In mathematics Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions and its philosophy The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of mathematics in our lives. The logical and structural nature, a mathematical object is an abstract object An abstract object is an object which does not exist at any particular time or place, but rather exists as a type of thing . In philosophy, an important distinction is whether an object is considered abstract or concrete. Abstract objects are sometimes called abstracta (sing. abstractum) and concrete objects are sometimes called concreta (sing arising in mathematics Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions. Commonly encountered mathematical objects include numbers A number is a mathematical object used in counting and measuring. A notational symbol which represents a number is called a numeral, but in common usage the word number is used for both the abstract object and the symbol, as well as for the word for the number. In addition to their use in counting and measuring, numerals are often used for labels ,, permutations In several fields of mathematics the term permutation is used with different but closely related meanings. They all relate to the notion of mapping the elements of a set to other elements of the same set, i.e., exchanging elements of a set, partitions In mathematics, a partition of a set X is a division of X into non-overlapping "parts" or "blocks" or "cells" that cover all of X. More formally, these "cells" are both collectively exhaustive and mutually exclusive with respect to the set being partitioned, matrices Entries of a matrix are often denoted by a variable with two subscripts, as shown on the right. Matrices of the same size can be added and subtracted entrywise and matrices of compatible size can be multiplied. These operations have many of the properties of ordinary arithmetic, except that matrix multiplication is not commutative, that is, AB and, sets A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Although it was invented at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In, functions In mathematics a function is a relation between a given set of elements and another set of elements (the range), which associates each element in the domain with exactly one element in the range. The elements so related can be any kind of thing (words, objects, qualities) but are typically mathematical quantities, such as real numbers, and relations In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subset of A × B. The terms dyadic relation and 2-place relation are synonyms for binary relations. Geometry Geometry is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the third century BC geometry was put into an axiomatic form by Euclid, whose treatment as a branch of mathematics has such objects as points In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue. Thus, a point is a 0-dimensional object. Because of their nature as one of the simplest geometric concepts, they are often used in one, lines In Euclidean geometry, a line is a straight curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height. Lines are an idealisation of such objects and have no width or height at all and are usually considered to be infinitely long. Lines are a fundamental concept in some, triangles A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ABC, circles A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are the same distance from a given point called the centre. The common distance of the points of a circle from its center is called its radius, spheres A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point. This distance r is known as the radius of the sphere. The, polyhedra A polyhedron is often defined as a geometric solid with flat faces and straight edges (the word polyhedron comes from the Classical Greek πολύεδρον, from poly-, stem of πολύς, "many," + -edron, form of εδρον, "base", "seat", or "face"), topological spaces Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology and manifolds In mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold. Thus a line and a circle are one-dimensional manifolds, a plane and the surface of a ball are two-dimensional manifolds,. Algebra Algebra is the branch of mathematics concerning the study of the rules of operations and the things which can be constructed from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, combinatorics, and number theory, algebra is one of the main branches of pure mathematics, another branch, has groups In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity and invertibility. While these are familiar from, rings In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations , where each operation combines two elements to form a third element. To qualify as a ring, the set together with its two operations must satisfy certain conditions; namely the set must be an Abelian group under addition, a monoid under, fields In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, various algebraic, group-theoretic lattices In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integral coefficients. A lattice may be viewed as a regular tiling of a space by a primitive cell and order-theoretic lattices In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum (the elements' least upper bound; called their join) and an infimum (greatest lower bound; called their meet). Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are. Categories are simultaneously homes to mathematical objects and mathematical objects in their own right.
The ontological status Ontology (from the Greek ὄν, genitive ὄντος: of being and -λογία, -logia: science, study, theory) is the philosophical study of the nature of being, existence or reality in general, as well as of the basic categories of being and their relations. Traditionally listed as a part of the major branch of philosophy known as metaphysics, of mathematical objects has been apart of a long dispute as the subject of much investigation and debate by philosophers of mathematics The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of mathematics in our lives. The logical and structural nature. On this debate, see the monograph by Burgess and Rosen (1997).
One view that emerged around the turn of the 20th century with the work of Cantor Georg Ferdinand Ludwig Phillip Cantor was a German mathematician, born in Russia. He is best known as the creator of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more is that all mathematical objects can be defined as sets A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Although it was invented at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In. The set {0,1} is a relatively clear-cut example. On the face of it the group In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity and invertibility. While these are familiar from Z2 of integers mod 2 is also a set with two elements. However it cannot simply be the set {0,1} because this does not mention the additional structure imputed to Z2 by its operations In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values. There are two common types of operations: unary and binary. Unary operations involve only one value, such as negation and trigonometric functions. Binary operations, on the other hand, take two values, of addition Addition is the mathematical process of putting things together. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples, since 3 + 2 = 5. Besides counts of fruit, addition can also represent and negation In logic and mathematics, negation is an operation on propositions. For example, in classical logic negation is normally interpreted by the truth function that takes truth to falsity and vice versa. In intuitionistic logic, according to the Brouwer-Heyting-Kolmogorov interpretation, the negation of a proposition P is the proposition whose proofs mod 2: how are we to tell which of 0 or 1 is the 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,, for example? To organize this group as a set it can first be coded as the quadruple ({0,1},+,−,0), which in turn can be coded using one of several conventions as a set representing that quadruple, which in turn entails encoding the operations + and − and the constant 0 as sets.
This approach to the ontology of mathematics raises a fundamental philosophical question of whether the ontology of mathematics needs to be beholden to either its practice or its pedagogy. Mathematicians do not work with such codings, which are neither canonical nor practical. They do not appear in any algebra texts, and neither students nor instructors in algebra courses have any familiarity with such codings. Hence if ontology is to reflect practice, mathematical objects cannot be reduced to sets in this way.
If however the goal of mathematical ontology is taken to be the internal consistency of mathematics, it is more important that mathematical objects be definable in some uniform way, for example as sets, regardless of actual practice in order to lay bare the essence In philosophy, essence is the attribute or set of attributes that make an object 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 object or substance has contingently, without which the substance can still retain its identity. The of its paradoxes A paradox is a statement or group of statements that leads to a contradiction or a situation which defies intuition; or, it can be an apparent contradiction that actually expresses a non-dual truth . Typically, either the statements in question do not really imply the contradiction, the puzzling result is not really a contradiction, or the. This has been the viewpoint taken by foundations of mathematics Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory. The search for foundations of mathematics is also a central question of the philosophy of mathematics: On what ultimate basis can mathematical, which has traditionally accorded the management of paradox higher priority than the faithful reflection of the details of mathematical practice as a justification for defining mathematical objects to be sets.
Much of the tension created by this foundational identification of mathematical objects with sets can be relieved without unduly compromising the goals of foundations by allowing two kinds of objects into the mathematical universe, sets and relations In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subset of A × B. The terms dyadic relation and 2-place relation are synonyms for binary relations, without requiring that either be considered merely an instance of the other. These form the basis of model theory In mathematics, model theory is the study of mathematical structures such as groups, fields, graphs, or even universes of set theory, using tools from mathematical logic. A structure that gives meaning to the sentences of a formal language is called a model for the language. If a model for a language moreover satisfies a particular sentence or as the domain of discourse The domain of discourse, sometimes called the universe of discourse, logical discourse, or simply discourse, is an analytic tool used in deductive logic, especially predicate logic. It indicates the relevant set of entities that are being dealt with by quantifiers of predicate logic In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulas contain variables which can be quantified. Two common quantifiers are the existential ∃ and universal. In this viewpoint mathematical objects are entities satisfying the axioms In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other truths of a formal theory expressed in the language of predicate logic.
A variant of this approach replaces relations with operations In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values. There are two common types of operations: unary and binary. Unary operations involve only one value, such as negation and trigonometric functions. Binary operations, on the other hand, take two values,, the basis of universal algebra Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes "the theory of groups" as an object of study. In this variant the axioms often take the form of equations An equation is a mathematical statement, in symbols, that two things are exactly the same . Equations are written with an equal sign, as in, or implications between equations.
A more abstract variant is category theory In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows, which abstracts sets as objects and the operations thereon as morphisms In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures between those objects. At this level of abstraction mathematical objects reduce to mere vertices In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces (typically triangles) in 3D models, where each such point is given as a vector of a graph In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges. Typically, a graph is depicted in diagrammatic form as a set whose edges as the morphisms abstract the ways in which those objects can transform and whose structure is encoded in the composition law for morphisms. Categories may arise as the models of some axiomatic theory and the homomorphisms between them (in which case they are usually concrete, meaning equipped with a faithful forgetful functor to the category Set or more generally to a suitable topos), or they may be constructed from other more primitive categories, or they may be studied as abstract objects in their own right without regard for their provenance.
References
| This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please improve this article by introducing more precise citations where appropriate. (June 2009) |
- Azzouni, J., 1994. Metaphysical Myths, Mathematical Practice. Cambridge University Press.
- Burgess, John, and Rosen, Gideon, 1997. A Subject with No Object. Oxford Univ. Press.
- Davis, Philip and Hersh, Reuben, 1999 [1981]. The Mathematical Experience. Mariner Books: 156-62.
- Gold, Bonnie, and Simons, Roger A., 2008. Proof and Other Dilemmas: Mathematics and Philosophy. Mathematical Association of America.
- Hersh, Reuben, 1997. What is Mathematics, Really? Oxford University Press.
- Sfard, A., 2000, "Symbolizing mathematical reality into being, Or how mathematical discourse and mathematical objects create each other," in Cobb, P., et al., Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools and instructional design. Lawrence Erlbaum.
- Stewart Shapiro, 2000. Thinking about mathematics: The philosophy of mathematics. Oxford University Press.
External links
- Stanford Encyclopedia of Philosophy: "Abstract Objects" -- by Gideon Rosen.
- Wells, Charles, "Mathematical Objects."
- Oldest Mathematical Object: the Lebombo Bone
- Second Oldest Mathematical Object: the Ishango Bone
- AMOF: The Amazing Mathematical Object Factory
- Mathematical Object Exhibit
Categories: Mathematics | Abstract objects
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Hydrodynamic Object Recognition: When Multipoles Count. Physical Review Letters, 2009; 102 (5): 058104 DOI: 10.1103/PhysRevLett.102.058104 Jan-Moritz P. ...
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March 20th 2007 Ever since 1887 when Norwegian mathematician Sophus Lie discovered the mathematical group called E8 researchers have been trying to understand the extraordinarily complex object described
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Okay, so whenever we define a new type of . mathematical object. , one of the first things we want to know is if there are any standard families of such . objects. . Here are a few standard graphs. (Note that from now on I usually won't label ...
