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# Oscar_Cunningham comments on Rationality Quotes June 2012 - Less Wrong

4 02 June 2012 05:14PM

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Comment author: 01 June 2012 02:31:58PM *  36 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: 01 June 2012 02:40:23PM *  39 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.

Comment author: 01 June 2012 03:12:50PM 35 points [-]

I recall a math teacher in high school explaining that often, in the course of doing a proof, one simply gets stuck and doesn't know where to go next, and a good thing to do at that point is to switch to working backwards from the conclusion in the general direction of the premise; sometimes the two paths can be made to meet in the middle. Usually this results in a step the two paths join involving doing something completely mystifying, like dividing both sides of an equation by the square root of .78pi.

"Of course, someone is bound to ask why you did that," he continued. "So you look at them completely deadpan and reply 'Isn't it obvious?'"

I have forgotten everything I learned in that class. I remember that anecdote, though.

Comment author: 09 June 2012 01:18:54AM 2 points [-]

The standard proof of the Product Rule in calculus has this form. You add and subtract the same quantity, and then this allows you to regroup some things. But who would have thought to do that?

Comment author: 09 June 2012 01:33:37AM 9 points [-]

One of the characteristics of successful scientists is having courage. Once you get your courage up and believe that you can do important problems, then you can. If you think you can't, almost surely you are not going to. Courage is one of the things that Shannon had supremely. You have only to think of his major theorem. He wants to create a method of coding, but he doesn't know what to do so he makes a random code. Then he is stuck. And then he asks the impossible question, ``What would the average random code do?'' He then proves that the average code is arbitrarily good, and that therefore there must be at least one good code. Who but a man of infinite courage could have dared to think those thoughts? That is the characteristic of great scientists; they have courage. They will go forward under incredible circumstances; they think and continue to think.

Comment author: 02 June 2012 06:10:39PM *  3 points [-]

IIRC there was an xkcd about that, but I don't remember enough of it to search for it.

EDIT: It was the alt test of 759.

Comment author: 02 June 2012 10:39:04PM 16 points [-]

Note that xkcd 759 is about something subtly different: you work from both ends and then, when they don't meet in the middle, try to write the "solution" in such a way that whoever's marking it won't notice the jump.

I know someone who did that in an International Mathematical Olympiad. (He used an advanced variant of the technique, where you arrange for the jump to occur between two pages of your solution.) He got 6/7 for that solution, and the mark he lost was for something else. (Which was in fact correct, but you will appreciate that no one was inclined to complain about it.)

Comment author: 02 June 2012 07:18:47PM *  2 points [-]

Is 759 the one you are thinking of? The alt-text seems to be relevant.

Comment author: 02 June 2012 08:19:32PM 0 points [-]

Yes.

Comment author: 09 June 2012 01:17:46AM 4 points [-]

This is also why I don't trust poets who claim that their works spring to them automatically from the Muse. Yes, it would be very impressive if that were so; but how do I know you didn't actually slave over revisions of that poem for weeks?

Comment author: 01 June 2012 02:57:55PM 3 points [-]

It's "Jaynes."

Comment author: 01 June 2012 03:19:10PM 3 points [-]

Fixed. Thanks.

Comment author: 03 June 2012 03:33:30AM 0 points [-]

Does anyone have a link to an ebook of this book?

Comment author: 03 June 2012 03:36:04AM 4 points [-]

libgen.info has a variety of versions.

Comment author: 03 June 2012 03:56:19AM 0 points [-]

Thank you! Looking forward to reading.

Comment author: 03 June 2012 08:10:09PM 1 point [-]

Honestly, I think PT:TLoS is probably best for those who already understand Bayesian statistics to a fair degree (and remember their calculus). I'm currently inching my way through Sivia's 2006 Data Analysis: A Bayesian Tutorial and hoping I'll do better with that than Jaynes.

Comment author: 06 June 2012 09:51:56AM 4 points [-]

I think PT:TLoS is probably best for those who understand frequentist statistics to a fair degree. He spends a whole load of the book arguing against them, so it helps to know what he's talking about (contrary to his recommendation that knowing no frequentist statistics will help). The Bayesian stuff he builds from the ground up, calculus is all that's needed to follow it.

Comment author: 05 June 2012 05:20:42PM 2 points [-]

Jaynes begins it with a caution that this is an upper undergrad to graduate level text, not knowing a great deal of probability in the first place, I stopped reading and picked up a more elementary text. What do you think are the core pre-reqs to reading Jaynes?

Comment author: 05 June 2012 06:09:12PM *  3 points [-]

I have no idea - I'll tell you when I manage to satisfy them!

Comment author: 04 June 2012 07:39:17PM 2 points [-]

I'd agree, with the exception that chapters one and five (and maybe other sections) are great for just about anybody to get a qualitative understanding of Jaynes-style bayesian epistemology.

Comment author: 04 June 2012 08:17:54PM 1 point [-]

Ah, yeah - chapter 5 is pretty good. (I recently inserted a long quote from it into my Death Note essay.)