A friend asked the following question:

Describe the contrast between the abstract and the concrete [in mathematics], citing examples. Some have claimed that the notion of abstraction is itself flawed. Argue either for or against this position.”

Here’s my take on it.

One example comes from modeling random situations, for example when using the Normal distribution. We really like it cause it’s this continuous, unbounded, distribution that’s central to our study of randomness. However, it also an abstract tool; due to physical constraints, real life distributions would be bounded, and even discrete, not continuous! This is one of those situations that abstraction actually adds value instead of subtracting, as modeling with a Normal distribution avoids all the different uncertainties we would have to quantify otherwise (what are the possible events? where does our “real” distribution stop?, etc., etc.).

IMO, one of the big strengths of mathematical knowledge lies in stripping away the superfluous, the unnecessary, the “natural” part of the problem, and leaving behind the source of the issue which, once resolved, can not only be tied back to the original example, but to any example that leads to the same issue! After all, from trying (and failing) to solve the quintic equation, the abstraction known as group theory was born by Galois, which lead to results regarding any polynomial equation!

One could argue that abstraction is also what makes mathematical knowledge difficult for students to assimilate. However, this is not a fault of the abstraction per se but instead a problem of how mathematics is sometimes taught and more often written. It’s rare for mathematicians to resolve to abstractions from the get-go—certainly not early on in research—and when they do, it’s the thing they jot down last. Strangely enough, when the time comes to publish the findings, it is the abstract result that is of interest first, and thus we tend to present the train of thought backwards; theorem-motivation-example instead of motivation-example-theorem.