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Mathematical Object
A mathematical object is an abstract object arising in mathematics. The concept is studied in philosophy of mathematics. In mathematical practice, an object is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs. Commonly encountered mathematical objects include numbers, permutations, partitions, matrices, sets, functions, and relations. Geometry
Geometry
as a branch of mathematics has such objects as hexagons, points, lines, triangles, circles, spheres, polyhedra, topological spaces and manifolds. Another branch—algebra—has groups, rings, fields, group-theoretic lattices, and order-theoretic lattices. Categories are simultaneously homes to mathematical objects and mathematical objects in their own right
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Abstract Object
Abstract and concrete are classifications that denote whether a term describes an object with a physical referent or one with no physical referents. They are most commonly used in philosophy and semantics. Abstract objects are sometimes called abstracta (sing. abstractum) and concrete objects are sometimes called concreta (sing. concretum)
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Foundations Of Mathematics
Foundations of mathematics is the study of the philosophical and logical[1] 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.[2] In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be quite vague. Foundations of mathematics can be conceived as the study of the basic mathematical concepts (number, geometrical figure, set, function, etc.) and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics (formulas, theories and their models giving a meaning to formulas, definitions, proofs, algorithms, etc.) also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics
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Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental.[2] Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises
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Ontology
Ontology
Ontology
(introduced in 1606) is the philosophical study of the nature of being, becoming, existence, or reality, as well as the basic categories of being and their relations.[1] Traditionally listed as a part of the major branch of philosophy known as metaphysics, ontology often deals with questions concerning what entities exist or may be said to exist and how such entities may be grouped, related within a hierarchy, and subdivided according to similarities and differences. A very simple definition of ontology is that it is the examination of what is meant by 'being'. In modern terms, the formal study of reality itself is in the domain of the physical sciences, while the study of personal "reality" is left to psychology
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Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor (/ˈkæntɔːr/ KAN-tor; German: [ˈɡeɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfɪlɪp ˈkantɔʁ]; March 3 [O.S. February 19] 1845 – January 6, 1918[1]) was a German mathematician. He invented set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. In fact, Cantor's method of proof of this theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic
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Operation (mathematics)
In mathematics, an operation is a calculation from zero or more input values (called operands) to an output value. The number of operands is the arity of the operation. The most commonly studied operations are binary operations of arity 2, such as addition and multiplication, and unary operations of arity 1, such as additive inverse and multiplicative inverse. An operation of arity zero, or 0-ary operation is a constant. The mixed product is an example of an operation of arity 3, or ternary operation. Generally, the arity is supposed to be finite, but infinitary operations are sometimes considered
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Addition
Addition
Addition
(often signified by the plus symbol "+") is one of the four basic operations of arithmetic; the others are subtraction, multiplication and division. The addition of two whole numbers is the total amount of those quantities combined. For example, in the adjacent picture, there is a combination of three apples and two apples together, making a total of five apples. This observation is equivalent to the mathematical expression "3 + 2 = 5" i.e., "3 add 2 is equal to 5". Besides counting items, addition can also be defined on other types of numbers, such as integers, real numbers and complex numbers. This is part of arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can be performed on abstract objects such as vectors and matrices. Addition
Addition
has several important properties
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Additive Inverse
In mathematics, the additive inverse of a number a is the number that, when added to a, yields zero. This number is also known as the opposite (number),[1] sign change, and negation.[2] For a real number, it reverses its sign: the opposite to a positive number is negative, and the opposite to a negative number is positive. Zero is the additive inverse of itself. The additive inverse of a is denoted by unary minus: −a (see the discussion below). For example, the additive inverse of 7 is −7, because 7 + (−7) = 0, and the additive inverse of −0.3 is 0.3, because −0.3 + 0.3 = 0 . The additive inverse is defined as its inverse element under the binary operation of addition (see the discussion below), which allows a broad generalization to mathematical objects other than numbers
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Additive Identity
In mathematics the additive identity of a set which is equipped with the operation of addition is an element which, when added to any element x in the set, yields x. One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings.Contents1 Elementary examples 2 Formal definition 3 Further examples 4 Proofs4.1 The additive identity is unique in a group 4.2 The additive identity annihilates ring elements 4.3 The additive and multiplicative identities are different in a non-trivial ring5 See also 6 References 7 External linksElementary examples[edit]The additive identity familiar from elementary mathematics is zero, denoted 0
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Essence
In philosophy, essence is the property or set of properties that make an entity or substance what it fundamentally is, and which it has by necessity, and without which it loses its identity. Essence is contrasted with accident: a property that the entity or substance has contingently, without which the substance can still retain its identity. The concept originates with Aristotle, who used the Greek expression to ti ên einai (τὸ τί ἦν εἶναι,[1] literally meaning "the what it was to be" and corresponding to the scholastic term quiddity) or sometimes the shorter phrase to ti esti (τὸ τί ἐστι,[2] literally meaning "the what it is" and corresponding to the scholastic term haecceity) for the same idea. This phrase presented such difficulties for its Latin translators that they coined the word essentia (English "essence") to represent the whole expression
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Paradox
A paradox is a statement that, despite apparently sound reasoning from true premises, leads to an apparently self-contradictory or logically unacceptable conclusion.[1][2] A paradox involves contradictory yet interrelated elements that exist simultaneously and persist over time.[3][4][5] Some logical paradoxes are known to be invalid arguments but are still valuable in promoting critical thinking.[6] Some paradoxes have revealed errors in definitions assumed to be rigorous, and have caused axioms of mathematics and logic to be re-examined
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Model Theory
In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic. The objects of study are models of theories in a formal language. A set of sentences in a formal language is called a theory; a model of a theory is a structure (e.g. an interpretation) that satisfies the sentences of that theory. Model theory recognises and is intimately concerned with a duality: it examines semantical elements (meaning and truth) by means of syntactical elements (formulas and proofs) of a corresponding language
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Lattice (group)
In geometry and group theory, a lattice in R n displaystyle mathbb R ^ n is a subgroup of the additive group R n displaystyle mathbb R ^ n which is isomorphic to the additive group Z n displaystyle mathbb Z ^ n , and which spans the real vector space R n displaystyle mathbb R ^ n . In other words, for any basis of R n displaystyle mathbb R ^ n , the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice. A lattice may be viewed as a regular tiling of a space by a primitive cell. Lattices have many significant applications in pure mathematics, particularly in connection to Lie algebras, number theory and group theory
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Domain Of Discourse
In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range.Contents1 Overview 2 Examples 3 Universe of discourse 4 Boole’s 1854 definition 5 See also 6 ReferencesOverview[edit]Giuseppe PeanoThe domain of discourse is usually identified in the preliminaries, so that there is no need in the further treatment to specify each time the range of the relevant variables.[1] Many logicians distinguish, sometimes only tacitly, between the domain of a science and the universe of discourse of a formalization of the science.[2] Giuseppe Peano formalized number theory (arithmetic of positive integers) taking its domain to be the positive integers and the universe of discourse to include all numbers, not just integers. Examples[edit] For example, in an interpretation of first-order logic, the domain of d
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Predicate Logic
First-order logic—also known as first-order predicate calculus and predicate logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic
First-order logic
uses quantified variables over non-logical objects and allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form "there exists X such that X is Socrates and X is a man" and there exists is a quantifier while X is a variable.[1] This distinguishes it from propositional logic, which does not use quantifiers or relations.[2] A theory about a topic is usually a first-order logic together with a specified domain of discourse over which the quantified variables range, finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of axioms believed to hold for those things
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