The mathematical logic for pricing financial securities was a trap. It had assumptions that ran contrary to the real world. The Saunterer pointed this out in the post of December 17, 2005 - “Gilbert K. Chesterton on how life is a trap for logicians” (to see, click here).
I want to give a simple illustration of how easy it is to fall into a logical trap. It involves a puzzle posed by Lewis Carroll (pen name of Charles Lutwidge Dodgson): You have a cup containing 50 spoonfuls of brandy, and another containing 50 spoonfuls of water. You take a spoonful of brandy from the first cup and mix it into the second cup. Then you take a spoonful of the mixture from the second cup and mix it into the first. Is there more, less, or the same amount of brandy in the water cup as there is water in the brandy cup? The mathematical logic (you can work it out; it’s easy) shows there is the same amount of brandy in the water cup as water in the brandy cup.
Yet that isn’t true in the real world, as Christopher S. Lange pointed out in a letter to The New York Book Review (March 1, 2009), saying:
The problem lies in the mixing of solutions. The molecules of water and alcohol each occupy a certain volume in pure solutions each. However, water and alcohol when mixed yield a sum that is more compact than the parts. Hence, 50 spoonfuls of water plus 50 spoonfuls of alcohol equals approximately 98 spoonfuls of mixture because of the interpenetration of each solution’s molecular spacing by the other. Thus, adding a spoonful of brandy to the same cup of water yields less than 51 spoonfuls. So taking a spoonful from the mixture involves sampling a more concentrated solution and leaves less than 50 spoonfuls in the cup of water; adding that more concentrated spoonful to the cup of brandy results in more brandy in the brandy cup that water in the water cup.
What we have here is a tidy object lesson about logical traps. Lange made it in response to the mathematician John Allen Paulos’ review (February 1, 2009, NYT Book Review) of Lewis Carroll in Numberland: His Fantastical Mathematical Logical Life, by Robin Wilson. W. W. Norton. 2008.)
For a brief biography of Charles Lutwidge Dodgson, click here. For John Allen Paulos’ home page, click here. For Christopher S. Lange’s home page, click here.
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