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Infinity
Given the old maxim about an infinite number of monkeys and
typewriters, one can assume that said simian digits will type
up the following line from Hamlet an infinite number of
times.
"I could confine myself to a nutshell and
declare myself king of infinity".
This quote could
almost be an epithet for the mathematician Georg Cantor, one
of the fathers of modern mathematics. Born in 1845, Cantor
obtained his doctorate from Berlin University at the
precocious age of 22. His subsequent appointment to the
University of Halle in 1867 led him to the evolution of Set
Theory and his involvement with the until-then taboo subject
of infinity.
Within Set Theory he defined infinity
as the size of the never-ending list of counting numbers (1,
2, 3, 4….). Within this he proved that sub-sets of numbers
that should be intuitively smaller (such as even numbers,
cubes, primes etc) had as many members as the counting numbers
and as such were of the same infinite size. By pairing off
counting and even numbers together, we see that the number of
counting and even numbers must be the same:
1 -> 2
2 -> 4
3 -> 6
4 ->
8
5 -> 10
6 -> 12
He then went on to
demonstrate the impossibility of pairing off all the real
numbers (those including irrational decimals like Pi) with the
counting numbers, concluding that one was larger than the
other. The result, confusing though it may seem, is that some
infinities are bigger than others!
Cantor's work
represented a threat to the entrenched complacency of the old
school mathematicians. Up until then infinity, to quote
mathematician Carl Friedrich Gauss, had been treated as a "way
of speaking and not as a mathematical value". This
stonewalling inevitably brought Cantor into conflict with his
less enlightened peers. His most vocal critic was Leopold
Kronecker (ironically one of Cantor's past professor) who
undertook a personal crusade to discredit his lapsed protégé.
Using his position at the University of Berlin he dedicated
himself to rubbishing Cantor's ideas and ruining him
personally. His coup de grace was blocking Cantor's lifelong
ambition of gaining an appointment at the University of
Berlin.
In 1884, consigned to a backwater University
and under constant attack from Kronecker, Cantor had his first
nervous breakdown. He spent the rest of his life in and out of
mental institutions, his serious work at an end. Cantor's
later years may have been defined by tragedy but his
contribution to modern mathematics is colossal. His one-time
collaborator David Hilbert once said of him in tribute "No one
will drive us from the paradise that Cantor has created."
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