The Unreasonable Effectiveness of Definitions in Mathematics

Recently, I have been thinking about the unreasonable effectiveness of definitions in mathematics. Without making any additional assumptions or knowing any additional information about a situation, by simply making judicious definitions and studying their theory, we can get a lot farther than we could without those definitions. (As you might have noticed, “unreasonable” here is a nod to Eugene Wigner’s classic piece.) Upon further thought, it seems somewhat baffling that this works so well, especially in certain cases.

To clarify further, there are a couple main kinds of definitions that we can make in a given situation. Some of them are not as baffling as to why they’d be so productive, while others are:

1. Abstraction: this is not surprising in its utility. The idea that, for example, we can abstract the properties of the rational numbers to a field, and then prove theorems for fields that can be applied multiple times in different contexts and ways, is not surprising, even when it comes to ultimately establishing results about rational numbers themselves (e.g., in algebraic number theory.) There are, for example, many appearances of fields beyond just the rational numbers themselves that are nevertheless connected to the rational numbers: for example, the finite fields, and with them connections to integers, modular arithmetic, and the rational numbers. That abstracted definitions yield additional value is not baffling, even though it is still exciting and incredible.
2. Constructions on top of given objects: This, on the other hand, is very surprising, and it warrants additional exploration. As an example, consider analytic number theory. For a very specific example, take the Prime Number Theorem. We can state everything contained in the Prime Number Theorem purely in terms of real numbers. The limit, written out, can become: “for any real , there exists real such that …,” and so on. (Incidentally, the limit is why the Prime Number Theorem can’t be formalized in just the integers. It also can’t be formalized in just the rationals: even though we can write “for any rational there exists rational such that …,” we still can’t formalize the logarithm without real numbers, at least with standard definitions.) However, for a while the best known proofs of this result used complex analysis. There was nothing in the statement of the Prime Number Theorem that assumed anything about structures beyond the real numbers, yet structures constructed on top of the real numbers (like complex numbers) proved to be invaluable in yielding this result. In some sense, this meant that all the “data” for implying the Prime Number Theorem was “already contained within” the real numbers. Why then was complex analysis so useful? What did complex numbers “add” to things?

In general, this is something I’ve been wondering: when a theorem is asserted for a particular object , and we prove it by performing constructions on top of , what exactly are these constructions “giving” us? There are no additional assumptions made, and all the “data” for the theorem to hold must already be contained in .

There are many examples of this phenomenon in math, and I think it is worth digging into these cases further to understand exactly what “data” these constructions are “pulling out.” Take representation theory, for example. Many results about say finite groups are much harder to prove without representations, but again the only assumptions being made are about the groups, not about any representations (which are just intermediaries.) What exactly is happening here?

Well, in the case of representations, we can say that the definition and theory of vector spaces constitutes a case of the first kind of definition (abstraction): we’re just applying that theory to a certain subset of group data. In other words, we can say that a certain subset of group data actually has analogous properties to vectors, and abstract linear algebra and representation theory is just exposing this fundamental fact.

Similarly, we can say that complex analysis just exposes the special properties of certain kinds of real functions — for example, complex-differentiable functions correspond exactly to real harmonic functions. Thus, complex analysis can just be seen as different notation and a proxy for picking out and using real harmonic functions. (Along with whatever other real functions the various other concepts of complex analysis would correspond to.)

So in some sense, every “new” definition done on top of the object we’re studying is just a proxy for picking out certain data about the object and possibly applying the first kind of definition (abstraction.)

It is still valuable then, as a line of research, to look more into what exactly the myriad constructions in math are “picking out” and which abstract theories are being applied to the “pickings.” For example, this has been done more with complex analysis, with researchers noting the connections between complex differentiable and real harmonic functions, but this is a useful thing to study in general whenever we introduce new constructions or auxiliary definitions.