The starting point for rigorous reasoning in mathematics is a system of axioms. An axiom is a statement that is assumed, without demonstration, to be true. It is usually self-evident, for example, "the whole is greater than the part". 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.
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.
In 1742, Christian Goldbach wrote a letter to his friend, the incomparable Leonhard Euler, proposing that every integer greater than two 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 more than 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.
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 doesn’t solve the problem: 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.
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!
Peter Lynch is emeritus professor at UCD school of mathematics and statistics, He blogs at thatsmaths.com