1596–1650

Painting of René Descartes, French philosopher and mathematician

René Descartes has been described as the first truly “modern” mathematician and philosopher. Certainly, his systematic, logical approach to knowledge was revolutionary, dominating philosophy for the next three centuries. Even more importantly, from the perspective of this article at least, it led to a new breakthrough which would greatly impact the future of mathematics and science.

Descartes initially gained a degree in law and spent several years in the military before eventually settling in Holland in 1628 where he composed all his great works. In 1649 he accepted a post as personal tutor to Queen Christina of Sweden. A lifelong late riser and lover of a warm bed – where Descartes claimed to have undertaken his most profound thinking – he succumbed to the harsh Swedish weather. Within months he had contracted pneumonia and died.

A Revelation of Philosophy

Three decades earlier, on the night of 10 November 1619, while campaigning with the army on the Danube, Descartes’ life had changed for forever when his influential journey began. He later claimed to have had several dreams on that date which formulated the principles behind his later work. It left him certain that he should pursue the theory that all knowledge could be gathered in a single, complete science and set about putting in place a system of thought by which this could be achieved. In turn, this left him to speculate on the source and truth of all existing knowledge. He began rejecting much of what was commonly accepted and vowed only to recognise facts which could be intuitively taken to be true beyond any doubt.

The full articulation of these processes came in Descartes’ 1641 work Meditations on First Philosophy. The book is centred around his famous maxim “Cogito, ergo sum” or “I think, therefore I am,” from which he pursued all “certainties” via a method of systematic, detailed mental analysis. This ultimately led him to a very detached, mechanistic interpretation of the natural world, reinforced in his 1644 metaphysical text the Principia Philosophiae or Principles of Philosophy. It is here in which he attempted to explain the universe according to the single system of logical, mechanical laws he had earlier envisaged and, although largely inaccurate, would have an important influence even after Newton’s more convincing explanations later in the century.

Descartes also regarded the human body as subject to the same mechanical laws as all matter, distinguished only by the mind which operated as a distinct, separate entity.

Mathematical Certainties

Descartes passionately believed in the logical certainty of mathematics and felt the subject could be applied to give a superior interpretation of the universe. It is through this reasoning that his greatest legacy to mathematics and science came. In his 1637 appendix to the Discourse, entitled La Geométrie, Descartes sought to describe the application of mathematics to the plotting of a single point in space. This led him to the invention of what are now known as Cartesian Coordinates, the ability to plot a position according to x and y (that is, perpendicular) axes (and in a 3D environment by adding in a third “depth” axis). Moreover, this method allowed geometric expressions such as curves to be written for the first time as algebraic equations (using the x, y and other elements from the graph).

The bringing together of geometry and algebra was a significant breakthrough and could, in theory at least, predict the future course of any object in space, given enough initial knowledge of its physical properties and movement. It is from his mathematical interpretation of the cosmos that Descartes would later claim, “Give me matter and motion and I will construct the universe.”

The “Cogito”

Perhaps the most famous of philosophical maxims, Cogito, ergo sum, was the result of a form of a thought experiment by Descartes, in which he resolved to cast doubt on any and all of his beliefs. This was done in order to discover that to which he was logically justified in holding. He argued that although all his experience could be the product of deception by an evil demon, the demon could not deceive him if he did not exist. That he can doubt his existence proves that he in fact exists.

Chronology

1596 – Descartes born in La Haye, France

1616 – Graduated in law from the University of Poitiers

1637 – Discours de la Méthode (Discourse on Method) published. La Geométrie (Geometry) also published as an appendix to Discours de la Méthode

1641 – Meditations on First Philosophy published