The very first mathematical document which still survives (in the Institute for Natural Science in Brussels) is the celebrated Ishango bone. This is the 37 000 year-old fibula of a baboon, excavated in central Africa in 1960. It is carved with a curious pattern of grooves and notches, now reliably interpreted to be a diagram representing the moon's phases over a period of several months. One may only speculate on the uses, both rational and religious, of such an object in a hunter--gatherer society. It has been claimed as evidence that the first mathematicians were women.
Much more sophisticated documents are associated with the Mesopotamian societies of some 3 500 years ago. These are numerous baked clay tablets used as astronomical and mathematical tables and school (University?) problem texts. Noteworthy mathematical advances revealed by these texts include linear interpolation to form tables of ephemerides, solution of all quadratic and some cubic equations, construction of "Pythagorean" triples and sophisticated approximations to irrational square roots.
The Han dynasty, (200BC-100AD by our calendar) was a period of remarkable intellectual growth in Ancient China. The only way for the common middle-class youth to advance was by working his (yes, this society had no anti-discrimination laws) way up the ladder of the Civil Service, and the only way to advance from grade to grade was by competitive examination. The four subjects studied were calligraphy, poetry (writing, not reciting), literature and mathematics. If only we were so wise! So remarkably long-lasting texts on arithmetic, geometry and algebra were written. For example an arithmetic text contains the so-called Chinese Remainder Theorem, which is an algorithm for solving a system of divisibilities of the form "x leaves a remainder of n_i when divided by m_i;, where i=1,2,..,n;" and an algebra text contains general methods for solving linear systems, which differ in no essential respect from Gaussian elimination.
Greek mathematics forms part of the ongoing tradition of Western mathematics, and some parts of it, notably Euclidean geometry, are still taught in schools. The most outstanding feature of Greek mathematics, one which remains at the centre of the mathematical canon, is a careful consideration of rigour in proofs. The beginnings of this rigorous method date back to the 'Father of Greek Mathematics' Thales of Miletus (c. 600BC). (The Mother was undoubtedly Hypatia, 200AD, the leading teacher of mathematics and pagan philosophy in Alexandria until murdered by an ignorant mob lead by a Christian fanatic). The outstanding exemplar was of course Euclid (c. 300 BC) whose texts still flourish. Archimedes (c. 250BC) remains a paragon of mathematics for his union of rigour with creative imagination.
In particular, Archimedes developed a form of limiting process called the method of exhaustion, with which he was able to compute areas and volumes to any desired accuracy. For example, he approximated the area and perimeter of a circle by inscribing and circumscribing a 96-sided regular polygon. Almost simultaneously, the Han methematicians in China, using an inscribed regular 128-sided polygon, approximated pi even more exactly.
One thing the Greeks never managed was to solve the problem of motion. The relationship between displacement, velocity and acceleration depends essentially on the methods of calculus, which in turn requires a rigorous treatment of limits. The first steps were taken by the scholastic philosophers of Europe, notably Oresme of Paris, but it was Galileo who first solved the problem of free fall and ballistic flight. He was followed by Kepler who found the laws of planetary motion, and finally Newton who showed they were the consequence of motion under a centrally directed force.
Newton invented Calculus, and simultaneously so did the German mathematician Leibniz . There followed an undignified squabble over priority, leading to a split in the development of mathematics between England and continental Europe. By far the leading developments in Calculus took place in Europe, due to the work of the Bernoulli family and Euler.
Algebra was invented by the Arabs around the year 100 ( by their calendar; ours puts the date at 722). At first it was entirely concerned with the solution of polynomial equations. It was as a result of his deep studies of such solutions that Galois invented group theory in 1832; romantically, this event is supposed to have happened on the eve of his death in a duel, at the age of 21. Since then, Group Theory has become one of the major branches of Algebra. The most interesting results concern finite groups. We now know all the finite simple groups, and all the ways they can be pasted together to construct finite groups.
Number theory as we know it today began with the Greek mathematician Diophantos of Alexandria. It was while reading a recent Latin translation of his work that Fermat thought of the famous result now known as Fermat's Last Theorem, really a conjecture, since he didn't find a proof. Much of the outstanding development in number theory in the 18th and 19th Centuries was triggered by attempts to confirm FLT, culminating as we know in Wiles' proof of 1994. On the other hand, number theory studies bore fruit elsewhere, notably in the invention by Dedekind of commutative ring theory. Non-commutative ring theory, on the other hand, is the offspring of the attempts by the Irish mathematician Hamilton to find a three-dimensional analogue of the Complex Numbers. He didn't; instead he discovered the quaternions, which were later developed by Grassmann into hypercomplex numbers, now called finite-dimensional linear algebras.
At about the same time, Cartan and Frobenius were analysing groups by means of their complex linear representations, obtaining important results on the group structures by studying the characters of their representations, which in turn meant the classification of finite dimensional semi-simple algebras. Wedderburn and Artin extended these results to arbitrary fields.
In addition to its connection to algebraic number theory, commutative ring theory arose also from the study of algebraic geometry, that is curves and surfaces described by their equations. The concrete tools used were the rings of real and complex polynomials in several variables, and their factor rings.
Dedekind was the first to describe such rings abstractly, but it was Emmy Noether who gave a complete axiomatic description of commutative rings with chain conditions, which ultimately lead to the great advances in ring theory in the 20th Century associated with the names of Krull, Schur and Prüfer.
The major tool used in the study of rings in general is the notion of module, whose prototype is the vector space. At present we know a great deal about modules over special types of rings, such as division rings, local rings and valuation rings. It is true to say however that in the case of abelian groups , which are the modules over the ring of integers, there are very few general classification theorems.
Author: Phill Schultz, schultz@maths.uwa.edu.au
Last updated September 7, 1999