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A mathematical object is an abstract object arising in philosophy of mathematics and mathematics itself. Commonly encountered mathematical objects include numbers, permutations, partitions, matrices, sets, functions, and relations. 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. The ontological status of mathematical objects has been the subject of much investigation and debate by philosophers of mathematics.〔Burgess, John, and Rosen, Gideon, 1997. ''A Subject with No Object: Strategies for Nominalistic Reconstrual of Mathematics''. Oxford University Press. ISBN 0198236158〕 ==Cantorian framework== One view that emerged around the turn of the 20th century with the work of Cantor is that all mathematical objects can be defined as sets. The set is a relatively clear-cut example. On the face of it the group Z2 of integers mod 2 is also a set with two elements. However, it cannot simply be the set , because this does not mention the additional structure imputed to Z2 by the operations of addition and negation mod 2: how are we to tell which of 0 or 1 is the additive identity, for example? To organize this group as a set it can first be coded as the quadruple (,+,−,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. Sets may include ordered denotation of the particular identities and operations that apply to them, indicating a group, abelian group, ring, field, or other mathematical object. These types of mathematical objects are commonly studied in abstract algebra. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「mathematical object」の詳細全文を読む スポンサード リンク
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