Abc conjecture — The Enormity of Math

Occam’s razor, named after the 14th century Franciscan friar William of Ockham, is a philosophical principle, usually applied to science, that states when given more than one possible explanation, the simpler hypothesis, as measured by the number of assumptions, is more likely the case. Does this heuristic apply figuratively to mathematical proofs?

In August 2012, mathematician Shinichi Mochizuki of Kyoto University published an over 500-page proof called the Inter-universal Teichmüller theory (IUT theory) of the abc conjecture, one of the unsolved problems in number theory.

The abc conjecture, or Oesterlé-Masser conjecture (1985), involves the equation a + b = c and the relationship between prime numbers. Proving or disproving the abc conjecture could impact many Diophantine (polynomial) math problems including Tijdeman’s theorem, Vojta’s conjecture, Erdős-Woods conjecture, Fermat’s last theorem, Wieferich prime and Roth’s theorem.

Polynomials are mathematical expressions with variables and coefficients that have positive integer exponents and the operations of addition, multiplication and subtraction, but not division. Polynomials are used in business, economics, engineering and science for many purposes, including forecasting, statistical analysis, encoding, pricing, demand analysis, and modeling.

Occam’s razor is not suitable for proving mathematical theorems, because like all heuristics, simplicity does not necessarily equate to truth. Long mathematical proofs have been published by mathematicians for centuries. Examples include Lafforgue’s theorem (600+ pages), Almgren regularity theorem (955 pages), Abel-Ruffini theorem (500 pages), and the Gorenstein-Harada theorem (464 pages). In 2011, Michael Aschbacher at the California Institute of Technology in Pasadena completed the longest proof in history that began in 1971, the Classification Theorem of Finite Simple Groups, or Enormous Theorem. This proof has 15,000 pages of calculations and a 1,200-page guide with contributions from a hundred mathematicians. In this context, a 500-page mathematical proof seems nearly minuscule.

Nearly five years later, Mochizuki’s proof of the abc conjecture has neither been widely accepted as proof nor disproved; it remains a mystery. Mochizuki has not indicated any plans to shorten the explanation. Is a 500-page mathematical proof unreasonable? Time will reveal the answer.

“Mathematics is written for mathematicians.” — Nicolaus Copernicus

Copyright © 2017 Cami Rosso All rights reserved.

Originally published at https://www.linkedin.com on February 24, 2017.