Mathematics

It seems unavoidable that mathematical thinking played a role in human theorizing from the start of the race, and in various ways. Arithmetic (as the later branch of mathematics became known) would have been one of them, motivated initially by forming integers in connection with counting. But other branches surely include geometry, linked to the appreciation of line, surface, and space; trigonometry, inspired by awareness of angles; mechanics, related to the motion of bodies large and small and the (in)stability of structures; part-whole theory, from consideration of collections of things; and probability, coming from judging and guessing about situations. In all cases the thinking would have started out as very intuitive, gradually becoming more explicit and less particular.

Some of the associated contexts would have been provided by study of the environment (such as days) and the heavens (such as new and full moons), which was a major concern of ancient cultures in all parts of the world; in those times mathematics and astronomy were linked closely. For example, the oldest recognized artifact is a bone from Africa, thought to be about thirty-seven thousand years old, upon which phases of the moon seem to have been recorded.

Among the various ancient cultures, the Babylonians have left the earliest extant evidence of their mathematical practice. They counted with tokens from the eighth millennium; and from the late fourth millennium they expressed numbers and properties of arithmetic in a numeral system to base 10 and handled fractions in expansions of powers of 1/60. Many surviving artifacts seem to relate to education—for example, exercises requiring calculations of unknown quantities, which correspond to the solution of equations but are not to be so identified. They also developed geometry, largely for terrestrial purposes. The Egyptians pursued similar studies, even also finding a formula (not the same one) for the volume of the rectangular base of a pyramid of given sides. They also took up the interesting mathematical problem of representing a fraction as the sum of reciprocals.

A major mathematical question for these cultures concerned the relationship between circles and spheres and rectilinear objects such as lines and cubes. They involve the quantity that we symbolize by, and ancient evidence survives of methods of approximating to its value. But it is not clear that these cultures knew that the same quantity occurs in all the relationships.