*The Mathematical Century*is broader in scope than that suggests.

Each "problem" is named after a key problem, discovery, innovation or conjecture, and even dated to a specific year, but is not narrowly focused on that. "Model Theory: Robinson's Hyperreal Numbers (1961)", for example, spends three pages on the history of infinitesimals, going back to Cusano, Fermat, Cavalieri, Leibniz, Newton, Cauchy, and Weierstrass, before a page on Robinson's hyperreal numbers and a closing paragraph on Conway's later surreal numbers.

The thirty eight "problems" are divided into five sections. The Foundations covers "The 1920s: Sets", "The 1940s: Structures", "The 1960s: Categories", and "The 1980s: Functions". There are fifteen problems covered within Pure Mathematics, from "Mathematical Analysis: Lebesgue Measure (1902)" to "Discrete Geometry: Hales's Solution of Kepler's Problem (1998)". Applied Mathematics has ten, from "Crystallography: Bieberbach's Symmetry Groups (1910)" to "Knot Theory: Jones Invariants (1984)". Mathematics and the Computer gets five, from "The Theory of Algorithms: Turing's Characterization (1936)" to "Fractals: The Mandelbrot Set (1980)". And four Open Problems feature, from "Arithmetic: The Perfect Numbers Problem (3000 B.C.)" to "Complexity Theory: The P = NP Problem (1970)".

Some extra structure is provided by Hilbert's problems and Fields Medals, Wolf Prizes and other awards. The result approaches a history of mathematics, albeit a massively multi-threaded one.

Links to physics are highlighted in several places, while other aspects of background or context that get a mention range from the iconoclastic interdiction of figurative art (in the discussion of symmetry groups) to central planning (in "Optimization Theory: Dantzig's Simplex Method (1947)") and the limitations of Walrasian equilibrium economics (in "General Equilibrium Theory: The Arrow-Debreu Existence Theorem (1954)". Links between different areas of mathematics are also touched on.

There's no actual mathematics in *The Mathematical Century* and it
doesn't assume much prior knowledge. Odifreddi does cover ideas quite
rapidly, however, in some more technical passages, probably too rapidly
for readers who don't have a handle on the material already. "Topology:
Brouwer's Fixed Point Theorem (1910)", for example, concludes:

"In a different direction, the conditions for the existence of fixed points haven been generalized in various ways, and, in particular, some very useful theorems have been proved: in 1922 by Banach, for contractions defined onThe most obvious audience forcomplete metric spaces(these space possess, unlike abstract topological spaces, a notion of distance); in 1928 by Knaster and Tarski, for monotonic functions defined oncomplete partial orders(in which every ascending chain of elements has an upper bound); in 1928 by Solomon Lefschetz, for continuous functions defined oncontractable compact complexes, instead of just on simplexes; and in 1941 by Kakutani, forsemicontinuous functionswhose image sets are all convex, instead of only for continuous functions."

*The Mathematical Century*will be those who have some formal training in the discipline, but lay readers should definitely take a look at it, especially if they have an interest in the history of mathematics.

Odifreddi's selection of topics obviously has some idiosyncrasies,
but he 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.

December 2010

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