The best and most understandable clarification and explanation of the subject matter I have ever had! It is not easy for one to present this complicated and very confusable matter to a level easily understood. But the Professor Mrtz has very smoothly and promptly succeeded.
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There is also a 4th law of thermodynamics as entropy is dual! From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics. All observers have a localized, focused or syntropic perspective. Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes or Plato's divided line. My syntropy is your entropy and your syntropy is my entropy -- duality. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics. Teleological physics (syntropy) is dual to non-teleological physics (entropy). Ellipsoids are dual to hyperboloids -- linear algebra & matrices, Gilbert Strang. Positive curvature is dual to negative curvature -- Gauss, Riemann geometry. Curvature or gravitation is dual -- gravitational energy is dual. There are patterns of duality hardwired into physics, mathematics & philosophy. Homology is dual to co-homology. Integration is dual to differentiation. Homology and integration reduce the number of dimensions or states -- a syntropic process. Co-homology and differentiation increase the number of dimensions or states -- an entropic process. Syntropy (convergent) is dual to increasing entropy (divergent) -- the 4th law of thermodynamics. Gaussian negative curvature is defined by at least two points -- non null homotopic. The big bang is a negative curvature singularity or a white hole -- Janus hole/point. Janus holes/points have two faces (duality) -- Julian Barbour, physicist. Points are dual to lines -- the principle of duality in geometry. Null homotopic is dual to non null homotopic. Topological holes cannot be shrunk down to zero -- non null homotopic. The "bridge" between elliptic curves & modular forms is duality! Injective is dual to surjective synthesizes bijective or isomorphism. Thesis is dual to anti-thesis leads to the converging thesis or synthesis -- the time independent Hegelian dialectic. Isomorphism is duality: two differing descriptions or aspects of the same thing. Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). "Always two there are" -- Yoda. Duality is hardwired into the laws of physics & mathematics!