Related Research Articles
In computability theory, the Church–Turing thesis is a thesis about the nature of computable functions. It states that a function on the natural numbers can be calculated by an effective method if and only if it is computable by a Turing machine. The thesis is named after American mathematician Alonzo Church and the British mathematician Alan Turing. Before the precise definition of computable function, mathematicians often used the informal term effectively calculable to describe functions that are computable by paper-and-pencil methods. In the 1930s, several independent attempts were made to formalize the notion of computability:
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules called a formal grammar.
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.
Alfred Tarski was a Polish-American logician and mathematician. A prolific author best known for his work on model theory, metamathematics, and algebraic logic, he also contributed to abstract algebra, topology, geometry, measure theory, mathematical logic, set theory, and analytic philosophy.
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.
Metalogic is the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths.
A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms by a set of inference rules.
Richard Merritt Montague was an American mathematician and philosopher who made contributions to mathematical logic and the philosophy of language. He is known for proposing Montague grammar to formalize the semantics of natural language. As a student of Alfred Tarski, he also contributed early developments to axiomatic set theory (ZFC). For the latter half of his life, he was a professor at the University of California, Los Angeles until his early death, believed to be a homicide, at age 40.
In mathematical logic, an axiom schema generalizes the notion of axiom.
Tarski's axioms are an axiom system for Euclidean geometry, specifically for that portion of Euclidean geometry that is formulable in first-order logic with identity. As such, it does not require an underlying set theory. The only primitive objects of the system are "points" and the only primitive predicates are "betweenness" and "congruence". The system contains infinitely many axioms.
In mathematics, Robinson arithmetic is a finitely axiomatized fragment of first-order Peano arithmetic (PA), first set out by Raphael M. Robinson in 1950. It is usually denoted Q. Q is almost PA without the axiom schema of mathematical induction. Q is weaker than PA but it has the same language, and both theories are incomplete. Q is important and interesting because it is a finitely axiomatized fragment of PA that is recursively incompletable and essentially undecidable.
In computer science, the shortest common supersequence of two sequences X and Y is the shortest sequence which has X and Y as subsequences. This is a problem closely related to the longest common subsequence problem. Given two sequences X = < x1,...,xm > and Y = < y1,...,yn >, a sequence U = < u1,...,uk > is a common supersequence of X and Y if items can be removed from U to produce X and Y.
In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation. The motivating example of a relation algebra is the algebra 2 X 2 of all binary relations on a set X, that is, subsets of the cartesian square X2, with R•S interpreted as the usual composition of binary relations R and S, and with the converse of R as the converse relation.
General set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include the Peano axioms.
In mathematical logic, predicate functor logic (PFL) is one of several ways to express first-order logic by purely algebraic means, i.e., without quantified variables. PFL employs a small number of algebraic devices called predicate functors that operate on terms to yield terms. PFL is mostly the invention of the logician and philosopher Willard Quine.
The Brouwer–Hilbert controversy was a debate in twentieth-century mathematics over fundamental questions about the consistency of axioms and the role of semantics and syntax in mathematics. L. E. J. Brouwer, a proponent of the constructivist school of intuitionism, opposed David Hilbert, a proponent of formalism. Much of the controversy took place while both were involved with Mathematische Annalen, the leading mathematical journal of the time, with Hilbert as editor-in-chief and Brouwer as a member of its editorial board. In 1920 Hilbert had Brouwer removed from the editorial board of Mathematische Annalen.
A timeline of mathematical logic; see also history of logic.
John Corcoran was an American logician, philosopher, mathematician, and historian of logic. He is best known for his philosophical work on concepts such as the nature of inference, relations between conditions, argument-deduction-proof distinctions, the relationship between logic and epistemology, and the place of proof theory and model theory in logic. Nine of Corcoran's papers have been translated into Spanish, Portuguese, Persian, and Arabic; his 1989 "signature" essay was translated into three languages. Fourteen of his papers have been reprinted; one was reprinted twice.
References
- ↑ John Corcoran and Matt Lavine, "Discovering string theory". Bulletin of Symbolic Logic. 19 (2013) 253–4.
- ↑ Alonzo Church, Introduction to Mathematical Logic, Princeton UP, Princeton, 1956
- ↑ John Corcoran, William Frank and Michael Maloney, "String theory", Journal of Symbolic Logic, vol. 39 (1974) pp. 625– 637
- ↑ Pages 173–4 of Alfred Tarski, The concept of truth in formalized languages, reprinted in Logic, Semantics, Metamathematics, Hackett, Indianapolis, 1983, pp. 152–278