higher-order logic See FORMAL LOGIC, PHILOSOPHY. OF LOGIC , SECOND -ORDER LOGI. Hilbert, David (1862–1943), German mathematician and philosopher of mathematics. Born in Königsberg, he also studied and served on the faculty there, accepting Weber’s chair in mathematics at Göttingen in 1895. He made important contributions to many different areas of mathematics and was renowned for his grasp of the entire discipline. His more philosophical work was divided into two parts. The focus of the first, which occupied approximately ten years beginning in the early 1890s, was the foundations of geometry and culminated in his celebrated Grundlagen der Geometrie (1899). This is a rich and complex work that pursues a variety of different projects simultaneously. Prominent among these is one whose aim is to determine the role played in geometrical reasoning by principles of continuity. Hilbert’s interest in this project was rooted in Kantian concerns, as is confirmed by the inscription, in the Grundlagen, of Kant’s synopsis of his critical philosophy: ‘Thus all human knowledge begins with intuition, goes from there to concepts and ends with ideas.’ Kant believed that the continuous could not be represented in intuition and must therefore be regarded as an idea of pure reason – i.e., as a device playing a purely regulative role in the development of our geometrical knowledge (i.e., our knowledge of the spatial manifold of sensory experience). Hilbert was deeply influenced by this view of Kant’s and his work in the foundations of geometry can be seen, in large part, as an attempt to test it by determining whether (or to what extent) pure geometry can be developed without appeal to principles concerning the nature of the continuous. To a considerable extent, Hilbert’s work confirmed Kant’s view – showing, in a manner more precise than any Kant had managed, that appeals to the continuous can indeed be eliminated from much of our geometrical reasoning. The same basic Kantian orientation also governed the second phase of Hilbert’s foundational work, where the focus was changed from geometry to arithmetic and analysis. This is the phase during which Hilbert’s Program was developed. This project began to take shape in the 1917 essay ‘Axiomatisches Denken.’ (The 1904 paper ‘Über die Grundlagen der Logik und Arithmetik,’ which turned away from geometry and toward arithmetic, does not yet contain more than a glimmer of the ideas that would later become central to Hilbert’s proof theory.) It reached its philosophically most mature form in the 1925 essay ‘Über das Unendliche,’ the 1926 address ‘Die Grundlagen der Mathematik,’ and the somewhat more popular 1930 paper ‘Naturerkennen und Logik.’ (From a technical as opposed to a philosophical vantage, the classical statement is probably the 1922 essay ‘Neubegründung der Mathematik. Erste Mitteilung.’) The key elements of the program are (i) a distinction between real and ideal propositions and methods of proof or derivation; (ii) the idea that the so-called ideal methods, though, again, playing the role of Kantian regulative devices (as Hilbert explicitly and emphatically declared in the 1925 paper), are nonetheless indispensable for a reasonably efficient development of our mathematical knowledge; and (iii) the demand that the reliability of the ideal methods be established by real (or finitary) means.
As is well known, Hilbert’s Program soon came under heavy attack from Gödel’s incompleteness theorems (especially the second), which have commonly been regarded as showing that the third element of Hilbert’s Program (i.e., the one calling for a finitary proof of the reliability of the ideal systems of classical mathematics) cannot be carried out.
See also GÖDEL’S INCOMPLETENESS THEO- REMS , HILBERT ‘s PROGRAM , PROOF THEOR. M.D.