Chapter 12: The Principle of Computational Equivalence

Section 6: Computational Irreducibility

[History of] exact solutions

Some notable cases where closed-form analytical results have been found in terms of standard mathematical functions include: quadratic equations (~2000 BC) (Sqrt); cubic, quartic equations (1530s) (x^(1/n)); 2-body problem (1687) (Cos); catenary (1690) (Cosh); brachistochrone (1696) (Sin); spinning top (1849; 1888; 1888) (JacobiSN; WeierstrassP; hyperelliptic functions); quintic equations (1858) (EllipticTheta); half-plane diffraction (1896) (FresnelC); Mie scattering (1908) (BesselJ, BesselY, LegendreP); Einstein equations (Schwarzschild (1916), Reissner-Nordström (1916), Kerr (1963) solutions) (rational and trigonometric functions); quantum hydrogen atom and harmonic oscillator (1927) (LaguerreL, HermiteH); 2D Ising model (1944) (Sinh, EllipticK); various Feynman diagrams (1960s-1980s) (PolyLog); KdV equation (1967) (Sech etc.); Toda lattice (1967) (Sech); six-vertex spin model (1967) (Sinh integrals); Calogero-Moser model (1971) (Hypergeometric1F1); Yang-Mills instantons (1975) (rational functions); hard-hexagon spin model (1979) (EllipticTheta); additive cellular automata (1984) (MultiplicativeOrder); Seiberg-Witten supersymmetric theory (1994) (Hypergeometric2F1). When problems are originally stated as differential equations, results in terms of integrals ("quadrature") are sometimes considered exact solutions—as occasionally are convergent series. When one exact solution is found, there often end up being a whole family—with much investigation going into the symmetries that relate them. It is notable that when many of the examples above were discovered they were at first expected to have broad significance in their fields. But the fact that few actually did can be seen as further evidence of how narrow the scope of computational reducibility usually is. Notable examples of systems that have been much investigated, but where no exact solutions have been found include the 3D Ising model, quantum anharmonic oscillator and quantum helium atom.

From Stephen Wolfram: A New Kind of Science [citation]