Michael Harris on Elster on Montaigne on Diagoras on Abraham Wald

Michael Harris — who is now blogging! — points out that Montaigne very crisply got to the point I make in How Not To Be Wrong about survivorship bias, Abraham Wald, and the missing bullet holes:

Here, for example, is how Montaigne explains the errors in reasoning that lead people to believe in the accuracy of divinations: “That explains the reply made by Diagoras, surnamed the Atheist, when he was in Samothrace: he was shown many vows and votive portraits from those who have survived shipwrecks and was then asked, ‘You, there, who think that the gods are indifferent to human affairs, what have you to say about so many men saved by their grace?’— ‘It is like this’, he replied, ‘there are no portraits here of those who stayed and drowned—and they are more numerous!’ ”

The quote is from Jon Elster, Reason and Rationality, p.26.

Abraham Wald and the volume of the 4-simplex

Abraham Wald is one of the figures who keeps popping up in HOW NOT TO BE WRONG.  I just learned yet another cool thing about him which doesn’t have a place in the book, so I’m putting it here.

You know Heron’s formula, which gives you the area of a triangle in terms of the lengths of its edges.  And there’s a generalization:  the Cayley-Menger determinant is a formula for the volume of an n-simplex in terms of its edge lengths.

What about other kinds of faces?  The volume of a tetrahedron is determined by the six edge lengths, but not by the areas of its four faces.  Imagine four identical isosceles right triangles formed into a flattened tetrahedron that’s really just one square laid on top of another; that tetrahedron has volume 0, but a regular tetrahedron with faces of the same area obviously has positive volume.

What about a 4-simplex?  The lengths of the 10 edges determine the volume.  So you might guess that the areas of the 10 2-faces would give you a way to compute the volume, too.  But nope!  Wald gave an example of two 4-simplices with the same face-areas but different volume.  I wonder what the space of 4-simplices with fixed face-areas looks like.