The big math story in 2006 was the publication of complete proofs of the Poincare conjecture, and subsequent events. In August, Grigori Perelman was awarded the Fields Medal for his role in the proof, which he turned down. At the same time the New Yorker published its famous article about Perelman and Shing-Tung Yau.

Ars Mathematica ran 224 posts for the year. The most popular post (judged by the number of comments) was Michael’s Who are youâ€¦who who, who who, which asked everyone to talk about their favorite subject. The second most popular was my unprovoked attack on category theory, Opinions of Category Theory. The third most popular, interestingly enough, was Hartry Field, which featured a detailed and substantive debate on Field’s interpretation of mathematics.

The post I am personally most proud of is Grete Hermann, which describes the contributions of an undeservedly obscure figure in twentieth century mathematics and physics. My New Year’s resolution for 2007 is to actually complete some of the partially-written posts I started in 2006 (I’m up to 60).

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On your Grete Hermann post, Walt, you said:

“Hermann did this before the invention of the computer, or even before the notion of an effective procedure had been formalized.”I wonder if anyone knows when exactly our modern notions of computation first arose and were named. After all, Babbage and, before him, Leibniz, had built machines capable of executing computations long before Hermann. And, we should not forget manufacturing industry.

In 1984, I saw in operation in a factory in Harare, Zimbabwe, a machine used to make wooden floor-brushes which used a ribbon with holes in it to guide the movements and placement of a drill which made holes in the wooden base for the brush bristles. By changing the ribbon, different patterns of bristle holes could be produced for different shaped brushes, all on the same machine. This machine had been made in Sheffield, England, in 1870. I believe there were similar machines in operation much earlier than this in the textile industry.

No doubt the engineers of Sheffield who designed this machine did not call the ribbon a program, nor perhaps even distinguish between what we would call “hardware” and “software”. But the concepts were clearly present, even if named differently or even not named.

People must have been familiar with the idea of computation using pen and paper for a long time. They must have had an intuitive sense of what it meant that long division, for example, would finish in a certain amount of steps as a function of the input. What probably compelled Hermann to write down a definition is that the algorithms she discovered were sufficiently complicated that it wasn’t obvious you would ever stop.