Oscar_Cunningham comments on Rationality Quotes June 2012 - Less Wrong

4 Post author: OpenThreadGuy 02 June 2012 05:14PM

You are viewing a comment permalink. View the original post to see all comments and the full post content.

Article Navigation

Comments (413)
Tags: quotes

Comments (413)

You are viewing a single comment's thread. Show more comments above.

Comment author: Oscar_Cunningham 01 June 2012 02:31:58PM *  37 points [-]

Two very different attitudes toward the technical workings of mathematics are found in the literature. Already in 1761, Leonhard Euler complained about isolated results which "are not based on a systematic method" and therefore whose "inner grounds seem to be hidden." Yet in the 20'th Century, writers as diverse in viewpoint as Feller and de Finetti are agreed in considering computation of a result by direct application of the systematic rules of probability theory as dull and unimaginative, and revel in the finding of some isolated clever trick by which one can see the answer to a problem without any calculation.

[...]

Feller's perception was so keen that in virtually every problem he was able to see a clever trick; and then gave only the clever trick. So his readers get the impression that:

  • Probability theory has no systematic methods; it is a collection of isolated, unrelated clever tricks, each of which works on one problem but not on the next one.
  • Feller was possessed of superhuman cleverness.
  • Only a person with such cleverness can hope to find new useful results in probability theory.

Indeed, clever tricks do have an aesthetic quality that we all appreciate at once. But we doubt whether Feller, or anyone else, was able to see those tricks on first looking at the problem. We solve a problem for the first time by that (perhaps dull to some) direct calculation applying our systematic rules. After seeing the solution, we may contemplate it and see a clever trick that would have led us to the answer much more quickly. Then, of course, we have the opportunity for gamesmanship by showing others only the clever trick, scorning to mention the base means by which we first found.

E. T. Jaynes "Probability Theory, The Logic of Science"

Comment author: Oscar_Cunningham 01 June 2012 02:40:23PM *  40 points [-]

Then there is the famous fly puzzle. Two bicyclists start twenty miles apart and head toward each other, each going at a steady rate of 10 m.p.h. At the same time a fly that travels at a steady 15 m.p.h. starts from the front wheel of the southbound bicycle and flies to the front wheel of the northbound one, then turns around and flies to the front wheel of the southbound one again, and continues in this manner till he is crushed between the two front wheels. Question: what total distance did the fly cover ?

The slow way to find the answer is to calculate what distance the fly covers on the first, northbound, leg of the trip, then on the second, southbound, leg, then on the third, etc., etc., and, finally, to sum the infinite series so obtained. The quick way is to observe that the bicycles meet exactly one hour after their start, so that the fly had just an hour for his travels; the answer must therefore be 15 miles.

When the question was put to von Neumann, he solved it in an instant, and thereby disappointed the questioner: "Oh, you must have heard the trick before!"

"What trick?" asked von Neumann; "all I did was sum the infinite series."

An anecdote concerning von Neumann, here told by Halmos.

Powered by Reddit Powered by Reddit