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.
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's lives.
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.
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be vague. Foundations of mathematics can be conceived as the study of the basic mathematical concepts and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics. The search for foundations of mathematics is a central question of the philosophy of mathematics; the abstract nature of mathematical objects presents special philosophical challenges.
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.
Legal formalism is both a descriptive theory and a normative theory of how judges should decide cases. In its descriptive sense, formalists maintain that judges reach their decisions by applying uncontroversial principles to the facts; formalists believe that there is an underlying logic to the many legal principles that may be applied in different cases. These principles, they claim, are straightforward and can be readily discovered by anyone with some legal expertise. Supreme Court Justice Oliver Wendell Holmes Jr., by contrast, believed that "The life of the law has not been logic: it has been experience". The formalist era is generally viewed as having existed from the 1870s to the 1920s, but some scholars deny that legal formalism ever existed in practice.
A formal system is an abstract structure and formalization of an axiomatic system used for inferring 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.
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.
New Formalism is a late 20th- and early 21st-century movement in American poetry that has promoted a return to metrical, rhymed verse and narrative poetry on the grounds that all three are necessary if American poetry is to compete with novels and regain its former popularity among the American people.
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 1920, Hilbert succeeded in having Brouwer, whom he considered a threat to mathematics, removed from the editorial board of Mathematische Annalen.
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.
