Analyzing (and synthesizing) analysis

Abstract
Equally surprisingly, Descartes’s paranoid belief was shared by several contemporary mathematicians, among them Isaac Barrow, John Wallis and Edmund Halley. (Huxley 1959, pp. 354-355.) In the light of our fuller knowledge of history it is easy to smile at Descartes. It has even been argued by Netz that analysis was in fact for ancient Greek geometers a method of presenting their results (see Netz 2000). But in a deeper sense Descartes perceived something interesting in the historical record. We are looking in vain in the writings of Greek mathematicians for a full explanation of what this famous method was. And I will argue for an answer to the question why this lacuna is there: Not because Greek geometers wanted to hide this method, but because they did not fully understand it. It is instructive to note the ambivalent attitude of the most rigorous mathematician of the period, Isaac Newton, to the method of analysis. He used it himself in his own mathematical work and in the expositions of that work. Yet when the mathematical push came to physical and cosmological shove, he formulated his Principia entirely in..
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