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The Mathematical Century: The 30 Greatest Problems of the Last 100 Years
The twentieth century was a time of unprecedented development in mathematics, as well as in all more theorems were proved and results found in a hundred years than in all of previous history. In The Mathematical Century , Piergiorgio Odifreddi distills this unwieldy mass of knowledge into a fascinating and authoritative overview of the subject. He concentrates on thirty highlights of pure and applied mathematics. Each tells the story of an exciting problem, from its historical origins to its modern solution, in lively prose free of technical details.
Odifreddi opens by discussing the four main philosophical foundations of mathematics of the nineteenth century and ends by describing the four most important open mathematical problems of the twenty-first century. In presenting the thirty problems at the heart of the book he devotes equal attention to pure and applied mathematics, with applications ranging from physics and computer science to biology and economics. Special attention is dedicated to the famous "23 problems" outlined by David Hilbert in his address to the International Congress of Mathematicians in 1900 as a research program for the new century, and to the work of the winners of the Fields Medal, the equivalent of a Nobel prize in mathematics.
This eminently readable book will be treasured not only by students and their teachers but also by all those who seek to make sense of the elusive macrocosm of twentieth-century mathematics.
Odifreddi opens by discussing the four main philosophical foundations of mathematics of the nineteenth century and ends by describing the four most important open mathematical problems of the twenty-first century. In presenting the thirty problems at the heart of the book he devotes equal attention to pure and applied mathematics, with applications ranging from physics and computer science to biology and economics. Special attention is dedicated to the famous "23 problems" outlined by David Hilbert in his address to the International Congress of Mathematicians in 1900 as a research program for the new century, and to the work of the winners of the Fields Medal, the equivalent of a Nobel prize in mathematics.
This eminently readable book will be treasured not only by students and their teachers but also by all those who seek to make sense of the elusive macrocosm of twentieth-century mathematics.
224 pages, Paperback
First published January 1, 2004
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About the author
Piergiorgio Odifreddi
123 books136 followersPiergiorgio Odifreddi is an Italian mathematician, logician and aficionado of the history of science, who is also extremely active as a popular science writer and essayist, especially in a perspective of philosophical atheism as a member of the Italian Union of Rationalist Atheists and Agnostics.
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Displaying 1 - 6 of 6 reviews
June 14, 2016
The informative value of the book is top notch. It delivers what it promises: the most important developments of 20th century mathematics in a nutshell, with neat historical introductions which explain related and very interesting problems. I will have to read it once or twice more to absorb everything. My only concern is with its expected audience: the author is not consistent with the level of expertise he requires from the reader. For example, he wastes one page explaining the commutativity and associativity properties of addition and multiplication, while in most parts he assumes a working knowledge of way more obscure concepts from analysis and differential geometry. Mathematicians which are not up to date with fundamental discoveries will make the most of this book. People wondering what serious math is about can get a lot of knowledge from it, but they should assume they are not going to fully understand some of its entries.
July 26, 2021
The Mathematical Century – The 30 Greatest Problems of the Last 100 Years is a book only for experts. I have a mathematical physics degree, I know my groups from my fields, I have spent a bit of time with tensors and delta distributions and assorted other mathematical flotsam and jetsam. But despite what I would think was a decent enough backing to at least get something out of a look back on the most interesting mathematical problems of the last century, I got virtually nothing out of this book.
I was struggling to know how strongly to criticise this book – am I simply not its audience; was I a fool for buying it, analogous to picking up an advanced textbook and expecting to understand it all?
Or at least, I was wondering deliberating over how much to soften my stance until I had a look at the blurb to remind myself why I might have bought this book in the first place (I picked it up many moons ago back in my idealistic days of thinking I’d ever teach myself some sexy pure mathematics like topology). To quote: “This eminently readable book will be treasured not only by students and their teachers but also by all those who seek to make sense of the elusive macrocosm of twentieth-century mathematics”.
I, presumably, qualify am one of the latter. Yet this book is so incredibly dense that I would be amazed at anyone below the level of a mathematics PhD really being able to get to grips with it. The scope is huge, covering seemingly all the developments in 20th century mathematics. From my recollection of mathematics courses at university, learning about the Cauchy-Reimann equations or basics of differential geometry, I cannot fathom how my mathematical undergraduate peers would be able to digest most of what is in this book.
A great example is the ultra-hard 4 pages on Milnor’s work in differential topology. It begins promisingly enough with a nice real-world motivation: “The fact that for a long time the Earth was believed to be flat …”. By paragraph 3 we’ve encountered the notion of differentiable structure on Riemann manifolds – so far so already quite difficult to follow. Two paragraphs later we are being informed about the 7-dimensional sphere and the existence of 28 differentiable structures on it – what are they, what does this mean? Via an extremely abstruse discussion of matrices that define topological invariants of “simply connected 4-dimensional manifolds” we arrive at the extraordinarily unhelpful sentence:
“Clifford Taubes and Robert Gompf then showed, in 1985, that the number of exotic structures on 4-dimensional Euclidean space is not only infinite but has the cardinality of the continuum”
This is all rounded off with a detour to very advanced undergraduate-level particle physics concepts, finishing with a doubling back to differentiable structure on 4-dimensional spheres.
My irritation with all this is that the book could have been so much more. Perhaps by taking more time, or opting for fewer examples, Piergiorgio Odifreddi could have provided a better intuition for what makes these problems interesting, what the results mean, how the proofs proceeded. There are glimmers of this every now and then; the authors writing isn’t bad per se. Rather, he has simply attempted to cover too much ground in far too little space and with too little care. This would be reasonable in a book advertised as a primer for experts on 20th century mathematical advances. But given its billing, I feel I can rightly criticise it as singularly failing to help me make “sense of the elusive macrocosm of 20th century mathematics”.
I was struggling to know how strongly to criticise this book – am I simply not its audience; was I a fool for buying it, analogous to picking up an advanced textbook and expecting to understand it all?
Or at least, I was wondering deliberating over how much to soften my stance until I had a look at the blurb to remind myself why I might have bought this book in the first place (I picked it up many moons ago back in my idealistic days of thinking I’d ever teach myself some sexy pure mathematics like topology). To quote: “This eminently readable book will be treasured not only by students and their teachers but also by all those who seek to make sense of the elusive macrocosm of twentieth-century mathematics”.
I, presumably, qualify am one of the latter. Yet this book is so incredibly dense that I would be amazed at anyone below the level of a mathematics PhD really being able to get to grips with it. The scope is huge, covering seemingly all the developments in 20th century mathematics. From my recollection of mathematics courses at university, learning about the Cauchy-Reimann equations or basics of differential geometry, I cannot fathom how my mathematical undergraduate peers would be able to digest most of what is in this book.
A great example is the ultra-hard 4 pages on Milnor’s work in differential topology. It begins promisingly enough with a nice real-world motivation: “The fact that for a long time the Earth was believed to be flat …”. By paragraph 3 we’ve encountered the notion of differentiable structure on Riemann manifolds – so far so already quite difficult to follow. Two paragraphs later we are being informed about the 7-dimensional sphere and the existence of 28 differentiable structures on it – what are they, what does this mean? Via an extremely abstruse discussion of matrices that define topological invariants of “simply connected 4-dimensional manifolds” we arrive at the extraordinarily unhelpful sentence:
“Clifford Taubes and Robert Gompf then showed, in 1985, that the number of exotic structures on 4-dimensional Euclidean space is not only infinite but has the cardinality of the continuum”
This is all rounded off with a detour to very advanced undergraduate-level particle physics concepts, finishing with a doubling back to differentiable structure on 4-dimensional spheres.
My irritation with all this is that the book could have been so much more. Perhaps by taking more time, or opting for fewer examples, Piergiorgio Odifreddi could have provided a better intuition for what makes these problems interesting, what the results mean, how the proofs proceeded. There are glimmers of this every now and then; the authors writing isn’t bad per se. Rather, he has simply attempted to cover too much ground in far too little space and with too little care. This would be reasonable in a book advertised as a primer for experts on 20th century mathematical advances. But given its billing, I feel I can rightly criticise it as singularly failing to help me make “sense of the elusive macrocosm of 20th century mathematics”.
December 27, 2012
An accessible gateway to the headline problems in 20th-century mathematics. Don't expect an in-depth treatment of the problems, or even a comprehensive overview. Instead, Odifreddi effectively weaves related historical and mathematical details into a general presentation of each problem. While at times this approach can upstage the problem under focus, the context is well-suited to such a text.
June 14, 2011
Some of the discussion is cursory or superficial but scope is impressive.
November 9, 2025
If you are not already familiar with most of the concepts covered in this book, you will not understand them by reading this book. It is terse; nothing is explained for the novice; there are mistakes; and the language is awkward from being translated poorly (needs an English proofreader/editor).
Foreword by pompous jackass Freeman Dyson who says he would have written the book differently. Who cares, Freeman?
Foreword by pompous jackass Freeman Dyson who says he would have written the book differently. Who cares, Freeman?
December 17, 2014
Love learning about evolution of certain subject be it a tool or scientific discipline.
3 stars only because of me being not enough versed in math to appreciate it fully.
3 stars only because of me being not enough versed in math to appreciate it fully.
Displaying 1 - 6 of 6 reviews