In the process of writing a history of mathematics it appears to be rather difficult to describe the beginnings. From a distant modern viewpoint the various influences and obstacles to long-lasting developments are often almost invisible. In addition, there do not exist sufficiently many sources about ancient researchers and their achievements. Consequently, it is quite difficult, one might even say impossible, to talk about the origins of mathematical thinking. Certainly, one can find in every ancient culture some early thoughts about counting. However, mathematics (as well as philosophy) is believed to have begun with Thales of Milet (who lived around 600 B.C.E.). Different from antecessors he invented the concepts of deduction and proof. His and most of the mathematics of the ancient Greeks was about geometrical objects and numbers. One of the earliest texts in mathematical literature is the Elements of Euclid (from around 300 B.C.E.); they provide us with a rather good understanding about the knowledge at that time and, most importantly, they include the first axioms, another concept that later on played a substantial role, in particular in the foundations of mathematical disciplines.

In this chapter, we do not aim to give a complete overview about the origins of math but focus on the early cultures of Babylonians, Greeks, Indians, and Chinese, mentioning just some of their discoveries and insights.