Descartes is read as the man who doubted everything, which makes him sound like a skeptic. He was the opposite. The doubt is a tool, used once, deliberately, to find something it cannot touch — and the reason he needed it is that the intellectual world he was born into was coming apart.

The scholastic account of nature had explained things by their natures and their ends: a stone falls because it seeks its natural place. Galileo and the new mechanics explained the same motions by matter, extension and mathematically stated laws, and explained them better. If that succeeds, the entire apparatus goes — substantial forms, final causes, the whole vocabulary in which the universities had done physics for four centuries.

Descartes's question is what can be rebuilt once you have thrown that out, and whether anything survives the throwing.

The method

Stated in the Discourse as four rules, and the first is the operative one: accept nothing as true that is not presented so clearly and distinctly that there is no occasion to doubt it. Divide each difficulty into as many parts as possible. Proceed from the simplest objects to the most complex. Make enumerations so complete that nothing is omitted.

It is geometry, generalized. Descartes had invented analytic geometry — the coordinate system is his, and it is the single most consequential thing in this entry — by showing that curves could be written as equations, and he believed the same procedure would work everywhere. Reduce to what is simple and certain, then rebuild in order.

The doubt is that first rule taken seriously. To find what is genuinely indubitable, he sets aside everything that admits of any doubt at all — the senses, which have deceived him; the external world, since dreams are indistinguishable from waking while they last; even arithmetic, if a powerful deceiver is arranging his thoughts. What survives is on the cogito page, and the sequence is in the Meditations.

Dualism, which is the legacy and the problem

He ends with two kinds of thing. Mind is thinking and unextended; body is extended and unthinking. They are really distinct — each can be conceived complete without the other — and he takes the distinction to be proved.

It solved his immediate problem completely. The new physics gets the whole of extended nature, to explain mechanically without remainder, and nothing about the soul is threatened by it, because the soul is not the kind of thing physics is about. It is a settlement, and it is why the argument mattered so much at the time.

The cost is the mind-body problem, which is his in its modern form. If mind and body are different substances with nothing in common, how does a decision move an arm? Descartes says they interact, locates the interaction at the pineal gland, and does not explain how two things sharing no property could act on each other at all.

The objection

Elisabeth of Bohemia put it to him directly in 1643, and it has never been answered on his terms. How can a thinking substance, having no extension, determine the animal spirits in the body to move? Moving something requires contact and contact requires extension. She is not raising a difficulty about the mechanism; she is pointing out that his definitions rule out any mechanism whatever.

His replies are evasive, and eventually he tells her that the union of mind and body is known by living rather than by the intellect — which is a confession. Everything afterwards in philosophy of mind is an attempt to avoid paying this bill: occasionalism, parallelism, Spinoza's one substance, and every materialism since.

The Cartesian circle is the other, and it is internal. He needs clear and distinct perception to be reliable, and he establishes that by proving a non-deceiving God — using premises he perceives clearly and distinctly. Arnauld pressed this in the Fourth Objections. Descartes's reply distinguishes what is currently perceived from what is remembered as having been perceived, and readers have disagreed ever since about whether that is an answer.

What survives

Not the physics, which is wrong throughout, and not the proofs.

What survives is the framing. Descartes made epistemology first philosophy — the question of what we can know, asked before any question about what there is — and everyone from Locke to Kant works inside that ordering whether or not they accept his answers. He also set the modern mind-body problem in a form that is still recognizably the one being argued about.

The coordinate geometry survives absolutely, and is used by more people every day than all the rest of the philosophy put together.