The notions of abstract and concrete change with time, and with one’s level of experience. As the picture above indicates, what seems to be *abstract* to people at one stage of their experience, is very concrete at another.

What does it really mean for one thing to be more “abstract” than another? As the examples above illustrate, the abstract thing deals with something more generic, more unknown, and more flexible. Let’s look at the examples shown here.

Kindergarten time:

*Concrete:* A picture of five people

*Abstract:* The notion of five

A picture of five people, after all, is just that — a picture of five people. But the number five carries with it a much greater richness of possible interpretation. Five could refer to five people, five boats, five senses, five birds. It could refer to the five fingers of the hand (including the thumb). It could refer to five as a quantitative measure (for instance, the volume of the jug is five times the volume of the cup). It could refer to five as an ordinal: I ended fifth in the horserace.

*Concrete*: 3 + 7 = 10

*Abstract*:

In the middle school example, the difference between the concrete and the abstract is less pronounced. Here, the abstract represents not so much a leap in generality as a leap in ignorance. While the concrete equation has only known quantities figuring in it (3, 7, 10), the abstract equation involves an unknown quantity, that we’ve denoted by . Abstraction (which, at this stage, is introduced with the word *algebra*) is the tool which allows us to talk of the unknown, without fearing it.

In high school:

*Concrete:*

*Abstract: *

*Concrete:*

This level of abstraction is akin to that in middle school. At the middle school stage, the idea of using variables for unknown, or arbitrary, numbers, is already well-established. But the idea of having *functions* as *unknown* quantities to be solved for, is still new, and somewhat puzzling.<

*What do these examples show?*

Abstraction is often introduced to unify existing concrete ideas, and to allow for the possibility of dealing with existing concrete ideas in a general fashion. This has the advantage of allowing us to *solve* concrete problems. Without the abstract notion of “five”, it is hard to systematically count a collection of objects and confirm that they are five in number, Without the abstract idea of an arbitrary unknown number, and the abstract study of how to manipulate equations involving unknowns, it would be hard to create systematic and general procedures for finding numbers satisfying certain equations. Similarly, without a general theory of functions, and an abstract study of how to manipulate general conditions on a function (as the theory of differential equations provides) it is hard to compute a specific function arising in a concrete situation, based on general conditions.

But there is something more to abstraction than simply putting a label on a general idea. That “something more” is figuring out general rules and laws of manipulation. Without such general laws, there’s little advantage in giving a generic all-encompassing name to everything. Algebra isn’t just about the use of the symbol for a variable: it is about the fact that there are general rules that hold for any . These general rules include commutativity and associativity of addition and multiplication, the distributive laws, the properties of zero and one with respect to addition and multiplication, and so on.

The importance of this point is often overlooked in general discussions of abstract versus concrete. Abstractness is often confused with the overuse of symbols, the absence of examples, or being in general hard to comprehend. “Abstract” is often confused with “abstruse” and people often indicate their difficulty in understanding something by saying *It’s just too abstract*.

In fact, symbol use, level of difficulty, and absence of examples have little to do with abstractness. Abstraction can be viewed as the art of identifying general and common patterns across similar objects (for instance, across all numbers, across all nice functions, across all groups, across all measure spaces). Sometimes, these common patterns are useful in solving concrete problems (for instance, solving an equation, solving a differential equation, finding a group or ring subject to certain constraints). At other times, it may give a feel for how objects in general behave, which may in itself be useful.

Symbol use has little to do with this. Saying that the total number of words in a document is isn’t abstraction. The abstraction lies in the act of identifying the “total number of words in the document” as a number, despite the number being unknown, and possibly subject to change. Using a single letter for this is merely a matter of notational convenience. In some contexts, this notational convenience is useful — for instance, if a person plans to solve equations or manipulate expressions involving the total number of words in the document, then using the letter may be more convenient than writing “total number of words in the document”.

In fact, there are contexts where the use of letters as variables is discouraged. One such context is computer programming, where variable names need to be chosen to be reflective of what they are representing. That’s so that different programmers can understand what a given variable name was for. So the total number of words in the document may be called “WordCount” or “NumWordsInDoc” rather than the uninformative .

The fact that, in a diagram of supply and demand curves, price is denoted by and quantity by , is completely incidental and irrelevant. The real abstraction lies in the conversion of the way people respond to economic incentives, to a property on the graph: namely, the demand drops as the price increases, while supply increases as price increases. The specific formula that relates price and demand (if such a formula exists) is also not as relevant. In fact, a specific and artificially imposed formula (for instance, saying that the demand is inversely proportional to price) goes against the grain of abstraction — because it puts numbers where they didn’t exist.

Mathematics does rely a lot on symbols, and sometimes, good symbols can aid in abstraction, as they allow for compacter and more revealing forms of expression of mathematical truths. However, a “symbol-free” way of expressing an idea carries with it its own power. Symbol-free expressions use natural language, and the connecting techniques of natural language, to convey the same idea in a more memorable fashion. For instance, saying that a “product of Hausdorff spaces is Hausdorff” is a more compact and expressive statement than “if are Hausdorff, so is their product “.

Next, does abstraction necessarily conflict with the ideal of “examples”? To consider this, we should consider what it means to give an “example”. An example should be something illustrative of an idea, but it should not have distracting features that make one think it is about another idea. Thus, for example, if one needs to describe what an American person is, then a list of people like Bill Gates, Steve Jobs, Steve Ballmer, is poor on the example front. All these people aren’t just American. They share a number of other similarities — they are rich, they own and run huge multibillion dollar technology-based enterprises.

Examples are not magic pills that can cure a boring definition or abstraction. Ironically, a number of educations seem to champion the use of examples, occasionally to the point of absurdity. For instance, in middle school, we were urged to give at least two examples whenever asked to define a term in an examination. But as pointed above, the real power of examples lies when they are used to illustrate the central idea, and when a sufficiently broad range of representative examples is chosen, which differ in other important respects.

This suggests that, rather than think of examples, we should think in terms of highlighting the common and crucial features behind the idea; features that might be brought out through a combination of existing examples, hypothetical examples, non-examples, and “abstract” definitions. At its core, an abstract definition is something that, once properly read and understood, gives *every* example and non-example.

Here’s an example. Suppose I gave you the sequence 2, 6, 20, 70, 252, … and asked you to fill it in. Unless you’ve had some experience with combinatorics, it is unlikely that you’ll guess where this sequence is headed. The “example” terms of the sequence don’t describe where it goes. But now here’s the abstract rule:

The term of the sequence is the number of ways of writing a string of length with an equal number of 0s and 1s.

Now, if you haven’t studied combinatorics, this definition may still not give you an insight into how the numbers grow. At least, though, you can in principle compute any term. If you’ve seen some combinatorics, you’ll identify the formula for this as . The formula then tells more: the sequence grows exponentially, roughly like . Looking at the sequence without knowing the formula may also have led you to a suspicion that it grows exponentially, but the formula gives a precise number — something that is simultaneously more abstract and more concrete.

The art of abstracting, “laying bare the essential features”, is closely allied with the art of choosing and selecting good and representative examples. Some educators prefer to first lay out an abstract definition, then guide students through representative examples, or let students figure out the representative examples. Some educator prefer to guide the students through representative examples, and let the students abstract out the definitions. Both approaches have their power and their place. Educators who want to be bad and boring can succeed in this equally well by giving abstractions and by giving examples.

Finally, is abstract stuff inherently more difficult to understand than concrete stuff? This is probably true, but it’s equally well true that one can never get to grips with certain aspects of the concrete stuff unless one has understood some of the abstract stuff behind it. A lot of the mathematical giants before the twentieth century (such as Poincare, trying to develop topology), who did great work, were hampered in their concrete stuff because some of the abstract frameworks underlying their stuff hadn’t been developed.

But the biggest and hardest abstractions that we need to make, were the ones we made in kindergarten — identifying the notion of a number. The leap in abstraction from collections of size five to the notion of five, is considerably bigger than the leap from a known number to an unknown variable. That said, the leap from arithmetic to algebra is probably much bigger than most of the leaps people are expected to take in high school, college and graduate school. Learning how to count is like learning how to walk; hard when one first tries, but simple and natural by the time we reach adulthood.

Next time you catch yourself calling something too “abstract”, or wishing there was more “concrete” stuff to mathematics, stop yourself and ask yourself: okay, what is the *real* problem with the stuff? Does it really use abstraction in a bad way? Does the problem lie with your perceptions, or expectations? Or is the problem withthe material really in a totally different direction, that you’ve confusedly lumped with “abstractness”?