डेविड हिल्बर्ट

कृतियाँ

• 1899ज्यामिति की नींव

  अपरिभाषित “बिंदुओं”, “रेखाओं” और “तलों” के लिए स्वयंसिद्धों के पाँच समूहों से यूक्लिडीय ज्यामिति का स्वयंसिद्ध निर्माण; स्वयंसिद्धों की संगति और स्वतंत्रता मॉडलों से सिद्ध की जाती है। फ़्रेगे ने आपत्ति की कि स्वयंसिद्धों को दिए गए विषयों के लिए सत्य होना चाहिए।

• 1900गणितीय समस्याएँ

  पेरिस में गणितज्ञों की अंतरराष्ट्रीय कांग्रेस में व्याख्यान; मुद्रित रूप में 23 खुली समस्याएँ रखी गई हैं (व्याख्यान में दस प्रस्तुत हुईं), जिनमें सातत्य परिकल्पना और अंकगणित के स्वयंसिद्धों की संगति हैं, इस दृढ़ विश्वास के साथ कि गणित में कोई “इग्नोराबिमुस” नहीं।

• 1928गणितीय तर्कशास्त्र के सिद्धांत

  हिल्बर्ट के व्याख्यानों से विल्हेल्म आकरमान के साथ लिखी पाठ्यपुस्तक: वह वाक्य-तर्कशास्त्र और प्रथम-क्रम तर्कशास्त्र रखती है और दो खुले प्रश्न उठाती है – क्या प्रथम-क्रम तर्कशास्त्र पूर्ण है और क्या कोई निर्णय-प्रक्रिया है – जिनका उत्तर गेडेल (1929) तथा चर्च और ट्यूरिंग (1936) ने दिया।

मत

Hilbert is on the map for the foundations of mathematics and logic. His “Foundations of Geometry” (1899) built Euclidean geometry on groups of axioms whose basic terms – point, line, plane – receive their meaning only from the axioms; he proved the independence of axioms by models and held that the consistency of axioms suffices for the existence of the objects they define. Frege objected in letters and two articles that axioms must be true statements about given objects; the exchange is a classic of the philosophy of mathematics. In response to the paradoxes of set theory and to Brouwer’s intuitionism, Hilbert proposed in the 1920s to formalise mathematics completely and to prove the consistency of these formal systems by “finitary” means, reasoning about finite strings of signs that he regarded as secure as elementary arithmetic. Infinitary mathematics would thus be justified as an instrument, and no one should drive mathematicians “from the paradise that Cantor created”. With Ackermann he also posed the decision problem: is there a mechanical procedure that decides whether a formula of logic is valid? Gödel’s incompleteness theorems (1931) showed that the consistency of arithmetic cannot be proved within arithmetic itself, so the programme could not succeed in its original form; Church and Turing (1936) answered the decision problem in the negative. Modified versions of the programme continue in proof theory. His Königsberg address of 1930 ended with the words “We must know – we will know”, against the “ignorabimus” of Emil du Bois-Reymond.

जीवन और संदर्भ

David Hilbert was born on 23 January 1862 in Königsberg in East Prussia (according to some sources in nearby Wehlau) and studied mathematics there, where Hermann Minkowski and Adolf Hurwitz became his friends; he took his doctorate in 1885 under Ferdinand Lindemann. He became professor in Königsberg in 1893 and moved to Göttingen in 1895, where he stayed until his retirement in 1930 and helped make the university a centre of mathematics; Edmund Husserl was his colleague there from 1901 to 1916. At the International Congress of Mathematicians in Paris in 1900 he spoke on mathematical problems; the printed version of the lecture set out 23 problems that shaped much research in the century that followed. He supported Emmy Noether’s habilitation against resistance in the faculty. In the 1920s he worked with Paul Bernays and Wilhelm Ackermann on the foundations of mathematics. After 1933 many of his colleagues and students were driven out by the Nazi regime. He died in Göttingen on 14 February 1943.

अन्य प्रमुख कृतियाँ

• Grundlagen der Geometrie (1899)
• Mathematische Probleme (Paris lecture, 1900)
• Neubegründung der Mathematik (1922)
• Über das Unendliche (1926)
• Grundzüge der theoretischen Logik (with W. Ackermann, 1928)
• Grundlagen der Mathematik (with P. Bernays, 2 vols., 1934–39)