How to explain things in the motivation section of a mathematical paper without proper definitions?

Question

I want to start a chapter in my dissertation by motivating a mathematical operator by showing why it is interesting to look at it and what I can contribute to understand it better. However, I actually need to introduce some mathematical objects in order to correctly state everything.

I think it is a rather bad Idea to start first with a section of introducing the mathematical concepts (like measure theory) and then start the actual motivation. But if I do it opposite, then I am at a loss for words.

For example, in my motivation I would need to use a additive-finite measure space, a operator, the space of mu-integrable functions and a stochastic process.

How would you suggest to cope with such a situation?

Answer 1

When I was a Ph.D. student working on my own dissertation, I went to the university writing center for help and had a revelatory experience. The person working with me sat down with the first page of my introduction and effectively dissected it to identify the problems without understanding any of my technical jargon. They did this by reading aloud as we discussed, substituting blank/nonsense words for every piece of jargon, e.g.:

Here we apply method X to determine whether adjective thingies can be made to wibble.

This type of substitution forces you to step back from the technical world that you have dedicated so much time and love to, and understand your narrative---or lack thereof.

In your motivation, you need to take a couple of steps back and ask: why does anybody care about additive-finite measure space ("frobs") and how it relates to the space of mu-integrable functions ("greebit-space") or a stochastic process ("wibbling").

You didn't pick these elements at random. There must be some reason why you picked them and how they relate to the bigger community. Are they intended to solve a puzzle that a lot of people care about? Or a small piece of such a puzzle? Do they unite two sets of concepts that people thought were different? Will they help understand string theory or give better tools for interpreting MRI imaging?

You want to be able to write something like this:

People have wondered about how to better understand frobs ever since Richard Feynman first used them to pick the locks in Los Alamos. Although X, Y, and Z attempts have been made, none of them got very far because they were all green-colored. In this dissertation, I examine an alternate path, reducing the problem of frobs to the simpler system of greebit-space by means of an innovative application of wibbling. These results bring us one step closer to solving the problem of frobs, and how they can be better used to quickly and cheaply pick locks.

Now, what I've written is pure gibberish, and your motivation will almost certainly be much longer. The point, however, is this: your goal in a motivation section is to motivate by explaining that there is a problem that people care about and that you have an approach that gives at least a piece of the solution. Explain it in a way that your jargon can just be placeholders in the reader's mind, and it will be fine to leave the complex definitions for later.

Answer 2

When I encounter this problem, I write the introduction as if the readers knew the concepts that I mention, but I include a parenthetical comment or a footnote, after such a concept, along the lines of "This and other concepts used in the introduction will be defined in Section 2."

Answer 3

If you go deeply enough into measure theory and stochastic processes to actually write your dissertation about it, it is safe to assume that readers will be familiar with common concepts. So just assume that people understand what you write about. Do some handwaving if necessary ("we examine an interesting class of operators that are distinguished in that...").

Worry less about correctness than about telling a good story. After all, this is a motivational section. Don't include any definitions, or no more than one if it is utterly necessary. (And then, if you find that a definition is necessary in an introduction section, I'd argue that you probably need to revisit what you want to write in that section, until the definition is not necessary any more.)

Worry about correctness in the main body of your chapter.

Answer 4

In addition to other good points made in the other answers, I think too often people overlook the question of the actual, likely audience/readership for a piece of technical writing. For example, it is unlikely that anyone without at least a rudimentary knowledge of your general subject would look at your thesis at all, so you can safely use the standard, basic terminology to give an introduction and overview of a given chapter. That is, it is not useful to imagine that you are explaining "from scratch" to someone who's completely unacquainted with the topic under discussion, since the reality would be that they'd not instantly assimilate "definitions" in any case.

In other words, contrary to what we sometimes may imagine, there is a context in which we write, and that context is most often richer than we acknowledge. Thus, the work is not to re-establish the basic context, but to make larger points. That is, as in the other answers, I don't want to hear delicate (and possibly pointless) semantic distinctions about word-use, but, rather, about why you are doing what you're doing, etc.

Answer 5

Mathematicians have a tendency to train to hide away they tracks they used to take to get to their goal (apologies to Simon Singh). This means that motivation is the thing they have been trained not to give. As compensation, they give examples, ranging from trivial to realistic to absurd following the definitions.

This is the situation on the ground. The reason is that mathematical objects are often obtained by so many steps of abstraction of originally natural-world concepts that their real-world origin is often obscured or very difficult to intuit (think the - very compact - definition of topology).

Therefore, it is useful to the reader to "recreate" the bridge to reality (which is often possible) and explain which of reality's features are required and which ones are discarded. Measure theory is not so bad in that respect. Basically, you are talking about a kind of "volume". In "nice" spaces, such a vector spaces, you could consider n-forms as volumes (almost literally), but if the space gets nastier, without a concept of tangent spaces and the associated structure, you have to look at which permits you to extend this concept to suitably selected subsets of your space. My favourite to asking the question what you miss if you have no measure is to respond with the Banach-Tarski paradox.

Now the game can also be played on a higher level if you talk to mathematicians who know already a lot of things. You now need to explain how your concepts will fit into what they already know. So, a group theorist may be motivated to look at semigroups by explaining which axioms you drop (and why). Or which phenomenon motivated your definition of semigroup (for instance attempting to model non-invertible operations).

In short: the point is to explain and to motivate what concepts and phenomena in "the universe of the reader" corresponds to properties discarded or generalised (abstractions) or newly studied phenomena in your universe.

Answer 6

It's a delicate balance. You say:

...in order to correctly state everything.

But why are you correctly stating everything if its just a motiational discussion? So you see you have a balancing act whereby you need to give up a little bit of space on the side of correctly stating everything in order to gain some space on the side of being able to flexibly discuss the concepts, ideas, history etc.

This is actually really hard and usually takes much more experience than it did to solve the research problem in the first place. So I think its common for e.g. a graduating PhD student to have the technical knowledge to solve the problem but to find it difficult to articulate where the problem lies within a much bigger field of inquiry.

As you gain more experience you will know when and how to lie. And you will also know much better what counts as standard. When you've just spent years learning the basics of a research field you often feel like things need definitions that don't really. Other experienced mathematicians are probably more comfortable than you think with not fully understanding every detail/remembering every definition but kind of vaguely knowing what such and such an object X is and vaguely what it does and just more or less getting the idea until the later point at which you define everytihng.

To try to give one piece of practical advice: Look for ways to not tell too big a lie. Find places you can say that 'an object X is essentially an object Y together with a parameterization of its involutions' (or whatever) where object Y is something you a sure is more standard.

One example that comes to mind from my education is distributions. I heard both of the following vagueries:

• Distributions are generalized functions.("OK right so I should think of them like functions")
• Distributions are like the abstract dual to functions. You pair a distribution with a function to get a number.

This confused me when I was younger. But after some experience I guess you know the ways in which these are both true and you get that different contexts call for different lies.

The readers who don't know the stuff well will essentially have no choice but to just swallow the lies. Then you get worried about the readers who do know the stuff well. Because then when you tell a lie, they might get offended, like "gah this writer has oversimplified and left out the crucial essence of object X; how will anyone get the important content from watered down motivational discussion!?" So like I said, it's a balancing act.