It is appropriate to note that often conducted by historians comparison of philosophy of Descartes and Plato on the grounds that both saw in mathematics the science that was the most reliable of the sciences and believed that only it can provide the basis for physics, that is why many of philosophers and historians lose sight of the differences between these thinkers in understanding what mathematics itself and its role in cognition.
First of all, Plato saw in mathematics primarily a means to prepare the mind to comprehend some super-sensible reality that was the intelligible world of ideas, whereas Descartes sees it as a means of knowledge of the empirical world.
Secondly, Plato contrasts sharply mathematics as a theoretical science as a technique for calculating the logistics, while Descartes, on the contrary, brings these two spheres, comparing the activity of mathematics with the work of a weaver. In Descartes, we often encounter an almost complete identification of the geometry with a calculator. (Plato. & Reeve, 2004)
Finally, Plato thinks that math is the science of content, because it has its own special subject of study: math is all about the numbers and their relationships, and geometry that is the ratio of figures. Descartes, in contrast, believed that mathematics was the science of formal that its rules and concepts appear to be the creation of the intellect, not having it is no reality, and that is why math is all the same, the science that counts the number of stars, sounds, etc. (Plato. & Reeve, 2004)
As a result of Descartes, like a calculator, or counters, offers negligible rigorous definition of concepts introduced by ancient mathematics. For example, a point that Euclid defines as “that which has no parts," Descartes offers thought of as "something that has the full sense of the word length and an infinite number of dimensions." Since geometric shapes and lines, triangles, rectangles, in analytic geometry created by Descartes, they play the role of characters indicating quite different connections and relationships, they will easily become a means for counting and lose its own value, so that, for example, a rectangle and a line, as indicated by Descartes, should no longer fundamentally differ. In the course of action often there are cases where a rectangle after it has been produced by multiplying the two lines, soon to other action is required to understand how the line. (Descartes & Tweyman, 1993)
Trying to follow the reasoning of Plato we understand that Plato was perhaps the first one to understand the world as a contradiction. At least in his philosophy of duality, the internal differences of the world for the first time act as a principle of both the world and philosophy. Let's say that all of us can remember the famous myth that Plato creates in his treatise the Republic. Plato proposes to understand such a situation: some people chained and held captive in a mountain cave, they are constrained so that they sit with their backs to the entrance in the cave, they sit in the cave for so long that they have forgotten that there is a huge external world outside this cage, and only sometimes when the sun costs low at sunset, the sun's rays penetrate through in the course of the cave and this appears to be all the world for that people who are locked in the cave. (Plato. & Reeve, 2004)
According to Plato the cave is an allegory, it is a hole that is an arousing world in which trapped prisoners live. As the detainees of the hollow, they accept that due to the sense they understand the genuine reality. On the other hand, this life is a simple deception of the prisoners. From the genuine universe of thoughts they achieve only dubious shadows. This is what matrix according to Plato appears to be.
One of the goals of the American researcher Brumbaugh that is the author of «Plato’s Mathematical imagination» was to show that a number of places in Plato's writings that can be adequately understood by assuming the presence of not surviving in the extant texts, drawings, diagrams, matrix. It was, above all, the texts that affect issues related to mathematics, and often looked like meaningless or mysterious. (Brumbaugh, 1954)
It is interesting to note that among those used by Plato as an auxiliary means of visual matrix has nine-cell matrix was a matrix consisting of three columns and three rows. For example, in one of the dialogues of Plato, Socrates conversing with sophist Paul talking about the different types of property, various kinds of corruption and about the different types of art, to get rid of corruption. Such a nine-cell matrix is also used in the «Republic» where it is said about the different types of people, the pleasures and the principles of the soul. (Brumbaugh, 1954)
The idea that everything we take for the real world, may be a dream, familiar to many who study philosophy, poetry and literature. Most of us sooner or later faced with the idea that we mistake a dream for reality, or reality of a dream. The most famous representative of this theory in the western philosophical school of seventeenth century is a French philosopher Rene Descartes. In an attempt to provide a solid rationale for knowledge, he founded the teachings that call into question everything that you can think about. This was done in order to determine the true knowledge, which, of course, had to endure so ardent and systematic skepticism. Descartes takes the first step to the formulation of the question, considering the likelihood that life is a dream.
Brumbaugh, R. (1954). Plato's mathematical imagination. Bloomington: Indiana University Press.
Descartes, R., & Tweyman, S. (1993). René Descartes "Meditations on first philosophy" in focus. London: Routledge.
Plato., & Reeve, C. (2004). Republic. Indianapolis: Hackett Pub. Co.