Introducing Mathematics
Icon Books (UK). Totem Books (USA). Published 1999 (ISBN
1-84046-011-3)
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...
Review section
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:
amazon.co.uk)
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.
Sections
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.
Quotes
"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. (http://traveltocaribbeanislands.com/1840460113.html)
J Y S Czhang
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.]
(http://listserv.brown.edu/archives/cgi-bin/wa?A2=ind0108c&L=conlang&F=&S=&P=2332)
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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