Leonhard Euler: Mathematical Genius

It is strange, but true, that not a lot of people enjoy reading about mathematics, or doing pretty much anything with it. This post is aimed at the exceptions, and especially those who are on the edges, just becoming interested in the subject.

In the early 1700s, mathematics had just taken off like a jet plane, shortly to be more like a rocket. Newton and Leibniz had just invented Calculus, and Johann Bernoulli, a brilliant Swiss mathematician, had turned his mind to inspect everything that the theory could do, and had asked the big theoretical questions that would drive the engine of mathematical discovery in the next hundred years or so. It was an exciting time in which to be a mathematician.

William Dunham, of Muhlenberg University has made a particular study of Leonhard Euler, one of the greatest mathematicians of all time, born in Basel, and a protegee of the famous Bernoulli. In his book "The Master of Us All", Dunham describes some of the most accessible and interesting of the discoveries of Euler, making a point to underscore the intuitive methods Euler used, which though not acceptable as formal proof today, reveal the amazing mind of Euler.

To the layman, the idea of an infinite series might be a little strange. An infinite series is simply an addition of an infinite number of terms. Generally, a sum of infinitely many terms will produce an infinite number. However, if the numbers are very small, the sum could be finite. Such finite-valued "infinite sums" can be highly useful; many useful numbers can be approximated as "truncated"infinite series (infinite sums of which only a finite number of terms have been added; what is omitted is carefully calculated to be less than an acceptable error tolerance).

Consider the infinite sum 1/2 + 1/4 + 1/8 + 1/16 + .... With a little thought, it can be seen that this sum can never be greater than 1. As Zeno observed, if you walk half a mile, then walk half of the remaining half mile, and then half of that, and keep doing this, you'll never walk more than a mile. In the chart below, the blue bars represent the terms that we want to add up, and the pink bars represent the running subtotals. As you can see, the subtotals approach 1, and rise no higher.


In contrast, the series 1 + 1/2 + 1/3 + 1/4 + ... can be seen to be infinite. Observe the sums climbing steadily. (They do slow down, but not enough to make the sums approach a finite value.)


It is harder to show, but the series 1 + 1/4 + 1/9 + 1/16 + ... , (the sum of the reciprocals of the square numbers, 1, 4, 9, 16, 25, ...) is a finite sum. (See below. The sum is about 1.64.) The story of how Euler showed that it adds up to the unlikely value of (Pi)^2/6 (Pi squared divided by 6) is entertainingly recounted by Dunham in his wonderful book. This derivation is characteristically Eulerian!

The discovery above is topped by a result that connects an infinite series consisting of the reciprocals of the squares of the positive integers on the left, with an infinite product of factors of the form (1 - 1/P^2) on the right:

Bill Dunham's book, beautifully written, combines stylish exposition with absolutely fascinating content. Euler, a contemporary of J.S.Bach, showed a similar brilliance and creativity to the musical genius, and both had similarly enormous influence over the development of their respective areas. Both had many children: Bach had 20, and Euler had 13, and in each case, only a handful lived to adulthood. Finally, both were afflicted with diseases of the eye. While Bach died at the age of 65, Euler lived to be nearly 80.

Arch

Harry Potter and The World of Children's Books

It seems clear, at this point, that the Harry Potter books, and the Harry Potter Phenomenon generally, has given children's books a fillip that was sadly needed for decades. You may disagree with Jane Rowling's metaphysics, or with her educational objectives or social consciousness or whatever, but you can't fault her saying that she writes a boring book. I'm willing to bet that there must be a million young readers out there for whom Harry Potter was the first book they read voluntarily. There must be a vast number who began reading Harry Potter under duress, but continued to read voluntarily. The hardest thing to do, and believe me, I know first hand, is to get a kid to start reading. You pick the wrong books, or she has the wrong interests, or there just aren't any interesting books, or her friends don't read, so she doesn't, either, or she despises her friends and they do read ... you can ring the changes on that ad nauseam. So even if kids are reading books that you'd rather they didn't, they are reading, dammit, which is a huge improvement over not reading at all.  

Is our adult preoccupation with reading a temporary 20th/21st -century phase? Will 25th century adults bemoan the fact that kids don't watch enough TV? Is there something special about books? Because I enjoy writing so much (hence this blog :) I assume that there is some benefit to reading and writing, both to the writer, and to the audience. When writing, you have to force your nebulous thought into words, and you begin to realize that you have linguistic resources that you did not dream you had! Language, especially written language, is a resource that one must never undervalue. I wish I had the skills to persuade everyone of the truth of this, but it is only an instinctive feeling, a gut conviction, at the moment.  

It's surprising how the Harry Potter books have caught on in America. Perhaps the wonderful movies have contributed to the overall phenomenon, but why did the books catch on at first? They rely so much on the British boarding-school phenomenon, to which most Americans are strangers (except for the few who have read boarding-school books, and I can't even think of a few offhand--I belong to the generation who read Billy Bunter and Psmith stories from books handed down from uncles and aunts). The idea of a mixed boarding school (girls and boys) is unfamiliar, but I imagine in this enlightened age such a thing is not impossible. Perhaps they have been around for decades; gosh, I'm such a dinosaur.  

Rowling is a master at characterization. There's absolutely nobody better. The plots are not as satisfying as I'd like (or rather, The Plot isn't as satisfying, I should say). It all goes to show that, really, the plot is only a minor component of the entire work of fiction. Brilliant books have been written with very sketchy plots. Rowling's characters are so real, that to this day it seems as though Hermione Granger in the books and Hermione in the movie (Emma Watson) are two different characters entirely, both of whom share Harry Potter. Honestly, I can't see Emma Watson in the role for the movie of the last book (Deathly Hallows). She just gives the impression of being too much in charge and unflappable, while the Hermione of the book seems at the end of her tether, and only hanging on by a thread. Let's see Emma Watson hanging on by a thread, then, and I'll believe she can do it. There are other authors who have been plugging away at the young adult audience over the years, with some success, if not as spectacular success as that of Rowling and Potter. Tamora Pierce is one, a particularly good writer, even if not as brilliantly gifted as Ms Rowling. Arguably, some of Pierce's stories are more satisfying than any given Harry Potter book, but the Harry Potter books are to Tamora Pierce's books as a Hollywood blockbuster is to a wonderful little Indie film. A lot of work has gone into the style and the stylishness of the Harry Potter books, and we can't all be J.K. Rowling. On the other hand, Tamora Pierce has written a score of books, many of them very, very good, and as far as we know, J.K. Rowling has not gone beyond Beadle the Bard, which is still squarely in the Potter universe. But she has plenty of time in which to prove that she is not just a 7-hit wonder.  

Archimedes