Maths and Hope

I know what you might be thinking after reading the title, probably something along the lines of "this guy picks the worst topics to write about". First it was statistics, then it was data analysis and now it's maths. It's hardly clickbait but hopefully this is more interesting than it sounds. 

I recently read Fermat's Last Theorem by Simon Singh and I thought it was amazing. Fermat's Last Theorem is a maths problem first posed by Pierre de Fermat in the 17th century. Fermat wrote that he had solved the problem but there wasn't enough space in the margin to explain it (an excuse that shouldn't be used in exams). Simon Singh's book begins in Ancient Greece, moves on to Fermat in 17th century France and then moves through the recent history of the problem including all the genius, confusion, rivalries and collaborations that occurred along the way. In this article I will focus on both individual genius and collaboration.

At first glance, it might seem contradictory that a story of a breakthrough contains both individual genius and collaboration. The book introduces Andrew Wiles as the man who solved the puzzle in the very first chapter. I expected the following chapters to read as a biography, the life story of a man who took on a problem that had bested the brightest minds of the last few hundred and beat it himself. That was not the case. In reality, Wiles' story doesn't start until the two final chapters of the book. Before this, the stars include Pythagoras (the bane of GCSE maths), Pierre de Fermat (obviously) and countless other mathematical savants of the 18th, 19th and 20th centuries. The reason for this is that in maths, as in life, every significant progress relies on the work done before them. Pythagoras and the Ancient Greeks set a foundation, it was built upon by "a handful of luminaries in India and Arabia" then becoming the hobby of bright minds worldwide as countries developed and education was prioritised. The point is that much of the progress relied on the work done before them. Negative numbers, for example, did not exist until they were 'invented' in the 7th century in India. While these being 'invented' was hard to get my head around, the prospect of maths, and life, without them seemed much harder. Every breakthrough is a building block for the future. In terms of Fermat's last theorem, Wiles' efforts are focused on proving something called the Taniyama-Shimura conjecture. This was the work of Japanese mathematicians Taniyama and Shimura, and their conjecture's role in Fermat's Last Theorem was proved by Ken Ribet, an American. So, while Wiles, a British man, made the crucial last step, his work was the result of collaboration that began in Ancient Greece. More immediately, he was using the work of two mathematicians from the other side of the globe in Taniyama and Shimura, as well as the work of one of his contemporaries in America. The beauty of this is twofold. The mathematicians previously, while obviously seeking to solve the questions themselves, contribute to the overall progress of the subject whether they are successful or not. Despite the presence of individual rivalries (see Cauchy and Lamé) and glory hunting, each attempt pushes the entire field in the right direction. The second aspect of this can be described as blind faith. Due to the timeframes involved, some of this collaboration spans centuries. This means that those mathematicians who made early progress did so not knowing the full significance of their work, but probably in hope that it would contribute to great things down the line. This is probably true of all the sciences and, for people more exposed to research than myself, might seem like a rather obvious thing to point out but it is one aspect of the story that stayed with me. The one massive exception to this rule is obviously Fermat himself as he refused to write down his proofs, meaning none of his genius could actually be relied upon, but we'll ignore that. 

The other side of the same coin is individual genius and it is one of the most inspiring things I have come across. At various stages of history, leading mathematicians and experts on the theorem had considered Fermat's puzzle impossible. Due to the number of attempts on it and the exhaustion of known methods in solving it, there was little confidence in the field that solving it was either possible or even worth spending time on. This general attitude to the problem did nothing to deter Wiles, however, and the results are there to see. It is often easy to slip into the mindset that a lot of what is possible has been done. The story of Wiles proves that, sometimes, a problem just needs one stroke of genius to be solved, regardless of how old a problem is or how many other people have attempted it. Obviously, Wiles is a maths genius and will go down in history in his field but, for me at least, it is nice to talk about contemporary geniuses or the possibility of future genius and innovation, instead of firmly facing the past and looking at those who came before us. It incites hope for the future, rather than reliance on those in the past. And, it's worth mentioning, Fermat himself was an amateur mathematician. 

To sum it up, the story of Fermat's Last Theorem can be used as an analogy for progress in general. Each seemingly unsurmountable problem has been overcome. Sometimes you might not reach the solution yourself, but progress towards it counts and will contribute to others in the future. You can't know for sure what will happen but you must have faith. At the same time, don't be intimidated by the failures of others. You never know where the next bit of groundbreaking genius might come from and, if you're lucky, you might get a book (and a blog) written about you.