Alex's Adventures in Numberland - Alex Bellos [175]
A devout Lutheran, Cantor wrote many letters to clergymen about the significance of his results. He believed that his approach to infinity showed that it could be contemplated by the human mind, and therefore bring one closer to God. Cantor had Jewish ancestry, which – it has been argued – may have influenced his choice of the aleph as the symbol for infinity, since he may have been aware that in the mystical Jewish tradition of Kabbalah, the aleph represents the oneness of God. Cantor said he was proud he chose the aleph since, as the first letter of the Hebrew alphabet, it was a perfect symbol for a new beginning.
The aleph is also a perfect place for an end to our journey. Mathematics, as I wrote in the opening chapters of this book, emerged as part of man’s desire to make sense of his own environment. By making notches on wood, or counting with fingers, our ancestors invented numbers. This was helpful for farming and trade, and ushered us into ‘civilization’. Then, as mathematics developed, the subject became less about real things and more about abstract ones. The Greeks introduced concepts such as a point and a line, and the Indians invented zero, which opened the door to even more radical abstractions like negative numbers. While these concepts were at first counter-intuitive, they were assimilated quickly and we now use them on a daily basis. By the end of the nineteenth century, however, the umbilical cord linking mathematics to our own experience snapped once and for all. After Riemann and Cantor, maths lost its connection to any intuitive appreciation of the world.
After finding c Cantor kept going, proving that there are even bigger infinities. As we saw, c is the number of points on a line. It is also equal to the number of points on a two-dimensional surface. (That’s another surprising result, which you’ll have to trust me on.) Let’s call d the number of all possible curves, lines and squiggles that can be drawn on a two-dimensional surface. Using set theory, we can prove that d is bigger than c. And we can go one step further, showing that there must be an infinity larger than d. Yet no one has so far been able to come up with a set of things that has a cardinality larger than d.
Cantor led us beyond the imaginable. It is a rather wonderful place and one that is amusingly opposite to the situation of the Amazonian tribe I mentioned at the beginning of this book. The Munduruku have many things, but not enough numbers to count them. Cantor has provided us with as many numbers as we like, but there are no longer enough things to count.
Glossary
Algorithm: a set of rules or instructions designed to solve a problem.
Ambigram: a word (or set of words) written in such a way as to conceal other words, often the same word (or set of words) written upside-down.
Amicable number: two numbers are amicable if the factors of one add up to the other and vice versa.
Axiom: a statement that is accepted without proof, usually because it is self-evident, and used as a foundation of a logical system.
Base: in a number system, the base is the size of the number-grouping which, when using Arabic numerals, is equal to the number of different digits allowed in the system. Binary, using 0 and 1, is base two, while decimal, using 0 to 9, is base ten.
Cardinality: the size of a set.
Circumference: the perimeter of a circle.
Combinatorics: the study of combinations and permutations.
Constant: a fixed value.
Continuum: the points on a continuous line.
Convergent series: an infinite series that adds up to a finite number.
Correlation: a measure of the interdependence