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.

Elementary
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.