Astounding World of Numbers

Mathematics as what we have in mind is always problem solving and finding its x and the other letters in the alphabet. And so Marcus du Sautoy, Oxford professor and pop-sci mathematician extraordinaire, takes a look at the history of maths and why it is so important.

The fourth episode, To Infinity and Beyond, concludes the series. After exploring Georg Cantor's work on infinity and Henri Poincare's work on chaos theory, he looks at how mathematics was itself thrown into chaos by the discoveries of Kurt Godel, who showed that the unknowable is an integral part of maths, and Paul Cohen, who established that there were several different sorts of mathematics in which conflicting answers to the same question were possible. He concludes his journey by considering the great unsolved problems of mathematics today, including the Riemann Hypothesis, a conjecture about the distribution of prime numbers. A million dollar prize and a place in the history books await anyone who can prove Riemann's theorem.

First top, du Sautoy discusses about David Hilbert who 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.And also 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.

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.

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.

Well I could then say that without the help of these wondrous guys, we won't be who we are today. Truly and indeed, mathematics has contributed starting from the minute ones up to vast and wide ones.