John Grieve: Dialectical analysis of Mathematics
 Dialectical analysis of Mathematics 4 comments
 12 Jan 2016 @ 10:23, by John Grieve Seven years ago I wrote an article entitled "Fixing Cartesianism" in which I put forward views to counter the alienating tendencies of Descartes ideas. This present article is a sequel and is slightly more ambitious. In an old book I found a definition of mathematics which isn't used much these days, though to me it seems spot-on: "Mathematics is the science of continuous and discrete magnitude". This is a dialectical statement in that continuity and discontinuity are opposites. Dialectics was all the rage two thousand years ago with Socrates and Plato and two hundred years ago with Hegel. It has fallen into disuse through its connection, in the public mind, with Marxism. I am one of those people who is trying to repopularize it. Mathematics, in essence, consists of two complementary parts: Geometry and Algebra. Geometry is continuous (primarily) in that it deals with space, while Algebra is discontinuous (primarily) as it deals with knowns and unknowns, sums and differences, which are discrete. In this respect Algebra is connected to Time, which is also primarily discontinuous. If we look at their secondary aspects however, we see the opposite. Geometry contains discontinuities such as infinities and singularities, which some think are connected to Black Holes. Algebra, in its secondary aspect, is continuous, and can be used as in co-ordinate geometry, to represent geometrical figures. In truth everything is both continuous and discontinuous, a famous example of this is Light, which is a wave (continuous) and also a particle (discontinuous). In my previous article I argued that Cartesianism could be fixed by using some of the mathematical forms and tools of Chaos theory, such as maps and iteration. It is my opinion that the spacetime continuum, based as it is on 4 dimensional Cartesian geometry, ignores discontinuities, and needs updating to include a more dialectical view. Infinities and singularities may yet prove to be very important.

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