Automated theorem proving is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a major motivating factor for the development of computer science.
Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics, and suffices for the everyday use of set theory concepts in contemporary mathematics.
In logic, a logical connective is a logical constant. Connectives can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary connective can be used to join the two atomic formulas and , rendering the complex formula .
In the philosophy of mathematics, intuitionism, or neointuitionism, is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.
In mathematical logic, Russell's paradox is a set-theoretic paradox published by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. The paradox had already been discovered independently in 1899 by the German mathematician Ernst Zermelo. However, Zermelo did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other academics at the University of Göttingen. At the end of the 1890s, Georg Cantor – considered the founder of modern set theory – had already realized that his theory would lead to a contradiction, as he told Hilbert and Richard Dedekind by letter.
Friedrich Ludwig Gottlob Frege was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philosophy, concentrating on the philosophy of language, logic, and mathematics. Though he was largely ignored during his lifetime, Giuseppe Peano (1858–1932), Bertrand Russell (1872–1970), and, to some extent, Ludwig Wittgenstein (1889–1951) introduced his work to later generations of philosophers. Frege is widely considered to be the greatest logician since Aristotle, and one of the most profound philosophers of mathematics ever.
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.
The history of logic deals with the study of the development of the science of valid inference (logic). Formal logics developed in ancient times in India, China, and Greece. Greek methods, particularly Aristotelian logic as found in the Organon, found wide application and acceptance in Western science and mathematics for millennia. The Stoics, especially Chrysippus, began the development of predicate logic.
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics owes itself to David Hilbert's attempt to secure the foundations of mathematics in the early part of the 20th century. Metamathematics provides "a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and logic". An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system. An informal illustration of this is categorizing the proposition "2+2=4" as belonging to mathematics while categorizing the proposition "'2+2=4' is valid" as belonging to metamathematics.
Hume's principle or HP says that the number of Fs is equal to the number of Gs if and only if there is a one-to-one correspondence between the Fs and the Gs. HP can be stated formally in systems of second-order logic. Hume's principle is named for the Scottish philosopher David Hume and was coined by George Boolos.
George Stephen Boolos was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.
In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that – for some coherent meaning of 'logic' – mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic. Bertrand Russell and Alfred North Whitehead championed this programme, initiated by Gottlob Frege and subsequently developed by Richard Dedekind and Giuseppe Peano.
Begriffsschrift is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book.
In the philosophy of language, the context principle is a form of semantic holism holding that a philosopher should "never ... ask for the meaning of a word in isolation, but only in the context of a proposition".
The Foundations of Arithmetic is a book by Gottlob Frege, published in 1884, which investigates the philosophical foundations of arithmetic. Frege refutes other idealist and materialist theories of number and develops his own platonist theory of numbers. The Grundlagen also helped to motivate Frege's later works in logicism.
In proof theory, a discipline within mathematical logic, double-negation translation, sometimes called negative translation, is a general approach for embedding classical logic into intuitionistic logic. Typically it is done by translating formulas to formulas that are classically equivalent but intuitionistically inequivalent. Particular instances of double-negation translations include Glivenko's translation for propositional logic, and the Gödel–Gentzen translation and Kuroda's translation for first-order logic.
In mathematical logic, the ancestral relation of a binary relation R is its transitive closure, however defined in a different way, see below.
Class logic is a logic in its broad sense, whose objects are called classes. In a narrower sense, one speaks of a class logic only if classes are described by a property of their elements. This class logic is thus a generalization of set theory, which allows only a limited consideration of classes.
