The Winding Road to Progress in Mathematics and Physics


Many textbooks that briefly describe the history of mathematics and physics leave you with the impression that progress in these fields is achieved through a direct march to the truth. People working in these domains, however, know that nothing can be further from the truth. The road to discovery is often a zigzag path, bedeviled by blind alleys and false starts. The last couple of weeks have provided two excellent examples of the tortuous process mathematicians and physicists have to go through, to confirm a major breakthrough.

About three years ago, the respected Japanese mathematician Shinichi Mochizuki posted a 500-page proof of the famous “ABC Conjecture.” In very simple terms, this 25-year-old conjecture states that if you have three positive integers a, b, and c, such that a + b = c (which gives the conjecture its name), then if a and b are composed from large powers of primes, then c is usually not divisible by large powers of primes. For instance: 32 + 81=113 can be written as 2 × 2 × 2 × 2 × 2 + 3 × 3 × 3 × 3 = 113, and 113 is a prime itself. The conjecture is thought to have significant implications for other mathematical theorems.

Unfortunately, Mochizuki’s proof involves a new branch of mathematics that he has created (“inter-universal Teichmuller theory”), and most mathematicians find his approach impenetrable. Two weeks ago dozens of mathematicians attended a workshop at Oxford, aimed at clarifying the proof.

This resulted only in further frustration, as no conclusion as to the validity of the proof emerged. The situation has not been helped by Mochizuki’s refusal to leave Japan to lecture about his proof. This represents a new paradigm of progress in mathematics, in which claimed proofs lie in a limbo of uncertainty. —> Read More