Understanding the term "Abstraction" in mathematics When the need for abstraction is asserted in mathematics is it generally meant that there is a need to apply a definition to n-dimensions such that n is an integer going to infinity?
 A: Abstraction in mathematics is usually the process of taking a familiar concept, selecting several properties of the concept you deem important, and then exploring all things that have those properties. The common theme is beginning with something familiar, and then asking about everything else that is "like" the object you are familiar with.
By focusing only on a specific set of properties, you can concentrate on exactly what follows from those properties, and other features, which had perhaps distracted you in the specific example, are cast aside.
For instance, the integers are a nonempty set which you can add, subtract and multiply in, and also the distributive property holds. The abstraction of those properties is called a ring.
Another example is this: "squares and triangles are finite strings of line segments in a plane which make a path that doesn't cross itself." Abstracting this, you would get the concept of simple polygons. 
In $\Bbb R^n$ you can add, subtract and scale vectors using the field $\Bbb R$, and addition and scaling distribute over each other. By replacing $\Bbb R$ with other fields $F$, and exploring other sets $V$ with the same properties allows us to abstract to vector spaces.
