27/08/2021

'Incomprehensible' proof sparks controversy in mathematics.

Imagem: Paddy Mills/ Flickr

A decade, over 500 pages, and four articles later, prominent Japanese mathematician Shinichi Mochizuki of Kyoto University was confident he had finally achieved a proof for the abc conjecture, one of the major open problems in arithmetic. In August 2012, he published his extensive proof on his personal page in PRIMS, the scientific journal where he is currently editor-in-chief. It only took a glance at the material from other colleagues in the field for his dream of achieving this great feat to crumble. Written in an enigmatic style, the proof exposed concepts unfamiliar to readers. The prestigious journal Nature classified it as "incomprehensible ," and since then, the proof has been causing a stir in the international mathematical community.

The complexity of the reasoning presented by Mochizuki impressed even renowned professionals in the field. "Looking at it (the proof), you feel a bit like you're reading an article from the future or outer space," wrote Jordan S. Ellenberg, a professor at the University of Wisconsin who was a plenary speaker at the 3Tuesdays Brazilian Mathematics Colloquium , on his personal blog at the time . Mochizuki himself estimated that a graduate mathematics student would take 10 years to understand it.

At the end of 2015, mathematicians from around the world gathered at a workshop at Oxford University in the United Kingdom to review the proof. Mochizuki declined the invitation, but the idea was that his collaborators present would explain the papers to the scientific community and answer their questions. This did not happen as expected. The resulting controversy led to an important reflection: if no one is able to understand a proof, it cannot be verified either.

Conjecture abc

Mochizuki's proof of the abc conjecture was paved by decades of research in noncommutative geometry, an area known for its high degree of difficulty. Formulated in 1985, the conjecture starts from a very simple equation: a + b = c. However, this apparent simplicity expresses a deep and hitherto unknown link between the sum and multiplication of integers.

Any integer can be factored into prime numbers, its "divisors". For example: 60 = 5 x 3 x 2 x 2. The conjecture states approximately that if many small primes divide two numbers a and b, then only a few, the large ones, divide their sum, c. A confirmed proof of the problem could revolutionize number theory, for example, by providing an innovative approach to proving Fermat's Last Theorem, the legendary problem formulated by Pierre de Fermat in 1637 and solved only in 1994.

Fields Medal winner Peter Scholze identified a flaw in the demonstration.

Five years passed between the publication of Mochizuki's articles and the demonstration beginning to be understood by the big names in mathematics. The young German Peter Scholze, already considered an authority on arithmetic geometry and who would later win the Fields Medal at the 2018 International Congress of Mathematicians (ICM) in Rio de Janeiro, was one of them. But the conclusion he reached was not very encouraging.

According to Quanta Magazine, Scholze usually refrained from discussing academic papers , except when directly asked about his opinion. In a post about the proof on the personal blog of mathematician Frank Calegari of the University of Chicago, he wrote a message stating that he was "completely unable to follow the logic after figure 3.8 in the proof of corollary 3.12."

He added emphatically that "those who claim to understand the demonstration are unwilling to admit that further explanation is needed at this point." This comment stirred the scientific community.

Articles have been published in scientific journals, but they still raise questions.

In March 2020, the scientific journal PRIMS published the Japanese mathematician's collection of articles on the abc conjecture. However, the section of corollary 3.12 criticized by Scholze remained unchanged. Despite being the journal's editor-in-chief, Mochizuki did not participate in the peer review process, something commonplace in this type of situation.

However, the mathematical community is pressing PRIMS to reveal who participated and what arguments they gave for its approval. Meanwhile, the mystery remains: will there be a confirmed proof for the abc conjecture?

Source: BBC Brazil ,Nature and Quanta Magazine