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.
Contents 1 Cantorian framework
2 Foundational paradoxes
3
Cantorian framework[edit]
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 0,1 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 0,1 , 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 ( 0,1 ,+,−,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.
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.
Foundational paradoxes[edit]
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
of its paradoxes. This has been the viewpoint taken by foundations of
mathematics, 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, without requiring
that either be considered merely an instance of the other. These form
the basis of model theory as the domain of discourse of predicate
logic. From this viewpoint, mathematical objects are entities
satisfying the axioms of a formal theory expressed in the language of
predicate logic.
Abstract object Mathematical structure References[edit] This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (June 2009) (Learn how and when to remove this template message) ^ Burgess, John, and Rosen, Gideon, 1997. A Subject with No Object: Strategies for Nominalistic Reconstrual of Mathematics. Oxford University Press. ISBN 0198236158 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 Reuben Hersh, 1999 [1981]. The Mathematical
Experience. Mariner Books: 156-62.
Gold, Bonnie, and Simons, Roger A., 2011. Proof and Other Dilemmas:
External links[edit] Stanford Encyclopedia of Philosophy: "Abstract Objects"—by Gideon Rosen. Wells, Charles, "Mathematical Objects." AMOF: The Amazing Mathematical Object Factory Mathematical Ob |