II.5 The Development of Mathematical Rigor in Mathematical Analysis: Weierstrass and His School, The Aftermath of Weierstrass and Riemann: Pages 125 – 128.

Weierstrass lectured in Berlin from the 1860’s to 1890 a series of 4 lectures on mathematical analysis.  These lectures were attended by many mathematicians and had indirect influence by circulation of the lecture notes.  Hermite and Jordan in France also incorporated aspects of his teachings.  Weierstrass’s approach followed that of Cauchy.  He banned the idea of motion from limit processes and considered functions of complex variables.  He did not incorporate set theory into his ideas.

He included the idea of uniform continuity and utilized the now common idea of epsilon – delta, ε – d, proofs.  To quote Weierstrass “After the assumption of an arbitrarily small quantity ε a bound d for x may be found, such that for every value of x for which |x| < d, the corresponding value of |y| will be less than ε“.

He arithmetized analysis with a very axiomatic and stepwise approach to the subject.  He generated counter-examples to illustrate difficulties and to distinguish between different kinds of analytical behavior.   One of his famous counterexamples was that of an everywhere continuous but nowhere differential function, namely, f(x) = S bn  cos(an x).    f(x) is uniformly convergent for b < 1 but is not differentiable for any x if ab > 1 +(3/2) p. He provided my other examples and counterexamples.  From the 1880’s this forced the form of rigorous argument as used by the Weierstrass school to be thrust upon mathematics generally.

The ideas of Weierstrass and Riemann forced a rigor into mathematical analysis.  Dedekind advanced their ideas using Dedekind cuts of the real numbers.  These divided the real line into three sections.  These ideas prompted discussion by the Cantor, E. Heine, Frege.  Leopold Kronecker denied the existence of the reals instead relying on finite sets.  These ideas influenced Brouwer and Kurt Hensel.

From 1890 to 1910 the foundational framework of analysis shifted toward the theory of sets.  The origin of this is in the work of Cantor a student of Weierstrass.  Cantor studied how to distinguish between different infinite sets.  He proved the rational and algebraic numbers have a countable infinite number while the reals have an uncountable infinite number.  This led to a hierarchy of infinite sets of different cardinality.

Cantor began to realize that set theory could come to be used as a foundational tool for all of mathematics.  Lebesgue and Borel around 1900 tied set theory to the calculus in a very concrete and intimate way. 

Hilbert in the 1890’s provided an axiomatic approach to geometries which helped to provide a new emphasis on mathematical theories as axiomatic structures in the foundations of analysis of the early twentieth century.  Peano in Italy had similar aims as Hilbert.  Hilbert’s student Ernst Zermelo worked on axiomatizing set theory.

 Russel had a famous paradox in set theory, if S is the set of all sets that do not contain themselves, then it is not possible for S to be in S, nor can it not be in S.  Zermelo’s theory sought to avoid this difficulty by not defining a set.

By 1910 Weyl referred to mathematics as the science of “set membership” rather than the science of “quantity”.

However, Zermelo’s axioms of set theory lacked a consistency proof. 

Weiner in 1914 provided a set-theoretic definition of an ordered pair.  This was followed by set-theoretic definition of a function. 

The basic elements of basic analysis, derivatives, integrals, series, existence and convergent behaviors of series, along with the axioms for the reals and set theory were well formulated in the early twentieth century.  Further debates about the infinitesimally small and infinitely large were mostly over.

There is a further research on this provided by Robinson in 1961.  He created differentials which are obtained by adjoining differentials to the real numbers.  The differentials satisfied the axioms of an ordered field.  His differentials include elements that are smaller than 1/n for every positive integer n.   These differentials seem to satisfy the ultimate goal of Leibniz to have a theory of infinitesimals which is part of the same structure as that of the reals.  This approach has never been widely accepted as a working foundation for analysis.

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