Beginning Mathematics/Introduction to Abstraction

Mathematics takes concrete things, and turns them into abstractions which help to generalize ideas. Take a simple example with numbers. Is quantity the property of a thing? We can have two cars or two shoes, but there are still two. We can also have one car or two cars, but we are still talking about a "car".

Think about it for a minute. Can you define the idea of number? There is no tangible thing as one. There is "one dollar", but there is not a one. There is a symbol for one, but again no tangible "one".

If you attempted to define numbers you probably came up with definitions that just start out listing 1,2,3,etc. or you can also look and see that one plus one is two. There is something inherent in this property.

This seemingly imperceptible condition of abstraction is the defining quality of mathematics. In geometry we consider enclosed shapes with three sides on them, and call them triangles. You may see a triangle, but you will never see the tangible idea of triangle. You merely see a representation of it.

Yet, you can discover such facts like a rectangle can be divided into two triangles. This is true regardless of the rectangle you choose, and you can reason this without actually seeing every rectangle, which would be impossible.

Mathematics is able to generalize various things into abstract ideas, upon which generalized statements can be made for classes of objects without considering each one individually. Triangles have certain properties. Two of anything has certain properties. These are all related in the abstract.

In understanding abstractions, mathematics begins to classify ways in which abstract objects are alike, and ways they are different. Some things are numbers, and others are shapes.

Mathematics deals with this problem by defining different types of equivalences. For any two things it covers, an equivalence class says they are either the same or different in a certain way. In ordinary arithmetic this might be expressed by something like $$3+3=6$$ which says that under the equal equivalence class 3+3 is the same as 6.