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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 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 the philosophy of mathematics, constructivism asserts that it is necessary to find a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation.
Stephen Cole Kleene was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of mathematical logic known as recursion theory, which subsequently helped to provide the foundations of theoretical computer science. Kleene's work grounds the study of computable functions. A number of mathematical concepts are named after him: Kleene hierarchy, Kleene algebra, the Kleene star, Kleene's recursion theorem and the Kleene fixed-point theorem. He also invented regular expressions in 1951 to describe McCulloch-Pitts neural networks, and made significant contributions to the foundations of mathematical intuitionism.
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic.
Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.
In programming language theory and proof theory, the Curry–Howard correspondence is the direct relationship between computer programs and mathematical proofs.
Computability logic (CoL) is a research program and mathematical framework for redeveloping logic as a systematic formal theory of computability, as opposed to classical logic, which is a formal theory of truth. It was introduced and so named by Giorgi Japaridze in 2003.
Game semantics is an approach to formal semantics that grounds the concepts of truth or validity on game-theoretic concepts, such as the existence of a winning strategy for a player, somewhat resembling Socratic dialogues or medieval theory of Obligationes.
Provability logic is a modal logic, in which the box operator is interpreted as 'it is provable that'. The point is to capture the notion of a proof predicate of a reasonably rich formal theory, such as Peano arithmetic.
In proof theory, ludics is an analysis of the principles governing inference rules of mathematical logic. Key features of ludics include notion of compound connectives, using a technique known as focusing or focalisation, and its use of locations or loci over a base instead of propositions.
Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic. It is also notable for its connections to theoretical computer science. In broad terms, categorical logic represents both syntax and semantics by a category, and an interpretation by a functor. The categorical framework provides a rich conceptual background for logical and type-theoretic constructions. The subject has been recognisable in these terms since around 1970.
Giorgi Japaridze is a Georgian-American researcher in logic and theoretical computer science. He currently holds the title of Full Professor at the Computing Sciences Department of Villanova University. Japaridze is best known for his invention of computability logic, cirquent calculus, and Japaridze's polymodal logic.
In mathematical logic, the Brouwer–Heyting–Kolmogorov interpretation, or BHK interpretation, of intuitionistic logic was proposed by L. E. J. Brouwer and Arend Heyting, and independently by Andrey Kolmogorov. It is also sometimes called the realizability interpretation, because of the connection with the realizability theory of Stephen Kleene. It is the standard explanation of intuitionistic logic.
Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.
In mathematical logic, realizability is a collection of methods in proof theory used to study constructive proofs and extract additional information from them. Formulas from a formal theory are "realized" by objects, known as "realizers", in a way that knowledge of the realizer gives knowledge about the truth of the formula. There are many variations of realizability; exactly which class of formulas is studied and which objects are realizers differ from one variation to another.
A timeline of mathematical logic; see also history of logic.
In mathematics, the effective topos introduced by Martin Hyland (1982) captures the mathematical idea of effectivity within the category theoretical framework.
References
- S.C. Kleene. On the interpretation of intuitionistic number theory . Journal of Symbolic Logic, 10:109-124, 1945.
- J.M.E. Hyland. The effective topos. In A. S. Troelstra and D. van Dalen, editors, The L.E.J. Brouwer Centenary Symposium, pages 165-216. North Holland Publishing Company, 1982.
- S. Awodey, L. Birkedal, and D.S. Scott. Local realizability toposes and a modal logic for computability . Mathematical Structures in Computer Science, 12(3):319-334, 2002.
- G. Japaridze, Introduction to computability logic . Annals of Pure and Applied Logic 123 (2003), pages 1–99.