“Mathematics” comes from Greek mathema knowledge, science and mathesis learning, understanding-Leibniz’ mathesis universalis. Deep knowledge?
Mathematics started with numbers (arithmetic) and shapes (geometry).* The concepts were related by “self-evident truths”, axioms, mirroring, inspired by empirical reality, counting, observing. And took on its own life: from primary concepts to new concepts by definition, from axioms to theorems by deduction. An abstract reality with one law: No Contradictions: a theorem and its negation cannot both be true, nor a theorem be both true and false. Forbidden, Streng verboten, Interdit. Many key mathematicians were German and French. Deductive cultures?
From the other side of the world came the daoist axiom that life is contradictory, yin/yang, with forces and counter-forces acting upon each other ad inf. Can mathematics only mirror dead nature? Does it through mirroring reduce life to contradiction-free, dead, matter? And what can we possibly learn from mathematics for peace, and how?
Something has already been said. Be careful with concepts. The definitions are contracts in public discourse; by “peace” I mean, by “violence” I mean. Stick to them, do not swerve. Be honest, make basic beliefs explicit, keeping in mind that what is self-evident to you may not be so to others. Draw conclusions, deductions in more or less stringent manners. In mathematics they are valid if stringent. But whether they mirror empirical truth in a reality replete with contradictions is another matter; that has to be tested empirically.
This is already a lot. Axioms provide economy of thought: “to reduce violence, solve and concile underlying conflicts and traumas”. And one more: “Solutions for simple conflicts between two goals are neither-nor, compromise or both-and (transcendence, “going beyond”). From this follow the first steps in mediation: identify the goals, the contradictions, and what “both-and” means in reality. Thanks, math.
But there is more to learn for mediation. Have a look at this, from Peace Mathematics (with my late friend Dietrich Fischer, p. 33):
The Evolution of Numbers
| Having | Adding | Getting | Permitting |
| natural numbers | negative numbers, zero, infinity |
integers | unlimited addition and unlimited subtraction |
| whole numbers | fractions | rational numbers | unlimited division |
| rational numbers | irrational numbers | real numbers | roots of positive numbers |
| real numbers | imaginary numbers | complex numbers | roots of negative numbers |
In the left column is an old mathematical reality. Then something is added to get a new mathematical reality overcoming a contradiction between an operation and what the old mathematical reality could accommodate. Happening over a hundred times, new mathematical languages compensate for losses in natural languages. Numeracy is as important as literacy.
Is mathematics as real as material reality, to be discovered; or a brain-child awaiting more inventions? Both-and is boring, but safe.
Called the Queen of sciences leading softly from behind, Symbolic Logic (Russell), directing how we think. But, who is the King? Maybe the purest parts of physics, mechanics and simple astronomy, not in the extremely chaotic reality of all kinds of things called “nature”. Their marriage in the heavens was written in mathematical scripts, by priests like Galileo, Newton, and Einstein. And has been in command.
Physics created new physical reality through engineering and architecture; chemistry through artificial compounds; biology through breeding; health sciences by transforming illness to wellness, health.
Humaniora have created new reality all the time, new words and other symbols, new ways of connecting them. But the social sciences seem more tied to empirical reality: social, political, economic innovators rarely include sociologists, politologists and economists.
So much for learning, directly, indirectly. How about teaching?
Numeracy, definitely; like the tricks for adding-subtracting in the head two digits numbers. The math of everyday life. Statistics! see Andrew Hacker, “The wrong way to teach math” (NYT 27/2/2016):
“figures cited on income distribution, climate change or whether cell-phones can damage your brain. What’s needed is a facility for sensing symptoms of bias, questionable samples and dubious sources of data”.
Calculus, derivatives-integrals, are not for daily life but for engineers etc.; leave it to them. Nor the obsession with curves by cutting cones: circles, ellipses, para/hyper-bolas. Or the related obsession with second degree equations, chasing the two X’s into the remotest corners; a mental sport related to nothing in daily life.
Boolean algebra makes and-or-either/or-neither/nor-both/and visual, and is helpful for thinking logically. Sets in general, relations, matrices. And, indeed, graphs, simple dots on sheet of paper, related by positive or negative lines or unrelated, for positive, negative and no interaction. Learning to identify center and periphery in a school class and in inter-state relations, and the minimum changes needed for maximum peace. In short, the mathematics of structures, combining simple arithmetic and geometry, proves very useful exactly in daily life. With exercises mirroring social reality.
But we should also teach about mathematics, says Edward Frenkel, in Love and Math: The Heart of Hidden Reality, reviewed by Jim Holt, “A Mathematical Romance” (NYRB, 5 Dec 2013). Like we teach about art without demanding that students become artists beyond some singing and drawing. Two huge edifices, Art and Math, waiting to be enjoyed, with many floors, and rooms, and views. And that marvelous sense conveyed by both when well done: Just right. Es stimmt. Ça-y-est.
To make the Queen boring is a crime against humanity. Stop it.
Johan Galtung
29 February 2016
