Euclidean vs Fractal approaches in the history of math and physics


450 B.C.E.: Euclid’s axioms vs Zeno’s paradoxes of infinite regress – Euclid wins. Infinity is banished.


1200 AD: Fibonacci shows recursive sequence useful in modeling nature.


1615 Kepler brings back concept of “infinitesimal” for convergence to a limit as infinity is approached, but “infinity itself” is still banished.


1831 Gauss: “I must protest most vehemently against your use of the infinite as something consummated, as this is never permitted in mathematics. The infinite is but a façon de parler, meaning a limit to which certain ratios may approach as closely as desired when others are permitted to increase indefinitely.


1877: Cantor creates first fractal, inventing transfinite set theory. Later Lebesgue, Hausdorf and others invent new measurements.


1900: David Hilbert: Cantor’s transfinite mathematics is both crisis and opportunity. Hilbert invents his own fractal, but also declares search for axiomatic proof of the completeness of all mathematics. “No one can expel us from the paradise created by Cantor.”


1920: Wiener proves that Brownian motion is a fractal. Later harmonic theory uses scaling structure in waveform decomposition.


1925 Hiesenberg shows that abstract algebraic description of quantum physics is possible.


1927 von Neumann proves that quantum physics can be reduced to a purely axiomatic system.


The foundation: chaos or logic????