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Kindly comments on Math is Subjunctively Objective - Less Wrong
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I am not Randolf, but I've met people who would answer this question thusly:
I don't think it's possible to use logic to convince someone of the importance of logic, unless he happens to be convinced already.
You can use naive logic to convince people of the importance of more rigorous logic, though, and 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 certain attitudes surrounding them. I've met a lot of people claiming to put more stock in gut feelings than clever reasoning, but I've never met one such that didn't have a handy store of justifications for their beliefs -- which seems to point to a certain trust even if it's unacknowledged.
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).
(nods) I generally summarize this as "reason is useful only for those topics where I'm confident I'm right or am willing to be corrected if wrong." To which my response is typically "how very convenient for you that it works out that way."