Wednesday, August 25, 2010

The Essence of Intuition and Its Role in Physical/Mathematical Discoveries

I have been thinking about the role of intuition in physical discoveries and mathematical reasoning. Intuition is a great tool that we have developed in order to navigate through the world around us. If one stops to think, intuition or common-sense is a crude and simple form of mathematical logic and a set of definitions that are based on what is around us. Our intuition can only comprehend finite and discrete things, as these are the things that we deal with in our lives. Furthermore, we build a conceptual framework by categorizing and naming how the things around us seem to us. Thus, say things like “this is a particle, that’s a wave.” And we use analogies to “understand” things by saying like “The particle acts like a bullet here, etc.” However, the whole universe is a much much more complicated mathematical structure which goes beyond the simple logic that evolution and experience have equipped us with. The remarkable fact, however, is that we have come to learn mathematical logic and discover nature through its most basic logic. Well maybe it is not that surprising after all since we are ourselves one of the products of this mathematical structure – the universe looking back at itself.

As we enter new realms of reality through experimentation or mathematical reasoning, we often discover “counter-intuitive” realities. If one does not have the right approach, the struggle to “understand” becomes really frustrating. One has to realize that our intuitive understanding is only designed for quick navigation for only a small portion of the vast mathematical reality. Therefore, none of our discoveries are counter-intuitive because we simply do not have any intuition for those contexts. For example, some people might find the fact that 1 = 0.99999… very counter-intuitive. But the point is that we do not have any intuition about something that is infinitely continued. I would only call it counter-intuitive if one shows that 0.9 = 1 or that a large sphere can fit in a smaller one, because those are the things that I have developed intuition for. But these things are mathematically false and so I don’t have to worry about them. The point is that intuition is a tool that has been developed out of mathematical logic and our intuitive understanding is limited to our context. Another example of the misuse of intuition is quantum mechanics. One cannot say the behaviour of things (I say “things” for the lack of a better word) is counter-intuitive at the quantum level, because we have not developed any intuition for that scale; its neither intuitive nor counter-intuitive.

Nevertheless, intuition can still be really helpful at times. After all, its our brains’ natural mode of thinking and reasoning. One very simple example of how intuition might be helpful could be analyzing the motion of a ball rolling down an incline. Generally-speaking, circular motion is not the most intuitive concept, but instead of taking the integral of the forces on all the points in and on the ball in its initial state in order to find the torque, we can quickly say that there is no torque due to gravity because of the symmetry of the ball. Note that this is still mathematical reasoning, but it’s the kind of reasoning that we are intuitively good at and it can save us a lot of time and effort.

All in all, it is important to realize the capabilities and limitations of our intuition. When used properly, it can boost our problem-solving and give us directions, and when used improperly it can prevent us from developing our understanding of nature.

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