Sunday, 2 December 2012

Box paradoxes

It's longer than I would have liked since I last posted - which is because I've spent the last two weekend at maths conferences.  First there was MathsJam - which was every bit as good as I expected when I wrote about it recently.  Then there was the IMA's Early Careers Mathematicians Conference at Greenwich - where we had the chance to play with lost of unusual puzzles and games from the collections of David Singmaster and Laurie Brokenshire CBE and to try out wonderful linkages with Danny Brown.

There is a lot from MathsJam I could write about - not least a wonderful piece of graph theory from Colin Wright, who showed that factorising large numbers can be reduced to a graph colouring problem!  My own short talk was about the two-box paradox(attributed to Schrodinger, in a slightly different form) that I wrote a blog post about in October.

This set me thinking about paradoxes involving boxes.  There are several: the Monty Hall problem, Newcomb's Paradox, and Schrodinger's Cat come immediately to mind.  What is nice is that they all tell us different things.

Schrodinger's Cat expresses concisely the quantum concept of the superposition of states.

The two-box problem I previously discussed tells us about the difficulty of selecting randomly from an infinite set wihtout being very specific about how you do it.

Newcomb's paradox gives the player the contents of two boxes.  They can choose to take box B only, or both boxes A and B.  Box A definitely contains one thousand pounds.  An infallible predictor has chosen the contents of Box B in advance.  If the predictor predicted that the player would take both boxes, they put nothing in Box B.  If they predicted that the player would take Box B only, they placed one million pounds in Box B.  What should the player do?  At the point at which they make their choice, the contents of both boxes are fixed, so logically they get more if they take both boxes. But should they ignore the predictor's infallibility.  This paradox, I think, shows us that no such predictor can exist.

The Monty Hall paradox  can be viewed as a disguised version of Martin Gardner's prisoners paradox.  Gardner wrote of three prisoners. held in solitary confinement, who know that two are to be executed the next day.  Each has a 2/3 chance of dying.  A thinks he can improve his odds - he points out to the guard that at least one of the other two must die, so that being told which of B or C will die isn't going to give him new information.  But when the guard says C will die, A now reckons that his chance of survival is now 1/2 since it is either him or B.  Of course, A is wrong (B's chance has improved) and I have a theory that it is memories of Gardner's article, where A's odds don't change, that led many mathematicians (myself included) to quickly jump to the wrong answer.

There is an excellent novel about mathematicians by Sue Woolfe, Leaning towards infinity, which contains a devastating account of the appalling treatment of women by male mathematicians at a fictional conference.  I have never seen such behaviour at a maths conference and I found the account implausible until I read in Jason Rosenhouse's account in The Monty Hall Problem of the treatment the journalist Marilyn Vos Savant received from (some) mathematicians when she wrote about this problem.  The responses would have been appalling even if Vos Savant had been wrong, but in fact she got it right and her critics didn't.  So one useful  lesson from the Monty Hall paradox is that on occasion even mathematicians can be jerks.

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