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Topic 3: Arithmetic and Geometry, René Descartes

It is clear why arithmetic and geometry are much more certain than the other sciences: it is because they alone deal with an object pure and simple enough to admit absolutely nothing that experience has made uncertain, and that they consist entirely of a series of consequences deduced by reasoning. They are therefore the easiest and clearest of all, and their object is such as we wish it to be, since, except by inattention, it seems impossible for man to make mistakes in them. And yet we should not be surprised if many minds spontaneously apply themselves rather to other studies or to philosophy: this is because each one gives himself more boldly the liberty of asserting things by divination in an obscure question than in an obvious one, and that it is much easier to make conjectures on any question than to arrive at the truth even on a question, however easy it may be.

From all this one must conclude, not, in truth, that one should learn only arithmetic and geometry, but only that those who seek the right path to truth should not concern themselves with any object, of which they cannot have a certainty equal to that of the demonstrations of arithmetic and geometry.

René Descartes,

Rules for the Direction of the Mind, 1628

Thesis of René Descartes in this excerpt from *Rules for the Direction of the Mind *

René Descartes himself gives the thesis he defends, in conclusion: “those who seek the right path of truth must not concern themselves with any object, of which they cannot have a certainty equal to that of the demonstrations of arithmetic and geometry”.

René Descartes applies himself in this excerpt to show that arithmetic and geometry are sciences, or pure demonstrations, which are not disproved by experience, and consequently which allow certainty.

Presentation of the *Rules for the Direction of the Mind*

The *Rules for* the Direction of the Mind were never completed by René Descartes. These 21 rules, commented by René Descartes until the 18th, should be a guide for any man who seeks the truth, avoiding the traps of reason or the senses.

Plan for the commentary

This outline is a suggested correction: there are other possible outlines for commenting on René Descartes’ text.

**I. The arithmetic and geometry of certain sciences**

According to René Descartes, arithmetic and geometry are always correct. Even when the conditions of experiments vary, as arithmetic and geometry are pure abstract sciences, the result is always the same. This is what makes it possible to establish certainties, and to be sure not to be mistaken. It is also what makes the knowledge of these sciences so desirable.

**II. The tendency of men to prefer easy conjectures**

Why then, if men can have certainties, do they think so much or get lost in reasoning about what they cannot be sure of? It is because it is easier to produce conjectures, i.e. propositions that cannot be rationally demonstrated. Man is far too tempted to assert things about what he does not know, about what is mysterious, like “soothsayers”.

**III. Truth can only be found in certainties**

For anyone who seeks only the truth and only the truth, there is no need to waste time in suppositions or conjectures. It is not a question of studying arithmetic or geometry exclusively, but at least of studying only those sciences which are as certain as these. Because they give clear and precise ideas, which cannot be disputed, these sciences are the means of access to a pure truth.

See also:

→ Explanations of the Discourse of Method – René Descartes

→ Letters to Elizabeth by René Descartes

→ All the correction to the BAC 2014