I sometimes find a mathematical insight in great literature or music. Bach's music has a lot of mathematical pattern, Haiku or some kind of lyrics also have a formal pattern. Some of Souseki Natume also have mathematical insight. These people sublimate these patterns to arts. I feel sometimes happy that I can see these aspect. Let me show you one example.
Do you know ``Meijinden (A master's story)'' by Atusi Nakajima? I like his ``Deshi (a student)'' and ``Riryou'', but my best story from him is Meijinden. A master of archery want to be the master of masters. One point he finally met the master of masters. He shot 20 birds down with one arrow. The master said to him, ``I see you can shoot an arrow. But that is just a shoot of shooting. You seem not to know a shoot of non-shooting.'' and the master shot 20 birds down without any arrows. The shooting world is limited if one uses an arrow, what is the substance of shooting? If we want to over the one world and look into the substance of it, sometimes you need to abandon an important but not substance component.
A few times I tried to explain this story to some people. This is a Japanese classic story based on Chinese old story. However, sometimes people think I am joking, even though I seriously think this is how new idea came. In Japanese, mathematics is ``数学 Suu-gaku(= study of numbers)''.(Some know a puzzle called ``数独 Suu-doku'', number alone: short from of Mathematics is good for a single person. Suu means number.) When mathematics reached at some level, it became not a study of numbers. The master might say: ``I see you know some of numbers. But that is just a number of numbering. You seem not to know a number of non-numbering.'' Mathematics started to count numbers, then found the operations between numbers (addition, subtraction, ...). But one point, mathematics stopped discussing numbers. Mathematics started discussing operators without numbers. That was one big point of mathematics. What is the operator can create. When the subtraction is invented, people found some of the subtraction can not be done. For instance, 3 - 5 can not have an answer until a minus number is invented. Some people doubted, can we add any numbers? If the number is very large, can we still add any two numbers? People started ``any'' numbers. This is a great one step. People knew the division has a similar problem, 3/5 can not have the answer until fraction is invented. Minus number was not enough. One started thought, when this will be end? Can we think about all the operators? Not only one operator, but all the operations we can think about. Like we can think not only one number, we can think about any numbers. Galois is one of the pioneer of this area, therefore, his work is important. Same of the computer programming, first, a function was put a number and get a new number. Then next level is: forget the number, put a function, get a new function. This idea of meta programming pushes the concept of programing to the next step. You might saw this year's Google's doodle about Turing [1]. He thought if there is a universal computer, which means a machine can simulate any computer, what is the limit of this machine. Incredibly, he proved what computer can do and what can not before nowadays electronic computer was invented. I can not stop these stories how the abstraction is powerful. I should back to the matrix story now.
Let's back to the inhabitants of Berlin and Potsdam next time.
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