The Lasting Influence of Pythagorism
During the three millennia BC, Egyptian and Mesopotamian mathematics developed fairly advanced computational techniques. Although they did not address the problem of a conceptual number determination (what are numbers?), there is every reason to believe t
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The Lasting Influence of Pythagorism
During the three millennia BC, Egyptian and Mesopotamian mathematics developed fairly advanced computational techniques. Although they did not address the problem of a conceptual number determination (what are numbers?), there is every reason to believe that numbers were then implicitly conceived as a property of numbered things. The fact that a number can be isolated from its material support is not evident, and the question will arise in the nineteenth century, when mathematicians will try to understand the exact nature of mathematical statements: why would the act of abstracting the number ten and the act of abstracting the colour white from the observation of a group of ten white marbles refer to two radically different types of experience and two radically different modes of conceptualization? Or, more abstractly and more generally, why and how would the nature of mathematical concepts be distinct from the nature of other concepts derived from experience? In fact, it must be recognized that it is much easier to think of numbers as having a more autonomous existence with respect to the things they serve to enumerate than colour, whose existence is difficult to conceive of outside a material medium. The autonomy of the rules of calculation with regard to the things that are numbered further accentuates this idea of a specificity of mathematical concepts. Thus Egyptian and Mesopotamian mathematics already hinted at the possibility of a development of calculation that would be independent of the concrete meanings at stake: distribution of rations to an army, distribution of wheat. . . It is difficult to go beyond these few observations without advancing theses with uncertain conclusions; if algebraic calculus has its own logic and brings into play in its functioning formal structures that can easily be used retrospectively to interpret ancient texts, the recognition of these structures was indeed very long to be established. The Greek theory of number contrasts with previous conceptions precisely because of its willingness to consider the nature of numbers beyond their roots in the practice of enumeration and calculation. The mathematical theory of number itself (arithmetic), the geometric theory of quantities, numerology (the mysticism of © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. Patras, The Essence of Numbers, Lecture Notes in Mathematics 2278, https://doi.org/10.1007/978-3-030-56700-2_2
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2 The Lasting Influence of Pythagorism
numbers), and the arithmetic features of philosophical questioning combine to form a complex and inseparable whole. These ideas were essential for the development of Greek thought as a whole, and of the later philosophical tradition. They continue, as we shall see, to influence our understanding of the role and meaning of mathematics. The idea that number can be defined independently of its rules of empirical use represents a considerable advance. It implies a change in
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