Dedekind ( 1888) proposed a different characterization, which lacked the formal logical character of Peano's axioms. Around the same time Richard Dedekind showed that the natural numbers are uniquely characterized by their induction properties. Peano was unaware of Frege's work at the time. Giuseppe Peano ( 1888) published a set of axioms for arithmetic that came to bear his name ( Peano axioms), using a variation of the logical system of Boole and Schröder but adding quantifiers. In logic, the term arithmetic refers to the theory of the natural numbers. Some concerns that mathematics had not been built on a proper foundation led to the development of axiomatic systems for fundamental areas of mathematics such as arithmetic, analysis, and geometry. This work summarized and extended the work of Boole, De Morgan, and Peirce, and was a comprehensive reference to symbolic logic as it was understood at the end of the 19th century. The two-dimensional notation Frege developed was never widely adopted and is unused in contemporary texts.įrom 1890 to 1905, Ernst Schröder published Vorlesungen über die Algebra der Logik in three volumes. Frege's work remained obscure, however, until Bertrand Russell began to promote it near the turn of the century. Gottlob Frege presented an independent development of logic with quantifiers in his Begriffsschrift, published in 1879, a work generally considered as marking a turning point in the history of logic. Their work, building on work by algebraists such as George Peacock, extended the traditional Aristotelian doctrine of logic into a sufficient framework for the study of foundations of mathematics ( Katz 1998, p. 686).Ĭharles Sanders Peirce built upon the work of Boole to develop a logical system for relations and quantifiers, which he published in several papers from 1870 to 1885. In the middle of the nineteenth century, George Boole and then Augustus De Morgan presented systematic mathematical treatments of logic. In the 18th century, attempts to treat the operations of formal logic in a symbolic or algebraic way had been made by philosophical mathematicians including Leibniz and Lambert, but their labors remained isolated and little known. Sophisticated theories of logic were developed in many cultures, including China, India, Greece and the Islamic world. The first half of the 20th century saw an explosion of fundamental results, accompanied by vigorous debate over the foundations of mathematics. Before this emergence, logic was studied with rhetoric, through the syllogism, and with philosophy. Mathematical logic emerged in the mid-19th century as a subfield of mathematics independent of the traditional study of logic ( Ferreirós 2001, p. 443). 12.4 Classical papers, texts, and collections.12.3 Research papers, monographs, texts, and surveys.7 Proof theory and constructive mathematics.6.1 Algorithmically unsolvable problems.Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems, rather than trying to find theories in which all of mathematics can be developed. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. Since its inception, mathematical logic has contributed to, and has been motivated by, the study of foundations of mathematics. In computer science (particularly in the ACM Classification) mathematical logic encompasses additional topics not detailed in this article see logic in computer science for those. These areas share basic results on logic, particularly first-order logic, and definability. Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics. Mathematical logic (also known as symbolic logic) is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic.