Descartes and the Philosophy of Mathematics Anthony Lo Bello

The Life and Times of Descartes

Descartes was born on 31 March 1596 at La Haye, near Tours; the French authorities have quite properly re- named the place La Haye Descartes. He was given the name RenG which was commonly bestowed on chil- dren whose mothers died in or shortly after childbirth. His was a well-to-do middle-class family; his father was a member of the Touraine parliament. This was during the convulsions of the French civil war, for at the assassination of Henry III in 1588, the throne fell to a Protestant, Henry de Bourbon, King of Navarre, and the French Catholics refused to accept a heretic to di- rect the affairs of the eldest daughter of the Roman Church. The whole business was soon settled, how- ever, when Henry of Navarre sensibly decided that Paris was well worth a Mass and converted to Catholi- cism; he went on to be the most popular king in French history. Descartes was sent away to school at the nearby Jesuit establishment at La Fl~che; there he first studied mathematics from the textbooks of Peter Clavius, the most famous mathematician of his time.

Clavius had acquired a great reputation from his edition of Euclid, in which he pointed out some omis- sions in the list of postulates and axioms. He was the first to use the decimal point. His chief accomplish- ment, however, and the real reason that Clavius is re- membered, was the reform of the Julian Calendar he undertook for Pope Gregory XIII; the calendar we use today, the "Gregorian" Calendar, is due to him. This got Clavius into a lot of trouble, for some significant mathematicians, Francois Vi6te for example, did not like Clavius's calendar. Furthermore, in order to catch up with the sun, Clavius had ordered that 4 October 1582 was to be fol lowed by 15 October 1582; the people of Frankfurt rioted against mathematicians and

the Pope, who were in cahoots to rob them of ten days. Clavius was sorely dismayed by the contro- versy, though he got the better of his adversaries.

While Descartes was learning mathematics from the books of Clavius, he developed a certain life-style at school that he was not to change until 1650, when the change resulted in his death. He petitioned the Most Reverend Head Master of the school, Fr. Charlet, that he be allowed to remain in bed until 11 o'clock and be exempted from all the morning's classes and activities. Surely this must strike a sympathetic chord in the hear ts of all s tuden t s who dislike early morning classes. Descartes' excuse was that he was too sickly to get out of bed so early, and that it was more beneficial for him to remain there and think until 11 a.m. The Jesuits granted him the requested dispensation, and

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 3 �9 1991 Spnnger-Verlag New York 35

for the rest of his life he stayed in bed thinking until 11 a.m., and it was during those hours that he produced all his mathematical and philosophical works. If one wants to accomplish anything in life, one has to make time for it, an inviolable block of time, and Descartes was of the opinion that his thinking was more impor- tant than attending the performances of his mediocre instructors. One makes time for wha t one really wants.

Descartes did not go on to attend a university. In those days it was not necessary to get a degree in order to function as a mathematician or anything else; if you wanted to be a doctor, you just put a sign on your door "I am a doctor" and that was sufficient. The mathematician Euler's first job was as a Professor of Medicine at St. Petersburg in Russia. The Master's De- gree was a license to teach, but Descartes had no in- tention of teaching, and I cannot think of many great mathematicians of his time who did. Teaching at a university then was entirely different from what it is today. It is even more amusing than instructive to point out some of the differences. In many cases, the professor was only paid by the students who attended his lectures; the greater his following, the greater his income. Since there were no grades, the students flocked to the better rather than the easier instructors; there was no need to weed out the poor teachers. They mostly starved, since everyone went to the cow that gave milk. In some places, like the University of Basel, certain positions were filled not by search com- mittees but by God; the names of all qualified appli- cants (those who had submitted a decent learned paper along with their application) were put into a bag, and a name was drawn out at random to fill the vacancy. This was why Euler had to leave Basel and go to Russia; the vacancy in the Physics Department for which he had applied was assigned to someone else by the laws of probability. Once you got a posi- tion, there was often no such thing as tenure to pro- tect you. In France, for example, no matter what your seniority, you could be challenged for your post by any newcomer, and if he defeated you in a public competition, you were out. The great Roberval, who found the area under one arch of the cycloid, held the record in France for the longest tenure under these trying circumstances; at the Royal Coll~ge de France he defeated all comers for about 40 years. In any case, Descartes determined not to become a teacher, and in- stead joined the army of the Dutch Prince of Orange.

As a result of this military career, Descartes was able to write the first book on dueling, which was his only work not to be put on the List of Prohibited Books after his death. When the Thirty Years War broke out in 1618, he transferred his allegiance to the Catholic army of the Duke of Bavaria. The war broke out be- cause people were itching for a fight; the immediate cause was an insult offered by some Protestant Bohe-

mians to the Catholic Holy Roman Emperor. They threw his emissaries out of a window. The emissaries were not physically harmed since they landed in a pile of manure , but this so-called "Defenes t ra t ion of Prague" was a sufficient casus belh. Descartes eventu- ally left the Bavarian army and joined the forces of Cardinal Richelieu, Prime Minister of France, which were reducing the Huguenot fortress of La Rochelle. After this, he retired from military life. It is not known how many people he personally killed.

Descartes spent some time travelling about on reli- gious pilgrimages; in particular, he went to inspect the Holy House of Loreto, and visited Rome to win the Jubilee indulgence of the Holy Year of 1625. He was a

Descartes moved his household at least once a year in order not to be bothered by the in- conveniencing courtesies of society.

pious Catholic, and it was his major concern later in life to provide a rational basis for his religious belief. When he returned to Paris, he took the advice of his mentor, Cardinal B6rulle, and decided to devote the rest of his life to learning. Since he was determined to begin by doubting everything, he thought it advisable to leave France, whose people love controversy and where it was dangerous to doubt, and go to some country where almost everything was tolerated. Just as today, so also in the seventeenth century, that country was Holland. The people there were too busy making money to care about his skepticism. Descartes removed thither in 1629 and stayed for twenty years. He was painted there by Hals, and the portraits are the most famous of any mathematician ever made. One can see from the Copenhagen and Louvre pic- tures that he dressed perpetually in dreary black, and that if the Dutch barbers prospered, it was not due to his business. One is reminded of Raphael's Plato. Des- cartes moved his household at least once a year in order not to be bothered by the inconveniencing cour- tesies of society. Only one man knew where to find him at all times, his friend Fr. Mersenne. He gave his address to no one else. He infuriated his visitors by refusing to get out of bed if they arrived before 11 a.m.

This all changed in 1649. In that year, Descartes re- ceived a message from Queen Christina of the Goths and Vandals (i.e., Sweden), who was fascinated by his books, to come and visit her in Stockholm. This was asking quite a lot, since Sweden was out of the world, though not so much as it is today. Queen Christina was in trouble because, having read in Descartes that she should doubt everything, she began to doubt whether the Swedish Lutheran state religion was right, and it was illegal for her to do so. Eventually she had to abdicate and go into exile in Rome. At any rate, she wanted Descartes to come to Stockholm, teach her

36 THE MATHEMATICAL INTELLIGENCER VOL 13, NO 3, 1991

phi losophy and geometry , and organize a Royal Academy of Science. Perhaps Descartes was flattered by this attention, because he accepted the invitation, and it was the end of him. The Queen wanted to study his Method and draw her tangent lines at 5 a.m. when she got up, not at 11 a.m. when he did. Descartes got run down by this new regime, and after catching pneumonia walking to the palace every early winter morning, he died on 11 February 1650.

I want to pause here to note that in his Short Account of the History of Mathematics, the English historian of science W. W. Rouse Ball wrote (p. 271) of Descartes, "In disposition, he was cold and selfish." I do not see how Ball could have known this. Probably he copied it from the reminiscences of one of Descartes's contem- poraries who did not like him. As a stubborn indi- vidual who knew what he wanted and went after it, Descartes must have aggravated many who were an- noyed that he had no time for them. These characters then called him "cold and selfish." More damaging is the fact that Descartes considered it right to enlist in the Dutch and Bavarian armies and take part in wars that did not concern him in the remotest way.

T h e D i s c o u r s e o n M e t h o d

Rather than dissipate my time in making a very few observations about each of Descartes's several works, I prefer to devote all the remaining time to one book, The Discourse on Method, the examination of which is sufficient for those who are content to know some- thing about Descartes and his philosophy of mathe- matics rather than everything about them. The book appeared anonymously, in case any of its doctrines should offend the authorities. It was written in French to underline the fact that it was revolutionary and had something to offer even those people who could not read Latin, for Descartes would not have agreed with Schopenhauer, who wrote in his Essay on Latin two centuries later that "he who does not know Latin is a fool, even if he is a virtuoso on the electric machine and has the base of hydrofluoric acid in his crucible." The title of the work, Discourse on Method, emphasized that he was offering a plan, a well thought out system- atic way of acquiring knowledge and then of orga- nizing that knowledge into science. Without the disci- pline of a method, one could not expect to find the truth. Descartes divided the Discourse into six books so that, he said, his readers could take it leisurely in six installments; however, it is not so long, a mere 50 printed pages. It is a masterpiece of seventeenth-cen- tury French prose.

In the first part, Descartes tells how, having studied the usual subjects at school and having travelled over much of Europe to read in what he calls "the great book of the world," he had concluded that among "the diverse actions and enterprises of all mankind, I

The portrait of Ren4 Descartes by Frans Hals (1580-1666) is on permanent loan to the Royal Museum of Fine Arts, Co- penhagen, from the Ny Carlsberg Glyptotek (Inv. No. Dep. 7). An inferior l ikeness of unknown authorship may be seen in the Louvre, Paris. Photo: Hans Petersen.

find scarcely any which do not seem to me vain and useless." He therefore decided to turn his mind in on itself and to make himself the object of his study. He was more at home in and by nature more suited to the mental world of ideas rather than the physical world without. He gave evidence of the Platonic predilection for mathemat ics and noted that of all his school studies,

Most of all was I delighted with mathematics because of the certainty of its demonstrations and the evidence of its reasoning.

So, the key to understanding Descartes is that he liked mathematics and that mathematics appeared to him not just one subject among many, nor even first among equals, but definitely special.

In Part II, he tells of his mystical experience in the stove-heated room, where God appeared to inspire him to begin from scratch:

As regards all the opinions which up to this time I had embraced, I thought I could not do better than endeavor once and for all to sweep them completely away

and to start all over. Descartes was one of those peo- ple who are obsessed with wanting to be absolutely certain. Such people must almost surely be disap-

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 3, 1991 37

pointed, and Descartes was careful not to recommend his plan for public consumption:

The simple resolve to strip oneself of all opinions and be- liefs formerly received is not to be regarded as an example that each man should follow.

He thought, though, that he might be the exception and end up the better for it, and he was at least sure that in going his own way he would not succumb to those errors mank ind had adopted by unan imous con- sent:

The voice of the majority does not afford a proof of any value in truths a little difficult to discover, because such truths are much more likely to have been discovered by one man than by a nation.

Descartes then goes on to explain the me thod of four parts that he adopted as an infallible procedure for discovering the truth. He came u p o n it by ob- serving how mathematicians go about their art; mathe- matics for h im provided the correct m e t h o d of rea- soning and seeking for truth in all subjects. The four parts are:

1) To accept noth ing as true that he did not clearly recognize to be so;

2) Divide and conquer; to break each big problem up into many smaller ones;

3) To proceed mathematically in solving the smaller problems, that is, from the simplest to the more com- plex, one at a t ime according to their order;

4) To check all his work to catch any error of omis- sion or commission.

This method was sure to work, he believed, because

Those long chains of reasoning, simple and easy as they are, of which geometricians make use in order to arrive at the most difficult demonstrations, had caused me to imagine that all those things which fall under the cogni- zance of man might very likely be mutually related in the same fashion.

He concludes the section by observing that he was twenty- three years old w h e n he came up wi th this plan.

Descartes begins Part III by observing that because he could not postpone living until he arrived at the truth he was after, he determined to live for the time being according to a reasonable moral code, which also had four parts:

1) To obey the laws and customs of his country, and to adhere to its religion;

2) Once he had decided to do something, to be firm and resolute in doing it;

3) To try always to conquer himself rather than for- tune, and to alter his desires rather than change the order of the world;

4) To review all the occupations of men in his life in order to determine the best for him, but meanwhile to continue in his own, namely, thinking.

He then describes how in his travels he viewed all the comedies that the wor ld displays before with- drawing to Holland to live as quietly as a hermit in deserts the most remote.

In Part IV, Descartes explained that though he could doubt everything else, he could not doubt that he who was thinking existed, and he arrived at the first prin- ciple of his ph i losophy , COGITO ERGO S U M - - I th ink , the re fo re I am. He t h e n p r o c e e d e d to the highest speculations:

I saw from the very fact that I thought of doubting the truth of other things, that it very evidently and certainly followed that I was; on the other hand if I had only ceased from thinking, even if all the rest of what I have ever imagined had really existed, I should have no reason for thinking that I had existed. From this I knew that I was a substance the whole essence or nature of which is to think, and that for its existence there is no need for any place, nor does it depend on any material thing; so that this "me," that is to say, the soul by which I am what I am, is entirely distract from body, and is even more easy to know than is the latter; and even if the body were not, the soul would not cease to be what it is.

He then describes how his mind conceived clearly and distinctly of an all-perfect being, and since for it not to exist would be an imperfection in it, it had to exist: The existence of the perfect being was implied in the idea of God just as, he says, the fact that the sum of the angles of a triangle is 180 ~ is implied in the idea of a triangle. The existence of God is therefore as certain as the results of mathematics; it is much more certain than the existence of the physical world, which may be an illusion, like something we see in a dream. In fact, instead of proving the existence of God from de- sign in nature (which John Stuart Mill said was the only a rgument with possibilities), he proved that the physical world existed from the existence of God, be- cause God would not deceive us. Thus, for Descartes, unlike for most phi losophers , the existence of the physical world is more difficult to establish than the immortal i ty of the soul and the existence of God, and in fact cannot be established wi thout first proving that Deity exists. He turns the usual order of things up- s ide-down.

Part V begins wi th a rev iew of all the theorems about the world that Descartes was able to prove using his me thod . The physical wor ld that we live in, he says, obeys laws tha t fol low direct ly f rom the at- tributes of God; they are necessary, so that, in a sense, we have here the idea that this is the only possible world:

Even if God had created other worlds, He could not have created any in which these laws would fail to be observed.

The laws of nature, then, follow from the perfection of Deity, a proposition that John Stuart Mill was to attack in his Essay on Nature. These laws are mathematical, and any other world that God created would turn out

3 8 THE MATHEMATICAL INTELLIGENCER VOL 13, NO 3, 1991

to be exactly like this one we now have. God did not need to create the world exactly as we now see it; it would have evolved thus even if God had only pro- duced the chaotic matter and allowed the laws to act upon it, but God did so in order to save time. Des- cartes then goes on to treat in some detail the func- tioning of the human heart, asking his reader to dis- sect the heart and lungs of a great mammal as they proceed through his description. The section ends with an account of how the soul of a man differs from that of an animal, namely, the man 's has reason, something independent of body and therefore not mortal, i.e., immortal.

For next to the error of those who deny God, which I think I have already sufficiently refuted, there is none which is more effectual in leading feeble spirits from the straight path of virtue, than to imagine that the soul of the brute is of the same nature as our own, and that in conse- quence, after this life we have nothing to fear or to hope for, any more than the flies and ants.

Like St. Paul, Descartes was one of those people who were obsessed with death. He just could not believe that his mind could stop thinking, any more than there could cease to be circles and triangles.

As for that reason which Descartes says distin- guishes the soul of man from that of an animal, what is the sign of it? The sign of reason, according to Aris- tippus, the Socratic philosopher, was mathematics.

Finally, in Part VI, Descartes tells how he had de- layed the publication of his scientific discoveries when he heard of the condemnation of Galileo, lest any of the opinions he expressed be found offensive by the authorities. He was tempted to change his mind when he realized that by keeping his method to himself, he was holding up the advancement of the human race, which would benefit from the truths his procedures made it possible to discover. Should he, for the good of humanity, allow his treatise to be published, and invite all men of learning to adopt his method and communicate to him the various discoveries that they should make by using it, so that he might circulate them to all? Indeed, he hoped for significant discov- eries, especially in medicine, which he considered the only real hope for the improvement of the human condition:

The mind depends so much on the temperament and dis- position of the bodily organs that, if it is possible to find a means of rendering men wiser and cleverer than they have hitherto been, I believe that it is in medicine that it must be sought.

No, Descartes finally decided he should not go public because 1) the inevi table cont rovers ies that his writings would arouse would disturb the peace and quiet he required for further progress, 2) the contribu- tions of others would probably be full of mistakes and superfluities, and 3) there is no better way to insure progress in science than to let the individual genius

alone and encourage him by protecting his precious leisure from the importunities of others. Nevertheless, as a sort of compromise, he relented and published three scientific appendices, on meteors, on optics, and on geometry, because 1) he did need to interest other scientists in helping him with necessary experiments and 2) he did not want to make people think that he was keeping quiet because he had something criminal to hide.

A Different Point of View

Descartes's thesis that the method of mathematics was universally valid and necessary for arriving at the truth in all the sciences was never received by all those competent to hold an opinion in such matters, nor have all philosophers ever agreed that mathematics was essential to strengthen, refine, and enrich the in- tellectual powers of students. Cardinal Newman, for example, in The Idea of a University, taught that the pe- rusal of the poets, historians, and philosophers of an- cient Greece and Rome, i.e., the Classics, will best ac- complish this latter purpose, and that each branch of knowledge has its own method of reasoning and in- quiring, that these methods are contrary the one to the other, and that controversy arises when the practi- tioners of one science attempt to impose its method on another. To prevent this aggression, he assigned to philosophy the authority to determine the method proper to each branch of knowledge and the precise boundaries of its subject matter. Mathematics did not play a conspicuous role in Newman's treatise, though the mathematical quadrivium provided four of the seven liberal arts of the ancient Roman system.

Those readers who are intrigued by the speculations discussed in this essay would do well to begin their investigations by reading the Republic, Timaeus, and Meno of Plato and, of course, the Discourse on Method by Descartes. For the history of philosophy, they might examine Will Durant's The Story of Philosophy, which made its author a millionaire.

Further Reading

W. W. Rouse Ball, A Short Account of the History of Mathe- rnatzcs, New York: Dover Publications, Inc. (1960).

Sir Kenneth Clark, Cwilizat~on, a Personal View (illustrated abridged transcript of the PBS television series), New York: Harper and Row (1969).

Ren6 Descartes, Discourse on Method, in Great Books of the Western World, vol. 31, translated by E. S. Haldane and G. R. T. Ross, Chicago: Encyclopedia Britannica Inc., (1952).

Will Durant, The Story of Phdosophy, New York: Simon and Schuster (1927), 20th printing.

Department of Mathematzcs Allegheny College Meadville, PA 16335 USA

THE MATHEMATICAL INTELLIGENCER VOL 13, NO 3, 1991 39