Descartes Fixes Mathematics

Descartes had already been convinced for a long time by his own experience of the little utility of mathematics, especially when one cultivates them only for themselves, without applying them to other things.

Since the year 1620 he had entirely neglected the rules of arithmetic. He even testifies that already before, he had so forgotten division and the extraction of the square root, that he would have been obliged to study them a second time in books, or to invent them by himself, if he had needed to use them. The attachments he had for geometry subsisted a little longer in his heart.

The mathematicians of Holland and Germany whom he had seen during his travels had contributed to retaining them until his return to France by the questions and problems that they had proposed for him to solve.

But they had already fallen in 1623, if it is true that in 1638, “it had been more than fifteen years since he made a profession of neglecting geometry, and of never stopping at the solution of any problem, except at the request of some friend.”

During his studies of mathematics he had been careful to read with attention the treatises that he could find.

He had applied himself particularly to arithmetic and to geometry, as much because of their simplicity, as because he had learned that they give great openings for the understanding of the other parts. But of all the authors who fell into his hands at the time, not one had the advantage of fully satisfying him. To tell the truth, he noticed in these authors many things concerning numbers, which were found to be true after the calculation he made of them. It was the same with regard to figures, and they represented several to him that his eyes could not deny.

But his mind demanded something else from them. He would have wished that they had shown him the reasons why it was so, and that they had produced for him the means of drawing the consequences from them.

This is what made him less surprised in the continuation to see that most of the able people, even among the most solid geniuses do not delay in neglecting or rejecting these kinds of sciences as vain and puerile amusements, as soon as they have made the first trials of them. Also he was very far from blaming those who having pre-sentiments of their uselessness, do not hesitate to renounce them early on, especially when they see themselves rebuffed by the difficulties and the embarrassments that are encountered from the beginning.

He found nothing indeed that seemed less solid to him than to occupy himself with simple numbers and imaginary figures, as if one had to stick to these “trifles” without carrying his sight beyond. He saw there even something more than useless: and he believed that it was dangerous to apply oneself too seriously to these superficial demonstrations, that industry and experience furnish less often than chance; and which are more of the resort of the eyes and of the imagination than of that of the understanding. His maxim was that this application makes us imperceptibly lose the habit of the use of our reason: and exposes us to lose “the path that its light traces for us.”

There are part of the motives that led him to renounce vulgar mathematics. But it appears that the respect he showed for the ancients prevented him from pushing the contempt he had for these sciences in the way of cultivating or teaching them.

For coming to reflect on the conduct of the ancient philosophers, who did not want to receive anyone into their schools who did not know mathematics, and particularly geometry, as if this science had appeared to them the easiest and the most necessary of all to prepare their minds for philosophy:

He preferred to believe that these ancients had a science of mathematics quite different from that which was taught in his time, than to confuse them among the moderns in the judgment he made of them. The prejudice he could have in favor of these ancients did not however go so far as to persuade him that they had a perfect knowledge of mathematics.

The immoderate rejoicings, and the sacrifices they made for the smallest discoveries were testimonies of the little progress they had yet made there, and of the coarseness of their century from which they were not exempt. The invention of certain machines that some historians have raised with so much praise and ostentation still contributed to confirm him in this thought: supposing that as simple and easy as they were, it was enough that they were new and unknown to the vulgar to attract public admiration.

The first seeds of truth, that nature has put in the mind of man, which make us still correct our errors every day by reading or conversation, and which had so much force in the mind of these ancients whose foundation was perhaps better prepared than ours, were able to produce, according to Mr. Descartes, great enough effects in these first philosophers, to give them the true ideas of philosophy and mathematics:

Although they could not yet have a perfect knowledge of them, and they did not have all the politeness of the later centuries. He perceived some traces of the true mathematics “in Pappus and in Diophantus,” who certainly had not been the first inventors of it.

But he did not believe these learned men exempt from the jealousy, which often prevents the communication of the best things. He judged them capable of having suppressed this science that they had received from the ancients, by the fear of making it despicable by divulging it, under the pretext that it was very simple and very easy.

And he held it against them for having wanted to substitute in place of this true science only dry and sterile truths, which they produced as demonstrations and consequences drawn from the principles of this true science, in order to make them admired as effects of their marvelous art: instead of showing the art in itself so as not to dupe anyone, and to make the admiration of the simple cease.

Descartes was not the first who noticed the bad state of this science of the ancients.

From the beginning of his century, very great minds had tried to revive it under the barbaric name “of algebra.

The thoughts that came to him on this subject made him abandon the particular study of arithmetic and geometry, to give himself entirely to the search for this general science, but true and infallible, that the Greeks judiciously named “mathesis,” and of which all mathematics are only parts.

After having solidly considered all the particular knowledge that one qualifies with the name of mathematics, he recognized that to deserve this name, it was necessary to have for object relationships, proportions, and measures. He judged from there that there was a general science destined to explain all the questions that one could ask concerning relationships, proportions and measures, by considering them as detached from all matter:

This general science could with very just title bear the name of “mathesis” or “universal mathematics”; since it contains all that can make the other knowledge deserve the name of science and of particular mathematics.

There is the unraveling of the difficulty that there would be to believe that Mr. Descartes had absolutely renounced mathematics in a time when it was no longer free for him to be ignorant of them. It will not be easier to believe that he wanted at the same time to do the same treatment to physics, if one does not find the twist that one can give to such a surprising resolution.

It must be admitted that finding himself sometimes discouraged by the little certainty that he noticed in his discoveries of physics, he had already tried more than once to abandon the researches, with the intention of no longer applying himself except to the science of living well.

In the middle of these laudable movements he had embraced the study of morality. He took it up again completely new since his return to Paris: and one can say that he continued it during his whole life. But it was without ostentation, and more to regulate his conduct than that of others. The man in the world who seems to have known him the most intimately, teaches us that morality was the object of his most ordinary meditations. But he was not long without returning to his observations on nature: and one can doubt that he ever seriously renounced physics, since he had stripped himself of the prejudices of the school. The satisfaction that his researches gave him on this point was ordinarily victorious over the little displeasures that were born to him from the inequality of the success in the beginnings.

He soon noticed that the study of physics was not useless to that of morality: and that nothing was more advantageous to him to regulate his actions than the steps he made in the discernment of the true and the false. This is what he recognized a long time since in a letter he wrote to Mr. Chanut, to whom he points out that he was entirely of his opinion, when he judged that the surest way to know how we should live, is to know beforehand what we are; what is the world in which we live; and who is the creator of this universe where we live. He declares to him, as a man persuaded of what he advances, that the knowledge that he had acquired well or badly of physics, had served him a lot to establish certain foundations in morality: and that it had been easier for him to find the satisfaction he was looking for on this point, than in several others that regarded medicine, although he had employed much more time there. So that he could not boast after all his researches of having found the means of preserving life; but only that of not fearing death, and of preparing himself for it without this sadness or this anxiety which is ordinary to those whose wisdom is all drawn from the teachings of others, and supported on foundations which depend only on the prudence and the authority of men.

Descartes was 2 months and a few days in Paris, entertaining his friends with this illusion in which he was regarding his pretended renunciation of mathematics and physics.

They often gave themselves the pleasure of denying his resolutions: and the slightest occasions that they presented to him to solve a problem or to make an experiment, were unavoidable traps for him. The embarrassments of his mind joined with the need he had to regulate his particular affairs made him leave the city towards the beginning of the month of May, to return to Brittany with his relatives.

After having spent a few days in Rennes, he took the consent of his father, to sell in Poitou some inheritances, of which he had had the kindness to put him in possession since he had become of age: and he went to Poitiers, then to Châtellerault towards the end of the month of May.

He employed in these negotiations the entire month of June and half of July. He disposed of the land of the Perron, which had fallen to him by the sharing of the goods of the succession of his mother; of two other farms which had been given to him around Châtellerault; and of a house in Poitiers.

The two farms, called one “the grand-maison,” and the other “le marchais,” were in the parish of Availle, which some call “poitevine,” so as not to confuse this place with Availle Limousine, which is beyond the Isle Jourdain on the limits of Poitou and Limousin. As for the land and lordship of “le perron”

it was one of the most noble fiefs of the Châtel-Heraudois or Duchy of Châtellerault, to the south of this city in the same parish of Availle, towards the confluence of the Clain and the Vienne. The two farms were sold by contract of June 5, 1623 to a rich merchant of Châtellerault; and the land of the Perron was to a qualified gentleman of the province, named Abel de Couhé, Sieur de Châtillon, and de la Tour-d’Asniére. He passed the contract with this gentleman before the notaries of Châtellerault on the 8th of July following. But he did not fail to retain the name of the land in accordance with their conventions, to satisfy the desire of his relatives; and he continued to call himself “Monsieur du Perron,” at least in his family.