21.
secret doctrines of the Pythagoreans: *
Plato, in the dialogue Timaeus (360 B.C) writes of the regular polyhedrons (today known as the 5 Platonic solids); he associated each of the four classical elements (earth, air, water, and fire) with a regular solid: Earth with the cube (the cube's being the only regular solid that tessellates Euclidean space, was believed to cause the solidity of the Earth); air with the octahedron (its minuscule components are so smooth that one can barely feel it); water with the icosahedron (flows out of one's hand when picked up, as if it is made of tiny little balls), and fire with the tetrahedron (the heat of fire feels sharp and stabbing). Of the 5th, the dodecahedron, Plato remarks "...the god used for arranging the constellations on the whole heaven".
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22.
theorems of regular polyhedrons: *
illustration of the 5 perfect symmetrical geometric solids.
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23.
Descartes' analysis of the infinite: *
Descartes noted that philosophy did not progress because philosophers disagreed about everything. He thought philosophy did not have the right method. He wanted to try to find the indubitable truth, the truth that cannot be doubted, and once it is found, it would be the foundation where he would deduce other truths of philosophy.
He begins by doubting everything. He cleared the table and started with a clean slate. He concluded the methodical doubt by saying that he can doubt everything, but in doubting I know I exist. ( ‘ I think, therefore I am.’), I cannot doubt: that I am, and I have ideas. Descartes is certain that he has ideas: the idea of himself, other people, and God. Where did I get the idea of God? He described this idea of God as an infinite substance, eternal, immutable, independent, omniscient, and all-powerful, a traditional idea in metaphysics. He concludes that the idea of the Infinite could only have been put in him. It couldn’t have been produced by him because the human being is a limited being, and as a limited being it cannot think of something greater than itself. The effect (the infinite) cannot be greater than the cause (finite human being). The infinite here must be a positive infinite, it is because of this positive idea of the infinite that I realize that I am imperfect.
With the idea of God, his essence necessarily includes existence. Descartes is saying that of all these ideas, the idea of God is the unique idea because its essence necessarily includes existence. This is how he proves that God exists. Knowing that the all-powerful and all-good God exists assures me that he will not deceive me.
