Timothy Y. Chow
MIT, USA

MATHEMATICS, in the minds of most people, consists mostly of algorithms for performing arithmetical computations and solving algebraic equations, although it is also understood to include two-dimensional and three-dimensional Euclidean geometry, including trigonometry.
Natural scientists, engineers, and social scientists have a more sophisticated view of mathematics, because their work makes heavy use of certain branches of higher mathematics, notably the differential and integral calculus; many scientific laws are most accurately expressed as differential equations.
However, scientists and engineers are primarily concerned only with applied mathematics, and usually emphasize calculational techniques for solving equations that arise in their own discipline.
Professional mathematicians have a broader view of mathematics, that includes not only applied mathematics but also pure mathematics, which in principle encompasses anything that can be studied in an exact, quantitative manner using rigorous logical reasoning.
Well-established branches of pure mathematics include arithmetic, algebra, analysis, geometry, logic and set theory, probability, and discrete mathematics.
An attempt is made below to define these terms, although it is impossible to give definitions that are both succinct and completely precise.
For a more detailed subdivision of mathematics, the reader is referred to the Mathematics Subject Classification of the American Mathematical Society.

• Arithmetic or number theory is the study of integers or whole numbers, and among other things includes the study of prime numbers and Diophantine equations (i.e., equations whose solutions are required to be integers).
• Algebra has its origins in the study of polynomial equations but has evolved into the study of abstract objects such as groups, rings, and fields, which are equipped with binary operations that have many of the same properties as ordinary addition and multiplication, such as the associative law and the distributive law.
• Analysis is the study of continuous change, including the calculus of real numbers (real analysis), the calculus of complex numbers (complex analysis), as well as partial differential equations and the abstract study of functions and function spaces (functional analysis and operator theory).
• Geometry is the study of spatial concepts such as dimension, shape, parallelism, distance, angle, curvature, and volume; topology is the branch of geometry that studies properties that are invariant under arbitrary continuous deformations (for example, a knot remains knotted even if it is stretched, twisted, or bent).
• Logic and set theory deal with the most fundamental concepts of mathematics such as axioms, valid proofs, and sets of unstructured objects.
• Probability is the quantitative study of chance, likelihood, and uncertainty, and since the mid-twentieth century has been closely related to measure theory, since the measure or size of a set provides a convenient formal representation of its probability.
• Discrete mathematics is the study of structures with discrete units having no continuous relationship to each other, and includes combinatorics or enumeration as well as the mathematics of theoretical computer science, including computational complexity theory.

Strikingly, modern mathematics exhibits great unity despite an explosive increase in mathematical knowledge and specialization. There are no sharp boundaries between the aforementioned branches of mathematics, and the solutions of important mathematical problems often use techniques from a wide variety of subfields.
Part of what enables this unity is the clarification of the foundations of mathematics that occurred in the twentieth century.
The work of Cantor, Hilbert, Frege, Russell, Whitehead, Zermelo, and others resulted in an extremely precise notion of formal proof as well as the demonstration that in principle, all known mathematics can be derived from a few set-theoretic axioms.
(Gödel later showed that any consistent, sufficiently powerful set of axioms for mathematics is incomplete, meaning that there are statements that are not settled by the axioms, but the Zermelo-Fraenkel-Choice or ZFC axiomatic system has so far been more than sufficient for all practical purposes.)
Putting all mathematics on a unified foundation allows easy translation of results from one subdomain to another, and the emphasis on absolutely precise statements and correct proofs eliminates any danger of misapplication.
Contrary to popular belief, mathematics is not a static discipline but is constantly expanding its boundaries.
Mathematics progresses both through the solution of concrete problems as well as through the development of comprehensive theories.
Unsolved problems in mathematics may arise from unsolved problems in science, or may arise within mathematics itself, such as when mathematicians pose conjectures that they suspect are true but that they cannot immediately prove.
Solution of a mathematical problem may involve a great deal of experimental mathematics, in which many special cases are calculated, possibly with the aid of a computer, and partial results and conjectures are obtained.
After a solution is found, it is typically written up in the form of theorems with rigorous proofs, and the methods and ideas developed en route may be extended and systematized into a body of theory.
The austere style in which theoretical results are written up is useful for providing a firm basis on which to make further progress, but has the effect of hiding the somewhat haphazard process by which most mathematical results are originally obtained, and may contribute to the myth that mathematics consists only of fixed, formal algorithms and facts.
Nothing generates more excitement in the mathematical world than the solution of a major, long-standing open problem.
This fascination with open problems may be illustrated by David Hilbert’s famous list of twenty-three open problems that he formulated in 1900 and that fueled much twentieth-century mathematical research, or by the list of seven Millennium Prize Problems, whose solution will earn the solver one million U.S. dollars each from the Clay Mathematics Institute.
The Fields Medal, which is the most prestigious award in mathematics, is often awarded for a spectacular proof of a major conjecture.
A remarkable number of long-standing problems in pure mathematics have been solved in recent decades, leading Keith Devlin to dub the present era a “new golden age” of mathematics.
It is impossible to give a comprehensive list of such problems, but here are four illustrative selections.

• Fermat’s Last Theorem is the statement made by Fermat in 1637 that for n > 2, there are no nonzero integer solutions to the equation xⁿ + yⁿ = zⁿ.
  Work of Gerhard Frey, Jean-Pierre Serre, and Ken Ribet showed that Fermat’s Last Theorem follows from the Shimura-Taniyama-Weil Conjecture, a.k.a. the Modularity Conjecture.
  A proof of a special case of the Modularity Conjecture (enough to imply Fermat’s Last Theorem) was found by Andrew Wiles, assisted by Richard Taylor, and published in 1995.
• The Poincaré Conjecture states that every compact simply-connected 3-manifold is homeomorphic to the 3-sphere.
  The first of the Millennium Prize Problems to fall, it was solved about a century after Poincaré posed it, in a series of preprints in 2002 and 2003 by Grigory Perelman, following a program started by Richard Hamilton.
  (When other experts scrutinized Perelman’s terse papers, some incorrect statements and incomplete arguments came to light, but they were all fixable.)
• The classification of finite simple groups states that every finite group with no nontrivial normal subgroup is either a cyclic group of prime order, an alternating group, a finite group of Lie type, or one of 26 sporadic groups. The proof, contained in numerous separate journal articles spanning thousands of pages, was announced in the early 1980’s, but contained a significant gap that was filled only in 2004 by a two-volume, 1300-page book by Aschbacher and Smith.
• The Kepler Conjecture, stated by Kepler in 1611, asserts that one cannot pack non-overlapping identical spheres in three-dimensional space more densely than in the “obvious” way.
  László Fejes Tóth reduced the problem to a finite search;
  Thomas Hales, assisted by Sam Ferguson, published a proof in a series of papers, with the final paper appearing in 2006.

It should be noted, however, that this recent dramatic progress in mathematics has come at a price.
As mathematics has grown in sophistication, verifying the correctness of published proofs has become increasingly difficult.
For example, Hales had previously criticized as incomplete a published proof of the Kepler Conjecture by Wu-Yi Hsiang, but Hales’s own intensively computer-assisted proof has defied independent verification.
A potential solution to this problem is the development of computerized proof checkers, that can formally verify the correctness of a proof provided it is encoded in a carefully specified machine-readable form.
Hales’s Flyspeck Project aims to produce a machine-checkable version of his proof of the Kepler Conjecture.
Future progress in mathematics will undoubtedly be catalyzed not only by internal developments but by advances in science.
Bioinformatics, string theory, quantum computing, and other new scientific developments have already stimulated much fruitful mathematical research and promise to continue doing so well into the twenty-first century.