The history of arithmetic and algebra illustrates one of the striking and curious features of the history of mathematics. Ideas that seem remarkably simple once explained were thousands of years in the making.

It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit.But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.

For a while he trampled with impunity on laws human and divine; but, as he was obsessed with the delusion that two and two make five, he fell, at last, a victim of the relentless rules of humble arithmetic. "Remember, O Stranger, Arithmetic is the first of the sciences and the mother of safety".

Algorithms existed for at least five thousand years, but people did not know that they were algorithmizing. Then came Turing (and Post and Church and Markov and others) and formalized the notion.

I think that mathematics will have to become more and more algorithmic if it is going to be active and vital in the creative life. This means it is necessary to rethink what we teach, in school, in college, and in graduate school. In our emphasis on deductive reasoning and rigor we have been following the Greek tradition, but there are other traditions—Babylonian, Hindu, Chinese, Mayan—and these have all followed a more algorithmic, more numerical procedure. After all, the word algorithm, like the word algebra, comes from Arabic. And the numerals we use come from Hindu mathematics via the Arabs. We can’t regard Greek mathematics as the only source of great mathematics, and yet somehow in the last half century there has been such emphasis on the greatness of “pure” mathematics that the other possible forms of mathematics have been put down. I don’t mean that it is necessary to put down the rigorous Greek style mathematics, but it is necessary to raise up the status of the numerical, the algorithmic, the discrete mathematics.

Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians.

Said the great and magnanimous Laplace: 'It is India that gave us the ingenious method of expressing all numbers by ten symbols, each receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity, the great ease which it has lent to all computations, puts our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.'

Seeing there is nothing, (right well beloved students of mathematics,) that is so troublesome to mathematical practice, nor that doth more molest and hinder calculations, that the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expence of time, are for the most part subject to many slippery errors, I began, therefore, to consider in my mind, by what certain and ready art I might remove these hindrances. And having thought upon many things to this purpose, I found at length some excellent brief rules to be treated of perhaps hereafter: But amongst all, none more profitable than this, which together with the hard and tedious multiplications, divisions, and extractions of roots, doth also cast away even the very numbers themselves that are to be multiplied, divided, and resolved into roots, and putteth other numbers in their place which perform as much as they can do, only by addition and substraction, division by two, or division by three. Which secret invention being, (as all other good things are,) so much the better as it shall be the more common, I thought good heretofore, to set forth in Latin for the public use of mathematicians.

The science of arithmetic may be called the science of exact limitation of matter and things in space, force, and time.

It was the ancient Egyptians who first figured out that numbers could, if you added and subtracted them, be used to form mathematics; this made it possible, for the first time, to build the pyramids as well as keep score in bowling.

For, among the world's incertitudes, this thing called arithmetic is established by a sure reasoning that we comprehend as we do the heavenly bodies. It is an intelligible pattern, a beautiful system, that both binds the heavens and preserves the earth. For is there anything that lacks measure, or transcends weight? It includes all, it rules all, and all things have their beauty because they are perceived under its standard.

Mathematics is an ancient discipline. For as long as we can reliably reach into the past, we find its development intimately connected with the development of the whole of our civilization. For as long as we have a record of man's curiosity and his quest for understanding, we find mathematics cultivated and cherished, practiced and taught. Throughout the ages it has stood as an ultimate in rational thought and as a monument to man's desire to probe the workings of his own mind.

Mathematics, from the earliest times to which the history of human reason can reach, has followed, among that wonderful people of the Greeks, the safe way of science. But it must not be supposed that it was as easy for mathematics as for logic, in which reason is concerned with itself alone, to find, or rather to make for itself that royal road. I believe, on the contrary, that there was a long period of tentative work (chiefly still among the Egyptians), and that the change is to be ascribed to a revolution, produced by the happy thought of a single man, whose experiments pointed unmistakably to the path that had to be followed, and opened and traced out for the most distant times the safe way of a science. The history of that intellectual revolution, which was far more important than the passage round the celebrated Cape of Good Hope, and the name of its fortunate author, have not been preserved to us. ... A new light flashed on the first man who demonstrated the properties of the isosceles triangle (whether his name was Thales or any other name), for he found that he had not to investigate what he saw hi the figure, or the mere concepts of that figure, and thus to learn its properties; but that he had to produce (by construction) what he had himself, according to concepts a priori, placed into that figure and represented in it, so that, in order to know anything with certainty a priori, he must not attribute to that figure anything beyond what necessarily follows from what he has himself placed into it, in accordance with the concept.

The mathematical activity of Ancient Greece reached its peak during the glorious era of Euclid, Eratosthenes, Archimedes and Apollonius, a time when Greek letters, art and philosophy were already on the decline. ...it was not Greece proper but its outposts in Asia Minor, in Lower Italy, in Africa that had contributed most to the development of mathematics.