Earlier this week, a colleague devised a clever algorithm for a particular problem but it relied on the following theorem. I would love to include it inline as they manage to do on the n-Category Café but, due to my incompetence, I will have to include as a picture:
He sketched this quickly on my whiteboard late one afternoon, it seemed intuitively correct and I promised to spend a few minutes on it at home that night. OK, I know now that I should have simply asked Messrs Prouhet, Tarry and Escott but I bet they didn’t know about Schubert’s Winterreise or how to fly an ILS approach to minimums.
I spent some time on it and found a remarkable number of “proofs” followed by subtleties that exploded them. Finally, after an hour or so, I found a proof that seemed to work, wrote it up in a short memorandum (from which the above picture comes) and went to bed satisfied. In the shower the next morning, while thinking of something else entirely, the flaw in my “proof” came to me. And, on arriving at work, my colleague mentioned that I should just have turned to my copy of Hardy and Wright (bought, according to the flyleaf, in September 1980 and waiting for such a moment as this) and turned to page 328.
So that resolved that, but I am still overwhelmingly impressed by the unconscious brain. I assume that I was really somewhat uncomfortable about my “proof” and, although my conscious mind was happy with it, my unconscious one kept working on it to uncover the cause of concern. With a lot of analytic philosophy turning to neuro-science, I wonder whether there’s a neuro-scientific explanation of these parallel brain activities. How many parallel activities can a brain handle? Is this number what distinguishes me from a genius?