I suspect that most of the people decrying logic, axiomatic systems, etc. aren't objecting to reasoning in general so much as certain levels of formality...

Or, in my experience, specific topics. For example, such a person would say that reasoning does apply to topics such as deciding which car to buy, or which stock to invest to, or what the sum of the angles in a triangle is. Reasoning does not, however, apply to other topics such as deciding what to eat for lunch, which deity to worship (if any), whom to date, and which topics are subject to reason in the first place.

The above is a real example, BTW (assuming I understood the person's position correctly).