The Math Book
Page 30
Probably no branch of mathematics has experienced a more surprising growth.
Raymond Louis Wilder
American mathematician
HENRI POINCARÉ
Born in 1854, in Nancy, France, Henri Poincaré showed such early promise that he was described by a teacher as a “monster of mathematics.” He graduated in the subject from the Paris École Polytechnique and earned his doctorate from the University of Paris. In 1886, he was appointed as chair of mathematical physics and probability at the Sorbonne in Paris, where he spent the rest of his career.
In 1887, Poincaré won a prize from King Oscar II of Sweden for his partial solution of the many variables involved in determining the stable orbit of three planets around one another. A self-confessed mistake threw his calculations for the stable orbit into doubt, but in turn paved the way for the study of “chaos theory.” He died in 1912.
Key works
1892–99 Les Méthodes nouvelles de la mécanique céleste (New Methods of Celestial Mechanics)
1895 Analysis Situs (Topology)
1903 La Science et l’hypothèse (Science and Hypothesis)
See also: Euclid’s Elements • Coordinates • The Möbius strip • Minkowski space • Proving the Poincaré conjecture
IN CONTEXT
KEY FIGURE
Jacques Hadamard (1865–1963)
FIELD
Number theory
BEFORE
1798 French mathematician Adrien-Marie Legendre offers an approximate formula to determine how many prime numbers there are below or equal to a given value.
1859 Bernhard Riemann outlines a possible proof for the prime number theorem, but the necessary mathematics to complete it does not yet exist.
AFTER
1903 German mathematician Edmund Landau simplifies Hadamard’s proof of the prime number theorem.
1949 Paul Erdős in Hungary and Atle Selberg in Norway both find a proof of the theorem using only number theory.
The prime numbers—those positive whole numbers that have only two factors, themselves and 1—have long fascinated mathematicians. If the first step was to find them, and they are frequent among the small numbers, the next step was to identify a pattern to describe their distribution. More than 2,000 years before, Euclid had proved that there are infinitely many primes, but it was only at the end of the 1700s that Legendre stated his conjecture—a formula to describe the distribution of primes. This became known as the prime number theorem. In 1896, Jacques Hadamard in France and Charles-Jean de la Vallée Poussin in Belgium both proved the theorem, quite independently.
It is evident that primes decrease in frequency as numbers get larger. Of the first 20 positive whole numbers, eight are prime— 2, 3, 5, 7, 11, 13, 17, and 19. Between the numbers 1,000 and 1,020, there are only three prime numbers (1,009, 1,013, 1,019), and between 1,000,000 and 1,000,020, the only prime is 1,000,003. This seems reasonable; the higher the number, the more numbers that could be divisors exist below it.
Many notable mathematicians have puzzled over how primes are distributed. In 1859, German mathematician Bernhard Riemann worked toward a proof in his paper On the Number of Primes Less Than a Given Magnitude. He believed that complex analysis, a branch of mathematics in which ideas of function are applied to complex numbers (combinations of real numbers, such as 1, and imaginary numbers, such as ), would lead to a resolution. He was right; the study of complex analysis developed, fueling the proofs of Hadamard and Poussin.
What the theorem says
The prime number theorem is designed to calculate how many primes there are less than or equal to a real number x. It states that π(x) is approximately equal to x ÷ ln(x) as x gets larger and tends to infinity. Here π(x) denotes the prime counting function (how many primes) and is unrelated to the number pi, and ln(x) is the natural logarithm of x. To explain the theorem slightly differently, for a large number x, the average gap between primes from 1 to x is approximately ln(x). Or, for any number between 1 and x, the probability of it being a prime is approximately 1 ÷ ln(x).
The prime numbers are the building blocks for numbers in mathematics, just as the elements are for compounds in chemistry. Fundamental to understanding this is the Riemann hypothesis—an unsolved conjecture—which, if true, could reveal a huge amount more about prime numbers.
Primes tend to decrease in frequency as numbers get larger. Although there are two primes between 30 and 40, and three between 40 and 50, the accuracy of the prime number theorem increases at higher numbers.
The prime numbers… grow like weeds among the natural numbers, seeming to obey no other law than that of chance.
Don Zagier
American mathematician
JACQUES HADAMARD
Born in Versailles, France, in 1865, Jacques-Salomon Hadamard became interested in mathematics thanks to an inspiring teacher. He obtained his doctorate in Paris in 1892 and the same year won the Grand Prix des Sciences Mathématiques for his work on primes. He moved to Bordeaux to lecture at the university, and there proved the prime number theorem.
In 1894, Alfred Dreyfus, a Jewish relative of Hadamard’s wife, was falsely accused of selling state secrets and was sentenced to life in prison. Hadamard, who was also Jewish, worked tirelessly on behalf of Dreyfus and he was eventually freed. Hadamard’s brilliant career was marred by personal loss; two of his sons died in World War I, and another in World War II. The death of his grandson Étienne in 1962 was a final blow. Hadamard died a year later.
Key works
1892 Determination of the Number of Primes Less than a Given Number
1910 Lesson on the Calculus of Variations
See also: Euclid’s Elements • Mersenne primes • Imaginary and complex numbers • The Riemann hypothesis
INTRODUCTION
In 1900, as the arms race that led to World War I intensified, German mathematician David Hilbert attempted to anticipate the directions that mathematics would take in the 20th century. His list of the 23 unsolved problems he considered crucial was influential in identifying the fields of mathematics that could be fruitfully explored by mathematicians.
New century, new fields
One area of exploration was the foundations of mathematics. In seeking to establish the logical basis of mathematics, Bertrand Russell described a paradox that highlighted a contradiction in Georg Cantor’s naive set theory, leading to a reappraisal of the subject. These ideas were taken up by André Weil and others, using the pseudonym Nicolas Bourbaki. Starting from the basics, they met in the 1930s and 40s, rigorously formalizing all branches of mathematics in terms of set theory.
Others, notably Henri Poincaré, explored the newly established field of topology, the offshoot of geometry dealing with surfaces and space. His famous conjecture concerns the 2-dimensional surface of a 3-dimensional sphere. Unlike many of his peers in the 1900s, Poincaré did not confine himself to any one single field of mathematics. As well as pure mathematics, he made significant discoveries in theoretical physics, including his proposed principle of relativity. Similarly, Hermann Minkowski—whose primary interest was in geometry and the geometrical method applied to problems in number theory—explored the notion of multiple dimensions, and suggested spacetime as a possible fourth dimension. Emmy Noether, one of the first female mathematicians of the modern era to gain recognition, came to the field of theoretical physics from a perspective of abstract algebra.
The computer age
In the first half of the 1900s, applied mathematics was largely concerned with theoretical physics, especially the implications of Einstein’s theories of relativity, but the latter part of the century was increasingly dominated by advances in computer sciences. Interest in computing had begun in the 1930s, in the search for a solution to Hilbert’s Entscheidungsproblem (decision problem) and the possibility of an algorithm to determine the truth or falsity of a statement. One of the first to tackle the problem was Alan Turing, who went on to develop code-cracking machines during World Wa
r II that were the forerunners of modern computers. He later proposed a test of artificial intelligence.
With the advent of electronic computers, mathematics was in demand to provide methods of designing and programming computer systems. But computers also provided a powerful tool for mathematicians. Hitherto unsolved mathematical problems such as the four-color theorem often involved lengthy calculations, which could now be done quickly and accurately by computer. Although Poincaré had laid the foundations of chaos theory, Edward Lorenz was able to establish the principles more firmly with the aid of computer models. His visual images of attractors and oscillators, along with Benoit Mandelbrot’s fractals, became icons of these new fields of study.
With the advent of computers, the secure transfer of data became an issue, and mathematicians devised complex cryptosystems using the factorization of large prime numbers. Launched in 1989, the World Wide Web facilitated the rapid transmission of knowledge, and computers became a part of everyday life, especially in the field of information technology.
New logic, new millennium
For a while, it seemed electronic computing could potentially provide answers to almost all problems. But computing science was based on a binary system of logic first proposed by George Boole in the 1800s, and the polar opposites of on-off, true-false, 0-1, and so on could not describe how things are in the real world. To overcome this, Lotfi Zadeh suggested a system of “fuzzy” logic, in which statements can be partly true or false, in a range between 0 (absolutely false) and 1 (absolutely true).
In 2000, 21st-century mathematics was heralded in a similar spirit to that of the 20th century, when the Clay Mathematics Institute announced seven Millennium Prize Problems, offering a $1 million prize for any of their solutions. As yet, only the Poincaré conjecture has been solved; Grigori Perelman’s proof was confirmed in 2006.
IN CONTEXT
KEY FIGURE
David Hilbert (1862–1943)
FIELDS
Logic, geometry
BEFORE
1859 Bernhard Riemann proposes the Riemann hypothesis, a famous problem that will later be Number 8 on Hilbert’s list and remains unresolved today.
1878 Georg Cantor advances the continuum hypothesis, later Number 1 on Hilbert’s list.
AFTER
2000 The Clay Institute issues a list of seven Millennium Prize mathematical problems, offering a million dollars for each problem solved.
2008 In a bid to stimulate major new mathematical breakthroughs, the US Defense Advanced Research Projects Agency (DARPA) announces its list of 23 unsolved problems.
It requires a special technical brilliance and self-confidence to predict relevant problems for the next hundred years, but this is what German mathematician David Hilbert did in 1900. Hilbert possessed a substantial grasp of most fields of mathematics. At the International Mathematical Congress in Paris in 1900, he confidently announced his choice of 23 questions that he believed should occupy mathematicians’ thoughts in the decades to come. This proved prescient; the math world rose to the challenge.
The range of problems
Many of Hilbert’s questions are highly technical, but some are more accessible. Number 3, for instance, asks if one of any two polyhedra of the same volume can always be cut into finitely many bits that can be reassembled to create the other polyhedron. This was soon resolved in 1900 by German-born American mathematician Max Dehn, who concluded that it could not.
The continuum hypothesis, the first problem on Hilbert’s list, pointed out that the set of natural numbers (the positive integers) was infinite, but so was the set of real numbers between 0 and 1. As a result of the work of German mathematician Georg Cantor, it was agreed that the first infinity was “smaller” than the second.
The continuum hypothesis also stated that there was no infinity lying between these two infinities. Cantor himself was sure this was true, but he could not prove it. In 1940, Austrian–American logician Kurt Gödel showed it could not be proved that such an infinity exists, and, in 1963, American mathematician Paul Cohen showed it could not be proved that such an infinity does not exist. Hilbert’s first problem is substantially resolved, although set theory (the study of the properties of sets) is a complex subject, and much more work on it remains to be done.
Of Hilbert’s 23 problems, 10 are considered resolved, seven have been partially solved, two have been classed as too vague to ever be definitively solved, three remain unsolved, and one (also unsolved) is really a physics problem. Among the unsolved problems is the Riemann hypothesis, which some observers think will remain unsolved for the foreseeable future.
The infinite! No other question has ever moved so profoundly the spirit of man.
David Hilbert
Challenges for the future
Hilbert’s remarkable achievement was to accurately predict what would concern mathematicians in the 1900s and beyond. When American mathematician and Fields Medal winner Steve Smale came up with his own list of 18 questions in 1998, it included Hilbert’s eighth and 16th problems. Two years later, the Riemann hypothesis was also one of the Clay Institute’s Millennium Prize problems. Today’s mathematicians face further challenges, but aspects of Hilbert’s problems – especially those that are still unsolved – remain relevant.
Problem solving and theory building go hand in hand. That’s why Hilbert risked offering a list of unsolved problems instead of presenting new methods or results.
Rüdiger Thiele
German mathematician
DAVID HILBERT
Born in Prussia in 1862 to German parents, David Hilbert entered the University of Königsberg in 1880 and later taught there before becoming professor of mathematics at the University of Göttingen in 1895. In this role, he turned Göttingen into one of the mathematical hubs of the world and taught a number of young mathematicians who later made their own mark.
Hilbert was renowned for his broad understanding of many areas of mathematics, and had a keen interest in mathematical physics, too. Exhausted by anemia, he retired in 1930, and Göttingen’s math faculty soon declined after the Nazi purges of Jewish colleagues. Despite his great contribution to mathematics, Hilbert’s death in 1943, during World War II, went largely unnoticed.
Key works
1897 Commentary on Numbers
1900 “The Problems of Mathematics” (Paris lecture)
1932–35 Collected Works
1934–39 Foundations of Mathematics (with Paul Bernays)
See also: Diophantine equations • Euler’s number • The Goldbach conjecture • The Riemann hypothesis • Transfinite numbers
IN CONTEXT
KEY FIGURE
Francis Galton (1822–1911)
FIELD
Number theory
BEFORE
1774 Pierre-Simon Laplace shows the expected pattern of distribution around the norm.
1809 Carl Friedrich Gauss develops the least squares method of finding the best fit line for a scatter of data.
1835 Adolphe Quetelet advocates the use of the bell curve to model social data.
AFTER
1900 Karl Pearson proposes the chi-squared test to determine the significance of differences between expected and observed frequencies.
Statistics is the branch of mathematics that is concerned with analyzing and interpreting large quantities of data. Its foundations were laid in the late 1800s, principally by British polymaths Francis Galton and Karl Pearson.
Statistics investigates whether the pattern of recorded data is significant or random. Its origins lie in the efforts of 18th-century mathematicians such as Pierre-Simon Laplace to identify observational errors in astronomy. In any set of scientific data, most errors are likely to be very small, and only a few are likely to be very large. So when observations are plotted on a graph, they create a bell-shaped curve with a peak created by the most likely result, or “norm,” in the middle. In 1835, Belgian mathematician Adolphe Quetelet posited that characteristics, such as body mas
s, within a human population follow a bell curve pattern, in which values around the mean are most frequent. Higher and lower values are less frequent. He devised the Quetelet Index (now called the BMI) to indicate body mass.
Typically, plotting two variables, such as height and age, on a graph creates a messy scatter of data points that cannot be linked by a neat line. However, in 1809, German mathematician Carl Friedrich Gauss found an equation to create a “best fit” line, which would show the relationship between the variables. Gauss used a method called “least squares,” which involves adding up the squares of the data; this is still used by statisticians. By the 1840s, mathematicians such as Auguste Bravais were looking at the level of error that could be accepted for this line, and tried to pin down the significance of the midpoint or “median” of a set of data.
Francis Galton invented the quincunx (sometimes called the Galton board) to model the bell curve. His original design had beads dropping over pegs.
Correlation and regression
It was first Galton, then Pearson, who began to draw these threads together. Galton was inspired by his cousin Charles Darwin’s work on evolution, and his aim was to show how likely it was that factors such as height, physiognomy, and even intelligence and criminal tendencies might be passed from one generation to the next. Galton and Pearson’s ideas are tainted by the doctrine of eugenics and racial improvement, but the techniques that they developed have found applications elsewhere.