Cartesian Method and the Problem of Reduction

Cartesian Method and the Problem of Reduction

Cartesian Method and the Problem of Reduction

Cartesian Method and the Problem of Reduction

Synopsis

The Cartesian method, construed as a way of organizing domains of knowledge according to the "order of reasons," was a powerful reductive tool. Descartes made significant strides in mathematics, physics, and metaphysics by relating certain complex items and problems back to more simple elements that served as starting points for his inquiries. But his reductive method also impoverished these domains in important ways, for it tended to restrict geometry to the study of straight line segments, physics to the study of ambiguously constituted bits of matter in motion, and metaphysics to the study of the isolated, incorporeal knower. This book examines in detail the negative and positive impact of Descartes's method on his scientific and philosophical enterprises, exemplified by the Geometry, the Principles, the Treatise of Man, and the Meditations.

Excerpt

Descartes in his mature work organizes a subject-matter as a linear progression from simples to complexes, according to the order of reasons, such that each item in the chain is known without the aid of succeeding items and all items are known solely on the basis of those that precede them. Thus a domain of human knowledge will arise from starting-points which are known in themselves and develop as a progression of successively more complex entities which are simples in some kind of association. a domain thus organized, Descartes believes, will not stray beyond the boundaries of our constructive intuition and can indeed lay claim to completeness. As I showed in the Introduction, Descartes and many of his commentators take his mathematical work to exemplify and vindicate this vision of method.

But what does this picture of an organized domain amount to exactly when Cartesian method reorganizes mathematics? in the next two chapters I will take issue with the dominant account of the relation of the method to the Geometry. While I will try to clarify the method's power to reshape and synthesize mathematical domains, I will also criticize its reductive tendency to check the growth of knowledge and obfuscate its own successes. the passages from the Discourse on Method examined above continue with explicit reference to mathematics, and serve to introduce Descartes's way of thinking about the restructuring of geometry according to the order of reasons. Having noted the success of mathematics in producing 'demonstrations', Descartes goes on to say,

I had no intention of trying to master all those particular sciences that receive in common the name of mathematics; but observing that, although their objects are different, they do not fail to agree in this, that they take nothing under consideration but the various relationships or proportions which are present in these objects, I thought that it would be better if I only examined these proportions in their general aspect, and without viewing them otherwise than in the objects which would serve most to facilitate a . . .

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