Please note: currently TUP does not take any new orders.
Peace Mathematics
ISBN: 978-82-300-0723-5
Year: 2012

Price: 20 €
(+ 10 € postage)

Peace Mathematics

This book is the joint work of two friends who have dedicated their lives to peace theory and practice, but also have a love for mathematics. It brings peace and mathematics closer, as peace mathematics; showing what such concepts as numbers, sets, probability, logic, relations, matrices, graphs, games, systems, calculus, chaos and catastrophe can contribute to peace theory and practice.


Johan Galtung and Dietrich Fischer

Peace Mathematics

Table of Contents


Prologue: Peace, Mathematics, and Peace Mathematics

Introduction: A Dialogue Between Enthusiast E and Skeptic S

[1] Numbers

Math: Transcendence, Primes, Goldbach Conjecture, Zero, Infinity

Peace: Camel Conflict, Transcendence, Dilemma-Tetralemma, Möbius

[2] Sets

Math: Intension-Extension, Zero and Product-Sets, Combinatorics

Peace: Commission-Omission, DPT, Polities, Economies, Mediation

[3] Probability

Math: Laplace-von Mises, Stochasticity, Parameters, Errors Type I&II

Peace: Equality Concepts and Measures, Peace as Disorder, Entropy

[4] Logic

Math: Implication, Heuristics Using Zero-Sets and Product-Sets

Peace: Approaches to Political-Economic Equality, Transformation

[5] Relations

Math: Attributes vs Relations, Types, Structure, Isomorphism

Peace: Building Equivalence, Subverting Dominance, Balance, Equity

[6] Matrices

Math: Representation of Relations, Stochastic Relation Matrices

Peace: Representations of Structures and Conflicts Dynamics

[7] Graphs

Math: Representation of Relations, (Im)Balance, Harary Theorem

Peace: Graphs of Direct and Structural Violence and Peace, Change

[8] Games

Math: Game Logic, Saddle Points, Pareto Optimum, Nash Equilibria Peace: Prisoner's Dilemma, Axelrod-Rapoport, Discourse Problems

[9] Change

Math: Calculus, Differential Equations, Stability-Instability

Peace: Richardson's Arms Races, Common Security, Defensive Defense

[10] Systems

Math: Control Feedbacks, Positive, Negative, Both

Peace: Feedbacks for Peace and Violence Processes, Examples

[11] Chaos

Math: Fractal vs Euclidean Geometry, Self-Similarity, Iteration

Peace: Normative Reality, Layers, Evolution as Normative Development

[12] Catastrophe

Math: Discontinuous Qualitative Change, Dialectics, Ruptures

Peace: Change of Structure, Structure of Change, Evolution

Epilogue: Enthusiast E and Skeptic S: Dialogue at a Higher Level




Johan Galtung and Dietrich Fischer

Peace Mathematics

Prologue: Peace, Mathematics, and Peace Mathematics

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 "mathematese".

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 of peace.

A basic point about peace to get started.

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:

negative-disharmonious: bad-good for one is good-bad for the other;

indifferent: non-relation, neither cares about good-bad for Other;

positive-harmonious: 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:

negative peace: the absence of violence, like cease-fire, or being apart; no negative, no positive, only indifferent, relations; and

positive 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:

negative peace studies: how to reduce-eliminate violent relations;

positive peace studies: how to make cooperative-harmonious relations;

violence-war-arms 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 human beings.

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.

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 answer, 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 good questions?3 Could "thinking mathematically" be a basis for thinking systematically, creatively, constructively? For instance, about relations, and about structures?

Could mathematics 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 reality?

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:

Arithmetic: patterns of counting and numbers

Geometry: patterns of shapes

Calculus: patterns of motion

Logic: patterns of reasoning

Probability: patterns of chance

Topology: 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 glittering example.

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 algebra7 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




1. Arithmetic, Geometry, Logic, Topology, Algebra

3. Probability


2. Calculus

4. Statistics

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 vastly underdeveloped.

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 not.

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". Too controversial?11

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 engineers, etc.

And yet this is what the PISA tests measure. Maybe Norway should be congratulated on scoring badly12, 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 fun.15 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 reality.

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 how.

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 puberty revolts.

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.


1. These particular PISA figures are from one year, 2003, reported in Der Spiegel, 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.

2. 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.

3. 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.

4. 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.

5. New York: Freeman & Co, 1998. The book is pedagogically brilliant, and so is the deeper, more technical follow-up, Keith Devlin, Mathematics: The New Golden Age, New York: Columbia University Press, 1999. There re also excellent books in French, like Richard Mankiewicz, L'Historie des Mathématiques, 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 Neuf Chapitres, Paris: Dunod, 2004.

How wonderful it would have been with such books giving depth and breadth to the study of mathematics 50 years ago!

6. Devlin 1998, p. 3.

7. 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 Selskab, Forhandlinger, Bd XXV, 1952, no. 18.

8. 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.

9. The word algebra comes from an Arab expression, "al iebr e al mokabala", restoration and reduction.

10. Formulas in Mathematics (Oslo: Gyldendal, 2001, 19th edition).

11. The Japan Times 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 [3] 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 day.

12. 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) when f(x)=-x2-2x, 1+2x32, (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 about gender.

13. 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.)

Alan Turing (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 disorders.

China Daily, 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".

14. 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 Nouvel Observateur, 17-23 February 2005).

15. Hans Magnus Enzensberger, in his fine Der Zahlenteufel, 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, 2006.

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.

16. 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.

17. For a detailed analysis of this see Johan Galtung, "Contradictory reality and mathematics: a contradiction?", Section 4.4. in Methodology and Development, Copenhagen: Ejlers, 1988, pp. 162-175.


© 2024 TRANSCEND International