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My LMS/BCS-FACS Seminar given on 15 January 2024. This is exactly the same recording as the one by the same name on the London Mathematical Society's channel, which I've linked to on my channel Abstract: The formalisation of mathematics is an ongoing process that started as early as the 19th century, intensified with the foundational crisis at the start of the 20th century, and since the 1970s has been conducted with the help of computers. Recent decades have seen the machine formalisation of lengthy and technically complicated proofs, but some have argued that even these were not representative of modern mathematics. Recent achievements by a number of different groups are starting to challenge this scepticism.