1900 – France

‘Mathematics is concerned with formal symbolic systems. It is an activity that uses a series of symbols and rearranges them according to various formal rules. This separates it from any concrete reality and consequently there is nothing external to its workings that can be used to validate it, so all of its arguments must be capable of justifying themselves’

The pursuit of formalistic ideas led to many developments within mathematics. Hilbert introduced an original approach to ways of considering mathematical invariants. An invariant is something that is left unchanged by some class of functions. In terms of a geometrical transformation, an invariant would be an object that does not alter its shape or size while it is being moved. He proved that all invariants could be expressed in terms of a finite number – a number that can actually be counted.

Hilbert spent the first two decades of the 20th century struggling to construct a self-justifying system of arguments that would prove that a finite number of steps of reasoning could not lead to a contradiction. This work was itself contradicted in 1931 when Czech born KURT GODEL published his incompleteness theorem, showing that every consistent theory must contain propositions that are undecidable. Godel pointed out that when proving statements about a mathematical system at least some of the rules and axioms must derive from outside that system. By doing this you create a larger system that will contain its own unprovable statements. The implication is that all logical systems of any complexity are, by definition, incomplete. In doing this Godel showed that truth is more important than provability.

Amongst the list of 23 problems presented to the Second International Congress of Mathematicians in Paris in 1900, was the question of decidability, the ENTSCHEIDUNGSPROBLEM: is it possible to find a definite method for deciding whether any given mathematical assertion was provable?

In modern terms this ‘definite method’ would be called an algorithm. ALAN TURING answered the question in the negative in April 1936, but may have been anticipated by the American logician Alonso Church. While Church appealed to contemporary mathematics to make his point, Turing had introduced a theoretical machine that could perform certain precisely defined elementary operations. By doing this, Turing created a foundation for modern theories of computation.

###### Related sites

- Maths is good for you (mathsisgoodforyou.com/)
- David Hilbert (cs.oberlin.edu/)