It's like shooting fish in a barrel, but here goes.
This was the final episode and Professor Marcus du Sautoy highlighted Cantor, Poincaré, Andre Weil, Kurt Gödel, and Alexander Grothendieck along with the familar melodrama of Galois's fatal duel. As usual, the programme had no idea of its audience, so talked about mathematicians rather than mathematics.
The explanation of Cantor's "diagonal argument", showing that the set of Reals is strictly larger than the set of Integers, was marred by the refusal to use the concept of"real number". Instead we had at various times "decimals" and "infinite decimals", which at best obscures the point.
Grothendieck was lauded for his work on the "structural reformulation of mathematics" without the phrase "category theory" being mentioned.
In a PC nod to the striking absence of top-level women mathematicians, a non-top-level woman mathematician was highlighted (Julia Robinson, who worked on Hilbert's tenth problem).
The theme of the programme was Hilbert's list of the 23 most important unsolved problems in mathematics as unveiled at the Paris conference of the International Congress of Mathematicians in 1900. Du Sautoy is himself focused on the Riemann hypothesis (problem 8 of Hilbert's 23) but this was never outlined, except to indicate it has 'something to do with the distribution of prime numbers'.
I did however think that the couple of sentence summary of Gödel's incompleteness theorem was probably as good as could realistically be achieved in the space available. And Paul Cohen seldom gets a name-check on TV.
I think the script team for these four programmes decided in advance that they would not attempt to convey any mathematics. In the default smug, patronising and faux high-mindedness of terminal Reithian broadcasting, they decided to make du Sautoy a regular bloke ("Look! He doesn't even know which way up Russian Cyrillic script should be - he's holding the St. Petersburg metro map upside down!") and show lots of holiday destinations, drinks in the local tavernas and and mini - racy if possible - bio-snippets of the most famous mathematicians (Gödel was saved by the love of a good woman ... right!).
So to be clear, that's a fail grade. There is still an audience of young talent out there who genuinely want to know more about real mathematics. Perhaps BBC 4 can stop patronising its audience and do it properly next time?
NOTE: Each of the earlier episodes has also been reviewed here in previous posts. Type "Story of Maths" in the search box, top left.