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.

Reviews

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). www.goodreads.comRunning 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). www.goodreads.com

A quick canter (should that be Cantor?) through the history of mathematics in a comic strip format.

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..."

_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).

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.]

(http://www.critiqueslibres.com/i.php/vcrit)

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...

(http://www.popularscience.co.uk/reviews/rev175.htm)

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