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 r^{2} <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:-

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

Peano:1858-1932 Gives axioms for arithmetic.