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.

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.""And?""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.

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.

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.

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.

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.

Modesty was not a condition from which Descartes suffered.

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

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

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.

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.

Descartes writes in a letter to Plempius in 1638 (page 81 of vol. 3 of his Philosophical Writings, ed. and trans. by Cottingham et al. [Cambridge University Press]): "For this is disproved by an utterly decisive experiment, which I was interested to observe several times before, and which I performed today in the course of writing this letter. First, I opened the chest of a live rabbit and removed the ribs to expose the heart and the trunk of the aorta. I then tied the aorta with a thread a good distance from the heart."

Starting with "I think," Descartes fixed his attention only on the "think," completely neglecting the "I." Now, this I is essential. For Man, and consequently the Philosopher, is not only Consciousness, but also—and above all—Self-Consciousness. Man is not only a being that thinks—i.e., reveals Being by Logos, by Speech formed of words that have a meaning. He reveals in addition—also by Speech—the being that reveals Being, the being that he himself is, the revealing being that he opposes to the revealed being by giving it the name Ich or Selbst, I or Self.

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.

The rationalism of René Descartes had a liberating effect on women because it assumed that the mind, not the body, was the instrument for sensation and knowledge and that men and women had the same potential for understanding. Cartesianism denied that formal education was the road to higher insight; anyone could think and reason logically. The effect of these ideas was not only to inspire a number of women, such as Mary Astell, Lady Damaris Masham, Marie de Gournay, to enter philosophical discourse with the outstanding male thinkers of their time, but it also helped them to create a new form for such a discourse through personal correspondence.

"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 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.

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."

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.

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

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.

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.

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?

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.

[http://descartes.cyberbrahma.com/life-sketch.html Brief bio]

[http://www.utm.edu/research/iep/d/descarte.htm Descartes] at the Internet Encyclopedia of Philosophy

[http://gutenberg.net/etext/59 Discourse On the Method] – at Project Gutenberg

[http://gutenberg.net/etext/4391 Selections from the Principles of Philosophy] – at Project Gutenberg

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

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

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

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

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

The 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?

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