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This essay traces the history of three interconnected strands. Firstly,changes in the concept of number, secondly, the study of the qualities ofnumber, which evolved into number theory, and thirdly, the nature ofmathematics itself, from early Greek mathematics to the 20th century. Thesewere embedded in philosophical shifts, from the classical Greek ontologiesthrough increasing pragmatism to formalism and logical positivism. Given thatGoedel demonstrated the limitations of the latter as a foundation formathematics, this essay explores phenomenology and Lakatosian ideas, whichtogether offer a more sound epistemological and ontological basis formathematics and a methodology for mathematical development.The question alsothen arises of the possible resurrection of earlier, neglected mathematicalprojects, including widening the domain of number theory to include integerqualities revealed in the growth of mathematics in general, which haspredominantly been the growth of quantitative mathematics or logistike, thecomplement of arithmetike in classical Greece.

Historical Changes in the Concepts of Number, Mathematics and Number Theory