Posts Tagged ‘Richard Feynman’
If, in some cataclysm, all scientific knowledge were to be destroyed, and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis (or atomic fact, or whatever you wish to call it) that all things are made of atoms — little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence you will see an enormous amount of information about the world, if just a little imagination and thinking are applied.
The Feynman Lectures on Physics, Vol. I (1964)
Is no one inspired by our present picture of the universe? This value of science remains unsung by singers, you are reduced to hearing not a song or poem, but an evening lecture about it.
This is not yet a scientific age.
What Do You Care What Other People Think? (1988)
I can live with doubt and uncertainty and not knowing. I think it’s much more interesting to live not knowing than to have answers that might be wrong. [...] I don’t feel frightened by not knowing things, by being lost in the mysterious universe without having any purpose.
The Pleasure of Finding Things Out (1999)
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)
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:
For a more intuitive geometric proof of this formula, imagine a square patch of land that measures on each side:
Its area is , 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 to the formula), two rectangles of dimensions 50 by x (each contributing an area of , for a combined total of ), and finally the small x by x square, which gives an area of , 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 is very much a dead skill today.
I have been re-reading Genius: The Life and Science of Richard Feynman by James Gleick (hence the recent Feynman-themed post), which reminded me of a very special formula in mathematics; one that Feynman himself described as follows in his famous Feynman Lectures on Physics:
In our study of oscillating systems, we shall have occasion to use one of the most remarkable, almost astounding, formulas in all of mathematics. From the physicists’ point of view, we could bring forth this formula in two minutes or so and be done with it. But science is as much for intellectual enjoyment as for practical utility, so instead of just spending a few minutes, we shall surround the jewel by its proper setting in the grand design of that branch of mathematics called elementary algebra.
This remarkable formula? Euler’s Identity:
In analytical mathematics, Euler’s identity (named for the pioneering Swiss-German mathematician, Leonhard Euler), is an equality renowned for its mathematical beauty, linking five fundamental mathematical constants:
- The number 0, the additive identity.
- The number 1, the multiplicative identity.
- The number , which is ubiquitous in trigonometry, the geometry of Euclidean space, and analytical mathematics ( = 3.14159265…)
- The number , the base of natural logarithms, which occurs widely in mathematical and scientific analysis ( = 2.718281828…); both and are transcendental numbers.
- The number , the imaginary unit of the complex numbers, whose study leads to deeper insights into many areas of algebra and calculus.
The identity is a special case of Euler’s Formula from complex analysis, which states that:
for any real number . The derivation to the identity follows, as and .
For me, Euler’s Identity reinforces the underlying beauty and interconnectedness of mathematics, pulling together three seemingly disparate fields into one simple formula; it certainly deserves being known as “the most remarkable formula in all of mathematics“.
When I am doing research, I often think of the Feynman Problem-Solving Algorithm, supposedly coined in jest by another Nobel Prize-winning physicist, Murray Gell-Mann, about Richard Feynman‘s innate problem-solving ability:
- Write down the problem.
- Think very hard.
- Write down the answer.
Feynman was renowned for his ability to develop innovative and creative solutions to hugely complex problems, without being able to give much insight into how this process worked. Nevertheless, the algorithm itself is much more helpful than I thought on first reading. I occasionally overlook how important it is to define and bound a problem and think about it in abstract terms before attempting to construct a solution. In fact, I try to instil this problem-solving ability in my students when I teach introductory programming, as they all rush head-first into writing code before actually thinking about the problem they are trying to solve.
I’m off to find a pen and some paper…
(Feynman was also known for frequently changing his mind during this problem-solving process; when he worked on the Manhattan Project, colleagues remarked that only when Feynman said something was true on three consecutive occasions, you could count on it.)