================== HES POSTING ===================== Roy, Friedman's claim for the centrality of Frege's new logic to the development of the modern picture of mathematics is a very strong one. To quote from Michael Friedman's essay: "In the face of the new developments, the Kantian conception of pure intuition can no loger be sustained. Indeed, during this same period, mathematicians are developing techniques that free pure mathematics from any dependence whatsoever on spatiotemporal intuition. Here I am referring to the so-called 'rigorization' of the calculus initiated by Bolzannno and Cauchy in the early eighteenth century that culminates in the 'arithematization' of analysis by Weierstrass. As a result of this work, the calculus is perged of all reference to intuitive ideas of motion and change and is instead given a purely 'formal' foundation in the modern ideas of function, convergence, and limit. Moreover, what makes this 'formal' conception of mathematics itself possible is the new perspective on logic and mathematical reasoning first adequately formulated by Frege. For, as suggested above, it is only this new logic that allows us to represetn ideas involving infinity (which of course are especially basic to the calculus) in a 'formal' or nonintuitive manner. In other words, it is the development of the new mathematical logic, above all, that makes possible the modern picture of mathematics as based on deductive systems involving strict logical inference from explicitly stated axioms -- axioms which therefore stand in no need whatever of an intuitive interpretation." (pp. 88-89) >From Michael Friedman, "Philosophy and the Exact Sciences", in John Earman, ed. _Inference, Explanation, and other Frustrations_, Los Angeles: U. of California Press, 1992. For the 'anti-formalist' take on some of these issues, I might recommend the books of S. Shanker on Wittgenstein and mathematics. Greg Ransom UC-Riverside [log in to unmask] http://members.gnn.com/logosapien/ransom.htm ============ FOOTER TO HES POSTING ============ For information, send the message "info HES" to [log in to unmask]