The Age of Critical Awareness and Foundations

Weierstrass: 1815- 1897


Cauchy sequences of rational numbers.

In his 1863/64 course on The general theory of analytic functions Weierstrass began to formulate his theory of the real numbers. In his 1863 lectures he proved that the complex numbers are the only commutative algebraic extension of the real numbers. Gauss had promised a proof of this in 1831 but had failed to give one.

Weierstrass's lectures developed into a four-semester course which he continued to give until 1890. The four courses were
1. Introduction to the theory of analytic functions,
2. Elliptic functions,
3. Abelian functions,
4. Calculus of variations or applications of elliptic functions.

Through the years the courses developed and a number of versions have been published such as the notes by Killing made in 1868 and those by Hurwitz from 1878. Weierstrass's approach still dominates teaching analysis today and this is clearly seen from the contents and style of these lectures, particularly the Introduction course. Its contents were: numbers, the function concept with Weierstrass's power series approach, continuity and differentiability, analytic continuation, points of singularity, analytic functions of several variables, in particular Weierstrass's "preparation theorem", and contour integrals.

The standards of rigour that Weierstrass set, defining, for example, irrational numbers as limits of convergent series, strongly affected the future of mathematics. He also studied entire functions, the notion of uniform convergence and functions defined by infinite products.

Dedekind :1831-1916


Real numbers characterized by "cuts". E.g. { rational numbers, r: where r2 <2}.

Dedekind made a number of highly significant contributions to mathematics and his work would change the style of mathematics into what is familiar to us today. One remarkable piece of work was his redefinition of irrational numbers in terms of Dedekind cuts which, as we mentioned above, first came to him as early as 1858. He published this in Stetigkeit und Irrationale Zahlen in 1872. In it he wrote:-

Now, in each case when there is a cut (A1, A2) which is not produced by any rational number, then we create a new, irrational number a, which we regard as completely defined by this cut; we will say that this number a corresponds to this cut, or that it produces this cut. As well as his analysis of the nature of number, his work on mathematical induction, including the definition of finite and infinite sets, and his work in number theory, particularly in algebraic number fields, is of major importance.

Frege:1848-1925 Tries to reduce mathematics to logic.

Peano:1858-1932 Gives axioms for arithmetic.