Single Review on past events

It was his second Jour Fixe talk at the Zukunftskolleg. While he spoke about “numbers” the first time, he now decided to inform his audience about “signs”: “This talk is not about a mathematical topic, it is about the role of notation for mathematics”, introduced Karim Becher. “It is based on my readings during the last two weeks, leading me into basic reactions about my own research field. It is nothing more than an accumulation of sketchy, possibly inspiring or entertaining thoughts.”

So he started his talk by telling a little anecdote. A colleague´s little daughter once said to her maths teacher at school: “My mother also does mathematics. But she calculates with letters rather than with numbers.” Karim explained that non-mathematicians often wonder how mathematicians can communicate orally about such an abstract and formal thing like mathematics. They ask if the mathematical signs and drawings actually mean anything or if they are just a fake. Karim admitted that he sometimes also asks himself these questions. But on the other hand he confessed that the oral communication about mathematics, the fact that it is possible to exchange and generate abstract ideas in the air by words, signs, and drawings, is what he likes about his subject.

On the basis of three points Karim illustrated the role and development of notation in different mathematical eras: antiquity, the beginning of modern age in the 16th/17th century and the turn of the 20th century. Around 600 B.C. many deep mathematical facts - procedures for computation and geometrical construction - in Arithmetics and Geometry had already been discovered, in particular by the Egyptians and Babylonians. But it wasn´t that mathematics as we understand it today. They introduced rigorous reasoning showing dependencies between different claims.

After an extremely prolific period in pure mathematics with an emphasis on Geometry in Ancient Greece, it took roughly 15 centuries for mathematics to enter into an area of continuous progress again. Nevertheless during these 15 centuries many important mathematical discoveries, in particular in Arithmetic, were done and began to form slowly the basis of a new discipline: Algebra. According to Karim the success story of Algebra is connected to the slowly growing acceptance to calculate with symbols that do not need to have a meaning at each step in the calculation.

The use of notation in mathematics, especially the use of symbols for basic operations and of letters for quantities involved, took until the 16th century. “The crucial importance of notation is not that a word like “unknown” is abbreviated by a letter, e.g. “X”, but rather that mathematical reasoning can be performed by the use of symbols”, explained the Fellow. He noted the turn of the 19th to 20th century as very important time, as mathematics was then quite developed. Furthermore notation and formalism in the mathematical language allowed at that time a new view at mathematical proofs, leading to Hilbert's program and Goedel's famous incompleteness theorems.