Soft question: What does "generally" mean in mathematics

Question

I've always used "generally" to mean "without exception under any circumstances" (e.g. "finding the general formula"): I don't think of "usually" as equivalent to "generally", although it is obviously used that way as well. However, I looked up the definition in several dictionaries, and they only have the "usually" definition. Am I simply incorrect, or is it used both ways in some fields?

Answer 1

I think the way that mathematicians usually express the meaning of "without exception" is to say something is "true in general". I wouldn't use "generally true", since that expression is common outside of mathematics, and is used to express "usually true". (I think it's a bit stronger than "usually", but certainly less strong than "always".)

Answer 2

I think some stuff can be a bit domain-specific. Transversal is called general position, properties that hold for residual sets in the space of continuous functions are said to hold "for the typical continuous function", and you have full measure being almost everywhere.

That said, "general" is usually taken to mean "in full generality", while I've sometimes heard "generic" in the sense of "usually".

Answer 3

When I see "generally" I don't even think "usually" but just "there are cases where it's not true".