The claim is often made that mathematical results are immutable. Once proven, they remain forever valid. But things are not so simple. There are problems at the very core of mathematics that cast a shadow of uncertainty. We can never be absolutely sure that the foundations of our subject are rock solid.
The ancient Greeks put geometry on a firm footing. Euclid set down a list of axioms, or basic intuitive assumptions. Upon these the entire edifice of Euclidean geometry is constructed. This axiomatic approach has been the model for mathematics ever since.
Euclid’s fifth axiom, the parallel postulate, stated that for any line and any point a unique line can be drawn through the point and parallel to the line. This was awkward and not self-evident like the other four axioms. For centuries mathematicians struggled to deduce the parallel postulate as a consequence of the remaining axioms. All such efforts failed and, in the early 19th century it was realised we could “take or leave” the parallel postulate. Taking it, we get Euclidean geometry. Leaving it, we get more exotic but rational and self-consistent systems known as non-Euclidean geometries.
‘We must know. We will know’
Minor gaps in Euclid's axiom system were noticed and rectified by the renowned German mathematician David Hilbert. In 1900, Hilbert listed 23 major problems in mathematics. This list had a strong influence on research during the 20th century. Hilbert had an unshakeable confidence that any valid mathematical problem could be solved. At a conference in Königsberg in 1930, he stated that there is no unsolvable problem, ending his speech with the words: "Wir müssen wissen. Wir werden wissen." These words – "We must know. We will know" – became his epitaph in 1943.
With a logically consistent system of axioms, it is impossible to derive mutually contradictory results: we cannot prove both a statement and its negation. By contrast, if the system is inconsistent, contradictory results are unavoidable. Such a situation would be catastrophic. The second problem on Hilbert’s list was to find a complete and consistent system of axioms for all of mathematics.
At the same conference, Kurt Gödel announced his “incompleteness theorem”. This showed that Hilbert’s goal to establish the completeness and consistency of mathematics is unattainable. The axiomatic approach is inherently limited: there are results that are true but that cannot be proved within the system. In essence, mathematics is incomplete!
Adding more axioms
Worse was to follow: a second theorem of Gödel showed that the consistency of such a system of axioms can never be proved by working within the system itself. Thus, we cannot be certain that the standard axiomatic systems of mathematics never lead to contradictions.
We might try to remove the incompleteness by adding more axioms, as is done in geometry. For example, in standard set theory an assumption called the continuum hypothesis can be added without affecting the consistency of the system. However, in any such augmented system there will always be unsolvable problems; like dirt under the carpet, the incompleteness just won’t go away.
Gödel’s theorem had seismic consequences for mathematical logic, and philosophers of mathematics continue to explore its extensions and implications. However, the majority of working mathematicians remain confident that our basic axiom systems are consistent and free from contradictions. They pay scant heed to Gödel’s result, ignoring the chasm separating provability from truth.
Peter Lynch is emeritus professor at the School of Mathematics and Statistics, University College Dublin. He blogs at thatsmaths.com. His book That's Maths has just been published by Gill Books.