5 The limitations of mathematics
This course has considered various scientific fields, and the possible limitations to our knowledge in each case. The key question throughout has been: what is it that we cannot know? Mathematics is different from the sciences as it is ‘axiomatic’ (based on a set of axioms), and all mathematical statements and results arise as logical conclusions from these axioms. So, does this mean that there’s nothing in mathematics that we can never know?
David Hilbert, whose ‘Infinite Hotel’ was discussed in Week 7, believed that this was the case. In 1900, at the International Congress of Mathematicians in Paris, he made a speech setting out the 23 greatest unsolved problems for mathematics in the twentieth century, in which he said:
This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus.
Here Hilbert used the Latin word ignorabimus, which means ‘we will not know’ (the English verb ‘ignore’ derives from the Latin verb). Hilbert strongly believed that there’s nothing in mathematics that we cannot ever know. He still held this view in 1930, giving a talk in the Prussian city of Königsberg where he said:
The true reason why [no one] has succeeded in finding an unsolvable problem is, in my opinion, that there is no unsolvable problem.
Hilbert didn’t know it, but just one day earlier, at a conference held in the same town, a young Austrian logician named Kurt Gödel had proved him wrong.