|Even in the old Arthur Conan Doyle stories, Sherlock Holmes' arch-nemesis was a mathematician. Moriarty was said to be a math professor who (when he wasn't being evil) worked on the binomial theorem and the dynamics of asteroids. This is the main reason that many Sherlock Holmes stories contain enough references to math to be included in this database of "mathematical fiction". (See here for some other examples.) But this relatively recent Holmes story by Barton and Capobianco takes it further into the realm of science fiction than most of the others. I first learned about this story from "William E. Emba" who summarized it for me by saying:
"William E. Emba"|
17 years after the death of Moriarty, Sherlock Holmes comes across some loose ends
involving Moriarty. Following these clues down into eastern Siberia with
Watson, a set of mathematical calculations that only Moriarty could do
proves essential. Highly recommended.
Now that I've had an opportunity to read this story, I'd like to comment further on Mr. "Emba"'s claim that "only Moriarty could do" these particular calculations. Let me warn you, however, that in doing so I will give away a bit of the plot. (I suspect that it was an attempt to avoid doing exactly that which led "Emba" to be so vague. However, unlike him, I have no problem with "spoilers" in my descriptions.) So, if you wish to read the story without having the surprises spoiled by my diatribe on sensitive dependence in dynamical systems stop reading now!
Okay. So, the plot of the story depends upon Moriarty's ability to predict the motion of celestial objects using mathematics. As we well know, his most famous paper was entitled The Dynamics of an Asteroid, after all. (See The Valley of Fear). However, I feel obligated to point out some bits of mathematical history that will suggest how difficult (in fact, impossible) it would be to do this with the precision suggested in this short story.
The true story of Poincare and his study of the dynamics of objects under the influence of Newtonian gravity is quite interesting...interesting enough to be turned into mathematical fiction although I've never seen it done. A great prize was offered to the mathematician who could work out the mathematics that would allow people to predict the motion of three objects (since one or two objects were trivially easy) under the laws of physics as described by Isaac Newton and his calculus. Poincare wrote a paper which won the prize, but it was only after it was published that he himself realized the fundamental problem with the paper. The simple fact is that even an extremely tiny change in the position or velocity of one of the objects in question would in a relatively short amount of time lead to a very drastic difference in their later behavior. This phenomenon, known colloquially as "the butterfly effect" and more technically as "sensitive dependence upon initial conditions" is at the heart of modern chaos theory, and its discovery here, although a triumph for mathematicians, was viewed as a failure by Poincare. This is because he realized that one had no realhope of predicting the future behavior of planets and asteroids in space as there was no way for the mathematician to know the actual positions and velocities of the objects with sufficient precision. Or, to quote Henri Poincare himself:
Henri Poincare (1903)|
If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. but even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation appproximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon.
(So far, the discovery that the universe doesn't quite follow Newton's laws of physics but something closer to a combination of general relativity and quantum field theory has thus far only made the problem worse, though we cannot rule out the possibility that some future discoveries in either math or physics will resolve this problem.) Nevertheless, the point remains that mathematicians and astronomers with today's most powerful computers do not claim to be able to predict with accuracy whether an asteroid will collide with the Earth 20 years in the future, let alone precisely where it will hit, and such a problem would certainly be beyond the powers of even the brilliant Moriarty. The "intersecting ellipses" described in the book does not capture the great sophistication of the problems one would have to overcome to make such a prediction.
Still, this is a cute story which ties the life of Sherlock Holmes to a historical event of scientific interest.