UnknownPosted: September 3, 2010
I read an interesting book on the life of Ramanujan, titled The Man Who Knew Infinity, by Robert Kanigel. The book gives an interesting idea of the concepts which Ramanujan presented to the world without needing an advanced degree in mathematics to understand. But thats not what i am writing about. There is an interesting aspect which is written about. While it is the logical, rational process to prove a theorem with the tools of mathematics available to the mathematician, what to prove is a less logical thing to find out. What is the theorem which should be proved. How to find out what to prove. And this is what i am writing about. As G. H. Hardy is quoted:
unconscious activity often plays a decisive part in discovery; that periods of ineffective effort are often followed, after intervals of rest or distraction, by moments of sudden illumination; that these flashes of inspiration are explicable only as a result of activities of which the agent has been unaware – the evidence for all this seems overwhelming.
This means that while the mathematical process is logical, there are aspects of mathematical discovery which are not completely rational, which depend on something which is beyond the human mind. I have written about this earlier, where i have asked how it is that scientists decide the questions they seek answers to. Or, how does a mathematician what should be the form of a theorem, which they can then go ahead and prove. Some of this comes from a part which seems to be inexplicable to the world of science. Inspiration, we may call it, or intuition. Or give it another name, but this is something which remains outside the domain of natural enquiry. As David Gurteen had pointed out, a quote from Poincare:
It is by logic that we prove but by intuition that we discover.
What this means is just that logic takes to a particular point, where something else takes over the process of discovery. What this is, i would not name, for we all have different names for this. But this is something which we need to recognize as the source of a number of great scientific discoveries.