Review of Archimedes’ The Sand-Reckoner

I found this short piece in my own library, in the Great Books of the Western World set that I’ve had since my grandfather gave it to me as a gift for my 16th birthday–the greatest gift, by far, that I have ever been given.  I ask only with this review that you refrain from comments on my sanity–because I prefer to learn things I don’t already know, and the fact that I’m reading a mathematical treatise is reflective of unadulterated madness.  In fact I have much to say about this short treatise, far more than I do about On the Equilibrium of Planes, which I read just before it.

The Sand-Reckoner is an attempt by Archimedes to refute those who said that if the whole universe was made of sand, the grains of sand involved could not be counted–they’d be infinite.  Given the information available to him, his proof is fascinating–he calculates the proportions of distances between the Earth, the sun, and the moon, hypothesizes orbits, and combines the geometrical astronomy with a heavy dose of number theory (especially dealing with powers of 10) and an assumption of the number of grains of sand you can hold in your hand to show that if all the space in the orbit was filled, the grains of sand would be calculable.  It is a work of true genius, one that has no modern parallel that I can think of.

The problem, however, should be obvious to the modern reader: to assume that the number of grains of sand contained in the universe is finite, we have to assume that the space contained in the universe is finite.  In Archimedes’ day this was generally (though not universally) assumed to be the case.  Today, however, we assume that the universe extends infinitely and that there is no calculable boundary at which space ends.  Thus the proof is discredited when the assumption on which it rests is conceived of as being inaccurate.

To what extent might this be true of our own scientific theories?  How many of us recognize what lies on assumption and what lies on fact?  How many know enough to make this recognition?  Consider the question as you read the results of studies reported in the news, listen to your doctor’s advice, and as your kids go through high school and college science courses.  You may be surprised at what you find.

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