Over the past weeks Snezana Lawrence, co-author of Mathematicians and their Gods, introduced us into a summer journey around the beauty of mathematics; including secret maths witches and wizards, and trying to answer the question: will we ever need maths after school? In this last post, Snezana tells us the story of bright amateurs in mathematics that had great influence on scientific discoveries, from multidimensionality to the fourth dimension.
A friend of mine picked an argument with me the other day about how people go on about the beauty of mathematics, but this is only not obvious to non-mathematicians, it cannot be accessed by those outside the field. Unlike, for example, modern art, which is also not always obvious, mathematical beauty is elusive to all but the mathematicians. Or so he said. He mused further that a non-mathematician can never bring anything new to mathematics, unlike art: in art, from time to time you get shifts, paradigm changes, contributions from people who don’t necessarily belong to the old establishment, who bring a new insight, that may change art and influence further developments at a profound level. This is how movements in art happen and so, the thinking goes, there is a possibility of an outsider making a contribution to bring about such a change or a shift. Furthermore, this in effect means that people generally engage with art on a much greater scale, as there is always a potential of making a contribution to it.
Is this really the case? Is mathematics really a discipline so insular that no amateur or admirer can ever play a role in its development? I wracked my brain to come up with a counter-example, hoping that, at least, I would be able to persuade him of the beauty of mathematical techniques. My master plan was to use the argument of the form reductio ad absurdum, an old trick that would finish by ‘aha!’ on my part.
But I think it ended up more nicely. I came up with the concept that I have investigated recently, of the amateurs making not only a huge contribution to the field, but actually enriching a view of mathematics itself. There was one development that took place at the turn of the twentieth century involving mathematicians and non-mathematicians alike in the development of the concept of multidimensionality. I am talking about, on one side, mathematicians such as Bernhard Riemann and Hermann Günther. Their work and lives were devoted to the development of this concept. On the other hand though, there were others whose work included philosophizing on the multidimensionality. People such as Edwin Abbott Abbott, the Shakespearean scholar and the London schoolmaster who wrote one of the most famous and popular novellas of all time, the Flatland, and Alicia Boole-Stott, whose three-dimensional models of four-dimensional polytopes contributed to the development of mathematics immensely for example. Alicia had no official mathematical education, apart from being the daughter of the famous George Boole (but earnt her honorary doctorate at the University of Gro¨ningenn in 1914).
How did this happen? By very unorthodox means, in fact. Abbott wrote Flatland at the time when he was still working at the City of London School and lived in Marylebone. Mary, Alicia’s mother, also lived in Marylebone and was writing mathematics books for children, but also contributed to theology and science. Her interests extended to Darwinian theory, philosophy and psychology, organizing discussion groups on all of these from time to time. Mary Boole was also at one time a personal secretary to James Hinton, a famous spiritualist, whose son Charles wrote some very interesting books on the fourth dimension, invented the term ‘tesseract,’ and also married Mary’s oldest daughter, also named Mary. Charles apparently believed in the multi-dimensionality of time too, which may explain his bigamous marriage within three years of his marriage to Mary (but to whom he later returned); he also taught Alicia to visualize four dimensional polytopes (polytope is an object in different dimensions – for example in two dimensions polytopes are triangles, squares etc., in three polytopes are cube, octahedron, etc. and so on) as they would pass through the third dimension. There is strong circumstantial evidence to show that there was a strong link with spiritualism that linked all of these people together, the belief that there is some other higher dimension, from which the dimensions of our world could be seen at once. Mary Boole and Edwin Abbott Abbott even wrote apologies about spiritualism, none of course forthcoming from Charles Hinton.
So there! I managed to come across a group of people who were not mathematicians – they had some links to it in their various ways, but none were actually mathematicians, apart from the weirdest of them all, Charles Hinton, who ended his life as a mathematics instructor at Princeton University. Yet their work, both in terms of writing and their social involvement and communication, made a lasting influence on the development of the concept of the fourth dimension in mathematics, and from there, the concept of multidimensionality.
Perhaps this type of mathematics comes very close to abstract art, but so be it. We can all enjoy the many representations of the tesseract, the word Charles Hinton coined, and Alicia Boole-Stott so beautifully represented with her many models. And we can certainly attempt to venture from the world of Flatland to the world of the fourth, and many more dimensions.
Featured image credit: Math Castle by Gabriel Molina. CC-BY-2.0 via Flickr.