David Hilbert Sets an Agenda for Twentieth-Century Mathematics
Overview
In 1900 David Hilbert (1862-1943), one of the acknowledged leaders of pure mathematics at the turn of the century, identified what he considered to be the most important problems facing contemporary mathematicians at an address to the Second International Congress of Mathematicians. While some of the questions concerned purely technical issues, a number addressed the foundations of mathematics. These led to major changes in the philosophy of mathematics and ultimately to the development of digital computers and artificial intelligence.
Background
The nineteenth century was a period of restructuring in mathematics. Once considered a study of self-evident truths about quantity and space, the paradoxes of set theory and the discovery of non-Euclidean geometries and generalized kinds of numbers motivated mathematicians to reexamine the most fundamental ideas in each area of mathematics. Many mathematicians thought that the best way to avoid any possible problems was to adopt the axiomatic method, allowing each field to have very few undefined concepts and unproved statements about it, called axioms, which were taken to be true. All other statements could only be considered true if they could be proved in a strictly logical way from the axioms.
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