I’ve recently been rethinking my approach to teaching abstraction in math. The approach I used to take — which I also think is the dominant one amongst math teachers, based on what I’ve often seen (not based on a scientifically analyzed comparison per se, but still seemingly so anecdotally and experientially) — is to present abstraction as enabling greater applicability: if I identify a few properties obeyed by the concrete system currently under my study and look at just the consequences of those properties, I’m now much more powerful, since my conclusions apply to any system that satisfies those properties. But if someone protests, “what is the use? what other system would we apply this to?”, the default response would be to label this as a lack of skill/imagination. We say that abstraction is an important skill for pursuing mathematics at a higher level, and you just need to have the imagination to see ahead that systems could exist that that abstract theory yields fruit for, without needing specific examples right now.
I do still believe that some amount of greater imagination in this vein is necessary to succeed in and truly avail of the fruits of mathematics, and that this is an important skill for people in general to develop regardless of whether they go on to actually be mathematicians. But that is a topic for a different post. With regards to abstraction, I now think a more effective way to justify it is based on clarity: the reason why we isolate certain properties is just to help us see what behavior is truly dependent on what characteristics. With a whole soup of functionality swirling around in a particular concrete system, this helps us truly “pin things down” and gain a greater conceptual understanding in the process. This is aligned with programs like reverse mathematics, which are “natural” in their desire to simply gain greater understanding, regardless of whether the increased abstraction is actually applied later. And the greater clarity has “real” applications not in the math itself, but in our cognitive understanding of it, as math is ultimately performed by humans. (And AI agents these days, but still in a way that was constructed by humans in the first place — but I digress. My thoughts on AI for math are best left to another post.)
As a concrete example, take model theory. I’ve often thought about what would be the best ways to teach various concepts in model theory, and earlier I would have emphasized the greater abstraction (for example, topics like the general definitions of homomorphism and isomorphism for a first-order structure) as enabling greater applicability. But while this is technically true, if I really look back at it, look back at my own model theory-oriented projects (like prototype model theory), that’s not truly the reason I even wanted to go into model theory in the first place. Instead, the reason was the clarity gained by working at that level and stripping away all the details that don’t matter in that context. When we see so many different instances of the concept of homomorphism or isomorphism in different abstract structures, it is a natural desire to pin down a more general formulation of the concept to really “capture” what it means, regardless of whether I actually have another system in mind I want to apply that general definition to right after.
This connects to philosophy too: why do humans study the subjects in math they choose to study? Now, there are multiple reasons and factors here, and part of the breadth and diversity of math comes from the fact that different topics can be reached and studied from a variety of underlying philosophical motivations. For example, historically, some abstractions did in fact arise from mathematicians seeing multiple examples for them, and it makes sense that some of these examples are just hard to cover in an introductory course where you still want to discuss the abstract theory to prepare the student better for later, so you would then just ask the student to suspend their disbelief. But another important reason, and one that I now think should be emphasized more pedagogically, is simply clarity, regardless of greater applicability. Part of the motivation for Bourbaki’s program, as well as other abstraction-oriented programs, can be seen as providing greater clarity on ideas rather than true application-enabling setup, and maybe presenting subjects with this motivation can help students for the same reasons that these mathematicians sought that very sort of presentation themselves.