Why history of maths?
Maths underpins our understanding of the universe and the development of much of our technology, but it has a reputation for being difficult. And advanced modern maths is, indeed, formidable to anyone but specialists! Yet even these difficult concepts were developed from simpler beginnings, so by looking at mathematical history, I can show readers how these simpler, underlying concepts arose. I think that understanding the basics helps us cope with the complexity of modern science and tech, for then we can have some sense, no matter how vague, of the ideas underpinning the increasingly sophisticated tools that give us the stories behind the headline discoveries.
But I also want to give readers an appreciation of the beauty of maths, as well as its power. For instance, in Vector I wanted to show not just how vectors and tensors evolved historically, but also the importance of mathematical symbolism – and how the right choice of notation can make equations more elegant and more meaningful. Back in the early 1890s, there was actually a ‘vector war’ over this notation question! That seems strange, in hindsight, but knowing about it helps underscore the importance of symbolism, which we often just take for granted.
I also love the fact that history includes stories about the people who developed these ideas. These stories not only show how the language of maths evolves to handle new concepts and to solve new practical problems; they also help make maths live, through the eyes of colourful characters who have loved and strived – and succeeded and failed – like all of us, and who had human weaknesses as well as genius. Maths may sometimes seem dry, and as immutable as if its rules had been carved in stone by the gods, but history shows that it is a wonderfully human endeavour.
Why this book?
Vectors and tensors enable equations to be written in an elegant and transparent way, as I indicated, so they are great subjects through which to share my love of mathematical language. I first wrote about the linguistic power of mathematics in my book Einstein’s Heroes, but in Vector I wanted to go more deeply into the actual maths than is usual in popular maths books. I felt I was taking a bit of a risk, but I was aiming to interest readers who enjoyed maths and/or physics in senior high school or first-year university courses and are curious to expand their understanding of key concepts – and/or who are fascinated by mathematical history and the people who made it.
There has been plenty of coverage of, say, the development of calculus in popular mathematics - why do you think there is far less about vectors, vector calculus and tensors?
I guess it comes down to what I said earlier, about popular books generally avoiding too much maths. I think readers need to be willing to engage in some higher-level mathematical thinking to see just how sophisticated these subtle tools really are – even though they seem simple at face value. I like to think of readers who like ‘brain candy!’ But I also wrote this book so that the historical and biographical contexts form a narrative about the basic ideas and why they matter, so that readers who want to skip over some of the maths can still come away with an appreciation for the subject.
What’s next?
That’s a challenging question, because I’m still thinking about vectors and tensors right now. But I’m looking forward to a summer break, and then to the chance to think about new directions. I have ideas swirling around, but nothing definitive yet.
What’s exciting you at the moment?
I’m really fascinated by recent research on new ways of thinking about meshing gravity and quantum theory – especially rethinking the foundational role of spacetime and developing ‘postquantum’ theories. After half a century of work on string theory and forty years of loop quantum gravity, it seems as though some really new thinking has been happening over the last decade or so.
Related to this kind of theoretical work, there are all the new observational experiments taking place, thanks to new and better telescopes and other equipment. I get a thrill every time I see a headline reporting that ‘Einstein was right, again,’ because general relativity (GR) is so beautiful. But I’m also excited to see what these new observations, at better and better accuracies, will reveal. For when there are serious discrepancies with the existing theories, new physics might emerge – along with new ideas for merging GR and quantum theory.
Closer to home, in addition to my own occasional mathematical dalliance with gravitomagnetic monopoles, I’m excited about local action to protect the environment. Just recently there’s been a significant step towards saving an ancient woodland near my area. This has taken years of effort by local community groups, so it’s thrilling to see some hopeful results. Related to this is the First Nations idea of ‘caring for Country,’ and I’m excited that First Nations environmental/scientific knowledge is finally being taken seriously in the broader community – especially among scientists and educators.
Interview by Brian Clegg - See all Brian's online articles or subscribe to a weekly email free here
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