Question

Apart from their technical statement and proof, a usual presentation of theorems is by leading up to them with a definite motivation or intuition, for example putting the results in the wider context of a research programme. This focus gives the author the ability to distinguish between "critical" and "technical" assumptions. By critical assumptions I mean the ones that are crucial to making the proof idea work in the first place. Technical assumptions are then the ones that just happen to be needed to straighten the details out.

For example, from this point of view, a topological space being compact Hausdorff could be said to be a `mere' technical assumption in Gelfand duality, but a critical assumption in Urysohn's lemma.

For another example, an irate referee once taught me not to speak about "mild assumptions", because "under mild assumptions every group is a ring".

But the line between technical and critical assumptions is vague and flexible at best. This question is about when the line moves.

Are there interesting examples of critical assumptions that later (e.g. via a new proof for a generalized setting) turn out to be technical? And, more interestingly, vice versa, are there good examples of technical assumptions that later (e.g. with a different motivation) turn out to be critical?

Answer 1

There is a certain division of labor, that makes technicality mainly an issue of perspective. Probability theory is a very good example of this. There are, on the one hand, people refining the existing major tools for the construction of regular conditional probabilities, laws of large numbers etc. and end users that employ these tools in their work. Of course, the intersection of these groups is nonempty in reality.

An end user may simply need the fact that a law of large numbers applies to a process, so she assumes the process to be iid. The assumption is purely technical, in that a lot of other, often much weaker, assumptions would have done the trick. But how the law of large numbers works is not central to the argument she makes.

But a toolmaker might well try to get a law of large numbers under weaker assumptions than the ones known to date. For her, the fine details of the assumptions are critical.