Lack of certainty about the future of this technology is dividing the world into zealots and sceptics. Moderates seem harder to come by nowadays. — My position on generative AI is best described as moderate. I have no doubt that recent developments and releases have transformed what it is possible to create without any form of human involvement, but I am also acutely aware of the current shortcomings of generative AI, the risks it poses…

Extolling the virtues of 30 minutes of math every morning — Every morning before I start work I tackle a math problem. I schedule 30 minutes to solve this problem, and if I don’t succeed I stick with it the next morning and subsequent mornings until I have solved it and I am ready to move on to the next. Why…

This Oxford University entrance exam question illustrates the systematic, first principles reasoning that all good mathematicians use, and which GPT4 still struggles with — Many of us are familiar with the rules of the sides of a right angled triangle. By the Pythagorean Theorem, if a and b are the two smaller sides, and c is the longest side, then a² + b² = c². Well known integer solutions to this are known as…

How mathematicians come at problems from different angles — One of the first things most mathematics undergraduates learn (and maybe even some smart and enthusiastic high school students) is a proof that there are infinitely many primes. In most cases the proof taught is the classic one developed by the Greek mathematician Euclid around 300BC and presented in Book…

The magic of DeMoivre’s Theorem — DeMoivre’s Theorem (written above) is a simple theorem relating complex numbers to trigonometry. It was proposed and proved for all positive integers n by Abraham DeMoivre (1667–1754), a French mathematician who was a religious exile in England from a young age and a friend of such greats as Isaac Newton…

I look at a Group Theory problem set to 18 year olds to demonstrate concepts of algebraic structure — If you were an enthusiastic and talented high school mathematics student in the UK in 1988, you would have studied elementary Group Theory as part of your Advanced Level (A-level) curriculum. Sadly this is not the case any more. …

How Graph Theory Helped Discover a King of England — One morning in August 2012 in a car park in Leicester in the English midlands, a mechanical digger was starting to cut into the concrete surface. A number of interested spectators were present, hoping against hope that something amazing but highly improbable might occur. Some rugged detective work had led…

Data Science and Statistics can drive highly personalized medical intervention — Early in 2021, I was diagnosed with Type 2 Diabetes. After a blood test, my HbA1c — a measure of average blood sugar levels over the past three months — came back as 49 (6.6 in the US metric). In the UK, a HbA1C level of 41 (5.9) or below…

A Guide to Abstract Thinking in Mathematics — We constantly use the word infinity in both our conversational language and to describe scientific concepts. However, infinity as a physical concept is impossible to visualize. One example is the statement ‘the Universe is infinite’. It’s a fine statement to make, but no human being can visualize an infinite universe…