mathematics

WELCOME TO THE STORY OF MATHEMATICS

WHAT IS MATHEMATICS? Mathematics may be defined as “the study of relationships among quantities, magnitudes and properties, and also of the logical operations by which unknown quantities, magnitudes, and properties may be deduced” (Microsoft Encarta Encyclopedia) or "the study of quantity, structure, space and change" (Wikipedia). Historically, it was regarded as the science of quantity, whether of magnitudes (as in geometry) or of numbers (as in arithmetic) or of the generalization of these two fields (as in algebra). Some have seen it in terms as simple as a search for patterns. During the 19th Century, however, mathematics broadened to encompass mathematical or symbolic logic, and thus came to be regarded increasingly as the science of relations or of drawing necessary conclusions (although some see even this as too restrictive). The discipline of mathematics now covers - in addition to the more or less standard fields of number theory, algebra, geometry, analysis (calculus), mathematical logic and set theory, and more applied mathematics such as probability theory and statistics - a bewildering array of specialized areas and fields of study, including group theory, order theory, knot theory, sheaf theory, topology, differential geometry, fractal geometry, graph theory, functional analysis, complex analysis, singularity theory, catastrophe theory, chaos theory, measure theory, model theory, category theory, control theory, game theory, complexity theory and many more.

The history of mathematics is nearly as old as humanity itself. Since antiquity, mathematics has been fundamental to advances in science, engineering, and philosophy. It has evolved from simple counting, measurement and calculation, and the systematic study of the shapes and motions of physical objects, through the application of abstraction, imagination and logic, to the broad, complex and often abstract discipline we know today.

From the notched bones of early man to the mathematical advances brought about by settled agriculture in Mesopotamia and Egypt and the revolutionary developments of ancient Greece and its Hellenistic empire, the story of mathematics is a long and impressive one.

The East carried on the baton, particularly China, India and the medieval Islamic empire, before the focus of mathematical innovation moved back to Europe in the late Middle Ages and Renaissance. Then, a whole new series of revolutionary developments occurred in 17th Century and 18th Century Europe, setting the stage for the increasing complexity and abstraction of 19th Century mathematics, and finally the audacious and sometimes devastating discoveries of the 20th Century.

Follow the story as it unfolds in this series of linked sections, like the chapters of a book. Read the human stories behind the innovations, and how they made - and sometimes destroyed - the men and women who devoted their lives to... THE STORY OF MATHEMATICS.

This is not intended as a comprehensive and definitive guide to all of mathematics, but as an easy-to-use summary of the major mathematicians and the developments of mathematical thought over the centuries. It is not intended for mathematicians, but for the interested laity like myself.   My intention is to introduce some of the major thinkers and some of the most important advances in mathematics, without getting too technical or getting bogged down in too much detail, either biographical or computational. Explanations of any mathematical concepts and theorems will be generally simplified, the emphasis being on clarity and perspective rather than exhaustive detail.   It is beyond the scope is this study to discuss every single mathematician who has made significant contributions to the subject, just as it is impossible to describe all aspects of a discipline as huge in its scope as mathematics. The choice of what to include and exclude is my own personal one, so please forgive me if your favourite mathematician is not included or not dealt with in any detail.

The main Story of Mathematics is supplemented by a List of Important Mathematicians and their achievements, and by an alphabetical Glossary of Mathematical Terms. You can also make use of the search facility at the top of each page to search for individual mathematicians, theorems, developments, periods in history, etc. Some of the many resources available for further study (of both included and excluded elements) are listed in the Sources section.

PREHISTORIC MATHEMATICS

The Ishango bone, a tally stick from central Africa, dates from about 20,000 years ago

Our prehistoric ancestors would have had a general sensibility about amounts, and would have instinctively known the difference between, say, one and two antelopes. But the intellectual leap from the concrete idea of two things to the invention of a symbol or word for the abstract idea of "two" took many ages to come about.

Even today, there are isolated hunter-gatherer tribes in Amazonia which only have words for "one", "two" and "many", and others which only have words for numbers up to five. In the absence of settled agriculture and trade, there is little need for a formal system of numbers.

Early man kept track of regular occurrences such as the phases of the moon and the seasons. Some of the very earliest evidence of mankind thinking about numbers is from notched bones in Africa dating back to 35,000 to 20,000 years ago. But this is really mere counting and tallying rather than mathematics as such.

Pre-dynastic Egyptians and Sumerians represented geometric designs on their artefacts as early as the 5th millennium BCE, as did some megalithic societies in northern Europe in the 3rd millennium BCE or before. But this is more art and decoration than the systematic treatment of figures, patterns, forms and quantities that has come to be considered as mathematics.

Mathematics proper initially developed largely as a response to bureaucratic needs when civilizations settled and developed agriculture - for the measurement of plots of land, the taxation of individuals, etc - and this first occurred in the Sumerian and Babylonian civilizations of Mesopotamia (roughly, modern Iraq) and in ancient Egypt.

According to some authorities, there is evidence of basic arithmetic and geometric notations on the petroglyphs at Knowth and Newgrange burial mounds in Ireland (dating from about 3500 BCE and 3200 BCE respectively). These utilize a repeated zig-zag glyph for counting, a system which continued to be used in Britain and Ireland into the 1st millennium BCE. Stonehenge, a Neolithic ceremonial and astronomical monument in England, which dates from around 2300 BCE, also arguably exhibits examples of the use of 60 and 360 in the circle measurements, a practice which presumably developed quite independently of the sexagesimal counting system of the ancient Sumerian and Babylonians
 

SUMERIAN/BABYLONIAN MATHEMATICS

Sumerian Clay Cones

Sumer (a region of Mesopotamia, modern-day Iraq) was the birthplace of writing, the wheel, agriculture, the arch, the plow, irrigation and many other innovations, and is often referred to as the Cradle of Civilization. The Sumerians developed the earliest known writing system - a pictographic writing system known as cuneiform script, using wedge-shaped characters inscribed on baked clay tablets - and this has meant that we actually have more knowledge of ancient Sumerian and Babylonian mathematics than of early Egyptian mathematics. Indeed, we even have what appear to school exercises in arithmetic and geometric problems.

As in Egypt, Sumerian mathematics initially developed largely as a response to bureaucratic needs when their civilization settled and developed agriculture (possibly as early as the 6th millennium BCE) for the measurement of plots of land, the taxation of individuals, etc. In addition, the Sumerians and Babylonians needed to describe quite large numbers as they attempted to chart the course of the night sky and develop their sophisticated lunar calendar.

They were perhaps the first people to assign symbols to groups of objects in an attempt to make the description of larger numbers easier. They moved from using separate tokens or symbols to represent sheaves of wheat, jars of oil, etc, to the more abstract use of a symbol for specific numbers of anything. Starting as early as the 4th millennium BCE, they began using a small clay cone to represent one, a clay ball for ten, and a large cone for sixty. Over the course of the third millennium, these objects were replaced by cuneiform equivalents so that numbers could be written with the same stylus that was being used for the words in the text. A rudimentary model of the abacus was probably in use in Sumeria from as early as 2700 - 2300 BCE.

Babylonian Numerals

Sumerian and Babylonian mathematics was based on a sexegesimal, or base 60, numeric system, which could be counted physically using the twelve knuckles on one hand the five fingers on the other hand. Unlike those of the Egyptians, Greeks and Romans, Babylonian numbers used a true place-value system, where digits written in the left column represented larger values, much as in the modern decimal system, although of course using base 60 not base 10. Thus, in the Babylonian system represented 3,600 plus 60 plus 1, or 3,661. Also, to represent the numbers 1 - 59 within each place value, two distinct symbols were used, a unit symbol () and a ten symbol () which were combined in a similar way to the familiar system of Roman numerals (e.g. 23 would be shown as ). Thus, represents 60 plus 23, or 83. However, the number 60 was represented by the same symbol as the number 1 and, because they lacked an equivalent of the decimal point, the actual place value of a symbol often had to be inferred from the context.

It has been conjectured that Babylonian advances in mathematics were probably facilitated by the fact that 60 has many divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60 - in fact, 60 is the smallest integer divisible by all integers from 1 to 6), and the continued modern-day usage of of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle, are all testaments to the ancient Babylonian system. It is for similar reasons that 12 (which has factors of 1, 2, 3, 4 and 6) has been such a popular multiple historically (e.g. 12 months, 12 inches, 12 pence, 2 x 12 hours, etc).

The Babylonians also developed another revolutionary mathematical concept, something else that the Egyptians, Greeks and Romans did not have, a circle character for zero, although its symbol was really still more of a placeholder than a number in its own right.

We have evidence of the development of a complex system of metrology in Sumer from about 3000 BCE, and multiplication and reciprocal (division) tables, tables of squares, square roots and cube roots, geometrical exercises and division problems from around 2600 BCE onwards. Later Babylonian tablets dating from about 1800 to 1600 BCE cover topics as varied as fractions, algebra, methods for solving linear, quadratic and even some cubic equations, and the calculation of regular reciprocal pairs (pairs of number which multiply together to give 60). One Babylonian tablet gives an approximation to √2 accurate to an astonishing five decimal places. Others list the squares of numbers up to 59, the cubes of numbers up to 32 as well as tables of compound interest. Yet another gives an estimate for π of 3 ¹⁄₈ (3.125, a reasonable approximation of the real value of 3.1416).

Babylonian Clay tablets from c. 2100 BCE showing a problem concerning the area of an irregular shape

The idea of square numbers and quadratic equations (where the unknown quantity is multiplied by itself, e.g. x²) naturally arose in the context of the meaurement of land, and Babylonian mathematical tablets give us the first ever evidence of the solution of quadratic equations. The Babylonian approach to solving them usually revolved around a kind of geometric game of slicing up and rearranging shapes, although the use of algebra and quadratic equations also appears. At least some of the examples we have appear to indicate problem-solving for its own sake rather than in order to resolve a concrete practical problem.

The Babylonians used geometric shapes in their buildings and design and in dice for the leisure games which were so popular in their society, such as the ancient game of backgammon. Their geometry extended to the calculation of the areas of rectangles, triangles and trapezoids, as well as the volumes of simple shapes such as bricks and cylinders (although not pyramids).

The famous and controversial Plimpton 322 clay tablet, believed to date from around 1800 BCE, suggests that the Babylonians may well have known the secret of right-angled triangles (that the square of the hypotenuse equals the sum of the square of the other two sides) many centuries before the Greek Pythagoras. The tablet appears to list 15 perfect Pythagorean triangles with whole number sides, although some claim that they were merely academic exercises, and not deliberate manifestations of Pythagorean triples.

ROMAN MATHEMATICS

Roman numerals

By the middle of the 1st Century BCE, the Roman had tightened their grip on the old Greek and Hellenistic empires, and the mathematical revolution of the Greeks ground to halt. Despite all their advances in other respects, no mathematical innovations occurred under the Roman Empire and Republic, and there were no mathematicians of note. The Romans had no use for pure mathematics, only for its practical applications, and the Christian regime that followed it (after Christianity became the official religion of the Roman empire) even less so.

Roman arithmetic

Roman numerals are well known today, and were the dominant number system for trade and administration in most of Europe for the best part of a millennium. It was decimal (base 10) system but not directly positional, and did not include a zero, so that, for arithmetic and mathematical purposes, it was a clumsy and inefficient system. It was based on letters of the Roman alphabet - I, V, X, L, C, D and M - combines to signify the sum of their values (e.g. VII = V + I + I = 7).

Later, a subtractive notation was also adopted, where VIIII, for example, was replaced by IX (10 - 1 = 9), which simplified the writing of numbers a little, but made calculation even more difficult, requiring conversion of the subtractive notation at the beginning of a sum and then its re-application at the end (see image at right). Due to the difficulty of written arithmetic using Roman numeral notation, calculations were usually performed with an abacus, based on earlier Babylonian and Greek abaci.

GREEK MATHEMATICS

Ancient Greek Herodianic numerals

As the Greek empire began to spread its sphere of influence into Asia Minor, Mesopotamia and beyond, the Greeks were smart enough to adopt and adapt useful elements from the societies they conquered. This was as true of their mathematics as anything else, and they adopted elements of mathematics from both the Babylonians and the Egyptians. But they soon started to make important contributions in their own right and, for the first time, we can acknowledge contributions by individuals. By the Hellenistic period, the Greeks had presided over one of the most dramatic and important revolutions in mathematical thought of all time.

The ancient Greek numeral system, known as Attic or Herodianic numerals, was fully developed by about 450 BCE, and in regular use possibly as early as the 7th Century BCE. It was a base 10 system similar to the earlier Egyptian one (and even more similar to the later Roman system), with symbols for 1, 5, 10, 50, 100, 500 and 1,000 repeated as many times needed to represent the desired number. Addition was done by totalling separately the symbols (1s, 10s, 100s, etc) in the numbers to be added, and multiplication was a laborious process based on successive doublings (division was based on the inverse of this process).

Thales' Intercept Theorem

But most of Greek mathematics was based on geometry. Thales, one of the Seven Sages of Ancient Greece, who lived on the Ionian coast of Asian Minor in the first half of the 6th Century BCE, is usually considered to have been the first to lay down guidelines for the abstract development of geometry, although what we know of his work (such as on similar and right triangles) now seems quite elementary.

Thales established what has become known as Thales' Theorem, whereby if a triangle is drawn within a circle with the long side as a diameter of the circle, then the opposite angle will always be a right angle (as well as some other related properties derived from this). He is also credited with another theorem, also known as Thales' Theorem or the Intercept Theorem, about the ratios of the line segments that are created if two intersecting lines are intercepted by a pair of parallels (and, by extension, the ratios of the sides of similar triangles).

To some extent, however, the legend of the 6th Century BCE mathematician Pythagoras of Samos has become synonymous with the birth of Greek mathematics. Indeed, he is believed to have coined both the words "philosophy" ("love of wisdom") and "mathematics" ("that which is learned"). Pythagoras was perhaps the first to realize that a complete system of mathematics could be constructed, where geometric elements corresponded with numbers. Pythagoras’ Theorem (or the Pythagorean Theorem) is one of the best known of all mathematical theorems. But he remains a controversial figure, as we will see, and Greek mathematics was by no means limited to one man.

The Three Classical Problems

Three geometrical problems in particular, often referred to as the Three Classical Problems, and all to be solved by purely geometric means using only a straight edge and a compass, date back to the early days of Greek geometry: “the squaring (or quadrature) of the circle”, “the doubling (or duplicating) of the cube” and “the trisection of an angle”. These intransigent problems were profoundly influential on future geometry and led to many fruitful discoveries, although their actual solutions (or, as it turned out, the proofs of their impossibility) had to wait until the 19th Century.

Hippocrates of Chios (not to be confused with the great Greek physician Hippocrates of Kos) was one such Greek mathematician who applied himself to these problems during the 5th Century BCE (his contribution to the “squaring the circle” problem is known as the Lune of Hippocrates). His influential book “The Elements”, dating to around 440 BCE, was the first compilation of the elements of geometry, and his work was an important source for Euclid's later work.

Zeno's Paradox of Achilles and the Tortoise

It was the Greeks who first grappled with the idea of infinity, such as described in the well-known paradoxes attributed to the philosopher Zeno of Elea in the 5th Century BCE. The most famous of his paradoxes is that of Achilles and the Tortoise, which describes a theoretical race between Achilles and a tortoise. Achilles gives the much slower tortoise a head start, but by the time Achilles reaches the tortoise's starting point, the tortoise has already moved ahead. By the time Achilles reaches that point, the tortoise has moved on again, etc, etc, so that in principle the swift Achilles can never catch up with the slow tortoise.

Paradoxes such as this one and Zeno's so-called Dichotomy Paradox are based on the infinite divisibility of space and time, and rest on the idea that a half plus a quarter plus an eighth plus a sixteenth, etc, etc, to infinity will never quite equal a whole. The paradox stems, however, from the false assumption that it is impossible to complete an infinite number of discrete dashes in a finite time, although it is extremely difficult to definitively prove the fallacy. The ancient Greek Aristotle was the first of many to try to disprove the paradoxes, particularly as he was a firm believer that infinity could only ever be potential and not real.

Democritus, most famous for his prescient ideas about all matter being composed of tiny atoms, was also a pioneer of mathematics and geometry in the 5th - 4th Century BCE, and he produced works with titles like "On Numbers", "On Geometrics", "On Tangencies", "On Mapping" and "On Irrationals", although these works have not survived. We do know that he was among the first to observe that a cone (or pyramid) has one-third the volume of a cylinder (or prism) with the same base and height, and he is perhaps the first to have seriously considered the division of objects into an infinite number of cross-sections.

However, it is certainly true that Pythagoras in particular greatly influenced those who came after him, including Plato, who established his famous Academy in Athens in 387 BCE, and his protégé Aristotle, whose work on logic was regarded as definitive for over two thousand years. Plato the mathematician is best known for his description of the five Platonic solids, but the value of his work as a teacher and popularizer of mathematics can not be overstated.

Plato’s student Eudoxus of Cnidus is usually credited with the first implementation of the “method of exhaustion” (later developed by Archimedes), an early method of integration by successive approximations which he used for the calculation of the volume of the pyramid and cone. He also developed a general theory of proportion, which was applicable to incommensurable (irrational) magnitudes that cannot be expressed as a ratio of two whole numbers, as well as to commensurable (rational) magnitudes, thus extending Pythagoras’ incomplete ideas.

Perhaps the most important single contribution of the Greeks, though - and Pythagoras, Plato and Aristotle were all influential in this respect - was the idea of proof, and the deductive method of using logical steps to prove or disprove theorems from initial assumed axioms. Older cultures, like the Egyptians and the Babylonians, had relied on inductive reasoning, that is using repeated observations to establish rules of thumb. It is this concept of proof that give mathematics its power and ensures that proven theories are as true today as they were two thousand years ago, and which laid the foundations for the systematic approach to mathematics of Euclid and those who came after him.

GREEK MATHEMATICS - PYTHAGORAS

Pythagoras of Samos (c.570-495 BCE)

It is sometimes claimed that we owe pure mathematics to Pythagoras, and he is often called the first "true" mathematician. But, although his contribution was clearly important, he nevertheless remains a controversial figure. He left no mathematical writings himself, and much of what we know about Pythagorean thought comes to us from the writings of Philolaus and other later Pythagorean scholars. Indeed, it is by no means clear whether many (or indeed any) of the theorems ascribed to him were in fact solved by Pythagoras personally or by his followers.

The school he established at Croton in southern Italy around 530 BCE was the nucleus of a rather bizarre Pythagorean sect. Although Pythagorean thought was largely dominated by mathematics, it was also profoundly mystical, and Pythagoras imposed his quasi-religious philosophies, strict vegetarianism, communal living, secret rites and odd rules on all the members of his school (including bizarre and apparently random edicts about never urinating towards the sun, never marrying a woman who wears gold jewellery, never passing an ass lying in the street, never eating or even touching black fava beans, etc) .

The members were divided into the "mathematikoi" (or "learners"), who extended and developed the more mathematical and scientific work that Pythagoras himself began, and the "akousmatikoi" (or "listeners"), who focused on the more religious and ritualistic aspects of his teachings. There was always a certain amount of friction between the two groups and eventually the sect became caught up in some fierce local fighting and ultimately dispersed. Resentment built up against the secrecy and exclusiveness of the Pythagoreans and, in 460 BCE, all their meeting places were burned and destroyed, with at least 50 members killed in Croton alone.

The over-riding dictum of Pythagoras's school was “All is number” or “God is number”, and the Pythagoreans effectively practised a kind of numerology or number-worship, and considered each number to have its own character and meaning. For example, the number one was the generator of all numbers; two represented opinion; three, harmony; four, justice; five, marriage; six, creation; seven, the seven planets or “wandering stars”; etc. Odd numbers were thought of as female and even numbers as male.

The Pythagorean Tetractys

The holiest number of all was "tetractys" or ten, a triangular number composed of the sum of one, two, three and four. It is a great tribute to the Pythagoreans' intellectual achievements that they deduced the special place of the number 10 from an abstract mathematical argument rather than from something as mundane as counting the fingers on two hands.

However, Pythagoras and his school - as well as a handful of other mathematicians of ancient Greece - was largely responsible for introducing a more rigorous mathematics than what had gone before, building from first principles using axioms and logic. Before Pythagoras, for example, geometry had been merely a collection of rules derived by empirical measurement. Pythagoras discovered that a complete system of mathematics could be constructed, where geometric elements corresponded with numbers, and where integers and their ratios were all that was necessary to establish an entire system of logic and truth.

He is mainly remembered for what has become known as Pythagoras’ Theorem (or the Pythagorean Theorem): that, for any right-angled triangle, the square of the length of the hypotenuse (the longest side, opposite the right angle) is equal to the sum of the square of the other two sides (or “legs”). Written as an equation: a² + b² = c². What Pythagoras and his followers did not realize is that this also works for any shape: thus, the area of a pentagon on the hypotenuse is equal to the sum of the pentagons on the other two sides, as it does for a semi-circle or any other regular (or even irregular( shape.

Pythagoras' (Pythagorean) Theorem

The simplest and most commonly quoted example of a Pythagorean triangle is one with sides of 3, 4 and 5 units (3² + 4² = 5², as can be seen by drawing a grid of unit squares on each side as in the diagram at right), but there are a potentially infinite number of other integer “Pythagorean triples”, starting with (5, 12 13), (6, 8, 10), (7, 24, 25), (8, 15, 17), (9, 40, 41), etc. It should be noted, however that (6, 8, 10) is not what is known as a “primitive” Pythagorean triple, because it is just a multiple of (3, 4, 5).

Pythagoras’ Theorem and the properties of right-angled triangles seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, and it was touched on in some of the most ancient mathematical texts from Babylon and Egypt, dating from over a thousand years earlier. One of the simplest proofs comes from ancient China, and probably dates from well before Pythagoras' birth. It was Pythagoras, though, who gave the theorem its definitive form, although it is not clear whether Pythagoras himself definitively proved it or merely described it. Either way, it has become one of the best-known of all mathematical theorems, and as many as 400 different proofs now exist, some geometrical, some algebraic, some involving advanced differential equations, etc.

It soom became apparent, though, that non-integer solutions were also possible, so that an isosceles triangle with sides 1, 1 and √2, for example, also has a right angle, as the Babylonians had discovered centuries earlier. However, when Pythagoras’s student Hippasus tried to calculate the value of √2, he found that it was not possible to express it as a fraction, thereby indicating the potential existence of a whole new world of numbers, the irrational numbers (numbers that can not be expressed as simple fractions of integers). This discovery rather shattered the elegant mathematical world built up by Pythagoras and his followers, and the existence of a number that could not be expressed as the ratio of two of God's creations (which is how they thought of the integers) jeopardized the cult's entire belief system.

Poor Hippasus was apparently drowned by the secretive Pythagoreans for broadcasting this important discovery to the outside world. But the replacement of the idea of the divinity of the integers by the richer concept of the continuum, was an essential development in mathematics. It marked the real birth of Greek geometry, which deals with lines and planes and angles, all of which are continuous and not discrete.

Among his other achievements in geometry, Pythagoras (or at least his followers, the Pythagoreans) also realized that the sum of the angles of a triangle is equal to two right angles (180°), and probably also the generalization which states that the sum of the interior angles of a polygon with n sides is equal to (2n - 4) right angles, and that the sum of its exterior angles equals 4 right angles. They were able to construct figures of a given area, and to use simple geometrical algebra, for example to solve equations such as a(a - x) = x² by geometrical means.

The Pythagoreans also established the foundations of number theory, with their investigations of triangular, square and also perfect numbers (numbers that are the sum of their divisors). They discovered several new properties of square numbers, such as that the square of a number n is equal to the sum of the first n odd numbers (e.g. 4² = 16 = 1 + 3 + 5 + 7). They also discovered at least the first pair of amicable numbers, 220 and 284 (amicable numbers are pairs of numbers for which the sum of the divisors of one number equals the other number, e.g. the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71, and 142, of which the sum is 220).

Pythagoras is credited with the discovery of the ratios between harmonious musical tones

Pythagoras is also credited with the discovery that the intervals between harmonious musical notes always have whole number ratios. For instance, playing half a length of a guitar string gives the same note as the open string, but an octave higher; a third of a length gives a different but harmonious note; etc. Non-whole number ratios, on the other hand, tend to give dissonant sounds. In this way, Pythagoras described the first four overtones which create the common intervals which have become the primary building blocks of musical harmony: the octave (1:1), the perfect fifth (3:2), the perfect fourth (4:3) and the major third (5:4). The oldest way of tuning the 12-note chromatic scale is known as Pythagorean tuning, and it is based on a stack of perfect fifths, each tuned in the ratio 3:2.

The mystical Pythagoras was so excited by this discovery that he became convinced that the whole universe was based on numbers, and that the planets and stars moved according to mathematical equations, which corresponded to musical notes, and thus produced a kind of symphony, the “Musical Universalis” or “Music of the Spheres”.
 

GREEK MATHEMATICS - PLATO

Plato (c.428-348 BCE)

Although usually remembered today as a philosopher, Plato was also one of ancient Greece’s most important patrons of mathematics. Inspired by Pythagoras, he founded his Academy in Athens in 387 BCE, where he stressed mathematics as a way of understanding more about reality. In particular, he was convinced that geometry was the key to unlocking the secrets of the universe. The sign above the Academy entrance read: “Let no-one ignorant of geometry enter here”.

Plato played an important role in encouraging and inspiring Greek intellectuals to study mathematics as well as philosophy. His Academy taught mathematics as a branch of philosophy, as Pythagoras had done, and the first 10 years of the 15 year course at the Academy involved the study of science and mathematics, including plane and solid geometry, astronomy and harmonics. Plato became known as the "maker of mathematicians", and his Academy boasted some of the most prominent mathematicians of the ancient world, including Eudoxus, Theaetetus and Archytas.

He demanded of his students accurate definitions, clearly stated assumptions, and logical deductive proof, and he insisted that geometric proofs be demonstrated with no aids other than a straight edge and a compass. Among the many mathematical problems Plato posed for his students’ investigation were the so-called Three Classical Problems (“squaring the circle”, “doubling the cube” and “trisecting the angle”) and to some extent these problems have become identified with Plato, although he was not the first to pose them.

Platonic Solids

Plato the mathematician is perhaps best known for his identification of 5 regular symmetrical 3-dimensional shapes, which he maintained were the basis for the whole universe, and which have become known as the Platonic Solids: the tetrahedron (constructed of 4 regular triangles, and which for Plato represented fire), the octahedron (composed of 8 triangles, representing air), the icosahedron (composed of 20 triangles, and representing water), the cube (composed of 6 squares, and representing earth), and the dodecahedron (made up of 12 pentagons, which Plato obscurely described as “the god used for arranging the constellations on the whole heaven”).

The tetrahedron, cube and dodecahedron were probably familiar to Pythagoras, and the octahedron and icosahedron were probably discovered by Theaetetus, a contemporary of Plato. Furthermore, it fell to Euclid, half a century later, to prove that these were the only possible convex regular polyhedra. But they nevertheless became popularly known as the Platonic Solids, and inspired mathematicians and geometers for many centuries to come. For example, around 1600, the German astronomer Johannes Kepler devised an ingenious system of nested Platonic solids and spheres to approximate quite well the distances of the known planets from the Sun (although he was enough of a scientist to abandon his elegant model when it proved to be not accurate enough).
 

ISLAMIC MATHEMATICS

Some examples of the complex symmetries used in Islamic temple decoration

The Islamic Empire established across Persia, the Middle East, Central Asia, North Africa, Iberia and parts of India from the 8th Century onwards made significant contributions towards mathematics. They were able to draw on and fuse together the mathematical developments of both Greece and India.

One consequence of the Islamic prohibition on depicting the human form was the extensive use of complex geometric patterns to decorate their buildings, raising mathematics to the form of an art. In fact, over time, Muslim artists discovered all the different forms of symmetry that can be depicted on a 2-dimensional surface.

The Qu’ran itself encouraged the accumulation of knowledge, and a Golden Age of Islamic science and mathematics flourished throughout the medieval period from the 9th to 15th Centuries. The House of Wisdom was set up in Baghdad around 810, and work started almost immediately on translating the major Greek and Indian mathematical and astronomy works into Arabic.

The outstanding Persian mathematician Muhammad Al-Khwarizmi was an early Director of the House of Wisdom in the 9th Century, and one of the greatest of early Muslim mathematicians. Perhaps Al-Khwarizmi’s most important contribution to mathematics was his strong advocacy of the Hindu numerical system (1 - 9 and 0), which he recognized as having the power and efficiency needed to revolutionize Islamic (and, later, Western) mathematics, and which was soon adopted by the entire Islamic world, and later by Europe as well.

Al-Khwarizmi's other important contribution was algebra, and he introduced the fundamental algebraic methods of “reduction” and “balancing” and provided an exhaustive account of solving polynomial equations up to the second degree. In this way, he helped create the powerful abstract mathematical language still used across the world today, and allowed a much more general way of analyzing problems other than just the specific problems previously considered by the Indians and Chinese.

Binomial Theorem

The 10th Century Persian mathematician Muhammad Al-Karaji worked to extend algebra still further, freeing it from its geometrical heritage, and introduced the theory of algebraic calculus. Al-Karaji was the first to use the method of proof by mathematical induction to prove his results, by proving that the first statement in an infinite sequence of statements is true, and then proving that, if any one statement in the sequence is true, then so is the next one.

Among other things, Al-Karaji used mathematical induction to prove the binomial theorem. A binomial is a simple type of algebraic expression which has just two terms which are operated on only by addition, subtraction, multiplication and positive whole-number exponents, such as (x + y)². The co-efficients needed when a binomial is expanded form a symmetrical triangle, usually referred to as Pascal’s Triangle after the 17th Century French mathematician Blaise Pascal, although many other mathematicians had studied it centuries before him in India, Persia, China and Italy, including Al-Karaji.

Some hundred years after Al-Karaji, Omar Khayyam (perhaps better known as a poet and the writer of the “Rubaiyat”, but an important mathematician and astronomer in his own right) generalized Indian methods for extracting square and cube roots to include fourth, fifth and higher roots in the early 12th Century. He carried out a systematic analysis of cubic problems, revealing there were actually several different sorts of cubic equations. Although he did in fact succeed in solving cubic equations, and although he is usually credited with identifying the foundations of algebraic geometry, he was held back from further advances by his inability to separate the algebra from the geometry, and a purely algebraic method for the solution of cubic equations had to wait another 500 years and the Italian mathematicians del Ferro and Tartaglia.

Al-Tusi was a pioneer in the field of spherical trigonometry

The 13th Century Persian astronomer, scientist and mathematician Nasir Al-Din Al-Tusi was perhaps the first to treat trigonometry as a separate mathematical discipline, distinct from astronomy. Building on earlier work by Greek mathematicians such as Menelaus of Alexandria and Indian work on the sine function, he gave the first extensive exposition of spherical trigonometry, including listing the six distinct cases of a right triangle in spherical trigonometry. One of his major mathematical contributions was the formulation of the famous law of sines for plane triangles, ^a⁄_{(sin A)} = ^b⁄_{(sin B)} = ^c⁄_{(sin C)}, although the sine law for spherical triangles had been discovered earlier by the 10th Century Persians Abul Wafa Buzjani and Abu Nasr Mansur.

Other medieval Muslim mathematicians worthy of note include:

• the 9th Century Arab Thabit ibn Qurra, who developed a general formula by which amicable numbers could be derived, re-discovered much later by both Fermat and Descartes(amicable numbers are pairs of numbers for which the sum of the divisors of one number equals the other number, e.g. the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71, and 142, of which the sum is 220);
• the 10th Century Arab mathematician Abul Hasan al-Uqlidisi, who wrote the earliest surviving text showing the positional use of Arabic numerals, and particularly the use of decimals instead of fractions (e.g. 7.375 insead of 7³⁄₈);
• the 10th Century Arab geometer Ibrahim ibn Sinan, who continued Archimedes' investigations of areas and volumes, as well as on tangents of a circle;
• the 11th Century Persian Ibn al-Haytham (also known as Alhazen), who, in addition to his groundbreaking work on optics and physics, established the beginnings of the link between algebra and geometry, and devised what is now known as "Alhazen's problem" (he was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily generalizable); and
• the 13th Century Persian Kamal al-Din al-Farisi, who applied the theory of conic sections to solve optical problems, as well as pursuing work in number theory such as on amicable numbers, factorization and combinatorial methods;
• the 13th Century Moroccan Ibn al-Banna al-Marrakushi, whose works included topics such as computing square roots and the theory of continued fractions, as well as the discovery of the first new pair of amicable numbers since ancient times (17,296 and 18,416, later re-discovered by Fermat) and the the first use of algebraic notation since Brahmagupta.

With the stifling influence of the Turkish Ottoman Empire from the 14th or 15th Century onwards, Islamic mathematics stagnated, and further developments moved to Europe.

ISLAMIC MATHEMATICS - AL-KHWARIZMI

Muhammad Al-Khwarizmi (c.780-850 CE)

One of the first Directors of the House of Wisdom in Bagdad in the early 9th Century was an outstanding Persian mathematician called Muhammad Al-Khwarizmi. He oversaw the translation of the major Greek and Indian mathematical and astronomy works (including those of Brahmagupta) into Arabic, and produced original work which had a lasting influence on the advance of Muslim and (after his works spread to Europe through Latin translations in the 12th Century) later European mathematics.

The word “algorithm” is derived from the Latinization of his name, and the word "algebra" is derived from the Latinization of "al-jabr", part of the title of his most famous book, in which he introduced the fundamental algebraic methods and techniques for solving equations.

Perhaps his most important contribution to mathematics was his strong advocacy of the Hindu numerical system, which Al-Khwarizmi recognized as having the power and efficiency needed to revolutionize Islamic and Western mathematics. The Hindu numerals 1 - 9 and 0 - which have since become known as Hindu-Arabic numerals - were soon adopted by the entire Islamic world. Later, with translations of Al-Khwarizmi’s work into Latin by Adelard of Bath and others in the 12th Century, and with the influence of Fibonacci’s “Liber Abaci” they would be adopted throughout Europe as well.

An example of Al-Khwarizmi’s “completing the square” method for solving quadratic equations

Al-Khwarizmi’s other important contribution was algebra, a word derived from the title of a mathematical text he published in about 830 called “Al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala” (“The Compendious Book on Calculation by Completion and Balancing”). Al-Khwarizmi wanted to go from the specific problems considered by the Indians and Chinese to a more general way of analyzing problems, and in doing so he created an abstract mathematical language which is used across the world today.

His book is considered the foundational text of modern algebra, although he did not employ the kind of algebraic notation used today (he used words to explain the problem, and diagrams to solve it). But the book provided an exhaustive account of solving polynomial equations up to the second degree, and introduced for the first time the fundamental algebraic methods of “reduction” (rewriting an expression in a simpler form), “completion” (moving a negative quantity from one side of the equation to the other side and changing its sign) and “balancing” (subtraction of the same quantity from both sides of an equation, and the cancellation of like terms on opposite sides).

In particular, Al-Khwarizmi developed a formula for systematically solving quadratic equations (equations involving unknown numbers to the power of 2, or x²) by using the methods of completion and balancing to reduce any equation to one of six standard forms, which were then solvable. He described the standard forms in terms of "squares" (what would today be "x²"), "roots" (what would today be "x") and "numbers" (regular constants, like 42), and identified the six types as: squares equal roots (ax² = bx), squares equal number (ax² = c), roots equal number (bx = c), squares and roots equal number (ax² + bx = c), squares and number equal roots (ax² + c = bx), and roots and number equal squares (bx + c = ax²).

Al-Khwarizmi is usually credited with the development of lattice (or sieve) multiplication method of multiplying large numbers, a method algorithmically equivalent to long multiplication. His lattice method was later introduced into Europe by Fibonacci.

In addition to his work in mathematics, Al-Khwarizmi made important contributions to astronomy, also largely based on methods from India, and he developed the first quadrant (an instrument used to determine time by observations of the Sun or stars), the second most widely used astronomical instrument during the Middle Ages after the astrolabe. He also produced a revised and completed version of Ptolemy's “Geography”, consisting of a list of 2,402 coordinates of cities throughout the known world.
 

MEDIEVAL MATHEMATICS

Medieval abacus, based on the Roman/Greek model

During the centuries in which the Chinese, Indian and Islamic mathematicians had been in the ascendancy, Europe had fallen into the Dark Ages, in which science, mathematics and almost all intellectual endeavour stagnated. Scholastic scholars only valued studies in the humanities, such as philosophy and literature, and spent much of their energies quarrelling over subtle subjects in metaphysics and theology, such as "How many angels can stand on the point of a needle?"

From the 4th to 12th Centuries, European knowledge and study of arithmetic, geometry, astronomy and music was limited mainly to Boethius’ translations of some of the works of ancient Greek masters such as Nicomachus and Euclid. All trade and calculation was made using the clumsy and inefficient Roman numeral system, and with an abacus based on Greek and Roman models.

By the 12th Century, though, Europe, and particularly Italy, was beginning to trade with the East, and Eastern knowledge gradually began to spread to the West. Robert of Chester translated Al-Khwarizmi's important book on algebra into Latin in the 12th Century, and the complete text of Euclid's “Elements” was translated in various versions by Adelard of Bath, Herman of Carinthia and Gerard of Cremona. The great expansion of trade and commerce in general created a growing practical need for mathematics, and arithmetic entered much more into the lives of common people and was no longer limited to the academic realm.

The advent of the printing press in the mid-15th Century also had a huge impact. Numerous books on arithmetic were published for the purpose of teaching business people computational methods for their commercial needs and mathematics gradually began to acquire a more important position in education.

Europe’s first great medieval mathematician was the Italian Leonardo of Pisa, better known by his nickname Fibonacci. Although best known for the so-called Fibonacci Sequence of numbers, perhaps his most important contribution to European mathematics was his role in spreading the use of the Hindu-Arabic numeral system throughout Europe early in the 13th Century, which soon made the Roman numeral system obsolete, and opened the way for great advances in European mathematics.

Oresme was one of the first to use graphical analysis

An important (but largely unknown and underrated) mathematician and scholar of the 14th Century was the Frenchman Nicole Oresme. He used a system of rectangular coordinates centuries before his countryman René Descartes popularized the idea, as well as perhaps the first time-speed-distance graph. Also, leading from his research into musicology, he was the first to use fractional exponents, and also worked on infinite series, being the first to prove that the harmonic series ¹⁄₁ + ¹⁄₂ + ¹⁄₃ + ¹⁄₄ + ¹⁄₅... is a divergent infinite series (i.e. not tending to a limit, other than infinity).

The German scholar Regiomontatus was perhaps the most capable mathematician of the 15th Century, his main contribution to mathematics being in the area of trigonometry. He helped separate trigonometry from astronomy, and it was largely through his efforts that trigonometry came to be considered an independent branch of mathematics. His book "De Triangulis", in which he described much of the basic trigonometric knowledge which is now taught in high school and college, was the first great book on trigonometry to appear in print.

Mention should also be made of Nicholas of Cusa (or Nicolaus Cusanus), a 15th Century German philosopher, mathematician and astronomer, whose prescient ideas on the infinite and the infinitesimal directly influenced later mathematicians like Gottfried Leibniz and Georg Cantor. He also held some distinctly non-standard intuitive ideas about the universe and the Earth's position in it, and about the elliptical orbits of the planets and relative motion, which foreshadowed the later discoveries of Copernicus and Kepler.

MEDIEVAL MATHEMATICS - FIBONACCI

Leonardo of Pisa (Fibonacci) (c.1170-1250)

The 13th Century Italian Leonardo of Pisa, better known by his nickname Fibonacci, was perhaps the most talented Western mathematician of the Middle Ages. Little is known of his life except that he was the son of a customs offical and, as a child, he travelled around North Africa with his father, where he learned about Arabic mathematics. On his return to Italy, he helped to disseminate this knowledge throughout Europe, thus setting in motion a rejuvenation in European mathematics, which had lain largely dormant for centuries during the Dark Ages.

In particular, in 1202, he wrote a hugely influential book called “Liber Abaci” ("Book of Calculation"), in which he promoted the use of the Hindu-Arabic numeral system, describing its many benefits for merchants and mathematicians alike over the clumsy system of Roman numerals then in use in Europe. Despite its obvious advantages, uptake of the system in Europe was slow (this was after all during the time of the Crusades against Islam, a time in which anything Arabic was viewed with great suspicion), and Arabic numerals were even banned in the city of Florence in 1299 on the pretext that they were easier to falsify than Roman numerals. However, common sense eventually prevailed and the new system was adopted throughout Europe by the 15th century, making the Roman system obsolete. The horizontal bar notation for fractions was also first used in this work (although following the Arabic practice of placing the fraction to the left of the integer).

The discovery of the famous Fibonacci sequence

Fibonacci is best known, though, for his introduction into Europe of a particular number sequence, which has since become known as Fibonacci Numbers or the Fibonacci Sequence. He discovered the sequence - the first recursive number sequence known in Europe - while considering a practical problem in the “Liber Abaci” involving the growth of a hypothetical population of rabbits based on idealized assumptions. He noted that, after each monthly generation, the number of pairs of rabbits increased from 1 to 2 to 3 to 5 to 8 to 13, etc, and identified how the sequence progressed by adding the previous two terms (in mathematical terms, F_n = F_n-1 + F_n-2), a sequence which could in theory extend indefinitely.

The sequence, which had actually been known to Indian mathematicians since the 6th Century, has many interesting mathematical properties, and many of the implications and relationships of the sequence were not discovered until several centuries after Fibonacci's death. For instance, the sequence regenerates itself in some surprising ways: every third F-number is divisible by 2 (F₃ = 2), every fourth F-number is divisible by 3 (F₄ = 3), every fifth F-number is divisible by 5 (F₅ = 5), every sixth F-number is divisible by 8 (F₆ = , every seventh F-number is divisible by 13 (F₇ = 13), etc. The numbers of the sequence has also been found to be ubiquitous in nature: among other things, many species of flowering plants have numbers of petals in the Fibonacci Sequence; the spiral arrangements of pineapples occur in 5s and 8s, those of pinecones in 8s and 13s, and the seeds of sunflower heads in 21s, 34s, 55s or even higher terms in the sequence; etc.

The Golden Ratio φ can be derived from the Fibonacci Sequence

In the 1750s, Robert Simson noted that the ratio of each term in the Fibonacci Sequence to the previous term approaches, with ever greater accuracy the higher the terms, a ratio of approximately 1 : 1.6180339887 (it is actually an irrational number equal to ^{(1 + √5)}⁄₂ which has since been calculated to thousands of decimal places). This value is referred to as the Golden Ratio, also known as the Golden Mean, Golden Section, Divine Proportion, etc, and is usually denoted by the Greek letter phi φ (or sometimes the capital letter Phi Φ). Essentially, two quantities are in the Golden Ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The Golden Ratio itself has many unique properties, such as ¹⁄_φ = φ - 1 (0.618...) and φ² = φ + 1 (2.618...), and there are countless examples of it to be found both in nature and in the human world.

A rectangle with sides in the ratio of 1 : φ is known as a Golden Rectangle, and many artists and architects throughout history (dating back to ancient Egypt and Greece, but particularly popular in the Renaissance art of Leonardo da Vinci and his contemporaries) have proportioned their works approximately using the Golden Ratio and Golden Rectangles, which are widely considered to be innately aesthetically pleasing. An arc connecting opposite points of ever smaller nested Golden Rectangles forms a logarithmic spiral, known as a Golden Spiral. The Golden Ratio and Golden Spiral can also be found in a surprising number of instances in Nature, from shells to flowers to animal horns to human bodies to storm systems to complete galaxies.

It should be remembered, though, that the Fibonacci Sequence was actually only a very minor element in “Liber Abaci” - indeed, the sequence only received Fibonacci's name in 1877 when Eduouard Lucas decided to pay tribute to him by naming the series after him - and that Fibonacci himself was not responsible for identifying any of the interesting mathematical properties of the sequence, its relationship to the Golden Mean and Golden Rectangles and Spirals, etc.

Fibonacci introduced lattice multiplication to Europe

However, the book's influence on medieval mathematics is undeniable, and it does also include discussions of a number of other mathematical problems such as the Chinese Remainder Theorem, perfect numbers and prime numbers, formulas for arithmetic series and for square pyramidal numbers, Euclidean geometric proofs, and a study of simultaneous linear equations along the lines of Diophantus and Al-Karaji. He also described the lattice (or sieve) multiplication method of multiplying large numbers, a method - originally pioneered by Islamic mathematicians like Al-Khwarizmi - algorithmically equivalent to long multiplication.

Neither was “Liber Abaci” Fibonacci’s only book, although it was his most important one. His “Liber Quadratorum” (“The Book of Squares”), for example, is a book on algebra, published in 1225 in which appears a statement of what is now called Fibonacci's identity - sometimes also known as Brahmagupta’s identity after the much earlier Indian mathematician who also came to the same conclusions - that the product of two sums of two squares is itself a sum of two squares e.g. (1² + 4²)(2² + 7²) = 26² + 15² = 30² + 1².

16TH CENTURY MATHEMATICS

The supermagic square shown in Albrecht Dürer's engraving "Melencolia I"

The cultural, intellectual and artistic movement of the Renaissance, which saw a resurgence of learning based on classical sources, began in Italy around the 14th Century, and gradually spread across most of Europe over the next two centuries. Science and art were still very much interconnected and intermingled at this time, as exemplified by the work of artist/scientists such as Leonardo da Vinci, and it is no surprise that, just as in art, revolutionary work in the fields of philosophy and science was soon taking place.

It is a tribute to the respect in which mathematics was held in Renaissance Europe that the famed German artist Albrecht Dürer included an order-4 magic square in his engraving "Melencolia I". In fact, it is a so-called "supermagic square" with many more lines of addition symmetry than a regular 4 x 4 magic square (see image at right). The year of the work, 1514, is shown in the two bottom central squares.

An important figure in the late 15th and early 16th Centuries is an Italian Franciscan friar called Luca Pacioli, who published a book on arithmetic, geometry and book-keeping at the end of the 15th Century which became quite popular for the mathematical puzzles it contained. It also introduced symbols for plus and minus for the first time in a printed book (although this is also sometimes attributed to Giel Vander Hoecke, Johannes Widmann and others), symbols that were to become standard notation. Pacioli also investigated the Golden Ratio of 1 : 1.618... (see the section on Fibonacci) in his 1509 book "The Divine Proportion", concluding that the number was a message from God and a source of secret knowledge about the inner beauty of things.

Basic mathematical notation, with dates of first use

During the 16th and early 17th Century, the equals, multiplication, division, radical (root), decimal and inequality symbols were gradually introduced and standardized. The use of decimal fractions and decimal arithmetic is usually attributed to the Flemish mathematician Simon Stevin the late 16th Century, although the decimal point notation was not popularized until early in the 17th Century. Stevin was ahead of his time in enjoining that all types of numbers, whether fractions, negatives, real numbers or surds (such as √2) should be treated equally as numbers in their own right.

In the Renaissance Italy of the early 16th Century, Bologna University in particular was famed for its intense public mathematics competitions. It was in just such a competion that the unlikely figure of the young, self-taught Niccolò Fontana Tartaglia revealed to the world the formula for solving first one type, and later all types, of cubic equations (equations with terms including x³), an achievement hitherto considered impossible and which had stumped the best mathematicians of China, India and the Islamic world.

Building on Tartaglia’s work, another young Italian, Lodovico Ferrari, soon devised a similar method to solve quartic equations (equations with terms including x⁴) and both solutions were published by Gerolamo Cardano. Despite a decade-long fight over the publication, Tartaglia, Cardano and Ferrari between them demonstrated the first uses of what are now known as complex numbers, combinations of real and imaginary numbers (although it fell to another Bologna resident, Rafael Bombelli, to explain what imaginary numbers really were and how they could be used). Tartaglia went on to produce other important (although largely ignored) formulas and methods, and Cardano published perhaps the first systematic treatment of probability.

With Hindu-Arabic numerals, standardized notation and the new language of algebra at their disposal, the stage was set for the European mathematical revolution of the 17th Century.

16TH CENTURY MATHEMATICS - TARTAGLIA, CARDANO & FERRARI

Niccolò Fontana Tartaglia (1499-1557)

In the Renaissance Italy of the early 16th Century, Bologna University in particular was famed for its intense public mathematics competitions. It was in just such a competition, in 1535, that the unlikely figure of the young Venetian Tartaglia first revealed a mathematical finding hitherto considered impossible, and which had stumped the best mathematicians of China, India and the Islamic world.

Niccolò Fontana became known as Tartaglia (meaning “the stammerer”) for a speech defect he suffered due to an injury he received in a battle against the invading French army. He was a poor engineer known for designing fortifications, a surveyor of topography (seeking the best means of defence or offence in battles) and a bookkeeper in the Republic of Venice.

But he was also a self-taught, but wildly ambitious, mathematician. He distinguised himself by producing, among other things, the first Italian translations of works by Archimedes and Euclid from uncorrupted Greek texts (for two centuries, Euclid's "Elements" had been taught from two Latin translations taken from an Arabic source, parts of which contained errors making them all but unusable), as well as an acclaimed compilation of mathematics of his own.

Cubic equations were first solved algebraically by del Ferro and Tartaglia

Tartaglia's greates legacy to mathematical history, though, occurred when he won the 1535 Bologna University mathematics competition by demonstrating a general algebraic formula for solving cubic equations (equations with terms including x³), something which had come to be seen by this time as an impossibility, requiring as it does an understanding of the square roots of negative numbers. In the competition, he beat Scipione del Ferro (or at least del Ferro's assistant, Fior), who had coincidentally produced his own partial solution to the cubic equation problem not long before. Although del Ferro's solution perhaps predated Tartaglia’s, it was much more limited, and Tartaglia is usually credited with the first general solution. In the highly competitive and cut-throat environment of 16th Century Italy, Tartaglia even encoded his solution in the form of a poem in an attempt to make it more difficult for other mathematicians to steal it.

Tartaglia’s definitive method was, however, leaked to Gerolamo Cardano (or Cardan), a rather eccentric and confrontational mathematician, doctor and Renaissance man, and author throughout his lifetime of some 131 books. Cardano published it himself in his 1545 book "Ars Magna" (despite having promised Tartaglia that he would not), along with the work of his own brilliant student Lodovico Ferrari. Ferrari, on seeing Tartaglia's cubic solution, had realized that he could use a similar method to solve quartic equations (equations with terms including x⁴).

In this work, Tartaglia, Cardano and Ferrari between them demonstrated the first uses of what are now known as complex numbers, combinations of real and imaginary numbers of the type a + bi, where i is the imaginary unit √-1. It fell to another Bologna resident, Rafael Bombelli, to explain, at the end of the 1560's, exactly what imaginary numbers really were and how they could be used.

Gerolamo Cardano (1501-1576)

Although both of the younger men were acknowledged in the foreword of Cardano's book, as well as in several places within its body, Tartgalia engaged Cardano in a decade-long fight over the publication. Cardano argued that, when he happened to see (some years after the 1535 competition) Scipione del Ferro's unpublished independent cubic equation solution, which was dated before Tartaglia's, he decided that his promise to Tartaglia could legitimately be broken, and he included Tartaglia's solution in his next publication, along with Ferrari's quartic solution.

Ferrari eventually came to understand cubic and quartic equations much better than Tartaglia. When Ferrari challenged Tartaglia to another public debate, Tartaglia initially accepted, but then (perhaps wisely) decided not to show up, and Ferrari won by default. Tartaglia was thoroughly discredited and became effectively unemployable.

Poor Tartaglia died penniless and unknown, despite having produced (in addition to his cubic equation solution) the first translation of Euclid’s “Elements” in a modern European language, formulated Tartaglia's Formula for the volume of a tetrahedron, devised a method to obtain binomial coefficients called Tartaglia's Triangle (an earlier version of Pascal's Triangle), and become the first to apply mathematics to the investigation of the paths of cannonballs (work which was later validated by Galileo's studies on falling bodies). Even today, the solution to cubic equations is usually known as Cardano’s Formula and not Tartgalia’s.

Ferrari, on the other hand, obtained a prestigious teaching post while still in his teens after Cardano resigned from it and recommended him, and was eventually able to retired young and quite rich, despite having started out as Cardano’s servant.

Cardano himself, an accomplished gambler and chess player, wrote a book called "Liber de ludo aleae" ("Book on Games of Chance") when he was just 25 years old, which contains perhaps the first systematic treatment of probability (as well as a section on effective cheating methods). The ancient Greeks, Romans and Indians had all been inveterate gamblers, but none of them had ever attempted to understand randomness as being governed by mathematical laws.

The circles used to generate hypocycloids are known as Cardano Circles

The book described the - now obvious, but then revolutionary - insight that, if a random event has several equally likely outcomes, the chance of any individual outcome is equal to the proportion of that outcome to all possible outcomes. The book was far ahead of its time, though, and it remained unpublished until 1663, nearly a century after his death. It was the only serious work on probability until Pascal's work in the 17th Century.

Cardano was also the first to describe hypocycloids, the pointed plane curves generated by the trace of a fixed point on a small circle that rolls within a larger circle, and the generating circles were later named Cardano (or Cardanic) circles.

The colourful Cardano remained notoriously short of money thoughout his life, largely due to his gambling habits, and was accused of heresy in 1570 after publishing a horoscope of Jesus (apparently, his own son contributed to the prosecution, bribed by Tartaglia).








 

20TH CENTURY MATHEMATICS

Fields of Mathematics

The 20th Century continued the trend of the 19th towards increasing generalization and abstraction in mathematics, in which the notion of axioms as “self-evident truths” was largely discarded in favour of an emphasis on such logical concepts as consistency and completeness.

It also saw mathematics become a major profession, involving thousands of new Ph.D.s each year and jobs in both teaching and industry, and the development of hundreds of specialized areas and fields of study, such as group theory, knot theory, sheaf theory, topology, graph theory, functional analysis, singularity theory, catastrophe theory, chaos theory, model theory, category theory, game theory, complexity theory and many more.

The eccentric British mathematician G.H. Hardy and his young Indian protégé Srinivasa Ramanujan, were just two of the great mathematicians of the early 20th Century who applied themselves in earnest to solving problems of the previous century, such as the Riemann hypothesis. Although they came close, they too were defeated by that most intractable of problems, but Hardy is credited with reforming British mathematics, which had sunk to something of a low ebb at that time, and Ramanujan proved himself to be one of the most brilliant (if somewhat undisciplined and unstable) minds of the century.

Others followed techniques dating back millennia but taken to a 20th Century level of complexity. In 1904, Johann Gustav Hermes completed his construction of a regular polygon with 65,537 sides (2¹⁶ + 1), using just a compass and straight edge as Euclid would have done, a feat that took him over ten years.

The early 20th Century also saw the beginnings of the rise of the field of mathematical logic, building on the earlier advances of Gottlob Frege, which came to fruition in the hands of Giuseppe Peano, L.E.J. Brouwer, David Hilbert and, particularly, Bertrand Russell and A.N. Whitehead, whose monumental joint work the “Principia Mathematica” was so influential in mathematical and philosophical logicism.

Part of the transcript of Hilbert’s 1900 Paris lecture, in which he set out his 23 problems

The century began with a historic convention at the Sorbonne in Paris in the summer of 1900 which is largely remembered for a lecture by the young German mathematician David Hilbert in which he set out what he saw as the 23 greatest unsolved mathematical problems of the day. These “Hilbert problems” effectively set the agenda for 20th Century mathematics, and laid down the gauntlet for generations of mathematicians to come. Of these original 23 problems, 10 have now been solved, 7 are partially solved, and 2 (the Riemann hypothesis and the Kronecker-Weber theorem on abelian extensions) are still open, with the remaining 4 being too loosely formulated to be stated as solved or not.

Hilbert was himself a brilliant mathematician, responsible for several theorems and some entirely new mathematical concepts, as well as overseeing the development of what amounted to a whole new style of abstract mathematical thinking. Hilbert's approach signalled the shift to the modern axiomatic method, where axioms are not taken to be self-evident truths. He was unfailingly optimistic about the future of mathematics, famously declaring in a 1930 radio interview “We must know. We will know!”, and was a well-loved leader of the mathematical community during the first part of the century.

However, the Austrian Kurt Gödel was soon to put some very severe constraints on what could and could not be solved, and turned mathematics on its head with his famous incompleteness theorem, which proved the unthinkable - that there could be solutions to mathematical problems which were true but which could never be proved.

Alan Turing, perhaps best known for his war-time work in breaking the German enigma code, spent his pre-war years trying to clarify and simplify Gödel’s rather abstract proof. His methods led to some conclusions that were perhaps even more devastating than Gödel’s, including the idea that there was no way of telling beforehand which problems were provable and which unprovable. But, as a spin-off, his work also led to the development of computers and the first considerations of such concepts as artificial intelligence.

With the gradual and wilful destruction of the mathematics community of Germany and Austria by the anti-Jewish Nazi regime in the 1930 and 1940s, the focus of world mathematics moved to America, particularly to the Institute for Advanced Study in Princeton, which attempted to reproduce the collegiate atmosphere of the old European universities in rural New Jersey. Many of the brightest European mathematicians, including Hermann Weyl, John von Neumann, Kurt Gödel and Albert Einstein, fled the Nazis to this safe haven.

Von Neumann’s computer architecture design

John von Neumann is considered one of the foremost mathematicians in modern history, another mathematical child prodigy who went on to make major contributions to a vast range of fields. In addition to his physical work in quantum theory and his role in the Manhattan Project and the development of nuclear physics and the hydrogen bomb, he is particularly remembered as a pioneer of game theory, and particularly for his design model for a stored-program digital computer that uses a processing unit and a separate storage structure to hold both instructions and data, a general architecture that most electronic computers follow even today.

The Soviet mathematician Andrey Kolmogorov is usually credited with laying the modern axiomatic foundations of probability theory in the 1930s, and he established a reputation as the world's leading expert in this field. He also made important contributions to the fields of topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and computational complexity.

André Weil was another refugee from the war in Europe, after narrowly avoiding death on a couple of occasions. His theorems, which allowed connections to be made between number theory, algebra, geometry and topology, are considered among the greatest achievements of modern mathematics. He was also responsible for setting up a group of French mathematicians who, under the secret nom-de-plume of Nicolas Bourbaki, wrote many influential books on the mathematics of the 20th Century.

Perhaps the greatest heir to Weil’s legacy was Alexander Grothendieck, a charismatic and beloved figure in 20th Century French mathematics. Grothendieck was a structuralist, interested in the hidden structures beneath all mathematics, and in the 1950s he created a powerful new language which enabled mathematical structures to be seen in a new way, thus allowing new solutions in number theory, geometry, even in fundamental physics. His “theory of schemes” allowed certain of Weil's number theory conjectures to be solved, and his “theory of topoi” is highly relevant to mathematical logic. In addition, he gave an algebraic proof of the Riemann-Roch theorem, and provided an algebraic definition of the fundamental group of a curve. Although, after the 1960s, Grothendieck all but abandoned mathematics for radical politics, his achievements in algebraic geometry have fundamentally transformed the mathematical landscape, perhaps no less than those of Cantor, Gödel and Hilbert, and he is considered by some to be one of the dominant figures of the whole of 20th Century mathematics.

Paul Erdös was another inspired but distinctly non-establishment figure of 20th Century mathematics. The immensely prolific and famously eccentric Hungarian mathematician worked with hundreds of different collaborators on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory. As a humorous tribute, an "Erdös number" is given to mathematicians according to their collaborative proximity to him. He was also known for offering small prizes for solutions to various unresolved problems (such as the Erdös conjecture on arithmetic progressions), some of which are still active after his death.

The Mandelbrot set, the most famous example of a fractal

The field of complex dynamics (which is defined by the iteration of functions on complex number spaces) was developed by two Frenchmen, Pierre Fatou and Gaston Julia, early in the 20th Century. But it only really gained much attention in the 1970s and 1980s with the beautiful computer plottings of Julia sets and, particularly, of the Mandelbrot sets of yet another French mathematician, Benoît Mandelbrot. Julia and Mandelbrot fractals are closely related, and it was Mandelbrot who coined the term fractal, and who became known as the father of fractal geometry.

The Mandelbrot set involves repeated iterations of complex quadratic polynomial equations of the form z_n+1 = z_n² + c, (where z is a number in the complex plane of the form x + iy). The iterations produce a form of feedback based on recursion, in which smaller parts exhibit approximate reduced-size copies of the whole, and which are infinitely complex (so that, however much one zooms in and magifies a part, it exhibits just as much complexity).

Paul Cohen is an example of a second generation Jewish immigrant who followed the American dream to fame and success. His work rocked the mathematical world in the 1960s, when he proved that Cantor's continuum hypothesis about the possible sizes of infinite sets (one of Hilbert’s original 23 problems) could be both true AND not true, and that there were effectively two completely separate but valid mathematical worlds, one in which the continuum hypothesis was true and one where it was not. Since this result, all modern mathematical proofs must insert a statement declaring whether or not the result depends on the continuum hypothesis.

Another of Hilbert’s problems was finally resolved in 1970, when the young Russian Yuri Matiyasevich finally proved that Hilbert’s tenth problem was impossible, i.e. that there is no general method for determining when polynomial equations have a solution in whole numbers. In arriving at his proof, Matiyasevich built on decades of work by the American mathematician Julia Robinson, in a great show of internationalism at the height of the Cold War.

In additon to complex dynamics, another field that benefitted greatly from the advent of the electronic computer, and particulary from its ability to carry out a huge number of repeated iterations of simple mathematical formulas which would be impractical to do by hand, was chaos theory. Chaos theory tells us that some systems seem to exhibit random behaviour even though they are not random at all, and conversely some systems may have roughly predictable behaviour but are fundamentally unpredictable in any detail. The possible behaviours that a chaotic system may have can also be mapped graphically, and it was discovered that these mappings, known as "strange attractors", are fractal in nature (the more you zoom in, the more detail can be seen, although the overall pattern remains the same).

An early pioneer in modern chaos theory was Edward Lorenz, whose interest in chaos came about accidentally through his work on weather prediction. Lorenz's discovery came in 1961, when a computer model he had been running was actually saved using three-digit numbers rather than the six digits he had been working with, and this tiny rounding error produced dramatically different results. He discovered that small changes in initial conditions can produce large changes in the long-term outcome - a phenomenon he described by the term “butterfly effect” - and he demonstrated this with his Lorenz attractor, a fractal structure corresponding to the behaviour of the Lorenz oscillator (a 3-dimensional dynamical system that exhibits chaotic flow).

Example of a four-colour map

1976 saw a proof of the four colour theorem by Kenneth Appel and Wolfgang Haken, the first major theorem to be proved using a computer. The four colour conjecture was first proposed in 1852 by Francis Guthrie (a student of Augustus De Morgan), and states that, in any given separation of a plane into contiguous regions (called a “map”) the regions can be coloured using at most four colours so that no two adjacent regions have the same colour. One proof was given by Alfred Kempe in 1879, but it was shown to be incorrect by Percy Heawood in 1890 in proving the five colour theorem. The eventual proof that only four colours suffice turned out to be significantly harder. Appel and Haken’s solution required some 1,200 hours of computer time to examine around 1,500 configurations.

Also in the 1970s, origami became recognized as a serious mathematical method, in some cases more powerful than Euclidean geometry. In 1936, Margherita Piazzola Beloch had shown how a length of paper could be folded to give the cube root of its length, but it was not until 1980 that an origami method was used to solve the "doubling the cube" problem which had defeated ancient Greek geometers. An origami proof of the equally intractible "trisecting the angle" problem followed in 1986. The Japanese origami expert Kazuo Haga has at least three mathematical theorems to his name, and his unconventional folding techniques have demonstrated many unexpected geometrical results.

The British mathematician Andrew Wiles finally proved Fermat’s Last Theorem for ALL numbers in 1995, some 350 years after Fermat’s initial posing. It was an achievement Wiles had set his sights on early in life and pursued doggedly for many years. In reality, though, it was a joint effort of several steps involving many mathematicans over several years, including Goro Shimura, Yutaka Taniyama, Gerhard Frey, Jean-Pierre Serre and Ken Ribet, with Wiles providing the links and the final synthesis and, specifically, the final proof of the Taniyama-Shimura Conjecture for semi-stable elliptic curves. The proof itself is over 100 pages long.

The most recent of the great conjectures to be proved was the Poincaré Conjecture, which was solved in 2002 (over 100 years after Poincaré first posed it) by the eccentric and reclusive Russian mathematician Grigori Perelman. However, Perelman, who lives a frugal life with his mother in a suburb of St. Petersburg, turned down the $1 million prize, claiming that "if the proof is correct then no other recognition is needed". The conjecture, now a theorem, states that, if a loop in connected, finite boundaryless 3-dimensional 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 a three-dimensional sphere. Perelman provided an elegant but extremely complex solution involving the ways in which 3-dimensional shapes can be “wrapped up” in even higher dimensions. Perelman has also made landmark contributions to Riemannian geometry and geometric topology.

John Nash, the American economist and mathematician whose battle against paranoid schizophrenia has recently been popularized by the Hollywood movie “A Beautiful Mind”, did some important work in game theory, differential geometry and partial differential equations which have provided insight into the forces that govern chance and events inside complex systems in daily life, such as in market economics, computing, artificial intelligence, accounting and military theory.

The Englishman John Horton Conway established the rules for the so-called "Game of Life" in 1970, an early example of a "cellular automaton" in which patterns of cells evolve and grow in a grid, which became extremely popular among computer scientists. He has made important contributions to many branches of pure mathematics, such as game theory, group theory, number theory and geometry, and has also come up with some wonderful-sounding concepts like surreal numbers, the grand antiprism and monstrous moonshine, as well as mathematical games such as Sprouts, Philosopher's Football and the Soma Cube.

Other mathematics-based recreational puzzles became even more popular among the general public, including Rubik's Cube (1974) and Sudoku (1980), both of which developed into full-blown crazes on a scale only previously seen with the 19th Century fads of Tangrams (1817) and the Fifteen puzzle (1879). In their turn, they generated attention from serious mathematicians interested in exploring the theoretical limits and underpinnings of the games.

Computers continue to aid in the identification of phenomena such as Mersenne primes numbers (a prime number that is one less than a power of two - see the section on 17th Century Mathematics). In 1952, an early computer known as SWAC identified 2²⁵⁷-1 as the 13th Mersenne prime number, the first new one to be found in 75 years, before going on to identify several more even larger.

Approximations for π

With the advent of the Internet in the 1990s, the Great Internet Mersenne Prime Search (GIMPS), a collaborative project of volunteers who use freely available computer software to search for Mersenne primes, has led to another leap in the discovery rate. Currently, the 13 largest Mersenne primes were all discovered in this way, and the largest (the 45th Mersenne prime number and also the largest known prime number of any kind) was discovered in 2009 and contains nearly 13 million digits. The search also continues for ever more accurate computer approximations for the irrational number π, with the current record standing at over 5 trillion decimal places.

The P versus NP problem, introduced in 1971 by the American-Canadian Stephen Cook, is a major unsolved problem in computer science and the burgeoning field of complexity theory, and is another of the Clay Mathematics Institute's million dollar Millennium Prize problems. At its simplest, it asks whether every problem whose solution can be efficiently checked by a computer can also be efficiently solved by a computer (or put another way, whether questions exist whose answer can be quickly checked, but which require an impossibly long time to solve by any direct procedure). The solution to this simple enough sounding problem, usually known as Cook's Theorem or the Cook-Levin Theorem, has eluded mathematicians and computer scientists for 40 years. A possible solution by Vinay Deolalikar in 2010, claiming to prove that P is not equal to NP (and thus such insolulable-but-easily-checked problems do exist), has attracted much attention but has not as yet been fully accepted by the computer science community.
 

20TH CENTURY MATHEMATICS

Fields of Mathematics

The 20th Century continued the trend of the 19th towards increasing generalization and abstraction in mathematics, in which the notion of axioms as “self-evident truths” was largely discarded in favour of an emphasis on such logical concepts as consistency and completeness.

It also saw mathematics become a major profession, involving thousands of new Ph.D.s each year and jobs in both teaching and industry, and the development of hundreds of specialized areas and fields of study, such as group theory, knot theory, sheaf theory, topology, graph theory, functional analysis, singularity theory, catastrophe theory, chaos theory, model theory, category theory, game theory, complexity theory and many more.

The eccentric British mathematician G.H. Hardy and his young Indian protégé Srinivasa Ramanujan, were just two of the great mathematicians of the early 20th Century who applied themselves in earnest to solving problems of the previous century, such as the Riemann hypothesis. Although they came close, they too were defeated by that most intractable of problems, but Hardy is credited with reforming British mathematics, which had sunk to something of a low ebb at that time, and Ramanujan proved himself to be one of the most brilliant (if somewhat undisciplined and unstable) minds of the century.

Others followed techniques dating back millennia but taken to a 20th Century level of complexity. In 1904, Johann Gustav Hermes completed his construction of a regular polygon with 65,537 sides (2¹⁶ + 1), using just a compass and straight edge as Euclid would have done, a feat that took him over ten years.

The early 20th Century also saw the beginnings of the rise of the field of mathematical logic, building on the earlier advances of Gottlob Frege, which came to fruition in the hands of Giuseppe Peano, L.E.J. Brouwer, David Hilbert and, particularly, Bertrand Russell and A.N. Whitehead, whose monumental joint work the “Principia Mathematica” was so influential in mathematical and philosophical logicism.

Part of the transcript of Hilbert’s 1900 Paris lecture, in which he set out his 23 problems

The century began with a historic convention at the Sorbonne in Paris in the summer of 1900 which is largely remembered for a lecture by the young German mathematician David Hilbert in which he set out what he saw as the 23 greatest unsolved mathematical problems of the day. These “Hilbert problems” effectively set the agenda for 20th Century mathematics, and laid down the gauntlet for generations of mathematicians to come. Of these original 23 problems, 10 have now been solved, 7 are partially solved, and 2 (the Riemann hypothesis and the Kronecker-Weber theorem on abelian extensions) are still open, with the remaining 4 being too loosely formulated to be stated as solved or not.

Hilbert was himself a brilliant mathematician, responsible for several theorems and some entirely new mathematical concepts, as well as overseeing the development of what amounted to a whole new style of abstract mathematical thinking. Hilbert's approach signalled the shift to the modern axiomatic method, where axioms are not taken to be self-evident truths. He was unfailingly optimistic about the future of mathematics, famously declaring in a 1930 radio interview “We must know. We will know!”, and was a well-loved leader of the mathematical community during the first part of the century.

However, the Austrian Kurt Gödel was soon to put some very severe constraints on what could and could not be solved, and turned mathematics on its head with his famous incompleteness theorem, which proved the unthinkable - that there could be solutions to mathematical problems which were true but which could never be proved.

Alan Turing, perhaps best known for his war-time work in breaking the German enigma code, spent his pre-war years trying to clarify and simplify Gödel’s rather abstract proof. His methods led to some conclusions that were perhaps even more devastating than Gödel’s, including the idea that there was no way of telling beforehand which problems were provable and which unprovable. But, as a spin-off, his work also led to the development of computers and the first considerations of such concepts as artificial intelligence.

With the gradual and wilful destruction of the mathematics community of Germany and Austria by the anti-Jewish Nazi regime in the 1930 and 1940s, the focus of world mathematics moved to America, particularly to the Institute for Advanced Study in Princeton, which attempted to reproduce the collegiate atmosphere of the old European universities in rural New Jersey. Many of the brightest European mathematicians, including Hermann Weyl, John von Neumann, Kurt Gödel and Albert Einstein, fled the Nazis to this safe haven.

Von Neumann’s computer architecture design

John von Neumann is considered one of the foremost mathematicians in modern history, another mathematical child prodigy who went on to make major contributions to a vast range of fields. In addition to his physical work in quantum theory and his role in the Manhattan Project and the development of nuclear physics and the hydrogen bomb, he is particularly remembered as a pioneer of game theory, and particularly for his design model for a stored-program digital computer that uses a processing unit and a separate storage structure to hold both instructions and data, a general architecture that most electronic computers follow even today.

The Soviet mathematician Andrey Kolmogorov is usually credited with laying the modern axiomatic foundations of probability theory in the 1930s, and he established a reputation as the world's leading expert in this field. He also made important contributions to the fields of topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and computational complexity.

André Weil was another refugee from the war in Europe, after narrowly avoiding death on a couple of occasions. His theorems, which allowed connections to be made between number theory, algebra, geometry and topology, are considered among the greatest achievements of modern mathematics. He was also responsible for setting up a group of French mathematicians who, under the secret nom-de-plume of Nicolas Bourbaki, wrote many influential books on the mathematics of the 20th Century.

Perhaps the greatest heir to Weil’s legacy was Alexander Grothendieck, a charismatic and beloved figure in 20th Century French mathematics. Grothendieck was a structuralist, interested in the hidden structures beneath all mathematics, and in the 1950s he created a powerful new language which enabled mathematical structures to be seen in a new way, thus allowing new solutions in number theory, geometry, even in fundamental physics. His “theory of schemes” allowed certain of Weil's number theory conjectures to be solved, and his “theory of topoi” is highly relevant to mathematical logic. In addition, he gave an algebraic proof of the Riemann-Roch theorem, and provided an algebraic definition of the fundamental group of a curve. Although, after the 1960s, Grothendieck all but abandoned mathematics for radical politics, his achievements in algebraic geometry have fundamentally transformed the mathematical landscape, perhaps no less than those of Cantor, Gödel and Hilbert, and he is considered by some to be one of the dominant figures of the whole of 20th Century mathematics.

Paul Erdös was another inspired but distinctly non-establishment figure of 20th Century mathematics. The immensely prolific and famously eccentric Hungarian mathematician worked with hundreds of different collaborators on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory. As a humorous tribute, an "Erdös number" is given to mathematicians according to their collaborative proximity to him. He was also known for offering small prizes for solutions to various unresolved problems (such as the Erdös conjecture on arithmetic progressions), some of which are still active after his death.

The Mandelbrot set, the most famous example of a fractal

The field of complex dynamics (which is defined by the iteration of functions on complex number spaces) was developed by two Frenchmen, Pierre Fatou and Gaston Julia, early in the 20th Century. But it only really gained much attention in the 1970s and 1980s with the beautiful computer plottings of Julia sets and, particularly, of the Mandelbrot sets of yet another French mathematician, Benoît Mandelbrot. Julia and Mandelbrot fractals are closely related, and it was Mandelbrot who coined the term fractal, and who became known as the father of fractal geometry.

The Mandelbrot set involves repeated iterations of complex quadratic polynomial equations of the form z_n+1 = z_n² + c, (where z is a number in the complex plane of the form x + iy). The iterations produce a form of feedback based on recursion, in which smaller parts exhibit approximate reduced-size copies of the whole, and which are infinitely complex (so that, however much one zooms in and magifies a part, it exhibits just as much complexity).

Paul Cohen is an example of a second generation Jewish immigrant who followed the American dream to fame and success. His work rocked the mathematical world in the 1960s, when he proved that Cantor's continuum hypothesis about the possible sizes of infinite sets (one of Hilbert’s original 23 problems) could be both true AND not true, and that there were effectively two completely separate but valid mathematical worlds, one in which the continuum hypothesis was true and one where it was not. Since this result, all modern mathematical proofs must insert a statement declaring whether or not the result depends on the continuum hypothesis.

Another of Hilbert’s problems was finally resolved in 1970, when the young Russian Yuri Matiyasevich finally proved that Hilbert’s tenth problem was impossible, i.e. that there is no general method for determining when polynomial equations have a solution in whole numbers. In arriving at his proof, Matiyasevich built on decades of work by the American mathematician Julia Robinson, in a great show of internationalism at the height of the Cold War.

In additon to complex dynamics, another field that benefitted greatly from the advent of the electronic computer, and particulary from its ability to carry out a huge number of repeated iterations of simple mathematical formulas which would be impractical to do by hand, was chaos theory. Chaos theory tells us that some systems seem to exhibit random behaviour even though they are not random at all, and conversely some systems may have roughly predictable behaviour but are fundamentally unpredictable in any detail. The possible behaviours that a chaotic system may have can also be mapped graphically, and it was discovered that these mappings, known as "strange attractors", are fractal in nature (the more you zoom in, the more detail can be seen, although the overall pattern remains the same).

An early pioneer in modern chaos theory was Edward Lorenz, whose interest in chaos came about accidentally through his work on weather prediction. Lorenz's discovery came in 1961, when a computer model he had been running was actually saved using three-digit numbers rather than the six digits he had been working with, and this tiny rounding error produced dramatically different results. He discovered that small changes in initial conditions can produce large changes in the long-term outcome - a phenomenon he described by the term “butterfly effect” - and he demonstrated this with his Lorenz attractor, a fractal structure corresponding to the behaviour of the Lorenz oscillator (a 3-dimensional dynamical system that exhibits chaotic flow).

Example of a four-colour map

1976 saw a proof of the four colour theorem by Kenneth Appel and Wolfgang Haken, the first major theorem to be proved using a computer. The four colour conjecture was first proposed in 1852 by Francis Guthrie (a student of Augustus De Morgan), and states that, in any given separation of a plane into contiguous regions (called a “map”) the regions can be coloured using at most four colours so that no two adjacent regions have the same colour. One proof was given by Alfred Kempe in 1879, but it was shown to be incorrect by Percy Heawood in 1890 in proving the five colour theorem. The eventual proof that only four colours suffice turned out to be significantly harder. Appel and Haken’s solution required some 1,200 hours of computer time to examine around 1,500 configurations.

Also in the 1970s, origami became recognized as a serious mathematical method, in some cases more powerful than Euclidean geometry. In 1936, Margherita Piazzola Beloch had shown how a length of paper could be folded to give the cube root of its length, but it was not until 1980 that an origami method was used to solve the "doubling the cube" problem which had defeated ancient Greek geometers. An origami proof of the equally intractible "trisecting the angle" problem followed in 1986. The Japanese origami expert Kazuo Haga has at least three mathematical theorems to his name, and his unconventional folding techniques have demonstrated many unexpected geometrical results.

The British mathematician Andrew Wiles finally proved Fermat’s Last Theorem for ALL numbers in 1995, some 350 years after Fermat’s initial posing. It was an achievement Wiles had set his sights on early in life and pursued doggedly for many years. In reality, though, it was a joint effort of several steps involving many mathematicans over several years, including Goro Shimura, Yutaka Taniyama, Gerhard Frey, Jean-Pierre Serre and Ken Ribet, with Wiles providing the links and the final synthesis and, specifically, the final proof of the Taniyama-Shimura Conjecture for semi-stable elliptic curves. The proof itself is over 100 pages long.

The most recent of the great conjectures to be proved was the Poincaré Conjecture, which was solved in 2002 (over 100 years after Poincaré first posed it) by the eccentric and reclusive Russian mathematician Grigori Perelman. However, Perelman, who lives a frugal life with his mother in a suburb of St. Petersburg, turned down the $1 million prize, claiming that "if the proof is correct then no other recognition is needed". The conjecture, now a theorem, states that, if a loop in connected, finite boundaryless 3-dimensional 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 a three-dimensional sphere. Perelman provided an elegant but extremely complex solution involving the ways in which 3-dimensional shapes can be “wrapped up” in even higher dimensions. Perelman has also made landmark contributions to Riemannian geometry and geometric topology.

John Nash, the American economist and mathematician whose battle against paranoid schizophrenia has recently been popularized by the Hollywood movie “A Beautiful Mind”, did some important work in game theory, differential geometry and partial differential equations which have provided insight into the forces that govern chance and events inside complex systems in daily life, such as in market economics, computing, artificial intelligence, accounting and military theory.

The Englishman John Horton Conway established the rules for the so-called "Game of Life" in 1970, an early example of a "cellular automaton" in which patterns of cells evolve and grow in a grid, which became extremely popular among computer scientists. He has made important contributions to many branches of pure mathematics, such as game theory, group theory, number theory and geometry, and has also come up with some wonderful-sounding concepts like surreal numbers, the grand antiprism and monstrous moonshine, as well as mathematical games such as Sprouts, Philosopher's Football and the Soma Cube.

Other mathematics-based recreational puzzles became even more popular among the general public, including Rubik's Cube (1974) and Sudoku (1980), both of which developed into full-blown crazes on a scale only previously seen with the 19th Century fads of Tangrams (1817) and the Fifteen puzzle (1879). In their turn, they generated attention from serious mathematicians interested in exploring the theoretical limits and underpinnings of the games.

Computers continue to aid in the identification of phenomena such as Mersenne primes numbers (a prime number that is one less than a power of two - see the section on 17th Century Mathematics). In 1952, an early computer known as SWAC identified 2²⁵⁷-1 as the 13th Mersenne prime number, the first new one to be found in 75 years, before going on to identify several more even larger.

Approximations for π

With the advent of the Internet in the 1990s, the Great Internet Mersenne Prime Search (GIMPS), a collaborative project of volunteers who use freely available computer software to search for Mersenne primes, has led to another leap in the discovery rate. Currently, the 13 largest Mersenne primes were all discovered in this way, and the largest (the 45th Mersenne prime number and also the largest known prime number of any kind) was discovered in 2009 and contains nearly 13 million digits. The search also continues for ever more accurate computer approximations for the irrational number π, with the current record standing at over 5 trillion decimal places.

The P versus NP problem, introduced in 1971 by the American-Canadian Stephen Cook, is a major unsolved problem in computer science and the burgeoning field of complexity theory, and is another of the Clay Mathematics Institute's million dollar Millennium Prize problems. At its simplest, it asks whether every problem whose solution can be efficiently checked by a computer can also be efficiently solved by a computer (or put another way, whether questions exist whose answer can be quickly checked, but which require an impossibly long time to solve by any direct procedure). The solution to this simple enough sounding problem, usually known as Cook's Theorem or the Cook-Levin Theorem, has eluded mathematicians and computer scientists for 40 years. A possible solution by Vinay Deolalikar in 2010, claiming to prove that P is not equal to NP (and thus such insolulable-but-easily-checked problems do exist), has attracted much attention but has not as yet been fully accepted by the computer science community.

LIST OF IMPORTANT MATHEMATICIANS

This is a chronological list of some of the most important mathematicians in history and their major achievments, as well as some very early achievements in mathematics for which individual contributions can not be acknowledged.

Where the mathematicians have individual pages in this website, these pages are linked; otherwise more information can usually be obtained from the general page relating to the particular period in history, or from the list of sources used. A more detailed and comprehensive mathematical chronology can be found at http://www-groups.dcs.st-and.ac.uk/~history/Chronology/full.html.