Tuesday, May 6, 2008
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Mathematics according to Descartes
Descartes contributed significantly to mathematical research by developing the conceptual foundations of analytical geometry, in addition, was the first mathematician who tried to classify the curves according to the type of equation associated to them, thus contributing to the theory of equations. Introduced the use of the last letters of the alphabet to designate the unknown and the first letters of the alphabet to designate the known terms. He also invented the method of indices (such as x2) to express powers of numbers. Also formulated the rule known as the Cartesian rule of signs, to find the number of positive and negative roots of any algebraic equation. 
Descartes contributed significantly to mathematical research by developing the conceptual foundations of analytical geometry, in addition, was the first mathematician who tried to classify the curves according to the type of equation associated to them, thus contributing to the theory of equations. Introduced the use of the last letters of the alphabet to designate the unknown and the first letters of the alphabet to designate the known terms. He also invented the method of indices (such as x2) to express powers of numbers. Also formulated the rule known as the Cartesian rule of signs, to find the number of positive and negative roots of any algebraic equation.
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Thought of Descartes
Descartes attempted to apply the philosophy of the deductive method of mathematics. Rejecting the method of education based on the principle of authority and the persuasiveness of the tradition, Descartes said that in the search path to knowledge would have accepted as true only to clear and distinct propositions, with evidence equal to that possessed by mathematical proofs and geometric. Therefore decided not to consider anything real before being absolutely certain and identified a method to discover truth in practice absolutely certain skepticism: he tried to doubt everything that felt real to investigate the possibility of doubt.
Pursuing this "method of doubt", he observes to be able to doubt everything, except their own existence: because the very act of thinking needed to be a thinking subject, he had to exist in order to think. Descartes expressed this conclusion in the famous motto: "Cogito, ergo sum" ("I think, therefore I am").
Based on the concept that thought can prove its own existence, he concluded that the essential characteristic of the subject was thinking. Descartes continued reflection claiming the existence of God and claiming that he had created two substances: the thinking substance (res cogitans), whose essential characteristic is the thinking and extended substance (res extensa), whose essential characteristic is that to occupy a particular physical extent, while the thinking substance shall comply with the laws of thought, the extended substance conforms to the mechanical laws of physics.
The bifurcation of reality into two substances, the physical and mental, is known as Cartesian dualism and extraordinarily influenced Western philosophy. 
Descartes attempted to apply the philosophy of the deductive method of mathematics. Rejecting the method of education based on the principle of authority and the persuasiveness of the tradition, Descartes said that in the search path to knowledge would have accepted as true only to clear and distinct propositions, with evidence equal to that possessed by mathematical proofs and geometric. Therefore decided not to consider anything real before being absolutely certain and identified a method to discover truth in practice absolutely certain skepticism: he tried to doubt everything that felt real to investigate the possibility of doubt.
Pursuing this "method of doubt", he observes to be able to doubt everything, except their own existence: because the very act of thinking needed to be a thinking subject, he had to exist in order to think. Descartes expressed this conclusion in the famous motto: "Cogito, ergo sum" ("I think, therefore I am").
Based on the concept that thought can prove its own existence, he concluded that the essential characteristic of the subject was thinking. Descartes continued reflection claiming the existence of God and claiming that he had created two substances: the thinking substance (res cogitans), whose essential characteristic is the thinking and extended substance (res extensa), whose essential characteristic is that to occupy a particular physical extent, while the thinking substance shall comply with the laws of thought, the extended substance conforms to the mechanical laws of physics.
The bifurcation of reality into two substances, the physical and mental, is known as Cartesian dualism and extraordinarily influenced Western philosophy.
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The Leviathan of Spinoza:
part by Spinoza 'hypothesis of a state of nature that precedes the civil society. In this condition the right of each is equal to its power, that is the force at its disposal to assert one's being, the strongest prevails over the weak. In fact, the power of one is not that the same power of nature, of which he is a particular expression. The state of nature is therefore a condition of insecurity and danger, since everyone is exposed to the possibility of having less power, less power, and therefore less natural rights of another. The reason, indicating their true good men, that their true value, thereby causing them to establish a social pact, under which the right and power of each is limited in order to ensure that all the security of person: it leaves part of their personal power to a 'higher instance, but the people who give up part of its power as an individual then it resets as a collective, in which lies the difference with Hobbes, according to which the waiver of its people individual power to give to a person single ruler. E 'then the same impulse to' self-preservation, the effort to persevere in their being, that 'man shares with all natural beings in a manner necessary to produce the shift from the state of nature to the civilian. In two places the political thought of Spinoza, however, differs from that of Hobbes, foreshadowing a completely different outcome. First, Spinoza does not believe in the covenant that the individual give up their natural right, but rather that they simply implement, through its limitation, the conditions necessary to preserve it. For this reason, as regards the amount of rights held by the individual, the condition for civil Spinoza must resemble as closely as possible the natural one. If the state of nature men were equal, must also be equal in the civil state. This leads Spinoza to prefer democracy to other forms of government (and Hobbes defended the superiority of the monarchy) but also for him the sovereign power, though democratic, it must necessarily be absolute. Second, Spinoza believes that among the natural rights of man 's can not give up in the transition to civil status should include freedom of thought and expression, too often denied to men. No government can then restrict this right, as long as it confines itself to 'rational analysis and has therefore, in itself, a value only theoretical. Freedom of thought can not in fact result in a right of resistance resulting in a 'practical political activity, as this would undermine the foundations of national security. This is where governments consider the liberal analysis of the subjects, and translate, in the case of a take, in political reality. 
part by Spinoza 'hypothesis of a state of nature that precedes the civil society. In this condition the right of each is equal to its power, that is the force at its disposal to assert one's being, the strongest prevails over the weak. In fact, the power of one is not that the same power of nature, of which he is a particular expression. The state of nature is therefore a condition of insecurity and danger, since everyone is exposed to the possibility of having less power, less power, and therefore less natural rights of another. The reason, indicating their true good men, that their true value, thereby causing them to establish a social pact, under which the right and power of each is limited in order to ensure that all the security of person: it leaves part of their personal power to a 'higher instance, but the people who give up part of its power as an individual then it resets as a collective, in which lies the difference with Hobbes, according to which the waiver of its people individual power to give to a person single ruler. E 'then the same impulse to' self-preservation, the effort to persevere in their being, that 'man shares with all natural beings in a manner necessary to produce the shift from the state of nature to the civilian. In two places the political thought of Spinoza, however, differs from that of Hobbes, foreshadowing a completely different outcome. First, Spinoza does not believe in the covenant that the individual give up their natural right, but rather that they simply implement, through its limitation, the conditions necessary to preserve it. For this reason, as regards the amount of rights held by the individual, the condition for civil Spinoza must resemble as closely as possible the natural one. If the state of nature men were equal, must also be equal in the civil state. This leads Spinoza to prefer democracy to other forms of government (and Hobbes defended the superiority of the monarchy) but also for him the sovereign power, though democratic, it must necessarily be absolute. Second, Spinoza believes that among the natural rights of man 's can not give up in the transition to civil status should include freedom of thought and expression, too often denied to men. No government can then restrict this right, as long as it confines itself to 'rational analysis and has therefore, in itself, a value only theoretical. Freedom of thought can not in fact result in a right of resistance resulting in a 'practical political activity, as this would undermine the foundations of national security. This is where governments consider the liberal analysis of the subjects, and translate, in the case of a take, in political reality.
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