Friday, April 21, 2017

Rene Descartes and the mathematical world


Rene Descartes (1596-1650) was an influential mathematician and philosopher best known for his contributions to geometry. Mathematician Eric Temple Bell said,
The implications of Descartes' analytic reformulation of geometry are obvious. Not only did the new method make possible a systematic investigation of known curves, but, what is of infinitely greater significance, it potentially created a whole universe of geometric forms beyond conception by the synthetic method. (The Development of Mathematics, 1940)
Descartes is also famous for influencing Isaac Newton,
The one book that turned out to be perhaps the most influential in guiding Newton's mathematical and scientific thought was none other than Descartes' La Géométrie. Newton read it in 1664 and re-read it several times until 'by degrees he made himself master of the whole'... Not only did analytic geometry pave the way for Newton's founding of calculus... but Newton's inner scientific spirit was truly set ablaze. (Is God a Mathematician? 2009)
This post is a collection of quotes from Descartes describing his philosophy of mathematics and knowledge.

Philosophy of mathematics


"I have resolved to quit only abstract geometry, that is to say, the consideration of questions which serve only to exercise the mind, and this, in order to study another kind of geometry, which has for its object the explanation of the phenomena of nature." (Letter to Marin Mersenne, 1638)

"The entire method consists in the order and arrangement of the things to which the mind’s eye must turn so that we can discover some truth." (Rules for the Direction of the Mind, 1628)

"With me, everything turns into mathematics." (Letter to Marin Mersenne, 1638)

"Thus, all unknown quantities can be expressed in terms if a single quantity, whenever the problem can be constructed by means of circles and straight lines, or by conic sections, or even by some other curve of degree not greater than the third or fourth." (La Géométrie, 1637)

Epistemology


"In order to seek truth, it is necessary once in the course of our life, to doubt, as far as possible, of all things." (Principles of Philosophy, 1644)

"So blind is the curiosity by which mortals are possessed, that they often conduct their minds along unexplored routes, having no reason to hope for success, but merely being willing to risk the experiment of finding whether the truth they seek lies there." (Rules for the Direction of the Mind, 1628)

"I think, therefore I am." (Le Discours de la Méthode, 1637)

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