trisecting and angle and all that)
Chapter 4: mainly projective geometry, remarks on non-euclidean geometries, discussion of the axiomatic method.
Chapter 5: Basics of Topology.
Chapter 6: functions and limits
Chapter 7: Maximum and minimum problems
Chapter 8 : Differential and integral calculus
-----------------------------------------
Q. how rigorous is the book?
A. varies chapter by chapter. Generally, the proofs are provided, but they are not absolutely rigorous. Sometimes, the proofs are merely hinted at, this can be seen particularly on the chapter on Topology.
--------------------------
Q. how difficult is the book?
A. Again, varies chapter by chapter. But this is mainly a consequence of the changes in mathematics education since the book was published. I think that the chapters on geometry, especially chapter 4, were quite difficult, but I think that this is partly because of the lack of proper education in geometry in current school curricula. Some other chapters, by contrast, were easy. Indeed, I only skimmed chapters 6 and 8 because I felt I was already familiar with most of the content.
--------------------------
The title might be misleading. One might think that this is a book that talks about mathematics, without any actual mathematics being done. This is not the case. This can mostly be thought of as an ordinary textbook on elementary mathematics. Now, by this I don't mean that the contents are easy or shallow, they are not. You could say that the book is a response to the question :
" What happens if you start thinking more deeply about certain areas of high-school mathematics?"
There are however, some small philosophical and historical sketches, but this does not make it a philosophy of mathematics book.
My main problem with the book was the over-emphasis on classical geometry. Although it is a beautiful subject, and I learnt many things that I'm certain I would not have known about in a standard undergraduate curriculum(say, projective geometry), It is far-removed from most of the modern undergraduate mathematics curriculum, and seems to me to be almost a dead subject.
This is not, by itself, a problem. But I was reading this book mostly to prepare for university mathematics, and I Don't think that it did a very good job in that.
However, in some ways, This was a type of book I had always wished to have read. I always had the suspicion that my knowledge of elementary mathematics was woefully incomplete, and there are certainly many deep theorems and beautiful results to be found there. Indeed, there are, and you can find them in this book
July 1, 2025
This has become something of a classic work in mathematics. No less a mind than Einstein himself recommended as an easily understandable overview of the entire field of mathematics. And it really is a great book. It’s a little dated in its approach, though a final chapter at the end concerning later developments does bring some of its content up to date, but if you’re looking for a single book to give you a taste of what mathematics is really all about, this would be one of my top several recommendations.
It's important to understand what the subtitle means by “elementary approach,” however. This is not elementary in the sense that many people might think. It’s not going to teach you basic arithmetic (which is, alas, all many people seem to know of mathematics). It’s not even going to teach you basic algebra. Indeed, its goal isn’t really to teach you how to DO mathematics of any sort. Instead, it intends to teach you about the breadth of mathematical ideas and methods, and to help you think like a mathematician. It is fairly elementary in the context of serious mathematics. For the most part, it assumes only that you have mastered basic algebra. A few chapters will require you to know how to manipulate the trigonometric functions. But that’s really all you need to be able to follow the book.
But at the same time, if you’re a budding mathematician who wants to learn all of the various disciplines it covers—graph theory, topology, calculus, you name it—you’ll need to supplement your education with other books. Its goal is to show you some fascinating results in each field and sort of give you a flavor of what each discipline is all about, not to replace an entire undergraduate degree in mathematics. Think of this as a tasting menu for math.
I would. However, highly recommend this for anyone embarking on (or considering) a degree in mathematics. It won’t tell you everything, but it’ll give you a good taste of what you’ll be doing in each course you’re likely to take and will likely put you ahead of the curve in many of them. For everyone else not currently studying mathematics in school, it remains a delightful one-stop-shop for anyone wondering about the kinds of things mathematicians spend their time thinking about.
It's important to understand what the subtitle means by “elementary approach,” however. This is not elementary in the sense that many people might think. It’s not going to teach you basic arithmetic (which is, alas, all many people seem to know of mathematics). It’s not even going to teach you basic algebra. Indeed, its goal isn’t really to teach you how to DO mathematics of any sort. Instead, it intends to teach you about the breadth of mathematical ideas and methods, and to help you think like a mathematician. It is fairly elementary in the context of serious mathematics. For the most part, it assumes only that you have mastered basic algebra. A few chapters will require you to know how to manipulate the trigonometric functions. But that’s really all you need to be able to follow the book.
But at the same time, if you’re a budding mathematician who wants to learn all of the various disciplines it covers—graph theory, topology, calculus, you name it—you’ll need to supplement your education with other books. Its goal is to show you some fascinating results in each field and sort of give you a flavor of what each discipline is all about, not to replace an entire undergraduate degree in mathematics. Think of this as a tasting menu for math.
I would. However, highly recommend this for anyone embarking on (or considering) a degree in mathematics. It won’t tell you everything, but it’ll give you a good taste of what you’ll be doing in each course you’re likely to take and will likely put you ahead of the curve in many of them. For everyone else not currently studying mathematics in school, it remains a delightful one-stop-shop for anyone wondering about the kinds of things mathematicians spend their time thinking about.
June 23, 2021
A nice book to get an overall understanding of the main fields of math. It's difficult to read sometimes, I read some sections without a full understanding, but it did help me have a better intuition of those topics. From this intuitive understanding now I will deepen my knowledge through other books and resources, this book is a good starting point.
I think more illustrations would make it easier to understand.
The kindle edition has display issues. Math expressions are sometimes too small o blurry.
I recommend it to those who have many unknowns in math, those who got through school solving problems but lacking real understanding, or locking knowledge of how the different math areas relate.
I think more illustrations would make it easier to understand.
The kindle edition has display issues. Math expressions are sometimes too small o blurry.
I recommend it to those who have many unknowns in math, those who got through school solving problems but lacking real understanding, or locking knowledge of how the different math areas relate.
May 20, 2023
Whoever said this is for high school students is lying. Or early university, lying. This is a book for those naturally interested in mathematics or who’ve really become inclined to study pure/applied math, not engineering or cs or anything else because there are far greater books for those disciplines and more interesting theoretical books for those disciplines. I find myself drawn back to this book and similar because I come from a background distant from math and now I wish to learn more about it and see if I want to pursue it fully and for that, as a hopeful phD seeker, I find it still daunting but also wonderful
February 28, 2022
Manuale stupendo, di una accessibilità sconvolgente.
Riesce a spiegare in un modo chiaro, ma non banale i pilastri della matematica moderna.
Ovviamente molte parti non si può che scorrerle e guardarle come fossero opere d'arte, come la sezione sulla topologia avanzata o sulla geometria non euclidea.
Ciò nonostante si riesce ad apprezzare la bellezza anche di queste parti più oscure.
La sezione più accattivante è quella sulla storia della matematica e sul sistema dei numeri, ma anche il calcolo ed i massimi-minimi sono scritte in modo più chiaro di molti libri di testo delle superiori.
Riesce a spiegare in un modo chiaro, ma non banale i pilastri della matematica moderna.
Ovviamente molte parti non si può che scorrerle e guardarle come fossero opere d'arte, come la sezione sulla topologia avanzata o sulla geometria non euclidea.
Ciò nonostante si riesce ad apprezzare la bellezza anche di queste parti più oscure.
La sezione più accattivante è quella sulla storia della matematica e sul sistema dei numeri, ma anche il calcolo ed i massimi-minimi sono scritte in modo più chiaro di molti libri di testo delle superiori.
August 16, 2020
+A clear explanation of Basic Math. it's sweet as a Novel, with enough Example(not less not more).
I think it's good for Bachelor's students in Engineering or computer science.
- while it's a good book, there are two negative points: 1. some subfields such as "Combination, Graph theory, Numerical Analysis, Optimizing, linear algebra, ..." are dismissed. 2.this is a little old-fashion and maybe today's teenagers avoid reading it.
I think it's good for Bachelor's students in Engineering or computer science.
- while it's a good book, there are two negative points: 1. some subfields such as "Combination, Graph theory, Numerical Analysis, Optimizing, linear algebra, ..." are dismissed. 2.this is a little old-fashion and maybe today's teenagers avoid reading it.
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