The Princeton Companion to Mathematics

 

by Imre Leader, June Barrow-Green, Timothy Gowers

Publisher: Princeton University Press

Release Date: July 2010

ISBN: 9781400830398

Topic: 

• Math & Science

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Book Description

This is a one-of-a-kind reference for anyone with a serious interest in mathematics. Edited by Timothy Gowers, a recipient of the Fields Medal, it presents nearly two hundred entries, written especially for this book by some of the world's leading mathematicians, that introduce basic mathematical tools and vocabulary; trace the development of modern mathematics; explain essential terms and concepts; examine core ideas in major areas of mathematics; describe the achievements of scores of famous mathematicians; explore the impact of mathematics on other disciplines such as biology, finance, and music--and much, much more.

Unparalleled in its depth of coverage, The Princeton Companion to Mathematics surveys the most active and exciting branches of pure mathematics. Accessible in style, this is an indispensable resource for undergraduate and graduate students in mathematics as well as for researchers and scholars seeking to understand areas outside their specialties.

• Features nearly 200 entries, organized thematically and written by an international team of distinguished contributors
• Presents major ideas and branches of pure mathematics in a clear, accessible style
• Defines and explains important mathematical concepts, methods, theorems, and open problems
• Introduces the language of mathematics and the goals of mathematical research
• Covers number theory, algebra, analysis, geometry, logic, probability, and more
• Traces the history and development of modern mathematics
• Profiles more than ninety-five mathematicians who influenced those working today
• Explores the influence of mathematics on other disciplines
• Includes bibliographies, cross-references, and a comprehensive index

 

VI.11 RENÉ DESCARTES

b. La Haye (now “Descartes”), France, 1596; d. Stockholm, 1650

Algebra; geometry; analytic geometry; foundations of mathematics

In 1637 Descartes published La Géométrie as an “essay” appended to his philosophical treatise Discours de la Méthode. It remained his only mathematical publication. No single early modern text shaped the development of mathematics between 1650 and 1700 as strongly as La Géométrie. It was the founding text of analytic geometry and it paved the way for the merging of algebra and geometry that made possible the development of the integral and differential calculus about fifty years later.

Descartes was educated at the Jesuit College at La Flèche. He spent his life mostly outside France, traveling through Europe in his early twenties and living in the Netherlands from 1628 until 1649; he then left for Sweden, invited by Queen Christina to her court. From an early age his interest in mathematics was tightly linked to his primary philosophical preoccupation: the certainty of knowledge. In a letter of 1619 he sketched a method, clearly inspired by arithmetic and geometry, for solving all problems in natural philosophy. Shortly afterward, his ideas grew into a passionate conviction that he could and should develop a philosophy along these problem-solving and mathematics-inspired lines. La Géométrie grew out of the mathematical part of his philosophical program; it was not a textbook on analytic geometry. Descartes offered little in the way of general principles, explaining his ideas by means of examples.

Descartes used a classical problem, Pappus’s problem, in order to explain coordinates and equations of curves, and showed that the defining property of a curve could be written as an equation. He introduced coordinates x and y, using oblique as well as rectangular coordinate axes, which he always adjusted to the problem at hand. He also introduced the now very common usage of employing x, y,and z for unknowns and a, b, and c for indeterminate fixed quantities.

For Descartes, a geometrical problem required a geometrical answer. The equation was at best an algebraic reformulation of the problem; the answer had to be a construction of the curve or of individual points. If, as in the particular case of Pappus’s problem in four lines, the equation was quadratic, then for any fixed value of y the x-coordinate was a root of a quadratic equation. Earlier in the book Descartes had shown how such a root could be constructed (using ruler and compass). Thus, the curve could be constructed “pointwise” by choosing a series of values for y and constructing the corresponding xs and points on the curve. Pointwise construction could not provide the whole curve. Therefore in Pappus’s problem Descartes used the equation to show that the solution curves were conic sections, and explained how to determine the nature of the conic, the location of its axes, and the values of its parameters. This was an impressive result; it was, in fact, the first classification of an algebraically defined class of curves.

René Descartes

A further influential result in La Géométrie, and the one of which Descartes himself said he was most proud, was his method to determine the normal (and thus also the tangent) at a given point on a curve with a given equation. It was a pre-calculus forerunner of differentiation.

There are three important differences between how Descartes treated curves and their equations and how they are treated in modern analytic geometry: he employed oblique as well as rectangular axes; he did not consider the equation as defining a curve—rather it represented a problem, namely to construct the curve itself, as well as its axes, tangents, etc.; and he did not consider the plane itself as a collection of points characterized by pairs of real numbers—for him the xs and ys were not dimensionless numbers but the lengths of line segments. (The term “Cartesian plane” for 2 is therefore anachronistic.)

Descartes supposed (too optimistically) that his procedures could be extended to polynomial equations of any degree (usually connected to Pappus’s problem with more than four lines) and that therefore he had shown how, in principle, all geometrical construction problems could be solved. For higher-order constructions he needed new algebraic techniques. The relevant section in La Géométrie constituted the first general theory of polynomial equations and their roots. It contained his “sign rule” about the number of positive and negative roots of a polynomial, various transformation rules, and methods to check equations for reducibility. He gave no proofs; his results were based on a conviction that polynomials could essentially be written as products of linear factors x − xi, in which the roots xi could be positive, negative, or “imaginary.”

It appears, then, that analytic geometry was not the primary goal of La Géométrie. Rather, its aim was to provide a universal method for solving geometrical problems, and to do so Descartes had to answer two urgent methodological questions. The first was how to solve geometrical problems not constructible by ruler and compass, and the second was how to use algebra as an analytic, i.e., solution-finding, tool in geometry.

For the first of these, Descartes allowed successively more complicated curves as means of construction. It was Descartes’s conviction that algebra, through the equations of these curves, could guide him to choose, among all such construction curves, the most appropriate for the problem, in particular the simplest, i.e., that of lowest degree.

The second question addressed serious conceptual difficulties that were felt at the time about using algebra in geometry. The transfer of algebraic operations to geometry was indeed problematic because multiplication in geometry was generally interpreted dimensionally: for example, a product of two lengths had to represent an area, and a product of three a volume. But until then algebra had dealt mostly with numbers and had routinely used products of more than three factors. Thus, a consistent and unrestricted geometrical interpretation of the operations of algebra was needed. Descartes did indeed provide such a reinterpretation. He introduced a unit line segment in such a way that multiplication no longer raised the dimension and inhomogeneous terms could be allowed in equations.

By 1637 he had given up on his earlier attempts to link philosophy and mathematics. Yet the preoccupation with certainty remained. As his concept of construction involved the use of curves, he had to consider which curves could be understood by the human mind with sufficient clarity to be acceptable in geometry. His answer was that all algebraic curves were acceptable (he called these “geometrical curves”) and all others were not (these he called “mechanical”). Few seventeenth-century mathematicians followed Descartes in this strict demarcation of geometry. This is typical of the reception of Descartes’s La Géométrie: the philosophical and methodological aspects of the book were largely ignored by his mathematical readers, but the technical mathematical aspects were eagerly accepted and used.

Further Reading

Bos, H. J. M. 2001. Redefining Geometrical Exactness: Descartes’ Transformation of the Early Modern Concept of Construction. New York: Springer.

Cottingham, J., ed. 1992. The Cambridge Companion to Descartes. Cambridge: Cambridge University Press.

Shea, W. R. 1991. The Magic of Numbers and Motion: The Scientific Career of René Descartes.Canton, MA: Watson Publishing.

Henk J. M. Bos

VI.13 BLAISE PASCAL

b. Clermont-Ferrand, France, 1623; d. Paris, 1662

Scientist and theologian

Pascal was the first to make a systematic study of the arithmetical triangle which now bears his name; although the triangle itself is found earlier, notably in the work of the Chinese mathematician Zhu Shijie (1303). “Pascal’s triangle”

a triangular array in which each number is the sum of the two immediately above it, provides a geometrical arrangement of the binomial coefficients , with  appearing as the (k + 1)st element in the (n + 1)st row. Here  is, as usual, the number of subsets of size k in a set of size n, so that

The number  is also the coefficient of ak bn − k in the binomial expansion of (a + b)n for any integer n ≥ 0 and 0 ≤ k ≤ n. In his Traité du Triangle Arithmétique (printed in 1654 but not distributed until 1665) Pascal was the first to connect binomial coefficients with the combinatorial coefficients that arise in probability. The Traité is famous too for its explicit statement of the principle of mathematical induction.

Pascal is also known for a theorem in projective geometry (given an arbitrary hexagon inscribed in any conic section, if the three pairs of opposite sides are continued until they meet, then the three points of intersection lie on a straight line) (1640); and for a two-function (addition and subtraction) mechanical calculating machine (1645).

VI.15 GOTTFRIED WILHELM LEIBNIZ

b. Leipzig, Germany, 1646; d. Hanover, Germany, 1716

Calculus; theory of linear equations and elimination theory; logic

Renowned among mathematicians for his invention of the calculus, Leibniz was a universal thinker who graduated in law and was self-taught in mathematics. In 1676 he became counselor and librarian in Hanover for the Duke Johann Friedrich of Braunschweig–Lüneburg, holding this position until the end of his life. Besides mathematics he occupied himself with technical, historiographical, political, religious, and philosophical questions. His philosophy distinguished between two areas of reality: the world of appearances and the world of substances. It was in developing his philosophy that he was led to declare that the real world is “the best of all possible worlds.” In 1700 he was appointed first president of the newly founded Brandenburg Society of Sciences established in Berlin.

Gottfried Wilhelm Leibniz

Most of his mathematical ideas and writings were not published during his lifetime, and consequently many of his results were rediscovered many years later. About a fifth of his mathematical papers have now been published. He was always more interested in general or even universal methods than in technical details, using analogy and inductive reasoning to develop the art of invention. For the same reason he became a key creator of mathematical notation: he knew how much a suitable notation could facilitate mathematical discoveries.

One of Leibniz’s earliest mathematical works was a treatise on infinitesimal geometry (written in 1675–76 but not published until 1993). In it he used his “quanta” concept of the infinite. In Leibniz’s eyes the actual infinite as well as indivisibles, in the strictest sense of the word, were not quantities and therefore not mathematical entities: hence, he used the notions “infinitely small” and “infinitely large.” These denoted, it is true, variable quantities, but nevertheless they were quantities of a sort, so they could be handled by mathematics. Among the results in this treatise is a rigorous proof, in the style of ARCHIMEDES [VI.3], of the existence of (what is today known as) the RIEMANN INTEGRAL [I.3 §5.5] of continuous functions, which is based on intermediary values of the function within subintervals. Only a few of these results were actually published by Leibniz, and even these mainly without proof: in 1682 the alternating series for π / 4; in 1691 some further results. In 1713 he communicated his alternating series test in a private letter to JOHANN BERNOULLI [VI.18].

The year 1675 was also the year in which Leibniz invented his version of the differential and integral calculus, although its publication did not begin until 1684. His calculus was based on the key concept of a variable (quantity) ranging over a sequence of values infinitely close to each other, with the differential, the difference between two successive values in the sequence, being itself a variable that could be manipulated in the usual manner. Differentiation was represented by the operator “d”, which assigned variables to variables. For example, if x is a line of variable length, then dx is a very short line, also of variable length. Integration meant summation. His notation (d and ∫) is still used today. He deduced the standard differentiation rules (the chain rule, the product rule, etc.) and successfully applied his calculus to the differentiation of families of curves, to differentiation under the integral sign, and to various types of differential equations.

Leibniz considered “combinatorial art” as a general qualitative science, which did not coincide with modern combinatorial analysis but included combinatorics and algebra: Leibniz considered it as “the inventive part of logic.” Here he found the Girard formula for the representation of sums of powers of roots of equations by means of elementary symmetric functions, and the so-called Waring formulas by which polynomial symmetric functions are reduced to power sums (these were rediscovered by WARING[VI.21] in 1762). He invented double and multiple indices in order to solve systems of linear equations and problems of elimination theory. Between 1678 and 1713 he laid the foundations for the theory of DETERMINANTS [III.15]. The method now known as Cramer’s rule, for solving simultaneous equations, which in modern terms is based on determinants, and which Cramer published in 1750, was in fact found in 1684 by Leibniz (but again not published by him). He also stated (without proof) several theorems in the theory of linear equations and elimination theory now attributed to EULER [VI.19], LAPLACE [VI.23], and SYLVESTER [VI.42].

Among Leibniz’s other mathematical interests was additive number theory. In 1673 he found a recursion formula for the number of tripartitions of a natural number (published in 1976) and discovered further rules of recurrence now attributed to Euler. He also developed a formalism for a positional calculus (calculus situs) in order to express positions in space: if the definitions of figures are completely expressed by this calculus, all of their properties can then be found by this calculus. This is closely linked with the modern notions of geometry and topology.

Leibniz was one of the pioneers of actuarial theory. Using mathematical models of human life he calculated the purchase price of life annuities both for single persons and for groups of men, and he applied such considerations to the liquidation of a state’s indebtedness.

From the very beginning of his scientific career Leibniz was deeply interested in logic. He conceived of a general science: that is, of an art of inventing and of judging all sciences by means of sufficient data and a suitable universal language or writing. Yet, his “characteristica universalis” and the ensuing logical calculi remained fragmentary projects. His “calculus ratiocinator” was meant to be a formalized deduction of truth. Given that Leibniz was interested in formalizing calculations, it is not surprising that he also constructed the first four-function calculating machine. In constructing this machine he invented a new technical device, which he developed in two different versions: the so-called pinwheel (before 1676) and the stepped drum (from 1693 or earlier).

Further Reading

Leibniz, G. W. 1990-. Sämtliche Schriften und Briefe, Reihe 7 Mathematische Schriften, four volumes (so far). Berlin: Akademie.

Eberhard Knobloch

VI.19 LEONHARD EULER

b. Basel, Switzerland, 1707; d. Saint Petersburg, Russia, 1783

Analysis; series; rational mechanics; number theory;

music theory; mathematical astronomy;

calculus of variations; differential equations

Euler was one of the most influential and prolific mathematicians in history. His first publication was a 1726 paper on mechanics, and his last was a collection published in 1862, seventy-nine years after his death. There are over eight hundred papers bearing his name, about three hundred of them appearing posthumously, and more than twenty books. His Opera Omnia fill over eighty volumes.

In number theory, Euler introduced the Euler phi function,  (n), to denote the number of positive integers less than n and relatively prime to n, and proved the FERMAT-EULER THEOREM [III.58] that ndivides a(n) - 1. He showed that the remainders relatively prime to n form what we now call a group under multiplication and he expanded the theory of quadratic and higher-order residues. He proved FERMAT’S LAST THEOREM [V.10] for n = 3. He stated that any real polynomial of degree n is a product of real and quadratic factors and has n complex roots, but was unable to give complete proofs. He was the first to use GENERATING FUNCTIONS [IV.18 §§2.4, 3] when he gave a generating function for Naudé’s partition problem: the question of how many different ways a given integer can be written as a sum of positive integers. He introduced the function σ(n), the sum of the divisors of an integer n, and used this function to increase the number of known pairs of amicable numbers (a pair m, n of numbers is called amicable if the sum of the proper divisors of m equals n, and vice versa) from 3 to over 100. He showed that any prime number of the form 4n + 1 is the sum of two rational squares. LAGRANGE [VI.22] later improved this result to show that such numbers are the sum of two integer squares. Euler factored the fifth Fermat number, F5 = 225 + 1, thus refuting FERMAT’S [VI.12] conjecture that all integers of the form Fn = 22n + 1 were prime. He made extensive studies of the binary quadratic forms x2 + y2, x2 + ny2, and mx2 + ny2, and proved a form of the LAW OF QUADRATIC RECIPROCITY [V.28].

Leonhard Euler

Euler was the first to use analytic methods in number theory. In the 1730s he calculated to several decimal places the so-called Euler-Mascheroni constant

and discovered many of its properties. Mascheroni added to those properties in the 1790s. Euler also discovered the sum-product formula for what we now call the Riemann zeta function,

and he evaluated the function for positive even values of s.

In analysis, Euler was largely responsible for shaping the modern calculus curriculum. He was also the first person to take a systematic approach to the solution of differential equations and to problems of the CALCULUS OF VARIATIONS [III.94]. He discovered a differential equation sometimes called the “Euler necessary condition and sometimes called the “Euler-Lagrange equation.” The equation tells us that if Jis defined by the integral equation J =  f(x, y, y′) dx, then a function y(x) that maximizes or minimizes J will satisfy the differential equation

Euler apparently thought that the condition was also sufficient. Very early in his career, he pioneered the use of integrating factors for solving differential equations, though the almost simultaneous published solution of Clairaut was more complete and more widely read, so credit for this innovation usually falls to Clairaut. He also did the first work using what are now called FOURIER SERIES [III.27] and LAPLACE TRANSFORMS [III.91], more than a generation before LAPLACE [VI.23] or FOURIER [VI.25] began doing mathematics, though they took the fields much farther than Euler had.

Much of Euler’s best work involved series. His first widely acclaimed result was when he solved one of the best-known problems of his age, the seventy-year-old “Basel problem.” The problem was to evaluate the sum of the reciprocals of the square integers, or ζ(2). Euler showed that

(For a sketch of a proof, see π [III.70].)

He developed the Euler-Maclaurin series to strengthen the relationships between series and integrals. The existence of the Euler-Mascheroni constant followed from these researches. Using techniques he called “interpolation of series,” he developed the GAMMA FUNCTION [III.31] and the beta function. He developed the first extensive theory of CONTINUED FRACTIONS [III.22], and derived series for the accurate and efficient calculation of LOGARITHMS [III.25 §4] and trigonometric tables, often to more than twenty decimal places.

He was the first to do calculus with complex numbers and to investigate logarithms of negative and complex numbers. This research led to a long and bitter controversy with D’ALEMBERT [VI.20].

Euler was not the first to prove that eiθ = cos θ + i sin θ or to know that eπi = −1, but he made so much more use of these facts than any of his predecessors that this last formula is generally known as Euler’s identity.

He is regarded as a pioneer in topology and graph theory for his necessary condition for a graph to have an Euler path, the so-called Königsburg bridge problem. This is to determine whether or not a graph has a path that traverses every edge exactly once. He also discovered and gave a flawed proof that, for a polyhedron “bounded by planes,” Euler’s words for what we now call “convex,” V − E + F = 2, where Vis the number of vertices, E is the number of edges, and F is the number of faces. (For details about the flaws in Euler’s proof, see Richeson and Francese (2006).)

Euler proved a form of the general addition theorem for elliptic integrals and gave a complete classification of elastic curves. At the command of his king, Frederick the Great of Prussia, he studied hydraulics, designed pumps and fountains, and evaluated the probabilities and combinatorics involved in the state lotteries.

In a triangle, the line on which the orthocenter, the centroid, and the circumcenter lie is the Euler line. The Euler method is an algorithm for giving numerical solutions to differential equations. The EULER DIFFERENTIAL EQUATION [III.23] is the partial differential equation that describes continuity of fluid flow.

Euler tried to use lunar and planetary theory to solve the problem of finding longitude at sea. In studying the orbit of a comet, he made the first steps in the statistics of observed data.

He left Switzerland in 1727 to work in the new academy of Peter the Great in Saint Petersburg. In 1741 he moved to Berlin and the academy of Frederick the Great, but returned to Saint Petersburg in 1766, after the ascension of Catherine the Great. He was blind for the last fifteen years of his life, during which time he nevertheless wrote over three hundred papers. He won the annual prize competition of the Paris Academy twelve times.

His series of calculus books, published in four volumes between 1755 and 1770, were the first successful calculus textbooks. It was the climax of a complete series of mathematical textbooks, including arithmetic (1738), algebra (1770), and the Introductio in Analysin Infinitorum (1748), a textbook on the mathematics Euler thought was necessary to understand calculus.

In the two volumes of the Mechanica (1736), Euler gave the first calculus-based treatment of the mechanics of point masses. He followed this with another two-volume work, Theoria Motus Corporum(1765), on the motions of solid bodies, including rotations.

Other books include Methodus Inveniendi (1744), the first unified treatment of the calculus of variations, Tentamen Novae Theoriae Musicae (1739), on the physics of music and including the first use of logarithms in the theory of pitch, three different books on celestial mechanics and lunar theory, two on the theory of shipbuilding, three on optics, and one on ballistics.

Our modern notion that FUNCTIONS [I.2 §2.2] are a fundamental object in mathematics is due to Euler.

Euler standardized the use of the symbols e, π, and i, as well as Σ for summations and Δ for finite differences.

His Letters to a German Princess in three volumes (1768–71) is regarded variously as the first work of popular science writing by a first-rate scientist and as an important work in the philosophy of science.

Laplace is reported to have advised, “Read Euler. Read Euler. He is the master of us all.” The words are probably not those of Laplace, but the misattribution does not affect the quality of the advice.

Further Reading

Bradley, R. E, and C. E. Sandifer, eds. 2007. Leonhard Euler: Life, Work and Legacy. Amsterdam: Elsevier.

Dunham, W. 1999. Euler: the Master of Us All. Washington, DC: Mathematical Association of America.

Euler, L. 1984. Elements of Algebra. New York: Springer. (Reprint of 1840 edition. London: Longman, Orme, and Co.)

———. 1988, 1990. Introduction to Analysis of the Infinite, books I and II, translated by J. Blanton. New York: Springer.

———. 2000. Foundations of Differential Calculus, translated by J. Blanton. New York: Springer.

Richeson, D., and C. Francese. 2007. The flaw in Euler’s proof of his polyhedral formula. American Mathematical Monthly 114(1):286-96.

Edward Sandifer

VI.82 SRINIVASA RAMANUJAN

b. Erode, India, 1887; d. Madras (now Chennai), India, 1920 Partitions; modular forms; mock theta functions

Ramanujan, a self-taught Indian genius, made monumental contributions to mathematics that set the stage for many of the breakthroughs in number theory in the twentieth century. He worked on analytic number theory, as well as on ELLIPTIC FUNCTIONS [V.31], hypergeometric series, and the theory of CONTINUED FRACTIONS [III.22]. Much of this work was carried out together with his friend, benefactor, and collaborator G. H. HARDY [VI.73].

Hardy and Ramanujan founded the powerful “circle method” in their remarkable paper that gave an exact formula for p(n), the number of integer partitions of n. Ramanujan independently discovered the two identities that came to be known as the Rogers–Ramanujan identities:

These have applications ranging from LIE THEORY [III.48] to statistical physics. The importance of these identities relates to the fact that the GENERATING FUNCTION [IV.18 §§2.4, 3] for p(n) is

Thus, for example, the second identity asserts that the number of partitions of n into parts all of which are 2 or 3 mod 5 is equal to the number of partitions into distinct parts, all greater than 1, in which no two parts are consecutive integers.

In his work on p(n) Ramanujan discovered and proved many divisibility properties, e.g., that 5 always divides p(5n + 4) and that 7 always divides p(7n + 6). His conjectures on these divisibility properties inspired the development of extensive methods in MODULAR FORMS [III.59], and his last conjecture was finally settled in 1969 by Oliver Atkin.

All Ramanujan’s studies involving p(n) concerned the modular form

The relevance of this is that q1/124/η (w) is the generating function for p (n). Of special interest to Ramanujan was the arithmetic function τ (n), defined by the 24th power of η (w): namely,

Ramanujan conjectured that | τ (p) | < 2ρ11/2 for every prime p. The study of this problem led to deep and extensive work on modular forms by H. Petersson, R. Rankin, and others. Eventually, the conjecture was proved by P. Deligne, who received the Fields Medal for his achievement in 1978.

The full story of Ramanujan’s life makes his achievements all the more amazing. As a child he was mathematically precocious. In high school he won prizes in mathematics. On the basis of his high school record, he won a scholarship to the Government College in Kumbakonam in 1904. At about this time, Ramanujan came into contact with the book A Synopsis of Elementary Results in Pure and Applied Mathematics by G. S. Carr. This rather eccentric book is essentially a huge collection of formulas and theorems compiled for students preparing for the celebrated Mathematical Tripos examination at Cambridge. This book fascinated Ramanujan, who became obsessed with mathematics. In college, he neglected his other subjects and gave his all to mathematics. Consequently, he failed some subjects and lost his scholarship. By 1913, Ramanujan seemed destined for obscurity—he was now a mere clerk in the Madras Port Trust. Friends encouraged him to write to English mathematicians about his mathematical discoveries. Eventually he wrote to G. H. Hardy, who was able to discern that Ramanujan was a truly extraordinary mathematician.

Hardy arranged for Ramanujan to travel to England, and between 1914 and 1918 the two of them produced the groundbreaking work described above.

In 1918, Ramanujan became ill with a sickness diagnosed as tuberculosis. He convalesced in England for a year. His health improved a little in 1919 and he was able to return to India. Unfortunately, his health worsened after his return, and he died in 1920. During this last year in India he penned the pages now known as Ramanujan’s Lost Notebook and therein laid the foundations of the theory of mock theta functions, a class of functions similar to but more general than the classical theta functions.

Further Reading

Berndt, B. 1985–98. Ramanujan’s Notebooks. New York: Springer.

Kanigel, R. 1991. The Man Who Knew Infinity. New York: Scribners.

George Andrews

VI.26 CARL FRIEDRICH GAUSS

b. Brunswick, Germany, 1777; d. Göttingen, Germany, 1855
Algebra; astronomy; complex function theory including elliptic
function theory; differential equations; differential geometry;
land surveying; number theory; potential theory; statistics

Gauss’s prodigious mathematical abilities brought him to the attention of the duke of Brunswick when he was fifteen, when the duke paid for his further education, lifting him out of near poverty. For the rest of his life Gauss felt a loyalty to the state and a strong desire to do useful work, which led him to become a professional astronomer. In 1801 he was the first person to manage to reobserve Ceres, the first asteroid to be discovered, after it had disappeared behind the Sun. Gauss produced a novel statistical analysis of the original observations, using the method of least squares, which he had invented but not published, to predict where Ceres would reappear. Gauss then assisted for many years in the analysis of the orbits of several more asteroids. He also wrote extensively on celestial mechanics and cartography, and did important work on telegraphy.

Nonetheless, it is as a pure mathematician that Gauss will always be remembered. In 1801 he published his Disquisitiones Arithmeticae, the book that created modern algebraic number theory. In it he gave the first rigorous proof of the law of QUADRATIC RECIPROCITY [V.28], going on to find seven more proofs over the years. Later he extended the theorem to higher powers, introducing the Gaussian integers for the purpose in 1831 (Gaussian integers are numbers of the form m + ni, where m and n are integers and . He did major work on differential equations, chiefly the hypergeometric equation, which is a second-order linear differential equation depending on three parameters and having two singular points, with the property that many of the familiar functions of analysis are related to its solutions. He showed that this equation played a significant role in the new theory of ELLIPTIC FUNCTIONS [V.31], but because most of this work was unpublished it had no influence on the dramatic and rapidly advancing publications of ABEL [VI.33] and JACOBI [VI.35]. This unpublished work showed that he was the first mathematician to see the need to create a theory of complex functions of a complex variable. He also gave four proofs of THE FUNDAMENTAL THEOREM OF ALGEBRA [V.13]. By the 1820s he was persuaded that physical space might not be Euclidean, but he confined his opinion to his circle of friends, most of them astronomers and sympathetic to the idea; the much more detailed accounts of BOLYAI [VI.34] and LOBACHEVSKII [VI.31] were published independently in the early 1830s. Credit for the first detailed, mathematical descriptions of a non-Euclidean space therefore rightly attaches to Bolyai and Lobachevskii (for further discussion of this, see GEOMETRY [II.2 §7]). In 1827 Gauss wrote his Disquisitiones Generales Circa Superficies Curvas, in which the concept of intrinsic (Gaussian) curvature of a surface was put forward for the first time, thus reformulating differential geometry.

Carl Friedrich Gauss

In statistics, he was one of the two or three discoverers of the NORMAL DISTRIBUTION [III.71 §5], and he was an expert in error analysis, bringing the levels of accuracy in astronomy to land surveying. In that context he invented the heliotrope, which couples a mirror to a telescope in order to transmit a precise beam of light, to improve precision measurement.

The sheer volume of Gauss’s work is overwhelming. The Werke run to twelve volumes, and there are several books, of which the Disquisitiones Arithmeticae stands out.

A truly original mathematician and scientist, Gauss was otherwise a conservative in his tastes and views. His first marriage ended after only four years with the death of his wife in 1809; he then married again. A number of Gauss’s descendants may now be found in the United States.

Gauss was the last great mathematician to be called the “Prince of mathematicians,” and he has been admired as much for his breadth as for the depth of his insights and the fertility of his ideas. His own view of mathematics and its importance is captured both in the much-quoted remark that “mathematics is the queen of the sciences and arithmetic the queen of mathematics” (which he did say) and in the apocryphal remark that “mathematics is the queen and the servant of science.”

Further Reading

Dunnington, G. W. 2003. Gauss: Titan of Science, new edition with additional material by J. J. Gray. Washington, DC: Mathematical Association of America.

Jeremy Gray

VI.83 RICHARD COURANT

b. Lublinitz, Silesia (then part of Germany, now Poland), 1888; d. New York, 1972

Mathematical physics; partial differential equations; minimal surfaces; compressible flow; shock waves

The long and eventful life of Courant was full of high achievements: in mathematical research, the applications of mathematics, as a teacher of many future mathematicians, as a writer of superb books on mathematics, and as an organizer and administrator of large institutions. The fact that Courant—an outsider in his native Germany and a refugee in the United States—could accomplish these things is a testament to his personality as well as to his scientific outlook.

Born in Lublinitz, Courant completed his high school training in Breslau, living on his own and supporting himself by tutoring. His older Breslau friends, Hellinger and Toeplitz, went on to Göttingen, then the mecca of mathematics, and in due course Courant followed them. There he was taken on as an assistant to HILBERT [VI.63], and he began a close friendship with Harald Bohr, which was later extended to Harald’s brother Niels.

Under Hilbert’s direction, Courant wrote his dissertation on the use of DIRICHLET’S PRINCIPLE [IV.12 §3.5] (on minimizing energy) for constructing conformal maps. Courant also used Dirichlet’s principle in several further mathematical studies.

During World War I Courant was drafted into the army as an officer; he fought on the western front and was seriously wounded. After returning to academic life he turned his energies to mathematics and proved some remarkable results: an isoperimetric inequality for the lowest frequency of a vibrating membrane; and the Courant max-min principle for the EIGENVALUES [I.3 §4.3] of a SELF-ADJOINT OPERATOR [III.50 §3.2], so useful in studying the distribution of eigenvalues of the operators of mathematical physics.

In 1920 Courant was named as KLEIN’s [VI.57] successor as professor in Göttingen; the appointment was pushed through by Klein and Hilbert, who saw, correctly, that he shared their vision of the relationship between mathematics and science, that he would strike a balance between research and education, and that he had the administrative energy and wisdom to push his mission to fruition.

Courant became close friends with the publisher Ferdinand Springer. One of the fruits of this relationship was the famous “Grundlehren” series of monographs, affectionately known as the “Yellow Peril.” The third volume in this series is Courant’s exposition of RIEMANN’s [VI.49] geometric view of the theory of analytic functions, combined with Hurwitz’s lectures on ELLIPTIC FUNCTIONS [V.31]. In 1924 the first volume of Courant-Hilbert on Mathematical Physics appeared; it contained, presciently, much of the mathematics needed for Schrödinger’s version of quantum mechanics. His influential calculus book appeared in 1927. His research did not languish; in 1928 he published, jointly with his students Friedrichs and Lewy, the basic paper on the difference equations of mathematical physics.

Under Courant’s leadership, Göttingen, where the lively international atmosphere had been destroyed by World War I, became once again an important center for mathematics, as well as physics: the list of visitors reads like a Who’s Who of mathematics. This was totally shattered when Hitler took over the government: Jewish professors, Courant among the first, were dismissed unceremoniously and had to flee or face annihilation. Courant and his family found refuge in New York, where he was invited to build a Graduate School of Mathematics at New York University (NYU). Without any foundation to build on, Courant succeeded in this task, with the help of his former student Friedrichs and of the American James Stoker, who shared Courant’s scientific ideals. Courant found New York a reservoir of talent, and attracted students such as Max Shiffman, and later Harold Grad, Joe Keller, Martin Kruskal, Cathleen Morawetz, Louis Nirenberg, and others, including the writer of this article.

In 1936, in a burst of creativity, Courant obtained several basic results about MINIMAL SURFACES [III.94 §3.1] using Dirichlet’s principle. In 1937 he finished the second volume of Courant–Hilbert. The immensely successful popular book he wrote jointly with Herb Robbins, What Is Mathematics?, appeared in 1940. In 1942 when federal financing for scientific research became available, Courant’s group embarked on an ambitious study of supersonic flow and shock waves.

Federal support did not stop after the war; this enabled Courant to vastly expand the scale of research and graduate instruction at NYU. The research combined, at a high intellectual level, theoretical mathematics with applications such as fluid dynamics, statistical mechanics, the theory of elasticity, meteorology, the numerical solution of partial differential equations, and other topics. Nothing like this had been attempted before at a university in the United States. The institute created by Courant, eventually named after him, is flourishing today and has served as a model for other centers around the world.

Courant hated the Nazis, but did not condemn all Germans; after the war he helped to rebuild mathematics in Germany and was instrumental in inviting talented young German mathematicians and physicists to the United States.

Courant received much help from friends of his youth, many of whom became leaders in their fields, as well as from science administrators in government and industry who admired his vision of mathematics and the gallant spirit that was demonstrated by his willingness to fight against seemingly insuperable odds.

Further Reading

Reid, C. 1976. Courant in Göttingen and New York: The Story of an Improbable Mathematician. New York: Springer.

Peter D. Lax

VI.94 ALAN TURING

b. London, 1912; d. Wilmslow, England, 1954

Logic; computing; cryptography; mathematical biology

In 1936, as a young Fellow of King’s College, Cambridge, Alan Turing made a crucial contribution to mathematical logic: he defined “computability” with what is now called the TURING MACHINES [IV.20 §1.1]. Although mathematically equivalent to a definition of effective calculability earlier given by CHURCH [VI.89], Turing’s concept was compelling because of his entirely original philosophical analysis. It won the endorsement of Church, and indeed also of GÖDEL [VI.92], whose 1931 INCOMPLETENESS THEOREM [V.15] underlay Turing’s investigation. Using his definition, Turing showed that first-order logic was undecidable, and thus dealt the final death blow to HILBERT’s [VI.63] formalist program. (See LOGIC AND MODEL THEORY [IV.23 §2] for more details.)

Computability is now fundamental in mathematics, in that it gives an exact meaning to the question of whether a method exists to solve a problem. As an illustration, HILBERT’S TENTH PROBLEM [V.20], on the general solubility of Diophantine equations, was completely resolved in 1970 by methods connected with Turing’s ideas. Turing himself pioneered extensions of his definition in mathematical logic, and applications of it in algebra. However, he was unusual as a mathematician in that he explored not only the mathematical uses of his ideas (in questions of decidability in algebra) but also the wider implications for philosophy, science, and engineering.

One factor in Turing’s breakthrough was his fascination with the problem of mind and matter. Turing’s analysis of mental states and operations has since become a point of departure for the cognitive sciences. Turing himself blazed this trail later by his advocacy of the possibility of artificial intelligence. His famous 1950 “Turing test” was part of an extensive range of research proposals in this field.

A more immediately applicable aspect of his 1936 work lay in his observation that a single “universal” machine could do the work of any Turing machine, by reading the description of that machine as a table of instructions. This is the essential principle of the modern digital computer, whose programs are themselves data structures. In 1945 Turing used this insight to plan a first electronic computer and its programming. He was preempted by VON NEUMANN [VI.91], but it can be argued that von Neumann had used Turing’s insight that computing must be primarily an application of logic. Thus, Turing laid the foundations of modern computer science.

Turing was able to bridge theory and practice because between 1938 and 1945 he was the chief scientific figure in British cryptography, with particular responsibility for decrypting German naval signals. His main contributions lay in a brilliant logical solution of the Enigma cipher, and in Bayesian information theory. The advanced electronics employed in British code breaking gave him the experience to become a pioneer of practical computing as well.

Turing had less success in postwar computer engineering, and increasingly withdrew from attempts to influence the course of computer development. Instead, at Manchester University after 1949 he concentrated on a theory of nonlinear partial differential equations applied to biological development. Like his 1936 work, this opened an entirely new field. It also illustrated his broad mathematical scope, which included important work on the RIEMANN ZETA FUNCTION [IV.2 §3]. He was busy working on biological theory and new ideas in physics at the time of his sudden death.

Turing’s short life combined the purest mathematics and the most practical applications. It was also marked by other contrasts. Although he promoted the theme of computer-based artificial intelligence, there was nothing mechanical about his thought or life. The wit and drama of the “Turing test” have made him a lasting figure in the popularization of mathematical ideas. The dramatization of his life, drawing on the extraordinary secrecy of his war work, and his subsequent persecution as a homosexual, have also attracted great public interest.

Further Reading

Hodges, A. 1983. Alan Turing: The Enigma. New York: Simon & Schuster.

Turing, A. M. 1992–2001. The Collected Works of A. M. Turing. Amsterdam: Elsevier.

Andrew Hodges

VI.91 JOHN VON NEUMANN

b. Budapest, 1903; d. Washington, District of Columbia, 1957

Axiomatic set theory;
quantum physics; measure theory; ergodic theory; operator theory; algebraic geometry;
theory of games; computer engineering; computer science

Raised as a Hungarian Jew in the Austrian Empire, Neumann János Lajos’s political outlook was strongly affected by the five-month reign of the communist Béla Kun’s regime after World War I. It formed his liberal and democratic political credo (although he did insist on retaining the title of nobility “margittai,” acquired by his father in 1913, which he later translated to the German “von”). He was a child prodigy, learning several languages and demonstrating an early enthusiasm for mathematics.

During the early 1920s von Neumann studied mathematics, physics, and chemistry in Berlin and Zürich, and was also enrolled to study mathematics in Budapest although he never attended any lectures there. He received a diploma in chemical engineering at the ETH Zürich and shortly afterward (in 1926) a doctorate in mathematics at the University of Budapest (his thesis was entitled “The axiomatic deduction of general set theory”). While engineering was considered a respectable profession for a brilliant young man with such wide-ranging interests, the theoretical challenges of mathematics and formal logic drove von Neumann to the more academic environment in Germany, where he immediately received attention from HILBERT [VI.63]. Although the sensible choice, academically speaking, would have been to stay with Hilbert at Göttingen—and he did spend six months there during 1926–27 on a Rockefeller Fellowship—he preferred the pulsating atmosphere of Berlin.

During the following years he published on the axiomatic foundations of set theory, on MEASURE THEORY[III.55], and on the mathematical foundations of quantum mechanics. He also wrote his first paper on game theory (“Zur Theorie der Gesellschaftsspiele,” published in Mathematische Annalen in 1928), proving the minimax theorem (the theorem that states that every two-person finite zero-sum game has optimal mixed strategies).

In 1927 von Neumann received his habilitation in mathematics from the Philosophical Faculty of Berlin University with a written thesis and a lecture on the foundations of set theory and mathematics, becoming one of the youngest Privatdozents in the history of the university. At this point he changed his name to the German Johann von Neumann. He gave lecture courses in Hamburg (1929–30) as well as in Berlin, but in 1933, with the Nazi seizure of power, he resigned from his appointment at Berlin. By that time he was already in Princeton, where his visiting status at the university, originally conferred in 1930, was transformed into a tenured position at the newly founded Institute for Advanced Study. He modified his name once again, this time to John von Neumann, receiving U.S. citizenship in 1937.

At Princeton he found a peaceful ivory tower. Much of his important mathematical work stems from that period in the mid 1930s: he published around six journal articles per year (a rate he maintained until his death), as well as several books. The Institute’s environment allowed him to expand his research scope, taking in, among other things, ERGODIC THEORY [V.9], Haar measure, certain spaces of operators on a HILBERT SPACE [III.37] (these spaces are now known as VON NEUMANN ALGEBRAS [IV.15 §2]), and “continuous geometry.”

Von Neumann was much too politically sensitive to ignore the European crisis that led to World War II. Having begun to investigate turbulent flow beyond the speed of sound in the mid 1930s, he was invited to the Ballistic Research Laboratory in 1937 as an expert on shock waves. Later he acted as a consultant to the Navy and the Air Force. Although he was not in the initial group of Los Alamos scientists, in 1943 he became an advisor to the Manhattan Project, where his mathematical treatment of shock waves became essential, leading to the “implosion lens,” an arrangement of explosives that started the uranium chain reaction.

In parallel with his war-related work, von Neumann pursued his interest in economics, which resulted in a collaboration with Oskar Morgenstern: their groundbreaking book The Theory of Games and Economic Behavior, partly based on his 1928 Mathematische Annalen paper, appeared in 1944.

In the 1940s von Neumann began to focus on computing as a result of two very different branches of his thinking: namely, the numerical approximation of solutions to otherwise unsolvable problems, and his proficiency in the foundations of mathematics. He had tried to enlist TURING [VI.94] as an assistant at Princeton and he was certainly aware of the importance of Turing’s seminal paper on computable numbers (1936). While Turing discussed an abstract machine in the form of a thought experiment, von Neumann also considered the problems arising from the actual construction of computers, such as those connected with the use of electronic hardware. His training as a mathematician allowed him to focus on the very essentials of computing machinery and avoid baroque designs like the Moore School’s ENIAC (Electronic Numerical Integrator And Computer). In 1945 he defined the essential components for the “Electronic Discrete Variable Computer.” His “First draft of a report on the EDVAC,” which summarized and focused ideas gathered from work on early electronic computers, provided a logical framework for the modern electronic computer, becoming a road map for computer architecture for the ensuing decades. While von Neumann probably did not consider this paper to have the same importance as his mathematical results, today it is considered the birth certificate of modern computers.

Von Neumann quickly recognized that programming computers (or “coding,” as he called it) was likely to be more demanding than building basic hardware. In essence he considered programming as a new branch of formal logic. In 1947 he coauthored (with Herman Goldstine) a three-part report, “Planning and coding of problems for an electronic computing instrument,” in which many insights on the novel and demanding art of software construction were collected together.

Von Neumann’s thinking went beyond the restrictions of calculating machines, and allowed him to venture into philosophical questions on the structure of the human brain and cellular automata and the idea of self-reproducing systems—questions that were forerunners to the disciplines now called “artificial intelligence” and “artificial life.” Consideration of these questions resulted in a series of lectures published as The Computer and the Brain (1958) and a book, Theory of Self-Reproducing Automata (1966), both of which appeared posthumously.

In 1954 von Neumann was appointed to the five-member U.S. Atomic Energy Commission and in 1956 he was awarded the Presidential Medal of Freedom by President Eisenhower.

Further Reading

Aspray, W. 1990. John von Neumann and the Origins of Modern Computing. Cambridge, MA: MIT Press.

Wolfgang Coy