  This essay traces the history of three interconnected strands. Firstly,
changes in the concept of number, secondly, the study of the qualities of
number, which evolved into number theory, and thirdly, the nature of
mathematics itself, from early Greek mathematics to the 20th century. These
were embedded in philosophical shifts, from the classical Greek ontologies
through increasing pragmatism to formalism and logical positivism. Given that
Goedel demonstrated the limitations of the latter as a foundation for
mathematics, this essay explores phenomenology and Lakatosian ideas, which
together offer a more sound epistemological and ontological basis for
mathematics and a methodology for mathematical development.The question also
then arises of the possible resurrection of earlier, neglected mathematical
projects, including widening the domain of number theory to include integer
qualities revealed in the growth of mathematics in general, which has
predominantly been the growth of quantitative mathematics or logistike, the
complement of arithmetike in classical Greece.
