We noted that the concept of the infinite was increasingly being used by mathematicians in areas such as the infinitesimal calculus of Newton.
It was a devout Lutheran mathematician, Georg Cantor (1845–1918), who was the first to investigate the concept of infinity systematically. To do so he invented the theory of sets. A set is simply a collection of distinct items; distinct in that there are no exact duplicates within a set. Cantor distinguished finite and infinite sets, with the “cardinality” of a set referring to the number of items it contained. The theory of sets became fundamental to many branches of mathematics.[1] German mathematician David Hilbert famously enthused that Cantor “created one of the most fertile and powerful branches of mathematics; a paradise from which no one can drive us out”.[2] Cantor classed the natural numbers[3] as countably infinite and was able to prove that the real numbers[4] were uncountably infinite, indicating at least two different classes of the infinite.
Just as Euclidean geometry had been revealed as just one specific type out of the set of all geometries, something similar was occurring with numbers themselves. The natural numbers were understood to be a subset of the rational numbers,[5] because the rational numbers included all natural numbers and all fractions. In turn, the rational numbers were known to be a subset of the real numbers, because the real numbers also contained the irrational numbers—numbers which could not be represented exactly as a ratio or fraction. But finally, mathematicians discovered that all known types of numbers were contained within a super-set called the complex numbers, in which each number is represented as a two-dimensional coordinate on the “complex plane”. Numbers were now understood to be anything which behaved like a number; a number did not need to take the “conventional” form. Complex numbers were the “numbers” that were found to behave most like numbers in all conceivable circumstances. They had both a numeric and a geometric representation.
Advances in calculus, probability, set theory and the study of infinity, non-Euclidean geometry and complex numbers indicated that mathematics in the Christian West had progressed far beyond that of the ancient Greeks. But if mathematicians with their non-Euclidean geometries were making a final break with ancient Greece, physicists were not far behind, and it was Scottish physicist James Maxwell who made some discoveries that would never have occurred to the ancient Greeks.
[1] Aleksandrov, Kolmogorov, & Lavrent’ev, 1999, p. 55 (Vol 1).
[2] Breuer, 2006, p. 2.
[3] The natural numbers are those used for counting, for example, 1, 2, 3, etc.
[4] Real numbers constitute a set including all integers, all fractional numbers, and all numbers that are impossible to represent as an exact fraction, such as the square root of two or pi.
[5] Rational numbers constitute a set containing all integers, and all fractional numbers, but excluding any number which cannot be represented as an exact fraction.