Gottlob frege contribution to logic
After Frege's graduation, they came into closer correspondence. His other notable university teachers were Christian Philipp Karl Snell —86; subjects: use of infinitesimal analysis in geometry, analytic geometry of planes , analytical mechanics, optics, physical foundations of mechanics ; Hermann Karl Julius Traugott Schaeffer —; analytic geometry, applied physics, algebraic analysis, on the telegraph and other electronic machines ; and the philosopher Kuno Fischer —; Kantian and critical philosophy.
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. Years later they adopted a son, Alfred. Little else is known about Frege's family life, however. Though his education and early mathematical work focused primarily on geometry, Frege's work soon turned to 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 logic , and in so doing, he devised techniques that separated him from the Aristotelian syllogistic but took him rather close to Stoic propositional logic. In effect, Frege invented axiomatic predicate logic , in large part thanks to his invention of quantified variables , which eventually became ubiquitous in mathematics and logic, and which solved the problem of multiple generality.
Previous logic had dealt with the logical constants and , or , if A frequently noted example is that Aristotle's logic is unable to represent mathematical statements like Euclid's theorem , a 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: from there on, the proof was to be purely logical and without gaps.
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 : unlike geometry, arithmetic was to be shown to have no basis in "intuition", and no need for non-logical axioms. Already in the Begriffsschrift important preliminary theorems, for example, a generalized form of law of trichotomy , were derived within what Frege understood to be pure logic.
Begriffsschrift gottlob frege biography
This idea was formulated in non-symbolic terms in his The Foundations of Arithmetic Die Grundlagen der Arithmetik , Most of these axioms were carried over from his Begriffsschrift , though not without some significant changes. 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: "Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr. 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 discourse , and 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, Dummett , but 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 — Frege is one of the founders of analytic philosophy , whose work on logic and language gave rise to the linguistic turn in philosophy. 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?
While conventional accounts of meaning took expressions to have just one feature reference , Frege introduced the view that expressions have two different aspects of significance: their sense and their reference. Reference or "Bedeutung" applied to proper names , where 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. It is indeed historically true that the Booleans had no quantification theory at that time, but this cannot be regarded as an essential difference between these variations of symbolic logic on a systematic level, because in the U.
Peirce and his student Oscar Howard Mitchell developed an almost equivalent quantification theory within the algebra of logic. Another essential difference can be seen in the fact that Frege aimed at giving a logical structure of judgeable contents, which implied an inherent semantics. The Booleans, in contrast, were interested in logical structures themselves, which could be applied in different domains.
Their systems allowed various interpretations. This required a supplementary external semantics. It was, thus, essentially philosophical. There is evidence that he was influenced by the philosophy of his contemporaries, especially by neo-Kantian approaches. Contrary to Immanuel Kant , who regarded mathematical arithmetical and geometrical propositions as examples for synthetic a priori propositions, that is, propositions that are not empirical, but enlarging knowledge, Frege wanted to prove that arithmetic could completely be founded on logic, that is, that each arithmetical concept, in particular the concept of number, could be derived from logical concepts.
Arithmetic was, thus, analytical. The logicist program is only sketched in the Begriffsschrift , where Frege gave purely logical definitions of equinumerousity and the successor relation. In his next book, the Grundlagen der Arithmetik There he further elaborated his earlier distinction between concept and object.
In particular he introduced in the Grundgesetze value-ranges considered as a special kind of objects. The identity criterion is given in Basic Law V, according to which the value-ranges of two functions are identical if the functions coincide in their values for every argument, with this giving the modern abstraction schema. In terms of concepts the law says that whatever falls under the concept F falls under the concept G and vice versa, if and only if the concepts F and G have the same extension.
Frege suggested an ad hoc solution forbidding that the extension of a concept may fall under the concept itself. In his latest publications Frege gave up logicism. The Nature of Axiomatics. After the failure of his logicistic program, Frege focused his research on geometry as the foundational discipline of mathematics.
In these papers Frege opposes the understanding of arithmetic as a purely formal game with calculations bare of any contents. The older formalism regards arithmetic as a game like chess. It starts from certain initial formulas, then derives new formulas using a fixed set of transformation rules. But, neither the initial formulas nor the transition rules are justified, so the derived formulas are not justified either.
Therefore, Frege concludes, these approaches could not provide any contribution to the foundations of arithmetic. In his Grundlagen der Geometrie of he gave an axiomatic presentation of Euclidean geometry. The geometrical concepts were not directly defined, but implicitly gained as concepts obeying the features set by some group of axioms, and justified by proving the independence of the axioms from one another, the completeness of the system, and its consistency.
The formalistic approach aims at a theory of structures. It became clear that Frege stuck to the traditional Aristotelian understanding of axioms in geometry, calling axioms sentences that are true but not proved, because they have emerged from a source of knowledge completely different from the logical, a source that can be called spatial intuition.
From the truth of the axioms follows that they do not contradict each other, so no consistency of proof was needed. Hilbert answered that if the arbitrarily set axioms do not contradict each other with all their implications, then they are true and the defined objects exist. For Hilbert consistency logical possibility is, thus, the criterion of truth and existence.
The two controversies mentioned show that Frege followed that traditional understanding of philosophy as all-embracing fundamental discipline that formed, along with logic, logical ontology, and epistemology the foundation for mathematics and sciences. He did not share the pragmatic attitude of some influential contemporaries in mathematics and sciences like Hilbert to keep philosophy away from their mathematical and scientific practice by simply fading out philosophical problems.
Nevertheless, Frege opened the way for directions like philosophy of science, which aimed at bridging the gap between philosophy on the one hand and mathematics and science on the other, and which became successful in the twentieth century. Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens.
Halle, Germany: L. Nebert, Oxford: Oxford University Press, In Hermes et al. Breslau, Germany: W. Koebner, Translated by John L. Austin as The Foundations of Arithmetic. Oxford: Blackwell, Review of E. Husserl, Philosophie der Arithmatik I. First Series. Second Series. Posthumous Writings. Chicago: University of Chicago Press, Philosophical and Mathematical Correspondence.
Edited by Gottfried Gabriel, et al. Abridged for the English edition by Brian McGuinness. Translated by Hans Kaal. Collected Papers on Mathematics, Logic, and Philosophy.
Gottlob frege contribution to logic
Edited by Brian McGuinness. Translated by Max Black et al. Kleine Schriften. Edited by Ignacio Angelelli. Hildesheim, Germany: G. Olms, Scholz and E. Gottlob Frege in the Time from 10 March to 9 April Inquiry 39 : — The Frege Reader.
Begriffsschrift gottlob frege biography images
Edited by Michael Beaney. Translated and edited by Erich H. Reck and Steve Awodey. Chicago: Open Court, Beaney, Michael. Frege: Making Sense. London: Duckworth, Beaney, Michael, and Erich H. Reck, eds. London: Routledge, Comprehensive collection of papers on Frege. Brady, Geraldine. Amsterdam: Elsevier, Stresses the significance of the algebraic tradition for the development of logic.
Demopoulos, William, ed. Frege: Philosophy of Mathematics. Gabriel, Gottfried, and Uwe Dathe, eds. Paderborn, Germany: Mentis, Grattan-Guinness, Ivor. See pp. Frege then went on to give his own definitions of the basic concepts of arithmetic based purely on logic, and from these he deduced, again using pure logic, the basic laws of arithmetic.
Dummett writes [ 39 ] :- The work is fascinating even for those quite uninterested in the philosophy of mathematics, since in the course of it many ideas are presented which are of significance for the whole of philosophy. What was the reaction to the Grundlagen from mathematicians and philosophers? One might have expected an enormous amount of interest, but this did not materialise.
The Grundlagen only received a single review and that was by Cantor. What did Cantor think of this brilliant book? Dummett writes that the review [ 2 ] The Grundlagen was a non-technical work, written without symbolism and with only sketches of proofs, which Frege saw as a first step towards the realisation of his goal of defining a precise logical framework in which to set up the basic concepts of arithmetic and to deduce the rules of arithmetic.
Although he was extremely disappointed at the reaction to the Grundlagen nevertheless in the following years he wrote a number of articles which polished and extended the ideas which he would need to carry out his project. Dummett calls these In this he gives his famous argument to show that sense and reference are distinct. His example concerns the planet Venus which was known as "the evening star" and as "the morning star" before it was realised that both were Venus.
However "the evening star" and "the morning star" refer to the same object so the reference of "the evening star" is distinct from its sense. Frege axiomatized arithmetic with an intuitive collection of axioms, and proofs of number theory results which he had only sketched earlier he now gave formally.
The main thrust of this volume was to develop the rules of number theory and in the later volumes Frege intended to extend the work to the real numbers. His bitter disappointment at the lack of reaction to his earlier work shows explicitly in the Preface to Volume 1 where he complains about other authors being unfamiliar with his ideas. He must have hoped that this first volume of what he viewed would be his greatest achievement would be well received, but except for one review by Peano , it was ignored by his contemporaries.
Frege, who had not allowed the previous lack of reaction to divert him from the tasks that he had set himself, decided to delay publication of the second of his three proposed volumes.
Gottlob frege sense and reference: Begriffsschrift (German for, roughly, "concept-writing") is a book on logic by Gottlob Frege, published in , and the formal system set out in that book. Begriffsschrift is usually translated as concept writing or concept notation ; the full title of the book identifies it as "a formula language, modeled on that of arithmetic, for pure.
During this period Frege was appointed ordinary honorary professor at Jena, a post funded by the Carl Zeiss Foundation with which Abbe was closely associated. In fact it would be ten years after the publication of Volume 1 of Die Grundgesetze der Arithmetik before Volume 2 appeared. This second volume gives Frege's development of the real numbers which he constructed straight from the integers without taking the route of first defining the rational numbers.
The bitterness which he now felt shows clearly in this volume with his attacks on the work of earlier mathematicians being abusive which it had never been before and there were clear signs that he was hitting back at those he felt had ignored his contributions. In particular he strongly criticised Cantor 's and Dedekind 's theories of irrational numbers.
After the work was written, but before it was published, Frege discovered that this volume, and Volume 1 , were based on inconsistent axioms. Russell pointed out, with great modesty, that the Russell paradox gave a contradiction in Frege's system of axioms. After many letters between the two, Frege modified one of his axioms and explains in an appendix to the book that this was done to restore the consistency of the system.
However with this modified axiom, many of the theorems of Volume 1 do not go through and Frege must have known this. He probably never realised that even with the modified axiom the system is inconsistent since this was only shown by Lesniewski after Frege's death. One often sees it stated that Frege's work was worthless because of the inconsistency pointed out by Russell.
In fact this is far from the truth and one must view Frege as the person who made one of the most important contributions to the foundations of mathematics that has ever been made.
Begriffsschrift gottlob frege biography wikipedia
In fact in many ways Russell is correct when he wrote in his History of Western Philosophy:- In spite of the epoch-making nature of [ Frege's ] discoveries, he remained wholly without recognition until I drew attention to him in Frege's influence in the short term came through the work of Peano , Wittgenstein , Husserl, Carnap and Russell.
In the longer term, however, Frege has become a major influence on the development of philosophical logic and the man who seems to have been largely ignored by his contemporaries has been avidly read by many in the second half of the twentieth century, particularly after his works were translated into English.
Another statement that one often reads is that Frege was so depressed after Russell 's letters that he gave up research. This is not entirely without foundation and it is certainly true that he never published the intended third volume of The Basic Laws of Arithmetic , but although he did indeed become very depressed the reasons are far more complex than this.
References [ edit ]. S2CID Bibliography [ edit ]. George Boolos , Ivor Grattan-Guinness , In Search of Mathematical Roots. Princeton University Press. External links [ edit ]. Wikimedia Commons has media related to Begriffsschrift. Categories : non-fiction books Books by Gottlob Frege Logic books Diagram algebras Analytic philosophy literature Philosophy of logic Classical logic Predicate logic.
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