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By Root 755 0

and Mad Stist have been for other eras: sort of our Prometheus, the one who goes to forbidden places and returns with gifts we can all use but he alone pays for.’ Literature and film are guilty of romanticizing a link between maths and insanity. It’s a cliché that suits the narrative requirements of a Hollywood script (exhibit A: A Beautiful Mind ) but is, of course, an unfair generalization. The great mathematician, however, for whom the archetype could have been invented is Cantor. The stereotype fits him especially well since he was grappling with infinity, a concept that links mathematics, philosophy and religion. Not only was he challenging mathematical doctrine, but he was also setting out a brand-new theory of knowledge and, in his mind, of human understanding of God. No wonder he upset a few people along the way.

Infinity is one of the most brain-mangling concepts in maths. We saw earlier, in our discussion of Zeno’s paradoxes, that envisaging an infinite number of ever-decreasing distances is full of mathematical and philosophical pitfalls. The Greeks tried to avoid infinity as much as they could. Euclid expressed ideas of infinity by making negative assertions. His proof that there is an infinite number of prime numbers, for instance, is actually a proof that there is no highest prime number. The ancients shied away from treating infinity as a self-contained concept, which is why the infinite series inherent in Zeno’s paradoxes were so problematic for them.

By the seventeenth century, mathematicians were willing to accept operations involving an infinite number of steps. The work of John Wallis, who in 1655 introduced the symbol 8 for infinity for the purpose of his work on infinitesimals (things that become infinitely small), paved the way for Isaac Newton’s calculus. The discovery of useful equations that involved an infinite number of terms, such as …showed that infinity was not an enemy, yet even so, it was still to be treated with care and suspicion. In 1831 Gauss stated the received wisdom when he said that infinity was ‘merely a way of speaking’ about a limit that one never reached, an idea that simply expressed the potential to carry on and on for ever. Cantor’s heresy was to treat infinity as an entity in itself.

The reason why mathematicians pre-Cantor were nervous about treating infinity like any other number was that it contained many conundrums, the most famous of which Galileo wrote about in Two New Sciences, and is known as Galileo’s paradox:

1. Some numbers are squares, such as 1, 4, 9 and 16, and some are not squares, such as 2, 3, 5, 6, 7, etc.

2. The totality of all numbers must be greater than the total of squares, since the totality of all numbers includes squares and non-squares.

3. Yet for every number, we can draw a one-to-one correspondence between numbers and their squares, for example:

4. So, there are, in fact, as many squares as there are numbers. Which is a contradiction, since we have said, in point 2, that there are more numbers than squares.

Galileo’s conclusion was that, when it comes to infinityostmerical concepts ‘more than’, ‘equal to’ and ‘less than’ do not make sense. These terms may be understandable and coherent when discussing finite amounts, but not with infinite ones. It is meaningless to say there are more numbers than there are squares, or that there is an equal number of numbers and squares, since the totality of both numbers and squares is infinite.

Georg Cantor devised a new way to think about infinity that made Galileo’s paradox redundant. Rather than thinking about individual numbers, Cantor considered collections of numbers, which he called ‘sets’. The cardinality of any set is the number of members in the collection. So {1, 2, 3} is a set with a cardinality of three and {17, 29, 5, 14} is a set with cardinality four. Cantor’s ‘set theory’ gets the pulse racing when considering sets with an infinite number of members. He introduced a new symbol for infinity, , (pronounced aleph-null), using the first letter of the Hebrew alphabet with