Tag Archives: Aesthetics

The beauty of a good mathematical proof

Throughout his life, John von Neumann, the Hungarian-American polymath, leaned towards pure mathematics, or pure mathematics with recognised applications. In his 1947 essay entitled The Mathematician, he described his personal concept of mathematics, showing him to be thoughtful and original concerning the philosophical underpinnings of the discipline. One word von Neumann repeatedly uses is aesthetical; he defends mathematics for mathematics’ sake, consciously posting analogies to the visual arts. For example, in listing the qualities of a good mathematical proof:

One also expects “elegance” in its “architectural,” structural make-up. Ease in stating the problem, great difficulty in getting hold of it and in all attempts at approaching it, then again some surprising twist by which the approach, or some part of the approach, becomes easy, etc. Also, if the deductions are lengthy or complicated, there should be some simple general principle involved, which ”explains” the complications and detours, reduces the apparent arbitrariness to a few simple guiding motivations, etc. These criteria are clearly those of any creative art, and the existence of some underlying empirical, worldly motif in the background — overgrown by aestheticizing developments and followed by a multitude of labyrinthine variants — all this is much more akin to the atmosphere of art pure and simple than to that of the empirical sciences.

 
Nevertheless, von Neumann insisted that the best mathematics was usually inspired by practical problems, perhaps in partial defence of game theory from fellow mathematicians who at the time deprecated it as an applied field.

(also see: Feynman, Bethe and the beauty of mathematics)

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Feynman, Bethe and the beauty of mathematics

To those who do not know mathematics, it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature…If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.

The Character of Physical Law (1965)
Richard Feynman

 
This term I have been teaching a new first year undergraduate module, Mathematics for Computing, in which I have been trying to impart a little bit of love for mathematics. While we have covered a number of underpinning topics relevant to computer science, such as propositional logic, set theory and number theory, I have also tried to show that there are a multitude of clever little tricks that can make arithmetic and performing seemingly complex calculations that little bit easier. And in doing so, I was reminded of the mathematical prowess of Richard Feynman as well as Hans Bethe, Nobel laureate in physics and Feynman’s mentor during the Manhattan Project. Bethe is one of the few scientists who can make the claim of publishing a major paper in his field every decade of his career, which spanned nearly 70 years; Freeman Dyson called Bethe the “supreme problem solver of the 20th century.

An example of Bethe’s mastery of mental arithmetic was the squares-near-fifty trick (taken from Genius: The Life and Science of Richard Feynman by James Gleick):

When Bethe and Feynman went up against each other in games of calculating, they competed with special pleasure. Onlookers were often surprised, and not because the upstart Feynman bested his famous elder. On the contrary, more often the slow-speaking Bethe tended to outcompute Feynman. Early in the project they were working together on a formula that required the square of 48. Feymnan reached across his desk for the Marchant mechanical calculator.

Bethe said, “It’s twenty-three hundred.”

Feynman started to punch the keys anyway. “You want to know exactly?” Bethe said. “It’s twenty-three hundred and four. Don’t you know how to take squares of numbers near fifty?” He explained the trick. Fifty squared is 2,500 (no thinking needed). For numbers a few more or less than 50, the approximate square is that many hundreds more or less than 2,500. Because 48 is 2 less than 50, 48 squared is 200 less than 2,500 — thus 2,300. To make a final tiny correction to the precise answer, just take that difference again — 2 — and square it. Thus 2,304.

 
Bethe’s trick is based on the following identity:

(50 + x)^2 = 2500 + 100x + x^2

For a more intuitive geometric proof of this formula, imagine a square patch of land that measures 50 + x on each side:

50xsquared

Its area is (50 + x)^2, which is the value we are looking for. As you can see in the diagram above, this area consists of a 50 by 50 square (which contributes the 2500 to the formula), two rectangles of dimensions 50 by x (each contributing an area of 50x, for a combined total of 100x), and finally the small x by x square, which gives an area of x^2, the final term in Bethe’s formula.

While Feynman had internalised an apparatus for handling far more difficult calculations (for which he became famous for at Los Alamos, such as summing the terms of infinite series or inventing a new and general method for solving third-order differential equations), Bethe impressed him with a mastery of mental arithmetic that showed he had built up a huge repertoire of these easy tricks, enough to cover the whole landscape of small numbers. Bethe knew instinctively that the difference between two successive squares is always an odd number (the sum of the numbers being squared); that fact, and the fact that 50 is half of 100, gave rise to the squares-near-fifty trick.

Unfortunately, the skill of mental arithmetic that did so much to establish Bethe’s (as well as Feynman’s) legend was doomed to a quick obsolescence in the age of machine computation — it appears to be a dead skill today.

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