Philosophers have a love-hate relationship with the imagination. René Descartes, for one, disparaged it as ‘more of a hindrance than a help’ in answering the most profound questions about the nature of existence. Trying to imagine one’s way towards metaphysical truth, he wrote in Meditations on First Philosophy (1641), is as foolish as falling asleep in the hope of obtaining a clearer picture of the world through dreams.
Yet Descartes also relied heavily on imagination in scientific and mathematical essays such as The World (1633), in which he tried to conjure up the details of the basic building blocks for structures such as humans, animals and machines. According to the philosopher Dennis Sepper at the University of Dallas, Descartes relied upon a kind of ‘biplanar’ imagination, pioneered by Plato, in which one level of reality could embody and display relations that existed on a different level, and vice versa.
The Scottish philosopher David Hume was equally conflicted about the imagination – especially when compared with perception and memory. ‘When we remember any past event, the idea of it flows in upon the mind in a forcible manner,’ he wrote in A Treatise of Human Nature (1738-40). But imagined images and sensations, he continued, are ‘faint and languid, and cannot without difficulty be preserved by the mind steady and uniform for any considerable time’. However, Hume also claimed that humans are most free when they’re engaging in imagination. Perception can show us only the actual, he said, but imagination can go beyond that, to the realm of the maybe, the what-if and if-only. Indeed, ‘nothing we imagine is absolutely impossible,’ Hume said.
Subscribe to Aeon’s Newsletter
What’s behind this apparent tension at the heart of the imagination? Hume put his finger on it when he talked about how our facility for fantasy helps us to move beyond and change our present reality. One need only think of how Leonardo da Vinci’s fantastical flying machines paved the way for the Wright brothers, or how H G Wells’s novel The War of the Worlds (1898) inspired the first liquid-fuelled space rocket, to see the truth of this insight. But imagination is also restricted by the extent of our previous perceptions and experiences, Hume said. ‘Let us chase our imagination to the heavens, or to the utmost limits of the Universe; we never really advance a step beyond ourselves,’ he wrote.
One way to resolve such ambivalence would be to divide the imagination into different kinds. Along these lines, towards the end of the 18th century, Immanuel Kant distinguished two forms of imagination: the productive imagination and reproductive imagination. The productive faculty is what helps to synthesise and transform sensory content into a meaningful whole. So the identification of something with pointy ears and fur, which meows and rubs itself against your legs, is brought together via the productive imagination into the form of a cat. This unifying tendency is implanted in every human mind irrespective of experience. For Kant, our productive imagination is what makes perception possible.
By contrast, the reproductive imagination is largely about recollection. When a story comes on the radio about a long-lost cat who has found its way home, you draw from the many cats you’ve seen before to picture the heartwarming scene; this would be the reproductive imagination at work. Because the reproductive faculty works only with materials previously provided to someone’s senses, it is subject to the kind of limits Hume discussed.
Kant’s bifurcation hints at why philosophers treat the imagination with both despair and delight. Perhaps the kind of imagination we despise is totally different from its more useful cousin. But in accepting this subdivision, we give up on the possibility of seeing the imagination as a unified mental faculty – which is perhaps more how we experience it.
When I think of all the wondrous things we can do with the imagination, I’m inclined towards a different way of unravelling its enigmatic duality. Rather than slicing up the imagination into distinct kinds, we might think about its distinct uses. I like to call these the transcendent and the instructive functions of the imagination. On the one hand, when we pretend, or fantasise, or escape into an engrossing work of literary fiction, imagination can take us beyond the here and now. On the other hand, when we imagine in an attempt to make sense of what other people are thinking, or to problem-solve or to make decisions, our speculations are used to help us understand the here and now. Whereas our transcendent uses of the imagination tend towards whimsy and fancy, its instructive functions point towards the practical and the concrete.
In both these modes, the secret to success seems to lie in the application of a kind of imaginative constraint. But what’s right for one use might not be fitting for the other. Perhaps the reason why philosophers have been conflicted about the imagination is that they haven’t grasped how limitations need to be tailored to circumstances. When we are writing fiction, or playing games of pretend, or making art, arguably we do our best imagining by setting the boundaries widely or removing the shackles entirely. In contrast, when we employ imagination in the context of scientific or technological discovery, or any other real-world problem-solving, we must allow our imaginations to be framed by the situation at hand.
Figuring out where to draw these lines isn’t easy. It can be extraordinarily tricky to know which factors should stay in play, and which should be eliminated. But by looking at how such constraints operate, not only can we see our way towards imaginative greatness – perhaps we can also purge philosophy of its anxiety about the idea. After all, as Hume observed, humans ‘are mightily govern’d by the imagination’.
MetaphysicsCognition & IntelligenceHistory of IdeasAll topics →
is professor of philosophy at Claremont McKenna College in California. She is the author of Persons and Personal Identity (2015).
René Descartes (March 31, 1596 – February 11, 1650) was a highly influential French philosopher, mathematician, physicist and writer. He is known for his influential arguments for substance dualism, where mind and body are considered to have distinct essences, one being characterized by thought, the other by spatial extension. He has been dubbed the "Father of Modern Philosophy" and the "Father of Modern Mathematics." He is also known as Cartesius.
- See also
- Discourse on the Method (1637)
- La Géométrie (1637)
- Meditations on First Philosophy (1641)
- Principles of Philosophy (1644)
- 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.
- Letter to Marin Mersenne (end of Feb., 1634) as quoted by Amir Aczel, Pendulum: Leon Foucault and the Triumph of Science (2003)
- 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.
- 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.
- 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.
- Letter to Pierre Chanut (Nov. 1, 1646) as quoted by Amir Aczel, Descartes' Secret Notebook (2005) citing René Descartes: Correspondance avec Elizabeth et autres lettres (1989) ed., Jean-Marie and M. Beysaade, pp. 245-246.
- 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.
- 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.
- Rules for the Direction of the Mind: IV
- 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.
- Rules for the Direction of the Mind: X.379
- As quoted in Clarke, Desmond M. (2006). Descartes : a Biography. Cambridge Press. p. 67. ISBN 978-0-521-82301-2.
- 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.
- Mais apud me omnia fiunt Mathematicè in Natura
- Loosely translated: With me, everything turns into mathematics.
- More closely translated as: but in my opinion, all things in nature occur mathematically.
- Note: "Mais" is French for "but" and the "but in my opinion" comes from the context of the original conversation. apud me omnia fiunt Mathematicè in Natura is in latin.
- Sometimes the Latin version is incorrectly quoted as Omnia apud me mathematica fiunt.
- Sources: Correspondence with Mersenne note for line 7 (1640), page 36, Die Wiener Zeit page 532 (2008); StackExchange Math Q/A Where did Descartes write...
Le Discours de la Méthode (1637)
- Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences
Main article: Discourse on the Method
- Je pense, donc je suis.
La Géométrie (1637)
- Translated as The Geometry of René_Descartes (1925) by David E. Smith and Marcia L. Lantham, unless otherwise noted.
- Thus, all unknown quantities can be expressed in terms if 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 principle 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.
- 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.
Principles of Philosophy (1644)
- In order to seek truth, it is necessary once in the course of our life, to doubt, as far as possible, of all things.
- Descartes, René (1644). Principles of Philosophy.
- Original Latin: Veritatem inquirenti, semel in vita de omnibus, quantum fieri potest, esse dubitandum
- Variant translation: If you would be a real seeker after truth, you must at least once in your life doubt, as far as possible, all things.
Quotes about Descartes
- Analytical geometry was invented by Descartes and the first exposition of it was given in 1637: that exposition was both difficult and obscure, and to most of his contemporaries, to whom the method was new, it must have been incomprehensible.Wallis made the method intelligible to all mathematicians.
- René Descartes is more widely known as a philosopher than as a mathematician, although his philosophy has been controverted while his mathematics has not. ...In accordance with the ideals of his age, when experimental science was first seriously challenging arrogant speculation, Descartes set a greater store by his philosophy than his mathematics. But he fully appreciated the power of his new method in geometry.
- The implications of Descartes' analytic reformulation of geometry are obvious. Not only did the new method make possible a systematic investigation of known curves, but, what is of infinitely greater significance, it potentially created a whole universe of geometric forms beyond conception by the synthetic method.
Descartes also saw that his method applies equally as well to surfaces... but he did not develop this. With the extension to surfaces, there was no reason why geometry should stop with equations in three variables; and the generalization to systems of equations in any finite number of variables was readily made in the nineteenth century. Finally, in the twentieth century, the farthest extension possible in this direction led to spaces of a non-denumerable infinity of dimensions. ...The path from Descartes to the creators of higher space is straight and clear; the remarkable thing is that it was not traveled earlier than it was.
- Eric Temple Bell, The Development of Mathematics (1940)
- 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.
- 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!
- 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 action at a distance. ...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.
- [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 inertia... 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 inertia 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...
- 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.
- 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.
- Tobias Dantzig, Number: The Language of Science (1930).
- 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.
- 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.
- 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.
- Bernard d'Espagnat, "My Interaction with John Bell", in Quantum [Un]speakables (2002) edited by R.A. Bertimann, A. Zeilinger
- 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.
- Michel Foucault, History of Madness (1961), Part Two: 2. The Transcendence of Delirium
- 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.
- John Freely, Before Galileo, The Birth of Modern Science in Medieval Europe (2012)
- 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.
- 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.
- 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.
- Friedrich Hayek, "Kinds of Rationalism", The Economic Studies Quarterly (1965)
- [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.
- 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.
- 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.
- Henry A. Kissinger, The White House Years
- 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.
- Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (1968) p. 206.
- 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.
- Fred S. Kleiner, Gardner’s Art through the Ages: A Global History (2009)
- 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.
- Morris Kline, Mathematics for the Nonmathematician (1967) pp. 255-256.
- As long as algebra and geometry travelled separate paths their advance was slow and their applications limited. But when these two sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace towards perfection. It is to Descartes that we owe the application of algebra to geometry,—an application which has furnished the key to the greatest discoveries in all branches of mathematics.
- He has given us only some beautiful beginnings, without getting to the bottom of things. ...he is still far from the true analysis and the general art of discovery. For I am convinced that his mechanics is full of errors, that his physics goes too fast, that his geometry is too narrow, and that his metaphysics is all these things.
- As for Descartes, this is... not the place to praise a man whose genius is elevated beyond all praise. He certainly began the true and right way through the ideas, and that which leads so far; but since he had aimed at his own excessive applause, he seems to have broken off the thread of his investigation and to have been content with metaphysical meditations and geometrical studies by which he could draw attention to himself. For the rest, he set out to discover the nature of bodies for the purposes of medicine, rightly indeed, if he had completed the task of ordering the ideas of the mind, for a greater light than can well be imagined would have arisen from these very experiments. His failure to apply his mind to this problem can be explained by no other cause than that he did not adequately think through the full reason and force of the thing. For had he seen a method of setting up a reasonable philosophy with the same unanswerable clarity as arithmetic, he would hardly have used any way other than this to establish a sect of followers, a thing which he so earnestly wanted. For by applying this method of philosophizing, a school would from its very beginning, and by the very nature of things, assert its supremacy in the realm of reason in a geometrical manner and could never perish nor be shaken until the sciences themselves die through the rise of a new barbarism among mankind.
- Gottfried Wilhelm Leibniz, "On the General Characteristic," (ca. 1679) as quoted in Philosophical Papers and Letters (1956) ed., Leroy E. Loemaker
- "There is one basis of science," says Descartes, "one test and rule of truth, namely, that whatever is clearly and distinctly conceived is true." A profound psychological mistake. It is true only of formal logic, wherein the mind never quits the sphere of its first assumptions to pass out into the sphere of real existences; no sooner does the mind pass from the internal order to the external order, than the necessity of verifying the strict correspondence between the two becomes absolute. The Ideal Test must be supplemented by the Real Test, to suit the new conditions of the problem.
- Functions are the bread and butter of of modern scientists, statisticians, and economists. Once many repeated... experiments and observations produce the same functional interrelationships, those may acquire the... status of laws of nature—mathematical descriptions... Descartes' ideas... opened the door for a systematic mathematization of everything—the very essence of the notion that God is a mathematician. ...[B]y establishing the equivalence of two perspectives of mathematics (algebraic and geometric) previously considered disjoint, Descartes expanded the horizons of mathematics and paved the way to the modern era of analysis, which allows [us] to comfortably cross from one mathematical discipline to another.
- While Descartes' theory of vortices was spectacularly wrong (as Newton ruthlessly pointed out later), it was still interesting, being the first serious attempt to formulate a theory of the universe as a whole based upon the same laws that apply on the Earth's surface. In other words, to Descartes there was no difference between terrestrial and celestial phenomena—the Earth was part of a universe that obeyed uniform physical laws.
- Mario Livio, Is God a Mathematician? (2009)
- The one book that turned out to be perhaps the most influential in guiding Newton's mathematical and scientific thought was none other than Descartes' La Géométrie. Newton read it in 1664 and re-read it several times until "by degrees he made himself master of the whole." ...Not only did analytic geometry pave the way for Newton's founding of calculus... but Newton's inner scientific spirit was truly set ablaze.
- By... confounding the properties of matter with those of space he arrives at the logical conclusion, that if the matter within a vessel could be entirely removed the space within the vessel would no longer exist. In fact he assumes that all space must be always full of matter.
- The primary property of matter was indeed distinctly announced by Descartes in what he calls the "First Law of Nature": "That every individual thing, so far as in it lies, perseveres in the same state, whether of motion or of rest."
- James Clerk Maxwell, Matter and Motion (1876)
- Descartes... fell back on his original confusion of matter with space—space being, according to him, the only form of substance, and all existing things but affections of space. This error... forms one of the ultimate foundations of the system of Spinoza.
- The first and typical example of the application of mathematics to the indirect investigation of truth, is within the limits of the pure science itself; the application of algebra to geometry, the introduction of which, far more than any of his metaphysical speculations, has immortalized the name of Descartes, and constitutes the greatest single step ever made in the progress of the exact sciences. Its rationale is simple. It is grounded on the general truth, that the position of every point, the direction of every line, and consequently the shape and magnitude of every enclosed space, may be fixed by the length of perpendiculars thrown down upon two straight lines, or (when the third dimension of space is taken into account) upon three plane surfaces, meeting one another at right angles in the same point. A consequence or rather a part of this general truth is that, curve lines and surfaces may be determined by their equations.
- As in Mathematicks, so in Natural Philosophy, the Investigation of difficult Things by the Method of Analysis, ought ever to precede the Method of Composition.
- “Do you know who first explained the true origin of the rainbow?” I asked.
“It was Descartes,” he said. After a moment he looked me in the eye.
“And what do you think was the salient feature of the rainbow that inspired Descartes’ mathematical analysis?” he asked.
“Well, the rainbow is actually a section of a cone that appears as an arc of the colors of the spectrum when drops of water are illuminated by sunlight behind the observer.”
“I suppose his inspiration was the realization that the problem could be analyzed by considering a single drop, and the geometry of the situation.”
“You’re overlooking a key feature of the phenomenon,” he said.
“Okay, I give up. What would you say inspired his theory?”
“I would say his inspiration was that he thought rainbows were beautiful.”
I looked at him sheepishly. He looked at me.
“How’s your work coming?” he asked.
I shrugged. “It’s not really coming.” I wished I was like Constantine. It all came so easily to him.
“Let me ask you something. Think back to when you were a kid. For you, that isn’t going too far back. When you were a kid, did you love science? Was it your passion?”
I nodded. “As long as I can remember.”
“Me, too,” he said. “Remember, it’s supposed to be fun.” And he walked on.
- Leonard Mlodinow, Feynman’s Rainbow: A Search for Beauty in Physics and in Life (2003)
- Descartes … is distinguished from Bacon in respect of the thoroughness of his education in the Scholastic philosophy and in the profound impression that geometrical demonstration had upon his mind, and the effect of these differences in education and inspiration is to make his formulation of the technique of inquiry more precise and in consequence more critical. His mind is oriented towards the project of an infallible and universal method or research, but since the method he propounds is modelled on that of geometry, its limitation when applied, not to possibilities but to things, is easily apparent. Descartes is more thorough than Bacon in doing his scepticism for himself and, in the end, he recognizes it to be an error to suppose that the method can ever be the sole means of inquiry. The sovereignty of technique turns out to be a dream and not a reality. Nevertheless, the lesson his successors believed themselves to have learned from Descartes was the sovereignty of technique and not his doubtfulness about the possibility of an infallible method.
- Michael Oakeshott, "Rationalism in Politics" (1947), published in Rationalism in Politics and other essays (1962)
- I would inquire of reasonable persons whether this principle: Matter is naturally wholly incapable of thought, and this other: I think, therefore I am, are in fact the same in the mind of Descartes, and in that of St. Augustine, who said the same thing twelve hundred years before. ...I am far from affirming that Descartes is not the real author of it, even if he may have learned it only in reading this distinguished saint; for I know how much difference there is between writing a word by chance without making a longer and more extended reflection on it, and perceiving in this word an admirable series of conclusions, which prove the distinction between material and spiritual natures, and making of it a firm and sustained principle of a complete metaphysical system, as Descartes has pretended to do. ...it is on this supposition that I say that this expression is as different in his writings from the saying in others who have said it by chance, as in a man full of life and strength, from a corpse.
- Descartes subscribed to the doctrine of instantaneous propagation, but with him something new emerged: for his was the first uncompromisingly mechanical theory that asserted the instantaneous propagation of light in a material medium... Indeed, mechanical analogies had been used to explain optical phenomena long before Descartes, but the Cartesian theory was the first clearly to assert that light itself was nothing but a mechanical property of the luminous object and of the transmitting medium. It is for this reason that we may regard Descartes' theory of light as legitimate starting point of modern physical optics.
- A. I. Sabra, Theories of Light, from Descartes to Newton (1981)
- In the theory of the state of the seventeenth century, the monarch is identified with God and has in the state a position exactly analogous to that attributed to God in the Cartesian system of the world.
- Descartes is rightly regarded as the father of modern philosophy primarily and generally because he helped the faculty of reason to stand on its own feet by teaching men to use their brains in place whereof the Bible, on the one hand, and Aristotle, on the other, had previously served.
- Arthur Schopenhauer, “Sketch for a history of the doctrine of the ideal and the real,” Parerga and Paralipomena, E. Payne, trans. (1974) Vol. 1, p. 3
- Descartes may have made a lot of mistakes, but he was right about this: you cannot doubt the existence of your own consciousness. That's the first feature of consciousness, it's real and irreducible. You cannot get rid of it by showing that it's an illusion in a way that you can with other standard illusions.
- 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.
- Descartes... clearly understood the power of algebraic methods in geometry. He wanted to withold 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..."
- 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.
- Mitch Stokes, Isaac Newton (2010)
- 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.
- It was left up to Newton to compute the detailed implications of the vortex-theory... and the result demolished the foundations of Descartes' cosmology.
- Stephen Toulmin, June Goodfield, The Architecture of Matter (1962)
- 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.
- 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?
- 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.
- Simone Weil, Lectures on Philosophy (1959), as translated by Hugh Price (Cambridge: 1978), p. 90
- 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.
- 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.
The Life of Monsieur Des Cartes (1693)
- Source, as translated by S. R., from Adrien Baillet, La Vie De Monsieur Des-Cartes (1691)
- [He] came back to Paris towards the middle of October . 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.
- BOOK VII. From 1644 till 1650, pp. 197-200.
- 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.
- BOOK VII. From 1644 till 1650, pp. 200-202.
A History of Mathematics (1893, 1919)
- Florian Cajori, source of 1919, 2nd edition, revised and enlarged
- 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.
- His philosophy has long since been superseded by other systems, but the analytical geometry of Descartes will remain a valuable possession forever.
- His Geometry is not easy reading. An edition appeared subsequently with notes by his friend De Beaune, which were intended to remove the difficulties.
- 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.
- 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.
- 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.
- 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.
A History of Elementary Mathematics (1898, 1901)
- Florian Cajori, Copyright 1898. Set up and electrotyped January, 1894; Reprinted March, 1895; October, 1897; November, 1901; source of 1901 reprint.
- 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 sometimes used algebra in connection with geometry.
- 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.
- An optimist may see a light where there is none, but why must the pessimist always run to blow it out?
- Michel de Saint-Pierre, as quoted in Cryptograms and Spygrams (1981) by Norma Gleason, p. 106; attributed to Descartes in The Athlete's Way : Training Your Mind and Body to Experience the Joy of Exercise (2008) by Christopher Bergland, p. 271.
- 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.
- The Latin version may be found in a footnote of the 1880 edition of "Publilii Syri mimi Sententiae".
It is not enough to have a good mind. The main thing is to use it well.