If a person asserts that they have solved a long-standing mathematical problem
and produces the written proof, who has the burden of proof – the creator
of the evidence of the proof to make it understandable or the cadre of
mathematicians to read, understand, and approve or reject the proof?
Japanese mathematician Shinichi Mochizuki has written a proof for the ABC
Conjecture using his inter-universal Teichmuller (IUT) theory, but
mathematicians say that they can’t understand it. Is it Mochizuki’s
responsibility (burden of proof) to make the proof understandable or is it the
rest of the mathematical community’s responsibility to understand it so that
they can accept that the long-standing ABC Conjecture has been solved.
The burden of proof (onus probandi)
in the legal sense is the necessity for a person(s) or entity to produce sufficient
evidence of a charge or claim. For example, a person stealing has the benefit
of assumption and the person accusing that person of stealing has the burden of
proof – i.e. to prove that the person in fact did the stealing. So the alleged
thief does not need to prove innocence, but the accuser must prove (beyond a
reasonable doubt) the guilt. This is where the statement ‘innocent until proven
guilty’ stems from.
The philosophic burden of proof is the same: the person who states a claim
– that another party disputes (or is unsure whether it is true or not) – has
the burden of proof to justify the claim. The obligation is on the person
presenting the idea to provide evidence to support its truth. Once the evidence
has been presented it is the responsibility of any opposing person(s) to prove
that the evidence is adequate or not adequate – accepted or not accepted.
Mathematically, or scientifically, a person can present their evidence of a
solution to a problem, and the mathematical community (the reviewers) may point
out that it is insufficient, requiring more data or analysis. This is usually
repeated until a sufficient number of experts agree that the solution (the
proof) has an acceptable level of data and explanation. All the while there are
standard procedures – community conventions – to meet the burden of proof.
From 7-11 December 2015, interested mathematicians met at the Mathematical
Institute, at the University of Oxford, England, to discuss the merits of
Shinichi Mochizuki’s inter-universal Teichmuller theory, which offered a
solution to the ABC Conjecture. The mathematicians found the work of Mochizuki
confusing (New Scientist, December
16, 2015).
Mochizuki posted the IUT theory, a 500-page proof of the ABC Conjecture,
online on August 31, 2012 in four papers – after working on the proof for ten
years since 2002. The ABC Conjecture, in its extreme brevity, is about the
nature of numbers commencing with a + b =
c, where a, b, and c are whole numbers, and a and b do not share any common factors (i.e. they cannot be divisible by
the same prime number). Conjectures come before theorems – they are
observations that mathematicians believe to be true, but they have not been
confirmed as true. Mathematicians create theories to explain the universe – and
hence the proof of the ABC Conjecture has to, in some way, explain the very
foundations of mathematics (addition and multiplication). If the proof is true,
it would open up the solution to other mathematical problems. Most students of
number theory have attempted the proof – as I have in my university days.
As Caroline Chen stated in her May 9, 2013 article, The Paradox of the
Proof, Mochizuki did not send his work to the Annals of Mathematics – the usual
convention when submitting a major proof. It may have gone unnoticed if it
weren’t for Jordan Ellenberg, a mathematics professor at the University of
Wisconsin-Madison who received an computerized email alert about the papers on
September 2. Ellenberg posted the papers on his blog, and they received media
attention.
That’s when the mathematical debate began – and the confusion – because the
mathematical community could not understand the proof – it was impossible to
read, they said. The problem was this: in writing the solution, Mochizuki
created new strands of mathematics, with their own terminology, that had to be
understood first, before people could determine whether his proof was adequate
and accepted. These new strands included anabelioids, Frobenioids, the etale
theta function, and log-shells.
After the four papers in 2012 (called IUTT-1 to IUTT-4) Mochizuki posted
progress reports in December 2013 and December 2014. His argument relies on
understanding his six previous works from 2005, such as the Hodge-Arakelov
Theory of elliptic curves (HAT), the Galois-Theoretic Kodaira-Spencer morphism
of an elliptic curve (GTKS), the HAT Survey 1, the HAT Survey 2, topics in
absolute anabelian geometry III (AbsTopIII), and the etale theta function and
its Frobenioid-theoretic manifestations (EtTh).
From 2012-2015, the mathematical community vented their frustration and
anger with Mochizuki’s ‘unorthodox’ approach and for not making his work
understandable. Mathematicians invited him to their universities so that he
could explain his work, but they were not prepared to go to him in Japan
because of time and work commitments. Mochizuki did not want to leave Japan.
Some mathematicians stated that his refusal to cooperate placed the burden on
them – because mathematics was, after all, a community of cooperative
colleagues – and that if he couldn’t explain his work, then the job was not
done. Several mathematicians made an effort to read his work, but were
discouraged by the terminology, original concepts, and the dense writing of the
proof.
Hence in December 2015 the Clay Mathematics Institute and the Mathematical
Institute hosted the workshop at the University of Oxford for mathematicians to
discuss Mochizuki’s proof (this is actually the third workshop in three years).
His proof involves elliptic curves, which is geometry (arithmetic geometry,
called anabelian geometry). The aim was, collectively, to become more familiar
with Mochizuki’s previous works and their integration and culmination that,
combined, lead to his complicated proof.
But what could the mathematicians hope for in five days?
Mochizuki, not wanting to leave Japan, was cooperative in answering
questions in emails and two videoconference sessions: one on December 8 and the
other on December 11, both for about two hours. One of the mathematicians at
the workshop last week was Go Yamashita, Mochizuki’s colleague at Kyoto
University, who presented his understanding of IUT. Others presented their
knowledge of the other new strands of mathematics in Mochizuki’s proof – his
‘magnum opus’ or ‘mammoth proof.’ The workshop led to some understanding by
some attendees of the ‘strikingly new ideas’ – yet still much frustration.
So some progress has been gained. Rather than confusion with the whole, there
is now only confusion with the parts – some ‘specific points.’
What do we know of Shinichi Mochizuki? He is shy, focused, travel-averse, unconventional, and unorthodox, but no crackpot. Born in Tokyo in March 1969, his family moved
to New York when he was five years old. He gained his doctorate at Princeton
University in America in 1992 at the age of 23, and worked at Harvard
University for two years. Known to be a brilliant mathematician, he moved to the
Research Institute for Mathematical Sciences (RIMS) in Kyoto, Japan, where he
has persistently and intensively studied the ABC Conjecture and created his own
world of mathematics in the process. At 33 he was promoted to professor. He
calls himself an ‘Inter-Universal Geometer.’
Brian Conrad, mathematics professor at Stanford, who wrote a summary of the
December 2015 workshop on the website, Mathbabe, said that despite no ‘aha’
moment, the work of Mochizuki should be taken seriously.
Much of the problem is in simplifying the complex, using the universal
language of mathematics. Conrad states that there has not been a significant
‘clean-up’ of Mochizuki’s work to give a more streamlined pathway into the work
with clearer terminology and notations. He concluded with ‘it is the efficient
communication of great ideas in written and oral form that inspires people to
invest the time to learn a difficult mathematical theory. What is needed is
‘instructive visible relevant examples and concise arguments that clearly
illustrate what is the point’ and if these are not provided, the people who say
they understand a theory ‘are not trying hard enough.’
Conrad indicated that the primary burden now is on those who understand IUT
to explain the substantial points to a wider community of arithmetic geometers
– specifically IUT, the relationship between number theory and arithmetic
geometry, and the recasting of the proof of Szpiro’s Conjecture (and the theory
of heights).
The burden of proof, as Conrad sees it, is through interloctors or
mediators who understand Mochizuki’s core concepts and his original mathematical
strands, and who can communicate it coherently to others in order to cross the
bridge between the creator and the mathematical community.
A teacher, for example, as someone who has mastered a mathematical concept,
has a responsibility to employ various techniques, explanations, and examples
in order for students to grasp the concepts. The language, the illustrations, streamlining
the steps, the patience, are all part of the process of teaching for learning.
At the same time, students need to be receptive – and it helps if they are
interested.
The mathematical community is
interested in Mochizuki’s proof because it is important and vital to
understanding more than just a mathematical concept. Just like an interested
student who wants to understand a concept, but the concept is presented in a
way that is difficult, time-consuming, and with new terminology, there is every
likelihood that frustration and anger will ensue – on both sides.
The next workshop to discuss the ABC Conjecture proof will be held in July
2016. Hopefully, the gap between knowledge and understanding can be reduced,
with the arrival of a long-awaited ‘aha’ moment.
http://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-workshop-by-brian-conrad/
Martina Nicolls is the author of Bardot's Comet (2011) about science and mathematics.
Martina Nicolls is the author of Bardot's Comet (2011) about science and mathematics.
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