I recently noticed a small article in a national daily newspaper reporting the views of a long-standing colleague, Brian Ripley. Intrigued to find the reason for his new-found fame, I read further. The article concerned the way that a new Labour government might change the National Lottery by drawing an extra number. Ripley, now professor of statistics at Oxford, was explaining the potential effect of this change.
Electronically scanned lottery tickets may not have been on the minds of the 17th-century French mathematicians Blaise Pascal and Pierre de Fermat when they wrote their seminal letters discussing games of chance, but had they been consulted they could surely have answered the hot questions for Ripley.
Excellent mathematics can be very permanent, with applications that extend way beyond its creators' objectives. In this case, our politicians required of Ripley a certain skill in counting permutations. The Art of Conjecturing, an early 18th century mathematical monograph on probability written by the Swiss mathematician Jakob Bernoulli, contains all that Ripley would have required to analyse a lottery.
The ideas of combinatorial probability, nurtured in simple and abstract discussions of games of chance, permeated quickly and soon established themselves in the developing insurance industry; they are still immutable, if quite elementary, today. A reminder that even the most applicable mathematical ideas are often more easily grasped and formalised in simple abstract settings. Could Bernoulli have guessed at the breadth and influence of these ideas from probability would have as they permeated into the more complex models provided by the 20th century; an influence spreading from genetics to information processing?
One might think that the permanence of excellent mathematics, together with the brilliant minds at work on probability in the 17th and 18th centuries (including also Laplace, Huygens, De Moivre and others of the Bernoulli dynasty) would, with the help of Gauss in the 19th century, have completed discussion of the subject. Perhaps 20th-century probabilists have only angels balanced on the heads of needles to consider. An extended study on lotteries would, despite its importance to those planning to replace the government, undoubtedly fall into this class.
Of all the mathematicians at the transition from the 17th to 18th century it was Daniel Bernoulli (1700-82, son of Johann and nephew of Jakob) who pointed the way. He was primarily interested in the evolution of systems of particles, and his contributions to probability were paralleled by his work in hydrodynamics and kinetic theory of gases.
Much of modern probability is trying to cope with the fact that the world exhibits many random phenomena that are truly of high dimension: systems that vary in space or time, or both. The snow on your television, and the hiss on your mobile phone are stochastic or random phenomena. A telephone company may regard your speech as a random phenomenon that is not like "white noise" and so can therefore be compressed to increase the flow through the system. Your computer treats your text documents in the same way. Statistical image reconstruction, and the robot trying to pick up a part despite inevitable small errors in measurement and movement, provide further examples of high dimensional stochastic systems.
Penetrating the associated mathematical problems can be hard to solve, requiring considerably more than a single person's life effort to make serious progress. The now-popular interacting particle system or cellular automata is an example of just such a development. A system of objects are situated on a lattice and set to randomly interact with their neighbours.The object is to understand how the macroscopic systems behave, and how they fluctuate from the expected state.
Probability and its cousin statistics are thriving and would be almost, but not quite, unrecognisable to our 17th-century predecessors. A quick glance at the 400 pages of this new journal, the official journal of the Bernoulli Society for Mathematical Statistics and Probability, would convince sceptics that probability is still an active area that knows few national boundaries. Looking at the established top-rank journals in probability and mathematical statistics one would find thousands of pages of research that excites other mathematicians.
Outsiders could be forgiven if they expressed bewilderment. Even practitioners find the spread of applications and the depth of some of the fundamental developments daunting. A scope that includes accommodating the noise when decoding an FM stereo signal, explaining the effect of the Josephson diode in terms of large deviations, and improving the quality of noisy visual images using wavelets. On the other hand, there are quite abstract papers on probability, as its fundamental tools are regarded as a subset of pure mathematical analysis.
The link between mathematics and application is as alive today as ever. One paper in this new journal studies the detection of unduly raised incidence levels of leukemia against various spatial factors. Analysing data of the latter type raises questions quite closely related to fundamental "pure" research on empirical processes and probability on Banach spaces.
Expansion and increasing complexity ensure that the gulf between the applications and the fundamental development is hard to bridge. So one of the exciting features of this new journal, with its very distinguished and capable board, is its attempt to place itself at the interface, accepting both theoretical and applied papers of quality. In this sense one can certainly say the journal points in an important strategic direction.
Will it succeed? It seems that the judges are still out. People in industry can learn mathematical topics regarded as rather difficult quite quickly if there is big money to be made. The construction of financial derivatives in major financial centres involves trillions of dollars and motivated many to learn stochastic calculus. Academics, under different but still significant pressures, are not always quick to feel comfortable with new ideas and approaches.
Terry J. Lyons is EPSRC senior fellow and professor of mathematics, Imperial College, London.
Bernoulli
Editor - Ole E. Barndorff-Nielson
ISBN - ISSN 1350 7265
Publisher - Chapman & Hall
Price - £128.00 print and internet £110.00 print
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