A Brief History of Mathematics
People seem compelled to organize. They also have a practical need to count certain things: cattle, cornstalks, and so on. There is the need to deal with simple geometrical situations in providing shelter and dealing with land. Once some form of writing is added into the mix, mathematics cannot be far behind. It might even be said that the symbolic approach precedes and leads to the invention of writing.
Archaeologists, anthropologists, linguists and others studying early societies have found that number ideas evolve slowly. There will typically be a different word or symbol for two people, two birds, or two stones. Only slowly does the idea of 'two' become independent from the things that there are two of. Similarly, of course, for other numbers. In fact, specific numbers beyond three are unknown in some lesser developed languages. A bit of this usage hangs on in our modern English when we speak, for example, of a flock of geese, but a school of fish.
The Maya, the Chinese, the Civilization of the Indus Valley, the Egyptians, and the region of Mesopotamia between the Tigris and Euphrates rivers -- all had developed impressive bodies of mathematical knowledge by the dawn of their written histories. In each case, what we know of their mathematics comes from a combination of archaeology, the references of later writers, and their own written record.
Mathematical documents from Ancient Egypt date back to 1900 B.C. The practical need to redraw field boundaries after the annual flooding of the Nile, and the fact that there was a small leisure class with time to think, helped to create a problem oriented, practical mathematics. A base-ten numeration system was able to handle positive whole numbers and some fractions. Algebra was developed only far enough to solve linear equations and, of course, calculate the volume of a pyramid. It is thought that only special cases of The Pythagorean Theorem were known; ropes knotted in the ratio 3:4:5 may have been used to construct right angles.
What we know of the mathematics of Mesopotamia comes from cuneiform writing on clay tablets which date back as far as 2100 B.C. Sixty was the number system base -- a system that we have inherited and preserve to this day in our measurement of time and angles. Among the clay tablets are found multiplication tables, tables of reciprocals, squares and square roots. A general method for solving quadratic equations was available, and a few equations of higher degree could be handled. From what we can see today, both the Egyptians and the Mesopotamians (or Babylonians) stuck to specific practical problems; the idea of stating and proving general theorems did not seem to arise in either civilization.
Chinese mathematics -- a vast and powerful body of knowledge --, although mainly practical and problem oriented, did contain general statements and proofs. A method similar to Gaussian Reduction with back-substitution for solving systems of linear equations was known two thousand years earlier in China than in the West. The value ofp was known to seven decimal places by 500 A.D., far in advance of the West.
In India mathematics was also mainly practical. Methods of solving equations were largely centered around problems in astronomy. Negative and irrational numbers were used. Of course, India is noted for developing the concept of zero, that was passed into Western mathematics via the Arabic tradition, and is so important as a place holder in our modern decimal number system.
The Classic Maya civilization (250 BC to 900 AD) also developed the zero and used it as a place holder in a base-twenty numeration system. Again, astronomy played a central role in their religion and motivated them to develop mathematics. It is noteworthy that the Maya calendar was more accurate than the European at the time the Spanish landed in The Yukatan Peninsula.
The axiomatic method came into full force in Ancient Greek times; it has characterized mathematics ever since. Geometry was center stage in ancient times. Mathematical models, or idealizations of the real world, were built around points, lines, and planes. Numbers were represented as lengths of line segments. Modern mathematics still relies on the axiomatic method, but tends to be more algebraically based.
Key to the axiomatic method are abstraction and proof. For example, the idea of a point as a pure location with no extension is an abstraction since a point cannot physically exist. A dot differs from a point in that a dot has extension, and represents only an approximate location. Nevertheless, since they can be seen, we use dots to represent points which cannot be seen. Lines, planes and circles are also abstract ideas. That is, they represent idealizations, rather than concrete objects which actually exist. After all, a plane has no thickness, and cannot be anything except a boundary between two regions in space.
An interest in investigating the properties of abstract objects characterizes Greek mathematics. Precise definitions; a small number of commonly accepted assumptions called axioms or postulates are made; then general results (lemmas, theorems, and corollaries) are proved using logic.
One of the best ways to learn more about the history of mathematics is by looking into the lives and work of mathematicians. What follows is a brief list of mathematicians. You can read about each by clicking on the hyperlinks. Use the back button on your browser to get back here.
The Middle Ages
In 476 A.D. The Roman Empire came to an end in the West; the last author of mathematical textbooks, Boethius, was executed in 524; the Eastern Roman Emperor, Justinian, closed the academies in Athens in 529 -- The Middle Ages had been born -- Mathematics, along with the rest of scholarly life, would fall into a decline which would last 1000 years.
Fortunately, during this period Chinese mathematics, the mathematics of India and The Arabic World would continue to flourish. Our modern base-ten number system featuring zero as a place holder was developed in the Eighth Century in India. The basis for algebra was developed in The Arabic World in the Eighth and Ninth Centuries. In fact, the word algebra comes from the Arabic al-jabr which refers to transposing a quantity from one side of an equation to the other.
One of the few bright spots in European mathematics during this period was the work of Fibonacci (1175-1250 A.D). He was the son of an Italian merchant who traveled widely and studied under a Muslim teacher. He helped to open Europe to the Arabic mathematical methods, including the use of 'Arabic Numerals,' which actually were invented in India, as we have seen. Many cegep students will have studied the Fibonacci sequence which has broad use in far-flung areas of mathematics.
By about 1500 A.D. the intellectual climate of Europe was changing. The Middle Ages were coming to an end and the Modern World was being born. Each century from that time until the present day would see the creation of powerful, new, mathematics.
The Sixteenth Century
The 1500's saw the emergence of what we recognize as mathematics in the modern world as opposed to the geometrical discussions of ancient times. Negative numbers were slowly gaining acceptance; the + and - signs made their debut; in accounting, Arabic numerals and double-entry book keeping came into use. Many people contributed to the more symbolic, algebraically based, mathematics that was coming into being.
Girolamo Cardano published his Ars Magne in 1545. In this work he presented for the first time a general solution of the cubic equation, and some special cases of the quartic, or fourth-degree polynomial, equation. This sparked a great deal of enthusiasm and the impetus lasted for centuries as mathematicians tried to solve fifth, and higher, degree equations in a general way. Only in the 1820's did Galois and Abel show that a general solution of fifth and higher degree equations was not possible. In the mean time a great deal of fresh mathematics was spun off as a by-product of the quest. Cardano, himself, lead an outrageous life full enough for several good biographies.
François Viète developed the first system of symbolic algebra. He introduced the use of braces and parentheses, and used the + and - signs along with a number of abbreviations for other operations. He was the first to make the crucial distinction between variable quantities and constant unknowns. In his work is found most of the methods of ordinary algebra as we know it today. He even foresaw the invention of logarithms in the next century by using the trig identity
The Seventeenth Century
The 1600's were an especially high point in scientific and mathematical history. This is the century of Kepler,Galileo, Descartes, Newton, and Leibniz . But, one of the most exciting advancements of the time was the introduction of Logarithms in 1614 by John Napier. Logarithms greatly reduced the labour involved in calculations, and was welcomed by a wider public than most mathematical ideas.
However, the two big new ideas are Descartes' founding of analytic geometry, or geometry based on algebra, and the simultaneous but independent invention of calculus by Newton and Leibniz. This is also the time when Fermatproposed his famous "Last Theorem" which has only just been proved by Andrew Wiles in the 1990's. As with the quest to find a general solution for the fifth degree (quintic) polynomial equation, the challenge presented by Fermat occupied great minds over a period of centuries and produced enormously rich benefits, but no solution until recently. Fermat's main contribution to mathematics was, however, the founding of number theory -- that branch of mathematics which deals with the arithmetic properties of the natural numbers.
Blaise Pascal worked closely with Fermat on number theory and also founded probability as we now know it. His name is commemorated in Pascal's Triangle as well as the Pascal programming language. In England, John Wallis, developed the analytic approach to the conic sections, an area dear to the hearts of many students even today. The Binomial Theorem, another favourite which dates to this period, was introduced by Newton himself. The Seventeenth Century, along with the works of Ancient Greece, establish the roots of the mathematical tradition which lives to the present day.
The Eighteenth Century
With calculus at its center, an ever widening body of knowledge began to take shape. The frontiers of mathematics in the Eighteenth Century included differential equations, infinite series, the study of planetary orbits, the theory of numbers, solutions to algebraic equations, probability theory, and complex numbers. It is possible here only to mention a few of the key figures involved.
Joseph Louis Lagrange (1736-1813) may have been France's greatest mathematician of the century. Like several of the top mathematicians of the era he was appointed to the Berlin Academy as Court Mathematician to Frederic the Great. A shy and quiet man, he extended greatly our understanding of solutions to algebraic equations, and of planetary orbits. Another great French mathematician was Pierre Simon de Laplace (1749-1827.) A more outgoing and practical person than Lagrange, his greatest contribution may have been in the area of probability theory. Laplace treated mathematics as a tool, as a means to an end; whereas Lagrange considered mathematics as a thing of great beauty -- like poetry -- and created mathematics as an end in itself. Another important figure wasD'Alembert (1717-1783), who did significant work in differential equations, sequences and series, mechanics, and astronomy; he was a member of the French Academie des Sciences.
At some point mention should be made of the Bernoulli's. This Swiss family produced at least thirteen mathematicians over a period of two centuries. In the Eighteenth Century two brothers, Jacob and Johann, played a major role. Another Bernoulli, Jean, tutored Leonard Euler (1707-1783) who was to become, without doubt, the century's greatest mathematician. Euler divided his career between the court of Frederic the Great, at Berlin, and the Russian Academy, at Saint Petersberg. His personal life was quiet and apparently happy despite much hardship, including blindness in old age. Amazingly, he was able to continue doing original mathematics even after the loss of his eyesight by dictating his work to others. Euler's output was truly amazing; his collected works fill 74 volumes. And, he frequently held back his results so that others could claim some credit. Besides developing the use of complex numbers and founding what we now know as topology, he introduced many of today's familiar notations, including p, S, e, log x, sin x, cos x, f(x) for functions, and others.
The Nineteenth Century
By the 1820's Augustin-Louis Cauchy (1789-1857) could state a formal definition for the limit equivalent to the modern d, e-definition; this advancement soothed centuries of haggling over the true meaning of differentials. Modern abstract algebra was getting started with the invention of group theory by Evariste Galois (1811-1832) and Niels Abel (1802-1829). Galois and Abel gave us some of the most beautiful mathematics ever written, but both men lead tragic lives -- Galois was killed in a senseless duel at age 22, and Abel died of disease and starvation brought on by extreme poverty at age 27.
Karl Frederic Gauss (1777-1854) was the greatest mathematician of the nineteenth century, and one of the greatest of all times. Despite having been born into a working-class family with a father who did not value education, he got his doctorate proving in his thesis The Fundamental Theorem of Algebra which states that an nth-degree polynomial has n roots in complex numbers.
Apart from extending calculus to the complex numbers and developing more abstract algebras, mathematics branched out in various ways. Non-Euclidean geometry is the study of geometries which result from modifications of Euclid's axioms. Along with Reimann, the names of Lobachevsky and Bolyai are impotortant here. George Boole and Georg Cantor were key in the foundations of set theory and mathematical logic. Karl Pearson (1857-1936) founded statistics as we know it today.
The Twentieth Century
By the beginning of The Twentieth Century mathematics had grown wide and deep, so vast that it is impossible to summarize the subject here. Let us just mention one thread.
The invention of mathematical logic lead to a deep analysis of the fundamentals underlying mathematics. It seemed that in the background there was a desire to mechanize intelligent thought itself. All of this was, of course, closely related to the impending introduction of computers. Then, in 1931, Kurt Gödel proved that statements can be formed that are neither provable nor disprovable in any complete and consistent axiom set. It follows that within a given mathematical system it is not possible to prove or disprove all of the statements that can be formed. Essentially, what Gödel did was to confirm that the human mind, and its spark of insight, can never be replaced by mechanical processes.
A few of the key figures in the subject this century are:
- Henri Poicaré
- John Von Neumann
- Bertrand Russell
- Kurt Gödel
- David Hilbert
- Andrew Wiles
- Luitzen Brouwer
|Alphabetical list of History Topics||History Topics Index|
Mathematics starts with counting. It is not reasonable, however, to suggest that early counting was mathematics. Only when some record of the counting was kept and, therefore, some representation of numbers occurred can mathematics be said to have started.
In Babylonia mathematics developed from 2000 BC. Earlier a place value notation number system had evolved over a lengthy period with a number base of 60. It allowed arbitrarily large numbers and fractions to be represented and so proved to be the foundation of more high powered mathematical development.
Number problems such as that of the Pythagorean triples (a,b,c) with a2+b2 = c2 were studied from at least 1700 BC. Systems of linear equations were studied in the context of solving number problems. Quadratic equations were also studied and these examples led to a type of numerical algebra.
Geometric problems relating to similar figures, area and volume were also studied and values obtained for π.
The Babylonian basis of mathematics was inherited by the Greeks and independent development by the Greeks began from around 450 BC. Zeno of Elea's paradoxes led to the atomic theory of Democritus. A more precise formulation of concepts led to the realisation that the rational numbers did not suffice to measure all lengths. A geometric formulation of irrational numbers arose. Studies of area led to a form of integration.
The theory of conic sections shows a high point in pure mathematical study by Apollonius. Further mathematical discoveries were driven by the astronomy, for example the study of trigonometry.
The major Greek progress in mathematics was from 300 BC to 200 AD. After this time progress continued in Islamic countries. Mathematics flourished in particular in Iran, Syria and India. This work did not match the progress made by the Greeks but in addition to the Islamic progress, it did preserve Greek mathematics. From about the 11th Century Adelard of Bath, then later Fibonacci, brought this Islamic mathematics and its knowledge of Greek mathematics back into Europe.
Major progress in mathematics in Europe began again at the beginning of the 16th Century with Pacioli, then Cardan, Tartaglia and Ferrari with the algebraic solution of cubic and quartic equations. Copernicus and Galileo revolutionised the applications of mathematics to the study of the universe.
The progress in algebra had a major psychological effect and enthusiasm for mathematical research, in particular research in algebra, spread from Italy to Stevinin Belgium and Viète in France.
The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a calculatory science with his discovery of logarithms. Cavalierimade progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry.
Progress towards the calculus continued with Fermat, who, together with Pascal, began the mathematical study of probability. However the calculus was to be the topic of most significance to evolve in the 17th Century.
Newton, building on the work of many earlier mathematicians such as his teacher Barrow, developed the calculus into a tool to push forward the study of nature. His work contained a wealth of new discoveries showing the interaction between mathematics, physics and astronomy. Newton's theory of gravitation and his theory of light take us into the 18th Century.
However we must also mention Leibniz, whose much more rigorous approach to the calculus (although still unsatisfactory) was to set the scene for the mathematical work of the 18th Century rather than that of Newton. Leibniz's influence on the various members of the Bernoulli family was important in seeing the calculus grow in power and variety of application.
The most important mathematician of the 18th Century was Euler who, in addition to work in a wide range of mathematical areas, was to invent two new branches, namely the calculus of variations and differential geometry. Euler was also important in pushing forward with research in number theory begun so effectively by Fermat.
Toward the end of the 18th Century, Lagrange was to begin a rigorous theory of functions and of mechanics. The period around the turn of the century sawLaplace's great work on celestial mechanics as well as major progress in synthetic geometry by Monge and Carnot.
The 19th Century saw rapid progress. Fourier's work on heat was of fundamental importance. In geometry Plücker produced fundamental work on analytic geometry and Steiner in synthetic geometry.
Non-euclidean geometry developed by Lobachevsky and Bolyai led to characterisation of geometry by Riemann. Gauss, thought by some to be the greatest mathematician of all time, studied quadratic reciprocity and integer congruences. His work in differential geometry was to revolutionise the topic. He also contributed in a major way to astronomy and magnetism.
The 19th Century saw the work of Galois on equations and his insight into the path that mathematics would follow in studying fundamental operations. Galois' introduction of the group concept was to herald in a new direction for mathematical research which has continued through the 20th Century.
Cauchy, building on the work of Lagrange on functions, began rigorous analysis and began the study of the theory of functions of a complex variable. This work would continue through Weierstrass and Riemann.
Algebraic geometry was carried forward by Cayley whose work on matrices and linear algebra complemented that by Hamilton and Grassmann. The end of the 19th Century saw Cantor invent set theory almost single handedly while his analysis of the concept of number added to the major work of Dedekind andWeierstrass on irrational numbers
Analysis was driven by the requirements of mathematical physics and astronomy. Lie's work on differential equations led to the study of topological groups and differential topology. Maxwell was to revolutionise the application of analysis to mathematical physics. Statistical mechanics was developed by Maxwell,Boltzmann and Gibbs. It led to ergodic theory.
The study of integral equations was driven by the study of electrostatics and potential theory. Fredholm's work led to Hilbert and the development of functional analysis.
Notation and communication
There are many major mathematical discoveries but only those which can be understood by others lead to progress. However, the easy use and understanding of mathematical concepts depends on their notation.
For example, work with numbers is clearly hindered by poor notation. Try multiplying two numbers together in Roman numerals. What is MLXXXIV times MMLLLXIX? Addition of course is a different matter and in this case Roman numerals come into their own, merchants who did most of their arithmetic adding figures were reluctant to give up using Roman numerals.
What are other examples of notational problems. The best known is probably the notation for the calculus used by Leibniz and Newton. Leibniz's notation lead more easily to extending the ideas of the calculus, while Newton's notation although good to describe velocity and acceleration had much less potential when functions of two variables were considered. British mathematicians who patriotically used Newton's notation put themselves at a disadvantage compared with the continental mathematicians who followed Leibniz.
Let us think for a moment how dependent we all are on mathematical notation and convention. Ask any mathematician to solve ax = b and you will be given the answer x = b/a. I would be very surprised if you were given the answer a = b/x, but why not. We are, often without realising it, using a convention that letters near the end of the alphabet represent unknowns while those near the beginning represent known quantities.
It was not always like this: Harriot used a as his unknown as did others at this time. The convention we use (letters near the end of the alphabet representing unknowns) was introduced by Descartes in 1637. Other conventions have fallen out of favour, such as that due to Viète who used vowels for unknowns and consonants for knowns.
Of course ax = b contains other conventions of notation which we use without noticing them. For example the sign "=" was introduced by Recorde in 1557. Also ax is used to denote the product of a and x, the most efficient notation of all since nothing has to be written!
It is quite hard to understand the brilliance of major mathematical discoveries. On the one hand they often appear as isolated flashes of brilliance although in fact they are the culmination of work by many, often less able, mathematicians over a long period.
For example the controversy over whether Newton or Leibniz discovered the calculus first can easily be answered. Neither did since Newton certainly learnt the calculus from his teacher Barrow. Of course I am not suggesting that Barrow should receive the credit for discovering the calculus, I'm merely pointing out that the calculus comes out of a long period of progress starting with Greek mathematics.
Now we are in danger of reducing major mathematical discoveries as no more than the luck of who was working on a topic at "the right time". This too would be completely unfair (although it does go some why to explain why two or more people often discovered something independently around the same time). There is still the flash of genius in the discoveries, often coming from a deeper understanding or seeing the importance of certain ideas more clearly.
How we view history
We view the history of mathematics from our own position of understanding and sophistication. There can be no other way but nevertheless we have to try to appreciate the difference between our viewpoint and that of mathematicians centuries ago. Often the way mathematics is taught today makes it harder to understand the difficulties of the past.
There is no reason why anyone should introduce negative numbers just to be solutions of equations such as x + 3 = 0. In fact there is no real reason why negative numbers should be introduced at all. Nobody owned -2 books. We can think of 2 as being some abstract property which every set of 2 objects possesses. This in itself is a deep idea. Adding 2 apples to 3 apples is one matter. Realising that there are abstract properties 2 and 3 which apply to every sets with 2 and 3 elements and that 2 + 3 = 5 is a general theorem which applies whether they are sets of apples, books or trees moves from counting into the realm of mathematics.
Negative numbers do not have this type of concrete representation on which to build the abstraction. It is not surprising that their introduction came only after a long struggle. An understanding of these difficulties would benefit any teacher trying to teach primary school children. Even the integers, which we take as the most basic concept, have a sophistication which can only be properly understood by examining the historical setting.
If you think that mathematical discovery is easy then here is a challenge to make you think. Napier, Briggs and others introduced the world to logarithms nearly 400 years ago. These were used for 350 years as the main tool in arithmetical calculations. An amazing amount of effort was saved using logarithms, how could the heavy calculations necessary in the sciences ever have taken place without logs.
Then the world changed. The pocket calculator appeared. The logarithm remains an important mathematical function but its use in calculating has gone for ever.
Here is the challenge. What will replace the calculator? You might say that this is an unfair question. However let me remind you that Napier invented the basic concepts of a mechanical computer at the same time as logs. The basic ideas that will lead to the replacement of the pocket calculator are almost certainly around us.
We can think of faster calculators, smaller calculators, better calculators but I'm asking for something as different from the calculator as the calculator itself is from log tables. I have an answer to my own question but it would spoil the point of my challenge to say what it is. Think about it and realise how difficult it was to invent non-euclidean geometries, groups, general relativity, set theory, .... .
General bibliography of about 700 items
General bibliography of about 700 items
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Astroseti (A Spanish translation of this article)
Astroseti (A Spanish translation of this article)
Article by: J J O'Connor and E F Robertson