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Meditations on First Philosophy (1641) and Principles of Philosophy (1644), main works of French mathematician and scientist René Descartes, considered the father of analytic geometry and the founder of modern rationalism, include the famous dictum "I think, therefore I am."
A set of two perpendicular lines in a plane or three in space intersect at an origin in Cartesian coordinate system. Cartesian coordinate, a member of the set of numbers, distances, locates a point in this system. Cartesian coordinates describe all points of a Cartesian plane.
From given sets, {X} and {Y}, one can construct Cartesian product, a set of all pairs of elements (x, y), such that x belongs to {X} and y belongs to {Y}.
René Descartes, a writer, highly influenced society. People continue to study closely his writings and subsequently responded in the west. He of the key figures in the revolution also apparently influenced the named coordinate system, used in planes and algebra.
Descartes frequently sets his views apart from those of his predecessors. In the opening section of the Passions of the Soul, a treatise on the early version of now commonly called emotions, he goes so far to assert that he writes on his topic "as if no one had written on these matters before." Many elements in late Aristotelianism, the revived Stoicism of the 16th century, or earlier like Saint Augustine of Hippo provide precedents. Naturally, he differs from the schools on two major points: He rejects corporeal substance into matter and form and any appeal to divine or natural ends in explaining natural phenomena. In his theology, he insists on the absolute freedom of act of creation of God.
Baruch Spinoza and Baron Gottfried Wilhelm von Leibniz later advocated Descartes, a major figure in 17th century Continent, and the empiricist school of thought, consisting of Thomas Hobbes, John Locke, George Berkeley, and David Hume, opposed him. Leibniz and Descartes, all well versed like Spinoza, contributed greatly. Descartes, the crucial bridge with algebra, invented the coordinate system and calculus. Reflections of Descartes on mind and mechanism began the strain of western thought; much later, the invention of the electronic computer and the possibility of machine intelligence impelled this thought, which blossomed into the Turing test and related thought. His stated most in §7 of part I and in part IV of Discourse on the Method.
THE FIRST VOLUME OF WORKS BY THE THINKER WHO CONCLUDED, "I THINK, THEREFORE I AM"
René Descartes (1596-1650) was a French philosopher, mathematician and writer who spent most of his life in the Dutch Republic. The Cartesian coordinate system--allowing reference to a point in space as a set of numbers--was named after him, and he is considered the "father of analytical geometry." The accompanying volume of his philosophical writings is The Philosophical Works of Descartes (v. II). (Incidentally, his philosophy is called "Cartesian" because medieval philosophers "Latinized" his name to "Renatus Cartesius.")
He said in his Rules for the Direction of the Mind, "Only those objects should engage our attention, to the sure and indubitable knowledge of which our mental powers seem to be adequate... In accordance with the above maxim, we reject all merely probable knowledge and make it a rule to trust only what is completely known and incapable of being doubted." (Pg. 3)
He outlines in the Second Discourse his famous four rules of method: "The first of these was to accept nothing as true which I did not clearly recognize to be so...carefully to avoid precipitancy and prejudice in judgments, and to accept in them nothing more than what was presented to my mind so clearly as distinctly that I could have no occasion to doubt it. The second was to divide up each of the difficulties which I examined into as many parts as possible, and as seemed requisite in order that it might be resolved in the best manner possible. The third was to carry on my reflections in due order, commencing with objects that were the most simple and easy to understand, in order to rise little by little... to knowledge of the most complex... The last was in all cases to make enumerations so complete and reviews so general that I should be certain of having omitted nothing." (Pg. 92)
I thought that it was necessary for me... to reject as absolutely false everything to which I could imagine the least ground of doubt, in order to see if afterwards there remained anything in my belief that was entirely certain. Thus, because our senses sometimes deceive us...and because there are men who deceive themselves in their reasoning... I rejected as false all the reasons formerly accepted by me as demonstrations. And since all the same thoughts and conceptions which we have while awake may also come to us in sleep, without any of them being at that time true, I resolved to assume that everything that ever entered into my mind was no more true than the illusions of my dreams.
"But immediately afterwards I noticed that whilst I thus wished to think all things false, it was absolutely essential the that `I' who thought this should be somewhat, and remarking that this truth I THINK, THEREFORE I AM ["Cogito, Ergo Sum" in Latin] was so certain and so assured that all the most extravagant suppositions brought forward by sceptics were incapable of shaking it, I came to the conclusion that I could receive it without scruple as the first principle of the Philosophy for which I was seeking... I saw that I could conceive that I have no body, and that there was no world... yet I could not for all that conceive that I was not... from the very fact that I thought of doubting the truth of other things, it very evidently and certainly followed that I was." (Pg. 101)
More controversially, he contends, "we may also recognize the difference that exists between man and brutes. For it is a very remarkable fact that there are none so depraved and stupid... that they cannot arrange different words together... there is no other animal... which can do the same... [animals] have no reason at all, and ... it is their nature which acts in them according to the disposition of their organs, just as a clock, which is only composed of wheels and weights is able to tell the hours and measure the time more correctly than we can do with all our wisdom." (Pg. 117)
He admits in the Preface to his Meditations, "although it is absolutely true that we must believe that there is a God, because we are so taught in the Holy Scriptures, and, on the other hand, that we must believe the Holy Scriptures because they come from God... we nevertheless could not place this argument before infidels, who might accuse us of reasoning in a circle." (Pg. 133) In Meditation VI, he concludes, "I can connect the perceptions ... [and] I am perfectly assured that these perceptions occur while I am waking and not during sleep... because God is in no wise a deceiver, it follows that I am not deceived in this." (Pg. 199)
Less presciently, in Principle XX of his Principles of Philosophy, he proposes to demonstrate the non-existence of atoms: "there cannot be any atoms or parts of matter which are indivisible of their own nature [as certain philosophers have imagined]. For however small the parts are supposed to be, yet because they are necessarily extended we are always able in thought to divide any one of them into two or more parts; and thus we know that they are divisible... And even though we should suppose that God had reduced some portion of matter to a smallness so extreme that it could not be divided into smaller, it would not ... properly be termed indivisible. For ... He could not deprive Himself of His power of division, because it is absolutely impossible that He should lessen His own omnipotence." (Pg. 264)
He concludes his Principles with the statement, "recalling my insignificance, I affirm nothing, but submit all these opinions to the authority of the Catholic Church, and to the judgment of the more sage; I wish no one to believe anything I have written, unless he is personally persuaded by the force and evidence of reason." (Pg. 302)
He also says of the pineal gland, "although the soul is joined to the whole body, there is yet in that a certain part in which it exercises its functions more particularly than in all the others... The reason which persuades me that the soul cannot have any other seat in all the body than this gland wherein to exercise its functions immediately, is that I reflect that the other parts of our brain are all of them double, just as we have two eyes, two hands, two ears, and finally all the organs of you outside senses are double." (Article XXXI-XXXII, pg. 345-346)
Descartes' works are part of the "Top Ten," "MUST READ" books of philosophy; anyone with even the most meager interest in philosophy should read his Discourse, and other works.