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, .... .
Article by: J J O'Connor and E F Robertson
3.1.4. Mathematics and Art
Today mathematicians frequently liken mathematics and its creations to music and art rather than to science. It is convenient to keep the old classification of mathematics as one of the sciences, but it is more just to call it an art or a game. Unlike the sciences, but like the art of music or a game of chess, mathematics is foremost a free creation of the human mind. Mathematics is the sister, as well as the servant of the arts and is touched with the same genius. In an age when specialization means isolation, a layman may be surprised to hear that mathematics and art are intimately related. Yet, they are closely identified from ancient times. To begin with, the visual arts are spatial by definition. It is therefore not surprising that geometry is evident in classic architecture or that the ruler and compass are as familiar to the artist as the artisan. Artists search for ideal proportions and mathematical principles of composition. Many trends and traditions in this search are mixed.
Mathematics and art are mutually indebted in the area of perspective and symmetry which express relations only now fully explained by the mathematical theory of groups, a development of the last centuries. But does not art, in breaking away from academic canons nowadays, also break with mathematics? On the contrary. In the last one hundred years mathematics also has its liberation. From the science of number and space, mathematics becomes the science of all relations, of structure in the broadest sense.
A mathematician, like a painter or a poet, is a maker of patterns. The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colours or the words must fit together in a harmonious way. Beauty and elegance are the true test for both. The revolutions in art and mathematics only deepen the relations between them. It is a common observation that the emotional drive for creation and the satisfaction from success are the same whether one constructs an object of art or a mathematical theory.
In ancient Greece mathematics was transformed from a tool for the advancement of other activities to an art. Arithmetic, geometry and astronomy were to the classical Greece music for the soul and the art of the mind; indeed, rational and aesthetic can hardly be separated in Greek thought. Mathematics and art were fused harmoniously in a single individual during the Renaissance. Though the further developments tended to weaken the connection, it was reinforced again in the last century and recent revolutionary changes in both fields open new possibilities for interaction without weakening the potential role of each as inspiration to the other. In both areas the creative process involves observation and experiment, judgement and rejection, intuition and feeling, careful calculation and analysis, sophistication, flashes of insight, and possibly results that are thrilling, satisfying and useful to both the artist and his audience. Patterns in either field may illustrate, explain, or inspire work in the other. The new mathematics and the new art are capable of an intimacy that we have not seen since the Renaissance.
3.1.5. Unsolved Problems of Antiquity
Greek mathematics is significant for the questions it raised and did not answer. Among such questions are three famous construction problems known to every amateur in mathematics. They are referred to as "squaring the circle", "doubling the cube" and "trisecting the angle". To square the circle means to construct a square, the area of which is equal to the area of a given circle. To double a cube means to construct the side of a cube whose volume shall be double that of a given cube. To trisect an angle means to divide any angle into three equal parts. These constructions are to be performed only with an unmarked ruler and a compass. No other instruments are to be used.
The reason for this restriction sheds light on the classic attitude towards mathematics. A ruler and a compass are the physical counterparts suggesting the concepts of a straight line and a circle. This restriction, self-imposed and arbitrary, was motivated by the desire to keep geometry simple and harmonious. The three construction problems were very popular in Greece. The first historical reference to them states that the philosopher Anaxagoras passed his time in prison trying to square the circle. Despite the repeated efforts of the best Greek mathematicians the problems were not solved. Nor were they to be solved for the next two thousand years. It was finally proved that the constructions cannot be performed under the conditions specified.