كورت غودل
في أي نظام قواعد متسق قويٍّ بما يكفي للحساب ثمة عبارات صادقة لا يستطيع إثباتها أبدًا – فالحقيقة الرياضية تتجاوز ما يبرهن عليه أي نظام واحد من هذا النوع.
التموضع
منطقي من فلك حلقة فيينا: كل نسق صوري متسق وغني بما يكفي للحساب يحوي جملًا صادقة لا يستطيع إثباتها، ولا يستطيع إثبات اتساقه الذاتي. ودافع في الفلسفة عن أفلاطونية في الأشياء الرياضية.
الأفكار الرئيسة
- مبرهنتا اللاإكمال
- اكتمال منطق الدرجة الأولى
- الأفلاطونية الرياضية
- الكون الدوّار
التأثير
تأثّر بـ 5
غوتفريد لايبنتس
تأثّر بـ
Gödel studied Leibniz intensively and, like him, sought to show first that God is possible, in a modal ontological proof built on positive properties (notes 1970; publ. 1987). K. Gödel, “Ontological Proof” (1970), in Collected Works III (1995); J. H. Sobel, “Gödel’s Ontological Proof”, in J. J. Thomson (ed.), On Being and Saying (1987); SEP “Kurt Gödel”
التوثيق: مسودة بالذكاء الاصطناعي
غوتفريد لايبنتس › 1646–1716
جورج كانتور
تأثّر بـ
Gödel’s paper on Cantor’s continuum problem takes up Cantor’s open question and presses the iterative conception of sets. SEP “The Early Development of Set Theory” (J. Ferreirós), §§3–4 (Gödel on Cantor’s continuum problem, iterative conception)
التوثيق: مسودة بالذكاء الاصطناعي
جورج كانتور › 1845–1918
إدموند هوسرل
تأثّر بـ
Gödel studied Husserl from 1959 and proposed phenomenology as a method for clarifying the meaning of mathematical concepts and intuition (draft, 1961; publ. 1995). K. Gödel, “The Modern Development of the Foundations of Mathematics in the Light of Philosophy” (1961), in Collected Works III (1995); SEP “Kurt Gödel”
التوثيق: مسودة بالذكاء الاصطناعي
إدموند هوسرل › 1859–1938
ألبرت أينشتاين
تأثّر بـ
For Einstein’s seventieth birthday Gödel found rotating-universe solutions of his field equations and argued that relativity supports an idealistic view of time (1949). K. Gödel, “An Example of a New Type of Cosmological Solutions of Einstein’s Field Equations of Gravitation”, Reviews of Modern Physics 21 (1949); “A Remark about the Relationship between Relativity Theory and Idealistic Philosophy”, in P. A. Schilpp (ed.), Albert Einstein: Philosopher-Scientist (1949); SEP “Kurt Gödel”
التوثيق: مسودة بالذكاء الاصطناعي
ألبرت أينشتاين › 1879–1955
موريتز شليك
تأثّر بـ
Gödel attended Schlick’s seminar on Russell’s Introduction to Mathematical Philosophy and, from 1926, meetings of the Vienna Circle, without adopting its positivism. J. W. Dawson, Logical Dilemmas: The Life and Work of Kurt Gödel (1997); SEP “Kurt Gödel”
التوثيق: مسودة بالذكاء الاصطناعي
موريتز شليك › 1882–1936
أثّر في 2
رودولف كارناب
أثّر في
Carnap used Gödel’s arithmetisation to state the syntax of a language within that language and, after the incompleteness results, separated derivability from consequence (Logische Syntax der Sprache, 1934). R. Carnap, Logische Syntax der Sprache (1934); K. Gödel, “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I”, Monatshefte für Mathematik und Physik 38 (1931); SEP “Rudolf Carnap”
التوثيق: مسودة بالذكاء الاصطناعي
رودولف كارناب › 1891–1970
آلان تورنغ
أثّر في
Turing related his unsolvability proof for the Entscheidungsproblem to Gödel’s incompleteness theorem and, like Gödel’s arithmetisation, coded machines as numbers (‘On Computable Numbers’, 1936). A. M. Turing, “On Computable Numbers, with an Application to the Entscheidungsproblem”, Proceedings of the London Mathematical Society 42 (1936–37); K. Gödel, Monatshefte für Mathematik und Physik 38 (1931); SEP “Alan Turing”
التوثيق: مسودة بالذكاء الاصطناعي
آلان تورنغ › 1912–1954
النقد
انتقده 1
لودفيغ فيتغنشتاين
انتقده
Wittgenstein questioned the philosophical significance of Gödel’s incompleteness theorem and doubted that ‘true but unprovable’ makes sense outside a system (Remarks on the Foundations of Mathematics, 1956). L. Wittgenstein, Bemerkungen über die Grundlagen der Mathematik (1956), part I, appendix I (3rd ed. 1978: appendix III); SEP “Wittgenstein’s Philosophy of Mathematics”
التوثيق: مسودة بالذكاء الاصطناعي
لودفيغ فيتغنشتاين › 1889–1951
ينتقد 2
ديفيد هلبرت تأثّر بـ
تأثّر بـ
Gödel answered the question posed in Hilbert and Ackermann’s Grundzüge der theoretischen Logik (1928) by proving the completeness of first-order logic (dissertation 1929; publ. 1930). D. Hilbert and W. Ackermann, Grundzüge der theoretischen Logik (1928); K. Gödel, “Die Vollständigkeit der Axiome des logischen Funktionenkalküls”, Monatshefte für Mathematik und Physik 37 (1930); SEP “Kurt Gödel”
التوثيق: مسودة بالذكاء الاصطناعي
انتقد
Gödel’s second incompleteness theorem (1931) is commonly read as defeating Hilbert’s programme of finitary consistency proofs; Gödel himself wrote that it did not contradict Hilbert’s formalist standpoint. K. Gödel, “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I”, Monatshefte für Mathematik und Physik 38 (1931), 173–198; SEP “Hilbert’s Program”
التوثيق: مسودة بالذكاء الاصطناعي
ديفيد هلبرت › 1862–1943
بيرتراند راسل تأثّر بـ
تأثّر بـ
Gödel took Principia Mathematica as the formal system of his incompleteness theorems (1931) and welcomed Russell’s early realism about classes and concepts (‘Russell’s Mathematical Logic’, 1944). K. Gödel, “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I”, Monatshefte für Mathematik und Physik 38 (1931); “Russell’s Mathematical Logic”, in P. A. Schilpp (ed.), The Philosophy of Bertrand Russell (1944); SEP “Kurt Gödel”
التوثيق: مسودة بالذكاء الاصطناعي
انتقد
Gödel criticised Russell’s vicious-circle principle and no-class theory as too restrictive for classical mathematics, defending classes and concepts as real objects (‘Russell’s Mathematical Logic’, 1944). K. Gödel, “Russell’s Mathematical Logic”, in P. A. Schilpp (ed.), The Philosophy of Bertrand Russell (1944), 123–153
التوثيق: مسودة بالذكاء الاصطناعي
بيرتراند راسل › 1872–1970
الأعمال
مبرهنتا عدم الاكتمال: لا يستطيع نسق متسق من هذا النوع أن يبرهن على جميع الحقائق الحسابية، ولا على اتساقه هو.
الكائنات الرياضية موجودة باستقلال عنا ونُدركها بضرب من الحدس؛ والمسائل غير المحسومة كفرضية المتصل لها جواب مع ذلك.
الموقف
The paper of 1931 showed that any consistent formal system that contains elementary arithmetic and whose axioms can be listed mechanically leaves some arithmetical sentences undecided: there is a true sentence that it cannot prove. A second theorem adds that such a system cannot prove its own consistency. Gödel achieved this by coding formulas and proofs as numbers, so that a sentence could in effect say of itself that it is unprovable. The result ended the hope of Hilbert’s programme to secure all of mathematics by finite consistency proofs, and his method of arithmetisation was taken up by Turing and Carnap. Gödel also showed that the axiom of choice and the continuum hypothesis are consistent with the standard axioms of set theory (1938–40). Philosophically he was a Platonist: mathematical objects exist independently of us and are grasped by a kind of intuition, so questions like the continuum hypothesis have a definite answer even if the axioms do not decide it. He criticised Russell’s no-class theory and, in drafts never published in his lifetime, Carnap’s view of mathematics as syntax. He studied Leibniz and from the late 1950s Husserl, found solutions to Einstein’s equations with rotating universes (1949), and left notes for a modal version of the ontological proof of God.
الحياة والسياق
Kurt Gödel was born in 1906 in Brünn (Brno) into a German-speaking family. From 1924 he studied physics and then mathematics in Vienna, where he attended Moritz Schlick’s seminar and meetings of the Vienna Circle without sharing its positivism. His dissertation under Hans Hahn (1929) proved the completeness of first-order logic; the incompleteness theorems followed in 1931. He visited the Institute for Advanced Study in Princeton several times from 1933 and suffered repeated depressive crises. After the annexation of Austria his position as a university lecturer lapsed, and in 1940 he and his wife Adele emigrated via the Trans-Siberian railway and Japan. In Princeton he was a daily walking companion of Einstein. He became a US citizen in 1948. Fearing poisoning, he ate almost nothing while Adele was in hospital, and he died in Princeton in January 1978.
أعمال رئيسة أخرى
- The Completeness of the Axioms of the Functional Calculus of Logic (dissertation 1929, published 1930)
- On Formally Undecidable Propositions of Principia Mathematica and Related Systems I (1931)
- The Consistency of the Continuum Hypothesis (1940)
- Russell’s Mathematical Logic (1944)
- An Example of a New Type of Cosmological Solutions of Einstein’s Field Equations (1949)
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