On Abstraction and Learning in Mathematics

Posted on December 17, 2016

Goal of this article is to describe the concept of Abstraction in an understandable manner. It should cover the process of learning and display how abstraction works in empirical mode and in self-contained (mathematical) mode, as well as providing information about the pattern recognition and noticing similarities that trigger abstraction.

What is abstraction in the big picture? It is a process of creating a framework to think about a class of problems or processes. In this framework are normally included three things — objects, relations and operations.

Below we provide a motivational example for empirical abstraction.

Humans noticed that different similar objects can be grouped and counted. Sheep or horses, amount of soldiers in an army, all different, but similar and we can calculate the precise quantity of those. This gives raise to natural numbers, such as $ 1 $, $ 2 $, $ 3 $ , $ \cdots{} $. Interestingly, when armies fight and the amount of soldiers decrease due to injury or, G-d forbid, death, the quantity changes, which gives raise to the operation of subtraction. When new cattle is born, the amount of sheep in the herd increases, which gives raise to addition.

Here is the summary of how empirical abstraction works:

  1. Objects that represent generalized version of entities that we work with while dealing with a class of problems or the process we abstract over. In the example with natural numbers, entities are things that are similar to each other — sheep, soldiers, et cetera. We abstract over the type of entity and the order in which we count those, assigning numbers to groups based on result of counting entities in those. Object in this example is a number.
  2. Relations that represent how objects compare to each other. In our example, relation would be “belonging to a group of similar things”, which is an abstraction over “soldiers belonging to west army, soldiers belonging to north army, sheep in Joe’s herd”.
  3. Operations on objects which allow us to get a desired result, which is either a solution for a problem or a useful description of a process. Operations in the example we’re working with abstract over events, such as birth of cattle or injury of soldiers.

Notice, that the example is abstraction that is purely based on experience. For example, it doesn’t give raise to negative numbers (such as -5, -42), or to real numbers, such as (9¾, 13.37).

HW #1: Which experience, that we should experience daily, can give raise to the abstract notion of real numbers?

This is a perfect example of empirical abstraction, the one that was made through experience.

Mathematics for at least a couple of last centuries, however, strives to bring the abstraction approach to the absolute, becoming self-contained, disregarding relation to the physical and social world. For instance, when we’re talking about natural numbers as mathematicians, we prefer to disregard things we can do with numbers, such as counting sheep or calculating losses in our army, but instead provide a system of objects and operations in the vacuum of itself. We will discuss how can it be so useful later on.

Now let’s ponder on the concept of this different sort of abstraction a little bit more (the explanation we provide here is almost completely taken from a paper by Michael Mitchelmore et al, that surveys the proceedings in understanding and defining abstraction in mathematics and mathematical learning[2]).

  1. Mathematics uses everyday words, but their meaning is defined precisely in relation to other mathematical terms and not by their everyday meaning. Even the syntax of mathematical argument (such as proofs) is different from the syntax of everyday language and is again quite precisely defined. These ideas are demonstrated pretty well in the first chapter of “Proofs and Fundamentals” by E. D. Bloch[1].
  2. Mathematics contains objects which are unique to itself. For example, although everyday language occasionally uses symbols like x and p, objects like x^p or sqrt(-1) are unknown outside mathematics.
  3. A large part of mathematics consists of rules for operating on mathematical objects and relationships. It is important that students learn to manipulate symbols using these rules and no others. For example, if we have a category that contains as objects things that aren’t necessarily numeric, and a custom operation “+” is defined on it, we can’t use operation “×” just because we feel like addition and multiplication are closely related. We can think of operations on mathematical objects as of rules of a board game, such as the way pieces move in chess.

These three aspects of mathematics make it unfeasible, or — even stronger — impossible to construct objects, relations and operations using empirical abstraction. Instead, an axiomatic approach is used, which we will illustrate, continuing the example of natural numbers.

Here is an accurate enough description of the most famous way in which natural numbers are defined in mathematics. It is called “Peano Axioms”.

  1. $ 0 $ is a number.
  2. If $ a $ is a number, $ S(a) $ is a number.
  3. There is no such number $ a $ that $ 0 = S(a) $.
  4. $ m = n $ if and only if $ S(m) = S(n) $ “If and only if” is often written as “iff” in mathematical texts.
  5. If a set $ I $ contains $ 0 $ and also successor of every number in $ I $ then $ I $ contains all numbers.

From those axioms, we can formally derive true statements. Those are called “lemmas” or “theorems” and the derivation is called “a proof”.

For instance, based on this list of axioms, we can prove that if $ m $ and $ n $ are equal then successor of successor of $ n $ is equal to successor of successor of $ m $.

HW#2 (optional): Prove it.

If we will run around, making an axiomatic out of every interesting abstraction, we’ll quickly get tangled with a bunch of different incompatible chunks of mathematics. However, we would like to have a way to develop axioms and prove statements with those (or, in other words, develop theories) completely independently. This is why in mathematics there is a notion of foundations. A foundation is a special bunch of axioms that provide mathematician with everything necessary to unify their new shiny theory with the work of other mathematicians on their shiny theories. The result of a process of specifying how exactly does a theory map onto a foundation is called “a model”. For example, a model of the Peano axioms is a 3-tuple $ (N, 0, S) $, where $ N $ is an infinite set, $ 0 \in N $, and $ S: N \to N $ satisfies Peano Axioms. This maps well to foundations such as ZF set theory.

Even though mathematics is developed as a thing in itself, it is useful.

Mathematics allows us to carry out computations. Starting from calculating change in a shop, ending in solving equations for some variable given parameters. For example, using calculations we can decide how many stops can we afford to make in order to travel from one country in Europe to another under a certain amount of time.

Mathematics is a powerful, if not the only proper tool for building models. Models are never 100%-precise, but useful to predict outcomes of processes in reality. For instance, we can have a model of a bet winning given historical data which is inherently probabilistic, or we can have a model of a rocket vertically landing on a launchpad, so that we don’t have to construct expensive rockets to check their properties.

Another important application of mathematics is reasoning. Example of mathematically-powered reasoning can be found in computer programming, where it allows us to extract interesting information about our programs. The kinds of this information are vast: starting from complexity analysis, ranging over simple type systems and type systems with dependent types (where you can write type-level programs), to complex and featureful proof assistants to generate proofs about the properties of software. Mathematical logic also might help in mundane reasoning.

Theories such as category theory show correspondence between different fields of knowledge, pointing how they map to each other, making results from one field applicable to another field.

Finally, let’s formulate informally the answer to the question “how to learn mathematics?”.

In order to learn mathematics, we have to see patterns and similarities in a problem domain. Often it is hard to see those, so we can use guided approach, where those similarities are suggested by an educator. As we are sure that those similarities aren’t noise, but are indeed characteristic to a given problem domain, we empirically abstract over those. Here we as students should try our best to come up with abstractions ourselves, keeping in mind the problems we want to solve. If we are studying with an educator, an educator should criticize our abstractions and point out some things that we might have missed. At the end of this back-and-forth, we should end up with a class of good-enough abstractions for the problem domain that help us solve practical problems. Next step is to play with those empirical abstractions and build up an intuition about how they work. With the intuition in place, we can now learn corresponding mathematical abstractions. This process will be reinforced with developed intuition, hence it should be relatively simple to fit mathematical abstractions in our head (simple does not necessarily mean easy!). In order to verify success in learning of mathematical abstractions, we should periodically solve exercises and mathematical problems. Now that we have mastered a mathematical theory or its part, vast opportunities lie as we proceed. We can decide to use it in practice, we can see how this theory correlates with other theories, check if there are generalizations of it. Most importantly though, have fun while learning and using your knowledge!

References:

[1] E.D.Bloch: Proofs and Fundamentals — http://www.unalmed.edu.co/~cemejia/doc/bloch.pdf

[2] Michael Mitchelmore et al: Abstraction In Mathematics and Mathematics Learning — http://www.emis.de/proceedings/PME28/RR/RR031_Mitchelmore.pdf