Infinite families

 This post is mostly a short complaint.  Hopefully more math coming soon now that the semester is over.  

Low-dimensional topology can be hard, especially in dimension 4.  As a result, pretty much every paper is of a "counterexample" structure (e.g. these manifolds are not diffeomorphic, this knot cannot bound a disk, etc).   Such papers are made stronger (i.e. get into better journals) if you can give an infinite family of counterexamples to the problem.  Sometimes, I really dislike this approach to papers (even though I am guilty of doing it because I still play the game).  Let me explain just a couple feelings about it.  

- Usually there's no reason that this infinite family comprises all possible counterexamples and probably does not really narrow down the possibilities more than having one example, so it is basically answering the same question.  (If it was expected this was the comprehensive list of counterexamples, then I would be stoked.)  Since the idea for the family of examples is basically the same, I feel like I am definitely more impressed with the author's technical work but that there is not as much extra mathematics added to the universe.  

- It tends to make reading the paper harder, especially for graduate students.  Rather than working through a single concrete example, indices explode and concrete objects are replaced by variable complexity.  

- This infinite list usually comes from finding a simple interesting example, and then realizing how to add one parameter of freedom in the construction.  This can make it harder to understand where that original example came from due to the increased complexity.  

- I don't uniformly think infinite families are a bad thing.  For example, there can be something really interesting coming out of the infinite family is the constructive aspect - here's a new way to cook up infinite families of knots, manifolds, etc - even if I think it doesn't add to the specific counterexample result itself.   

In my opinion, a good solution could be to have a section where one does a specific example that makes the ideas really easy to follow, and then there is another section or appendix where the general case is done.  (Referees probably wouldn't like it, but I'd even be fine to have one of these two parts be a bit less rigorous as long as the other part is.)   

I was talking with someone recently who felt completely opposite to me (which is great), so don't take this as a general sentiment in the community about infinite families.  (I think I'm probably an outlier.)  I'm curious to hear what everyone's thoughts are on this since maybe I'm missing something.  I doubt I'm risking a flame war here - please comment below!