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When we define an abstract structure, establish theories or theorems, or build models, etc., we tend to want them to be "general" enough so they can be applied to a variety of situations. However, of course, more generality is not always better. Once generality passes a certain threshold, things become vague and useless.

Of course, we could treat this on a case-by-case, topic-by-topic basis. However, I wonder, is there or can there ever be a generalized / systematic approach to decide the optimal level of generality?

Here is an example in math. The following paragraph is from Kreyszig's Functional Analysis on how to generalize the concept of "distance",

In functional analysis we shall study more general "spaces" and "functions" defined on them. We arrive at a sufficiently general and flexible concept of a "space" as follows. We replace the set of real numbers underlying R on an abstract set X (set of elements whose nature is left unspecified) and introduce on X a "distance function" which has only a few of the most fundamental properties of the distance function on R. But what do we mean by "most fundamental"? This question is far from being trivial. In fact, the choice and formulation of axioms in a definition always needs experience, familiarity with practical problems and a clear idea of the goal to be reached. In the present case, a development of over sixty years has led to the following concept which is basic and very useful in functional analysis and its applications.

1.1-1 Definition (Metric space, metric). A metric space is a pair (X, d), where X is a set and d is a metric on X (or distance function on X), that is, a function defined on X × X such that for all x, y, z ∈ X we have:

(M1) d is real-valued, finite, and nonnegative
(M2) d(x, y) = 0 if and only if x = y
(M3) d(x, y) = d(y, x)     [Symmetry]
(M4) d(x, y) ≤ d(x, z) + d(z, y)     [Triangle Inequality]

Erin
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  • It is ironic that you want a general rule to "optimize" generality without "optimizing" the generality of such a rule. When we define a useful abstract structure it is only after many of its instances came up in different applications. The "rule" for introducing abstractions is that there are similarities in reasoning across them and non-trivial parts of it can be reproduced uniformly in abstraction. This has to clarify it by dispensing with inessential concrete details, and lead to new non-trivial results to be useful. And there can be multiple successful variants, so no "optimization". – Conifold Mar 29 '21 at 06:31
  • No. Each organism does its best, and nature selects who remains alive. This applies to any possible level of instinct and reason. Nature has no rules, we create and attribute them. – RodolfoAP Nov 29 '23 at 11:20

2 Answers2

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The 'optimal' level of generality is the one satisfying the following constraints:

  • it enables you to reach a conceptual goal, such as a result, a theorem or a desirable property and
  • it includes the example(s) which you want that goal to apply to.

Suppose your conceptual goal is finding the solutions of a polynomial. A physicist will likely only care about solutions in the complex numbers, while a number theorist might wonder about solutions in other fields or rings. The number theorist will work at a higher level of generality, but will have to do more work to find out when solutions even exist; the physicist can rest easy thanks to the Fundamental Theorem of Algebra. [I apologise that my examples are all extremely mathematical, but hopefully they convey what I am hoping to convey.]

There is not a context-independent optimal level of generality, but with the context fixed there are criteria that one can apply to evaluate what the right level is:

  • Efficiency: finding common features of your examples is necessary to finding a generalization, but not all of the common features are necessarily relevant to your goal. Extraneous features can be eliminated.
  • Occam's razor: in tension with the above, simpler features may be preferable. The extreme of efficiency would be to identify precise necessary and sufficient conditions for your conceptual goal to apply, but perhaps your intended examples have common features that are easier to state that are sufficient for your goal!
  • Conceptual clarity: This is a somewhat aesthetic feature, but it is important. A generalization which only serves to artificially tie some examples together without isolating the concepts which make reaching the goal possible is rarely an "optimal" one. Getting this right may involve a novel perspective on your examples.

All of the above applies to finding the right level of generality "locally", such as in the context of a single book (or book chapter) or university course. In that sort of setting, the level of generality is a choice that one must make; shifting levels can be disorienting, so it is best avoided or only done in a very controlled way, which restricts how much we can systematize.

If your setting is broad enough, however, you might want to be interested in shifting between levels. Returning to algebra, a first course in Commutative Algebra focusses on the study of commutative rings. Rings were conceived as a generalization of the collection integers $\mathbb{Z}$: they come equipped with addition, subtraction and multiplication operations. Key examples include the "ring of integers modulo k" which are the focus of modular arithmetic, the real numbers, the complex numbers, the Gaussian integers...

From this starting point, one is led to wonder how much like the integers a ring is. We use additional properties of the integers to define classes of rings:

  • If we have integers x and y such that x.y = 0, then x = 0 or y = 0. A ring with this property is called an integral domain. We can prove further properties such as cancellability using just the fact that a ring is an integral domain.
  • We can uniquely (up to multiplying by -1) factor an integer into powers of prime numbers. A ring with this property is called a unique factorization domain. We can show that every such is a principal ideal domain...

The details of this example are not important. Rather, I want to highlight the structure of this area of study: we have identified a broad category* of structures of interest (rings) by abstraction of one or more structure of specific interest (the integers). From this starting point, we adjust the level of generality by imposing further properties of our generative example(s). A conceptual goal in the form of a desired property determines a level of generality: a specific subcategory of these structures. It may not be easy to show that a given example (of a ring) is a member of this subcategory. As such, we may need to build a chain of properties, each implying the next, to build up to proving membership. Any point in this chain constitutes a level of generality that one could have taken as a starting point, and we should keep them around because once we move onto another conceptual goal we may be able to use one of them as our starting point!

*The word "category" here is deliberate, although the relevance of category theory for systematizing generalization is subtle and deserves a more nuanced discussion than I can sensibly provide here.

Morgan Rogers
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In your case above, the goal is clear which is to require a concept (definition) only containing the most fundamental properties of a general space associated with a general real-valued function in your studied domain of functional analysis. This leads to the most basic fundamental definition of "metric space". So your focused goal is a major yardstick to determine the optimal level of generality, of course your own domain knowledge and philosophizing skills are other factors too. However, in reality, to even make clear sense of your goal to generalize within a relatively young research domain may be extremely obscure and hard, such as abstract geometry, category theory, etc. That's why the above author said it took mathematicians over sixty years to arrive at this "correct" foundational concept in functional analysis.

Further upon your case, there may be more general and non-trivial goal worth to pursue, for example, in real analysis you can easily form a similar but more general concept - topological space, which can arise out of any metric space. However, there are non-metrizable topological spaces and thus it's more basic compared to metric space.

Btw, regarding your "Once generality passes a certain threshold, things become vague and useless", it may not be so for more generality (actually I only form a propositional attitude of the opposite). For example, classic syllogism is extremely general, but it's also extremely clear and useful - being applied all the time everywhere...

Double Knot
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