II.7 The Crisis in the Foundation of Mathematics, Early Foundational Questions and Around 1900: Pages 143 – 144

The crisis in the foundation of mathematics is related to the three viewpoints of “logicism”, “formalism”, and “intuitionism” and the status of mathematical knowledge in light of Godel’s incompleteness results.  This crisis is often considered to be localized around 1920 as a debate between the classical mathematics advocated by Hilbert and his critics led by Brouwer.  There are aspects of this crisis which have been ongoing over a longer period of time.

“Logicism” is the thesis that the basic concepts of mathematics are definable by means of logical notions and the key principles of mathematics are deducible from logical principles alone.  Hilbert endorsed this view starting around 1899.  This view was consistent with that of Dedekind, Gauss and Dirichlet.  This view was challenged by Riemann, Dedekind, Cantor, Hilbert and others.  Opposition to this new direction came from Kronecker and Weierstrass.

The most characteristic traits of this modern approach were: i. acceptance of an arbitrary function proposed by Dirichlet, ii. infinite sets and the higher infinite, iii. put thoughts in place of calculations and a concentration on axiomatic structures and iv. purely existential methods of proof.

Dedekind utilized these traits in algebraic number theory with his set definition of number fields and ideals and in proving the fundamental theroem of unique factorization.  Dedekind was able to prove that in any ring of algebraic integers ideals possess a unique decomposition into prime ideals.

Kronecker complained that these proofs were purely existential not allowing one to calculate, for example, the relevant divisors or ideals.

Dedekind and Riemann had ideas and methods that were formulated on general concepts rather than on formulas and calculations.

Weierstrass analytic or holomorphic functions as as power series of a certain form.   He used analytic continuation to connect these with one another.  Riemann defined a function that satisfies the Cauchy-Riemann differentiability conditions.  Weierstrass offered examples of continuous functions that are nowhere differentiable.   

The “conceptual approach” in 19-th century mathematics was that of Dirichlet’s abstract idea of a function as an arbitrary way of associating with each x some y = f(x).  This was just the right framework to define and analyze general concepts such as continuity and integration.

The Bolzano-Weierstrass theorem which states that an infinite bounded set of real numbers has an accumulation point.  Kronecker criticized this theorem because the accumulation point cannot be be constructed by elementary operations from the rational numbers.

Hilbert around 1890 had an existential proof of the basis theorem that ever ideal in a polynomial ring is finitely generated.  Paul Gordan remarked humorously that this was theology and not mathematics. 

Eventually the modern camp enrolled new members such as Hurwitz, Minkowski, Hilbert, Volterra, Peano and Hadamard and was defended by Klein. 

After this followed the logical paradoxes discovered by Cantor, Russell, Zermelo, and others.   The set theoretic paradoxes were arguments showing that assumptions that certain sets exist lead to contradictions.  The semantic paradoxes showed difficulties with the notions of truth and definability.  Around 1900 these paradoxes destroyed the attractive view of logicism.

Around 1900 Hilbert set down his famous list of problems.  These included Cantor’s continuum problem and whether every set can be well ordered.  These two problems and the axiom of choice were employed by Zermelo to show that the real numbers, R, can be well ordered.

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