Reaching for the Extreme has a common theme, but the chapters are essentially independent, so you can dip in anywhere without losing the thread. The theme is ‘extremes’ in mathematics — a very broad term that covers a wide variety of topics and applications. By ‘extremes’ I mean things like the shortest curve between two points on a surface, the most efficient way to pack spheres together, the fewest colours needed to colour a map so that adjacent countries have different colours, and so on.

In a more metaphorical sense, I’ve included things like the quest for the biggest possible number and the smallest possible number. Neither of those things exists — which is the point. Sometimes when you reach for the extreme you miss. Historically those questions led to deep and important concepts of infinity and infinitesimals in calculus, taxed mathematicians for centuries, and underpin much modern technology. The most challenging chapter is ‘weirdest symmetry’, about an amazing series of discoveries that began as a numerical coincidence and eventually revealed unexpected links between 19th century number theory, 20the century algebra, and 21st century physics.

You don’t need specialist mathematical knowledge to follow these stories. They’re told in an accessible way, with analogies, simple examples, plenty of pictures. The aim is to provide a glimpse of today’s mathematical research. Sometimes its origins go back into the deep past — the legendary founding of the city of Carthage, for instance. (Chapter 2, ‘biggest area’). Sometimes it’s very recent, such as the  solution in 2018 of the protein folding problem (Chapter 15, ‘best fold’). A team at Google DeepMind applied Artificial Intelligence and machine learning to predict a protein’s folding pattern from its DNA sequence. The AlphaFold Protein Structure Database, launched in 2021, contained predicted models of protein structures for almost all proteins found in humans and twenty other organisms — more than 365,000 proteins.

A novel feature of this book is to sample a huge diversity of mathematical areas in one package, along with their history, theory, and applications. Today’s mathematics is far-ranging, diverse, and creative — not remotely like its typical public image. Roughly a million pages of new math is produced every year by the world’s research mathematicians. If the mathematics that makes today’s world possible were suddenly taken away, everything would collapse. A 2024 survey revealed that one in six UK workers were in mathematical science occupations, generating one fifth of the country’s gross value added: £489 billion. 

If you want financial reasons for the importance of mathematics, that’s a pretty clear answer. But there are other reasons too, which I think are more important — among them the subject’s intrinsic beauty, the element of surprise when unexpected links are forged between different areas, and the insights mathematics gives us into the natural world.

Ongoing thread. More from Ian Stewart to follow.

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