Johan Galtung and Dietrich Fischer
Prologue: Peace, Mathematics, and
The program for this book is in the
subtitle. Like government of the people, by the people, for the
people--from Abraham Lincoln's Gettysburg Address--there can be
mathematics of peace, by peace and for peace.
Mathematics of peace is a
mathematized syntax and semantics of peace, as absence of violence,
as cooperation and general harmony. In short, normal human relations;
not perfectly good, but very far from perfectly bad. And they have a
form to be explored, formulated, and possibly reformulated in
Mathematics by peace: there
are ways in which peace may give rise to mathematics, like the
physical world certainly has done. However, for several reasons,
peace and health have not attracted enough attention to inspire
mathematicians, but disease, war and violence are reflected in
mathematics of epidemics and arms races.
Mathematics for peace may open
for new peace possibilities; like it did for physics. There may be
hidden, or not-so-obvious, worlds of peace that a little mathematics
can make available.
But, where do we start? We had a
choice. We could start with peace theory, and call on, or try to
invent, adequate mathematics. Or, we could start with branches of
mathematics, and then develop applications to peace theory. The
former would be an "of" and "by" approach, the
latter would be a "for" approach.
We chose the latter because we also
had an other agenda: a new approach to mathematics for elementary and
high schools, and for early years in college. More open to
abstractions, to philosophy, and with applications to conflict and
peace in daily and world affairs. The book not only explores peace
but also mathematics, favoring some branches for teaching at school
more than others. But the major advocacy is for peace mathematics;
the by, of, and for approaches. For that purpose, first a mini-theory
A basic point about peace to
Peace is a relation. Peace is
between two or more parties. The parties may be inside or
among persons, states or nations, regions or civilizations, pulling
in compatible directions--or not.
Peace is not an attribute of one
party alone, but an attribute of the relation between the
parties. That in no sense belittles the significance of the parties'
intent and capability to build peaceful relations. But the relation
between them is essential, which is why we find lovely people related
in a less-than-lovely marriage, and less-than-lovely people
having a reasonable marriage.
What kind of relations can we have?
Three types, it seems:
bad-good for one is good-bad for the other;
non-relation, neither cares about good-bad for Other;
bad-good for one is bad-good for the other.
In the real world relations may be
mixes of all three.
When the relation is intended, the
party is an actor. And if it is negative we talk about harm, direct
violence, and about war if the actor is collective. If the violence
is not intended--but watch out for acts of omission, they can be
intended!--it may be referred to as indirect, often caused by
inequitable structures producing harm, structural violence.
And then the role of culture legitimizing either or both types of
violence, cultural violence.
From this follow two concepts of
peace: the absence of violence, like cease-fire, or being
apart; no negative, no positive, only indifferent, relations; and
peace: the presence of equity, in cooperation, harmony.
They are as different as negative
health, absence of (symptoms of) illness and positive health, the
feeling of physical, mental and social well-being in the World Health
Organization's definition, with capacity to handle illness (like an
intact immune system). Capacity to handle violence might be added to
the definition of positive peace.
From this, then, follow three types
of peace studies:
peace studies: how to reduce-eliminate violent relations;
peace studies: how to make cooperative-harmonious relations;
studies: of intent and capability to inflict harm.
The third, often found, may be useful
when coupled with studies of the intent and capability to reduce
violence and build harmony.
One approach to negative peace
studies opens for peace and conflict studies, seeing
violence-war as the smoke signals from the underlying fire of an
unresolved conflict. And that leads to a major approach to negative
peace: remove the conflict by solving it, or transforming it
so that the parties can handle it in a nonviolent way, with empathy
for each other, and with creativity.
The root of a conflict is by
definition a contradiction, an incompatibility, clash of goals, which
then translates into a clash of parties and violent behavior. At any
stage in the process negative attitudes may enter.
Attitudes-Behavior-Contradictions, ABC, feed into each other, in
vicious cycles or triads (the ABC-triangle). In the wake of those
processes are traumatized parties, and actors with festering wounds
on body, mind and spirit.
That leads us to the two key tasks in
search of, as a minimum, negative peace: mediation to solve or
transform incompatibilities; and conciliation, healing the
traumas of the past, removing them from the relation between the
parties for closure. But, if closure is brought about without
conflict resolution we have pacification rather than conciliation. A
non-starter, likely to erupt, sooner or later.
A conciliation metaphor is to turn a
page in the history of their relations, opening a blank page. But if
that page remains blank, only bland indifference has been obtained.
Arguably better than hatred and harm, but much below our potential as
To inscribe that page with ideas of
positive peace and put them into practice gives us the other side of
peace: positive peace, cooperation, harmony. In a marriage the
harmony of body, mind and spirit; even fusion. An indicative term is
cooperation, another a joint project, cognitive ideas,
supported by positive emotions. A joint project is spiritual, giving
the relation new meaning; something to live for, together. For the
content the sky is the limit.
So, exactly what do we mean by
cooperation, by a joint project?
Peace studies would answer with two
words: structural peace, meaning, to start equity . All
parties benefit from the relation, and those benefits, if not exactly
equal, are at least not too unequal; that would be exploitation. And
it means reciprocity, as opposed to mental conditioning of one
by the other. And holism, the use of many faculties in all, as
opposed to segmentation. It implies integration in the sense
of all relating to all, as opposed to fragmentation. And inclusion
of them all, as opposed to exclusion, marginalization. There is a
whole structure involved. Add up all the negatives, exploitation,
conditioning etc., and we get structural violence.
Structural peace is what friendship,
close kinship, neighborship, good families and marriages, good
relations of worship and workship are about. Bring in structural
violence, and we are in deep trouble.
At the level of a multi-national
state this is what a community of nations is about. At the
macro-level this would point to a community, even a union, of
countries. And at the mega-level to the mobilization of both genders,
the three generations, the five or so races, all classes, the 2000
nations and 200 states, in a joint project of human dignity for all.
As a bulwark, as an immune system against violence. As a concrete and
feasible, utopia. As peace.
We note the centrality of relation,
and structure. Mathematics.
is much talked about nowadays. But the conversation is not
necessarily a good one. The point of departure is often how countries
are ranked in the PISA-investigations of school achievement, by OECD,
the Organization for Economic Cooperation and Development, of rich
countries. Typical rankings: Hong Kong no. 1, Finland no. 2,
Switzerland No. 10, Norway no. 22, USA no. 281,
right ahead of Russia. Mathematics is identified with the ability to
solve problems, to have a correct
and one more way of ranking nations.
The result has
become some kind of educational Olympic Games.2
Ability to read, and natural science knowledge, are also included,
with Switzerland as no. 13 and 12 and Norway as no. 12 and 28.
But imagine now
that mathematics is not only about correct answers but also about
Could "thinking mathematically" be a basis for thinking
systematically, creatively, constructively? For instance, about
relations, and about structures?
be an entry gate to using our mental abilities better? Another entry
gate is art4;
a third knowledge, and a fourth would be experience; to new
questions, about changing relations, structures, realities? And to
fruitful, not only correct answers?
But first some answers to the
question "What is mathematics?" Mathematics is symbolic,
abstract; nothing concrete in our hands. OK, but mathematics also
represents, mirrors, some concrete reality outside ourselves, does it
not? Yes, it often does, like language, music and arts also can
mirror something "out there". But we must in addition give
space for the possibility of mirroring something inside the
mathematician. The painter Jackson Pollock, for instance, was of the
opinion that his painting mirrored his inner reality and nothing "out
there". Psychologists and many others may have some doubts about
that; how did the "inner reality" come about? As subjective
Mathematics represents, mirrors,
something. But what?
The answer given
by Keith Devlin in his brilliant The
Language of Mathematics: Making the Invisible Visible5,
is also ours: patterns.
"The science of patterns,"
Devlin says, is the science of structures. A mathematician is
somebody who goes around with a huge amount of patterns imprinted on
his brain, like all of us are walking around with huge quantities of
experiences, sedimented as memories and emotions. The patterns could
be something that person or others have observed, for instance
triangles with a right angle, or squares. Some people then want to
know more about them. But they could also be patterns the
mathematician is intuiting, in the center of imagination and creative
ability. He knows how to explore them further, inside himself, needs
no reference to outside reality, only to mathematics.
Devlin organizes his book around
different types of patterns that form the basis for different
branches of mathematics6:
patterns of counting and numbers
patterns of shapes
patterns of motion
patterns of reasoning
patterns of chance
patterns of closeness and position
Around 1900 there were about 12 such
branches of mathematics, today 60–70 branches. Needless to say, no
single mathematician can be competent in all, nor do they agree on
their ranking in importance.
There is much human history in
general, and cultural history in particular, in this list. We sense
the ability to master a static world by counting numbers, and drawing
shapes. But dynamics enters this image of the world, and a different
mathematics of change, and movement, is introduced with Newton and
Leibniz, adding calculus to arithmetic and geometry. Then mathematics
itself, and thinking in general, become major subjects of
exploration, with Russell-Whitehead Principia Mathematica as a
Probability opens for uncertainty,
the random, non-deterministic, also known as the stochastic,
later on followed by the mathematics of chaos and catastrophe. The
mirrors become more adequate to reality.
To this add
as the pure discipline of patterns, of the patterns of patterns. And
Devlin adds patterns of beauty, like symmetry, and patterns of the
universe from Egypt-Babylon to Einstein.
Appetizing? Should be. But school
mathematics has not been good at conveying such messages. An enormous
field of culture is waiting, like an empty museum, for visitors to
learn, reflect and enjoy.
We can only give
some glimpses of this landscape, sharing some tools, some sources of
joy, and above all our efforts to explore the peace-mathematics
interface. But Devlin's list overlaps to a large extent with our
independently developed table of contents.8
Toward that end let us classify these
branches of mathematics:
Table 1. A Classification of
Branches of Mathematics
Arithmetic, Geometry, Logic, Topology, Algebra
The beginning is in
Category 1: discrete. deterministic. Numbers, and shapes like points,
lines and planes, circles, triangles and quadrangles, with a clear,
immediate application to what can be counted, and owned; to shapes
that can be measured, and owned. This is deepened in logic and
topology, generalized in algebra.9
And then two major expansions into
new territory, from discrete to continuous, and from deterministic to
stochastic. And both.
Where is peace mathematics? All over,
but being a relation above all in category 1, maybe in logic and
algebra, rather than with buying-selling and ownership. But like them
making excursions into the continuous, the stochastic, and both. Only
Let us contrast
this with a high school mathematics booklet, 23 pages, from Teaching
Center, Directorate of Education, Oslo.10
The level is high school, so there is no useful arithmetics in the
basic sense of being able to check bills and the ins and outs of
personal budgets; home economics. The booklet is above that.
The mathematics of land property is
well represented as geometry and trigonometry, highly Euclidean,
boring, and probably useless. People want to know the price tag, the
number of m2 and then figure out price per unit, for
comparisons. Shape is less interesting, and sin, cos, tg and cotg are
as antiquated as logarithms in the age of computers, hand-held or
The most useful
formulas in the booklet deal with price index, real salary, growth,
interest; all applicable in state capitalist societies. But a little
touch of social democracy, discussing the many measures of inequality
in income and wealth distribution, how some conceal and some reveal,
would have been useful, maybe in connection with some probability and
statistics. And what a chance to get closer to politics guided by the
mathematics of inequality, some might even say "injustice".
We are treated to millennia old
equations of first and second degree, centuries old derivatives and
integrals, vectors, polar coordinates, analytical geometry. How many
have ever encountered a parabola after the ceremonial farewell at the
final examination table, except for the few heading for IT, for
physics-engineering, and for the eternalization of useless
mathematics as teachers?
But maybe parabolas teach them to
think mathematically!? The evidence of that dubious hypothesis is
missing. Thinking can be taught, but not by mathematics so detached
from life as lived by the majority of students as opposed to
And yet this is
what the PISA tests measure. Maybe Norway should be congratulated on
and Finland deplored? This type of mathematics is autistic, like life
in a bubble. Maybe there is also something autistic about Finns
sitting on a bench, winter time, with a cap down over the ears, and a
bottle at hand? Maybe autistic meets autistic?13
Maybe a typical student reaction in Norway, "so boring, I was
never touched by it, I get afraid" is a sign of an advanced
mental health missing in a Directorate never reflecting on what they
are forcing upon innocent students?14
What is missing in
this approach largely based on 18th century mathematics? Maybe
mathematics as discovery,
as a wonderful tool to be shaped by the user, as a source of delight
And, without making that a major point: a source of peace.
That does not rule out mathematics for money in and out, accounting
and budgeting, astronomy and astrology for units of time, like lunar
and solar months, but it does rule out most of the other mathematics
mentioned. It may also marginalize calculus with its impressive
derivatives and integrals, based on continuous variables encountered
only in some parts of physics, and from there imposed upon economic
But it rules in the little there is
of algebra with its discrete variables, and probability-statistics
with stochastic, as opposed to deterministic, variables. The general
thesis is that discrete and stochastic mathematics is closer to
reality as lived by the students than the continuous, deterministic
variables serving the physical sciences of the Enlightenment.
We have the mathematics teaching we
deserve, not because everybody is infatuated with
physics-engineering, some accounting, and real estate. But because we
had it yesterday, and because we imported it from countries often
imitated. Two strong forces.
So much about what mathematics is
taught. Being in love with the field, what would we recommend? What
to teach, for instance at the TRANSCEND Peace University? Set theory,
cartesian products, relation theory, matrices and graphs, probability
and statistics, game, system, change, chaos and catastrophe theory.
As this book is an effort in that direction, something about why and
Soft mathematics is a heuristic, a
wonderful tool to help us think systematically, maybe even to think
at all, deeper, more basic than puzzles to be solved, searching for
the correct answer.
A good point of departure is the
double definition of a set, in extension as a list of
elements, and in intension as a concept. Other terms:
denotation for the list; connotation for the meaning.
Take "democracy": there is a list of countries and of
meanings. Do they tally? Same meanings produce the same lists.
But how about different meanings?
Fair and free elections, FAFE, is one meaning of democracy; human
rights, HR, is another. That gives us two sets of meanings, FAFE or
not, HR or not; four combinations. The trick is to explore all,
asking which countries are both, one but not the other, neither one
nor the other. Same lists? Could FAFE and HR be synonyms? Or, do we
have a law; they produce each other?
Imagine we find no country "HR,
not FAFE". Why? The empty cell is a gift forcing us to think
deeper--like the number 0--opening for something new: under what
conditions could this combination be found? Bhutan some time ago,
"enlightened monarchy", high on HR low on FAFE?
On top of that we
also speak the word "democracy". We have three corners of
the so-called Ogden's triangle separating concrete cases, their
meanings, and the words; reality, thought and language; things,
concepts and terms16;
states of affairs, propositions and sentences. All of them can be
contradictory, or they can be consistent.17
Relations take this to a higher
level. Love is one, sex another, marriage a third. That makes for
eight combinations, may be worth exploring, all the time comparing
empirical cases and meanings. And training in distinguishing between
meanings, attributes, predicates, of the elements, and their
relations. The elements, a man, a woman, may be "good", but
their relation, the marriage "bad", and vice versa. A
relation cannot be reduced to two attributes, predicates. Conflict,
violence and peace are relations, not predicates of people, nations
or states, even if all of them may have more or less capacity to
solve conflict, avoid violence, build peace. The job is to build
relations; improving the elements is not enough.
Hatred is inside a person, violence
is on the outside. The clash-contradiction-incompatibility is neither
inside, nor outside the parties, but in-between. Like 3 in 2<3<4.
That is where the repair work has to start. And the point made here
is that the jump from elements to relations between them--could be
many, not only 2--is crucial. Many people never make that jump but
see conflict as due to Other, that bad one, the bully; at school, in
marriage, at work. Some familiarity with basic mathematical, and
logical, tools will clarify all that. And the examples will be
considerably more exciting than those that can be constructed around
a second degree equation, or sin-cos-tg-cotg.
Love, sex, marriage; a set with three
elements. What comes first, second, third? Some combinatorics gives
us six orders, with moral connotations. Systematics makes us look at
all; see later.
Combine elements and relations, and
we get structure. With structure we get that major tool of
thought, isomorphism: two or more structures where
corresponding relations relate corresponding elements. Thus, a good
map is isomorphic with the terrain it maps.
Teach that concept with many and
fascinating examples and mathematics has already proved itself. The
students realize that with mathematics they see more, and more
deeply, than with their naked eyes. Like isomorphisms between their
family and USA-Latin America, with obvious politics corresponding to
From there we go on to matrices and
graphs. With matrices we can calculate, identify chains and groups of
positive and negative relations in a class of pupils or countries.
With graphs we are back to geometry--mathematics for the eyes--a
great tool to map what happens when the elements, the points, are
human beings and the "edges", the lines, are their
relations. A week is enough for such key parameters of graphs as
degree of connectivity, distances, associated numbers, center and
periphery, levels of horizontality and verticality (for structural
violence), level of polarization (for readiness for direct violence),
and so on. A fine tool, and immediately applicable to daily life
situations. Like game theory and systems theory; basic tools carrying
a long way. Like a little logic and a little probability making us
discover new worlds.
Chaos theory is actually neither
about chaos nor a theory. It reproduces what looks highly disorderly
by iterating some simple processes. It is a general discourse, a way
of looking at shapes that are non-euclidean, opening our eyes for
more than lines, squares, circles, etc. And catastrophe theory opens
for collapse; through tensions leading to ruptures and new realities.
High school mathematics mirrors an
orderly world; dynamic but according to knowable, even known laws,
continuous, deterministic. This is dramatic indoctrination. The world
is changing and changeable, dialectic, jumpy, chaotic, at times
catastrophic. Some other mathematics may indicate how. High time to
make that change.
These particular PISA figures are from one year, 2003, reported in
50/2004. PISA is not a city in Italy but the OECD "Programme
for International Student Assessment". We sense a focus on
students as "human capital"; as opposed to social skills,
critical ability, creativity. For
a superb analysis from that angle see Sandra Tresch, "Aus der
Schönen neuen Bildungswelt XXI", Zeitfragen
(Zürich), 14 February 2005. The
ranking of countries becomes ranking in competitiveness.
The only focus is on the ranking list, the gold-silver-bronze
medals, the losers, and how they are doing relative to last year.
There is as little concern for what high ranks do to the minds as
for what sports games do to the bodies.
Children incessantly ask questions;
at school they learn answers.
Einstein said that children have natural curiosity, and attributed
his own discoveries to this childish curiosity. He asked what the
world would be like if the speed of light was the speed of the
street car; leading to the theory of relativity.
The Swiss mathematician Hugo Hadwiger (1908-1981) called
mathematics "the art of the mind", arguing that an elegant
proof has as much beauty as a piece of music or a painting.
New York: Freeman & Co, 1998. The book is pedagogically
brilliant, and so is the deeper, more technical follow-up, Keith
The New Golden Age,
New York: Columbia University Press, 1999. There re also excellent
books in French, like Richard Mankiewicz, L'Historie
Paris: Seuil, 2001 originally in English; and David Berlinski, La
Vie Revée des Maths,
Paris: Saint-Simon, 2001, also originally in English. And, "La
bible chinoise des maths", Les
Paris: Dunod, 2004.
How wonderful it would
have been with such books giving depth and breadth to the study of
mathematics 50 years ago!
Galtung had the good luck of studying algebra under the famous
professor Th. Skolem, cherishing, for instance, his "A Theorem
on some Semi-groups", Det Kongelige Norske Videnskabers
Bd XXV, 1952, no. 18.
We did not include "geometry" and "topology"
explicitly, but they are present in Graphs, Chaos, and Catastrophe.
And much of Devlin's "logic" is in Sets, Relations and
Matrices. Devlin's "calculus" is in Change, and our Games
relates to Devlin's "probability". Some beauty in the
sense of symmetry has found its way into Logic.
The word algebra comes from an Arab expression, "al iebr e al
mokabala", restoration and reduction.
(Oslo: Gyldendal, 2001, 19th edition).
reported 26-02-2012 that "25% of college students miss basic
math problem". They were told that the average height of 100
students was 163.5 cm and then given a choice of implications. The
correct answer, chosen by 76%, was "the total height of the 100
students was 16,350 cm. But 24% chose "students around 163,5
cm formed the largest group among the 100", or that "the
number of students taller or shorter than the average was the same".
Experts were alarmed.
What a stupid problem,
mirroring nothing in reality. Acrobats may pile 3, 4 or 5 people on
top of each other, not 100. Substitute for cm the average earning
from "arubaito", a job, and average, total and dispersion
are highly meaningful.
Paulo Freire used
teaching of the alphabet to promote consciousness about social
reality. Teaching mathematics can be done the same way, not as
indoctrination of any particular policy but as awareness, but maybe
somebody prefers ignorance? In chapter  below measures of
dispersion and skewness are explored, and if the variable had ben
yen rather than cm the two wrong answers would have been socially
right even if mathematically wrong. Japanese students are paid less
than before and in the present crisis may have to skip one meal a
This may also apply to gender differences. Girls-women, less
interested in mathematics, may get lower grades and be
under-represented among the inventors. Nature vs culture has
dominated the debate about that difference; another approach might
be interest. Maybe girls find problems like "calculate f(-2)
(2/21)x(7/4)" (from a test in mathematics, Aftenposten,
Oslo, 11 April 2007) meaningless? Thus, when there are few women on
top of economic science, this may also say more about economics than
Mathematics is so captivating, demanding and rewarding that it is
not strange if top mathematicians live in a "bubble" and
find the more ordinary world secondary. Thus, Fondation Cartier in
Paris organized an exhibition dedicated to "Mathématiques"
October 2011-March 2012, and one of their texts is about the "four
mysteries of the world": the nature of the laws of physics, the
mystery of life, the mystery of the role of the brain--and the
fourth: the structure of mathematics. On top of them all; a good
place for some autism.
In the structure of
mathematics are conjectures, theorems intuited but not (yet?)
proved. The great German mathematician David Hilbert listed in 1900,
at the Second International Mathematics Conference in Paris, 23 of
them, as tasks for the century. Several of them, and others, have
been solved, like Fermat's Last Theorem, that the equation an+bn=cn
has no solution in natural numbers if n>2; by the Briton Arthur
Wiles, 357 years later, in 1994. These break-throughs bring
mathematics into the media, not only the PISA reports.
The Russian Grigori
Perelman solved the Poincaré conjecture from 1904 in 2002; but what
fascinated the media was his rejection of the Nobel of mathematics,
the Fields Medal, $1 million. Massa Geshen, in Perfect Rigor: A
Genius and the Breakthrough of the Century (New York: Houghton
Mifflin, 2010) traced his life, finds him suffering from Asperger's
syndrome, "autism-lite", limited social skills, trouble
communicating, speaking oddly. Extraordinarily good at
systematizing, extraordinarily poor at empathizing (John Allen
Paulos, "He Conquered the Conjecture" in The New York
Review of Books, April 20 2010.)
(1912-1954), the computer science and artificial intelligence
genius, Fields Medal, leaves behind a reputation for mental disorder
(Der Spiegel, 2/2012). Which does not detract from his
tremendous contributions to the most important international medium
today: the computer, Internet. As did two Norwegian pioneers, also
Fields Medal, Kristen Nygård (a friend of Galtung's) and Ole Johan
Dahl for the work on SIMULA, both inspired by Turing, with no
September 9 2010, reports how the French-Vietnamese Ngo Bao Chau,
who solves "a mathematical conundrum known as the fundamental
lemma", and accepted the Fields Medal under the headline "Math
whiz prefers to stay out of the limelight". And also reports
that "Top student shuns studies for prayers". Liu Zhiyu
won a gold medal in the Mathematical Olympiad in 2006; then turned
down a full scholarship from MIT and headed for Longquan Temple,
Beijing to become a monk.
A different story: the
Rumanian mathematician Preda Mihailescu who in 2002 solved the 158
years old Catalan (a Spanish mathematician) conjecture--that the
equation xp-yq=1 has only one solution in
natural numbers >1--tells Der Spiegel (27/2002), how he is
inspired by music: each step in a proof is like a toccata, "I
follow the variables with rhythms and melodies and try in that dance
to find a structure that brings me further".
But the media attention was on admiration ("Hong Kong and
Finland found to have top math students", IHT,
7 December 2004), and what can we learn ("Was man von Finland
lernen kann" Zeitfragen,
Zürich 2 May 2005; "Ecole: La leçon finlandaise:, Le
17-23 February 2005).
Hans Magnus Enzensberger, in his fine Der
München: Carl Hanser Verlag, 1997; for children, imaginatively
illustrated, manages to make numbers as such (and a little more)
fun. Enzensberger's book is a triple piece of art: mathematics,
literature, and drawings.
Another book in the
same category of superb pedagogy, mainly about numbers, is Lawrence
Potter, Mathematics Minus Fear: All You Ever Wanted to Know About
Mathematics But Were Too Afraid to Ask, London: Marion Boyers,
In Japan Jin Akiyama
directs a TV program Mathematical Circus since 1991, with
five million viewers every week, and textbooks, using visual methods
to make mathematics more attractive and to stimulate creativity.
That raises an important question: are the terms closer to the
things or to the concepts? Ancient written languages used
pictograms close to the things; modern languages join letters into
words with no resemblance. Spoken language have some onomatopoetic
words simulating things, like "sus" (soos)
in Norwegian for air softly blowing, but generally phonemes are
joined into words resembling nothing. That is why we point and use
body and spoken language to be understood.
Chinese characters put
Chinese and Japanese in in-between positions. Some things can still
be recognized, bringing Chinese closer to the ground,
"down-to-earth"; a factor in the famous Chinese
empiricism-pragmatism. Westerners, Germans and the French in
particular, can go on for hours about, say, the relation between
"freedom" and "equality" with no example or
evidence; for Chinese and Japanese examples are needed. Pure
conceptual reasoning is as meaningless as to demand examples of
German or French professors suspended in thin conceptual air.
For a detailed analysis of this see Johan Galtung, "Contradictory
reality and mathematics: a contradiction?", Section 4.4. in
Copenhagen: Ejlers, 1988, pp. 162-175.