The starting point for rigorous reasoning in maths is a system of axioms. An axiom is a statement that is assumed, without demonstration, to be true. The Greek mathematician Thales is credited with introducing the axiomatic method, in which each statement is deduced either from axioms or from previously proven statements, using the laws of logic. The axiomatic method has dominated mathematics ever since [TM206 or search for “thatsmaths” at irishtimes.com].
An axiom is usually self-evident: “the whole is greater than the part” is an example. But Euclid’s fifth axiom, or postulate, concerning parallel lines, is far from obvious. For two thousand years, mathematicians struggled to deduce it from the initial four postulates, but all attempts ended in failure. We now know that the parallel postulate is independent of the remaining axioms of Euclid. It may be assumed to be true or false and a self-consistent system of geometry follows from either choice.
A proposition that is believed to be true, but for which no proof has been found, is called a conjecture. Number theory abounds with intriguing conjectures: the Riemann conjecture, the twin primes conjecture and Goldbach’s conjecture. The proof of any of these would bring enduring fame to the discoverer.
Goldbach’s letter
In 1742, Christian Goldbach wrote a letter to his friend, the incomparable Leonhard Euler, proposing that every integer greater than 2 is the sum of three prime numbers. Euler responded that this would follow from the simpler statement that “every even integer greater than 2 is the sum of two primes.”
Goldbach’s Conjecture is one of the best-known unsolved problems in mathematics. It is a simple matter to check the conjecture for a few cases: 8 = 5+3, 16 = 13+3, 36 = 29+7. It has been confirmed for numbers up to over a million million million. But there is an infinite number of possibilities, so this approach can never prove the conjecture. Many brilliant mathematicians have tried and failed to prove it. If a proof is found, it will likely involve some radically new idea or approach.
“Uncle Petros and Goldbach’s Conjecture”
The Goldbach Conjecture is the central theme of a novel by Apostolos Doxiadis, “Uncle Petros and Goldbach’s Conjecture”. The hero is Petros Papachristos, a gifted, reclusive Greek mathematician who has spent most of his career trying to prove Goldbach’s Conjecture. The narrator is his nephew, who tells the story of how, when he was a young teenager, his eccentric Uncle Petros set him the task of proving the conjecture.
The novel describes aspects of the recent history of mathematics, and gives some brilliant insights into the mental state and methods of a research mathematician. Although it is a work of fiction, Doxiadis gets the mathematical details right. He gives a great feeling for the passion that drives a research mathematician, and a good flavour of the nature of pure mathematics.
Uncle Petros realises the implication of advances in mathematical logic: Goldbach’s Conjecture may be unprovable; the goal of his life’s work may be unattainable.
Hilbert’s Dream
Can every true mathematical statement be proved? The great German mathematician David Hilbert believed so and in 1928 he posed a challenge, asking for an algorithm to establish the validity or otherwise of any conjecture. He was destined to be disappointed.
In 1931, logician Kurt Gödel proved that mathematics is incomplete: whatever system of axioms we assume, there are statements that are true but that cannot be proved using only these axioms; in a nutshell, provability is a weaker concept than truth. Adding additional axioms may make such statements true, but then new true-but-unprovable statements inevitably arise.
But what if we have a conjecture that we wish to prove, starting from the usual axioms of mathematics? Can we know in advance whether a mathematical proof is possible, or whether the conjecture is unprovable? Hilbert’s Decision Problem asked, in essence, if there is a way to determine – in the absence of a proof – whether any given mathematical statement or proposition is true or false.
In 1936, the American logician Alonzo Church showed that there can be no positive answer to the decision problem. Independently, Alan Turing reached the same conclusion. The implication is that, within a given system of axioms, there is no way to tell, ahead of time, whether a given conjecture can or cannot be proved. Hilbert’s dream was shattered.
When Uncle Petros learned of these results, he too was devastated. He realised that a proof of Goldbach’s Conjecture, on which he had laboured for decades, might not be possible. His life’s work could have been in vain.
There is no solid reason for suggesting that Goldbach’s Conjecture cannot be proved on the basis of the usual axioms of mathematics; the only justification for such a claim is that the problem has been around for almost 280 years. But let us suppose the conjecture is unprovable. Then it must be true!
Why? Because, if it were false, there would be some finite even number that is not the sum of two primes. A finite search could confirm this, making the conjecture “provably false”! In other words, falsehood of the conjecture is incompatible with unprovability. This contradiction forces us to an ineluctable conclusion: if Goldbach’s Conjecture is unprovable, it must be true!
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That’s Maths II: A Ton of Wonders
by Peter Lynch now available.
Full details and links to suppliers at
http://logicpress.ie/2020-3/
>> Review in The Irish Times <<
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This article is based on an article in the Bulletin of the Irish Mathematical Society: Lynch, Peter, 2020: Goldbach’s Conjecture: if it’s unprovable, it must be true. Bull. Irish Math. Soc., 86, 103-106. PDF.
Sources:
Doxiadis, Apostolos, 2000: Uncle Petros and Goldbach’s Conjecture. Faber & Faber, London. ISBN: 978-0-5712-0511-0.
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