When we learn math, a typical approach is the following. We discuss some motivation, based on what we have already done. This motivation leads, in a possibly informal way, to a formal definition. (It could also be a set of formal axioms, but here we encompass that, as well as any other type of formal “framing,” under the term “definition.”) We then work with our definition using formal logic to produce theorems.
This can be a great approach. However, there are important caveats that we must keep in mind, which I call attention to in this post.
Math is, at its heart, what I like to call a “formally logical art form.” We construct concepts and definitions from all sorts of inspirations. Sometimes, a definition could follow logically from some invariance result. Other times, it could come from some connection to an already known concept, even though it may not lead to results immediately for this connected concept. And still other times, it may not be born out of a clear connection at all. All these possibilities exist and are present, and the diversity and complexity of them all is what gives mathematics its breathtaking nature.
Thus, we need to be very careful in delineating how much motivation goes into a definition and what the type or the nature of the motivation is. We should do this in order to set the reader’s expectations appropriately. We should include as much motivation as makes sense for the concept that we want to lead up to, but we should not expect that level of motivation to be uniform for every concept we start studying.
This setting of expectations is crucial. These expectations in turn can connect to implications of the neurobiological structures of our brains. However, at the time of writing, I do not believe I understand those structures well enough to make further comments on these ideas.
These expectations also connect to the fact that the process of leading from motivation to formal definition may very well be informal, and therefore may not have a singular “right answer.” For example, a particular intuition could have multiple possible formalizations. Mathematicians choose one of them, where this choice may be influenced by what was published historically as well as many other possible factors. We may assign this definition a term used in informal language, but where the term is now given a technical meaning. But there is no guarantee that this had to be the “only” definition of that term, and it can be misleading to imply that to students.
In fact, it can be a great experience for the students to investigate their own formalizations of their intuition, and how those may logically connect to the “standard” formalizations. This can be an outlet for their creativity, and beyond that it could even result in new research!
My personal experiences have informed these points. Earlier, when self-studying math subjects, I would give myself a “theoretical check.” This was a self-challenge to come up with the definitions and derive the standard theorems of the subject myself. One of the subjects I tried this for was multivariable calculus. I attempted initially to come up with the definitions by asking myself how I would appropriately generalize the single-variable concepts. However, my definitions were different than the standard ones that we accept! I saw that there are many possible ways to generalize the single-variable ideas, some of which may not even be part of the “standard” subject of multivariable calculus. Thus, the choices of which definitions were chosen for the standard subject may involve many factors, and some aspects of these choices could very well be arbitrary.
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EDIT (11/08/2023): I still agree with the general ideas discussed in this post, although now I would rephrase them slightly.
To provide more context on this: in general, while I would still consider the study of any logical system to be within the scope of math, ultimately the systems we do consider tend to be motivated by some connection to concepts already being studied. I further think this motivation is often essential for a student’s ability to “wrap their head around” a concept; for example, I remember finding it hard myself to break into topology until I wrote out the proofs that arbitrary unions and finite intersections of real open sets are open. As a result, I would say that every new concept has some “motivating thread” from existing concepts, and that’s the thread we should follow to teach the concept most effectively.
Of course, a particular set of existing concepts can motivate many different new concepts, if we follow different such “threads.” For example, this is the case with multivariable calculus, as I had already pointed out, where there are multiple different generalizations of the single-variable derivative and integral to the multivariable setting. And I do fully agree with one point I made earlier, concerning terminology: we often appropriate non-technical terms in math and assign them technical meanings, but we need to be really careful and avoid asserting that that was the only possible formalization of the informal idea. That is in fact often not true.
