Mathematics and culture 2.pdf

Mathematics

and Culture II

Michele Emmer

Mathematics

and Culture II

Visual Perfection:

Mathematics and Creativity

Editor

Michele Emmer

Dipartimento di Matematica “G. Castelnuovo”

Università degli Studi “La Sapienza”, Roma, Italy

Piazzale Aldo Moro 2

00185 Roma, Italy

e-mail: emmer@mat.uniroma1.it

Library of Congress Control Number: 2003064905

A catalog record for this book is available from the Library of Congress.

Bibliographic information published by Die Deutsche Bibliothek

Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;

detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>.

The articles by Apostolos Doxiadis Euclid’s Poetics: An examination of the similarity between nar-

rative and proof, Simon Singh The Rise of Narrative Non-Fiction and Robert Osserman Mathe-

matics Takes Center Stage were originally published in Italian in Matematica e Cultura 2002,

88-470-0154-4 by Springer Italia, Milano 2002.

The Articles Mathematics and Culture: Ever New Ideas; Mathematics, Art and Architecture;

Mathland: From Topology to Virtual Architecture ; Mathematics, Literature and Cinem aand

Mathematics and Raymond Queneau by Michele Emmer were translated from the Italian Lan-

guage by Gianfranco Marletta.

e-mail: gianfranco@marletta.co.uk

Mathematics Subject Classification (2000): 00Axx, 00B10, 01-XX

ISBN 3-540-21368-6 Springer Berlin Heidelberg New York

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Mathematics and Culture: Ever New Ideas

Michele Emmer

One of the interesting questions in the study of links between mathematics, art

and creativity is whether a mathematician’s creativity leads him to invent a new

world, or rather makes him discover one that already exists in its own right. It

could seem like a superfluous question of very little interest. Though it might

seem so to non mathematicians, it certainly isn’t so for many mathematicians like

Roger Penrose, who has dedicated a part of the book The Emperor’s New Mind [1]

to this subject. “In mathematics, should one talk of invention or of discovery?”

asks Roger Penrose. There are two possible answers to the question: when the

mathematician obtains new results, he creates only some elaborate mental con-

structions which, although have no relation to physical reality whatsoever, never-

theless possess such power and elegance that they are able to make the researcher

believe that these “mere mental constructions” have their very own reality. Alter-

natively, do mathematicians discover that “these mere mental constructions” are

already there, a truth whose existence is completely independent of their work-

ings out? Penrose is inclined toward the second point of view, even though he

adds that the problem is not as simple as it seems. His opinion is that in mathe-

matics one determines situations for which the term discovery is certainly more

appropriate than the term invention. There are cases in which the results essen-

tially derive from the structure itself, more than from the input of mathemati-

cians. Penrose cites the example of complex numbers: “Later we find many other

magical properties that these complex numbers possess, properties that we had

no inkling about at first. These properties are just there. They were not put there

by Cardano, nor by Bombelli, nor Wallis, nor Coates, nor Euler, nor Wessel, nor

Gauss, despite the undoubted farsightedness of these, and other, great mathema-

ticians; such magic was inherent in the very structure that they gradually uncov-

ered.”

When mathematicians discover a structure of this kind, it means that they

stumbled upon that which Penrose calls “works of God”. So, are mathematicians

mere explorers? Fortunately not all mathematical structures are so strictly prede-

termined. There are cases in which “the results are obtained equally by merit of

the structure and of the mathematicians’ calculations”; in this case, Penrose says,

it is more appropriate to use the word invention than the word discovery. Hence

there is room for what he calls works of man, though he notes that the discoveries

of structures that are works of God are of vastly greater importance than the

‘mere’ inventions that are the works of man.

Mathematics and Culture II

One can make analogous distinctions in the arts and in engineering. “Great

works of art are indeed ‘closer to God’ than are lesser ones.”

Among artists, Penrose claims, the idea that their most important works reveal

eternal truths is not uncommon, that they have “some kind of prior ethereal exis-

tence”, while the lesser important works have a more personal character, and are

arbitrary, mortal constructions. In mathematics, this need to believe in an imma-

terial and eternal existence, at least of the most profound mathematical concepts

(the works of God), is felt even more strongly. Penrose observes that “there is a

compelling uniqueness and universality in such mathematical ideas which seems

to be of quite a different order from that which one could expect in the arts or or

engineering.” A work of art can be appreciated or brought into question in differ-

ent epochs, but no-one can put in doubt a correct proof of a mathematical result.

Penrose explains very explicitly that mathematicians think of their discipline

as a highly creative activity that has nothing to envy of the creativity of artists and

indeed, because of the uniqueness and universality of mathematical creation, one

that should be regarded as superior to the artistic discipline; Penrose doesn’t

write it explicitly, but many mathematicians think that mathematics is the true

Art; a difficult, laborious art, with it’s very own language and symbolism, that

produces universally accepted results.

Penrose takes up the question of the existence in its own right of the world of

mathematical ideas, a question that has dogged the science of mathematics since

its inception. It comes from the book Matière à pensée , co-authored by a mathe-

matician, Alain Connes, winner of the Fields medal, and a neurobiologist, Jean-

Pierre Changeux [2].

In one of the chapters in the book, entitled Invention ou découverte the

neurobiologist, speaking of the nature of mathematical objects, recalls that there

is a realist attitude directly inspired by Plato, an attitude that can be summarised

in the phrase: the world is populated by ideas that have a reality separate from

physical reality. Supporting his observation, Changeux quotes a claim of Dieu-

donné, according to whom “mathematicians accept that mathematical objects

possess a reality distinct from physical reality”, a reality that can be compared to

that which Plato accords to his Ideas. From this point of view it is of secondary im-

portance whether or not the mathematical world is a divine creation, as the math-

ematician Cantor (1874–1918) believed: “The highest perfection of God is the pos-

sibility of creating an infinite set, and his immense bounty leads him to do so.”

We are in complete divine mathesis, in complete metaphysics. This is surpris-

ing to serious scientists, comments Changeux. Connes, the mathematician, not at

all bothered by the biologist’s arguments, replies very clearly that he identifies

strongly with the realist point of view. After emphasising that the sequence of

prime numbers has a more stable reality than the material reality surrounding us,

he notes that “one can compare the work of a mathematician to that of an ex-

plorer discovering the world.” If practical experience leads us to discover pure

and simple facts such as, for example, that the sequence of prime numbers appear

to have no end, the work of a mathematician consists of proving that there exists

an infinite number of primes. Once this property has been proved, no-one can

ever claim to having found the largest prime of all. It would be easy to show them

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