Arithmetization of analysis

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The arithmetization of analysis was a research program in the foundations of mathematics carried out in the second half of the 19th century which aimed to abolish all geometric intuition from the proofs in analysis. For the followers of this program, the fundamental concepts of calculus should also not make references to the ideas of motion and velocity. This ideal was pursued by Augustin-Louis Cauchy, Bernard Bolzano, Karl Weierstrass, among others, who considered that Isaac Newton's calculus lacked rigor.

Contents

History

Kronecker originally introduced the term arithmetization of analysis, by which he meant its constructivization in the context of the natural numbers (see quotation at bottom of page). The meaning of the term later shifted to signify the set-theoretic construction of the real line. Its main proponent was Weierstrass, who argued the geometric foundations of calculus were not solid enough for rigorous work.

Research program

The highlights of this research program are:

Legacy

An important spinoff of the arithmetization of analysis is set theory. Naive set theory was created by Cantor and others after arithmetization was completed as a way to study the singularities of functions appearing in calculus.

The arithmetization of analysis had several important consequences:

Quotation

Related Research Articles

Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics, and suffices for the everyday use of set theory concepts in contemporary mathematics.

<span class="mw-page-title-main">David Hilbert</span> German mathematician (1862–1943)

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<span class="mw-page-title-main">Georg Cantor</span> German mathematician (1845–1918)

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<span class="mw-page-title-main">Cantor's first set theory article</span> First article on transfinite set theory

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<span class="mw-page-title-main">Richard Dedekind</span> German mathematician (1831–1916)

Julius Wilhelm Richard Dedekind was a German mathematician who made important contributions to number theory, abstract algebra, and the axiomatic foundations of arithmetic. His best known contribution is the definition of real numbers through the notion of Dedekind cut. He is also considered a pioneer in the development of modern set theory and of the philosophy of mathematics known as Logicism.

The mathematical concept of a function dates from the 17th century in connection with the development of the calculus; for example, the slope of a graph at a point was regarded as a function of the x-coordinate of the point. Functions were not explicitly considered in antiquity, but some precursors of the concept can perhaps be seen in the work of medieval philosophers and mathematicians such as Oresme.

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