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.
In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously-defined concepts. It is often motivated informally, usually by an appeal to intuition and everyday experience. In an axiomatic theory, relations between primitive notions are restricted by axioms. Some authors refer to the latter as "defining" primitive notions by one or more axioms, but this can be misleading. Formal theories cannot dispense with primitive notions, under pain of infinite regress.
Semiotic literary criticism, also called literary semiotics, is the approach to literary criticism informed by the theory of signs or semiotics. Semiotics, tied closely to the structuralism pioneered by Ferdinand de Saussure, was extremely influential in the development of literary theory out of the formalist approaches of the early twentieth century.
Classical logic or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy.
Foundations of mathematics are the logical and mathematical framework that allows the development of mathematics without generating self-contradictory theories, and, in particular, to have reliable concepts of theorems, proofs, algorithms, etc. This may also include the philosophical study of the relation of this framework with reality.
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 mathematics and logic, an axiomatic system is any set of primitive notions and axioms to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system. A formal theory is an axiomatic system that describes a set of sentences that is closed under logical implication. A formal proof is a complete rendition of a mathematical proof within a formal system.
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.
In programming language theory, semantics is the rigorous mathematical study of the meaning of programming languages. Semantics assigns computational meaning to valid strings in a programming language syntax. It is closely related to, and often crosses over with, the semantics of mathematical proofs.
Russian formalism was a school of literary theory in Russia from the 1910s to the 1930s. It includes the work of a number of highly influential Russian and Soviet scholars such as Viktor Shklovsky, Yuri Tynianov, Vladimir Propp, Boris Eichenbaum, Roman Jakobson, Boris Tomashevsky, Grigory Gukovsky who revolutionised literary criticism between 1914 and the 1930s by establishing the specificity and autonomy of poetic language and literature. Russian formalism exerted a major influence on thinkers like Mikhail Bakhtin and Juri Lotman, and on structuralism as a whole. The movement's members had a relevant influence on modern literary criticism, as it developed in the structuralist and post-structuralist periods. Under Stalin it became a pejorative term for elitist art.
In art history, formalism is the study of art by analyzing and comparing form and style. Its discussion also includes the way objects are made and their purely visual or material aspects. In painting, formalism emphasizes compositional elements such as color, line, shape, texture, and other perceptual aspects rather than content, meaning, or the historical and social context. At its extreme, formalism in art history posits that everything necessary to comprehending a work of art is contained within the work of art. The context of the work, including the reason for its creation, the historical background, and the life of the artist, that is, its conceptual aspect is considered to be external to the artistic medium itself, and therefore of secondary importance.
Certainty is the epistemic property of beliefs which a person has no rational grounds for doubting. One standard way of defining epistemic certainty is that a belief is certain if and only if the person holding that belief could not be mistaken in holding that belief. Other common definitions of certainty involve the indubitable nature of such beliefs or define certainty as a property of those beliefs with the greatest possible justification. Certainty is closely related to knowledge, although contemporary philosophers tend to treat knowledge as having lower requirements than certainty.
In mathematical logic, a proof calculus or a proof system is built to prove statements.
Formalism is a school of literary criticism and literary theory having mainly to do with structural purposes of a particular text. It is the study of a text without taking into account any outside influence. Formalism rejects or sometimes simply "brackets" notions of culture or societal influence, authorship, and content, and instead focuses on modes, genres, discourse, and forms.
A coordinative definition is a postulate which assigns a partial meaning to the theoretical terms of a scientific theory by correlating the mathematical objects of the pure or formal/syntactical aspects of a theory with physical objects in the world. The idea was formulated by the logical positivists and arises out of a formalist vision of mathematics as pure symbol manipulation.
In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings using established manipulation rules. A central idea of formalism "is that mathematics is not a body of propositions representing an abstract sector of reality, but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess." According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they aren't "about" anything at all. Rather, mathematical statements are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation. In contrast to mathematical realism, logicism, or intuitionism, formalism's contours are less defined due to broad approaches that can be categorized as formalist.
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 1928, Hilbert had Brouwer removed from the editorial board of Mathematische Annalen.
A mathematical object is an abstract concept arising in mathematics. Typically, a mathematical object can be a value that can be assigned to a symbol, and therefore can be involved in formulas. Commonly encountered mathematical objects include numbers, expressions, shapes, functions, and sets. Mathematical objects can be very complex; for example, theorems, proofs, and even theories are considered as mathematical objects in proof theory.
In linguistics, the term formalism is used in a variety of meanings which relate to formal linguistics in different ways. In common usage, it is merely synonymous with a grammatical model or a syntactic model: a method for analyzing sentence structures. Such formalisms include different methodologies of generative grammar which are especially designed to produce grammatically correct strings of words; or the likes of Functional Discourse Grammar which builds on predicate logic.
