Conjecture

Conjecture (pronounced kuhn-jek-cher)

(1) The formation or expression of an opinion or theory without sufficient evidence for proof; an opinion or theory so formed or expressed; guess; speculation; to conclude or suppose from grounds or evidence insufficient to ensure reliability.

(2) The interpretation of signs or omens (obsolete though still used in some superstitious circles and a common phrase among occultists).

(3) In mathematics and philology, a technical term for a statement which, based on available evidence, is likely to be true but for which there’s no formal proof.

1350–1400: From the Middle English conjecturen (infer, predict, form (an opinion or notion) upon probabilities or slight evidence), from the Old and Middle French from the Latin conjectūra (a guess; inferring, an assembling of facts; reasoning), the construct being conject(us), past participle of conjicere (to throw together; to form a conclusion).  The late Middle English verb conjecturen was a direct borrowing from the Middle French, from the Late Latin conjecturāre, derivative of the noun.  The Latin conjicere is a combining form jacere (to throw) + -ūra –ure (the Latin suffix used to form nouns of quality from adjectives).  The Latin coniectūra is derived from coniectus, perfect passive participle of cōniciō (throw or cast together; guess), the construct being con- (together) + iaciō (throw, hurl).  In Middle English, there were also peacefully co-existing forms, the noun conjecte & the verb conjecten.

Derived forms include the adjective conjecturable, the adverb conjecturably, the noun conjecturer and the verbs (used with our without the object) conjectured and conjecturing.  The verbs misconjecture & misconjectured and the noun misconjecturing are valid words but so rare that some dictionaries list them as obscure.  Indeed, given the meaning of the root, it can be argued there’s little difference between conjecture and misconjecture although it could be useful in describing things in retrospect.  For those times when conjecture seems not quite right, there’s surmise, inference, supposition, theory, hypothesis, suppose, presume, guesswork, hunch, presumption, guess, fancy, opinion, conclusion, notion, guesstimate, gather, figure, conclude, feel, deem & expect.

The Oesterlé–Masser Conjecture

The Oesterlé–Masser conjecture, a problem in number theory, is named after the mathematicians Joseph Oesterlé (b 1954) and David Masser (b 1948) who first published their speculation in the 1980s and popularly known as the abc conjecture, based on the equation which underlies it all.  The conjecture postulates that if a lot of small prime numbers divide two numbers (a) and (b), then only a few large ones divide their sum (c); basically, if you add lots of primes together the result is divisible only by a few large numbers.  Mathematicians concur that intuitively this seems likely because of the nature of prime numbers but a proof has proved elusive.  It’s of interest to the profession because it might resolve some of the fundamental problems in Diophantine geometry, a typically arcane fork of number theory but beyond the implications for mathematics, given the importance of prime numbers in commerce, ICT and diplomacy (primes underpin encryption), other fields may be significantly affected. 

Japanese mathematician Shinichi Mochizuki san (b 1969) has been working on the problem for some thirty years and, over the decades has circulated within the community many un-published papers, none of which garnered much support.  Not discouraged, Mochizuki San persisted and in 2012 posted on his website, four papers 500 pages in length, claiming they contained the definitive proof (including a new theory called inter-universal Teichmüller theory (IUTT)).  While some of his peers actively disagreed with his methods or conclusions, most either ignored his work or said it couldn’t be understood, one recently commenting his experience was something like “reading a paper from the future, or from outer space”.

Several years later, despite conferences staged to explain Mochizuki san’s work to other mathematicians, there is no consensus and he has been accused of not doing enough to communicate (in the sense of explaining) his ideas.  While there are some who claim to have both read his work (that alone an achievement) and understood it (more admirable still given how much that depends on knowledge of other work he has developed over decades), they're a small sub-set of number theorists, most of whom remain sceptical or dismissive .  Interest was stirred in 2018 when two noted German mathematicians, Peter Scholze (b 1987) and Jakob Stix (b 1974), published a paper in which they asserted a critical part of Mochizuki san’s work (said to be central to the proof), was wrong.  Unusually in this matter, their work was based not only on analysis but a face-to-face meeting with Mochizuki san.  The discussion however concluded with neither sided able to persuade the other, something like three pocket calculators sitting on a table, unable to agree on the best method of determining a number without knowing that number.

In April 2020, it was announced the claimed proof would be published in the Japanese journal Publications of the Research Institute for Mathematical Sciences (RIMS).  Although Mochizuki san was RIMS's chief editor, the institution noted he was “…not involved in the review” or the decision to publish.  There was scepticism but in 2021, the material appeared in RIMS and the number theory community awaits with interest to see if there are defections from the tiny “proven” faction or the more populated “unproven”.

It's not just number theorists who have engaged with Mochizuki san.  Ted Nelson (b 1937), a US sociologist who as long ago as 1963 invented the term hypertext, thinks the controversial Japanese professor may be the inventor of Bitcoin.  Dr Nelson noted that that Bitcoin creator "Satoshi Nakamoto san" appears to have existed long enough only to (1) introduce Bitcoin, (2) stimulate excitement and (3) disappear and thought this similar behavior to that of Mochizuki san who has some history of making mathematical discoveries and posting them to the internet to be found, rather than publishing.  Not many share the suspicion, noting that while a grasp of high-level mathematics would have been essential to build the Blockchain, Mochizuki san is not known to have any background in software development although, given Bitcoin may have been developed by a team, that may not be significant.  Dr Nelson remained convinced and in 2013 offered to donate one Bitcoin (then trading at $US123) to charity were Mochizuki san to deny being the inventor.  It's not known if Dr Nelson revised the terms of his offer as the Bitcoin price moved.