The Ancient Greece path to knowledge is the path of the thinker. It is that of logic and measurement, of deduction and step-by-step reasoning. It is about creating a cohesive mental model of the world and "running simulations" inside that model, then, adjusting the model (and deepening understanding).
The Indian school, is that of intuitive knowledge, the path of the dreamer. It is about contemplation and visualization, about looking at things without taking them apart, about sudden flashes of understanding, being in the now; about seeing the flow without interrupting the flow, about dismissing all mental simulations and seeing the world as it really is.
The two schools correspond to the types of knowledge controlled by the two hemispheres of the brain: you have the left-side of the brain, the logical side, and the right side, the intuitive one.
For us Westerners, it's difficult to grasp the oriental path, as our entire education process skips over the capabilities of the right side of the brain. When taught, you are given information, dates, numbers, sentences, the how-to of doing things, methods and rationales.
The intuitive side is completely ignored. You will have a chance to develop your intuitive side if you're arts-oriented as a career, but even then, intuition and intuitive knowledge is not explored as a path to understanding, but merely develops as a side effect.
More than that, some right-sided brain activities are seldom spoken of in polite company. You will get one reaction if you speak of the scientific process and a completely different reaction if you speak of the state of inner peace obtained through deep meditation; if you get to claim that this clearness of mind brings you closer to reality, depending on your audience, you might even find yourself in trouble.
I believe it is very difficult for us ("us" meaning westerners) to understand how far the intuitive mind can go; it is very easy to state that "well ... you can tell if your significant other is cheating on you", or to generalize this kind of statement to affirmations like "intuition is just the mind coming to a conclusion without being consciously aware of all the clues it has available". Unfortunately, that just a rationalization of intuition, and as far as understanding goes, it just scratches the surface.
To understand how deep the intuitive mind can go, one has to actually go with it. We don't do that though. We keep shunning it, dismissing it, even though clues to the abilities of the intuitive mind are there.
Out of the top of my head:
Wade Davis spoke of a people in the Amazon rain-forest, that were locating particular plants, "because the plants told them where they were". We have no idea what that means, but rush to consider such people "primitives".
Meditation has noticeable effects in people suffering from lots of things, from ADHD to epilepsy. It also has effects on attention span, and attention distribution and is among the most useful tools in reducing stress. Efforts to understand what is happening in meditation are being made, but it's a fringe effort, virtually unknown. Meanwhile, the stigma is there.
To pull an Argumentum ad Auctoritatem, here's a quote:
The intuitive mind is a sacred gift and the rational mind is a faithful servant. We have created a society that honors the servant and has forgotten the gift.
That's all I had to say about it.
I'll probably say more about it, later.