In the book, I note that a number of ‘conditions’ which economists impose upon their models are actually ‘proofs by contradiction’ that their theories contain errors. The most blatant example of this are what are known as the ‘Sonnenshein-Mantel-Debreu’ conditions. These say that, in order to be able to aggregate the individual utility which different consumers derive from consuming different commodities:

Ÿ      (a) all consumers indifference curves must have the same slope; and

Ÿ      (b) the slope must be such that spending doesn’t change with income—therefore all Engels curves must be straight lines.

But condition (a) means that all consumers must be identical—so that there is really only one consumer. Condition (b) means that all commodities must be identical—so that there is really only one commodity. This means that utility can’t be aggregated across different consumers and different commodities—it is a ‘proof by contradiction’ that what economists are trying to do is impossible.

This discovery should have caused economists to abandon the notion that they can model society simply by adding up all the individuals in it, but they were so wedded to this idea from the time of Bentham that they refused to see a contradiction when it stared them in the face.

In the book, I promised to give an example of how mathematicians used ‘proof by contradiction’ properly to transcend an old belief that all numbers were rational, and ushered in one of many great leaps in our understanding of mathematical logic. So here goes:

Irrational Numbers

This proof, though mathematical, is very easy to understand. Start by assuming that the square root of 2 is rational: that it can be accurately expressed as the ratio of two integers. Therefore “the square root of 2 equals m divided by n”, where we assume that m and n are the smallest integers for which this ratio is true.

This means that m and n have no factors in common apart from one. Squaring both sides and rearranging gives you  “2 times n squared equals m squared”. This means that m squared is an even integer, since it’s equal to two times n, which must be an even number, whether n itself is even or odd. Therefore m itself must be even; it can’t be odd because an odd number times an odd number yields another odd number. Therefore m is divisible by two.

Let’s call this number which is half of m “k”, so that “k equals m divided by 2”: . Then m squared, which we already know equals two times n, is the same thing as four times k squared. This means in turn that “m squared equals 2 times n squared equals 4 times k squared”, since four times k squared is equal to m squared, Which by our assumption that the square root of two is rational–equals two times n squared, so that “n squared equals 2 times k squared”.

But this means that n has to be even, since it is equal to two times another integer, and two times any number, even or odd, is an even number. Thus m and n share the common factor two; but we began by assuming that they had no factors in common apart from one.

We have a contradiction, and therefore have proven that the square root of two can’t be expressed as the ratio of two integers. Therefore it is an irrational number.