# Abstraction

According to a timeline of the history of mathematics by William Richardson at Wichita State University, the earliest known evidence of mathematics is the evidence of counting around 50,000 B.C. Since then mathematics has advanced tremendously. The advancement of mathematics is experienced by school children, yet very often it goes unnoticed and unappreciated. Students begin by learning to count and getting a general idea of what numbers “mean”. Once they have a grasp on this, they start learning how these numbers interact with each other through addition, subtraction, multiplication, and division. And these are all concrete ideas. For example, addition/subtraction can be described as gaining or losing. Multiplication can be described as gaining or losing the same amount regularly. So if I produce 50 apples a day, after 6 days I will have 50*6 = 300 apples. Division can be thought of as a separation of sorts. For example, if I have 20 people and I want to split them evenly into 4 groups then I need 20/4 = 5 people in each group. This idea of concreteness is what solidified mathematics as a necessity in civilization. But civilization today is nothing like “civilization” in 50,000 B.C. So one important question arises.

What spurs the advancement of civilization? This is easily a very complex question, however the simple answer we will use is technology. Civilizations advance as their technologies advance. Where once there were hunters and gatherers, tools and other knowledge allowed for the possibility of crop cultivation. This sort of technological innovation can be seen during every major change of civilization. At the heart of this innovation sits mathematics. But we couldn’t have gotten to civilization as we know it today just with basic geometry and counting.

So, what happened? Abstraction is an important key in the advancement of mathematics. Back to our students, as they progress through learning how numbers interact, they eventually reach the true first level of abstraction. Instead of learning how numbers interact with each other through examples, students learn how number interact with each other through ideas. It is at this point, our students have reached algebra. In algebra we focus on the ideas of how numbers interact with each other. We learn things like the distributive property, polynomials, graphing, solving equations, etc. These theorems that are developed and taught in algebra classes, show the students that numbers are predictable. Now instead treating each multiplication problem independently, they get to see that it is truly the same across the board. It is at this point our story splits and follows those students that delve further into the depths of mathematics.

Our math loving students will next make it in to calculus, where they have their first true dance with the real numbers. But the ideas of calculus are not restricted to just the real numbers as these students will find out. In fact, the real numbers are a part of a larger system (and our next level of abstraction) of metric spaces. Students begin learning the structure of metric spaces and how metric spaces interact with one another via maps. Often these ideas are imparted to the students in a mathematical analysis course. Many of the ideas of metric spaces are still not restricted to this system. In fact, we have reached the next level of abstraction, topology. Here, students are no longer concerned with numbers and distances at all. Instead they simply have a notion of “closeness” in terms of open sets. They begin working with highly abstract spaces, often those that can’t even be visualized, yet the ideas of topology allow us to assess the deeper structure of mathematics. For example, manifolds helps us identify types of spaces that would, on the small scale, “look” the same as the some product of the real numbers. This might say something about the shape of the universe itself. For example, we could be sitting in a manifold right now, but  it “looks” like 3-space on our very tiny scale. But there are problems in topology that are quite hard to solve. Namely, determining which spaces are homemorphic (can “morph” into) to each other. But there are many tools in abstract algebra that, if phrased in the right way, will be able to help. There must be a way to marry these two fields.

This is exactly what Saunders Mac Lane and Samuel Eilenberg achieved with

General Theory of Natural Equivalences, Transactions of the American Mathematical Society Vol. 58, No. 2 (Sep., 1945), pp. 231-294 (JSTOR).

Though it was not widely accepted (and in fact rejected), this article was published as seen above. Mac Lane went on to publish Category Theory for the Working Mathematician in 1971, which is widely accepted as THE introductory book for the subject. As described in an earlier post Categories and Functors, category theory looks at the most basic relationships between mathematical spaces. Many common patterns emerge as proofs in one specific setting are seen applied quite easily to the universal setting. This in effect can prove many important theorems at once.  But is this as far as abstraction takes us? Can we go farther? The answer is most certainly yes. See higher category theory. There may be no end to our abstraction as subject. One day, it might even lead us to the very nature of Mathematics itself, and thus the nature of the language of the universe. Who knows what we’ll discover!