I have returned from my blog vacation.
A few days ago I saw one of my college student patients who usually visit me during their break from school. This particular patient did not present a difficult treatment challenge as the first intervention worked well several years ago and continues to work well without complications, side effects or adjustments. He originally wanted to go to medical school but after some time at college and an experience with organic chemistry, he decided to change his field of concentration to mathematics. He talked about the proposed aggressive reduction in the Math Department budget and its anticipated effects. He shared his new view of mathematics, that is, he talked about real mathematics as distinct from applied or "school" mathematics. Applied, school, or useful mathematics includes arithmetic, elementary algebra, elementary Euclidean geometry, elementary differential and integral calculus. Real mathematics do not appear to have any real application to the physical work but represent and underlying beauty and pattern. This includes such areas as modern algebra and geometry, number theory, theory of aggregates and functions, relativity, quantum mechanics, and the theory of matrices and groups. Who could have predicted that matrices and groups would have application to physics?
This brief story of my last visit with this patient contains many important strands to pull and fondle. First, I am reminded that the cultivation of a platform that supports and encourages creative new solutions to the problems that abound around us is hugely consequential for the future of the American economy. Successful scientists, philosophers, and inventors have weighed in on this issue from the ancient Greeks to contemporaries like Richard Feynman. Feynman's advice to be irreverent toward established ideas is mentioned on the home page of this site. Probably the best advice for fostering creative thinking comes from G.H. Hardy, the Brittish mathematician, who worked in analytic number theory and developed the Hardy-Weinberg principle which states that both allele and genotype frequencies remain constant in a population. This is an ideal state (not found in nature because of a variety of disturbing features such as mutations, non-random mating etc.) and can be used to compare actual observation to this ideal state. In any event he wrote a rather intriguing and provocative article in 1940 entitled A Mathematicians Apology in which he compared real mathematics to poetry and art.
Hardy talks about the mathematician as a creative artist, as a maker of patterns fashioned from ideas. Painters use shapes and colors while poets use words. Just as the painters colors and poet's words must be beautiful so must the mathematicians ideas be beautiful by fitting together harmoniously. These mathematical ideas must also be serious, that is, it can be connected in a natural way with a large number of other complex mathematical ideas. The seriousness lies in its content not its consequences just as Shakespeare enormously influenced the English language because of the superiority of his poetry and its deep content. In order for the mathematical idea to be significant, it must have beauty, seriousness, and depth but also be general, that is, be a constituent in multiple mathematical constructs used to prove theorems in different areas of mathematics. The final quality that Hardy discusses is that of unexpectedness combined with inevitability and economy. The arguments are unusual, simple (often in the extreme) and produce far reaching results.
What are the qualities of creative thinking? Often in award ceremonies one hears the words creativity, importance, and impact to describe the winners scientific contribution. One could just as well use Hardy's words significance, generalizability, and unexpectedness.
In my opinion, the decision by administrators to cut the Mathematics Department budget was short-sighted and harmful to the American economy because it will likely reduce creative thinking in areas we need most.