# Mathematical object

(Redirected from Mathematical objects)

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:

Combinatorics (a branch of mathematics) has such objects as:

Set theory (a branch of mathematics) has such objects as:

Geometry (a branch of mathematics) has such objects as:

Graph theory (a branch of mathematics) has such objects as:

Topology (a branch of mathematics) has such objects as:

Linear algebra (a branch of mathematics) has such objects as:

Abstract algebra (a branch of mathematics) has such objects as:

Categories are simultaneously homes to mathematical objects and mathematical objects in their own right. In proof theory, proofs and theorems are also mathematical objects.

The ontological status of mathematical objects has been the subject of much investigation and debate by philosophers of mathematics.

## 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 {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.