The abc of Formal Thinking

Some people consider gender to be a cultural construct, others are convinced that intelligence is determined biologically. Some people believe in life after death, other people think that they deserve all the money they earn. Some people are Catholic, some are unemployed, some die young, while others grow old, some people are racists, some are scientists, and they all have different opinions, views, beliefs and convictions on virtually everything. This has been the case at all times and in all places, with one exception.

Mathematics is different. One plus one is two. We all agree on that. Three plus three is six, there is no doubt about this, as long as you know how to calculate. But even if someone does not know how to count, she or he will realise at a glance that seven apples are more than six. Even animals have an understanding of quantities, a natural number sense if you wish. A cat with seven kittens realises that one is missing if only six show up for feeding. There is a common, universal, unique mathematical understanding that we all share, which makes mathematics different from all other things.

Everybody is a mathematician. This sounds like a cheap rephrasing of “Everyone is an artist” (Joseph Beuys). But I mean it. I remember that I had to be taught how to tie my shoelaces. But nobody had to show me which shoe belongs to which foot. When putting on our shoes, we simply feel which shoe fits the right foot, and that the other shoe does not. We tried to do it wrongly once or twice as kids, and since then we just know. We understood how the symmetric shape of our feet is mirrored by the symmetry of our pair of shoes. Some people confuse the words “left” and “right”, but everybody knows which shoe of a given pair fits the left foot. When you stand up and want to leave a room, you do not attempt to walk through the wall, - you rather pass through an open door. Everyone does that, although it clearly requires some unconscious calculations. Everybody can tell whether an elephant is bigger than a mouse or not. Everybody can clap the rhythm of his or her national anthem. Without counting, one knows that a bucket full of nuts contains more nuts than one hand can grasp. If someone is told to set the table for four persons and is handed over four plates and four glasses, then everybody manages to place one plate in front of each of four chairs and to associate one glass with each plate. This is what mathematics is about: Shape, symmetry, space, quantity, numbers and relationships. Everyone has direct access to this elementary or fundamental mathematics, as we may call it now. It is about structure, order and rhythm; it is about life. Number is not all (to paraphrase a well known saying of the Pythagorean Brotherhood), but all is nothing without numbers.

Fundamental mathematics is much more elementary than what is taught at secondary school. The scholarly, secondary school mathematics deals with sines and cosines, with integrals and logarithms and all that stuff which many of us have never fully understood. But you do not have to study literature to enjoy a novel. You do not have to study English grammar to understand and speak some English. You do not have to study scholarly mathematics to put your shoes on, to set a table, to sing in a choir, to enjoy the shape of a pebble, the geometry of a pyramid, or to wonder why 123 is divisible by 3 (but not by 9 without giving a remainder).

Edition Rechenheft is not scholarly, or is only so in a very playful way - a forum for Gaia Scienza (“Fröhliche Wissenschaft”). This platform is dedicated to figures arising from figures, to forms obtained from formulae. It is about the art, basis and critique (abc) of formal thinking.