I am thinking on posting two "big-list" questions and I would like to discuss them first here.

1) Generalized notions of solutions in various areas of mathematics

In many areas of mathematics (PDE, Algebra, combinatorics, geometry) when we have difficulty in coming with a solution to a problem we consider various notions of "generalized solutions". (There are also other reasons to generalize the notion of a solution in various contexts.) I would like to collect a list of "generalized solutions" concepts in various areas, hoping that looking at these various concepts side-by side can be useful and interesting.

Let me demonstrate what I mean by an example from graph theory: A perfect matching in a graph is a set of disjoint edges such that every vertex is included in precisely one edge. A fractional perfect matching is an assignment of non negative weights to the edges so that for every vertex, the sum of weights is 1. In combinatorics, moving from a notion described by a 0-1 solution for a linear programming problem to the solution over the reals is called LP relaxation of a problem and it is quite important in various contexts.


2) Successful heuristic arguments in mathematics.

In mathematics there are quite a few things that we know without being able to prove and there are various heuristic arguments and heuristic computations that are valuable but not rigorous. Give example of a heuristic mathematical argument/calculation.

A few comments regarding the big list issue in the context of these questions. Big list are indeed side show of MO like survey articles (compared to research articles) are side show in published mathematics. But big list questions like survey articles can be valuable. (And the main issue is their academic merit.) Presenting such questions over MO rather than on a blog has several advantages: First MO attracts much more attention and questions in MO attracts much more responses than posts in blogs. A more important (related) reason is that in MO the answers are at least as important (and perhaps even more important) then the questions. (And the graphic of MO supports that.) This is not the case for comments in blogs, including successful blogs that attract many comments.

In order to be able to pay attention to answers (and also in order not to contribute to proliferation of big list questions) I plan to ask problem 1 in October/November and problem 2 in April or so. (I still want to put some work in my old example question. (To make it a more useful source I would like to add a designated answer to collect relations between the various examples.)

Of course, this post is NOT meant to be a recommendation to people to discuss first on meta questions they plan to post. (This may have a chilling effect on questions and on answers.)

@Gil, For #2 I would suggest mentioning the Poisson clumping heuristic (if you don't, then I will answer with it).

Gil,

I have a general interest in this. The study of heuristics, "methods of discovery", whatever you want to call them, is linked to the "problem of abduction" and non-demonstrative or synthetic reasoning in general, for instance, as it comes up in pragmatic approaches to inquiry. Peirce, in particular, not only made a study of abductive and inductive reasoning in relation to deductive inference but employed a number of very general heuristics all throughout his logical and mathematical work. I don't see much chance of exploring those issues on the main board, or even on this "meta"-board, but it might be possible to collect a few interested parties on one or another of the wikis that I know about.

Jon Awbrey