The Foundations of Arithmetic is a book by Gottlob Frege, published in , which Title page of Die Grundlagen der Title page of the original . Friedrich Ludwig Gottlob Frege was a German philosopher, logician, and mathematician. He is .. Grundgesetze der Arithmetik, Band I (); Band II ( ), Jena: Verlag Hermann Pohle (online version). In English (translation of selected. Die Grundlagen der Arithmetik. Eine logisch mathematische Untersuchung über den Begriff der Zahl von. Dr. G. Frege,. a. o. Professor an der Universität Jena.
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He is understood by many to be the father of analytic philosophyconcentrating on the philosophy of language and mathematics. Though largely ignored during his lifetime, Giuseppe Peano — and Bertrand Russell — introduced his work to later generations of logicians and philosophers.
His contributions include the development of modern logic in the Begriffsschrift and work in the foundations of mathematics. His book the Foundations of Arithmetic is the seminal text of the logicist project, and is arithmetij by Michael Dummett as where to pinpoint the linguistic turn. His father Carl Karl Alexander Frege — was the co-founder and headmaster of a girls’ high school until his death. In childhood, Frege encountered philosophies that would guide his future scientific career.
Grundgesetze der arithmetik
Hinstorff,the first section of which dealt with the structure and logic of language. Frege studied at a gymnasium in Wismar and graduated in His teacher Gustav Adolf Leo Sachse 5 November — 1 Septemberwho was a poet, played the most important role in determining Frege’s future scientific career, encouraging him to continue his studies at the University of Jena.
Frege matriculated at the University of Jena in the spring of as aritmhetik citizen of the North German Confederation. In the four semesters of his studies he attended approximately twenty courses of lectures, most of them on mathematics and physics.
His most important teacher was Ernst Karl Abbe —; physicist, mathematician, and inventor. Abbe gave lectures on theory of gravity, galvanism and electrodynamics, complex analysis theory of functions of a complex variable, applications of physics, selected divisions of mechanics, and mechanics of solids. Abbe was more than a teacher to Frege: After Frege’s graduation, they came into closer correspondence. His other notable university teachers were Christian Philipp Karl Snell —86; subjects: Many of the philosophical doctrines of the mature Frege have parallels in Lotze; it has been the subject of scholarly debate whether or not there was a direct influence on Frege’s views arising from his attending Lotze’s lectures.
Though his education and early mathematical work focused primarily on geometry, Frege’s work soon turned to logic. His Begriffsschrifteine der arithmetischen nachgebildete Formelsprache des reinen Denkens [ Concept-Script: Verlag von Louis Nebert, marked a turning point in the history of logic. The Begriffsschrift broke new ground, including a rigorous treatment of the ideas of functions and variables.
Frege’s goal was to show that mathematics grows out of logicand in so doing, he devised techniques that took him far beyond the Aristotelian syllogistic and Stoic propositional logic that had come down to him in the logical tradition.
In effect, Frege invented axiomatic predicate logicin large part thanks to his invention of quantified variableswhich eventually became ubiquitous in mathematics and logic, and which solved the problem of multiple generality. Previous logic had dealt with the logical constants andorif A frequently noted example is that Aristotle’s logic is unable to represent mathematical statements like Euclid’s theorema fundamental statement of number theory that there are an infinite number of prime numbers.
Frege’s “conceptual notation” however can represent such inferences. One of Frege’s stated purposes was to isolate genuinely logical principles of inference, so that in the proper representation of mathematical proof, one would at no point appeal to “intuition”. If there was an intuitive element, it was to be isolated and represented separately as an axiom: Having exhibited this possibility, Frege’s larger purpose was to defend the view that arithmetic is a branch of logic, a view known as logicism: Already in the Begriffsschrift important preliminary theorems, for example a generalized form of law of trichotomywere derived within what Frege understood to be pure logic.
This idea was formulated in non-symbolic terms in his The Foundations of Arithmetic Later, in his Basic Laws of Arithmetic vol.
Most of these axioms were carried over from his Begriffsschriftthough not without some significant changes. The one truly new principle was one he called the Basic Law V: The crucial case of the law may be formulated in modern notation as follows. The case is special because what is here being called the extension of a predicate, or a set, is only one type of “value-range” of a function. In a famous episode, Bertrand Russell wrote to Frege, just as Vol. It is easy to define the relation of membership of a set or extension in Frege’s system; Russell then drew attention to “the set of things x that are such that x is not a member of x “.
The system of the Grundgesetze entails that the set thus characterised both is and is not a member of itself, and is thus inconsistent. Frege wrote a hasty, last-minute Appendix to Vol. Frege opened the Appendix with the exceptionally honest comment: This was the position I was placed in by a letter of Mr.
Gottlob Frege, Grundgesetze der Arithmetik Begriffsschriftlich Abgeleitet – PhilPapers
Bertrand Russell, just when the printing of this volume was nearing its completion. Frege’s proposed remedy was subsequently shown to imply that there is but one object in the universe of discourseand hence is worthless indeed, this would make for a contradiction in Frege’s system if he had axiomatized the idea, fundamental to his discussion, that the True and the False are distinct objects; see, for example, Dummettbut recent work has shown that much of the program of the Grundgesetze might be salvaged in other ways:.
Frege’s work in logic had little international attention until when Russell wrote an appendix to The Principles of Mathematics stating his differences with Frege.
The diagrammatic notation that Frege used had no antecedents and has had no imitators since. Moreover, until Russell and Whitehead’s Principia Mathematica 3 vols. Frege’s logical ideas nevertheless spread through the writings of his student Rudolf Carnap — and other admirers, particularly Bertrand Russell and Ludwig Wittgenstein arihmetik Frege is one of the founders of analytic philosophywhose work on logic and language gave rise to the linguistic turn in philosophy.
Gottlob Frege – Wikipedia
His contributions to the philosophy of language include:. As a philosopher of mathematics, Frege attacked the psychologistic appeal to mental explanations of the content of judgment of the meaning of sentences. His original purpose was very far from answering general questions about meaning; instead, he devised his logic to explore the foundations of arithmetic, undertaking to answer questions such as “What is a number?
It should be kept in mind that Frege was employed as a frebe, not a philosopher, and he published his philosophical papers in scholarly journals that often were hard to access outside of the German-speaking world.
He never published a philosophical monograph other than The Foundations of Arithmeticmuch of which was mathematical in content, and the first collections of his writings appeared only after World War II. A volume of English translations of Frege’s philosophical essays first appeared inedited by students of Wittgenstein, Peter Geach and Max Black —88with the bibliographic assistance of Wittgenstein see Geach, ed.
Despite the generous praise of Russell and Wittgenstein, Frege was little known as a philosopher during his lifetime. His ideas spread chiefly through those he influenced, such as Russell, Wittgenstein, and Carnap, and through work on logic and semantics by Polish logicians. While conventional accounts of meaning took expressions to have just one feature referenceFrege introduced the view that expressions have two different aspects of significance: Referenceor, “Bedeutung” applied to proper nameswhere a given expression say the expression “Tom” simply refers to the entity bearing the name the person named Tom.
Frege also held that propositions had a referential relationship with their truth-value in other words, a statement “refers” to the truth-value it takes. By contrast, the sense or “Sinn” associated with a complete sentence is the thought it expresses. The sense of an expression is said to be the “mode of presentation” of the item referred to, and there can be multiple modes of representation for the same referent. The distinction can be illustrated thus: But the sense of the word “Wales” is a part of the sense of the latter expression, but no part of the sense of the “full name” of Prince Charles.
These distinctions were disputed by Bertrand Russell, especially in his paper ” On Denoting “; the controversy has continued into the present, fueled especially by Saul Kripke ‘s famous lectures ” Naming and Necessity “. Frege’s published philosophical writings were of a very technical nature and divorced from practical issues, so much so that Frege scholar Dummett expresses his “shock to discover, while reading Frege’s diary, that his hero was an anti-Semite.
In the last year of his life, at the age of 76, his diary contains extreme right-wing political opinions, opposing the parliamentary system, democrats, liberals, Catholics, the French and Jews, who he thought ought to be deprived of political rights and, preferably, expelled from Germany. Some interpretations have been written about that time. Frege had friendly relations with Jews in real life: Frege was described by his students as a highly introverted person, seldom entering into dialogue, mostly facing the blackboard while lecturing though being witty and sometimes bitterly sarcastic.
Die Grundlagen der Arithmetik: Verlag Hermann Pohle online version. Frege intended that the following three papers be published together in a book titled Logische Untersuchungen Logical Investigations. Though the German book never appeared, the papers were published together in Logische Untersuchungened. Peter Geach, Blackwell, From Wikipedia, the free encyclopedia. WismarMecklenburg-SchwerinGermany. Bad KleinenMecklenburg-SchwerinGermany. LeibnizBernard Bolzano . Philosophy portal Logic portal.
Frege as Idealist and then Realist,” Inquiry 22 1—4: Harvard University Press, pp. Essays on the Philosophical and Foundational Work of G. Hintikka, Synthese Library, D. From Frege to Wittgenstein: Random House Webster’s Unabridged Dictionary.
Oxford University Press,p. Philosophy of Languagep. Philosophy and Politics in Nazi Germanypp. Sluga’s source was an article by Eckart Menzler-Trott: Introduction by the editors on pp. This article has been translated into English, in: Gottlob Frege at Wikipedia’s sister projects. Aretaic turn Australian realism Communitarianism Ordinary language philosophy Philosophical logic Philosophy of language Philosophy of science Postanalytic philosophy. Index of language articles.
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