Mathematicians From France

"One of my conjectures was solved in six months, a second in five years, a third in ten. But the basic conjecture, despite heroic efforts rewarded by two Fields Medals, remains a conjecture, now called MLC: the Mandelbrot Set is locally connected. The notion that these conjectures might have been reached by pure thought — with no picture — is simply inconceivable."

"For many years I had been hearing the comment that fractals make beautiful pictures, but are pretty useless. I was irritated because important applications always take some time to be revealed. For fractals, it turned out that we didn't have to wait very long. In pure science, fads come and go. To influence basic big-budget industry takes longer, but hopefully also lasts longer."

"My efforts over the years had been successful to the extent, to take an example, that fractals made many mathematicians learn a lot about physics, biology, and economics. Unfortunately, most were beginning to feel they had learned enough to last for the rest of their lives. They remained mathematicians, had been changed by considering the new problems I raised, but largely went their own way."

"My ambition was not to create a new field, but I would have welcomed a permanent group of people having interests close to mine and therefore breaking the disastrous tendency towards increasingly well-defined fields. Unfortunately, I failed on this essential point, very badly. Order doesn't come by itself."

"There is a saying that every nice piece of work needs the right person in the right place at the right time. For much of my life, however, there was no place where the things I wanted to investigate were of interest to anyone. So I spent much of my life as an outsider, moving from field to field, and back again, according to circumstances. Now that I near 80, write my memoirs, and look back, I realize with wistful pleasure that on many occasions I was 10, 20, 40, even 50 years "ahead of my time."

"The Mandelbrot set covers a small space yet carries a large number of different implications. Is it a fitting epitaph? Absolutely."

"My work is more varied than at any other point in my life. I am still carrying out research in pure mathematics. And I am working on an idea that I had several years ago on negative dimensions. … Negative dimensions are a way of measuring how empty something is. In mathematics, only one set is called empty. It contains nothing whatsoever. But I argued that some sets are emptier than others in a certain useful way. It is an idea that almost everyone greets with great suspicion, thinking I've gone soft in the brain in my old age. Then I explain it and people realise it is obvious. Now I'm developing the idea fully with a colleague. I have high hopes that once we write it down properly and give a few lectures about it at suitable places that negative dimensions will become standard in mathematics."

"In a different era, I would have called myself a natural philosopher. All my life, I have enjoyed the reputation of being someone who disrupted prevailing ideas. Now that I'm in my 80th year, I can play on my age and provoke people even more."

"There is a problem that is specific to financial markets. In most fields of research, when someone makes an important finding, they publish it. In the case of prices, they set up a firm and sell advice about their discovery. If they can make money from it, they will. So the research into market dynamics is a closed field."

"The most important thing I have done is to combine something esoteric with a practical issue that affects many people. In this spirit, the stock market is one of the most attractive things imaginable. Stock-market data is abundant so I can check everything. Financial markets are very influential and I want to be part of this field now that it is maturing."

"What motivates me now are ideas I developed 10, 20 or 30 years ago, and the feeling that these ideas may be lost if I don't push them a little bit further."

"My life seemed to be a series of events and accidents. Yet when I look back I see a pattern."

"How Long Is the Coast of Britain?"

"The Mandelbrot set is the modern development of a theory developed independently in 1918 by Gaston Julia and Pierre Fatou. Julia wrote an enormous book — several hundred pages long — and was very hostile to his rival Fatou. That killed the subject for 60 years because nobody had a clue how to go beyond them. My uncle didn't know either, but he said it was the most beautiful problem imaginable and that it was a shame to neglect it. He insisted that it was important to learn Julia's work and he pushed me hard to understand how equations behave when you iterate them rather than solve them. At first, I couldn't find anything to say. But later, I decided a computer could take over where Julia had stopped 60 years previously."

"There is no single rule that governs the use of geometry. I don't think that one exists."

"There is nothing more to this than a simple iterative formula. It is so simple that most children can program their home computers to produce the Mandelbrot set. … Its astounding complication was completely out of proportion with what I was expecting. Here is the curious thing: the first night I saw the set, it was just wild. The second night, I became used to it. After a few nights, I became familiar with it. It was as if somehow I had seen it before. Of course I hadn't. No one had seen it. No one had described it. The fact that a certain aspect of its mathematical nature remains mysterious, despite hundreds of brilliant people working on it, is the icing on the cake to me."

"I always felt that science as the preserve of people from Oxbridge or Ivy League universities — and not for the common mortal — was a very bad idea."

"Many creative minds overrate their most baroque works, and underrate the simple ones."

"Fractal geometry is not just a chapter of mathematics, but one that helps Everyman to see the same world differently."

"CARTESIAN, adj. Relating to Descartes, a famous philosopher, author of the celebrated dictum, Cogito ergo sum -- whereby he was pleased to suppose he demonstrated the reality of human existence. The dictum might be improved, however, thus: Cogito cogito ergo cogito sum -- "I think that I think, therefore I think that I am;" as close an approach to certainty as any philosopher has yet made."

"Doubt is the origin of wisdom and Latin: Dubium sapientiae initium. This has been attributed to Descartes, including here previously, but no original attribution has been found. Descartes Meditationes de prima philosophia has been cited as the source of Dubium sapientiae initium, but this quote is not found in this work."

"An optimist may see a light where there is none, but why must the pessimist always run to blow it out?"

"Methods of drawing tangents were invented by Roberval and Fermat... Descartes gave a third method. Of all the problems which he solved by his geometry, none gave him as great pleasure as his mode of constructing tangents. It is profound but operose, and, on that account, inferior to Fermat's. His solution rests on the method of Indeterminate Coefficients, of which he bears the honour of invention. Indeterminate coefficients were employed by him also in solving bi-quadratic equations."

"It is frequently stated that Descartes was the first to apply algebra to geometry. This statement is inaccurate, for Vieta and others had done this before him. Even the Arabs sometimes used algebra in connection with geometry."

"In the Greek geometry the idea of motion was wanting but with Descartes it became a very fruitful conception. ...This geometric idea of co-ordinate representation, together with the algebraic idea of two variables in one equation having an indefinite number of simultaneous values, furnished a method for the study of loci, which is admirable for the generality of its solutions. Thus the entire conic sections of Apollonius is wrapped up and contained in a single equation of the second degree."

"It is frequently stated that Descartes was the first to apply algebra to geometry. This statement is inaccurate, for Vieta and others had done this before him. Even the Arabs some times used algebra in connection with geometry. The new step that Descartes did take was the introduction into geometry of an analytical method based on the notion of variables and constants, which enabled him to represent curves by algebraic equations. In the Greek geometry, the idea of motion was wanting, but with Descartes it became a very fruitful conception. By him a point on a plane was determined in position by its distances from two fixed right lines or axes. These distances varied with every change of position in the point. This geometric idea of co-ordinate representation, together with the algebraic idea of two variables in one equation having an indefinite number of simultaneous values, furnished a method for the study of loci, which is admirable for the generality of its solutions. Thus the entire conic sections of Apollonius is wrapped up and contained in a single equation of the second degree."

"In mechanics Descartes can hardly be said to have advanced beyond Galileo. ...His statement of the first and second laws of motion was an improvement in form, but his third law is false in substance. The motions of bodies in their direct impact was imperfectly understood by Galileo, erroneously given by Descartes, and first correctly stated by Wren, Wallis, and Huygens."

"The first important example solved by Descartes in his geometry is the "problem of Pappus"... Of this celebrated problem the Greeks solved only the special case... By Descartes it was solved completely, and it afforded an excellent example of the use which can be made of his analytical method in the study of loci. Another solution was given later by Newton in the Principia."

"Descartes' geometry was called "analytical geometry," partly because unlike the synthetic geometry of the ancients it is actually analytical in the sense that the word is used in logic; and partly because the practice had then already arisen, of designating by the term analysis the calculus [i.e., symbolic calculation or computation] with general quantities."

"His Geometry is not easy reading. An edition appeared subsequently with notes by his friend De Beaune, which were intended to remove the difficulties."

"His philosophy has long since been superseded by other systems, but the analytical geometry of Descartes will remain a valuable possession forever."

"Among the earliest thinkers of the seventeenth and eighteenth centuries, who employed their mental powers toward the destruction of old ideas and the up-building of new ones, ranks Rene Descartes. Though he professed orthodoxy in faith all his life, yet in science he was a profound sceptic. He found that the world's brightest thinkers had been long exercised in metaphysics, yet they had discovered nothing certain; nay, they had even flatly contradicted each other. ...The certainty of the conclusions in geometry and arithmetic brought out in his mind the contrast between the true and false ways of seeking the truth. He thereupon attempted to apply mathematical reasoning to all sciences."

"He dedicated his Book of Principles to his most Illustrious Disciple, Elizabeth, Princess Palatine of the Rhine... The Princess had been Educated in the Knowledge of abundance of Languages, and in whatsoever Learning is comprised under the name of Litterae humaniores, or Politiores; but the elevation of, and profoundness of her genius and natural parts, would not suffer her to dwell long upon these Arts, by which the greatest Wits of her Sex, who are satisfied with desiring to seem somebody, are commonly limited. She desir'd to proceed to those parts of Learning, that the strongest Application of Men had advanced, and accomplish'd her self with, and became a great proficient in Philosophy and Mathematicks; till such time as seeing the Essays of Monsieur Des Cartes his Philosophy, she conceived such high esteem and affection for his Doctrine, that she look'd upon all she had learn'd till that time as good as nothing; and so put her self under his Tuition for to raise a new Structure upon his Principles. Thereupon she sends to him, to come and see her, that she might drink in the true Phi∣osophy at the Fountain Head; and the great desire to do her Service nearer, was one of the reasons that drew him to Leiden & to Eindegeest. Never did Master more happily improve the docibility, aptness, penetration, and withal the solidity of a Scholar's Mind. Having accustomed her insensibly to the profound Meditation of the grand Mysteries of Nature, and sufficiently exercising of her in the most abstracted Questions of Geometry, and the most sublime ones of Metaphysicks. There was no longer any thing abstruse or mysterious to her; and he ingeniously confesseth and owneth, that he had not yet met with any besides her (he excepted Regius in another place) that ever arrived at a perfect understanding of the Works he had published till that time. By this Testimony that he bore to the extraordinary Capacity of the Princess, he intended to distinguish her from those who were not able to apprehend his Metaphysicks, altho' they might have some insight into Geometry; and from those that were not able to understand his Geometry, altho' they might be pretty well vers'd in Metaphysical Truths. She continued to Philosophise with him Viva voce, till a certain Accident obliged her to absent herself from the Presence of the Queen of Bohemia her Mother, and to quit her abode in Holland for Germany; then she changed her Acquaintance into an Intelligence by Letter, which she kept afoot with him, by the Ministery of the Princesses her Sisters."

"[He] came back to Paris towards the middle of October [1644]. At his Arrival, An Edition of his Principles of philosophy... and the Latine Translation of his Essays [he found] finished, and the Copies came out of Holland. The Treatise of Principles did not come out, neither did that Piece he called his World, nor his Course of Philosophy, both of which were suppress'd. He had a mind to divide them into other Parts: The First of which contains the Principles of Humane Knowledge, which one may call the first Philosophy or Metaphysicks: wherein it hath very much relation and connexion with his Meditations. The Second contains what is most general in Philosophy, and the Explanation of the first Laws of Nature, and of the principles of natural things, the Proprieties of Bodies, Space, and Motion, &c.The Third contains a particular Explanation, of the System of the World, and more especially of what we mean by the Heavens and Celestial Bodies.The Fourth contains whatsoever belongs to the Earth. That which is most remarkable in this Work, is, That the Author after having first of all established the distinction and difference he puts between the Soul and the Body, when he hath laid down, for the Principles of corporeal things, bigness, figure and local motion; all which are things in themselves so clear and intelligible, that they are granted and received by every one whatsoever; he hath found out a way to explain all Nature in a manner, and to give a reason of the most wonderful Effects, without altering the Principles; yea, and without being inconsistent with himself in any thing whatsoever. Yet... he [had] not the presumption for all that to believe he had hit upon the explication of all natural things, especially such that do not fall under our senses, in the same manner as they really and truly are in themselves. He should do something indeed, if he could but come the nearest that it was possible to likelihood or verisimilitude, to which others before him could never reach; and if he could bring the matter about, that, whatsoever he had written should exactly agree with all the Phenomena's of Nature, this he judged sufficient for the use of Life, the profit and benefit of which seems to be the main and only end one ought to propose to himself in Mechanicks, Physick, or Medicine; and in all Arts that may be brought to perfection by the help of Physick or natural Philosophy. But of all things he hath explained, there is not one of them that doth not seem at least morally certain in respect of the profit of life, notwithstanding they may be uncertain in respect of the absolute Power of God. Nay, there are several of them that are absolutely, or more than morally certain; such as are Mathematical Demonstrations, and those evident ratiocinations he hath framed concerning the existence of material things. Nevertheless, he was indued with that Modesty, as no where to assume the authority of positively deciding, or ever to assert any thing for undeniable. Altho' what he intended to offer, under the Name of Principles of Philosophy, was brought to that Conclusion, that one could not lawfully nor reasonably require more for the perfecting his design; yet did it give some cause to his Friends, to hope to see the Explication of all other things, which made people say, That his Physick was not compleat. He promised himself likewise to explain after the same manner, the nature of other more particular Bodies, that belong to the Terrestrial Globe; as, Minerals, Plants, Animals, and Man in particular; After which, he proposed to himself (according as God should please to lengthen out his days) to treat with the same exactness of all Physick or Medicine, of Mechanicks, and of the whole Doctrine of Morality or Ethicks; whereby to present the World with an entire Body of Philosophy."

"Man occupies a special place in the Cartesian scheme. He alone is endowed with mind. Descartes believed that animals did not possess one, that they were simply extremely complicated automatons. Other thinkers have rejected this point of view and proposed to endow all matter in the universe—living or inanimate—with consciousness. This "panpsychism" has been promoted by, among others, Teilhard de Chardin and, more recently by the British-American physicist Freeman Dyson, who holds that mind is present in every particle of matter."

"After Bruno's death, during the first half of the seventeenth century, Descartes seemed about to take the leadership of human thought... in promoting an evolution doctrine as regards the mechanical formation of the solar system... but his constant dread of persecution, both from Catholics and Protestants, led him steadily to veil his thoughts and even to suppress them. The execution of Bruno had occurred in his childhood, and in the midst of his career he had watched the Galileo struggle in all its stages. He had seen his own works condemned by university after university under the direction of theologians and placed upon the Roman Index. ...Since Roger Bacon, perhaps, no great thinker had been so completely abased and thwarted by theological oppression."

"The first thing that we know about ourselves is our imperfection. This is what Descartes meant when he said: 'I know God before I know myself.' The only mark of God in us is that we feel that we are not God."

""Catatau" by Paulo Leminski captures that impulse as he imagines the French philosopher René Descartes driven mad by Brazil, a country where everything gets mixed and defies categorization."

"The truth is sum, ergo cogito — I am, therefore I think, although not everything that is thinks. Is not consciousness of thinking above all consciousness of being? Is pure thought possible, without consciousness of self, without personality? Can there exist pure knowledge without feeling, without that species of materiality which feelings lends to it? Do we not perhaps feel thought, and do we not feel ourselves in the act of knowing and willing? Could not the man in the stove [Descartes] have said: "I feel, therefore I am"? or "I will, therefore I am"? And to feel oneself, is it not perhaps to feel oneself imperishable?"

"Having laid down his rules of method, Descartes proceeded to deviate from them. We expect a demonstration of the truth after the principles have already been found, such as been prescribed by Plato. But we get a different way of accounting for the facts: "the way of hypothesis" proscribed by Plato. Unable to proceed further deductively, he resorted to the invention of and choice among different hypothesis, the choice being determined by crucial experiments. Thus he gave up the certainty of the a priori method in favor of the conjectural a posteriori. This meant that in his practice, the synthesis preceded analysis, for it preceded that form of it known as inductive testing."

"The mechanical philosophy is a case of being victimized by metaphor. I choose Descartes and Newton as excellent examples of metaphysicians of mechanism malgré eux, that is to say, as unconscious victims of the metaphor of the great machine. Together they have founded a church, more powerful than that founded by Peter and Paul, whose dogmas are now so entrenched that anyone who tries to reallocate the facts is guilty of more than heresy."

"His decipherment of Nature might be crude, yet he had the courage to insist that the mechanical sense could be made of the workings of Nature, throughout the realms of physics, chemistry, and even physiology. By reasserting the unity and rationality of Nature, he did as much as any man to put seventeenth-century scientists back on the intellectual road first trodden by the Greeks."

"It was left up to Newton to compute the detailed implications of the vortex-theory... and the result demolished the foundations of Descartes' cosmology."

"How you picked your hypotheses (he argued) was of no importance whatever... even if picked at random. ...[A] hypothesis was to be judged by its fruits. ...Descartes' analogy between code-breaking and theory-making is excellent."

"Algebra had already been applied to geometry by other writers, as we have seen. The wholly new contribution made by Descartes was in importing the idea of motion into geometry. It is said that the idea came to him while lying in bed and watching the movements of a fly crawling near an angle of the room. He saw that its position at any moment could be defined by its perpendicular distance from the ceiling and two adjacent walls. Thus he saw a curve as described by a moving point, the point being the point of intersection of two moving lines which were always parallel to two fixed lines at right angles. As the moving point described the curve, its distances from the two fixed axes would vary in a manner characteristic of the curve, and an equation between these distances could be formed which would express some property of the curve. Algebraical transformations of this equation would then reveal other properties of the curve."

"Despite Newton's belated appreciation of Euclid's geometry, he set it aside as an undergraduate and immediately turned to Descartes' Geometrie, a much more difficult text. Newton read a few pages... and immediately got stuck. ...The second time through, he progressed a page or two further before running into more difficulties. Again, he read it from the beginning, this time getting further still. He continued this process until he mastered Descartes' text. Had Newton mastered Euclid first, Descartes' analytic geometry would have been much easier to understand. Newton later advised others not to make the same mistake. But Descartes had ignited Newton's interest in mathematics, an interest that bordered on obsession."

"[E]arly analytic geometers—Descartes in particular—did not accept that geometry could be based on numbers or algebra. Perhaps the first to take the idea of arithmetizing geometry seriously was Wallis..."

"Descartes... clearly understood the power of algebraic methods in geometry. He wanted to withhold this power from his contemporaries, however, particularly... Roberval... La Géométrie was written to boast about his discoveries, not to explain them. There is little systematic development, and proofs are frequently omitted with a sarcastic remark such as, "I shall not stop to explain... because I should deprive you of the pleasure...""

"Descartes's so-called dualism is often taken to represent a fundamental revolution in ideas and the starting point of modern philosophy. ...but in substance his work is... better understood as an attempt to conserve the old truths in the face of new threats. His dualism was in essence an armistice... between the established religion and the emerging science of his time. ...isolating the mind from the physical world... ensured that many of the central doctrines of orthodoxy—immortality of the soul, the freedom of will, and, in general, the "special" status of humankind—were rendered immune to any possible contravention by the scientific investigation of the physical world. ... For men such as Descartes, Malebranche, and Leibniz, solving the mind-body problem was vital to preserving the theological and political order inherited from the Middle Ages... For Spinoza, it was a means of destroying that same order and discovering a new foundation for human worth."

"Modesty was not a condition from which Descartes suffered."