The more I learn, the more it seems that mathematics is the key to understanding most everything, at least to the best that we can. Plus, math is an art, not a science. That is why BA's and not BS's are granted in the study of pure math. Also, key in math, are infinity and mathematical paradoxes. They allow both infinite possibilities and also opposite realities to exist at the same time, ie. dualities.
What is important about the "art" part, is that it encourages deductive thinking in addition to just inductive thinking. There have been times when science centered on inductive thinking, basically observing things and coming up with laws based on them. However, great advances have come from just envisioning something creatively, then seeing if it can be proved by observances.
So, one is encouraged to think about anything, then see where that leads. In fact, the real purpose of encouraging the study of pure math, is basically to teach a student "how" to think, not "what" to think. It is about wiring the brain to think analytically, deductively. Thinking that way is the key to critical thinking. That is also why research universities focus on the theoretical side of math, to prepare students toward doctoral programs which are aimed at discovering new things, thoughts, etc.
So, I'll begin there, but will refer back to math as I go on to other areas.
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