Fundamentals of Accounting List #76709
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The Foundations of Arithmetic. By Gottlob Frege. Translated by Kristján Kristjánsson. Introduction by Guðmundur Heiðar Frímannsson.
The philosopher and mathematician Gottlob Frege is considered the father of modern logic and the one who laid the foundation for philosophical semantics as it is practiced today. His work attracted surprisingly little attention during his lifetime, but in recent decades, philosophers have increasingly realized Frege's importance and his work has therefore begun to receive the respect it deserves. The most notable are his discoveries of declarative logic and quantifiers, i.e. methods of logic for stating what conclusions can be drawn from the internal structure of sentences and not only the relationship between entire sentences as propositional logic had previously done.
In this book, as in Frege's other major works, he attempts to show that mathematics can be reduced to logic. This theory is now generally considered to have been disproven. However, Frege's efforts bore abundant fruit, including the previously known declarative logic, as well as a new system of symbols for logic. The Foundations of Arithmetic, published in 1884, is, however, free from foreign symbols and formulas, as the author intended it to be an accessible and easily understood presentation of his mathematical philosophy. In the book, he argues for the theory of logic and the so-called objectivism about the nature of numbers, and argues against the theory, on the one hand, that mathematics is a game with symbols, independent of external reality (literalism), and on the other hand, that the laws of mathematics and logic are actually psychological laws (psychology). Frege's objectivism, in its simplest form, states that numerals serve as singulars in mathematical statements, and that in true statements, singulars refer to some existing thing. His arguments for this include the key role of equality or identity in our understanding of both numbers and the concept of number, but also of the concept of thing. The theory of logical sufficiency claims that only the fundamental rules and definitions of logic are necessary to prove mathematical statements, and thus they are logical sufficiency.
Both theories are fraught with problems, as Guðmundur Heiðar Frímannsson explains in the introduction to The Foundations of Arithmetic. Nevertheless, Frege's arguments undoubtedly have great philosophical weight, they are indispensable for anyone who wants to study the philosophy of mathematics and are ultimately attempts to answer the great riddles about objective reality and the relationship of language to the world. (www.hib.is)
The philosopher and mathematician Gottlob Frege is considered the father of modern logic and the one who laid the foundation for philosophical semantics as it is practiced today. His work attracted surprisingly little attention during his lifetime, but in recent decades, philosophers have increasingly realized Frege's importance and his work has therefore begun to receive the respect it deserves. The most notable are his discoveries of declarative logic and quantifiers, i.e. methods of logic for stating what conclusions can be drawn from the internal structure of sentences and not only the relationship between entire sentences as propositional logic had previously done.
In this book, as in Frege's other major works, he attempts to show that mathematics can be reduced to logic. This theory is now generally considered to have been disproven. However, Frege's efforts bore abundant fruit, including the previously known declarative logic, as well as a new system of symbols for logic. The Foundations of Arithmetic, published in 1884, is, however, free from foreign symbols and formulas, as the author intended it to be an accessible and easily understood presentation of his mathematical philosophy. In the book, he argues for the theory of logic and the so-called objectivism about the nature of numbers, and argues against the theory, on the one hand, that mathematics is a game with symbols, independent of external reality (literalism), and on the other hand, that the laws of mathematics and logic are actually psychological laws (psychology). Frege's objectivism, in its simplest form, states that numerals serve as singulars in mathematical statements, and that in true statements, singulars refer to some existing thing. His arguments for this include the key role of equality or identity in our understanding of both numbers and the concept of number, but also of the concept of thing. The theory of logical sufficiency claims that only the fundamental rules and definitions of logic are necessary to prove mathematical statements, and thus they are logical sufficiency.
Both theories are fraught with problems, as Guðmundur Heiðar Frímannsson explains in the introduction to The Foundations of Arithmetic. Nevertheless, Frege's arguments undoubtedly have great philosophical weight, they are indispensable for anyone who wants to study the philosophy of mathematics and are ultimately attempts to answer the great riddles about objective reality and the relationship of language to the world. (www.hib.is)