Descartes: A Biography

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Magic, Mathematics, and Mechanics 

order algebraic equations; and ultimately, whether discrete and continu-

ous quantity could be mapped perfectly onto each other.

This whole episode illustrates an important feature of Descartes’ ambi-

tious claims. The letter to Beeckman in Marchbetrayed an exagger-

ated confidence in the extent to which the new compasses would resolve

long-standing problems in mathematics. During the following nine years,

Descartes began to appreciate the significance of applying algebraic meth-

ods to the solution of problems in geometry. However, before he had made

the kind of progress that might have supported his claims on behalf of a

generalmathesis,hewas once again promising to have discovered a method

bywhich all problems in any discipline could be resolved. This involved

two major moves, neither of which was justified by his work to date. The

first was a reduction of problems in every discipline to the form or struc-

ture of a mathematical problem. Then, within mathematics, the ambition

was to integrate arithmetic and geometry fully, so that the only remaining

task was to find the appropriate equation for any given problem. This

assumed that the solution of such equations was a relatively minor matter.

Having promised too much, Descartes had to acknowledge that he could

not deliver on his overly ambitious plans. It simply was not the case that

all problems are similar enough to mathematical problems that we could

translate one into the other without significant loss. Even within mathe-

matics proper, Descartes had made little progress over Vi`ete’s application

of algebraic methods to geometry. It was time, therefore, to leave aside the

misleading allure of a general solution to all problems and to look to the

solution of particular problems, however that might be achieved.

A Mechanical World

This formative period in Descartes’ life was dominated by uncertainty,

both personal and professional. He was unsure of what career to follow,

or where to live. He had rather vague ambitions about contributing to

aradically new perspective on nature that many contemporary writers

mentioned, but he was unsure about how best to get started on that project.

He was briefly attracted by the promises of an arcane wisdom that was

discovered, not by scientific work that any intelligent person could learn,

butbyinitiation into a mystical tradition that reflected features of the

early Christian Church as described by religious reformers. In the midst

of this intellectual confusion, he had no conceptual resources with which