Borin Van Loon: Maths Guide coverBorin Van Loon: Teacher 2aaIntroducing Mathematics

Icon Books (UK). Totem Books (USA). Published 1999

by Ziauddin Sardar & Jerry Ravetz, Illustrated / designed by Borin Van Loon
Most mathematicians are born not made. I fall into the latter category (as so often in these abstruse subjects, I represent "the beginner" in the making of the book). Having done very badly at Pure Maths at 'A' Level when at school I feel amply qualified to be appalled by the rarified and mystifying mental contortions of the mathematician. Hence my pride at the end result of what must rank as one of my most taxing 'Introducing' titles of the lot.

Here's a sample spread from the book...


This is a guide to maths as a subject, If maths ever seemed boring and pointless when you were at school this book will illuminate the subject and give you a glimpse of this fascinating and sadly out of vogue subject. Sadly although it will get you interested, unless you have already covered the maths, it is likely to remain obscure. Frightful_elk (Sept 5 2008).

Running the full gamut of the history of math, "Intro" takes you on an exciting, illustrated, and often humorous voyage through just about every sort of number crunching aneurysm inducing math conundrum that has presented itself before humanity in the past three thousand years or so.
Baudolino (Dec 27, 2010).

A quick canter (should that be Cantor?) through the history of mathematics in a comic strip format.
A surprising amount of information is included, but there are also rather too many printing mistakes. The latter could cause significant confusion. Not one of the best of the "Introducing..." series. (Reviewer: elgar22 from Haslemere, Surrey, England:

What to do with the time you used to spend on long division. [An amazing, lengthy posting from a discussion website.]
Shelley Walsh, Tue, 30 Sep 2003
I just ran across a book that I just realized embodies my idea really quite closely of what the non-specialist should learn about mathematics. The book is Introducing Mathematics by Ziauddin Sardar, Jerry Ravetz and Borin Van Loon. It is 171 pages with by far more space taken up by pictures than words, but nonetheless gives a far better idea idea of what mathematics is about than any curriculum anyone has dared to present.
Why Maths?, Counting, Written Numbers, The Zero, Special Numbers, Large Numbers, Powers, Logarithms, Calculation, Equations, Measurement, Greek Mathematics, Pythagoras, Zeno's Paradoxes, Euclid, Chinese Mathematics, The Chu Chang, Four Chinese Mathematicians, Vedic Geometry, Brahmagupta, Jain Numbers, Vedic and Jain Combinations, Mathematical Verse, Ramanujan, Islamic Mathematics, Al-Khwarazmi, Development of Algebra, The Discovery of Trigonometry, Al-Battani, Abu Wafa, Ibn Yunu and Thabit Ibn Qurra, Al-Tusi, Solutions of Problems Involving Integers, Emergence of European Mathematics,
Rene Descartes, Analytic Geometry, Functions, The Calculus, Differentiation, Integration, Berkeley's Questions, Euler's God, Non-Euclidean Geometries, N-Dimension Spaces, Evariste Galois, Groups, Boolean Algebra, Cantor and Sets, Crisis in Mathematics, Russel and Mathematical Truth, Godel's Theorem, The Turing Machine, Fractals, Chaos Theory, Topology, Number Theory, Statistics, P-Values and Outliers, Probability, Uncertainty, Policy Numbers,Mathematics and Eurocentrism, Ethnomathematics, Mathematics and Gender, Where Now?, Further Reading
Now as you may guess, the book came from the history of mathematics section of the bookstore rather than the mathematics section. But nonetheless there is really a lot of mathematics taught in it, and the mathematics that you can learn from a book like this is, I think, of a far more important kind that the boring stuff we teach in required mathematics general education courses, and definitely far more interesting, important, and possibly even more mental strengthening--whatever that is--than the drilling of long division. And, by the way, there is absolutely nothing in that book, even in the final pages that can be any better understood by someone who can do division by pencil and paper than by one who needs a calculator for it.
"The best way to systematize naming and counting is to have a "base", a number that marks the beginning of counting again. The simplest base is just two. For example, the Gumulgal, an Australian indigenous people, counted like this:"... "This may seem primitive and tedious. But the base two, in form of 0's and 1's is built into digital computers as the foundation of all their calculations."
"Just how easily we can reach large numbers can be well illustrated by that old evil, the chain letter."
"In order to multiply or divide two logarithmic expressions, we use the fact that multiplication and division of powers of a number corresponds to addition and subtraction of these powers."
"Counting and calculation concern separate, discrete quantities, involving exact numbers. Measurement, by contrast, concerns continuous magnitudes. No measurement is exact. When we compare the object being measured against a standard, we always interpolate between the points on the finest scale. And every report of a complex measurement has (or should have!) an "error bar" to indicate the "fringe" of uncertainty associated with it.
"once curves were perceived as graphs of functions, then the problems of areas could be seen in a double perspective. On the one hand, areas could be "exhausted" by thin vertical strips; and the other, the area as a new function is just the one whose derivative equals the original function."
[Berkeley callout] "I observe that forming a quotient with the increments makes sense only if it is not zero; otherwise we are dividing by zero, and that is illegitimate. Is the increment always non-zero, or is it "the ghost
of a vanished quantity? And apart from that, sirrah, Mr. Newton is naked."
"Euler's formula is a mysterious, transcendent expression that connects the five more fundamental numbers in the universe:"
[uncountability of real numbers] "How could we possibly construct a number that is not on that list? Well suppose we have one that is different from the first number in the first place, different from the second number in the second place, third in the third, fourth in the fourth, and son on, and on."
"When one is talking about "sets" in such a general way, there is nothing to stop one from referring to "the set of all sets"--it makes grammatical sense, doesn't it? Now, that must be the biggest set of all, and its size
will be a certain Aleph, let's call it Aleph F for final. But, like any other set, it will have a power set, whose number can be defined as 2 to the Aleph F. So what we defined as truly biggest set, the set of all sets, can
generate an even bigger one."
[Godel callout] "My theorem proved that any consistent mathematical system must by incomplete..."

I enjoyed this treatment of the history of mathematics, and, in particular, I liked the sections on Islamic mathematics and Cantor's theories on infinity. The book is quite thought-provoking, and I recommend it highly. The book was a good but very, very basic ntroduction to mathematics, including the areas of basic research. The best part is at the end, discussing mathematics, cultural theory and where they intersect. Not something you'd EVER find in a classical math text. (

A really good basic introduction to ethnomathematics is contained in _Introducing Mathematics_ by Ziauddin Sardar, Jerry Ravetz, & Borin Van Loon (Icon Books in the UK, Totem Books in the USA, 1999, ISBN 1-84046-011-3). Cited in this beginners' book is the following for a book-length treatment:
_Ethnomathematics_ by M. Ascher (Brooks/Cole Publishing, Pacific Grove, 1990, ISBN not given!).
_Introducing Mathematics_ does a good job of giving the basics about ancient Hindu, Chinese, Mayan, Babylonian, Egyptian, Greek, Native American, African, aboriginal (Australian), etc. math systems and their attendent viewpoints. Good antidote to Eurocentricism and great inspiration to constantly see the world anew from some other's worldviewpoint (IMHO a much needed skill in not only conlanging, but in this 21st Century). [Eh? -Ed.]
J Y S Czhang  (

Maths sans aspirine (Les) de Ziauddin Sardar, Jerry Ravetz, Borin Van Loon
critiqué par Kinbote, le 19 décembre 2001 (Jumet - 46 ans) 4 STAR REVIEW
La plus grande création de l'intelligence humaine depuis l'eau chaude
 Voici un livre traduit de l'anglais qui, sous couvert d ’humour et de légèreté, nous conduit à l’essentiel des mathématiques. Le livre est truffé de collages de vieilles illustrations réalisés par Borin Van Loon qui font passer par les bulles la pilule du texte, jamais amère, faut-il le préciser.
 D'abord les auteurs décrivent les divers systèmes numériques (des Aztèques, Mayas, Egyptiens, Babyloniens et Chinois) qui ont assuré de par le globe la préhistoire des mathématiques avant la théorisation effectuée par Euclide dans ses fameux Livres. Cette diversité des approches mathématiques est maintenue tout au long du manuel. Les mathématiciens mis à l’honneur sont ceux qui ont fait bifurquer leur discipline, lui ont ouvert de nouveaux horizons ou l'ont profondément mise en question comme Zénon d’Elée avec ses paradoxes (au V ème siècle avant Jésus-Christ) dont le plus célèbre reste celui de la course d'Achille avec la tortue.
 On sait peu que les mathématiques se sont, dans les premiers siècles de notre ère, développées surtout en Chine et en Inde (ces deux civilisations trouvant des approximations très correctes de Pi et découvrant le triangle de Pascal bien avant le penseur français), assurant ainsi le lien entre l’arrêt des recherches grecques, faute de civilisation hellénique, et la reprise de toutes les traditions existantes par les Arabes dès le IX ème siècle, avant leur transmission à l'Europe de la Renaissance. L’apport arabe concernera surtout l’algèbre (application systématique des opérations de l’arithmétique élémentaire aux expressions algébriques) et de la trigonométrie. Descartes sera ensuite celui qui va fusionner l’algèbre et la géométrie parce qu'il jugeait la première « obscure et confuse » et la seconde « trop restrictive » pour fonder la géométrie analytique.
 Les auteurs nous donnent une illustration intéressante, à partir d'une automobile en mouvement , de la dérivation et de l’intégration, les deux opérations qui sont à la base du calcul infinitésimal. On apprend que Berkeley,philosophe et évêque anglican, au XVIII ème siècle, visera à démontrer que les libres penseurs, dans leur science, reproduisaient le dogmatisme et l’obscurité dont on accusait à l'époque les pires théologiens, critiques à l'encontre de la Raison qui seront reprises au XX ème siècle par T.S. Kuhn.
 Les auteurs citent à l’occasion un ouvrage de fiction amusant de E.A. Abbott décrivant une société de polygones vivant dans un plan.
 Evariste Galois, au destin tragique, est celui qui va ouvrir la voie, avec sa théorie des groupes, à une mathématique structurelle, libérée des nombres et portant sur de nouveaux objets. L’utilisation de l'algèbre booléenne (des ensembles) permettra, entre autres choses, le fonctionnement des moteurs de recherche sur le web. On voit aussi la méthode de classement employée par Cantor pour énumérer tous les nombres rationnels (les fractions) qui va, lui aussi, mettre au jour des paradoxes et des incomplétudes qui vont sérieusement ébranler les mathématiques, avant les travaux de logiciens comme Bertrand Russell ou Kurt Gödel qui ne feront que les mettre un peu plus en péril. On nous explique en quelques mots de quoi retourne la théorie du chaos, des fractales ou la topologie et aussi à nous méfier de l'usage que la politique fait des chiffres et des résultats statistiques. Ils distinguent les trois types de probabilités, souvent confondus: géométrique empirique et d'estimation. Ils relèvent pour terminer la mainmise de la civilisation occidentale sur les mathématiques et les sciences, qui s’est appropriée ou a négligé les formes de savoir des peuples non européens (Inde, Chine ou Islam) qui ne répondaient pas à la conception platonicienne des mathématiques (savoir affranchi de la pratique qui atteint la Vérité et ignore la contradiction) ou au vieux projet de Descartes de rendre tout mathématisable. [Merci beaucoup, Kinbote, mon ami!- Ed.]

It is almost impossible to rate these relentlessly hip books - they are pure marmite*. The huge  Introducing ... series (about 80 books covering everything from Quantum Theory to Islam), previously known as ... for Beginners, puts across the message in a style that owes as much to Terry Gilliam and pop art as it does to popular science. Pretty well every page features large graphics with speech bubbles that are supposed to emphasise the point.
Does it work in practice? In this case it's a mixed bag. The illustrations are rather less mind boggling than in many of the series relying a lot on what look like old magazine illustrations - they are wonderful, but perhaps not quite as good at shocking the illustration into your brain as the weird and wonderful images these books usually contain.
The maths itself is fine, though there are a few worrying omissions, uncomfortable changes of speed and perhaps a rather excessive political correctness. For example, talking about powers, it doesn't bother to explain how multiplying two powered numbers together, you add the powers. (e.g. 105 x 103 = 108 because 5+3=8). If we had been told that it would make a lot more sense of the negative powers, and particular of something to the power zero (e.g. 100=1), the explanation for which in the book is absolutely feeble.
Changes of speed are evidenced in the way it shoots into some stuff with very little explanation (tedious pages of trigonometry, for example), the spends ages over a triviality. And the political correctness is clear in the excessive attempts to allocate priority wherever possible to a non-Western source. E.g. the Jain idea that there are three types of infinity "near infinite, truly infinite and infinitely infinite" receives the comment "European mathematics did not scale those heights until just a century ago, in the work of Cantor." Apart from the fact that Cantor was dealing with proofs on the nature of infinite sets, not vague woffly statements that could mean almost anything, this overlooks the fact that Galileo had already made much more specific comments on different infinity well before Cantor was around.
Don't take it that this book's all bad. As it says on the cover, there just isn't another book around that can precis a subject the way this does, but it just could have done the job better.
*Marmite? If you are puzzled by this assessment, you probably aren't from the UK. Marmite is a yeast-based product (originally derived from beer production waste) that is spread on bread/toast. It's something people either love or hate, so much so that the company has run very successful TV ad campaigns showing people absolutely hating the stuff...

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