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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
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http://members.gnn.com/logosapien/ransom.htm
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