The gradual construction of number systems took time. In the fourth millennium BCE, the Sumerians recorded their accounting transactions using a system of tokens shaped from clay, known as calculi (Latin for "small stones"), by analogy with the pebbles used to teach arithmetic. These were placed inside a hollow sphere, the bulla, which was then sealed. If there was any doubt about the number of animals entrusted to a shepherd, one need only break open the bulla to check that none were missing. Although the shapes impressed on the bulla did not necessarily correspond to those of the calculi inside it, the numbers always matched. Around 3300 BCE, it was agreed that a summary of the contents, in the form of a kind of docket, should be impressed on the bulla beside the seal, making the contents themselves redundant. The bullae were flattened into tablets. Writing and the earliest representations of numbers had arisen from the need to count. But what a long road still lay ahead before the modern notion of number could emerge!
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The power of axiomatization
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In the beginning were the integers known as natural numbers. Formally, they can be constructed from set theory, and more specifically from the empty set. The first person to propose a satisfactory axiomatic system for them, in 1889, was Giuseppe Peano (1858–1932). Like any axiomatic system, it consists of assumptions about the existence of objects and the relations between them. Like any axiomatic system, it reflects the structure of our brains, which consist of neurons (objects) and synapses (relations between those objects). Peano considers a set containing a particular element a and postulates the existence of a map
