Why would you consider unfair to call logic thinking as mathematical thinking? Maybe it is the engineering background indeed, at least it is for myself. Maybe it was all those "engineering math" mandatory courses... you know, calculus, differential equations, linear algebra, statistics. If we keep thinking about math in terms of some sort of computation which ought to result in a numeric answer... then, absolutely, I agree that has nothing to do with logic.
However, least see what happens when one does not apply mathematical thinking into his own reasoning. I do it myself from time to time and refer to it as my "rat intelligence". You go over and over the facts, one at a time, trying to figure out the way out... It feels pretty much like a rat walking through a maze inside of my brain.
But it comes to a point, when the problem is very hard, that no matter how many times the rat goes over and over the same data, it does not find the answer. In that case you need a superior type of reasoning. You need to step aside and look at the problem as a whole. To find patterns that you can abstract out and rephrase the problem in a way more tractable. And I think in this broader sense logic and math use the same type of mental tools (principles) to attack the same issues.
Actually, all of those computations "which ought to result in a numeric answer" are done according to the principles of logic. Maybe I see it more clearly having taught myself calculus and more advanced math without having someone to "explain" things to me, but all mathematical problem solving is also logical reasoning.
However, least see what happens when one does not apply mathematical thinking into his own reasoning. I do it myself from time to time and refer to it as my "rat intelligence". You go over and over the facts, one at a time, trying to figure out the way out... It feels pretty much like a rat walking through a maze inside of my brain.
But it comes to a point, when the problem is very hard, that no matter how many times the rat goes over and over the same data, it does not find the answer. In that case you need a superior type of reasoning. You need to step aside and look at the problem as a whole. To find patterns that you can abstract out and rephrase the problem in a way more tractable. And I think in this broader sense logic and math use the same type of mental tools (principles) to attack the same issues.