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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 
[log in to unmask] 
http://members.gnn.com/logosapien/ransom.htm 
 
 
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