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"Odifreddi clearly and concisely describes important 20th-century developments in pure and applied mathematics. . . . Unlike similar volumes, this book keeps descriptions general and contains a short section on the philosophical foundations of mathematics to help non-mathematicians easily navigate the material."—Library Journal
"This is an astonishingly readable, succinct, and wonderful account of twentieth-century mathematics! It is a great book for mathematics majors, students in liberal-arts courses in mathematics, and the general public. I am amazed at how easily the author has set out the achievements in a broad array of mathematical fields. The writing appears effortless."—Paul Campbell, Mathematics Magazine
"Piergiogio Odifreddi's book successfully portrays the major developments in 20th century mathematics by an examination of the mathematical problems that have gained prominence during the past 100 years. . . . [T]he literary style is such that the contents are made accessible to a very wide readership, but with no hint of oversimplification."—P.N. Ruane, MathDL
"Odifreddi . . . has an engaging and effective style and a knack for compact but comprehensible summaries, making his presentation seem effortless. The Mathematical Century can be dabbled in, read through, or perhaps even used as a quick reference."—Danny Yee, Danny Reviews
Modern science has nonetheless undermined the naive vision of the external world. Scientific research has extended its reach to the vastness of the cosmos as well as to the infinitesimally small domain of the particles, making a direct sensorial perception of galaxies and atoms impossible-or possible only indirectly, through technological means-and thus reducing them in effect to mathematical representations. Likewise, modern mathematics has also extended its domain of inquiry to the rarefied abstractions of structures and the meticulous analysis of the foundations, freeingitself completely from any possible visualization.
Twentieth-century science and mathematics thus share a common difficulty to explain their achievements in terms of classical concepts. But these difficulties can be overcome: often it is only the superficial and futile abstractions that are difficult to justify, while the profound and fruitful ones are rooted in concrete problems and intuitions. In other words, a good abstraction is never an end in itself, an art-for-art's-sake conception, but it is always a necessity, an art-for-humans creation.
A second difficulty in any attempt to survey twentieth-century science and mathematics is the production explosion. Mathematicians, once a small group that often had to earn their living by means other than their trade, are today legion. They survive by producing research that too often has neither interest nor justification, and the university circles in which the majority of mathematicians work unwisely encourage them to "publish or perish," according to an unfortunate American motto. As a result of all this, there are now hundreds of specialized journals in which year after year hundreds of thousands of theorems are published, the majority of them irrelevant.
A third kind of difficulty is due to the fragmentation of mathematics that began in the 1700s, and which became pathological in the 1900s. The production explosion is one of its causes, but certainly not the only one. Another, perhaps even more significant cause is the very progress of mathematical knowledge. The problems that are simple and easy to solve are few, and once they have been solved a discipline can only grow by tackling complex and difficult problems, requiring the development of specific techniques, and hence of specialization. This is indeed what happened in the twentieth century, which has witnessed a hyperspecialization of mathematics that resulted in a division of the field into subfields of ever narrower and strictly delimited borders.
The majority of these subfields are no more than dry and atrophied twigs, of limited development in both time and space, and which die a natural death. But the branches that are healthy and thriving are still numerous, and their growth has produced a unique situation in the history of mathematics: the extinction of a species of universal mathematicians, that is, of those individuals of an exceptional culture who could thoroughly dominate the entire landscape of the mathematics of their time. The last specimen of such a species appears to have been John von Neumann, who died in 1957.
For all these reasons, it is neither physically possible nor intellectually desirable to provide a complete account of the activities of a discipline that has clearly adopted the typical features of the prevailing industrial society, in which the overproduction of low-quality goods at low cost often takes place by inertia, according to mechanisms that pollute and saturate, and which are harmful for the environment and the consumer.
The main problem with any exposition of twentieth-century mathematics is, therefore, as in the parable, to separate the wheat from the chaff, burning up the latter and storing the former away in the barn. The criteria that might guide us in a selection of results are numerous and not at all unambiguous: the historical interest of the problem, the seminal or final nature of a result, the intrinsic beauty of the proposition or of the techniques employed, the novelty or the difficulty of the proof, the mathematical consequences or the practical usefulness of the applications, the potential philosophical implications, and so on.
The choice we propose to the reader can only be a subjective one, with both its positive and its negative aspects. On the one hand, this choice must be made within the bounds of a personal knowledge that is inevitably restricted from a general point of view. And on the other hand, the choice must result from a selection dictated by the author's particular preferences and taste.
The subjective aspects of our choice can nevertheless be minimized by trying to conform to criteria that are in some sense "objective." In the present case, our task has been facilitated by two complementary factors that have marked the development of mathematics throughout the century. Both are related, as we shall see, to the International Congresses of Mathematicians. As in the case of the Olympic Games, these meetings take place every four years, and those invited to present their work are the ones whom the mathematical community considers to be its most distinguished representatives.
The first official congress took place in Zurich in 1897, and the opening address was given by Henri Poincaré, who devoted it to the connections between mathematics and physics. Paris hosted the second congress in 1900, and this time David Hilbert was chosen to open the meeting. The numerological factor prevailed over his desire to reply, three years later in time, to Poincaré's speech, and Hilbert chose rather to "indicate probable directions for mathematics in the new century."
In his inspired address, he gave, first of all, certain implicit clues that shall guide our choice of topics: the important results are those that exhibit a historical continuity with the past, bring together different aspects of mathematics, throw a new light on old knowledge, introduce profound simplifications, are not artificially complicated, admit meaningful examples, or are so well understood that they can be explained to the person in the street.
But Hilbert's address became famous above all for his explicit list of twenty-three open problems that he considered crucial for the development of mathematics in the new century. As if to confirm his lucid foresight, many of those problems really turned out to be fruitful and stimulating, especially during the first half of the century-and we shall examine some of these in detail. In the second half of the century, the thrust from Hilbert's problems petered out, and mathematics often followed paths that did not even exist at the beginning of the century.
To guide us during this period it is useful to turn our attention to a prize created in 1936 and awarded at the International Congress to mathematicians under age forty who have obtained the most important results in the past few years. The age limit is not particularly restrictive, given that most significant results are in fact obtained during a mathematician's youth. As Godfrey Hardy put it in A Mathematician's Apology: "No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game."
The prize was established in memory of John Charles Fields, the mathematician who came up with the idea and obtained the necessary funds. It consists of a medal bearing an engraving of Archimedes' head and the inscription Transire suum pectus mundoque potiri, "to transcend human limitations and to master the universe" (fig. I.1). For this reason the prize is nowadays known as the Fields Medal.
This award is considered the equivalent of the Nobel Prize in mathematics, which does not exist. What does exist is a story, widely circulated in mathematical circles, according to which the absence of a Nobel Prize in mathematics would have been due to Alfred Nobel's intention to prevent the Swedish mathematician Gösta Mittag-Leffler from obtaining it. In fact, the two men hardly knew each other, and the latter was certainly not the lover of the former's wife, as the story goes, since Nobel was not married. The real reason is simply that the five original prizes (physics, chemistry, medicine, literature, and peace) were dedicated to the disciplines in which Nobel had had a lifelong interest, and mathematics was not one of them.
In the twentieth century, forty-two Fields medals were awarded, two in 1936 and the rest between 1950 and 1998. Since the winners include some of the best mathematicians of the second half of the century, and the results for which the medals were granted are among the top mathematical achievements of the time, we shall often come back to the subject.
A complement to the Fields Medal is the Wolf Prize, a kind of Oscar for life achievement in a field established in 1978 by Ricardo Wolf, a Cuban philanthropist of German origin who was ambassador to Israel from 1961 to 1973. As is the case for the Nobel Prize, the Wolf Prize has no age restriction, is awarded in various fields (physics, chemistry, medicine, agriculture, mathematics, and art), is presented by the head of state in the awarding country's capital (the king of Sweden in Stockholm in one case, and the president of Israel in Jerusalem in the other) and involves a substantial sum of money ($100,000, compared to $10,000 for a Fields Medal, and $1 million for a Nobel Prize).
To prevent any misunderstanding, I wish to emphasize that the solutions to Hilbert's problems, and the results for which the Fields Medal or the Wolf Prize were awarded, are only significant landmarks and do not exhaust the landscape of twentieth-century mathematics. It will thus be necessary to go beyond them in order to give as complete an account as possible, within the limits previously established, of the variety and depth of contemporary mathematics.
The decision to focus on the great results which, furthermore, constitute the essence of mathematics, determines the asynchronous character of the book's exposition, which will inevitably take the form of a collage. This approach has the advantage of allowing a largely independent reading of the various sections, and the disadvantage of resulting in a loss of unity. This inconvenience could be removed on a second reading, which would allow the reader, having already an overall view of the whole, to revisit the various parts.
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|Ch. 1||The Foundations||8|
|1.1||The 1920s: Sets||10|
|1.2||The 1940s: Structures||14|
|1.3||The 1960s: Categories||17|
|1.4||The 1980s: Functions||21|
|Ch. 2||Pure Mathematics||25|
|2.1||Mathematical Analysis: Lebesgue Measure (1902)||29|
|2.2||Algebra: Steinitz Classification of Fields (1910)||33|
|2.3||Topology: Brouwer's Fixed-Point Theorem (1910)||37|
|2.4||Number Theory: Gelfand Transcendental Numbers (1929)||39|
|2.5||Logic: Godel's Incompleteness Theorem (1931)||43|
|2.6||The Calculus of Variations: Douglas's Minimal Surfaces (1931)||47|
|2.7||Mathematical Analysis: Schwartz's Theory of Distributions (1945)||52|
|2.8||Differential Topology: Milnor's Exotic Structures (1956)||56|
|2.9||Model Theory: Robinson's Hyperreal Numbers (1961)||59|
|2.10||Set Theory: Cohen's Independence Theorem (1963)||63|
|2.11||Singularity Theory: Thom's Classification of Catastrophes (1964)||66|
|2.12||Algebra: Gorenstein's Classification of Finite Groups (1972)||71|
|2.13||Topology: Thurston's Classification of 3-Dimensional Surfaces (1982)||78|
|2.14||Number Theory: Wiles's Proof of Fermat's Last Theorem (1995)||82|
|2.15||Discrete Geometry: Hales's Solution of Kepler's Problem (1998)||87|
|Ch. 3||Applied Mathematics||92|
|3.1||Crystallography: Bieberbach's Symmetry Groups (1910)||98|
|3.2||Tensor Calculus: Einstein's General Theory of Relativity (1915)||104|
|3.3||Game Theory: Von Neumann's Minimas Theorem (1928)||108|
|3.4||Functional Analysis: Von Neumann's Axiomatization of Quantum Mechanics (1932)||112|
|3.5||Probability Theory: Kolmogorov's Axiomatization (1933)||116|
|3.6||Optimization Theory: Dantzig's Simplex Method (1947)||120|
|3.7||General Equilibrium Theory: The Arrow-Debreu Existence Theorem (1954)||122|
|3.8||The Theory of Formal Languages: Chomsky's Classification (1957)||125|
|3.9||Dynamical Systems Theory: The KAM Theorem (1962)||128|
|3.10||Knot Theory: Jones Invariants (1984)||132|
|Ch. 4||Mathematics and the Computer||139|
|4.1||The Theory of Algorithms: Turing's Characterization (1936)||145|
|4.2||Artificial Intelligence: Shannon's Analysis of the Game of Chess (1950)||148|
|4.3||Chaos Theory: Lorenz's Strange Attractor (1963)||151|
|4.4||Computer-Assisted Proofs: The Four-Color Theorem of Appel and Haken (1976)||154|
|4.5||Fractals: The Mandelbrot Set (1980)||159|
|Ch. 5||Open Problems||165|
|5.1||Arithmetic: The Perfect Numbers Problem (300 B.C.)||166|
|5.2||Complex Analysis: The Riemann Hypothesis (1859)||168|
|5.3||Algebraic Topology: The Poincare Conjecture (1904)||172|
|5.4||Complexity Theory: The P = NP Problem (1972)||176|
|References and Further Reading||187|
"The Mathematical Century is both popular and scholarly. Piergiorgio Odifreddi clearly and accurately covers many important mathematical problems and the contributions that leading mathematicians have made to their solutions. Offering a personal but very balanced perspective, his book is one that amateur and professional alike can learn from."—Sir Michael Atiyah, Fields Medalist 1966, and former President of the Royal Society
"Piergiorgio Odifreddi has done a superb job, telling the story of twentieth-century mathematics in one short and readable volume."—Freeman Dyson, Institute for Advanced Study, Princeton
"Odifreddi's book successfully portrays the major developments in 20th century mathematics by an examination of the mathematical problems that have gained prominence during the past 100 years. . . . [It] comes very near to being that intangible entity—a history of modern mathematics. Moreover, the literary style is such that the contents are made accessible to a very wide readership, but with no hint of oversimplification."—P.N. Ruane, MAA Online