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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?

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closed as not a real question by Felipe Voloch, Martin Brandenburg, Bruce Westbury, Ryan Budney, Andrés E. Caicedo Nov 15 '11 at 16:37

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

Going beyond mere technical assumptions, the following question addresses a very related concern:… – Suvrit Nov 14 '11 at 13:02
I'm sorry, but I don't understand the difference between something that is "crucial to making the proof idea work in the first place" and the ones "that just happen to be needed to straighten the details out". It seems to me that both of these categories are essential for the proof to work. Until you have a proof that shows that a technical assumption is not really necessary, it remains crucial. Once you have such a proof, then that assumption is gone and is equivalent to any other assumption you have not made. Of course, there are assumptions that seem to be more natural than others,... – Sándor Kovács Nov 15 '11 at 2:41
but I doubt that there is a good way to actually make sense of this feeling. – Sándor Kovács Nov 15 '11 at 2:41
Sandor: of course every theorem worth its salt will use all of its assumptions. Usually some assumptions nevertheless feel more important than others, at least to me. Perhaps a way to make sense of the feeling that some assumptions are "merely technical" is that they hold in virtually all known ("important") cases. So if such a technical assumption becomes critical or vice versa, that indicates an extension of the class of known examples, or a shift in perspective. This is why I'm interested in the, admittedly somewhat vague, question – Chris Heunen Nov 15 '11 at 11:01

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.

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