Descartes, René (1596-1650)

Rules for the Direction of the Mind [Regulae ad Directionem Ingenii]

Amsterdam

P. and J. Blaeu

w. 1628, published in 1684, 1701

"The writers displayed many geometrical truths before my very eyes, as it were, and derived them by means of logical arguments"

When I first applied my mind to the mathematical disciplines, I at once read most of the customary lore which mathematical writers pass on to us. I paid special attention to arithmetic and geometry, for these were said to be the simplest and, as it were, to lead into the rest. But in neither subject did I come across writers who fully satisfied me. I read much about numbers which I found to be true once I had gone over the calculations for myself. The writers displayed many geometrical truths before my very eyes, as it were, and derived them by means of logical arguments. But they did not seem to make it sufficiently clear to my mind why these things should be so and how they were discovered. So I was not surprised to find that even many clever and learned men, after dipping into these arts, either quickly lay them aside as childish and pointless or else take them to be so very difficult and complicated that they are put off at the outset from learning them. For there is really nothing more futile than so busying ourselves with bare numbers and imaginary figures that we seem to rest content in the knowledge of such trifles. And there is nothing more futile than devoting our energies to those superficial proofs which are discovered more through chance than method and which have more to do with our eyes and imagination than our intellect; for the outcome of this is that, in a way, we get out of the habit of using our reason. At the same time there is nothing more complicated than using such a method of proof to resolve new problems which are beset with numerical disorder. Later on I wondered why the founders of philosophy would admit no one to the pursuit of wisdom who was unversed in mathematics †1 - as if they thought that this discipline was the easiest and most indispensable of all for cultivating and preparing the mind to grasp other more important sciences. I came to suspect that they were familiar with a kind of mathematics quite different from the one which prevails today; not that I thought they had a perfect knowledge of it, for their wild exultations and thanksgivings for trivial discoveries clearly show how rudimentary their knowledge must have been. I am not shaken in this opinion by those machines†2 of theirs which are so much praised by historians. These mechanical devices may well have been quite simple, even though the ignorant and wonder-loving masses may have raised them to the level of marvels. But I am convinced that certain primary seeds of truth naturally implanted in human minds thrived vigorously in that unsophisticated and innocent age - seeds which have been stifled in us through our constantly reading and hearing all sorts of errors. So the same light of the mind which enabled them to see (albeit without knowing why) that virtue is preferable to pleasure, the good preferable to the useful, also enabled them to grasp true ideas in philosophy and mathematics, although they were not yet able fully to master such sciences. Indeed, one can even see some traces of this true mathematics, I think, in Pappus and Diophantus who, though not of that earliest antiquity, lived many centuries before our time. But I have come to think that these writers themselves, with a kind of pernicious cunning, later suppressed this mathematics as, notoriously, many inventors are known to have done where their own discoveries were concerned. They may have feared that their method, just because it was so easy and simple, would be depreciated if it were divulged; so to gain our admiration, they may have shown us, as the fruits of their method, some barren truths proved by clever arguments, instead of teaching us the method itself, which might have dispelled our admiration. In the present age some very gifted men have tried to revive this method, for the method seems to me to be none other than the art which goes by the outlandish name of 'algebra' - or at least it would be if algebra were divested of the multiplicity of numbers and incomprehensible figures which overwhelm it and instead possessed that abundance of clarity and simplicity which I believe the true mathematics ought to have. It was these thoughts which made me turn from the particular studies of arithmetic and geometry to a general investigation of mathematics. I began my investigation by inquiring what exactly is generally meant by the term 'mathematics'†1 and why it is that, in addition to arithmetic and geometry, sciences such as astronomy, music, optics, mechanics, among others, are called branches of mathematics. To answer this it is not enough just to look at the etymology of the word, for, since the word 'mathematics' has the same meaning as 'discipline',†2 these subjects have as much right to be called 'mathematics' as geometry has. Yet it is evident that almost anyone with the slightest education can easily tell the difference in any context between what relates to mathematics and what to the other disciplines. When I considered the matter more closely, I came to see that the exclusive concern of mathematics is with questions of order or measure and that it is irrelevant whether the measure in question involves numbers, shapes, stars, sounds, or any other object whatever. This made me realize that there must be a general science which explains all the points that can be raised concerning order and measure irrespective of the subject-matter, and that this science should be termed mathesis universalis †3 - a venerable term with a well-established meaning - for it covers everything that entitles these other sciences to be called branches of mathematics. How superior it is to these subordinate sciences both in utility and simplicity is clear from the fact that it covers all they deal with, and more besides; and any difficulties it involves apply to these as well, whereas their particular subject-matter involves difficulties which it lacks. Now everyone knows the name of this subject and without even studying it understands what its subject-matter is. So why is it that most people painstakingly pursue the other disciplines which depend on it, and no one bothers to learn this one? No doubt I would find that very surprising if I did not know that everyone thinks the subject too easy, and if I had not long since observed that the human intellect always bypasses subjects which it thinks it can easily master and directly hurries on to new and grander things.
(Rule 4, p. 17-20)

Body::Eye

Past Masters

Reading Descartes, René. The Philosophical Writings of Descartes, trans. John Cottingham, Robert Stoothof, and Dugald Murdoch (Cambridge: Cambridge UP, 1985).

See Opuscula posthuma, physica et mathematica (Amsterdam: P. and J. Blaeu, 1701).

Not published in Descartes' lifetime. Dutch translation in 1684; published in Latin in 1701.

Mind's Eye

10/01/2003