Customer Review

December 20, 2009
Mathematics is the model for all knowledge. "In seeking the correct path to truth we should be concerned with nothing about which we cannot have a certainty equal to that of the demonstrations of arithmetic and geometry." (II, p. 366) "And although I speak a good deal here of figures and numbers ... nevertheless anyone who pays close attention to my meaning will easily observe that I am not thinking at all of common mathematics, but I am setting forth a certain new discipline ... broad enough to bring out the truths of any subject whatsoever." (IV, p. 374) The power of mathematics stems from "certain basic roots of truth implanted in the human mind by nature, which we extinguish in ourselves daily by reading and hearing many varied errors" (IV, p. 377).

Proofs should be intuited as wholes. Deductions "may sometimes be accomplished through such a long chain of inferences that when we have arrived at the conclusions we do not easily remember the whole procedure which led us to them ... Because of this, I have learned to consider each of these steps by a certain continuous process of the imagination ... Thus I go from first to last so quickly that by entrusting almost no parts of the process to the memory, I seem to grasp the whole series at once." (VII, pp. 388-389) "In this way our knowledge is made much more certain and the capacity of our minds is increased as much as possible." (XI, p. 408)

To achieve this end "we must make use of every assistance of the intellect, the imagination, the senses, and the memory" (XII, p. 411). "By the aid of each faculty ... human efforts can serve to repair the deficiencies of the mind." (XII, p. 417). For example, "if the intellect proposes to examine something which can be related to the body it should produce in the imagination the most distinct idea of it possible; and in order to do this more readily, the object which this idea represents should be exhibited to the external senses." (XII, pp. 417-418). "We are to do nothing from this point on without the aid of the imagination." (XIV, p. 444)

Algebra and analytic geometry is intended to be precisely such an aid to the intuition. For having recorded the steps of a proof in algebraic terms, "we can run through all of them in a very rapid movement of thought and grasp as many as possible at the same time" (XVI, p. 456). Intuiting the whole in this way is important to us "who are seeking evident and distinct knowledge of things; but not the arithmeticians, who are satisfied if the have discovered the number sought even though they have not noticed how it depends upon the given facts, although this latter is the only point in which science truly lies" (XVI, p. 459).

Another benefit of algebra in this regard. By algebra "we translate what we understand to be affirmed about magnitudes in general into that particular magnitude that we can most easily and distinctly picture in our imagination" (XIV, p. 442); specifically, "a magnitude should never be regarded in the imagination otherwise than as a line or a surface, even though it may be called a 'cube' or a 'biquadratic'" (XVI, p. 457).

A quip on why we should denote the answers we seek by a letter such as x. "It frequently happens that individuals are so eager to investigate problems that they apply their capricious intelligence to finding a solution before they have determined by what signs they will recognize the object of their search, if they should stumble upon it by accident; these persons are no less foolish that would be a boy, sent somewhere by his master, who was so eager to obey that he started to run without waiting for instructions, and without knowing where he was ordered to go." (XIII, p. 435)
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