The British Society for the History of Mathematics
The impact of developments in mathematics is often: taken up in other areas of science, engineering and technology; felt over the long term; and quite different to what, if anything, was intended. These issues can lead to a view that mathematics is not a subject which provides 'value for money' when funded through public research grants. Consequently, there is a mood among mathematicians to turn to history for examples to refute this view. In this a relatively few, well known examples are often cited. I believe BSHM can have a role to play in arming mathematicians with further historical examples of the impact – long term, unexpected or unplanned – of mathematics. I hope this will be useful to colleagues needing to defend their discipline and to provide BSHM an opportunity to increase interest, awareness and knowledge of the history of mathematics among a wider community.
We would like to collect 400-word short articles on a variety of topics. These could be examples of historical mathematics that has an unexpected impact on the modern world and those for which the impact is historical as well. These could be examples of curiosity-driven mathematics that later had an impact as well as work intended for one purpose that happened to have a different, unplanned impact.
If you are willing to help with this initiative, please send me an email briefly describing an idea that could be worked into a 400-word piece. I will coordinate these to make sure we don’t waste time and effort by writing multiple submissions on the same topic. Below are two example articles on the Theory of Numbers and the Mathematics of Insurance so you can get a feel for the length and style, although you should not feel limited by these.
Peter Rowlett, 19 June 2010
p.rowlett@bham.ac.uk
Martin Campbell-Kelly, University of Warwick
In 1940 G. H. Hardy, Sadlerian Professor of Pure Mathematics at Cambridge University, wrote one of the most famous books about doing mathematics, A Mathematician’s Apology. It was said at the time to be a masterpiece that conveyed “the excitement of the creative artist”—and this is just as true today. Hardy’s central argument was that mathematics was a wonderful realm for the imagination and creativity but one should not expect it to be useful—if it was, that was all to the good, but one should not expect it. He was a world famous researcher in the Theory of Numbers, but he considered this topic to be one of “supreme uselessness”. Of course, he was proved wrong, though not for many years.
In 1975 three American mathematicians, Martin Hellman, Whitfield Diffie, and Ralph Merkle, invented public-key cryptography. This was based on the idea of a “one-way function”. In a computer, a message is really just a big binary number, so applying a function to that number would just transform it into a different big binary number—but one that scrambled the original message. The property of a one-way function was that it would be fast and easy to encode a message, but extremely difficult to reverse the process. Ideally, it would take just a few milliseconds to do the encryption, but it would take centuries using the fastest available computers to decode the message by inverting the function. So even if people knew the function by which the message was encrypted, they would never have sufficient computing horse-power to reverse the process. However, the recipient of the message would have a private key so that the message could be decoded rapidly. The only problem with the Hellman-Diffie-Merkle scheme was that no one knew of such a function.
The following year at the Massachusetts Institute of Technology three mathematicians, Ronald Rivest, Adi Shamir, and Leonard Adleman, came up with a solution known as the RSA algorithm. The algorithm, which involved the use of very large primes, relied on the Theory of Numbers—the subject Hardy considered to be one of “supreme uselessness”. At first the RSA technique was useful for secure electronic communications, but with the explosion of the Internet in the mid 1990s it became the basis for virtually all small consumer transactions. It was used by web browsers such as Internet Explorer, by on-line banks, and by Internet merchants such as amazon.com, eBay, and PayPal.
Peter Rowlett, University of Nottingham
In the 16th Century, Girolamo Cardano was a mathematician and a compulsive gambler. Tragically for Cardano, this led to him squandering large parts of the money he inherited and earned. Fortunately for us, this combination led to a mathematical treatment of probability which is considered the first work in modern probability theory. Around a century after Cardano had collected his work, but not published it, gambler Chevalier de Méré found himself with a dilemma. He had been offering a game in which he bet he could throw a six in four rolls of a die, and had done well out of this. He varied the game in a way that seemed sensible, betting he could throw a double six with two dice in 24 rolls. He had calculated the chances of winning in both games was equivalent, but found he lost money in the long run playing the second game. Confused, he asked his friend Blaise Pascal for an explanation. Pascal wrote to Pierre de Fermat and a correspondence ensued which laid the foundations for probability theory.
In the late 17th and early 18th centuries, Jakob Bernoulli recognised that the probability theory could be applied much more widely than to games of chance. Part of what Cardano had discovered was the idea that with sufficiently many rolls of a fair, six-sided die we can expect each outcome to appear around one sixth of the time, but that if we roll one die six times we shouldn't expect to see each outcome precisely once. Bernoulli extended this idea and gave a proof of the Law of Large Numbers, which says the larger a sample, the more closely the sample characteristics match those of the parent population. Without an understanding of this Law, insurance companies had been limiting the number of policies they sold. Since policies were based on probabilities, each policy sold seemed to incur an additional risk, the cumulative effect of which could ruin the company. With an understanding of the Law of Large Numbers, we see that the more policies a company can sell, the more likely the predictions are to be accurate. The roots of actuarial science owed a debt to mathematical investigations of games of chance motivated by questions from gamblers over the preceding two hundred years.
The British Society for the History of Mathematics is registered as a company limited by guarantee, no. 3326816, and as a charity, no. 1061229. Its registered office is c/o Andrew Thurburn & Co, 38 Tamworth Road, Croydon, Surrey CR0 1XU, UK.