The Story of Maths

The final episode considers the great unsolved problems that confronted mathematicians in the 20th century. On 8 August 1900 David Hilbert gave a historic talk at the International Congress of Mathematicians in Paris. Hilbert posed twenty-three then unsolved problems in mathematics which he believed were of the most immediate importance. Hilbert succeeded in setting the agenda for 20thC mathematics and the programme commenced with Hilbert's first problem.

Georg Cantor considered the infinite set of whole numbers 1, 2, 3 ... ∞ which he compared with the smaller set of numbers 10, 20, 30 ... ∞. Cantor showed that these two infinite sets of numbers actually had the same size as it was possible to pair each number up; 1 - 10, 2 - 20, 3 - 30 ... etc.

If fractions now are considered there are an infinite number of fractions between any of the two whole numbers, suggesting that the infinity of fractions is bigger than the infinity of whole numbers. Yet Cantor was still able to pair each such fraction to a whole number 1 - ¹/₁; 2 - ²/₁; 3 - ¹/₂ ... etc. through to ∞; i.e. the infinities of both fractions and whole numbers were shown to have the same size.

But when the set of all infinite decimal numbers was considered, Cantor was able to prove that this produced a bigger infinity. This was because, no matter how one tried to construct such a list, Cantor was able to provide a new decimal number that was missing from that list. Thus he showed that there were different infinities, some bigger than others.

However, there was a problem that Cantor was unable to solve: Is there an infinity sitting between the smaller infinity of all the fractions and the larger infinity of the decimals? Cantor believed, in what became known as the Continuum Hypothesis, that there is no such set. This would be the first problem listed by Hilbert.[2]

Next Marcus discusses Henri Poincaré's work on the discipline of 'Bendy geometry'. If two shapes can be moulded or morphed to each other's shape then they have the same topology. Poincaré was able to identify all possible two-dimensional topological surfaces; however in 1904 he came up with a topological problem, the Poincaré conjecture, that he could not solve; namely what are all the possible shapes for a 3D universe.[2]

According to the programme, the question was solved in 2002 by Grigori Perelman who linked the problem to a different area of mathematics. Perelman looked at the dynamics of the way things can flow over the shape. This enabled him to find all the ways that 3D space could be wrapped up in higher dimensions.[2]

The achievements of David Hilbert were now considered. In addition to Hilbert's problems, Hilbert space, Hilbert Classification and the Hilbert Inequality, du Sautoy highlights Hilbert's early work on equations as marking him out as a mathematician able to think in new ways. Hilbert showed that, while there were an infinity of equations, these equations could be constructed from a finite number of building block like sets. Hilbert could not construct that list of sets; he simply proved that it existed. In effect Hilbert had created a new more abstract style of Mathematics.[2]

For 30 years Hilbert believed that mathematics was a universal language powerful enough to unlock all the truths and solve each of his 23 Problems. Yet, even as Hilbert was stating We must know, we will know, Kurt Gödel had shattered this belief; he had formulated the Incompleteness Theorem based on his study of Hilbert's second problem:

This statement cannot be proved

Using a code based on prime numbers, Gödel was able to transform the above into a pure statement of arithmetic. Logically, the above cannot be false and hence Gödel had discovered the existence of mathematical statements that were true but were incapable of being proved.[2]

In 1950s American mathematician Paul Cohen took up the challenge of Cantor's Continuum Hypothesis which asks "is there is or isn't there an infinite set of number bigger than the set of whole numbers but smaller than the set of all decimals". Cohen found that there existed two equally consistent mathematical worlds. In one world the Hypothesis was true and there did not exist such a set. Yet there existed a mutually exclusive but equally consistent mathematical proof that Hypothesis was false and there was such a set. Cohen would subsequently work on Hilbert's eighth problem, the Riemann hypothesis, although without the success of his earlier work.[2]

Hilbert's tenth problem asked if there was some universal method that could tell whether any equation had whole number solutions or not. The growing belief was that no so such method was possible yet the question remained, how could you prove that, no matter how ingenious you were, you would never come up with such a method. He mentions Paul Cohen. To answer this Julia Robinson, who created the Robinson Hypothesis which stated that to show that there was no such method all you had to do was cook up one equation whose solutions were a very specific set of numbers: The set of numbers needed to grow exponentially yet still be captured by the equations at the heart of Hilbert's problem. Robinson was unable to find this set. This part of the solution fell to Yuri Matiyasevich who saw how to capture the Fibonacci sequence using the equations at the heart of Hilbert's tenth.[2]

The final section briefly covers algebraic geometry. Évariste Galois had refined a new language for mathematics. Galois believed mathematics should be the study of structure as opposed to number and shape. Galois had discovered new techniques to tell whether certain equations could have solutions or not. The symmetry of certain geometric objects was the key. Galois' work was picked up by André Weil who built algebraic geometry, a whole new language. Weil's work connected number theory, algebra, topology and geometry.

Finally du Sautoy mentions Weil's part in the creation of the fictional mathematician Nicolas Bourbaki and another contributor to Bourbaki's output - Alexander Grothendieck.[2]