“And when it comes to mathematics, you must realize that this is the human mind at the extreme limit of its capacity.”(H. Robbins)
“ … so reduce the use of the brain and calculate!”(E. W. Dijkstra)
“The fact that a brain can do it seems to suggest that the difficulties [of trying with a machine] may not really be so bad as they now seem.”(A. Turing)
§1. Computer calculation.
1.1. A panorama of the status quo. Where stands the mathematical endeavor?
In 2012, many mathematical utilities are reaching consolidation. It is an age of large aggregates and large repositories of mathematics: the arXiv, Math Reviews, and euDML, which promises to aggregate the many European archives such as Zentralblatt Math and Numdam. Sage aggregates dozens of mathematically oriented computer programs under a single Python-scripted front-end.
Book sales in the U.S. have been dropping for the past several years. Instead, online sources such as Wikipedia and Math Overflow are rapidly becoming students' preferred math references. The Polymath blog organizes massive mathematical collaborations. Other blogs organize previously isolated researchers into new fields of research. The slow, methodical deliberations of referees in the old school are giving way; now in a single stroke, Tao blogs, gets feedback, and publishes.
Machine Learning is in its ascendancy. Log Answer and Wolfram Alpha answer our elementary questions about the quantitative world; Watson our Jeopardy questions.