It is shown how the historiographic purport of Lakatosian methodology of mathematics is structured on the theme of analysis and synthesis. This theme is explored and extended to the revolutionary phase around 1800. On the basis of this historical investigation it is argued that major innovations, crucial to the appraisal of mathematical progress, defy reconstruction as irreducibly rational processes and should instead essentially be understood as processes of social-cognitive interaction. A model of conceptual change is developed whose essential ingredients are (...) the variability of rational responses to new intellectual and practical challenges arising in the cultural environment of mathematics, and the shifting selective pressure of society. The resulting view of mathematical development is compared with Kuhn's theory of scientific paradigms in the light of some personal communications. (shrink)

The ArgumentThis paper deals with the achievements of those French mathematicians active in the period 1800–1830 who oriented their work specifically around the needs of engineering and technology. In addition to a review of their achievements, the principal organizations and institutions are noted, as is their importance as sources of employment and influence.The argument is centered on the word ‘neglected“ in the title. A case is made that a mass of work was produced which made considerable impact at the time (...) but has been overlooked or even completely ignored by historians since. The paper begins with a general discussion of the notion of context, both for the historical figures and for their supposed historians, and several examples of historical distortion are given.Regarding France itself, we see a professional and research profile rather different from that in other countries. The question of national differences in the organization and prosecution of science is thereby sharply exposed. (shrink)

Recent philosophical accounts of mathematics increasingly focus on the quasi-Empirical rather than the formal aspects of the field, The praxis of how mathematics is done rather than the idealized logical structure and foundations of the theory. The ultimate test of any philosophy of mathematics, However idealized, Is its ability to account adequately for the factual development of the subject in real time. As a text case, The works and views of felix klein (1849-1925) were studied. Major advances in mathematics turn (...) out to be most adequately understood as shifts to new conceptualizations at different levels of idealization and abstraction. The implications of the model are explored with special reference to modern philosophical views of the nature and the warrants of mathematical claims to knowledge, And the methodology of the development of this knowledge. (shrink)

In this paper I explore possibilities of bringing post-positivist philosophies of empirical science to bear on the dynamics of mathematical development. This is done by way of a convergent accommodation of a mathematical version of Lakatos's methodology of research programmes, and a version of Kuhn's account of scientific change that is made applicable to mathematics by cleansing it of all references to the psychology of perception. The resulting view is argued in the light of two case histories of radical conceptual (...) innovations. (shrink)

Recent philosophical accounts of mathematics increasingly focus on the quasi-Empirical rather than the formal aspects of the field, The praxis of how mathematics is done rather than the idealized logical structure and foundations of the theory. The ultimate test of any philosophy of mathematics, However idealized, Is its ability to account adequately for the factual development of the subject in real time. As a text case, The works and views of felix klein (1849-1925) were studied. Major advances in mathematics turn (...) out to be most adequately understood as shifts to new conceptualizations at different levels of idealization and abstraction. The implications of the model are explored with special reference to modern philosophical views of the nature and the warrants of mathematical claims to knowledge, And the methodology of the development of this knowledge. (shrink)