Malazil Here they focus on the fact that, for Frege, thinking is an act which serves as a general precondition for any act of knowing any truths, since grasping a thought is a general precondition to knowing its truth. This has lead to a very important debate in the philosophy of language, which, unfortunately, we cannot fully discuss here. Jacquette in Jacquette . Frege suggests also that this confusion would have the absurd result that numbers simply are the numerals, the signs on the page, and that we should be able to study their geanke with a microscope. Olms, ; reprinted in Thiel with Corrigenda six pages and some textual criticism. Exposition of the System.
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Childhood —69 [ edit ] Frege was born in in Wismar , Mecklenburg-Schwerin today part of Mecklenburg-Vorpommern. Frege studied at a grammar school in Wismar and graduated in Studies at University —74 [ edit ] Frege matriculated at the University of Jena in the spring of as a 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. His other notable university teachers were Christian Philipp Karl Snell —86; subjects: use of infinitesimal analysis in geometry, analytical geometry of planes , analytical mechanics, optics, physical foundations of mechanics ; Hermann Karl Julius Traugott Schaeffer —; analytical geometry, applied physics, algebraic analysis, on the telegraph and other electronic machines ; and the philosopher Kuno Fischer —; Kantian and critical philosophy.
The Begriffsschrift broke new ground, including a rigorous treatment of the ideas of functions and variables. Title page to Begriffsschrift 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 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.
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. 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 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. 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.
The best-known way is due to philosopher and mathematical logician George Boolos — , who was an expert on the work of Frege. This principle, too, is consistent if second-order arithmetic is, and suffices to prove the axioms of second-order arithmetic.
Predicative second-order logic plus Basic Law V is provably consistent by finitistic or constructive methods, but it can interpret only very weak fragments of arithmetic. The diagrammatic notation that Frege used had no antecedents and has had no imitators since. Philosopher[ edit ] 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: Function and argument analysis of the proposition ;.
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