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Suppose I am writing an article or a book wherein I wish to introduce a mathematical object O. There is an intuitive image I have in mind when thinking about object O, and this intuitive image lends itself to an obvious sounding definition.

However, upon playing around with the obvious-sounding definition, it becomes clear that there is a subtlety that does not seem apparent at first glance, or a technical point that makes proofs more burdensome than they could be. In order to fix it, I must mangle the original definition to something less obvious.

For example, suppose I am introducing a reader to the notion of product topology. The obvious definition for a topology on a cartesian product is actually the definition for the box topology; the notion of what we now call product topology traces its roots to Tychonoff in 1935. As such, when introducing the product topology, it might be fruitful to begin with the obvious definition, but, after running into issues dealing with infinite products, back up a bit and think about what is actually important about our topology (projections are continuous), what properties we would like our topology to have (pointwise convergence implies convergence in the product topology), and how we could mangle our definition to make it have these properties.

Of course, doing it in this order is problematic, because it means that some pages of work are rendered useless and the writer must now either rewrite them in this new context, or handwave and assert that the proofs are "adaptable without much effort". On the other hand, I feel like it might be a worthwhile sacrifice, as those wasted pages might be crucial in understanding why the definition was made this way.

How would one handle this tradeoff nicely?

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  • $\begingroup$ This is a question of expository style. Often what is appropriate for good teaching is not appropriate for research articles, and vice versa. Providing motivation for a technical definition by indicating a problem when the technicality is ignored can be very instructive. $\endgroup$
    – Ted Shifrin
    Jan 12, 2020 at 19:02
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    $\begingroup$ I would recommend looking at how actual textbooks (i.e., Munkres) discuss box/product topology. Munkres, for example, discusses finite cartesian products first, and discusses the box topology, and then discusses the infinite cartesian product box topology without defining it as the product topology. After showing why it isn't particularly useful, the product topology is introduced. $\endgroup$
    – Rushabh Mehta
    Jan 12, 2020 at 19:02

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Never write something incorrect, even if you're going to correct the error in the next paragraph. The problem is that someone may just be looking for the definition of $O$, and stop reading your work after finding the incorrect definition.

If you want to introduce a "provisional" definition of $O$, you might give it a different name (pseudo-$O$?).

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    $\begingroup$ +1. To complement this, note that you can always introduce new headers like "Wrong definition $2.1$" or "Naive definition $3.4$." It's very useful to clearly present ideas that don't work, but "clearly" there applies both to the idea and the fact that it doesn't work: basically, always work to prevent possible confusion. (And always assume your reader is fairly lazy!) $\endgroup$
    – Noah Schweber
    Jan 12, 2020 at 19:15

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