MATHEMATICS
In the wake of the Renaissance, the 17th Century saw an unprecedented
explosion of mathematical and scientific ideas across Europe, a period
sometimes called the Age of Reason. Hard on the heels of the “Copernican
Revolution” of Nicolaus Copernicus in the 16th Century, scientists
like Galileo Galilei, Tycho Brahe and Johannes Kepler were making
equally revolutionary discoveries in the exploration of the Solar
system, leading to Kepler’s formulation of mathematical laws of
planetary motion.
The invention of the logarithm in the early 17th Century by John Napier (and later improved by Napier and Henry Briggs) contributed to the advance of science, astronomy and mathematics by making some difficult calculations relatively easy. It was one of the most significant mathematical developments of the age, and 17th Century physicists like Kepler and Newton could never have performed the complex calculatons needed for their innovations without it. The French astronomer and
mathematician Pierre Simon Laplace remarked, almost two centuries later, that Napier, by halving the labours of astronomers, had doubled their lifetimes.
The logarithm of a number is the exponent when that number is expressed as a power of 10 (or any other base). It is effectively the inverse of exponentiation. For example, the base 10 logarithm of 100 (usually written log10 100 or lg 100 or just log 100) is 2, because 102 = 100. The value of logarithms arises from the fact that multiplication of two or more numbers is equivalent to adding their logarithms, a much simpler operation. In the same way, division involves the subtraction of logarithms, squaring is as simple as multiplying the logarithm by two (or by three for cubing, etc), square roots requires dividing the logarithm by 2 (or by 3 for cube roots, etc).
Although base 10 is the most popular base, another common base for logarithms is the number e which has a value of 2.7182818... and which has special properties which make it very useful for logarithmic calculations. These are known as natural logarithms, and are written loge or ln. Briggs produced extensive lookup tables of common (base 10) logarithms, and by 1622 William Oughted had produced a logarithmic slide rule, an instrument which became indispensible in technological innovation for the next 300 years.
Napier also improved Simon Stevin's decimal notation and popularized the use of the decimal point, and made lattice multiplication (originally developed by the Persian mathematician Al-Khwarizmi and introduced into Europe by Fibonacci) more convenient with the introduction of “Napier's Bones”, a multiplication tool using a set of numbered rods.
Although not principally a mathematician, the role of the Frenchman
Marin Mersenne as a sort of clearing house and go-between for
mathematical thought in France during this period was crucial. Mersenne
is largely remembered in mathematics today in the term Mersenne primes -
prime numbers that are one less than a power of 2, e.g. 3 (22-1), 7 (23-1), 31 (25-1), 127 (27-1), 8191 (213-1),
etc. In modern times, the largest known prime number has almost always
been a Mersenne prime, but in actual fact, Mersenne’s real connection
with the numbers was only to compile a none-too-accurate list of the
smaller ones (when Edouard Lucas devised a method of checking them in
the 19th Century, he pointed out that Mersenne had incorrectly included 267-1 and left out 261-1, 289-1 and 2107-1 from his list).
The Frenchman René Descartes is sometimes considered the first of the modern school of mathematics. His development of analytic geometry and Cartesian coordinates in the mid-17th Century soon allowed the orbits of the planets to be plotted on a graph, as well as laying the foundations for the later development of calculus (and much later multi-dimensional geometry). Descartes is also credited with the first use of superscripts for powers or exponents.
Two other great French mathematicians were close contemporaries of Descartes: Pierre de Fermat and Blaise Pascal. Fermat formulated several theorems which greatly extended our knowlege of number theory, as well as contributing some early work on infinitesimal calculus. Pascal is most famous for Pascal’s Triangle of binomial coefficients, although similar figures had actually been produced by Chinese and Persian mathematicians long before him.
It was an ongoing exchange of letters between Fermat and Pascal that led to the development of the concept of expected values and the field of probability theory. The first published work on probability theory, however, and the first to outline the concept of mathematical expectation, was by the Dutchman Christiaan Huygens in 1657, although it was largely based on the ideas in the letters of the two Frenchmen.
The French mathematician and engineer Girard Desargues is considered
one of the founders of the field of projective geometry, later developed
further by Jean Victor Poncelet and Gaspard Monge. Projective geometry
considers what happens to shapes when they are projected on to a
non-parallel plane. For example, a circle may be projected into an
ellipse or a hyperbola, and so these curves may all be regarded as
equivalent in projective geometry. In particular, Desargues developed
the pivotal concept of the “point at infinity” where parallels actually
meet. His perspective theorem states that, when two triangles are in
perspective, their corresponding sides meet at points on the same
collinear line.
By “standing on the shoulders of giants”, the Englishman Sir Isaac Newton was able to pin down the laws of physics in an unprecedented way, and he effectively laid the groundwork for all of classical mechanics, almost single-handedly. But his contribution to mathematics should never be underestimated, and nowadays he is often considered, along with Archimedes and Gauss, as one of the greatest mathematicians of all time.
Newton and, independently, the German philosopher and mathematician Gottfried Leibniz, completely revolutionized mathematics (not to mention physics, engineering, economics and science in general) by the development of infinitesimal calculus, with its two main operations, differentiation and integration. Newton probably developed his work before Leibniz, but Leibniz published his first, leading to an extended and rancorous dispute. Whatever the truth behind the various claims, though, it is Leibniz’s calculus notation that is the one still in use today, and calculus of some sort is used extensively in everything from engineering to economics to medicine to astronomy.
Both Newton and Leibniz also contributed greatly in other areas of mathematics, including Newton’s contributions to a generalized binomial theorem, the theory of finite differences and the use of infinite power series, and Leibniz’s development of a mechanical forerunner to the computer and the use of matrices to solve linear equations.
However, credit should also be given to some earlier 17th Century mathematicians whose work partially anticipated, and to some extent paved the way for, the development of infinitesimal calculus. As early as the 1630s, the Italian mathematician Bonaventura Cavalieri developed a geometrical approach to calculus known as Cavalieri's principle, or the “method of indivisibles”. The Englishman John Wallis, who systematized and extended the methods of analysis of Descartes and Cavalieri, also made significant contributions towards the development of calculus, as well as originating the idea of the number line, introducing the symbol ∞ for infinity and the term “continued fraction”, and extending the standard notation for powers to include negative integers and rational numbers. Newton's teacher Isaac Barrow is usually credited with the discovery (or at least the first rigorous statrement of) the fundamental theorem of calculus, which essentially showed that integration and differentiation are inverse operations, and he also made complete translations of Euclid into Latin and English
DESCARTES
René Descartes has been dubbed the "Father of Modern Philosophy", but
he was also one of the key figures in the Scientific Revolution of the
17th Century, and is sometimes considered the first of the modern school
of mathematics.
As a young man, he found employment for a time as a soldier (essentially as a mercenary in the pay of various forces, both Catholic and Protestant). But, after a series of dreams or visions, and after meeting the Dutch philosopher and scientist Isaac Beeckman, who sparked his interest in mathematics and the New Physics, he concluded that his real path in life was the pursuit of true wisdom and science.
Back in France, the young Descartes soon came to the conclusion that the key to philosophy, with all its uncertainties and ambiguity, was to build it on the indisputable facts of mathematics. To pursue his rather heretical ideas further, though, he moved from the restrictions of Catholic France to the more liberal environment of the Netherlands, where he spent most of his adult life, and where he worked on his dream of merging algebra and geometry.
In 1637, he published his ground-breaking philosophical and mathematical treatise "Discours de la méthode" (the “Discourse on Method”), and one of its appendices in particular, "La Géométrie", is now considered a landmark in the history of mathematics. Following on from early movements towards the use of symbolic expressions in mathematics by Diophantus, Al-Khwarizmi and François Viète, "La Géométrie" introduced what has become known as the standard algebraic notation, using lowercase a, b and c for known quantities and x, y and z for unknown quantities. It was perhaps the first book to look like a modern mathematics textbook, full of a's and b's, x2's, etc.
It was in "La Géométrie" that Descartes first proposed that each
point in two dimensions can be described by two numbers on a plane, one
giving the point’s horizontal location and the other the vertical
location, which have come to be known as Cartesian coordinates. He used
perpendicular lines (or axes), crossing at a point called the origin, to
measure the horizontal (x) and vertical (y) locations, both positive and negative, thus effectively dividing the plane up into four quadrants.
Any equation can be represented on the plane by plotting on it the solution set of the equation. For example, the simple equation y = x yields a straight line linking together the points (0,0), (1,1), (2,2), (3,3), etc. The equation y = 2x yields a straight line linking together the points (0,0), (1,2), (2,4), (3,6), etc. More complex equations involving x2, x3, etc, plot various types of curves on the plane.
As a point moves along a curve, then, its coordinates change, but an equation can be written to describe the change in the value of the coordinates at any point in the figure. Using this novel approach, it soon became clear that an equation like x2 + y2 = 4, for example, describes a circle; y2 - 16x a curve called a parabola; x2⁄a2 + y2⁄b2 = 1 an ellipse; x2⁄a2 - y2⁄b2 = 1 a hyperbola; etc.
Descartes’ ground-breaking work, usually referred to as analytic geometry or Cartesian geometry, had the effect of allowing the conversion of geometry into algebra (and vice versa). Thus, a pair of simultaneous equations could now be solved either algebraically or graphically (at the intersection of two lines). It allowed the development of Newton’s and Leibniz’s subsequent discoveries of calculus. It also unlocked the possibility of navigating geometries of higher dimensions, impossible to physically visualize - a concept which was to become central to modern technology and physics - thus transforming mathematics forever.
Although analytic geometry was far and away Descartes’ most important
contribution to mathematics, he also: developed a “rule of signs”
technique for determining the number of positive or negative real roots
of a polynomial; "invented" (or at least popularized) the superscript
notation for showing powers or exponents (e.g. 24 to show 2 x
2 x 2 x 2); and re-discovered Thabit ibn Qurra's general formula for
amicable numbers, as well as the amicable pair 9,363,584 and 9,437,056
(which had also been discovered by another Islamic mathematician, Yazdi, almost a century earlier).
For all his importance in the development of modern mathematics, though, Descartes is perhaps best known today as a philosopher who espoused rationalism and dualism. His philosophy consisted of a method of doubting everything, then rebuilding knowledge from the ground, and he is particularly known for the often-quoted statement “Cogito ergo sum”(“I think, therefore I am”).
He also had an influential rôle in the development of modern physics, a rôle which has been, until quite recently, generally under-appreciated and under-investigated. He provided the first distinctly modern formulation of laws of nature and a conservation principle of motion, made numerous advances in optics and the study of the reflection and refraction of light, and constructed what would become the most popular theory of planetary motion of the late 17th Century. His commitment to the scientific method was met with strident opposition by the church officials of the day.
His revolutionary ideas made him a centre of controversy in his day, and he died in 1650 far from home in Stockholm, Sweden. 13 years later, his works were placed on the Catholic Church's "Index of Prohibited Books"
FERMAT
Another Frenchman of the 17th Century, Pierre de Fermat, effectively
invented modern number theory virtually single-handedly, despite being a
small-town amateur mathematician. Stimulated and inspired by the
“Arithmetica” of the Hellenistic mathematician Diophantus,
he went on to discover several new patterns in numbers which had
defeated mathematicians for centuries, and throughout his life he
devised a wide range of conjectures and theorems. He is also given
credit for early developments that led to modern calculus, and for early
progress in probability theory.
Although he showed an early interest in mathematics, he went on study law at Orléans and received the title of councillor at the High Court of Judicature in Toulouse in 1631, which he held for the rest of his life. He was fluent in Latin, Greek, Italian and Spanish, and was praised for his written verse in several languages, and eagerly sought for advice on the emendation of Greek texts.
Fermat's mathematical work was communicated mainly in letters to friends, often with little or no proof of his theorems. Although he himself claimed to have proved all his arithmetic theorems, few records of his proofs have survived, and many mathematicians have doubted some of his claims, especially given the difficulty of some of the problems and the limited mathematical tools available to Fermat.
One example of his many theorems is the Two Square Theorem, which
shows that any prime number which, when divided by 4, leaves a remainder
of 1 (i.e. can be written in the form 4n + 1), can always be re-written as the sum of two square numbers (see image at right for examples).
His so-called Little Theorem is often used in the testing of large prime numbers, and is the basis of the codes which protect our credit cards in Internet transactions today. In simple (sic) terms, it says that if we have two numbers a and p, where p is a prime number and not a factor of a, then a multiplied by itself p-1 times and then divided by p, will always leave a remainder of 1. In mathematical terms, this is written: ap-1 = 1(mod p). For example, if a = 7 and p = 3, then 72 ÷ 3 should leave a remainder of 1, and 49 ÷ 3 does in fact leave a remainder of 1.
Fermat identified a subset of numbers, now known as Fermat numbers, which are of the form of one less than 2 to the power of a power of 2, or, written mathematically, 22n + 1. The first five such numbers are: 21 + 3 = 3; 22 + 1 = 5; 24 + 1 = 17; 28 + 1 = 257; and 216 + 1 = 65,537. Interestingly, these are all prime numbers (and are known as Fermat primes), but all the higher Fermat numbers which have been painstakingly identified over the years are NOT prime numbers, which just goes to to show the value of inductive proof in mathematics.
Fermat's pièce de résistance, though, was his famous Last Theorem, a
conjecture left unproven at his death, and which puzzled mathematicians
for over 350 years. The theorem, originally described in a scribbled
note in the margin of his copy of Diophantus' “Arithmetica”, states that no three positive integers a, b and c can satisfy the equation an + bn = cn for any integer value of n
greater than two (i.e. squared). This seemingly simple conjecture has
proved to be one of the world’s hardest mathematical problems to prove.
There are clearly many solutions - indeed, an infinite number - when n = 2 (namely, all the Pythagorean triples), but no solution could be found for cubes or higher powers. Tantalizingly, Fermat himself claimed to have a proof, but wrote that “this margin is too small to contain it”. As far as we know from the papers which have come down to us, however, Fermat only managed to partially prove the theorem for the special case of n = 4, as did several other mathematicians who applied themselves to it (and indeed as had earlier mathematicians dating back to Fibonacci, albeit not with the same intent).
Over the centuries, several mathematical and scientific academies offered
substantial prizes for a proof of the theorem, and to some extent it
single-handedly stimulated the development of algebraic number theory in
the 19th and 20th Centuries. It was finally proved for ALL numbers only
in 1995 (a proof usually attributed to British mathematician Andrew
Wiles, although in reality it was a joint effort of several steps
involving many mathematicians over several years). The final proof made
use of complex modern mathematics, such as the modularity theorem for
semi-stable elliptic curves, Galois representations and Ribet’s epsilon
theorem, all of which were unavailable in Fermat’s time, so it seems
clear that Fermat's claim to have solved his last theorem was almost
certainly an exaggeration (or at least a misunderstanding).
In addition to his work in number theory, Fermat anticipated the development of calculus to some extent, and his work in this field was invaluable later to Newton and Leibniz. While investigating a technique for finding the centres of gravity of various plane and solid figures, he developed a method for determining maxima, minima and tangents to various curves that was essentially equivalent to differentiation. Also, using an ingenious trick, he was able to reduce the integral of general power functions to the sums of geometric series.
Fermat’s correspondence with his friend Pascal also helped mathematicians grasp a very important concept in basic probability which, although perhaps intuitive to us now, was revolutionary in 1654, namely the idea of equally probable outcomes and expected values
PASCAL
The Frenchman Blaise Pascal was a prominent 17th Century scientist,
philosopher and mathematician. Like so many great mathematicians, he was
a child prodigy and pursued many different avenues of intellectual
endeavour throughout his life. Much of his early work was in the area of
natural and applied sciences, and he has a physical law named after him
(that “pressure exerted anywhere in a confined liquid is transmitted
equally and undiminished in all directions throughout the liquid”), as
well as the international unit for the meaurement of pressure. In
philosophy, Pascals’ Wager is his pragmatic approach to believing in God
on the grounds that is it is a better “bet” than not to.
But Pascal was also a mathematician of the first order. At the age of sixteen, he wrote a significant treatise on the subject of projective geometry, known as Pascal's Theorem, which states that, if a hexagon is inscribed in a circle, then the three intersection points of opposite sides lie on a single line, called the Pascal line. As a young man, he built a functional calculating machine, able to perform additions and subtractions, to help his father with his tax calculations.
He is best known, however, for Pascal’s Triangle, a convenient
tabular presentation of binomial co-efficients, where each number is the
sum of the two numbers directly above it. A binomial is a simple type
of algebraic expression which has just two terms operated on only by
addition, subtraction, multiplication and positive whole-number
exponents, such as (x + y)2. The co-efficients produced when a binomial is expanded form a symmetrical triangle (see image at right).
Pascal was far from the first to study this triangle. The Persian mathematician Al-Karaji had produced something very similar as early as the 10th Century, and the Triangle is called Yang Hui's Triangle in China after the 13th Century Chinese mathematician, and Tartaglia’s Triangle in Italy after the eponymous 16th Century Italian. But Pascal did contribute an elegant proof by defining the numbers by recursion, and he also discovered many useful and interesting patterns among the rows, columns and diagonals of the array of numbers. For instance, looking at the diagonals alone, after the outside "skin" of 1's, the next diagonal (1, 2, 3, 4, 5,...) is the natural numbers in order. The next diagonal within that (1, 3, 6, 10, 15,...) is the triangular numbers in order. The next (1, 4, 10, 20, 35,...) is the pyramidal triangular numbers, etc, etc. It is also possible to find prime numbers, Fibonacci numbers, Catalan numbers, and many other series, and even to find fractal patterns within it.
Pascal also made the conceptual leap to use the Triangle to help solve problems in probability theory. In fact, it was through his collaboration and correspondence with his French contemporary Pierre de Fermat and the Dutchman Christiaan Huygens on the subject that the mathematical theory of probability was born. Before Pascal, there was no actual theory of probability - notwithstanding Gerolamo Cardano’s early exposition in the 16th Century - merely an understanding (of sorts) of how to compute “chances” in dice and card games by counting equally probable outcomes. Some apparently quite elementary problems in probability had eluded some of the best mathematicians, or given rise to incorrect solutions.
It fell to Pascal (with Fermat's help) to bring together the separate threads of prior knowledge (including Cardano's early work) and to introduce entirely new mathematical techniques for the solution of problems that had hitherto resisted solution. Two such intransigent problems which Pascal and Fermat applied themselves to were the Gambler’s Ruin (determining the chances of winning for each of two men playing a particular dice game with very specific rules) and the Problem of Points (determining how a game's winnings should be divided between two equally skilled players if the game was ended prematurely). His work on the Problem of Points in particular, although unpublished at the time, was highly influential in the unfolding new field.
The Problem of Points at its simplest can be illustrated by a simple
game of “winner take all” involving the tossing of a coin. The first of
the two players (say, Fermat
and Pascal) to achieve ten points or wins is to receive a pot of 100
francs. But, if the game is interrupted at the point where Fermat, say, is winning 8 points to 7, how is the 100 franc pot to divided? Fermat
claimed that, as he needed only two more points to win the game, and
Pascal needed three, the game would have been over after four more
tosses of the coin (because, if Pascal did not get the necessary 3
points for your victory over the four tosses, then Fermat must have gained the necessary 2 points for his victory, and vice versa. Fermat
then exhaustively listed the possible outcomes of the four tosses, and
concluded that he would win in 11 out of the 16 possible outcomes, so he
suggested that the 100 francs be split 11⁄16 (0.6875) to him and 5⁄16 (0.3125) to Pascal.
Pascal then looked for a way of generalizing the problem that would avoid the tedious listing of possibilities, and realized that he could use rows from his triangle of coefficients to generate the numbers, no matter how many tosses of the coin remained. As Fermat needed 2 more points to win the game and Pascal needed 3, he went to the fifth (2 + 3) row of the triangle, i.e. 1, 4, 6, 4, 1. The first 3 terms added together (1 + 4 + 6 = 11) represented the outcomes where Fermat would win, and the last two terms (4 + 1 = 5) the outcomes where Pascal would win, out of the total number of outcomes represented by the sum of the whole row (1 + 4 + 6 +4 +1 = 16).
Pascal and Fermat had grasped through their correspondence a very important concept that, though perhaps intuitive to us today, was all but revolutionary in 1654. This was the idea of equally probable outcomes, that the probability of something occurring could be computed by enumerating the number of equally likely ways it could occur, and dividing this by the total number of possible outcomes of the given situation. This allowed the use of fractions and ratios in the calculation of the likelhood of events, and the operation of multiplication and addition on these fractional probabilities. For example, the probability of throwing a 6 on a die twice is 1⁄6 x 1⁄6 = 1⁄36 ("and" works like multiplication); the probability of throwing either a 3 or a 6 is 1⁄6 + 1⁄6 = 1⁄3 ("or" works like addition).
Later in life, Pascal and his sister Jacqueline strongly identified with the extreme Catholic religious movement of Jansenism. Following the death of his father and a "mystical experience" in late 1654, he had his "second conversion" and abandoned his scientific work completely, devoting himself to philosophy and theology. His two most famous works, the "Lettres provinciales" and the "Pensées", date from this period, the latter left incomplete at his death in 1662. They remain Pascal’s best known legacy, and he is usually remembered today as one of the most important authors of the French Classical Period and one of the greatest masters of French prose, much more than for his contributions to mathematics
NEWTON
In the heady atmosphere of 17th Century England, with the expansion
of the British empire in full swing, grand old universities like Oxford
and Cambridge were producing many great scientists and mathematicians.
But the greatest of them all was undoubtedly Sir Isaac Newton.
Physicist, mathematician, astronomer, natural philosopher, alchemist and theologian, Newton is considered by many to be one of the most influential men in human history. His 1687 publication, the "Philosophiae Naturalis Principia Mathematica" (usually called simply the "Principia"), is considered to be among the most influential books in the history of science, and it dominated the scientific view of the physical universe for the next three centuries.
Although largely synonymous in the minds of the general public today with gravity and the story of the apple tree, Newton remains a giant in the minds of mathematicians everywhere (on a par with the all-time greats like Archimedes and Gauss), and he greatly influenced the subsequent path of mathematical development.
Over two miraculous years, during the time of the Great Plague of 1665-6, the young Newton developed a new theory of light, discovered and quantified gravitation, and pioneered a revolutionary new approach to mathematics: infinitesimal calculus. His theory of calculus built on earlier work by his fellow Englishmen John Wallis and Isaac Barrow, as well as on work of such Continental mathematicians as René Descartes, Pierre de Fermat, Bonaventura Cavalieri, Johann van Waveren Hudde and Gilles Personne de Roberval. Unlike the static geometry of the Greeks, calculus allowed mathematicians and engineers to make sense of the motion and dynamic change in the changing world around us, such as the orbits of planets, the motion of fluids, etc.
The initial problem Newton was confronting was that, although it was
easy enough to represent and calculate the average slope of a curve (for
example, the increasing speed of an object on a time-distance graph),
the slope of a curve was constantly varying, and there was no method to
give the exact slope at any one individual point on the curve i.e.
effectively the slope of a tangent line to the curve at that point.
Intuitively, the slope at a particular point can be approximated by taking the average slope (“rise over run”) of ever smaller segments of the curve. As the segment of the curve being considered approaches zero in size (i.e. an infinitesimal change in x), then the calculation of the slope approaches closer and closer to the exact slope at a point (see image at right).
Without going into too much complicated detail, Newton (and his contemporary Gottfried Leibniz independently) calculated a derivative function f ‘(x) which gives the slope at any point of a function f(x). This process of calculating the slope or derivative of a curve or function is called differential calculus or differentiation (or, in Newton’s terminology, the “method of fluxions” - he called the instantaneous rate of change at a particular point on a curve the "fluxion", and the changing values of x and y the "fluents"). For instance, the derivative of a straight line of the type f(x) = 4x is just 4; the derivative of a squared function f(x) = x2 is 2x; the derivative of cubic function f(x) = x3 is 3x2, etc. Generalizing, the derivative of any power function f(x) = xr is rxr-1. Other derivative functions can be stated, according to certain rules, for exponential and logarithmic functions, trigonometric functions such as sin(x), cos(x), etc, so that a derivative function can be stated for any curve without discontinuities. For example, the derivative of the curve f(x) = x4 - 5p3 + sin(x2) would be f ’(x) = 4x3 - 15x2 + 2xcos(x2).
Having established the derivative function for a particular curve, it is then an easy matter to calcuate the slope at any particular point on that curve, just by inserting a value for x. In the case of a time-distance graph, for example, this slope represents the speed of the object at a particular point.
The “opposite” of differentiation is integration or integral calculus
(or, in Newton’s terminology, the “method of fluents”), and together
differentiation and integration are the two main operations of calculus.
Newton’s Fundamental Theorem of Calculus states that differentiation
and integration are inverse operations, so that, if a function is first
integrated and then differentiated (or vice versa), the original
function is retrieved.
The integral of a curve can be thought of as the formula for calculating the area bounded by the curve and the x axis between two defined boundaries. For example, on a graph of velocity against time, the area “under the curve” would represent the distance travelled. Essentially, integration is based on a limiting procedure which approximates the area of a curvilinear region by breaking it into infinitesimally thin vertical slabs or columns. In the same way as for differentiation, an integral function can be stated in general terms: the integral of any power f(x) = xr is xr+1⁄r+1, and there are other integral functions for exponential and logarithmic functions, trigonometric functions, etc, so that the area under any continuous curve can be obtained between any two limits.
Newton chose not to publish his revolutionary mathematics straight away, worried about being ridiculed for his unconventional ideas, and contented himself with circulating his thoughts among friends. After all, he had many other interests such as philosophy, alchemy and his work at the Royal Mint. However, in 1684, the German Leibniz published his own independent version of the theory, whereas Newton published nothing on the subject until 1693. Although the Royal Society, after due deliberation, gave credit for the first discovery to Newton (and credit for the first publication to Leibniz), something of a scandal arose when it was made public that the Royal Society’s subsequent accusation of plagiarism against Leibniz was actually authored by none other Newton himself, causing an ongoing controversy which marred the careers of both men.
Despite being by far his best known contribution to mathematics,
calculus was by no means Newton’s only contribution. He is credited with
the generalized binomial theorem, which describes the algebraic
expansion of powers of a binomial (an algebraic expression with two
terms, such as a2 - b2); he made substantial contributions to the theory of finite differences (mathematical expressions of the form f(x + b) - f(x + a));
he was one of the first to use fractional exponents and coordinate
geometry to derive solutions to Diophantine equations (algebraic
equations with integer-only variables); he developed the so-called
“Newton's method” for finding successively better approximations to the
zeroes or roots of a function; he was the first to use infinite power
series with any confidence; etc.
In 1687, Newton published his “Principia” or “The Mathematical Principles of Natural Philosophy”, generally recognized as the greatest scientific book ever written. In it, he presented his theories of motion, gravity and mechanics, explained the eccentric orbits of comets, the tides and their variations, the precession of the Earth's axis and the motion of the Moon.
Later in life, he wrote a number of religious tracts dealing with the literal interpretation of the Bible, devoted a great deal
of time to alchemy, acted as Member of Parliament for some years, and
became perhaps the best-known Master of the Royal Mint in 1699, a
position he held until his death in 1727. In 1703, he was made President
of the Royal Society and, in 1705, became the first scientist ever to
be knighted. Mercury poisoning from his alchemical pursuits perhaps
explained Newton's eccentricity in later life, and possibly also his
eventual death
LEIBNIZ
The German polymath Gottfried Wilhelm Leibniz occupies a grand place in the history of philosophy. He was, along with René Descartes
and Baruch Spinoza, one of the three great 17th Century rationalists,
and his work anticipated modern logic and analytic philosophy. Like many
great thinkers before and after him, Leibniz was a child prodigy and a
contributor in many different fields of endeavour.
But, between his work on philosophy and logic and his day job as a politician and representative of the royal house of Hanover, Leibniz still found time to work on mathematics. He was perhaps the first to explicitly employ the mathematical notion of a function to denote geometric concepts derived from a curve, and he developed a system of infinitesimal calculus, independently of his contemporary Sir Isaac Newton. He also revived the ancient method of solving equations using matrices, invented a practical calculating machine and pioneered the use of the binary system.
Like Newton, Leibniz was a member of the Royal Society in London, and was almost certainly aware of Newton’s work on calculus. During the 1670s (slightly later than Newton’s early work), Leibniz developed a very similar theory of calculus, apparently completely independently. Within the short period of about two months he had developed a complete theory of differential calculus and integral calculus (see the section on Newton for a brief description and explanation of the development of calculus).
Unlike Newton, however, he was more than happy to publish his work,
and so Europe first heard about calculus from Leibniz in 1684, and not
from Newton
(who published nothing on the subject until 1693). When the Royal
Society was asked to adjudicate between the rival claims of the two men
over the development of the theory of calculus, they gave credit for the
first discovery to Newton, and credit for the first publication to Leibniz. However, the Royal Society, by then under the rather biassed presidency of Newton himself, later also accused Leibniz of plagiarism, a slur from which Leibniz never really recovered.
Ironically, it was Leibniz’s mathematics that eventually triumphed, and his notation and his way of writing calculus, not Newton’s more clumsy notation, is the one still used in mathematics today.
In addition to calculus, Leibniz re-discovered a method of arranging linear equations into an array, now called a matrix, which could then be manipulated to find a solution. A similar method had been pioneered by Chinese mathematicians almost two millennia earlier, but had long fallen into disuse. Leibniz paved the way for later work on matrices and linear algebra by Carl Friedrich Gauss. He also introduced notions of self-similarity and the principle of continuity which foreshadowed an area of mathematics which would come to be called topology.
During the 1670s, Leibniz worked on the invention of a practical calculating machine, which used the binary system and was capable of multiplying, dividing and even extracting roots, a great improvement on Pascal’s rudimentary adding machine and a true forerunner of the computer. He is usually credited with the early development of the binary
number system (base 2 counting, using only the digits 0 and 1),
although he himself was aware of similar ideas dating back to the I
Ching of Ancient China. Because of the ability of binary
to be represented by the two phases "on" and "off", it would later
become the foundation of virtually all modern computer systems, and
Leibniz's documentation was essential in the development process.
Leibniz is also often considered the most important logician between Aristotle in Ancient Greece and George Boole and Augustus De Morgan in the 19th Century. Even though he actually published nothing on formal logic in his lifetime, he enunciated in his working drafts the principal properties of what we now call conjunction, disjunction, negation, identity, set inclusion and the empty set.
MATHEMATICS
Most of the late 17th Century and a good part of the early 18th were taken up by the work of disciples of Newton and Leibniz, who applied their ideas on calculus to solving a variety of problems in physics, astronomy and engineering.
The period was dominated, though, by one family, the Bernoulli’s of Basel in Switzerland, which boasted two or three generations of exceptional mathematicians, particularly the brothers, Jacob and Johann. They were largely responsible for further developing Leibniz’s infinitesimal calculus - paricularly through the generalization and extension of calculus known as the "calculus of variations" - as well as Pascal and Fermat’s probability and number theory.
Basel was also the home town of the greatest of the 18th Century mathematicians, Leonhard Euler, although, partly due to the difficulties in getting on in a city dominated by the Bernoulli family, Euler spent most of his time abroad, in Germany and St. Petersburg, Russia. He excelled in all aspects of mathematics, from geometry to calculus to trigonometry to algebra to number theory, and was able to find unexpected links between the different fields. He proved numerous theorems, pioneered new methods, standardized mathematical notation and wrote many influential textbooks throughout his long academic life.
In a letter to Euler in 1742, the German mathematician Christian Goldbach proposed the Goldbach Conjecture, which states that every even integer greater than 2 can be expressed as the sum of two primes (e.g. 4 = 2 + 2; 8 = 3 + 5; 14 = 3 + 11 = 7 + 7; etc) or, in another equivalent version, every integer greater than 5 can be expressed as the sum of three primes. Yet another version is the so-called “weak” Goldbach Conjecture, that all odd numbers greater than 7 are the sum of three odd primes. They remain among the oldest unsolved problems in number theory (and in all of mathematics), although the weak form of the conjecture appears to be closer to resolution than the strong one. Goldbach also proved other theorems in number theory such as the Goldbach-Euler Theorem on perfect powers.
Despite Euler’s and the Bernoullis’ dominance of 18th Century mathematics, many of the other important mathematicians were from France. In the early part of the century, Abraham de Moivre is perhaps best known for de Moivre's formula, (cosx + isinx)n = cos(nx) + isin(nx), which links complex numbers and trigonometry. But he also generalized Newton’s famous binomial theorem into the multinomial theorem, pioneered the development of analytic geometry, and his work on the normal distribution (he gave the first statement of the formula for the normal distribution curve) and probability theory were of great importance.
France became even more prominent towards the end of the century, and a handful of late 18th Century French mathematicians in particular deserve mention at this point, beginning with “the three L’s”.
Joseph Louis Lagrange collaborated with Euler in an important joint work on the calculus of variation, but he also contributed to differential equations and number theory, and he is usually credited with originating the theory of groups, which would become so important in 19th and 20th Century mathematics. His name is given an early theorem in group theory, which states that the number of elements of every sub-group of a finite group divides evenly into the number of elements of the original finite group.
Lagrange is also credited with the four-square theorem, that any
natural number can be represented as the sum of four squares (e.g. 3 = 12 + 12 + 12 + 02; 31 = 52 + 22 + 12 + 12; 310 = 172 + 42 + 22 + 12;
etc), as well as another theorem, confusingly also known as Lagrange’s
Theorem or Lagrange’s Mean Value Theorem, which states that, given a
section of a smooth continuous (differentiable) curve, there is at least
one point on that section at which the derivative (or slope) of the
curve is equal (or parallel) to the average (or mean) derivative of the
section. Lagrange’s 1788 treatise on analytical mechanics offered the most comprehensive treatment of classical mechanics since Newton, and formed a basis for the development of mathematical physics in the 19th Century.
Pierre-Simon Laplace, sometimes referred to as “the French Newton”, was an important mathematician and astronomer, whose monumental work “Celestial Mechanics” translated the geometric study of classical mechanics to one based on calculus, opening up a much broader range of problems. Although his early work was mainly on differential equations and finite differences, he was already starting to think about the mathematical and philosophical concepts of probability and statistics in the 1770s, and he developed his own version of the so-called Bayesian interpretation of probability independently of Thomas Bayes. Laplace is well known for his belief in complete scientific determinism, and he maintained that there should be a set of scientific laws that would allow us - at least in principle - to predict everything about the universe and how it works.
Adrien-Marie Legendre also made important contributions to
statistics, number theory, abstract algebra and mathematical analysis in
the late 18th and early 19th Centuries, athough much of his work (such
as the least squares method for curve-fitting and linear regression, the
quadratic reciprocity law, the prime number theorem and his work on elliptic functions) was only brought to perfection - or at least to general notice - by others, particularly Gauss. His “Elements of Geometry”, a re-working of Euclid’s
book, became the leading geometry textbook for almost 100 years, and
his extremely accurate measurement of the terrestrial meridian inspired
the creation, and almost universal adoption, of the metric system of
measures and weights.
Yet another Frenchman, Gaspard Monge was the inventor of descriptive geometry, a clever method of representing three-dimensional objects by projections on the two-dimensional plane using a specific set of procedures, a technique which would later become important in the fields of engineering, architecture and design. His orthographic projection became the graphical method used in almost all modern mechanical drawing.
After many centuries of increasingly accurate approximations, Johann Lambert, a Swiss mathematician and prominent astronomer, finally provided a rigorous proof in 1761 that π is irrational, i.e. it can not be expressed as a simple fraction using integers only or as a terminating or repeating decimal. This definitively proved that it would never be possible to calculate it exactly, although the obsession with obtaining more and more accurate approximations continues to this day. (Over a hundred years later, in 1882, Ferdinand von Lindemann would prove that π is also transcendental, i.e. it cannot be the root of any polynomial equation with rational coefficients). Lambert was also the first to introduce hyperbolic functions into trigonometry and made some prescient conjectures regarding non-Euclidean space and the properties of hyperbolic triangles
BERNOULLI BROTHERS
Unusually in the history of mathematics, a single family, the
Bernoulli’s, produced half a dozen outstanding mathematicians over a
couple of generations at the end of the 17th and start of the 18th
Century.
The Bernoulli family was a prosperous family of traders
and scholars from the free city of Basel in Switzerland, which at that
time was the great commercial hub of central Europe.The brothers, Jacob
and Johann Bernoulli, however, flouted their father's wishes for them to
take over the family spice business or to enter respectable professions
like medicine or the ministry, and began studying mathematics together.
After Johann graduated from Basel University, the two developed a rather jealous and competitive relationship. Johann in particular was jealous of the elder Jacob's position as professor at Basel University, and the two often attempted to outdo each other. After Jacob's early death from tuberculosis, Johann took over his brother's position, one of his young students being the great Swiss mathematician Leonhard Euler. However, Johann merely shifted his jealousy toward his own talented son, Daniel (at one point, Johann published a book based on Daniel's work, even changing the date to make it look as though his book had been published before his son's).
Johann received a taste of his own medicine, though, when his student Guillaume de l'Hôpital published a book in
his own name consisting almost entirely of Johann's lectures, including
his now famous rule about 0 ÷ 0 (a problem which had dogged
mathematicians since Brahmagupta's initial work on the rules for dealing
with zero back in the 7th Century). This showed that 0 ÷ 0 does not
equal zero, does not equal 1, does not equal infinity, and is not even
undefined, but is "indeterminate" (meaning it could equal any number).
The rule is still usually known as l'Hôpital's Rule, and not Bernoulli's
Rule.
Despite their competitive and combative personal relationship, though, the brothers both had a clear aptitude for mathematics at a high level, and constantly challenged and inspired each other. They established an early correspondence with Gottfried Leibniz, and were among the first mathematicians to not only study and understand infinitesimal calculus but to apply it to various problems. They became instrumental in disseminating the newly-discovered knowledge of calculus, and helping to make it the cornerstone of mathematics it has become today.
But they were more than just disciples of Leibniz,
and they also made their own important contributions. One well known
and topical problem of the day to which they applied themselves was that
of designing a sloping ramp which would allow a ball to roll from the
top to the bottom in the fastest possible time. Johann Bernoulli
demonstrated through calculus that neither a straight ramp or a curved
ramp with a very steep initial slope were optimal, but actually a less
steep curved ramp known as a brachistochrone curve (a kind of
upside-down cycloid, similar to the path followed by a point on a moving
bicycle wheel) is the curve of fastest descent.
This application was an example of the “calculus of variations”, a generalization of infinitesimal calculus that the Bernoulli brothers developed together, and has since proved useful in fields as diverse as engineering, financial
investment, architecture and construction, and even space travel.
Johann also derived the equation for a catenary curve, such as that
formed by a chain hanging between two posts, a problem presented to him
by his brother Jacob.
Jacob Bernoulli’s book “The Art of Conjecture”, published
posthumously in 1713, consolidated existing knowledge on probability
theory and expected values, as well as adding personal contributions,
such as his theory of permutations and combinations, Bernoulli trials
and Bernoulli distribution, and some important elements of number
theory, such as the Bernoulli Numbers sequence. He also published papers
on transcendental curves, and became the first person to develop the
technique for solving separable differential equations (the set of
non-linear, but solvable, differential equations are now named after
him). He invented polar coordinates (a method of describing the location
of points in space using angles and distances) and was the first to use
the word “integral” to refer to the area under a curve.
Jacob Bernoulli also discovered the appropximate value of the irrational number e while exploring the compound interest on loans. When compounded at 100% interest annually, $1.00 becomes $2.00 after one year; when compounded semi-annually it ppoduces $2.25; compounded quarterly $2.44; monthly $2.61; weekly $2.69; daily $2.71; etc. If it were to be compounded continuously, the $1.00 would tend towards a value of $2.7182818... after a year, a value which became known as e. Alegbraically, it is the value of the infinite series (1 + 1⁄1)1.(1 + 1⁄2)2.(1 + 1⁄3)3.(1 + 1⁄4)4...
Johann’s sons Nicolaus, Daniel and Johann II, and even his grandchildren Jacob II and Johann III, were all accomplished mathematicians and teachers. Daniel Bernoulli, in particular, is well known for his work on fluid mechanics (especially Bernoulli’s Principle on the inverse relationship between the speed and pressure of a fluid or gas), as much as for his work on probability and statistics
EULER
Leonhard Euler was one of the giants of 18th Century mathematics. Like the Bernoulli’s, he was born in Basel, Switzerland, and he studied for a while under Johann Bernoulli at Basel University. But, partly due to the overwhelming dominance of the Bernoulli
family in Swiss mathematics, and the difficulty of finding a good
position and recognition in his hometown, he spent most of his academic
life in Russia and Germany, especially in the burgeoning St. Petersburg
of Peter the Great and Catherine the Great.
Despite a long life and thirteen children, Euler had more than his fair share of tragedies and deaths, and even his blindness later in life did not slow his prodigious output - his collected works comprise nearly 900 books and, in the year 1775, he is said to have produced on average one mathematical paper every week - as he compensated for it with his mental calculation skills and photographic memory (for example, he could repeat the Aeneid of Virgil from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last).
Today, Euler is considered one of the greatest mathematicians of all time. His interests covered almost all aspects of mathematics, from geometry to calculus to trigonometry to algebra to number theory, as well as optics, astronomy, cartography, mechanics, weights and measures and even the theory of music.
Much of the notation used by mathematicians today - including e, i, f(x) , ∑, and the use of a, b and c as constants and x, y and z
as unknowns - was either created, popularized or standardized by Euler.
His efforts to standardize these and other symbols (including π and the trigonometric functions) helped to internationalize mathematics and to encourage collaboration on problems.
He even managed to combine several of these together in an amazing feat of mathematical alchemy to produce one of the most beautiful of all mathematical equations, eiπ = -1, sometimes known as Euler’s Identity. This equation combines arithmetic, calculus, trigonometry and complex analysis into what has been called "the most remarkable formula in mathematics", "uncanny and sublime" and "filled with cosmic beauty", among other descriptions. Another such discovery, often known simply as Euler’s Formula, is eix = cosx + isinx. In fact, in a recent poll of mathematicians, three of the top five most beautiful formulae of all time were Euler’s. He seemed to have an instinctive ability to demonstrate the deep relationships between trigonometry, exponentials and complex numbers.
The discovery that initially sealed Euler’s reputation was announced in 1735 and concerned the calculation of infinite sums. It was called the Basel problem after the Bernoulli’s had tried and failed to solve it, and asked what was the precise sum of the of the reciprocals of the squares of all the natural numbers to infinity i.e. 1⁄12 + 1⁄22 + 1⁄32 + 1⁄42 ... (a zeta function using a zeta constant of 2). Euler’s friend Daniel Bernoulli had estimated the sum to be about 13⁄5, but Euler’s superior method yielded the exact but rather unexpected result of π2⁄6. He also showed that the infinite series was equivalent to an infinite product of prime numbers, an identity which would later inspire Riemann’s investigation of complex zeta functions.
Also in 1735, Euler solved an intransigent mathematical and logical
problem, known as the Seven Bridges of Königsberg Problem, which had
perplexed scholars for many years, and in doing so laid the foundations
of graph theory and presaged the important mathematical idea of
topology. The city of Königsberg in Prussia (modern-day Kaliningrad in
Russia) was set on both sides of the Pregel River, and included two
large islands which were connected to each other and the mainland by
seven bridges. The problem was to find a route through the city that
would cross each bridge once and only once.
In fact, Euler proved that the problem has no solution, but in doing so he made the important conceptual leap of pointing out that the choice of route within each landmass is irrelevant and the only important feature is the sequence of bridges crossed. This allowed him to reformulate the problem in abstract terms, replacing each land mass with an abstract node and each bridge with an abstract connection. This resulted in a mathematical structure called a “graph”, a pictorial representation made up of points (vertices) connected by non-intersecting curves (arcs), which may be distorted in any way without changing the graph itself. In this way, Euler was able to deduce that, because the four land masses in the original problem are touched by an odd number of bridges, the existence of a walk
traversing each bridge once only inevitably leads to a contradiction.
If Königsberg had had one fewer bridges, on the other hand, with an even
number of bridges leading to each piece of land, then a solution would
have been possible.
The list of theorems and methods pioneered by Euler is immense, and
largely outside the scope of an entry-level study such as this, but
mention could be made of just some of them:
the demonstration of geometrical properties such as Euler’s Line and Euler’s Circle;
the definition of the Euler Characteristic χ (chi) for the
surfaces of polyhedra, whereby the number of vertices minus the number
of edges plus the number of faces always equals 2 (see table at right);
a new method for solving quartic equations;
the Prime Number Theorem, which describes the asymptotic distribution of the prime numbers;
proofs (and in some cases disproofs) of some of Fermat’s theorems and conjectures;
the discovery of over 60 amicable numbers (pairs of numbers for
which the sum of the divisors of one number equals the other number),
although some were actually incorrect;
a method of calculating integrals with complex limits (foreshadowing the development of modern complex analysis);
the calculus of variations, including its best-known result, the
Euler-Lagrange equation; a proof of the infinitude of primes, using the
divergence of the harmonic series;
the integration of Leibniz's differential calculus with Newton's
Method of Fluxions into a form of calculus we would recognize today, as
well as the development of tools to make it easier to apply calculus to
real physical problems;
etc, etc.
In 1766, Euler accepted an invitation from Catherine the Great to
return to the St. Petersburg Academy, and spent the rest of his life in
Russia. However, his second stay in the country was marred by tragedy,
including a fire in 1771 which cost him his home (and almost his life),
and the loss in 1773 of his dear wife of 40 years, Katharina. He later
married Katharina's half-sister, Salome Abigail, and this marriage would
last until his death from a brain hemorrhage in 1783
MATHEMATICS
The 19th Century saw an unprecedented increase in the breadth and
complexity of mathematical concepts. Both France and Germany were caught
up in the age of revolution which swept Europe in the late 18th
Century, but the two countries treated mathematics quite differently.
After the French Revolution, Napoleon emphasized the practical usefulness of mathematics and his reforms and military ambitions gave French mathematics a big boost, as exemplified by “the three L’s”, Lagrange, Laplace and Legendre (see the section on 18th Century Mathematics), Fourier and Galois.
Joseph Fourier's study, at the beginning of the 19th Century, of infinite sums in which the terms are trigonometric functions were another important advance in mathematical analysis. Periodic functions that can be expressed as the sum of an infinite series of sines and cosines are known today as Fourier Series, and they are still powerful tools in pure and applied mathematics. Fourier (following Leibniz, Euler, Lagrange and others) also contributed towards defining exactly what is meant by a function, although the definition that is found in texts today - defining it in terms of a correspondence between elements of the domain and the range - is usually attributed to the 19th Century German mathematician Peter Dirichlet.
In 1806, Jean-Robert Argand published his paper on how complex numbers (of the form a + bi, where i is √-1) could be represented on geometric diagrams and manipulated using trigonometry and vectors. Even though the Dane Caspar Wessel had produced a very similar paper at the end of the 18th Century, and even though it was Gauss who popularized the practice, they are still known today as Argand Diagrams.
The Frenchman Évariste Galois proved in the late 1820s that there is no general algebraic method for solving polynomial equations of any degree greater than four, going further than the Norwegian Niels Henrik Abel who had, just a few years earlier, shown the impossibility of solving quintic equations, and breaching an impasse which had existed for centuries. Galois' work also laid the groundwork for further developments such as the beginnings of the field of abstract algebra, including areas like algebraic geometry, group theory, rings, fields, modules, vector spaces and non-commutative algebra.
Germany, on the other hand, under the influence of the great educationalist Wilhelm von Humboldt, took a rather different approach, supporting pure mathematics for its own sake, detached from the demands of the state and military. It was in this environment that the young German prodigy Carl Friedrich Gauss, sometimes called the “Prince of Mathematics”, received his education at the prestigious University of Göttingen. Some of Gauss’ ideas were a hundred years ahead of their time, and touched on many different parts of the mathematical world, including geometry, number theory, calculus, algebra and probability. He is widely regarded as one of the three greatest mathematicians of all times, along with Archimedes and Newton.
Later in life, Gauss
also claimed to have investigated a kind of non-Euclidean geometry
using curved space but, unwilling to court controversy, he decided not
to pursue or publish any of these avant-garde ideas. This left the field
open for János Bolyai and Nikolai Lobachevsky
(respectively, a Hungarian and a Russian) who both independently
explored the potential of hyperbolic geometry and curved spaces.
The German Bernhard Riemann worked on a different kind of non-Euclidean geometry called elliptic geometry, as well as on a generalized theory of all the different types of geometry. Riemann,
however, soon took this even further, breaking away completely from all
the limitations of 2 and 3 dimensional geometry, whether flat or
curved, and began to think in higher dimensions. His exploration of the
zeta function in multi-dimensional complex numbers revealed an
unexpected link with the distribution of prime numbers, and his famous
Riemann Hypothesis, still unproven after 150 years, remains one of the
world’s great unsolved mathematical mysteries and the testing ground for
new generations of mathematicians.
British mathematics also saw something of a resurgence in the early and mid-19th century. Although the roots of the computer go back to the geared calculators of Pascal and Leibniz in the 17th Century, it was Charles Babbage in 19th Century England who designed a machine that could automatically perform computations based on a program of instructions stored on cards or tape. His large "difference engine" of 1823 was able to calculate logarithms and trigonometric functions, and was the true forerunner of the modern electronic computer. Although never actually built in his lifetime, a machine was built almost 200 years later to his specifications and worked perfectly. He also designed a much more sophisticated machine he called the "analytic engine", complete with punched cards, printer and computational abilities commensurate with modern computers.
Another 19th Century Englishman, George Peacock, is usually credited with the invention of symbolic algebra, and the extension of the scope of algebra beyond the ordinary systems of numbers. This recognition of the possible existence of non-arithmetical algebras was an important stepping stone toward future developments in abstract algebra.
In the mid-19th Century, the British mathematician George Boole devised an algebra (now called Boolean algebra or Boolean logic), in which the only operators were AND, OR and NOT, and which could be applied to the solution of logical problems and mathematical functions. He also described a kind of binary
system which used just two objects, "on" and "off" (or "true" and
"false", 0 and 1, etc), in which, famously, 1 + 1 = 1. Boolean algebra
was the starting point of modern mathematical logic and ultimately led
to the development of computer science.
The concept of number and algebra was further extended by the Irish
mathematician William Hamilton, whose 1843 theory of quaternions (a
4-dimensional number system, where a quantity representing a
3-dimensional rotation can be described by just an angle and a vector).
Quaternions, and its later generalization by Hermann Grassmann, provided
the first example of a non-commutative algebra (i.e. one in which a x b does not always equal b x a), and showed that several different consistent algebras may be derived by choosing different sets of axioms.
The Englishman Arthur Cayley extended Hamilton's quaternions and developed the octonions. But Cayley was one of the most prolific mathematicians in history, and was a pioneer of modern group theory, matrix algebra, the theory of higher singularities, and higher dimensional geometry (anticipating the later ideas of Klein), as well as the theory of invariants.
Throughout the 19th Century, mathematics in general became ever more complex and abstract. But it also saw a re-visiting of some older methods and an emphasis on mathematical rigour. In the first decades of the century, the Bohemian priest Bernhard Bolzano was one of the earliest mathematicians to begin instilling rigour into mathematical analysis, as well as giving the first purely analytic proof of both the fundamental theorem of algebra and the intermediate value theorem, and early consideration of sets (collections of objects defined by a common property, such as "all the numbers greater than 7" or "all right triangles", etc). When the German mathematician Karl Weierstrass discovered the theoretical existence of a continuous function having no derivative (in other words, a continuous curve possessing no tangent at any of its points), he saw the need for a rigorous “arithmetization” of calculus, from which all the basic concepts of analysis could be derived.
Along with Riemann and, particularly, the Frenchman Augustin-Louis Cauchy, Weierstrass completely reformulated calculus in an even more rigorous fashion, leading to the development of mathematical analysis, a branch of pure mathematics largely concerned with the notion of limits (whether it be the limit of a sequence or the limit of a function) and with the theories of differentiation, integration, infinite series and analytic functions. In 1845, Cauchy also proved Cauchy's theorem, a fundamental theorem of group theory, which he discovered while examining permutation groups. Carl Jacobi also made important contributions to analysis, determinants and matrices, and especially his theory of periodic functions and elliptic functions and their relation to the elliptic theta function.
August Ferdinand Möbius is best known for his 1858 discovery of the
Möbius strip, a non-orientable two-dimensional surface which has only
one side when embedded in three-dimensional Euclidean space (actually a
German, Johann Benedict Listing, devised the same object just a couple
of months before Möbius, but it has come to hold Möbius' name). Many
other concepts are also named after him, including the Möbius
configuration, Möbius transformations, the Möbius transform of number
theory, the Möbius function and the Möbius inversion formula. He also
introduced homogeneous coordinates and discussed geometric and
projective transformations.
Felix Klein also pursued more developments in non-Euclidean geometry, include the Klein bottle, a one-sided closed surface which cannot be embedded in three-dimensional Euclidean space, only in four or more dimensions. It can be best visualized as a cylinder looped back through itself to join with its other end from the "inside". Klein’s 1872 Erlangen Program, which classified geometries by their underlying symmetry groups (or their groups of transformations), was a hugely influential synthesis of much of the mathematics of the day, and his work was very important in the later development of group theory and function theory.
The Norwegian mathematician Marius Sophus Lie also applied algebra to the study of geometry. He largely created the theory of continuous symmetry, and applied it to the geometric theory of differential equations by means of continuous groups of transformations known as Lie groups.
In an unusual occurrence in 1866, an unknown 16-year old Italian, Niccolò Paganini, discovered the second smallest pair of amicable numbers (1,184 and 1210), which had been completely overlooked by some of the greatest mathematicians in history (including Euler, who had identified over 60 such numbers in the 18th Century, some of them huge).
In the later 19th Century, Georg Cantor established the first foundations of set theory, which enabled the rigorous treatment of the notion of infinity, and which has since become the common language of nearly all mathematics. In the face of fierce resistance from most of his contemporaries and his own battle against mental illness, Cantor explored new mathematical worlds where there were many different infinities, some of which were larger than others.
Cantor’s
work on set theory was extended by another German, Richard Dedekind,
who defined concepts such as similar sets and infinite sets. Dedekind
also came up with the notion, now called a Dedekind cut which is now a
standard definition of the real numbers. He showed that any irrational
number divides the rational numbers into two classes or sets, the upper
class being strictly greater than all the members of the other lower
class. Thus, every location on the number line continuum contains either
a rational or an irrational number, with no empty locations, gaps or
discontinuities. In 1881, the Englishman John Venn introduced his “Venn
diagrams” which become useful and ubiquitous tools in set theory.
Building on Riemann’s deep ideas on the distribution of prime numbers, the year 1896 saw two independent proofs of the asymptotic law of the distribution of prime numbers (known as the Prime Number Theorem), one by Jacques Hadamard and one by Charles de la Vallée Poussin, which showed that the number of primes occurring up to any number x is asymptotic to (or tends towards) x⁄log x.
Hermann Minkowski, a great friend of David Hilbert
and teacher of the young Albert Einstein, developed a branch of number
theory called the "geometry of numbers" late in the 19th Century as a
geometrical method in multi-dimensional space for solving number theory
problems, involving complex concepts such as convex sets, lattice points
and vector space. Later, in 1907, it was Minkowski who realized that
the Einstein’s 1905 special theory of relativity could be best
understood in a four-dimensional space, often referred to as Minkowski
space-time.
Gottlob Frege’s 1879 “Begriffsschrift” (roughly translated as “Concept-Script”) broke new ground in the field of logic, including a rigorous treatment of the ideas of functions and variables. In his attempt to show that mathematics grows out of logic, he devised techniques that took him far beyond the logical traditions of Aristotle (and even of George Boole). He was the first to explicitly introduce the notion of variables in logical statements, as well as the notions of quantifiers, universals and existentials. He extended Boole's "propositional logic" into a new "predicate logic" and, in so doing, set the stage for the radical advances of Giuseppe Peano, Bertrand Russell and David Hilbert in the early 20th Century.
Henri Poincaré came to prominence in the latter part of the 19th Century with at least a partial solution to the “three body problem”, a deceptively simple problem which had stubbornly resisted resolution since the time of Newton, over two hundred years earlier. Although his solution actually proved to be erroneous, its implications led to the early intimations of what would later become known as chaos theory. In between his important work in theoretical physics, he also greatly extended the theory of mathematical topology, leaving behind a knotty problem known as the Poincaré conjecture which remined unsolved until 2002.
Poincaré was also an engineer and a polymath, and perhaps the last of the great mathematicians to adhere to an older conception of mathematics, which championed a faith in human intuition over rigour and formalism. He is sometimes referred to as the “Last Univeralist” as he was perhaps the last mathematician able to shine in almost all of the various aspects of what had become by now a huge, encyclopedic and incredibly complex subject. The 20th Century would belong to the specialists
GALOIS
Évariste Galois was radical republican and something of a romantic
figure in French mathematical history. He died in a duel at the young
age of 20, but the work he published shortly before his death made his
name in mathematical circles, and would go on to allow proofs by later
mathematicians of problems which had been impossible for many centuries.
It also laid the groundwork for many later developments in mathematics,
particularly the beginnings of the important fields of abstract algebra
and group theory.
Despite his lacklustre performance at school (he twice failed entrance exams to the École Polytechnique), the young Galois devoured the work of Legendre and Lagrange in his spare time. At the tender age of 17, he began making fundamental discoveries in the theory of polynomial equations (equations constructed from variables and constants, using only the operations of addition, subtraction, multiplication and non-negative whole-number exponents, such as x2 - 4x + 7 = 0). He effectively proved that there can be no general formula for solving quintic equations (polynomials including a term of x5), just as the young Norwegian Niels Henrik Abel had a few years earlier, although by a different method. But he was also able to prove the more general, and more powerful, idea that there is no general algebraic method for solving polynomial equations of any degree greater than four.
Galois achieved this general proof by looking at whether or not the
“permutation group” of its roots (now known as its Galois group) had a
certain structure. He was the first to use the term “group” in its
modern mathematical sense of a group of permutations (foreshadowing the
modern field of group theory), and his fertile approach, now known as
Galois theory, was adapted by later mathematicians to many other fields
of mathematics besides the theory of equations.
Galois’ breakthrough in turn led to definitive proofs (or rather disproofs) later in the century of the so-called “Three Classical Problems” problems which had been first formulated by Plato and others back in ancient Greece: the doubling of the cube and the trisection of an angle (both were proved impossible in 1837), and the squaring of the circle (also proved impossible, in 1882).
Galois was a hot-headed political firebrand (he was arrested several times for political acts), and his political affiliations and activities as a staunch republican during the rule of Louis-Philippe continually distracted him from his mathematical work. He was killed in a duel in 1832, under rather shady circumstances, but he had spent the whole of the previous night outlining his mathematical ideas in a detailed letter to his friend Auguste Chevalier, as though convinced of his impending death.
Ironically, his young contemporary Abel also had a promising career cut short. He died in poverty of tubercolosis at the age of just 26, although his legacy lives on in the term “abelian” (usually written with a small "a"), which has since become commonplace in discussing concepts such as the abelian group, abelian category and abelian variety
GAUSS
Carl Friedrich Gauss is sometimes referred to as the "Prince of
Mathematicians" and the "greatest mathematician since antiquity". He has
had a remarkable influence in many fields of mathematics and science
and is ranked as one of history's most influential mathematicians.
Gauss was a child prodigy. There are many anecdotes concerning his precocity as a child, and he made his first ground-breaking mathematical discoveries while still a teenager.
At just three years old, he corrected an error in his father payroll calculations, and he was looking after his father’s accounts on a regular basis by the age of 5. At the age of 7, he is reported to have amazed his teachers by summing the integers from 1 to 100 almost instantly (having quickly spotted that the sum was actually 50 pairs of numbers, with each pair summing to 101, total 5,050). By the age of 12, he was already attending gymnasium and criticizing Euclid’s geometry.
Although his family was poor and working class, Gauss' intellectual abilities attracted the attention of the Duke of Brunswick, who sent him to the Collegium Carolinum at 15, and then to the prestigious University of Göttingen (which he attended from 1795 to 1798). It was as a teenager attending university that Gauss discovered (or independently rediscovered) several important theorems.
At 15, Gauss was the first to find any kind of a pattern in the occurrence of prime numbers, a problem which had exercised
the minds of the best mathematicians since ancient times. Although the
occurrence of prime numbers appeared to be almost competely random,
Gauss approached the problem from a different angle by graphing the
incidence of primes as the numbers increased. He noticed a rough pattern
or trend: as the numbers increased by 10, the probability of prime
numbers occurring reduced by a factor of about 2 (e.g. there is a 1 in 4
chance of getting a prime in the number from 1 to 100, a 1 in 6 chance
of a prime in the numbers from 1 to 1,000, a 1 in 8 chance from 1 to
10,000, 1 in 10 from 1 to 100,000, etc). However, he was quite aware
that his method merely yielded an approximation and, as he could not
definitively prove his findings, and kept them secret until much later
in life.
In Gauss’s annus mirabilis of 1796, at just 19 years of age, he
constructed a hitherto unknown regular seventeen-sided figure using only
a ruler and compass, a major advance in this field since the time of Greek
mathematics, formulated his prime number theorem on the distribution of
prime numbers among the integers, and proved that every positive
integer is representable as a sum of at most three triangular numbers.
Although he made contributions in almost all fields of mathematics, number theory was always Gauss’ favourite area, and he asserted that “mathematics is the queen of the sciences, and the theory of numbers is the queen of mathematics”. An example of how Gauss revolutionized number theory can be seen in his work with complex numbers (combinations of real and imaginary numbers).
Gauss gave the first clear exposition of complex numbers and of the
investigation of functions of complex variables in the early 19th
Century. Although imaginary numbers involving i (the imaginary unit, equal to the square root of -1) had been used since as early as the 16th Century to solve equations that could not be solved in any other way, and despite Euler’s ground-breaking work on imaginary and complex numbers in the 18th Century,
there was still no clear picture of how imaginary numbers connected
with real numbers until the early 19th Century. Gauss was not the first
to intepret complex numbers graphically (Jean-Robert Argand produced his
Argand diagrams in 1806, and the Dane Caspar Wessel had described
similar ideas even before the turn of the century), but Gauss was
certainly responsible for popularizing the practice and laos formally
introduced the standard notation a + bi
for complex numbers. As a result, the theory of complex numbers
received a notable expansion, and its full potential began to be
unleashed.
At the age of just 22, he proved what is now known as the Fundamental Theorem of Algebra (although it was not really about algebra). The theorem states that every non-constant single-variable polynomial over the complex numbers has at least one root (although his initial proof was not rigorous, he improved on it later in life). What it also showed was that the field of complex numbers is algebraically "closed" (unlike real numbers, where the solution to a polynomial with real co-efficients can yield a solution in the complex number field).
Then, in 1801, at 24 years of age, he published his book “Disquisitiones Arithmeticae”, which is regarded today as one of the most influential mathematics books ever written, and which laid the foundations for modern number theory. Among many other things, the book contained a clear presentation of Gauss’ method of modular arithmetic, and the first proof of the law of quadratic reciprocity (first conjectured by Euler and Legendre).
For much of his life, Gauss also retained a strong interest in
theoretical astrononomy, and he held the post of Director of the
astronomical observatory in Göttingen for many years. When the planetoid
Ceres was in the process of being identified in the late 17th Century,
Gauss made a prediction of its position which varied greatly from the
predictions of most other astronomers of the time. But, when Ceres was
finally discovered in 1801, it was almost exacly where Gauss had
predicted. Although he did not explain his methods at the time, this was
one of the first applications of the least squares approximation
method, usually attributed to Gauss, although also claimed by the
Frenchman Legendre. Gauss claimed to have done the logarithmic
calculations in his head.
As Gauss’ fame spread, though, and he became known throughout Europe as the go-to man for complex mathematical questions, his character deteriorated and he became increasingly arrogant, bitter, dismissive and unpleasant, rather than just shy. There are many stories of the way in which Gauss had dismissed the ideas of young mathematicians or, in some cases, claimed them as his own.
In the area of probability and statistics, Gauss introduced what is
now known as Gaussian distribution, the Gaussian function and the
Gaussian error curve. He showed how probability could be represented by a
bell-shaped or “normal” curve, which peaks around the mean or expected
value and quickly falls off towards plus/minus infinity, which is basic
to descriptions of statistically distributed data.
He also made ths first systematic study of modular arithmetic - using integer division and the modulus - which now has applications in number theory, abstract algebra, computer science, cryptography, and even in visual and musical art.
While engaged on a rather banal surveying job for the Royal House of Hanover in the years after 1818, Gauss was also looking into the shape of the Earth, and starting to speculate on revolutionary ideas like shape of space itself. This led him to question one of the central tenets of the whole of mathematics, Euclidean geometry, which was clearly premised on a flat, and not a curved, universe. He later claimed to have considered a non-Euclidean geometry (in which Euclid's parallel axiom, for example, does not apply), which was internally consistent and free of contradiction, as early as 1800. Unwilling to court controversy, however, Gauss decided not to pursue or publish any of his avant-garde ideas in this area, leaving the field open to Bolyai and Lobachevsky, although he is still considered by some to be a pioneer of non-Euclidean geometry.
The Hanover survey work also fuelled Gauss' interest in differential geometry (a field of mathematics dealing
with curves and surfaces) and what has come to be known as Gaussian
curvature (an intrinsic measure of curvature, dependent only on how
distances are measured on the surface, not on the way it is embedded in
space). All in all, despite the rather pedestrian nature of his
employment, the responsibilities of caring for his sick mother and the
constant arguments with his wife Minna (who desperately wanted to move
to Berlin), this was a very fruitful period of his academic life, and he
published over 70 papers between 1820 and 1830.
Gauss’ achievements were not limited to pure mathematics, however. During his surveying years, he invented the heliotrope, an instrument that uses a mirror to reflect sunlight over great distances to mark positions in a land survey. In later years, he collaborated with Wilhelm Weber on measurements of the Earth's magnetic field, and invented the first electric telegraph. In recognition of his contributions to the theory of electromagnetism, the international unit of magnetic induction is known as the gauss
BOLYAI AND LOBACHEVSKY
János Bolyai was a Hungarian mathematician who spent most of his life
in a little-known backwater of the Hapsburg Empire, in the wilds of the
Transylvanian mountains of modern-day Romania, far from the mainstream
mathematical communities of Germany, France and England. No original
portrait of Bolyai survives, and the picture that appears in many
encyclopedias and on a Hungarian postage stamp is known to be
unauthentic.
His father and teacher, Farkas Bolyai, was himself an accomplished mathematician and had been a student of the great German mathematician Gauss for a time, but the cantankerous Gauss refused to take on the young prodigy János as a student. So, he was forced to join the army in order to earn a living and support his family, although he persevered with his mathematics in his spare time. He was also a talented linguist, speaking nine foreign languages, including Chinese and Tibetan.
In particular, Bolyai became obsessed with Euclid's
fifth postulate (often referred to as the parallel postulate), a
fundamental principle of geometry for over two millennia, which
essentially states that only one line can be drawn through a given point
so that the line is parallel to a given line that does not contain the
point, along with its corollary that the interior angles of a triangle
sum to 180° or two right angles. In fact, he became obsessed to such an
extent that his father warned him that it may take up all his time and
deprive him of his "health, peace of mind and happiness in life", a
tragic irony given the unfolding of subsequent events.
Bolyai, however, persisted in his quest, and eventually came to the radical conclusion that it was in fact possible to have consistent geometries that were independent of the parallel postulate. In the early 1820s, Bolyai explored what he called “imaginary geometry” (now known as hyperbolic geometry), the geometry of curved spaces on a saddle-shaped plane, where the angles of a triangle did NOT add up to 180° and apparently parallel lines were NOT actually parallel. In curved space, the shortest distance between two points a and b is actually a curve, or geodesic, and not a straight line. Thus, the angles of a triangle in hyperbolic space sum to less than 180°, and two parallel lines in hyperbolic space actually diverge from each other. In a letter to his father, Bolyai marvelled, “Out of nothing I have created a strange new universe”.
Although it is easy to visualize a flat surface and a surface with positive curvature (e.g. a sphere, such as a the Earth), it is impossible to visualize a hyperbolic surface with negative curvature, other than just over a small localized area, where it would look like a saddle or a Pringle. So the very concept of a hyperbolic surface appeared to go against all sense of reality. It certainly represented a radical departure from Euclidean geometry, and the first step along the road which would lead to Einstein’s Theory of Relativity among other applications (although it still fell well short of the multi-dimensional geometry which was to be later realized by Riemann). Between 1820 and 1823, Bolyai prepared, but did not immediately publish, a treatise on a complete system of non-Euclidean geometry.
His work was, however, only published in 1832, and then only a short exposition in the appendix of a textbook by his father. On reading this, Gauss clearly recognized the genius of the younger Bolyai’s ideas, but he refused to encourage the young man, and even tried to claim his ideas as his own. Further disheartened by the news that the Russian mathematician Lobachevski had published something quite similar two years before his own paper, Bolyai became a recluse and gradually went insane. He died in obscurity in 1860. Although he only ever published the 24 pages of the appendix, Bolyai left more than 20,000 pages of mathematical manuscripts when he died (including the development of a rigorous geometric concept of complex numbers as ordered pairs of real numbers).
Completely independent from Bolyai, in the distant provincial Russian
city of Kazan, Nikolai Ivanovich Lobachevsky had also been working,
along very similar lines as Bolyai, to develop a geometry in which Euclid’s
fifth postulate did not apply. His work on hyperbolic geometry was
first reported in 1826 and published in 1830, although it did not have
general circulation until some time later.
This early non-Euclidean geometry is now often referred to as Lobachevskian geometry or Bolyai-Lobachevskian geometry, thus sharing the credit. Gauss’ claims to have originated, but not published, the ideas are difficult to judge in retrospect. Other much earlier claims are credited to the 11th Century Persian mathematician Omar Khayyam, and to the early 18th Century Italian priest Giovanni Saccheri, but their work was much more speculative and inconclusive in nature.
Lobachevsky also died in poverty and obscurity, nearly blind and unable to walk.
Among his other mathematical achievements, largely unknown during his
lifetime, was the development of a method for approximating the roots of
algebraic equations (a method now known as the Dandelin-Gräffe method,
named after two other mathematicians who discovered it independently),
and the definition of a function as a correspondence between two sets of
real numbers (usually credited to Dirichlet, who gave the same
definition independently soon after Lobachevsky)
RIEMANN
Bernhard Riemann was another mathematical giant hailing from northern
Germany. Poor, shy, sickly and devoutly religious, the young Riemann
constantly amazed his teachers and exhibited exceptional mathematical
skills (such as fantastic mental calculation abilities) from an early
age, but suffered from timidity and a fear of speaking in public. He
was, however, given free rein of the school library by an astute
teacher, where he devoured mathematical texts by Legendre and others,
and gradually groomed himself into an excellent mathematician. He also
continued to study the Bible intensively, and at one point even tried to
prove mathematically the correctness of the Book of Genesis.
Although he started studying philology and theology in order to become a priest and help with his family's finances, Riemann's father eventually managed to gather enough money
to send him to study mathematics at the renowned University of
Göttingen in 1846, where he first met, and attended the lectures of, Carl Friedrich Gauss. Indeed, he was one of the very few who benefited from the support and patronage of Gauss,
and he gradually worked his way up the University's hierarchy to become
a professor and, eventually, head of the mathematics department at
Göttingen.
Riemann developed a type of non-Euclidean geometry, different to the hyperbolic geometry of Bolyai and Lobachevsky, which has come to be known as elliptic
geometry. As with hyperbolic geometry, there is no such thing as
parallel lines, and the angles of a triangle do not sum to 180° (in this
case, however, they sum to more than 180º). He went on to develop
Riemannian geometry, which unified and vastly generalized the three
types of geometry, as well as the concept of a manifold or mathematical
space, which generalized the ideas of curves and surfaces.
A turning point in his career occurred in 1852 when, at the age of 26, have gave a lecture on the foundations of geometry and outlined his vision of a mathematics of many different kinds of space, only one of which was the flat, Euclidean space which we appear to inhabit. He also introduced one-dimensional complex manifolds known as Riemann surfaces. Although it was not widely understood at the time, Riemann’s mathematics changed how we look at the world, and opened the way to higher dimensional geometry, a potential which had existed, unrealized, since the time of Descartes.
With his “Riemann metric”, Riemann completely broke away from all the
limitations of 2 and 3 dimensional geometry, even the geometry of
curved spaces of Bolyai and Lobachevsky, and began to think in higher dimensions, extending the differential geometry of surfaces into n
dimensions. His conception of multi-dimensional space (known as
Riemannian space or Riemannian manifold or simply “hyperspace”) enabled
the later development of general relativity, and is at the heart of much
of today’s mathematics, in geometry, number theory and other branches
of mathematics.
He introduced a collection of numbers (known as a tensor) at every point in space, which would describe how much it was bent or curved. For instance, in four spatial dimensions, a collection of ten numbers is needed at each point to describe the properties of the mathematical space or manifold, no matter how distorted it may be.
Riemann’s big breakthrough occurred while working on a function in the complex plane called the Riemann zeta function (an extension of the simpler zeta function first explored by Euler in the previous century). He realized that he could use it to build a kind of 3-dimensional landscape, and furthermore that the contours of that imaginary landscape might be able to unlock the Holy Grail of mathematics, the age-old secret of prime numbers.
Riemann noticed that, at key places, the surface of his 3-dimensional
graph dipped down to height zero (known simply as “the zeroes”) and was
able to show that at least the first ten zeroes inexplicably appeared
to line up in a straight line through the 3-dimensional landscape of the
zeta-function, known as the critical line, where the real part of the
value is equal to ½.
With a huge imaginative leap, Riemann realized that these zeroes had a completely unexpected connection with the way the prime numbers are distributed. It began to seem that they could be used to correct Gauss’ inspired guesswork regarding the number of primes as numbers as one counts higher and higher.
The famous Riemann Hypothesis, which remains unproven, suggests that ALL the zeroes would be on the same straight line. Although he never provided a definitive proof of this hypothesis, Riemann’s work did at least show that the 15-year-old Gauss’ initial approximations of the incidence of prime numbers were perhaps more accurate than even he could have known, and that the primes were in fact distributed over the universe of numbers in a regular, balanced and beautiful way.
The discovery of the Riemann zeta function and the relationship of its zeroes to the prime numbers brought Riemann instant fame when it was published in 1859. He too, though, died young at just 39 years of age, in 1866, and many of his loose papers were accidentally destroyed after his death, so we will never know just how close he was to proving his own hypothesis. Over 150 years later, the Riemann Hypothesis is still considered one of the fundamental questions of number theory, and indeed of all mathematics, and a prize of $1 million has been offered for the final solution
BOOLE
The British mathematician and philosopher George Boole, along with
his near contemporary and countryman Augustus de Morgan, was one of the
few since Leibniz to give any serious thought to logic and its mathematical implications. Unlike Leibniz, though, Boole came to see logic as principally a discipline of mathematics, rather than of philosophy.
His extraordinary mathematical talents did not manifest themselves in early life. He received his early lessons in mathematics from his father, a tradesman with an amateur interest in in mathematics and logic, but his favourite subject at school was classics. He was a quiet, serious and modest young man from a humble working class background, and largely self-taught in his mathematics (he would borrow mathematical journals from his local Mechanics Institute).
It was only at university and afterwards that his mathematical skills began to be fully realized, although, even then, he was all but unknown in his own time, other than for a few insightful but rather abstruse papers on differential equations and the calculus of finite differences. By the age of 34, though, he was well respected enough in his field to be appointed as the first professor of mathematics of Queen's College (now University College) in Cork, Ireland.
But it was his contributions to the algebra of logic which were later to be viewed as immensely important and influential. Boole began to see the possibilities for applying his algebra to the solution of logical problems, and he pointed out a deep analogy between the symbols of algebra and those that can be made to represent logical forms and syllogisms. In fact, his ambitions stretched to a desire to devise and develop a system of algebraic logic that would systematically define and model the function of the human brain. His novel views of logical method were due to his profound confidence in symbolic reasoning, and he speculated on what he called a “calculus of reason” during the 1840s and 1850s.
Determined to find a way to encode logical arguments into a language
that could be manipulated and solved mathematically, he came up with a
type of linguistic algebra, now known as Boolean algebra. The three most
basic operations of this algebra were AND, OR and NOT, which Boole saw
as the only operations necessary to perform comparisons of sets of
things, as well as basic mathematical functions.
Boole’s use of symbols and connectives allowed for the simplification of logical expressions, including such important algebraic identities as: (X or Y) = (Y or X); not(not X) = X; not(X and Y) = (not X) or (not Y); etc.
He also developed a novel approach based on a binary
system, processing only two objects (“yes-no”, “true-false”, “on-off”,
“zero-one”). Therefore, if “true” is represented by 1 and “false” is
represented by 0, and two propositions are both true, then it is
possible under Boolean algebra for 1 + 1 to equal 1 ( the “+” is an
alternative representation of the OR operator)
Despite the standing he had won in the academic community by that time, Boole’s revolutionary ideas were largely criticized or just ignored, until the American logician Charles Sanders Peirce (among others) explained and elaborated on them some years after Boole’s death in 1864.
Almost seventy years later, Claude Shannon made a major breakthrough in realizing that Boole's work could form the basis of mechanisms and processes in the real world, and particularly that electromechanical relay circuits could be used to solve Boolean algebra problems. The use of electrical switches to process logic is the basic concept that underlies all modern electronic digital computers, and so Boole is regarded in hindsight as a founder of the field of computer science, and his work led to the development of applications he could never have imagined
CANTOR
The German Georg Cantor was an outstanding violinist, but an even
more outstanding mathematician. He was born in Saint Petersburg, Russia,
where he lived until he was eleven. Thereafter, the family moved to
Germany, and Cantor received his remaining education at Darmstradt,
Zürich, Berlin and (almost inevitably) Göttingen before marrying and
settling at the University of Halle, where he was to spend the rest of
his career.
He was made full professor at Halle at the age of just 34, a notable accomplishment, but his ambitions to move to a more prestigious university, such as Berlin, were largely thwarted by Leopold Kronecker, a well-established figure within the mathematical community and Cantor's former professor, who fundamentally disagreed with the thrust of Cantor's work.
Cantor’s first ten papers were on number theory, after which he turned his attention to calculus (or analysis as it had become known by this time), solving a difficult open problem on the uniqueness of the representation of a function by trigonometric series. His main legacy, though, is as perhaps the first mathematician to really understand the meaning of infinity and to give it mathematical precision.
Back in the 17th Century, Galileo had tried to confront the idea of infinity and the apparent contradictions thrown up by comparisons of different infinities, but in the end shied away from the problem. He had shown that a one-to-one correspondence could be drawn between all the natural numbers and the squares of all the natural numbers to infinity, suggesting that there were just as many square numbers as integers, even though it was intuitively obvious there were many integers that were were not squares, a concept which came to be known as Galileo’s Paradox. He had also pointed out that two concentric circles must both be comprised of an infinite number of points, even though the larger circle would appear to contain more points. However, Galileo had essentially dodged the issue and reluctantly concluded that concepts like less, equals and greater could only be applied to finite sets of numbers, and not to infinite sets. Cantor, however, was not content with this compromise.
Cantor's starting point was to say that, if it was possible to add 1
and 1, or 25 and 25, etc, then it ought to be possible to add infinity
and infinity. He realized that it was actually possible to add and
subtract infinities, and that beyond what was normally thought of as
infinity existed another, larger infinity, and then other infinities
beyond that. In fact, he showed that there may be infinitely many sets
of infinite numbers - an infinity of infinities - some bigger than
others, a concept which clearly has philosophical, as well as just
mathematical, significance. The sheer audacity of Cantor’s theory set
off a quiet revolution in the mathematical community, and changed
forever the way mathematics is approached.
His first intimations of all this came in the early 1870s when he considered an infinite series of natural numbers (1, 2, 3, 4, 5, ...), and then an infinite series of multiples of ten (10, 20 , 30, 40, 50, ...). He realized that, even though the multiples of ten were clearly a subset of the natural numbers, the two series could be paired up on a one-to-one basis (1 with 10, 2 with 20, 3 with 30, etc) - a process known as bijection - to show that they were the same “sizes” of infinite sets, in that they had the same number of elements.
This clearly also applies to other subsets of the natural numbers, such as the even numbers 2, 4, 6, 8, 10, etc, or the squares 1, 4, 9, 16, 25, etc, and even to the set of negative numbers and integers. In fact, Cantor realized that he could, in the same way, even pair up all the fractions (or rational numbers) with all the whole numbers, thus showing that rational numbers were also the same sort of infinity as the natural numbers, despite the intuitive feeling that there must be more fractions than whole numbers.
However, when Cantor considered an infinite series of decimal numbers, which includes irrational numbers like π, e
and √2, this method broke down. He used several clever arguments (one
being the "diagonal argument" explained in the box on the right) to show
how it was always possible to construct a new decimal number that was
missing from the original list, and so proved that the infinity of
decimal numbers (or, technically, real numbers) was in fact bigger than
the infinity of natural numbers.
He also showed that they were “non-denumerable” or "uncountable" (i.e. contained more elements than could ever be counted), as opposed to the set of rational numbers which he had shown were technically (even if not practically) “denumerable” or "countable". In fact, it can be argued that there are an infinite number of irrational numbers in between each and every rational number. The patternless decimals of irrational numbers fill the "spaces" between the patterns of the rational numbers.
Cantor coined the new word “transfinite” in an attempt to distinguish these various levels of infinite numbers from an absolute infinity, which the religious Cantor effectively equated with God (he saw no contradiction between his mathematics and the traditional concept of God). Although the cardinality (or size) of a finite set is just a natural number indicating the number of elements in the set, he also needed a new notation to describe the sizes of infinite sets, and he used the Hebrew letter aleph (
). He defined 0 (aleph-null or aleph-nought) as the cardinality of the countably infinite set of natural numbers; 1
(aleph-one) as the next larger cardinality, that of the uncountable set
of ordinal numbers; etc. Because of the unique properties of infinite
sets, he showed that 0 + 0 = 0, and also that 0 x 0 = 0.
All of this represented a revolutionary step, and opened up new possibilities in mathematics. However, it also opened up the possibility of other infinities, for instance an infinity - or even many infinities - between the infinity of the whole numbers and the larger infinity of the decimal numbers. This idea is known as the continuum hypothesis, and Cantor believed (but could not actually prove) that there was NO such intermediate infinite set. The continuum hypothesis was one of the 23 important open problems identified by David Hilbert in his famous 1900 Paris lecture, and it remained unproved - and indeed appeared to be unprovable - for almost a century, until the work of Robinson and Matiyasevich in the 1950s and 1960s.
Just as importantly, though, this work of Cantor's between 1874 and
1884 marks the real origin of set theory, which has since become a
fundamental part of modern mathematics, and its basic concepts are used
throughout all the various branches of mathematics. Although the concept
of a set had been used implicitly since the beginnings of mathematics,
dating back to the ideas of Aristotle, this was limited to everyday
finite sets. In contradistinction, the “infinite” was kept quite
separate, and was largely considered a topic for philosophical, rather
than mathematical, discussion. Cantor, however, showed that, just as
there were different finite sets, there could be infinite sets of
different sizes, some of which are countable and some of which are
uncountable.
Throughout the 1880s and 1890s, he refined his set theory, defining well-ordered sets and power sets and introducing the concepts of ordinality and cardinality and the arithmetic of infinite sets. What is now known as Cantor's theorem states generally that, for any set A, the power set of A (i.e. the set of all subsets of A) has a strictly greater cardinality than A itself. More specificially, the power set of a countably infinite set is uncountably infinite.
Despite the central position of set theory in modern mathematics, it was often deeply mistrusted and misunderstood by other mathematicians of the day. One quote, usually attributed to Henri Poincaré, claimed that "later generations will regard Mengenlehre (set theory) as a disease from which one has recovered". Others, however, were quick to see the value and potential of the method, and David Hilbert declared in 1926 that "no one shall expel us from the Paradise that Cantor has created".
Cantor had few other mathematicians with whom he could discuss his ground-breaking work, and most were distinctly unnerved by his contemplation of the infinite. During the 1880s, he encountered resistance, sometimes fierce resistance, from mathematical contemporaries such as his old professor Leopold Kronecker and Henri Poincaré, as well as from philosophers like Ludwig Wittgenstein and even from some Christian theologians, who saw Cantor's work as a challenge to their view of the nature of God. Cantor himself, a deeply religious man, noted some annoying paradoxes thrown up by his own work, but some went further and saw it as the wilful destruction of the comprehensible and logical base on which the whole of mathematics was based.
As he aged, Cantor suffered from more and more recurrences of mental illness, which some have directly linked to his constant contemplation of such complex, abstract and paradoxical concepts. In the last decades of his life, he did no mathematical work at all, but wrote extensively on his two obsessions: that Shakespeare’s plays were actually written by the English philosopher Sir Francis Bacon, and that Christ was the natural son of Joseph of Arimathea. He spent long periods in the Halle sanatorium recovering from attacks of manic depression and paranoia, and it was there, alone in his room, that he finally died in 1918, his great project still unfinished.
POINCARÉ
Paris was a great centre for world mathematics towards the end of the
19th Century, and Henri Poincaré was one of its leading lights in
almost all fields - geometry, algebra, analysis - for which he is
sometimes called the “Last Universalist”.
Even as a youth at the Lycée in Nancy, he showed himself to be a polymath, and he proved to be one of the top students in every topic he studied. He continued to excel after he entered the École Polytechnique to study mathematics in 1873, and, for his doctoral thesis, he devised a new way of studying the properties of differential equations. Beginning in 1881, he taught at the Sorbonne in Paris, where he would spend the rest of his illustrious career. He was elected to the French Academy of Sciences at the young age of 32, became its president in 1906, and was elected to the Académie française in 1909.
Poincaré deliberately cultivated a work habit that has been compared to a bee flying from flower to flower. He observed a strict work regime of 2 hours of work in the morning and two hours in the early evening, with the intervening time left for his subconscious to carry on working on the problem in the hope of a flash of inspiration. He was a great believer in intuition, and claimed that "it is by logic that we prove, but by intuition that we discover".
It was one such flash of inspiration that earned Poincaré a generous prize from the King of Sweden in 1887 for his partial solution to the “three-body problem”, a problem that had defeated mathematicians of the stature of Euler, Lagrange and Laplace. Newton had long ago proved that the paths of two planets orbiting around each other would remain stable, but even the addition of just one more orbiting body to this already simplified solar system resulted in the involvement of as many as 18 different variables (such as position, velocity in each direction, etc), making it mathematically too complex to predict or disprove a stable orbit. Poincaré’s solution to the “three-body problem”, using a series of approximations of the orbits, although admittedly only a partial solution, was sophisticated enough to win him the prize.
But he soon realized that he had actually made a mistake, and that
his simplifications did not indicate a stable orbit after all. In fact,
he realized that even a very small change in his initial conditions
would lead to vastly different orbits. This serendipitous discovery,
born from a mistake, led indirectly to what we now know as chaos theory,
a burgeoning field of mathematics most familiar to the general public
from the common example of the flap of a butterfly’s wings leading to a
tornado on the other side of the world. It was the first indication that
three is the minimum threshold for chaotic behaviour.
Paradoxically, owning up to his mistake only served to enhance Poincaré’s reputation, if anything, and he continued to produce a wide range of work throughout his life, as well as several popular books extolling the importance of mathematics.
Poincaré also developed the science of topology, which Leonhard Euler had heralded with his solution to the famous Seven Bridges of Königsberg problem. Topology is a kind of geometry which involves one-to-one correspondence of space. It is sometimes referred to as “bendy geometry” or “rubber sheet geometry” because, in topology, two shapes are the same if one can be bent or morphed into the other without cutting it. For example, a banana and a football are topologically equivalent, as are a donut (with its hole in the middle) and a teacup (with its handle); but a football and a donut, are topologically different because there is no way to morph one into the other. In the same way, a traditional pretzel, with its two holes is topological different from all of these examples.
In the late 19th Century, Poincaré described all the possible
2-dimensional topological surfaces but, faced with the challenge of
describing the shape of our 3-dimensional universe, he came up with the
famous Poincaré conjecture, which became one of the most important open
questions in mathematics for almost a century. The conjecture looks at a
space that, locally, looks like ordinary 3-dimensional space but is
connected, finite in size and lacks any boundary (technically known as a
closed 3-manifold or 3-sphere). It asserts that, if a loop in that
space can be continuously tightened to a point, in the same way as a
loop drawn on a 2-dimensional sphere can, then the space is just a
three-dimensional sphere. The problem remained unsolved until 2002, when
an extremely complex solution was provided by the eccentric and
reclusive Russian mathematician Grigori Perelman, involving the ways in
which 3-dimensional shapes can be “wrapped up” in higher dimensions.
Poincaré’s work in theoretical physics was also of great significance, and his symmetrical presentation of the Lorentz transformations in 1905 was an important and necessary step in the formulation of Einstein’s theory of special relativity (some even hold that Poincaré and Lorentz were the true discoverers of relativity). He also made important contribution in a whole host of other areas of physics including fluid mechanics, optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory and cosmology
 |
 Logarithms were invented by John Napier, early in the 17th Century |
The invention of the logarithm in the early 17th Century by John Napier (and later improved by Napier and Henry Briggs) contributed to the advance of science, astronomy and mathematics by making some difficult calculations relatively easy. It was one of the most significant mathematical developments of the age, and 17th Century physicists like Kepler and Newton could never have performed the complex calculatons needed for their innovations without it. The French astronomer and
mathematician Pierre Simon Laplace remarked, almost two centuries later, that Napier, by halving the labours of astronomers, had doubled their lifetimes.
The logarithm of a number is the exponent when that number is expressed as a power of 10 (or any other base). It is effectively the inverse of exponentiation. For example, the base 10 logarithm of 100 (usually written log10 100 or lg 100 or just log 100) is 2, because 102 = 100. The value of logarithms arises from the fact that multiplication of two or more numbers is equivalent to adding their logarithms, a much simpler operation. In the same way, division involves the subtraction of logarithms, squaring is as simple as multiplying the logarithm by two (or by three for cubing, etc), square roots requires dividing the logarithm by 2 (or by 3 for cube roots, etc).
Although base 10 is the most popular base, another common base for logarithms is the number e which has a value of 2.7182818... and which has special properties which make it very useful for logarithmic calculations. These are known as natural logarithms, and are written loge or ln. Briggs produced extensive lookup tables of common (base 10) logarithms, and by 1622 William Oughted had produced a logarithmic slide rule, an instrument which became indispensible in technological innovation for the next 300 years.
Napier also improved Simon Stevin's decimal notation and popularized the use of the decimal point, and made lattice multiplication (originally developed by the Persian mathematician Al-Khwarizmi and introduced into Europe by Fibonacci) more convenient with the introduction of “Napier's Bones”, a multiplication tool using a set of numbered rods.
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Graph of the number of digits in the known Mersenne primes |
The Frenchman René Descartes is sometimes considered the first of the modern school of mathematics. His development of analytic geometry and Cartesian coordinates in the mid-17th Century soon allowed the orbits of the planets to be plotted on a graph, as well as laying the foundations for the later development of calculus (and much later multi-dimensional geometry). Descartes is also credited with the first use of superscripts for powers or exponents.
Two other great French mathematicians were close contemporaries of Descartes: Pierre de Fermat and Blaise Pascal. Fermat formulated several theorems which greatly extended our knowlege of number theory, as well as contributing some early work on infinitesimal calculus. Pascal is most famous for Pascal’s Triangle of binomial coefficients, although similar figures had actually been produced by Chinese and Persian mathematicians long before him.
It was an ongoing exchange of letters between Fermat and Pascal that led to the development of the concept of expected values and the field of probability theory. The first published work on probability theory, however, and the first to outline the concept of mathematical expectation, was by the Dutchman Christiaan Huygens in 1657, although it was largely based on the ideas in the letters of the two Frenchmen.
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Desargues’ perspective theorem |
By “standing on the shoulders of giants”, the Englishman Sir Isaac Newton was able to pin down the laws of physics in an unprecedented way, and he effectively laid the groundwork for all of classical mechanics, almost single-handedly. But his contribution to mathematics should never be underestimated, and nowadays he is often considered, along with Archimedes and Gauss, as one of the greatest mathematicians of all time.
Newton and, independently, the German philosopher and mathematician Gottfried Leibniz, completely revolutionized mathematics (not to mention physics, engineering, economics and science in general) by the development of infinitesimal calculus, with its two main operations, differentiation and integration. Newton probably developed his work before Leibniz, but Leibniz published his first, leading to an extended and rancorous dispute. Whatever the truth behind the various claims, though, it is Leibniz’s calculus notation that is the one still in use today, and calculus of some sort is used extensively in everything from engineering to economics to medicine to astronomy.
Both Newton and Leibniz also contributed greatly in other areas of mathematics, including Newton’s contributions to a generalized binomial theorem, the theory of finite differences and the use of infinite power series, and Leibniz’s development of a mechanical forerunner to the computer and the use of matrices to solve linear equations.
However, credit should also be given to some earlier 17th Century mathematicians whose work partially anticipated, and to some extent paved the way for, the development of infinitesimal calculus. As early as the 1630s, the Italian mathematician Bonaventura Cavalieri developed a geometrical approach to calculus known as Cavalieri's principle, or the “method of indivisibles”. The Englishman John Wallis, who systematized and extended the methods of analysis of Descartes and Cavalieri, also made significant contributions towards the development of calculus, as well as originating the idea of the number line, introducing the symbol ∞ for infinity and the term “continued fraction”, and extending the standard notation for powers to include negative integers and rational numbers. Newton's teacher Isaac Barrow is usually credited with the discovery (or at least the first rigorous statrement of) the fundamental theorem of calculus, which essentially showed that integration and differentiation are inverse operations, and he also made complete translations of Euclid into Latin and English
DESCARTES
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René Descartes (1596-1650) |
As a young man, he found employment for a time as a soldier (essentially as a mercenary in the pay of various forces, both Catholic and Protestant). But, after a series of dreams or visions, and after meeting the Dutch philosopher and scientist Isaac Beeckman, who sparked his interest in mathematics and the New Physics, he concluded that his real path in life was the pursuit of true wisdom and science.
Back in France, the young Descartes soon came to the conclusion that the key to philosophy, with all its uncertainties and ambiguity, was to build it on the indisputable facts of mathematics. To pursue his rather heretical ideas further, though, he moved from the restrictions of Catholic France to the more liberal environment of the Netherlands, where he spent most of his adult life, and where he worked on his dream of merging algebra and geometry.
In 1637, he published his ground-breaking philosophical and mathematical treatise "Discours de la méthode" (the “Discourse on Method”), and one of its appendices in particular, "La Géométrie", is now considered a landmark in the history of mathematics. Following on from early movements towards the use of symbolic expressions in mathematics by Diophantus, Al-Khwarizmi and François Viète, "La Géométrie" introduced what has become known as the standard algebraic notation, using lowercase a, b and c for known quantities and x, y and z for unknown quantities. It was perhaps the first book to look like a modern mathematics textbook, full of a's and b's, x2's, etc.
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Cartesian Coordinates |
Any equation can be represented on the plane by plotting on it the solution set of the equation. For example, the simple equation y = x yields a straight line linking together the points (0,0), (1,1), (2,2), (3,3), etc. The equation y = 2x yields a straight line linking together the points (0,0), (1,2), (2,4), (3,6), etc. More complex equations involving x2, x3, etc, plot various types of curves on the plane.
As a point moves along a curve, then, its coordinates change, but an equation can be written to describe the change in the value of the coordinates at any point in the figure. Using this novel approach, it soon became clear that an equation like x2 + y2 = 4, for example, describes a circle; y2 - 16x a curve called a parabola; x2⁄a2 + y2⁄b2 = 1 an ellipse; x2⁄a2 - y2⁄b2 = 1 a hyperbola; etc.
Descartes’ ground-breaking work, usually referred to as analytic geometry or Cartesian geometry, had the effect of allowing the conversion of geometry into algebra (and vice versa). Thus, a pair of simultaneous equations could now be solved either algebraically or graphically (at the intersection of two lines). It allowed the development of Newton’s and Leibniz’s subsequent discoveries of calculus. It also unlocked the possibility of navigating geometries of higher dimensions, impossible to physically visualize - a concept which was to become central to modern technology and physics - thus transforming mathematics forever.
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Descartes' Rule of Signs |
For all his importance in the development of modern mathematics, though, Descartes is perhaps best known today as a philosopher who espoused rationalism and dualism. His philosophy consisted of a method of doubting everything, then rebuilding knowledge from the ground, and he is particularly known for the often-quoted statement “Cogito ergo sum”(“I think, therefore I am”).
He also had an influential rôle in the development of modern physics, a rôle which has been, until quite recently, generally under-appreciated and under-investigated. He provided the first distinctly modern formulation of laws of nature and a conservation principle of motion, made numerous advances in optics and the study of the reflection and refraction of light, and constructed what would become the most popular theory of planetary motion of the late 17th Century. His commitment to the scientific method was met with strident opposition by the church officials of the day.
His revolutionary ideas made him a centre of controversy in his day, and he died in 1650 far from home in Stockholm, Sweden. 13 years later, his works were placed on the Catholic Church's "Index of Prohibited Books"
FERMAT
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Pierre de Fermat (1601-1665) |
Although he showed an early interest in mathematics, he went on study law at Orléans and received the title of councillor at the High Court of Judicature in Toulouse in 1631, which he held for the rest of his life. He was fluent in Latin, Greek, Italian and Spanish, and was praised for his written verse in several languages, and eagerly sought for advice on the emendation of Greek texts.
Fermat's mathematical work was communicated mainly in letters to friends, often with little or no proof of his theorems. Although he himself claimed to have proved all his arithmetic theorems, few records of his proofs have survived, and many mathematicians have doubted some of his claims, especially given the difficulty of some of the problems and the limited mathematical tools available to Fermat.
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Fermat’s Theorem on Sums of Two Squares |
His so-called Little Theorem is often used in the testing of large prime numbers, and is the basis of the codes which protect our credit cards in Internet transactions today. In simple (sic) terms, it says that if we have two numbers a and p, where p is a prime number and not a factor of a, then a multiplied by itself p-1 times and then divided by p, will always leave a remainder of 1. In mathematical terms, this is written: ap-1 = 1(mod p). For example, if a = 7 and p = 3, then 72 ÷ 3 should leave a remainder of 1, and 49 ÷ 3 does in fact leave a remainder of 1.
Fermat identified a subset of numbers, now known as Fermat numbers, which are of the form of one less than 2 to the power of a power of 2, or, written mathematically, 22n + 1. The first five such numbers are: 21 + 3 = 3; 22 + 1 = 5; 24 + 1 = 17; 28 + 1 = 257; and 216 + 1 = 65,537. Interestingly, these are all prime numbers (and are known as Fermat primes), but all the higher Fermat numbers which have been painstakingly identified over the years are NOT prime numbers, which just goes to to show the value of inductive proof in mathematics.
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Fermat’s Last Theorem |
There are clearly many solutions - indeed, an infinite number - when n = 2 (namely, all the Pythagorean triples), but no solution could be found for cubes or higher powers. Tantalizingly, Fermat himself claimed to have a proof, but wrote that “this margin is too small to contain it”. As far as we know from the papers which have come down to us, however, Fermat only managed to partially prove the theorem for the special case of n = 4, as did several other mathematicians who applied themselves to it (and indeed as had earlier mathematicians dating back to Fibonacci, albeit not with the same intent).
Over the centuries, several mathematical and scientific academies offered
substantial prizes for a proof of the theorem, and to some extent it
single-handedly stimulated the development of algebraic number theory in
the 19th and 20th Centuries. It was finally proved for ALL numbers only
in 1995 (a proof usually attributed to British mathematician Andrew
Wiles, although in reality it was a joint effort of several steps
involving many mathematicians over several years). The final proof made
use of complex modern mathematics, such as the modularity theorem for
semi-stable elliptic curves, Galois representations and Ribet’s epsilon
theorem, all of which were unavailable in Fermat’s time, so it seems
clear that Fermat's claim to have solved his last theorem was almost
certainly an exaggeration (or at least a misunderstanding).In addition to his work in number theory, Fermat anticipated the development of calculus to some extent, and his work in this field was invaluable later to Newton and Leibniz. While investigating a technique for finding the centres of gravity of various plane and solid figures, he developed a method for determining maxima, minima and tangents to various curves that was essentially equivalent to differentiation. Also, using an ingenious trick, he was able to reduce the integral of general power functions to the sums of geometric series.
Fermat’s correspondence with his friend Pascal also helped mathematicians grasp a very important concept in basic probability which, although perhaps intuitive to us now, was revolutionary in 1654, namely the idea of equally probable outcomes and expected values
PASCAL
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Blaise Pascal (1623-1662) |
But Pascal was also a mathematician of the first order. At the age of sixteen, he wrote a significant treatise on the subject of projective geometry, known as Pascal's Theorem, which states that, if a hexagon is inscribed in a circle, then the three intersection points of opposite sides lie on a single line, called the Pascal line. As a young man, he built a functional calculating machine, able to perform additions and subtractions, to help his father with his tax calculations.
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The table of binomial coefficients known as Pascal’s Triangle |
Pascal was far from the first to study this triangle. The Persian mathematician Al-Karaji had produced something very similar as early as the 10th Century, and the Triangle is called Yang Hui's Triangle in China after the 13th Century Chinese mathematician, and Tartaglia’s Triangle in Italy after the eponymous 16th Century Italian. But Pascal did contribute an elegant proof by defining the numbers by recursion, and he also discovered many useful and interesting patterns among the rows, columns and diagonals of the array of numbers. For instance, looking at the diagonals alone, after the outside "skin" of 1's, the next diagonal (1, 2, 3, 4, 5,...) is the natural numbers in order. The next diagonal within that (1, 3, 6, 10, 15,...) is the triangular numbers in order. The next (1, 4, 10, 20, 35,...) is the pyramidal triangular numbers, etc, etc. It is also possible to find prime numbers, Fibonacci numbers, Catalan numbers, and many other series, and even to find fractal patterns within it.
Pascal also made the conceptual leap to use the Triangle to help solve problems in probability theory. In fact, it was through his collaboration and correspondence with his French contemporary Pierre de Fermat and the Dutchman Christiaan Huygens on the subject that the mathematical theory of probability was born. Before Pascal, there was no actual theory of probability - notwithstanding Gerolamo Cardano’s early exposition in the 16th Century - merely an understanding (of sorts) of how to compute “chances” in dice and card games by counting equally probable outcomes. Some apparently quite elementary problems in probability had eluded some of the best mathematicians, or given rise to incorrect solutions.
It fell to Pascal (with Fermat's help) to bring together the separate threads of prior knowledge (including Cardano's early work) and to introduce entirely new mathematical techniques for the solution of problems that had hitherto resisted solution. Two such intransigent problems which Pascal and Fermat applied themselves to were the Gambler’s Ruin (determining the chances of winning for each of two men playing a particular dice game with very specific rules) and the Problem of Points (determining how a game's winnings should be divided between two equally skilled players if the game was ended prematurely). His work on the Problem of Points in particular, although unpublished at the time, was highly influential in the unfolding new field.
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Fermat and Pascal’s solution to the Problem of Points |
Pascal then looked for a way of generalizing the problem that would avoid the tedious listing of possibilities, and realized that he could use rows from his triangle of coefficients to generate the numbers, no matter how many tosses of the coin remained. As Fermat needed 2 more points to win the game and Pascal needed 3, he went to the fifth (2 + 3) row of the triangle, i.e. 1, 4, 6, 4, 1. The first 3 terms added together (1 + 4 + 6 = 11) represented the outcomes where Fermat would win, and the last two terms (4 + 1 = 5) the outcomes where Pascal would win, out of the total number of outcomes represented by the sum of the whole row (1 + 4 + 6 +4 +1 = 16).
Pascal and Fermat had grasped through their correspondence a very important concept that, though perhaps intuitive to us today, was all but revolutionary in 1654. This was the idea of equally probable outcomes, that the probability of something occurring could be computed by enumerating the number of equally likely ways it could occur, and dividing this by the total number of possible outcomes of the given situation. This allowed the use of fractions and ratios in the calculation of the likelhood of events, and the operation of multiplication and addition on these fractional probabilities. For example, the probability of throwing a 6 on a die twice is 1⁄6 x 1⁄6 = 1⁄36 ("and" works like multiplication); the probability of throwing either a 3 or a 6 is 1⁄6 + 1⁄6 = 1⁄3 ("or" works like addition).
Later in life, Pascal and his sister Jacqueline strongly identified with the extreme Catholic religious movement of Jansenism. Following the death of his father and a "mystical experience" in late 1654, he had his "second conversion" and abandoned his scientific work completely, devoting himself to philosophy and theology. His two most famous works, the "Lettres provinciales" and the "Pensées", date from this period, the latter left incomplete at his death in 1662. They remain Pascal’s best known legacy, and he is usually remembered today as one of the most important authors of the French Classical Period and one of the greatest masters of French prose, much more than for his contributions to mathematics
NEWTON
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Sir Isaac Newton (1643-1727) |
Physicist, mathematician, astronomer, natural philosopher, alchemist and theologian, Newton is considered by many to be one of the most influential men in human history. His 1687 publication, the "Philosophiae Naturalis Principia Mathematica" (usually called simply the "Principia"), is considered to be among the most influential books in
the history of science, and it dominated the scientific view of the physical universe for the next three centuries.Although largely synonymous in the minds of the general public today with gravity and the story of the apple tree, Newton remains a giant in the minds of mathematicians everywhere (on a par with the all-time greats like Archimedes and Gauss), and he greatly influenced the subsequent path of mathematical development.
Over two miraculous years, during the time of the Great Plague of 1665-6, the young Newton developed a new theory of light, discovered and quantified gravitation, and pioneered a revolutionary new approach to mathematics: infinitesimal calculus. His theory of calculus built on earlier work by his fellow Englishmen John Wallis and Isaac Barrow, as well as on work of such Continental mathematicians as René Descartes, Pierre de Fermat, Bonaventura Cavalieri, Johann van Waveren Hudde and Gilles Personne de Roberval. Unlike the static geometry of the Greeks, calculus allowed mathematicians and engineers to make sense of the motion and dynamic change in the changing world around us, such as the orbits of planets, the motion of fluids, etc.
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Differentiation (derivative) approximates the slope of a curve as the interval approaches zero |
Intuitively, the slope at a particular point can be approximated by taking the average slope (“rise over run”) of ever smaller segments of the curve. As the segment of the curve being considered approaches zero in size (i.e. an infinitesimal change in x), then the calculation of the slope approaches closer and closer to the exact slope at a point (see image at right).
Without going into too much complicated detail, Newton (and his contemporary Gottfried Leibniz independently) calculated a derivative function f ‘(x) which gives the slope at any point of a function f(x). This process of calculating the slope or derivative of a curve or function is called differential calculus or differentiation (or, in Newton’s terminology, the “method of fluxions” - he called the instantaneous rate of change at a particular point on a curve the "fluxion", and the changing values of x and y the "fluents"). For instance, the derivative of a straight line of the type f(x) = 4x is just 4; the derivative of a squared function f(x) = x2 is 2x; the derivative of cubic function f(x) = x3 is 3x2, etc. Generalizing, the derivative of any power function f(x) = xr is rxr-1. Other derivative functions can be stated, according to certain rules, for exponential and logarithmic functions, trigonometric functions such as sin(x), cos(x), etc, so that a derivative function can be stated for any curve without discontinuities. For example, the derivative of the curve f(x) = x4 - 5p3 + sin(x2) would be f ’(x) = 4x3 - 15x2 + 2xcos(x2).
Having established the derivative function for a particular curve, it is then an easy matter to calcuate the slope at any particular point on that curve, just by inserting a value for x. In the case of a time-distance graph, for example, this slope represents the speed of the object at a particular point.
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Integration approximates the area under a curve as the size of the samples approaches zero |
The integral of a curve can be thought of as the formula for calculating the area bounded by the curve and the x axis between two defined boundaries. For example, on a graph of velocity against time, the area “under the curve” would represent the distance travelled. Essentially, integration is based on a limiting procedure which approximates the area of a curvilinear region by breaking it into infinitesimally thin vertical slabs or columns. In the same way as for differentiation, an integral function can be stated in general terms: the integral of any power f(x) = xr is xr+1⁄r+1, and there are other integral functions for exponential and logarithmic functions, trigonometric functions, etc, so that the area under any continuous curve can be obtained between any two limits.
Newton chose not to publish his revolutionary mathematics straight away, worried about being ridiculed for his unconventional ideas, and contented himself with circulating his thoughts among friends. After all, he had many other interests such as philosophy, alchemy and his work at the Royal Mint. However, in 1684, the German Leibniz published his own independent version of the theory, whereas Newton published nothing on the subject until 1693. Although the Royal Society, after due deliberation, gave credit for the first discovery to Newton (and credit for the first publication to Leibniz), something of a scandal arose when it was made public that the Royal Society’s subsequent accusation of plagiarism against Leibniz was actually authored by none other Newton himself, causing an ongoing controversy which marred the careers of both men.
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Newton's Method for approximating the roots of a curve by successive interations after an initial guess |
In 1687, Newton published his “Principia” or “The Mathematical Principles of Natural Philosophy”, generally recognized as the greatest scientific book ever written. In it, he presented his theories of motion, gravity and mechanics, explained the eccentric orbits of comets, the tides and their variations, the precession of the Earth's axis and the motion of the Moon.
Later in life, he wrote a number of religious tracts dealing with the literal interpretation of the Bible, devoted a great deal
of time to alchemy, acted as Member of Parliament for some years, and
became perhaps the best-known Master of the Royal Mint in 1699, a
position he held until his death in 1727. In 1703, he was made President
of the Royal Society and, in 1705, became the first scientist ever to
be knighted. Mercury poisoning from his alchemical pursuits perhaps
explained Newton's eccentricity in later life, and possibly also his
eventual deathLEIBNIZ
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Gottfried Leibniz (1646-1716) |
But, between his work on philosophy and logic and his day job as a politician and representative of the royal house of Hanover, Leibniz still found time to work on mathematics. He was perhaps the first to explicitly employ the mathematical notion of a function to denote geometric concepts derived from a curve, and he developed a system of infinitesimal calculus, independently of his contemporary Sir Isaac Newton. He also revived the ancient method of solving equations using matrices, invented a practical calculating machine and pioneered the use of the binary
system.Like Newton, Leibniz was a member of the Royal Society in London, and was almost certainly aware of Newton’s work on calculus. During the 1670s (slightly later than Newton’s early work), Leibniz developed a very similar theory of calculus, apparently completely independently. Within the short period of about two months he had developed a complete theory of differential calculus and integral calculus (see the section on Newton for a brief description and explanation of the development of calculus).
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Leibniz’s and Newton’s notation for Calculus |
Ironically, it was Leibniz’s mathematics that eventually triumphed, and his notation and his way of writing calculus, not Newton’s more clumsy notation, is the one still used in mathematics today.
In addition to calculus, Leibniz re-discovered a method of arranging linear equations into an array, now called a matrix, which could then be manipulated to find a solution. A similar method had been pioneered by Chinese mathematicians almost two millennia earlier, but had long fallen into disuse. Leibniz paved the way for later work on matrices and linear algebra by Carl Friedrich Gauss. He also introduced notions of self-similarity and the principle of continuity which foreshadowed an area of mathematics which would come to be called topology.
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Binary Number System |
system and was capable of multiplying, dividing and even extracting roots, a great improvement on Pascal’s rudimentary adding machine and a true forerunner of the computer. He is usually credited with the early development of the binary
number system (base 2 counting, using only the digits 0 and 1),
although he himself was aware of similar ideas dating back to the I
Ching of Ancient China. Because of the ability of binary
to be represented by the two phases "on" and "off", it would later
become the foundation of virtually all modern computer systems, and
Leibniz's documentation was essential in the development process.Leibniz is also often considered the most important logician between Aristotle in Ancient Greece and George Boole and Augustus De Morgan in the 19th Century. Even though he actually published nothing on formal logic in his lifetime, he enunciated in his working drafts the principal properties of what we now call conjunction, disjunction, negation, identity, set inclusion and the empty set.
MATHEMATICS
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Calculus of variations |
The period was dominated, though, by one family, the Bernoulli’s of Basel in Switzerland, which boasted two or three generations of exceptional mathematicians, particularly the brothers, Jacob and Johann. They were largely responsible for further developing Leibniz’s infinitesimal calculus - paricularly through the generalization and extension of calculus known as the "calculus of variations" - as well as Pascal and Fermat’s probability and number theory.
Basel was also the home town of the greatest of the 18th Century mathematicians, Leonhard Euler, although, partly due to the difficulties in getting on in a city dominated by the Bernoulli family, Euler spent most of his time abroad, in Germany and St. Petersburg, Russia. He excelled in all aspects of mathematics, from geometry to calculus to trigonometry to algebra to number theory, and was able to find unexpected links between the different fields. He proved numerous theorems, pioneered new methods, standardized mathematical notation and wrote many influential textbooks throughout his long academic life.
In a letter to Euler in 1742, the German mathematician Christian Goldbach proposed the Goldbach Conjecture, which states that every even integer greater than 2 can be expressed as the sum of two primes (e.g. 4 = 2 + 2; 8 = 3 + 5; 14 = 3 + 11 = 7 + 7; etc) or, in another equivalent version, every integer greater than 5 can be expressed as the sum of three primes. Yet another version is the so-called “weak” Goldbach Conjecture, that all odd numbers greater than 7 are the sum of three odd primes. They remain among the oldest unsolved problems in number theory (and in all of mathematics), although the weak form of the conjecture appears to be closer to resolution than the strong one. Goldbach also proved other theorems in number theory such as the Goldbach-Euler Theorem on perfect powers.
Despite Euler’s and the Bernoullis’ dominance of 18th Century mathematics, many of the other important mathematicians were from France. In the early part of the century, Abraham de Moivre is perhaps best known for de Moivre's formula, (cosx + isinx)n = cos(nx) + isin(nx), which links complex numbers and trigonometry. But he also generalized Newton’s famous binomial theorem into the multinomial theorem, pioneered the development of analytic geometry, and his work on the normal distribution (he gave the first statement of the formula for the normal distribution curve) and probability theory were of great importance.
France became even more prominent towards the end of the century, and a handful of late 18th Century French mathematicians in particular deserve mention at this point, beginning with “the three L’s”.
Joseph Louis Lagrange collaborated with Euler in an important joint work on the calculus of variation, but he also contributed to differential equations and number theory, and he is usually credited with originating the theory of groups, which would become so important in 19th and 20th Century mathematics. His name is given an early theorem in group theory, which states that the number of elements of every sub-group of a finite group divides evenly into the number of elements of the original finite group.
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Lagrange’s Mean value Theorem |
the most comprehensive treatment of classical mechanics since Newton, and formed a basis for the development of mathematical physics in the 19th Century. Pierre-Simon Laplace, sometimes referred to as “the French Newton”, was an important mathematician and astronomer, whose monumental work “Celestial Mechanics” translated the geometric study of classical mechanics to one based on calculus, opening up a much broader range of problems. Although his early work was mainly on differential equations and finite differences, he was already starting to think about the mathematical and philosophical concepts of probability and statistics in the 1770s, and he developed his own version of the so-called Bayesian interpretation of probability independently of Thomas Bayes. Laplace is well known for his belief in complete scientific determinism, and he maintained that there should be a set of scientific laws that would allow us - at least in principle - to predict everything about the universe and how it works.
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The first six Legendre polynomials (solutions to Legendre’s differential equation) |
functions) was only brought to perfection - or at least to general notice - by others, particularly Gauss. His “Elements of Geometry”, a re-working of Euclid’s
book, became the leading geometry textbook for almost 100 years, and
his extremely accurate measurement of the terrestrial meridian inspired
the creation, and almost universal adoption, of the metric system of
measures and weights.Yet another Frenchman, Gaspard Monge was the inventor of descriptive geometry, a clever method of representing three-dimensional objects by projections on the two-dimensional plane using a specific set of procedures, a technique which would later become important in the fields of engineering, architecture and design. His orthographic projection became the graphical method used in almost all modern mechanical drawing.
After many centuries of increasingly accurate approximations, Johann Lambert, a Swiss mathematician and prominent astronomer, finally provided a rigorous proof in 1761 that π is irrational, i.e. it can not be expressed as a simple fraction using integers only or as a terminating or repeating decimal. This definitively proved that it would never be possible to calculate it exactly, although the obsession with obtaining more and more accurate approximations continues to this day. (Over a hundred years later, in 1882, Ferdinand von Lindemann would prove that π is also transcendental, i.e. it cannot be the root of any polynomial equation with rational coefficients). Lambert was also the first to introduce hyperbolic functions into trigonometry and made some prescient conjectures regarding non-Euclidean space and the properties of hyperbolic triangles
BERNOULLI BROTHERS
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Jacob (1654-1705) and Johann Bernoulli (1667-1748) |
The Bernoulli family was a prosperous family of traders
and scholars from the free city of Basel in Switzerland, which at that
time was the great commercial hub of central Europe.The brothers, Jacob
and Johann Bernoulli, however, flouted their father's wishes for them to
take over the family spice business or to enter respectable professions
like medicine or the ministry, and began studying mathematics together.
After Johann graduated from Basel University, the two developed a rather jealous and competitive relationship. Johann in particular was jealous of the elder Jacob's position as professor at Basel University, and the two often attempted to outdo each other. After Jacob's early death from tuberculosis, Johann took over his brother's position, one of his young students being the great Swiss mathematician Leonhard Euler. However, Johann merely shifted his jealousy toward his own talented son, Daniel (at one point, Johann published a book based on Daniel's work, even changing the date to make it look as though his book had been published before his son's).
Johann received a taste of his own medicine, though, when his student Guillaume de l'Hôpital published a book in
his own name consisting almost entirely of Johann's lectures, including
his now famous rule about 0 ÷ 0 (a problem which had dogged
mathematicians since Brahmagupta's initial work on the rules for dealing
with zero back in the 7th Century). This showed that 0 ÷ 0 does not
equal zero, does not equal 1, does not equal infinity, and is not even
undefined, but is "indeterminate" (meaning it could equal any number).
The rule is still usually known as l'Hôpital's Rule, and not Bernoulli's
Rule.
Despite their competitive and combative personal relationship, though, the brothers both had a clear aptitude for mathematics at a high level, and constantly challenged and inspired each other. They established an early correspondence with Gottfried Leibniz, and were among the first mathematicians to not only study and understand infinitesimal calculus but to apply it to various problems. They became instrumental in disseminating the newly-discovered knowledge of calculus, and helping to make it the cornerstone of mathematics it has become today.
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The Bernoulli’s first derived the brachistrochrone curve, using his calculus of variation method |
This application was an example of the “calculus of variations”, a generalization of infinitesimal calculus that the Bernoulli brothers developed together, and has since proved useful in fields as diverse as engineering, financial
investment, architecture and construction, and even space travel.
Johann also derived the equation for a catenary curve, such as that
formed by a chain hanging between two posts, a problem presented to him
by his brother Jacob.
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Bernoulli Numbers |
Jacob Bernoulli also discovered the appropximate value of the irrational number e while exploring the compound interest on loans. When compounded at 100% interest annually, $1.00 becomes $2.00 after one year; when compounded semi-annually it ppoduces $2.25; compounded quarterly $2.44; monthly $2.61; weekly $2.69; daily $2.71; etc. If it were to be compounded continuously, the $1.00 would tend towards a value of $2.7182818... after a year, a value which became known as e. Alegbraically, it is the value of the infinite series (1 + 1⁄1)1.(1 + 1⁄2)2.(1 + 1⁄3)3.(1 + 1⁄4)4...
Johann’s sons Nicolaus, Daniel and Johann II, and even his grandchildren Jacob II and Johann III, were all accomplished mathematicians and teachers. Daniel Bernoulli, in particular, is well known for his work on fluid mechanics (especially Bernoulli’s Principle on the inverse relationship between the speed and pressure of a fluid or gas), as much as for his work on probability and statistics
EULER
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Leonhard Euler (1707-1783) |
Despite a long life and thirteen children, Euler had more than his fair share of tragedies and deaths, and even his blindness later in life did not slow his prodigious output - his collected works comprise nearly 900 books and, in the year 1775, he is said to have produced on average one mathematical paper every week - as he compensated for it with his mental calculation skills and photographic memory (for example, he could repeat the Aeneid of Virgil from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last).
Today, Euler is considered one of the greatest mathematicians of all time. His interests covered almost all aspects of mathematics, from geometry to calculus to trigonometry to algebra to number theory, as well as optics, astronomy, cartography, mechanics, weights
and measures and even the theory of music.
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Mathematical notation created or popularized by Euler |
He even managed to combine several of these together in an amazing feat of mathematical alchemy to produce one of the most beautiful of all mathematical equations, eiπ = -1, sometimes known as Euler’s Identity. This equation combines arithmetic, calculus, trigonometry and complex analysis into what has been called "the most remarkable formula in mathematics", "uncanny and sublime" and "filled with cosmic beauty", among other descriptions. Another such discovery, often known simply as Euler’s Formula, is eix = cosx + isinx. In fact, in a recent poll of mathematicians, three of the top five most beautiful formulae of all time were Euler’s. He seemed to have an instinctive ability to demonstrate the deep relationships between trigonometry, exponentials and complex numbers.
The discovery that initially sealed Euler’s reputation was announced in 1735 and concerned the calculation of infinite sums. It was called the Basel problem after the Bernoulli’s had tried and failed to solve it, and asked what was the precise sum of the of the reciprocals of the squares of all the natural numbers to infinity i.e. 1⁄12 + 1⁄22 + 1⁄32 + 1⁄42 ... (a zeta function using a zeta constant of 2). Euler’s friend Daniel Bernoulli had estimated the sum to be about 13⁄5, but Euler’s superior method yielded the exact but rather unexpected result of π2⁄6. He also showed that the infinite series was equivalent to an infinite product of prime numbers, an identity which would later inspire Riemann’s investigation of complex zeta functions.
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The Seven Bridges of Königsberg Problem |
In fact, Euler proved that the problem has no solution, but in doing so he made the important conceptual leap of pointing out that the choice of route within each landmass is irrelevant and the only important feature is the sequence of bridges crossed. This allowed him to reformulate the problem in abstract terms, replacing each land mass with an abstract node and each bridge with an abstract connection. This resulted in a mathematical structure called a “graph”, a pictorial representation made up of points (vertices) connected by non-intersecting curves (arcs), which may be distorted in any way without changing the graph itself. In this way, Euler was able to deduce that, because the four land masses in the original problem are touched by an odd number of bridges, the existence of a walk
traversing each bridge once only inevitably leads to a contradiction.
If Königsberg had had one fewer bridges, on the other hand, with an even
number of bridges leading to each piece of land, then a solution would
have been possible.
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The Euler Characteristic |
MATHEMATICS
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Approximation of a periodic function by the Fourier Series |
After the French Revolution, Napoleon emphasized the practical usefulness of mathematics and his reforms and military ambitions gave French mathematics a big boost, as exemplified by “the three L’s”, Lagrange, Laplace and Legendre (see the section on 18th Century Mathematics), Fourier and Galois.
Joseph Fourier's study, at the beginning of the 19th Century, of infinite sums in which the terms are trigonometric functions were another important advance in mathematical analysis. Periodic functions that can be expressed as the sum of an infinite series of sines and cosines are known today as Fourier Series, and they are still powerful tools in pure and applied mathematics. Fourier (following Leibniz, Euler, Lagrange and others) also contributed towards defining exactly what is meant by a function, although the definition that is found in texts today - defining it in terms of a correspondence between elements of the domain and the range - is usually attributed to the 19th Century German mathematician Peter Dirichlet.
In 1806, Jean-Robert Argand published his paper on how complex numbers (of the form a + bi, where i is √-1) could be represented on geometric diagrams and manipulated using trigonometry and vectors. Even though the Dane Caspar Wessel had produced a very similar paper at the end of the 18th Century, and even though it was Gauss who popularized the practice, they are still known today as Argand Diagrams.
The Frenchman Évariste Galois proved in the late 1820s that there is no general algebraic method for solving polynomial equations of any degree greater than four, going further than the Norwegian Niels Henrik Abel who had, just a few years earlier, shown the impossibility of solving quintic equations, and breaching an impasse which had existed for centuries. Galois' work also laid the groundwork for further developments such as the beginnings of the field of abstract algebra, including areas like algebraic geometry, group theory, rings, fields, modules, vector spaces and non-commutative algebra.
Germany, on the other hand, under the influence of the great educationalist Wilhelm von Humboldt, took a rather different approach, supporting pure mathematics for its own sake, detached from the demands of the state and military. It was in this environment that the young German prodigy Carl Friedrich Gauss, sometimes called the “Prince of Mathematics”, received his education at the prestigious University of Göttingen. Some of Gauss’ ideas were a hundred years ahead of their time, and touched on many different parts of the mathematical world, including geometry, number theory, calculus, algebra and probability. He is widely regarded as one of the three greatest mathematicians of all times, along with Archimedes and Newton.
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Euclidean, hyperbolic and elliptic geometry |
The German Bernhard Riemann worked on a different kind of non-Euclidean geometry called elliptic
geometry, as well as on a generalized theory of all the different types of geometry. Riemann,
however, soon took this even further, breaking away completely from all
the limitations of 2 and 3 dimensional geometry, whether flat or
curved, and began to think in higher dimensions. His exploration of the
zeta function in multi-dimensional complex numbers revealed an
unexpected link with the distribution of prime numbers, and his famous
Riemann Hypothesis, still unproven after 150 years, remains one of the
world’s great unsolved mathematical mysteries and the testing ground for
new generations of mathematicians.British mathematics also saw something of a resurgence in the early and mid-19th century. Although the roots of the computer go back to the geared calculators of Pascal and Leibniz in the 17th Century, it was Charles Babbage in 19th Century England who designed a machine that could automatically perform computations based on a program of instructions stored on cards or tape. His large "difference engine" of 1823 was able to calculate logarithms and trigonometric functions, and was the true forerunner of the modern electronic computer. Although never actually built in his lifetime, a machine was built almost 200 years later to his specifications and worked perfectly. He also designed a much more sophisticated machine he called the "analytic engine", complete with punched cards, printer and computational abilities commensurate with modern computers.
Another 19th Century Englishman, George Peacock, is usually credited with the invention of symbolic algebra, and the extension of the scope of algebra beyond the ordinary systems of numbers. This recognition of the possible existence of non-arithmetical algebras was an important stepping stone toward future developments in abstract algebra.
In the mid-19th Century, the British mathematician George Boole devised an algebra (now called Boolean algebra or Boolean logic), in which the only operators were AND, OR and NOT, and which could be applied to the solution of logical problems and mathematical functions. He also described a kind of binary
system which used just two objects, "on" and "off" (or "true" and
"false", 0 and 1, etc), in which, famously, 1 + 1 = 1. Boolean algebra
was the starting point of modern mathematical logic and ultimately led
to the development of computer science.
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Hamilton’s quaternion |
The Englishman Arthur Cayley extended Hamilton's quaternions and developed the octonions. But Cayley was one of the most prolific mathematicians in history, and was a pioneer of modern group theory, matrix algebra, the theory of higher singularities, and higher dimensional geometry (anticipating the later ideas of Klein), as well as the theory of invariants.
Throughout the 19th Century, mathematics in general became ever more complex and abstract. But it also saw a re-visiting of some older methods and an emphasis on mathematical rigour. In the first decades of the century, the Bohemian priest Bernhard Bolzano was one of the earliest mathematicians to begin instilling rigour into mathematical analysis, as well as giving the first purely analytic proof of both the fundamental theorem of algebra and the intermediate value theorem, and early consideration of sets (collections of objects defined by a common property, such as "all the numbers greater than 7" or "all right triangles", etc). When the German mathematician Karl Weierstrass discovered the theoretical existence of a continuous function having no derivative (in other words, a continuous curve possessing no tangent at any of its points), he saw the need for a rigorous “arithmetization” of calculus, from which all the basic concepts of analysis could be derived.
Along with Riemann and, particularly, the Frenchman Augustin-Louis Cauchy, Weierstrass completely reformulated calculus in an even more rigorous fashion, leading to the development of mathematical analysis, a branch of pure mathematics largely concerned with the notion of limits (whether it be the limit of a sequence or the limit of a function) and with the theories of differentiation, integration, infinite series and analytic functions. In 1845, Cauchy also proved Cauchy's theorem, a fundamental theorem of group theory, which he discovered while examining permutation groups. Carl Jacobi also made important contributions to analysis, determinants and matrices, and especially his theory of periodic functions and elliptic
functions and their relation to the elliptic theta function.
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Non-orientable surfaces with no identifiable "inner" and "outer" sides |
Felix Klein also pursued more developments in non-Euclidean geometry, include the Klein bottle, a one-sided closed surface which cannot be embedded in three-dimensional Euclidean space, only in four or more dimensions. It can be best visualized as a cylinder looped back through itself to join with its other end from the "inside". Klein’s 1872 Erlangen Program, which classified geometries by their underlying symmetry groups (or their groups of transformations), was a hugely influential synthesis of much of the mathematics of the day, and his work was very important in the later development of group theory and function theory.
The Norwegian mathematician Marius Sophus Lie also applied algebra to the study of geometry. He largely created the theory of continuous symmetry, and applied it to the geometric theory of differential equations by means of continuous groups of transformations known as Lie groups.
In an unusual occurrence in 1866, an unknown 16-year old Italian, Niccolò Paganini, discovered the second smallest pair of amicable numbers (1,184 and 1210), which had been completely overlooked by some of the greatest mathematicians in history (including Euler, who had identified over 60 such numbers in the 18th Century, some of them huge).
In the later 19th Century, Georg Cantor established the first foundations of set theory, which enabled the rigorous treatment of the notion of infinity, and which has since become the common language of nearly all mathematics. In the face of fierce resistance from most of his contemporaries and his own battle against mental illness, Cantor explored new mathematical worlds where there were many different infinities, some of which were larger than others.
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Venn diagram |
Building on Riemann’s deep ideas on the distribution of prime numbers, the year 1896 saw two independent proofs of the asymptotic law of the distribution of prime numbers (known as the Prime Number Theorem), one by Jacques Hadamard and one by Charles de la Vallée Poussin, which showed that the number of primes occurring up to any number x is asymptotic to (or tends towards) x⁄log x.
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Minkowski space-time |
Gottlob Frege’s 1879 “Begriffsschrift” (roughly translated as “Concept-Script”) broke new ground in the field of logic, including a rigorous treatment of the ideas of functions and variables. In his attempt to show that mathematics grows out of logic, he devised techniques that took him far beyond the logical traditions of Aristotle (and even of George Boole). He was the first to explicitly introduce the notion of variables in logical statements, as well as the notions of quantifiers, universals and existentials. He extended Boole's "propositional logic" into a new "predicate logic" and, in so doing, set the stage for the radical advances of Giuseppe Peano, Bertrand Russell and David Hilbert in the early 20th Century.
Henri Poincaré came to prominence in the latter part of the 19th Century with at least a partial solution to the “three body problem”, a deceptively simple problem which had stubbornly resisted resolution since the time of Newton, over two hundred years earlier. Although his solution actually proved to be erroneous, its implications led to the early intimations of what would later become known as chaos theory. In between his important work in theoretical physics, he also greatly extended the theory of mathematical topology, leaving behind a knotty problem known as the Poincaré conjecture which remined unsolved until 2002.
Poincaré was also an engineer and a polymath, and perhaps the last of the great mathematicians to adhere to an older conception of mathematics, which championed a faith in human intuition over rigour and formalism. He is sometimes referred to as the “Last Univeralist” as he was perhaps the last mathematician able to shine in almost all of the various aspects of what had become by now a huge, encyclopedic and incredibly complex subject. The 20th Century would belong to the specialists
GALOIS
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Évariste Galois (1811-1832) |
Despite his lacklustre performance at school (he twice failed entrance exams to the École Polytechnique), the young Galois devoured the work of Legendre and Lagrange in his spare time. At the tender age of 17, he began making fundamental discoveries in the theory of polynomial equations (equations constructed from variables and constants, using only the operations of addition, subtraction, multiplication and non-negative whole-number exponents, such as x2 - 4x + 7 = 0). He effectively proved that there can be no general formula for solving quintic equations (polynomials including a term of x5), just as the young Norwegian Niels Henrik Abel had a few years earlier, although by a different method. But he was also able to prove the more general, and more powerful, idea that there is no general algebraic method for solving polynomial equations of any degree greater than four.
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An example of Galois’ rather undisciplined notes |
Galois’ breakthrough in turn led to definitive proofs (or rather disproofs) later in the century of the so-called “Three Classical Problems” problems which had been first formulated by Plato and others back in ancient Greece: the doubling of the cube and the trisection of an angle (both were proved impossible in 1837), and the squaring of the circle (also proved impossible, in 1882).
Galois was a hot-headed political firebrand (he was arrested several times for political acts), and his political affiliations and activities as a staunch republican during the rule of Louis-Philippe continually distracted him from his mathematical work. He was killed in a duel in 1832, under rather shady circumstances, but he had spent the whole of the previous night outlining his mathematical ideas in a detailed letter to his friend Auguste Chevalier, as though convinced of his impending death.
Ironically, his young contemporary Abel also had a promising career cut short. He died in poverty of tubercolosis at the age of just 26, although his legacy lives on in the term “abelian” (usually written with a small "a"), which has since become commonplace in discussing concepts such as the abelian group, abelian category and abelian variety
GAUSS
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Carl Friedrich Gauss (1777-1855) |
Gauss was a child prodigy. There are many anecdotes concerning his precocity as a child, and he made his first ground-breaking mathematical discoveries while still a teenager.
At just three years old, he corrected an error in his father payroll calculations, and he was looking after his father’s accounts on a regular basis by the age of 5. At the age of 7, he is reported to have amazed his teachers by summing the integers from 1 to 100 almost instantly (having quickly spotted that the sum was actually 50 pairs of numbers, with each pair summing to 101, total 5,050). By the age of 12, he was already attending gymnasium and criticizing Euclid’s geometry.
Although his family was poor and working class, Gauss' intellectual abilities attracted the attention of the Duke of Brunswick, who sent him to the Collegium Carolinum at 15, and then to the prestigious University of Göttingen (which he attended from 1795 to 1798). It was as a teenager attending university that Gauss discovered (or independently rediscovered) several important theorems.
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Graphs of the density of prime numbers |
the minds of the best mathematicians since ancient times. Although the
occurrence of prime numbers appeared to be almost competely random,
Gauss approached the problem from a different angle by graphing the
incidence of primes as the numbers increased. He noticed a rough pattern
or trend: as the numbers increased by 10, the probability of prime
numbers occurring reduced by a factor of about 2 (e.g. there is a 1 in 4
chance of getting a prime in the number from 1 to 100, a 1 in 6 chance
of a prime in the numbers from 1 to 1,000, a 1 in 8 chance from 1 to
10,000, 1 in 10 from 1 to 100,000, etc). However, he was quite aware
that his method merely yielded an approximation and, as he could not
definitively prove his findings, and kept them secret until much later
in life.
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17-sided heptadecagon constructed by Gauss |
Although he made contributions in almost all fields of mathematics, number theory was always Gauss’ favourite area, and he asserted that “mathematics is the queen of the sciences, and the theory of numbers is the queen of mathematics”. An example of how Gauss revolutionized number theory can be seen in his work with complex numbers (combinations of real and imaginary numbers).
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Representation of complex numbers |
At the age of just 22, he proved what is now known as the Fundamental Theorem of Algebra (although it was not really about algebra). The theorem states that every non-constant single-variable polynomial over the complex numbers has at least one root (although his initial proof was not rigorous, he improved on it later in life). What it also showed was that the field of complex numbers is algebraically "closed" (unlike real numbers, where the solution to a polynomial with real co-efficients can yield a solution in the complex number field).
Then, in 1801, at 24 years of age, he published his book “Disquisitiones Arithmeticae”, which is regarded today as one of the most influential mathematics books ever written, and which laid the foundations for modern number theory. Among many other things, the book contained a clear presentation of Gauss’ method of modular arithmetic, and the first proof of the law of quadratic reciprocity (first conjectured by Euler and Legendre).
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Line of best fit by Gauss’ least squares method |
As Gauss’ fame spread, though, and he became known throughout Europe as the go-to man for complex mathematical questions, his character deteriorated and he became increasingly arrogant, bitter, dismissive and unpleasant, rather than just shy. There are many stories of the way in which Gauss had dismissed the ideas of young mathematicians or, in some cases, claimed them as his own.
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Gaussian, or normal, probability curve |
He also made ths first systematic study of modular arithmetic - using integer division and the modulus - which now has applications in number theory, abstract algebra, computer science, cryptography, and even in visual and musical art.
While engaged on a rather banal surveying job for the Royal House of Hanover in the years after 1818, Gauss was also looking into the shape of the Earth, and starting to speculate on revolutionary ideas like shape of space itself. This led him to question one of the central tenets of the whole of mathematics, Euclidean geometry, which was clearly premised on a flat, and not a curved, universe. He later claimed to have considered a non-Euclidean geometry (in which Euclid's parallel axiom, for example, does not apply), which was internally consistent and free of contradiction, as early as 1800. Unwilling to court controversy, however, Gauss decided not to pursue or publish any of his avant-garde ideas in this area, leaving the field open to Bolyai and Lobachevsky, although he is still considered by some to be a pioneer of non-Euclidean geometry.
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Gaussian curvature |
with curves and surfaces) and what has come to be known as Gaussian
curvature (an intrinsic measure of curvature, dependent only on how
distances are measured on the surface, not on the way it is embedded in
space). All in all, despite the rather pedestrian nature of his
employment, the responsibilities of caring for his sick mother and the
constant arguments with his wife Minna (who desperately wanted to move
to Berlin), this was a very fruitful period of his academic life, and he
published over 70 papers between 1820 and 1830.Gauss’ achievements were not limited to pure mathematics, however. During his surveying years, he invented the heliotrope, an instrument that uses a mirror to reflect sunlight over great distances to mark positions in a land survey. In later years, he collaborated with Wilhelm Weber on measurements of the Earth's magnetic field, and invented the first electric telegraph. In recognition of his contributions to the theory of electromagnetism, the international unit of magnetic induction is known as the gauss
BOLYAI AND LOBACHEVSKY
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János Bolyai (1802-1860) and Nikolai Lobachevsky (1792-1856) |
His father and teacher, Farkas Bolyai, was himself an accomplished mathematician and had been a student of the great German mathematician Gauss for a time, but the cantankerous Gauss refused to take on the young prodigy János as a student. So, he was forced to join the army in order to earn a living and support his family, although he persevered with his mathematics in his spare time. He was also a talented linguist, speaking nine foreign languages, including Chinese and Tibetan.
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Euclid's parallel postulate |
Bolyai, however, persisted in his quest, and eventually came to the radical conclusion that it was in fact possible to have consistent geometries that were independent of the parallel postulate. In the early 1820s, Bolyai explored what he called “imaginary geometry” (now known as hyperbolic geometry), the geometry of curved spaces on a saddle-shaped plane, where the angles of a triangle did NOT add up to 180° and apparently parallel lines were NOT actually parallel. In curved space, the shortest distance between two points a and b is actually a curve, or geodesic, and not a straight line. Thus, the angles of a triangle in hyperbolic space sum to less than 180°, and two parallel lines in hyperbolic space actually diverge from each other. In a letter to his father, Bolyai marvelled, “Out of nothing I have created a strange new universe”.
Although it is easy to visualize a flat surface and a surface with positive curvature (e.g. a sphere, such as a the Earth), it is impossible to visualize a hyperbolic surface with negative curvature, other than just over a small localized area, where it would look like a saddle or a Pringle. So the very concept of a hyperbolic surface appeared to go against all sense of reality. It certainly represented a radical departure from Euclidean geometry, and the first step along the road which would lead to Einstein’s Theory of Relativity among other applications (although it still fell well short of the multi-dimensional geometry which was to be later realized by Riemann). Between 1820 and 1823, Bolyai prepared, but did not immediately publish, a treatise on a complete system of non-Euclidean geometry.
His work was, however, only published in 1832, and then only a short exposition in the appendix of a textbook by his father. On reading this, Gauss clearly recognized the genius of the younger Bolyai’s ideas, but he refused to encourage the young man, and even tried to claim his ideas as his own. Further disheartened by the news that the Russian mathematician Lobachevski had published something quite similar two years before his own paper, Bolyai became a recluse and gradually went insane. He died in obscurity in 1860. Although he only ever published the 24 pages of the appendix, Bolyai left more than 20,000 pages of mathematical manuscripts when he died (including the development of a rigorous geometric concept of complex numbers as ordered pairs of real numbers).
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Hyperbolic Bolyai-Lobachevskian geometry |
This early non-Euclidean geometry is now often referred to as Lobachevskian geometry or Bolyai-Lobachevskian geometry, thus sharing the credit. Gauss’ claims to have originated, but not published, the ideas are difficult to judge in retrospect. Other much earlier claims are credited to the 11th Century Persian mathematician Omar Khayyam, and to the early 18th Century Italian priest Giovanni Saccheri, but their work was much more speculative and inconclusive in nature.
Lobachevsky also died in poverty and obscurity, nearly blind and unable to walk
.
Among his other mathematical achievements, largely unknown during his
lifetime, was the development of a method for approximating the roots of
algebraic equations (a method now known as the Dandelin-Gräffe method,
named after two other mathematicians who discovered it independently),
and the definition of a function as a correspondence between two sets of
real numbers (usually credited to Dirichlet, who gave the same
definition independently soon after Lobachevsky)RIEMANN
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Bernhard Riemann (1826-1866) |
Although he started studying philology and theology in order to become a priest and help with his family's finances, Riemann's father eventually managed to gather enough money
to send him to study mathematics at the renowned University of
Göttingen in 1846, where he first met, and attended the lectures of, Carl Friedrich Gauss. Indeed, he was one of the very few who benefited from the support and patronage of Gauss,
and he gradually worked his way up the University's hierarchy to become
a professor and, eventually, head of the mathematics department at
Göttingen.
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Elliptic geometry |
geometry. As with hyperbolic geometry, there is no such thing as
parallel lines, and the angles of a triangle do not sum to 180° (in this
case, however, they sum to more than 180º). He went on to develop
Riemannian geometry, which unified and vastly generalized the three
types of geometry, as well as the concept of a manifold or mathematical
space, which generalized the ideas of curves and surfaces.A turning point in his career occurred in 1852 when, at the age of 26, have gave a lecture on the foundations of geometry and outlined his vision of a mathematics of many different kinds of space, only one of which was the flat, Euclidean space which we appear to inhabit. He also introduced one-dimensional complex manifolds known as Riemann surfaces. Although it was not widely understood at the time, Riemann’s mathematics changed how we look at the world, and opened the way to higher dimensional geometry, a potential which had existed, unrealized, since the time of Descartes.
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2-D representation of Riemann’s zeta function |
He introduced a collection of numbers (known as a tensor) at every point in space, which would describe how much it was bent or curved. For instance, in four spatial dimensions, a collection of ten numbers is needed at each point to describe the properties of the mathematical space or manifold, no matter how distorted it may be.
Riemann’s big breakthrough occurred while working on a function in the complex plane called the Riemann zeta function (an extension of the simpler zeta function first explored by Euler in the previous century). He realized that he could use it to build a kind of 3-dimensional landscape, and furthermore that the contours of that imaginary landscape might be able to unlock the Holy Grail of mathematics, the age-old secret of prime numbers.
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3-D representation of Riemann’s zeta function and Riemann’s Hypothesis |
With a huge imaginative leap, Riemann realized that these zeroes had a completely unexpected connection with the way the prime numbers are distributed. It began to seem that they could be used to correct Gauss’ inspired guesswork regarding the number of primes as numbers as one counts higher and higher.
The famous Riemann Hypothesis, which remains unproven, suggests that ALL the zeroes would be on the same straight line. Although he never provided a definitive proof of this hypothesis, Riemann’s work did at least show that the 15-year-old Gauss’ initial approximations of the incidence of prime numbers were perhaps more accurate than even he could have known, and that the primes were in fact distributed over the universe of numbers in a regular, balanced and beautiful way.
The discovery of the Riemann zeta function and the relationship of its zeroes to the prime numbers brought Riemann instant fame when it was published in 1859. He too, though, died young at just 39 years of age, in 1866, and many of his loose papers were accidentally destroyed after his death, so we will never know just how close he was to proving his own hypothesis. Over 150 years later, the Riemann Hypothesis is still considered one of the fundamental questions of number theory, and indeed of all mathematics, and a prize of $1 million has been offered
for the final solutionBOOLE
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George Boole (1815-1864) |
His extraordinary mathematical talents did not manifest themselves in early life. He received his early lessons in mathematics from his father, a tradesman with an amateur interest in in mathematics and logic, but his favourite subject at school was classics. He was a quiet, serious and modest young man from a humble working class background, and largely self-taught in his mathematics (he would borrow mathematical journals from his local Mechanics Institute).
It was only at university and afterwards that his mathematical skills began to be fully realized, although, even then, he was all but unknown in his own time, other than for a few insightful but rather abstruse papers on differential equations and the calculus of finite differences. By the age of 34, though, he was well respected enough in his field to be appointed as the first professor of mathematics of Queen's College (now University College) in Cork, Ireland.
But it was his contributions to the algebra of logic which were later to be viewed as immensely important and influential. Boole began to see the possibilities for applying his algebra to the solution of logical problems, and he pointed out a deep analogy between the symbols of algebra and those that can be made to represent logical forms and syllogisms. In fact, his ambitions stretched to a desire to devise and develop a system of algebraic logic that would systematically define and model the function of the human brain. His novel views of logical method were due to his profound confidence in symbolic reasoning, and he speculated on what he called a “calculus of reason” during the 1840s and 1850s.
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Boolean logic |
Boole’s use of symbols and connectives allowed for the simplification of logical expressions, including such important algebraic identities as: (X or Y) = (Y or X); not(not X) = X; not(X and Y) = (not X) or (not Y); etc.
He also developed a novel approach based on a binary
system, processing only two objects (“yes-no”, “true-false”, “on-off”,
“zero-one”). Therefore, if “true” is represented by 1 and “false” is
represented by 0, and two propositions are both true, then it is
possible under Boolean algebra for 1 + 1 to equal 1 ( the “+” is an
alternative representation of the OR operator)Despite the standing he had won in the academic community by that time, Boole’s revolutionary ideas were largely criticized or just ignored, until the American logician Charles Sanders Peirce (among others) explained and elaborated on them some years after Boole’s death in 1864.
Almost seventy years later, Claude Shannon made a major breakthrough in realizing that Boole's work could form the basis of mechanisms and processes in the real world, and particularly that electromechanical relay circuits could be used to solve Boolean algebra problems. The use of electrical switches to process logic is the basic concept that underlies all modern electronic digital computers, and so Boole is regarded in hindsight as a founder of the field of computer science, and his work led to the development of applications he could never have imagined
CANTOR
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Georg Cantor (1845-1918) |
He was made full professor at Halle at the age of just 34, a notable accomplishment, but his ambitions to move to a more prestigious university, such as Berlin, were largely thwarted by Leopold Kronecker, a well-established figure within the mathematical community and Cantor's former professor, who fundamentally disagreed with the thrust of Cantor's work.
Cantor’s first ten papers were on number theory, after which he turned his attention to calculus (or analysis as it had become known by this time), solving a difficult open problem on the uniqueness of the representation of a function by trigonometric series. His main legacy, though, is as perhaps the first mathematician to really understand the meaning of infinity and to give it mathematical precision.
Back in the 17th Century, Galileo had tried to confront the idea of infinity and the apparent contradictions thrown up by comparisons of different infinities, but in the end shied away from the problem. He had shown that a one-to-one correspondence could be drawn between all the natural numbers and the squares of all the natural numbers to infinity, suggesting that there were just as many square numbers as integers, even though it was intuitively obvious there were many integers that were were not squares, a concept which came to be known as Galileo’s Paradox. He had also pointed out that two concentric circles must both be comprised of an infinite number of points, even though the larger circle would appear to contain more points. However, Galileo had essentially dodged the issue and reluctantly concluded that concepts like less, equals and greater could only be applied to finite sets of numbers, and not to infinite sets. Cantor, however, was not content with this compromise.
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Cantor’s procedure of bijection or one-to-one correspondence to compare infinite sets |
His first intimations of all this came in the early 1870s when he considered an infinite series of natural numbers (1, 2, 3, 4, 5, ...), and then an infinite series of multiples of ten (10, 20 , 30, 40, 50, ...). He realized that, even though the multiples of ten were clearly a subset of the natural numbers, the two series could be paired up on a one-to-one basis (1 with 10, 2 with 20, 3 with 30, etc) - a process known as bijection - to show that they were the same “sizes” of infinite sets, in that they had the same number of elements.
This clearly also applies to other subsets of the natural numbers, such as the even numbers 2, 4, 6, 8, 10, etc, or the squares 1, 4, 9, 16, 25, etc, and even to the set of negative numbers and integers. In fact, Cantor realized that he could, in the same way, even pair up all the fractions (or rational numbers) with all the whole numbers, thus showing that rational numbers were also the same sort of infinity as the natural numbers, despite the intuitive feeling that there must be more fractions than whole numbers.
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Cantor’s diagonal argument for the existence of uncountable sets |
He also showed that they were “non-denumerable” or "uncountable" (i.e. contained more elements than could ever be counted), as opposed to the set of rational numbers which he had shown were technically (even if not practically) “denumerable” or "countable". In fact, it can be argued that there are an infinite number of irrational numbers in between each and every rational number. The patternless decimals of irrational numbers fill the "spaces" between the patterns of the rational numbers.
Cantor coined the new word “transfinite” in an attempt to distinguish these various levels of infinite numbers from an absolute infinity, which the religious Cantor effectively equated with God (he saw no contradiction between his mathematics and the traditional concept of God). Although the cardinality (or size) of a finite set is just a natural number indicating the number of elements in the set, he also needed a new notation to describe the sizes of infinite sets, and he used the Hebrew letter aleph (
). He defined 0 (aleph-null or aleph-nought) as the cardinality of the countably infinite set of natural numbers; 1
(aleph-one) as the next larger cardinality, that of the uncountable set
of ordinal numbers; etc. Because of the unique properties of infinite
sets, he showed that 0 + 0 = 0, and also that 0 x 0 = 0.All of this represented a revolutionary step, and opened up new possibilities in mathematics. However, it also opened up the possibility of other infinities, for instance an infinity - or even many infinities - between the infinity of the whole numbers and the larger infinity of the decimal numbers. This idea is known as the continuum hypothesis, and Cantor believed (but could not actually prove) that there was NO such intermediate infinite set. The continuum hypothesis was one of the 23 important open problems identified by David Hilbert in his famous 1900 Paris lecture, and it remained unproved - and indeed appeared to be unprovable - for almost a century, until the work of Robinson and Matiyasevich in the 1950s and 1960s.
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Modern set theory notation |
Throughout the 1880s and 1890s, he refined his set theory, defining well-ordered sets and power sets and introducing the concepts of ordinality and cardinality and the arithmetic of infinite sets. What is now known as Cantor's theorem states generally that, for any set A, the power set of A (i.e. the set of all subsets of A) has a strictly greater cardinality than A itself. More specificially, the power set of a countably infinite set is uncountably infinite.
Despite the central position of set theory in modern mathematics, it was often deeply mistrusted and misunderstood by other mathematicians of the day. One quote, usually attributed to Henri Poincaré, claimed that "later generations will regard Mengenlehre (set theory) as a disease from which one has recovered". Others, however, were quick to see the value and potential of the method, and David Hilbert declared in 1926 that "no one shall expel us from the Paradise that Cantor has created".
Cantor had few other mathematicians with whom he could discuss his ground-breaking work, and most were distinctly unnerved by his contemplation of the infinite. During the 1880s, he encountered resistance, sometimes fierce resistance, from mathematical contemporaries such as his old professor Leopold Kronecker and Henri Poincaré, as well as from philosophers like Ludwig Wittgenstein and even from some Christian theologians, who saw Cantor's work as a challenge to their view of the nature of God. Cantor himself, a deeply religious man, noted some annoying paradoxes thrown up by his own work, but some went further and saw it as the wilful destruction of the comprehensible and logical base on which the whole of mathematics was based.
As he aged, Cantor suffered from more and more recurrences of mental illness, which some have directly linked to his constant contemplation of such complex, abstract and paradoxical concepts. In the last decades of his life, he did no mathematical work at all, but wrote extensively on his two obsessions: that Shakespeare’s plays were actually written by the English philosopher Sir Francis Bacon, and that Christ was the natural son of Joseph of Arimathea. He spent long periods in the Halle sanatorium recovering from attacks of manic depression and paranoia, and it was there, alone in his room, that he finally died in 1918, his great project still unfinished.
POINCARÉ
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Henri Poincaré (1854-1912) |
Even as a youth at the Lycée in Nancy, he showed himself to be a polymath, and he proved to be one of the top students in every topic he studied. He continued to excel after he entered the École Polytechnique to study mathematics in 1873, and, for his doctoral thesis, he devised a new way of studying the properties of differential equations. Beginning in 1881, he taught at the Sorbonne in Paris, where he would spend the rest of his illustrious career. He was elected to the French Academy of Sciences at the young age of 32, became its president in 1906, and was elected to the Académie française in 1909.
Poincaré deliberately cultivated a work habit that has been compared to a bee flying from flower to flower. He observed a strict work regime of 2 hours of work in the morning and two hours in the early evening, with the intervening time left for his subconscious to carry on working on the problem in the hope of a flash of inspiration. He was a great believer in intuition, and claimed that "it is by logic that we prove, but by intuition that we discover".
It was one such flash of inspiration that earned Poincaré a generous prize from the King of Sweden in 1887 for his partial solution to the “three-body problem”, a problem that had defeated mathematicians of the stature of Euler, Lagrange and Laplace. Newton had long ago proved that the paths of two planets orbiting around each other would remain stable, but even the addition of just one more orbiting body to this already simplified solar system resulted in the involvement of as many as 18 different variables (such as position, velocity in each direction, etc), making it mathematically too complex to predict or disprove a stable orbit. Poincaré’s solution to the “three-body problem”, using a series of approximations of the orbits, although admittedly only a partial solution, was sophisticated enough to win him the prize.
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Computer representation of the paths generated by Poincaré’s analysis of the three body problem |
Paradoxically, owning up to his mistake only served to enhance Poincaré’s reputation, if anything, and he continued to produce a wide range of work throughout his life, as well as several popular books extolling the importance of mathematics.
Poincaré also developed the science of topology, which Leonhard Euler had heralded with his solution to the famous Seven Bridges of Königsberg problem. Topology is a kind of geometry which involves one-to-one correspondence of space. It is sometimes referred to as “bendy geometry” or “rubber sheet geometry” because, in topology, two shapes are the same if one can be bent or morphed into the other without cutting it. For example, a banana and a football are topologically equivalent, as are a donut (with its hole in the middle) and a teacup (with its handle); but a football and a donut, are topologically different because there is no way to morph one into the other. In the same way, a traditional pretzel, with its two holes is topological different from all of these examples.
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A 2-dimensional representation of the 3-dimensional problem in the Poincaré conjecture |
Poincaré’s work in theoretical physics was also of great significance, and his symmetrical presentation of the Lorentz transformations in 1905 was an important and necessary step in the formulation of Einstein’s theory of special relativity (some even hold that Poincaré and Lorentz were the true discoverers of relativity). He also made important contribution in a whole host of other areas of physics including fluid mechanics, optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory and cosmology
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