Physicists From France

"René Descartes famously argued that a dog howling pitifully when hit by a carriage does not feel pain. The dog is simply a broken machine, devoid of the res cogitans or cognitive substance that is the hallmark of people. For those who argue that Descartes didn't truly believe that dogs and other animals had no feelings, I present the fact that he, like other natural philosophers of his age, performed vivisection on rabbits and dogs. That's live coronary surgery without anything to dull the agonizing pain. As much as I admire Descartes as a revolutionary thinker, I find this difficult to stomach."

"Descartes ... complained that Greek geometry was so much tied to figures "that is can exercise the understanding only on condition of greatly fatiguing the imagination." Descartes also deplored that the methods of Euclidean geometry were exceedingly diverse and specialized and did not allow for general applicability. Each theorem required a new kind of proof... What impressed Descartes especially was that algebra enables man to reason efficiently. It mechanizes thought, and hence produces almost automatically results that may otherwise be difficult to establish. ...historically it was Descartes who clearly perceived and called attention to this feature. Whereas geometry contained the truth about the universe, algebra offered the science of method. It is ... paradoxical that great thinkers should be enamored with ideas that mechanize thought. Of course, their goal is to get at more difficult problems, as indeed they do."

"As a symbol of the power of absolutism, Versailles has no equal. It also expresses, in the most monumental terms of its age, the rationalistic creed—based on scientific advances, such as the physics of Sir Isaac Newton (1642–1727) and the mathematical philosophy of René Descartes (1596–1650)—that all knowledge must be systematic and all science must be the consequence of the intellect imposed on matter. The whole spectacular design of Versailles proudly proclaims the mastery of human intelligence (and the mastery of Louis XIV) over the disorderliness of nature."

"The essential difference between Descartes and Vieta is not in the least that Descartes unites "arithmetic" and "geometry" into a single science while Vieta retains their separation. ...both have in mind a universal science: Descartes' "mathesis universalis" corresponds completely to Vieta's "zetetic," by means of which is realized, with the aid of "logistica speciosa," the "new" and "pure" algebra, interpreted as a general "analytic art." But whereas Vieta sees the most important part of analytic in "rhetoric" or "exegetic" in which the numerical computations and the geometric constructions indeed represent two different possibilities of application (so that the traditional conception of geometry is preserved), Descartes begins by understanding geometric "figures" as structures whose "being" is determined solely by their symbolic character. The truth is that Descartes does not, as is often thoughtlessly said, identify "arithmetic" and "geometry"—rather he identifies "algebra" understood as symbolic logistic with geometry interpreted by him for the first time as a symbolic science."

"Thus was the Nixon Administration first exposed to the maddening diplomatic style of the North Vietnamese. It would have been impossible to find two societies less intended by fate to understand each other than the Vietnamese and the American. On the one side, Vietnamese history and Communist ideology combined to produce almost morbid suspicion and ferocious self-righteousness. This was compounded by a legacy of Cartesian logic from French colonialism that produced an infuriatingly doctrinaire technique of advocacy."

"Aristotle remarks in his Poetics that poetry is superior to history, because history presents only what has occurred, poetry what could and ought to have occurred, poetry has possibility at its disposal. Possibility, poetic and intellectual, is superior to actuality; the esthetic and the intellectual are disinterested. But there is only one interest, the interest in existing; disinterestedness is the expression for indifference to actuality. The indifference is forgotten in the Cartesian Cogito-ergo sum, which disturbs the disinterestedness of the intellectual and offends speculative thought, as if something else should follow from it. I think, ergo I think; whether I am or it is (in the sense of actuality, where I means a single existing human being and it means a single definite something) is infinitely unimportant. That what I am thinking is in the sense of thinking does not, of course, need any demonstration, nor does it need to be demonstrated by any conclusion, since it is indeed demonstrated. But as soon as I begin to want to make my thinking teleological in relation to something else, interest enters the game. As soon as it is there, the ethical is present and exempts me from further trouble with demonstrating my existence, and since it obliges me to exist, it prevents me from making an ethically deceptive and metaphysically unclear flourish of a conclusion."

"There is no doubt of the enormous influence Descartes has exercised from his own day to ours. But his relation to modern philosophy is not that of father to son, nor of architect to palace, nor of planner to city. Rather, in the history of philosophy his position is like that of the waist of an hourglass. As the sand in the upper chamber of such a glass reaches its lower chamber only through the slender passage between the two, so too ideas that had their origin in the Middle Ages have reached the modern world through a narrow filter: the compressing genius of Descartes."

"Descartes was an eminent mathematician, and it would seem that the bent of his mind led him to overestimate the value of deductive reasoning from general principles, as much as Bacon had underestimated it."

"[I]n the Copenhagen interpretation of quantum theory we can indeed proceed without mentioning ourselves as individuals, but we cannot disregard the fact that natural science is formed by men. Natural science does not simply describe and explain nature; it is part of the interplay between nature and ourselves; it describes nature as exposed to our nature of questioning. This was a possibility of which Descartes could not have thought, but it makes a sharp separation between the world and the I impossible. If one follows the great difficulty which even eminent scientists like Einstein had in understanding and accepting the Copenhagen interpretation... one can trace the roots... to the Cartesian partition....it will take a long time for it [this partition] to be replaced by a really different attitude toward the problem of reality."

"There seems to me to exist a sort of rationalism which, by not recognizing these limits of the powers of individual reason, in fact tends to make human reason a less effective instrument than it could be. ... This sort of rationalism is a comparatively new phenomenon, though its roots go back to ancient Greek philosophy. Its modern influence, however, begins only in the sixteenth and seventeenth century and particularly with the formulation of its main tenets by the French philosopher, René Descartes."

"It should be remembered that it was Descartes who systematised our Mathematical notation. He used the letters at the end of the alphabet as variables, and those at the beginning as constants, and he brought into general use our present system of indices. He also introduced the method of indeterminate co-efficients for the solution of equations."

"The Geometry is divided into three books. In the first book Descartes briefly explains his method. He says that every geometric problem may be reduced to a problem of straight lines; and he points out that, in order to find these lines, nothing more advanced is required than the five fundamental operations of Arithmetic, viz.: Addition, Subtraction, Multiplication, Division, and Root Extraction. He advises:— (1) That the problem should be imagined as done. (2) That then lines, whether known or unknown, which appear necessary in its solution, should be named. (3) And finally, that their relation to each other should be sought, and expressed by means of an equation or equations. Descartes strongly advocates this analytic treatment of Geometry as giving greater clearness and more continuity of argument, qualities lacking in the work of the Ancients, who probably did not understand where such reasoning would carry them, and whose isolated proofs were necessarily wanting in connection and generality."

"Newton's proof of the law of refraction is based on an erroneous notion that light travels faster in glass than in air, the same error that Descartes had made. This error stems from the fact that both of them thought that light was corpuscular in nature."

"Descartes, I think, broke with this when he said, "To accede to truth, it suffices that I be any subject that can see what is evident." Evidence is substituted for ascesis at the point where the relationship to the self intersects the relationship to others and the world. The relationship to the self no longer needs to be ascetic to get into relation to the truth. It suffices that the relationship to the self reveals to me the obvious truth of what I see for me to apprehend the truth definitively. Thus, I can be immoral and know the truth. I believe this is an idea that, more or less explicitly, was rejected by all previous culture. Before Descartes, one could not be impure, immoral, and know the truth. With Descartes, direct evidence is enough. After Descartes, we have a nonascetic subject of knowledge. This change makes possible the institutionalization of modern science."

"To say that madness is dazzlement is to say that the madman sees the day, the same day that rational men see, as both live in the same light, but that when looking at that very light, nothing else and nothing in it, he sees it as nothing but emptiness, night and nothingness. Darkness for him is another way of seeing the day. Which means that in looking at the night and the nothingness of the night, he does not see at all. And that in the belief that he sees, he allows the fantasies of his imagination and the people of his nights to come to him as realities. For that reason, delirium and dazzlement exist in a relation that is the essence of madness, just as truth and clarity, in their fundamental relation, are constitutive of classical reason. In that sense, the Cartesian progression of doubt is clearly the great exorcism of madness. Descartes closes his eyes and ears the better to see the true light of the essential day, thereby ensuring that he will not suffer the dazzlement of the mad, who open their eyes and only see night, and not seeing at all, believe that they see things when they imagine them. In the uniform clarity of his closed senses, Descartes has broken with all possible fascination, and if he sees, he knows he really sees what he is seeing. Whereas in the madman's gaze, drunk on the light that is night, images rise up and multiply, beyond any possible self-criticism, since the madman sees them, but irremediably separated from being, since the madman sees nothing. Unreason is to reason as dazzlement is to daylight."

"This momentous finding of nonlocality has, in common with that of Einstein concerning time, the additional feature that it disproves the validity, not of a "view of the World", but of a deeply ingrained concept. And this brings me to the second reason. It is that this disproof of a deeply ingrained concept pointed in fact in a direction quite consonant with my own line of thought. In a way, it brings us back to Descartes, for, as we all know, Descartes was the first scientist who dared to question our common views, including even all the notions that had always seemed so primitive and obvious that thinkers, scientists and so on never hesitated in making use of them. He found out that, at the start, he could doubt of everything but his own thinking, and in this, according to Hegel, he was a hero. Unfortunately he then constructed a grand metaphysical argument that led him to the view that, after all, since God is not a liar, the "obvious" realistic concepts must apply. He thereby founded mechanicism, which is the theory that, apart from thought, everything has to be described by the exclusive means of familiar concepts. We know, of course, how deficient such a view is. But I think Descartes' really significant contribution on these matters is not mechanicism. It is what I just said. In other words, it is the realization that a sharp distinction has to be made between rationality on the one hand and the use of seemingly obvious concepts on the other hand. And that therefore, if you are a rational person you cannot demand that science should be based exclusively on seemingly obvious concepts without first logically justifying this demand. This Descartes tried to do but, since his argument based on God not being a liar is now considered as not convincing enough, we are not bound to his conclusion. Indeed, I consider that we are not even bound to the idea that physics should be expressed in an ontological language, which was more or less Einstein's view. The older Einstein seems to have considered that Reality, and even physical objects in the plural, can be described as they really are, if not by familiar concepts, at least by unfamiliar ones, such as those borrowed from mathematics. I do not think this is necessarily true. I consider that mere predictive rules, such as the Born rule in quantum mechanics, count as fully fledged explanations, or, more precisely, as constituting, when all taken together, a first step in an explanation, the second step being the philosophical idea that these predictive rules dimly reflect some existing, largely hidden structures of Mind-Independent Reality. Of course, these speculations of mine go much further than nonlocality. But you understand that they receive some support from it."

"René Descartes played his part in the world as a man in a mask. The phrase is his own. It implied no conscious duplicity but a certain apartness rendering him incapable of taking his share unreservedly in the game of life. His attitude was that of an unimpassioned spectator. He looked on and learned while others struggled for the stakes. A born recluse, he remained solitary even in the throng of social intercourse. ...The transport felt by Bacon at the glorious vision of a future in which the harvest he had sown would be reaped was not shared by Descartes. He indeed held himself to be the sower and the reaper in one. Calmly, and with settled conviction, he claimed to have virtually expounded all the phenomena of nature. He had crossed the frontier of the new world of knowledge; those who desired to follow him needed only to purge their eyes with the euphrasy of methodic doubt in order to obtain those crystal-clear intuitions from which, by sure reasoning, universal science could be deduced. ...He was a mathematician of first rate originality and power. ...Unfortunately, however, he was misled by false analogies of thought; he sought to universalise what was by its nature special and restricted; and thus pursued the flitting vision of a deceptive unity in knowledge. As the upshot, he gave to his philosophy a pseudo-mathematical character, and sacrificed the solid achievement of much by aiming at the illusory certainty of everything."

"The famous problems of antiquity doubling the cube, angle trisection, and squaring the circle]... were now disposed of by Descartes in a matter-of-fact statement that any problem which leads to an equation of the first degree is capable of a geometric solution by straight-edge only; that a straight-edge [and] compass construction is equivalent to the solution of a quadratic equation; but that if a problem leads to an irreducible equation of degree higher than the second, its geometrical solution is not possible by means of a ruler and compass only."

"Descartes implicitly assumed a complete correspondence existed between the real numbers and the points of a fixed axis. ...tacitly, because it seemed so natural as to go without saying, he accepted it as axiomatic that between the points of a plane and the aggregate of all pairs of real numbers there can be established a perfect correspondence. Thus the Dedekind-Cantor axiom, extended to two dimensions, was tacitly incorporated in a discipline which was created two hundred years before Dedekind and Cantor saw the day. This discipline became the [tool and] proving-grounds for all achievements of the following two centuries: the calculus, the theory of functions, mechanics, and physics. Nowhere did this discipline, analytic geometry, strike any contradictions; and such was its power to suggest new problems and forecast the results that wherever applied it would soon become the indispensable tool of investigation."

"The great invention... Descartes gave to the world, the analytical diagram, ...gives at a glance a graphical picture of the law governing a phenomenon, or of the correlation which exists between dependent events, or of the changes which a situation undergoes in the course of time. ...the invention of Descartes not only created the important discipline of analytic geometry, but it gave Newton, Leibnitz, Euler, and the Bernoullis that weapon for the lack of which Archimedes and later Fermat had to leave inarticulate their profound and far-reaching thoughts."

"[B]elievers in the plenum... the Cartesian idea of a space entirely filled with matter—contributed... to what was to become the Newtonian synthesis. Descartes himself achieved the modern formulation of the law of ... working it out by a natural deduction from his theory of the conservation of momentum, his theory that the amount of motion in the universe always remains the same. It was he rather than Galileo who fully grasped this principle of inertia and formulated it in all its clarity. ...The modern doctrine of had been put forward by Descartes and was quickly gaining acceptance, though people like Borelli... still seemed to think that they had to provide a force actually pushing the planets along..."

"Descartes was liable to be misled by too easy an acceptance of data that had been handed down by scholastic writers. ...two grand Aristotelian principles helped to condition the form of the universe as he reconstructed it—first, the view that a vacuum is impossible, and secondly, the view that objects could only influence one another if they actually touched—there could be no such thing as attraction, no such thing as . ...Descartes insisted that every fraction of space should be fully occupied all the time by continuous matter... infinitely divisible. The particles were... packed so tightly that one of them could not move without communicating the commotion to the rest. The matter formed whirlpools in the skies, and it was because the planets were caught each in its own whirlpool that they were carried around... all similarly caught in a larger whirlpool, which had the sun as its centre... Gravity itself was the result of these whirlpools of invisible matter which had the effect of sucking things down towards their centre. ...In the time of Newton the system of Descartes and the theory of vortices or whirlpools proved to be vulnerable to both mathematical and experimental attack."

"Descartes maintained his confidence in the instantaneity of light. ...Yet in his derivation of the law of refraction, Descartes reasoned that light traveled faster in a dense medium than in one less dense. He seems to have had no qualms about comparing infinite magnitudes!"

"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."

"Descartes devised the notation x, x2, x3, x4,... for powers, and made the final break with the Greek tradition of admitting only the first, second, and third powers ('lengths,' 'areas,' and 'volumes') in geometry. After Descartes, geometers freely used powers higher than the third without a qualm, recognizing that representability as figures in Euclidean space for all of the terms in an equation is irrelevant to the geometrical interpretation of the analysis. The principle of undetermined coefficients was also stated by Descartes. A second outstanding addition to algebra was the famous rule of signs... the first universally applicable criterion for the nature of the roots of an algebraic equation. ...it admirably represents Descartes' flair for generality which made him the mathematician that he was."

"In order to seek truth, it is necessary once in the course of our life, to doubt, as far as possible, of all things."

"I could give here several other ways of tracing and conceiving a series of curved lines, each curve more complex than any preceding one, but I think the best way to group together all such curves and them classify them in order, is by recognizing the fact that all points of those curves which we may call "geometric," that is, those which admit of precise and exact measurement, must bear a definite relation to all points of a straight line, and that this relation must be expressed by a single equation. If this equation contains no term of higher degree than the rectangle of two unknown quantities, or the square of one, the curve belongs to the first and simplest class, which contains only the circle, the parabola, the hyperbola, and the ellipse; but when the equation contains one or more terms of the third or fourth degree in one or both of the two unknown quantities (for it requires two unknown quantities to express the relation between two points) the curve belongs to the second class; and if the equation contains a term of the fifth or sixth degree in either or both of the unknown quantities the curve belongs to the third class, and so on indefinitely."

"Thus, all unknown quantities can be expressed in terms of a single quantity, whenever the problem can be constructed by means of circles and straight lines, or by conic sections, or even by some other curve of degree not greater than the third or fourth. But I shall not stop to explain this in more detail, because I should deprive you of the pleasure of mastering it yourself, as well as of the advantage of training your mind by working over it, which is in my opinion the principal benefit to be derived from this science. Because, I find nothing here so difficult that it cannot be worked out by anyone at all familiar with ordinary geometry and with algebra, who will consider carefully all that is set forth in this treatise."

"Je pense, donc je suis."

"Le bon sens est la chose du monde la mieux partagée : car chacun pense en être si bien pourvu, que ceux même qui sont les plus difficiles à contenter en toute autre chose, n'ont point coutume d'en désirer plus qu'ils en ont."

"Mais apud me omnia fiunt Mathematicè in Natura"

"No more useful inquiry can be proposed than that which seeks to determine the nature and the scope of human knowledge. ... This investigation should be undertaken once at least in his life by anyone who has the slightest regard for truth, since in pursuing it the true instruments of knowledge and the whole method of inquiry come to light. But nothing seems to me more futile than the conduct of those who boldly dispute about the secrets of nature ... without yet having ever asked even whether human reason is adequate to the solution of these problems."

"The entire method consists in the order and arrangement of the things to which the mind's eye must turn so that we can discover some truth."

"So blind is the curiosity by which mortals are possessed, that they often conduct their minds along unexplored routes, having no reason to hope for success, but merely being willing to risk the experiment of finding whether the truth they seek lies there."

"Me tenant comme je suis, un pied dans un pays et l'autre en un autre, je trouve ma condition très heureuse, en ce qu'elle est libre."

"Mr. Clerselier has written me that you are expecting from him my Meditations... in order to present them to the queen of the land. ...If I had only been as wise as they say the savages persuaded themselves that the monkeys were, I never would have become known as a maker of books: Since it is said that they imagined that the monkeys could indeed speak, if they wanted to, but that they chose not to so lest they be forced to work. And since I had not the same prudence to abstain from writing, I now have neither as much liesure nor as much peace as I would have had if I had kept quiet. But since the mistake has already been made, and since I am now known by an infinity of people at the academy, who look askance at my writings and scour them for means of harming me, I do have great hope of being known to persons of great merit, whose power and virtue could protect me."

"M. Desargues puts me under obligations on account of the pains that it has pleased him to have in me, in that he shows that he is sorry that I do not wish to study more in geometry, but I have resolved to quit only abstract geometry, that is to say, the consideration of questions which serve only to exercise the mind, and this, in order to study another kind of geometry, which has for its object the explanation of the phenomena of nature... You know that all my physics is nothing else than geometry."

"What I have given in the second book on the nature and properties of curved lines, and the method of examining them, is, it seems to me, as far beyond the treatment in the ordinary geometry, as the rhetoric of Cicero is beyond the a, b, c of children."

"No doubt you know that Galileo had been convicted not long ago by the Inquisition, and that his opinion on the movement of the Earth had been condemned as heresy. Now I will tell you that all things I explain in my treatise, among which is also that same opinion about the movement of the Earth, all depend on one another, and are based upon certain evident truths. Nevertheless, I will not for the world stand up against the authority of the Church. ...I have the desire to live in peace and to continue on the road on which I have started."

"C'est même des hypothèses simples qu'il faut le plus se défier, parce que ce sont celles qui ont le plus de chances de passer inaperçues."

"Later generations will regard set theory as a disease from which one has recovered."

"Later generations will regard Mengenlehre as a disease from which one has recovered."

"Science is a system of relations. Poincaré, saying so, says also, "It is before all a classification, a manner of bringing together facts which appearances separate, though they are bound together by some natural and hidden kinship...."It is in the relations alone that objectivity must be sought; it would be vain to seek it in beings considered as isolated from one another...."External objects...for which the word object was invented, are really objects and not fleeting and fugitive appearances, because they are not only groups of sensations, but groups cemented by a constant bond. It is this bond, and this bond alone, which is the object in itself, and this bond is a relation. "Therefore, when we ask what is the objective value of science, that does not mean: Does science teach us the true nature of things? but it means: Does it teach us the true relations of things?""

"I rely for my information about mathematical creation on such sources as Poincaré, who speaks of the mind seeming to act only of itself and on itself, selecting, making only the useful combinations, choosing, and finally being struck, as by strong light, with certainty."

"With the disappearance of the great French mathematician has disappeared the one man whose thought could carry all other thoughts, the one mind who, through a sort of rediscovery, could penetrate to its very depth all the knowledge which the mind of man can comprehend. And that is why the demise of this man at the peak of his intellectual strength is such a disaster. Discoveries will lag, groping efforts will be drawn out; for, the potent luminous brain will not be there to coordinate disjointed research, or to cast the daring plummet of a new theory into a world of obscure facts suddenly revealed by experience."

"One of the enduring legacies of Napoleon was the French system of grandes écoles, the elite schools that train the country's top technocratic and managerial students even today. ... Biographies of French mathematicians often begin with awed accounts of how well they did on the entrance tests, and how they fared on various national exams and competitions. Poincaré was no exception. He obtained the first prize in several national competitions, and was among the highest-ranking applicants to the École Polytechnique and the École Normale Supérieure in Paris, schools especially famous for the quality of their mathematics."

"Poincaré has justly emphasized the fact that we distinguish two kinds of alterations of the bodily object, "changes of state" and "changes of position." The latter, he remarked, are alterations which we can reverse by arbitrary motions of our bodies."

"Pure mathematicians... more than all others, have been led to realise how cautious we must be of the dictates of intuition and so-called common sense. They know that the fact that we can conceive or imagine a certain thing only in a certain way is no criterion of the correctness of our judgement. Examples in mathematics abound. ...Mathematicians, as a whole, refused to question the soundness of Einstein's theory on the sole plea that it conflicted with our traditional intuitional concepts of space and time, and we need not be surprised to find Poincaré... lending full support to Einstein when the theory was so bitterly assailed in its earlier days."

"Le plus grand hasard est la naissance d’un grand homme. Ce n’est que par hasard que se sont rencontrées deux cellules génitales, de sexe différent, qui contenaient précisément, chacune de son côté, les éléments mystérieux dont la réaction mutuelle devait produire le génie. On tombera d’accord que ces éléments doivent être rares et que leur rencontre est encore plus rare. Qu’il aurait fallu peu de chose pour dévier de sa route le spermatozoïde qui les portait ; il aurait suffi de le dévier d’un dixième de millimètre et Napoléon ne naissait pas et les destinées d’un continent étaient changées. Nul exemple ne peut mieux faire comprendre les véritables caractères du hasard."

"La logique parfois engendre des monstres. Depuis un demi-siècle on a vu surgir une foule de fonctions bizarres qui semblent s’efforcer de ressembler aussi peu que possible aux honnêtes fonctions qui servent à quelque chose. Plus de continuité, ou bien de la continuité, mais pas de dérivées, etc. Bien plus, au point de vue logique, ce sont ces fonctions étranges qui sont les plus générales, celles qu’on rencontre sans les avoir cherchées n’apparaissent plus que comme un cas particulier. Il ne leur reste qu’un tout petit coin."