Ana
[Compact Textbooks in Mathematics] Turning Points in the History of Mathematics 
[Compact Textbooks in Mathematics] Turning Points in the History of Mathematics 
Grant, Hardy, Kleiner, IsraelBu kitabı ne kadar beğendiniz?
İndirilen dosyanın kalitesi nedir?
Kalitesini değerlendirmek için kitabı indirin
İndirilen dosyaların kalitesi nedir?
Cilt:
10.1007/97
Yıl:
2015
Dil:
english
DOI:
10.1007/9781493932641
Dosya:
PDF, 3.28 MB
Etiketleriniz:
Sorun bildir
This book has a different problem? Report it to us
Check Yes if
Check Yes if
Check Yes if
Check Yes if
you were able to open the file
the file contains a book (comics are also acceptable)
the content of the book is acceptable
Title, Author and Language of the file match the book description. Ignore other fields as they are secondary!
Check No if
Check No if
Check No if
Check No if
 the file is damaged
 the file is DRM protected
 the file is not a book (e.g. executable, xls, html, xml)
 the file is an article
 the file is a book excerpt
 the file is a magazine
 the file is a test blank
 the file is a spam
you believe the content of the book is unacceptable and should be blocked
Title, Author or Language of the file do not match the book description. Ignore other fields.
Are you sure you want to report this book? Please specify the reason below
Change your answer
Thanks for your participation!
Together we will make our library even better
Together we will make our library even better
Dosya 15 dakika içinde epostanıza teslim edilecektir.
Dosya 15 dakika içinde sizin kindle'a teslim edilecektir.
Not: Kindle'a gönderdiğiniz her kitabı doğrulamanız gerekir. Amazon Kindle Support'tan gelen bir onay epostası için eposta gelen kutunuzu kontrol edin.
Not: Kindle'a gönderdiğiniz her kitabı doğrulamanız gerekir. Amazon Kindle Support'tan gelen bir onay epostası için eposta gelen kutunuzu kontrol edin.
Conversion to is in progress
Conversion to is failed
0 comments
Kitap hakkında bir inceleme bırakabilir ve deneyiminizi paylaşabilirsiniz. Diğer okuyucular, okudukları kitaplar hakkındaki düşüncelerinizi bilmek isteyeceklerdir. Kitabı beğenip beğenmediğinize bakılmaksızın, onlara dürüst ve detaylı bir şekilde söylerseniz, insanlar kendileri için ilgilerini çekecek yeni kitaplar bulabilecekler.
1


2


Compact Textbooks in Mathematics For further volumes: http://www.springer.com/series/11225 Compact Textbooks in Mathematics This textbook series presents concise introductions to current topics in mathematics and mainly addresses advanced undergraduates and master students. The concept is to offer small books covering subject matter equivalent to 2 or 3hour lectures or seminars which are also suitable for selfstudy. The books provide students and teachers with new perspectives and novel approaches. They feature examples and exercises to illustrate key concepts and applications of the theoretical contents. The series also includes textbooks specifically speaking to the needs of students from other disciplines such as physics, computer science, engineering, life sciences, finance. • compact: small books presenting the relevant knowledge • learning made easy: examples and exercises illustrate the application of the contents • useful for lecturers: each title can serve as basis and guideline for a 23 hours course/lecture/seminar Hardy Grant Israel Kleiner Turning Points in the History of Mathematics Hardy Grant Department of Mathematics and Statistics York University Toronto, Ontario Canada Israel Kleiner Department of Mathematics and Statistics York University Toronto, Ontario Canada ISSN 22964568 ISBN 9781493932634 DOI 10.10.1007/9781493932641 Springer New York Heidelberg Dordrecht London ISSN 2296455X (electronic) ISBN 9781493932641 (eBook) Library of Congress Control Number: 2015952852 Birkhäuser © Springer Science+Business Media, LLC 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or h; ereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acidfree paper Springer Science+Business Media LLC New York is part of Springer Science+Business Media (www.springer.com) v Preface The development of mathematics has not followed a smooth or continuous curve, although in hindsight we may think so. As the mathematician and historian of mathematics Eric Temple Bell (1883–1960) said: “Nothing is easier … than to fit a deceptively smooth curve to the discontinuities of mathematical invention” [1, p. viii]. In fact, there have been dramatic insights and breakthroughs in mathematics throughout its history, as well as what seemed for a time to be insurmountable stumbling blocks—both leading to major shifts in the subject. And then—for the most part—there have been relatively “routine” developments, from whose importance we do not wish to detract. Here are two “nonroutine” examples: a. The invention (discovery?) of noneuclidean geometry—a breakthrough which was about two millennia in the making (ca 300 BC—ca 1830), and which culminated in the resolution of “the problem of the fifth postulate.” This brought about a reevaluation of the nature of geometry and its relationship to the physical world and to philosophy, as well as a reconsideration of the nature of axiomatic systems. See 7 Chapter 7. b. The introduction, around the mideighteenth century, of “foreign objects”, such as irrational and complex numbers, into number theory, to be followed in the late nineteenth century by the founding of a new subject—algebraic number theory. These developments paved the way for splendid achievements of modern mathematics, including, to take a familiar example, the resolution of the problem, stated in the 1630s, concerning the unsolvability in integers of Fermat’s equation xn + yn = zn, n > 2. The proof of unsolvability, given by Andrew Wiles in 1994, required most of the grand ideas which number theory had evolved during the twentieth century. See 7 Chapter 6 and [4]. We aim in this book to discuss some of these major turning points—transitions, shifts, breakthroughs, discontinuities, revolutions (if you will)—in the history of mathematics, ranging from ancient Greece to the present [2, 3]. Among those which we consider are the rise of the axiomatic method (7 Chapter 1), the wedding of algebra and geometry (7 Chapter 4), the taming of the infinitely small and the infinitely large (7 Chapter 5), the passage from algebra to algebras (7 Chapter 8), and the revolutions resulting in the late nineteenth and early twentieth centuries from Cantor’s creation of transfinite set theory (7 Chapters 9 and 10). The historical origin of each turning point is discussed, as well as some of the resulting mathematics. The above examples, and others discussed in this book, highlight the great drama inherent in the evolution of mathematics. Teachers of this grand subject will benefit from reflecting on this important aspect of it, focusing on the big ideas in its development—though not, of course, to the neglect of “routine” mathematics. They should pass on to students—at some point in their studies—at least the spirit, if not always the content, of these ideas. In particular, students should be made aware that not every fact, technique, idea, or theory is as important, and should receive as much emphasis, as every other. If this thought is not conveyed to them, our teaching will do justice neither to the students nor to the subject. vi Preface The book contains ten chapters, more or less of equal length, though not of equal difficulty. They describe only a small number of “turning points” in the history of mathematics, and we have appended an 11th chapter which suggests “Some Further Turning Points” to pursue. Each chapter contains about ten “problems and projects”, most of which are intended to deepen or extend the material in the text. At the end of each chapter there is a substantial list of references, whose aim is to elaborate, enhance, and exemplify the material in the text proper. Finally, the book has a comprehensive index. This book can be read by a person with some mathematical background who is interested in getting a nontraditional look at aspects of the history of mathematics. It can also be used in historyofmathematics courses, especially those centered around the important idea of “turning points.” Moreover, since appreciation of the historical development of the central ideas of mathematics enhances, we strongly believe, one’s understanding and appreciation of the subject, this book can serve as a text in a capstone course for mathematics majors, a course that will integrate and “humanize” at least some of their knowledge of mathematics by placing it in historical perspective. In any such course our book will probably need to be supplemented by additional technical material; a teacher will know best when and how to use this “extra” material in his or her particular classroom setting. Teachers are resourceful and will likely use the book in ways we have not anticipated. One of the reviewers of our book said the following: “I see the value of the manuscript in its role as a ‘starter’ to ignite love for the history of maths and to give a first overview. It is a good ‘teaser’.” We hope that readers’ experiences will justify this assessment. We want to thank Chris Tominich, Assistant Editor, Birkhäuser Science, for his outstanding cooperation in seeing this book to completion, and Ben Levitt, Birkhäuser Science Editor, for his cordial and efficient support. I (HG) want to thank three generations of my family—my dear sisters Nancy and Kathy, and my cherished nephews, niecesinlaw, and greatnieces Ross, Sevaun, Zada, and Alyn, and Ian, Lynn, and Charlotte—for love, inspiration, and many good times. And I (IK) want to thank my dear wife of 50 years, Nava, for her support and encouragement over these many years. I have also gained invaluable perspective in seeing our children and grandchildren—Ronen, Leeor, Tania, Ayelet, Howard, Tamir, Tia, Jordana, Jake, and Elise—grow, mature, and thrive. References 1. Bell, E.T.: The Development of Mathematics, 2nd ed. McGrawHill, New York (1945) 2. Gillies, D. (ed.): Revolutions in Mathematics. Clarendon Press, Oxford (1992) 3. Laugwitz, D.: Bernhard Riemann 1826–1866: Turning Points in the Conception of Mathematics. Birkhauser, New York (2008) 4. Singh, S.: Fermat’s Enigma: The Quest to Solve the World’s Greatest Mathematical Problem. Penguin, New York (1997) Hardy Grant and Israel Kleiner vii Contents 1 Axiomatics—Euclid’s and Hilbert’s: From Material to Formal���������������� 1.1 Euclid’s Elements ���������������������������������������������������������������������������������������������������������������������� 1.2 Hilbert’s Foundations of Geometry������������������������������������������������������������������������������������ 1.3 The Modern Axiomatic Method ���������������������������������������������������������������������������������������� 1.4 Ancient vs. Modern Axiomatics������������������������������������������������������������������������������������������ Problems and Projects ���������������������������������������������������������������������������������������������������������� References ���������������������������������������������������������������������������������������������������������������������������������� Further Reading������������������������������������������������������������������������������������������������������������������������ 1 1 3 5 6 7 7 8 2 Solution by Radicals of the Cubic: From Equations to Groups and from Real to Complex Numbers ���������������������������������������������������������������������� 2.1 Introduction ������������������������������������������������������������������������������������������������������������������������������ 2.2 Cubic and Quartic Equations ���������������������������������������������������������������������������������������������� 2.3 Beyond the Quartic: Lagrange�������������������������������������������������������������������������������������������� 2.4 Ruffini, Abel, Galois ���������������������������������������������������������������������������������������������������������������� 2.5 Complex Numbers: Birth ������������������������������������������������������������������������������������������������������ 2.6 Growth������������������������������������������������������������������������������������������������������������������������������������������ 2.7 Maturity���������������������������������������������������������������������������������������������������������������������������������������� Problems and Projects ��������������������������������������������������������������������������������������������������������� References ���������������������������������������������������������������������������������������������������������������������������������� Further Reading����������������������������������������������������������������������������������������������������������������������� 9 9 10 11 13 13 15 16 17 18 18 3 nalytic Geometry: From the Marriage of Two Fields to the A Birth of a Third ���������������������������������������������������������������������������������������������������������������������� 19 3.1 Introduction ������������������������������������������������������������������������������������������������������������������������������ 3.2 Descartes ������������������������������������������������������������������������������������������������������������������������������������ 3.3 Fermat ������������������������������������������������������������������������������������������������������������������������������������������ 3.4 Descartes’ and Fermat’s Works from a Modern Perspective ���������������������������������� 3.5 The Significance of Analytic Geometry �������������������������������������������������������������������������� Problems and Projects ��������������������������������������������������������������������������������������������������������� References ���������������������������������������������������������������������������������������������������������������������������������� Further Reading����������������������������������������������������������������������������������������������������������������������� 19 19 21 22 23 24 25 25 4 Probability: From Games of Chance to an Abstract Theory �������������������� 4.1 The Pascal–Fermat Correspondence�������������������������������������������������������������������������������� 4.2 Huygens: The First Book on Probability�������������������������������������������������������������������������� 4.3 Jakob Bernoulli’s Ars Conjectandi (The Art of Conjecturing)������������������������������������ 4.4 De Moivre’s The Doctrine of Chances�������������������������������������������������������������������������������� 4.5 Laplace’s Théorie Analytique des Probabilités���������������������������������������������������������������� 4.6 Philosophy of Probability ���������������������������������������������������������������������������������������������������� 4.7 Probability as an Axiomatic Theory���������������������������������������������������������������������������������� 4.8 Conclusion���������������������������������������������������������������������������������������������������������������������������������� Problems and Projects ��������������������������������������������������������������������������������������������������������� References ���������������������������������������������������������������������������������������������������������������������������������� Further Reading����������������������������������������������������������������������������������������������������������������������� 27 27 29 30 32 32 33 33 34 35 35 35 viii Contents 5 Calculus: From Tangents and Areas to Derivatives and Integrals �������������������������������������������������������������������������������������������������������������������������� 5.1 Introduction ������������������������������������������������������������������������������������������������������������������������������ 5.2 SeventeenthCentury Predecessors of Newton and Leibniz���������������������������������� 5.3 Newton and Leibniz: The Inventors of Calculus ���������������������������������������������������������� 5.4 The Eighteenth Century: Euler�������������������������������������������������������������������������������������������� 5.5 A Look Ahead: Foundations ������������������������������������������������������������������������������������������������ Problems and Projects ��������������������������������������������������������������������������������������������������������� References ���������������������������������������������������������������������������������������������������������������������������������� Further Reading����������������������������������������������������������������������������������������������������������������������� 37 37 37 40 43 45 46 46 47 6 Gaussian Integers: From Arithmetic to Arithmetics ������������������������������������ 6.1 Introduction ������������������������������������������������������������������������������������������������������������������������������ 6.2 Ancient Times���������������������������������������������������������������������������������������������������������������������������� 6.3 Fermat ������������������������������������������������������������������������������������������������������������������������������������������ 6.4 Euler and the Bachet Equation x2 + 2 = y3 ���������������������������������������������������������������������� 6.5 Reciprocity Laws, Fermat’s Last Theorem, Factorization of Ideals ���������������������� 6.6 Conclusion���������������������������������������������������������������������������������������������������������������������������������� Problems and Projects ��������������������������������������������������������������������������������������������������������� References ���������������������������������������������������������������������������������������������������������������������������������� Further Reading����������������������������������������������������������������������������������������������������������������������� 49 49 49 49 51 51 55 55 56 56 7 Noneuclidean Geometry: From One Geometry to Many �������������������������� 7.1 Introduction ������������������������������������������������������������������������������������������������������������������������������ 7.2 Euclidean Geometry���������������������������������������������������������������������������������������������������������������� 7.3 Attempts to Prove the Fifth Postulate ���������������������������������������������������������������������������� 7.4 The Discovery (Invention) of Noneuclidean Geometry�������������������������������������������� 7.5 Some Implications of the Creation of Noneuclidean Geometry �������������������������� Problems and Projects ��������������������������������������������������������������������������������������������������������� References ���������������������������������������������������������������������������������������������������������������������������������� Further Reading����������������������������������������������������������������������������������������������������������������������� 57 57 57 58 60 62 65 66 66 8 Hypercomplex Numbers: From Algebra to Algebras ���������������������������������� 8.1 Introduction ������������������������������������������������������������������������������������������������������������������������������ 8.2 Hamilton and Complex Numbers�������������������������������������������������������������������������������������� 8.3 The Quaternions ���������������������������������������������������������������������������������������������������������������������� 8.4 Beyond the Quaternions ������������������������������������������������������������������������������������������������������ Problems and Projects ��������������������������������������������������������������������������������������������������������� References ���������������������������������������������������������������������������������������������������������������������������������� Further Reading����������������������������������������������������������������������������������������������������������������������� 67 67 68 69 70 71 72 73 9 The Infinite: From Potential to Actual�������������������������������������������������������������������� 9.1 The Greeks���������������������������������������������������������������������������������������������������������������������������������� 9.2 Before Cantor ��������������������������������������������������������������������������������������������������������������������������� 9.3 Cantor������������������������������������������������������������������������������������������������������������������������������������������ 9.4 Paradoxes Lost������������������������������������������������������������������������������������������������������������������������� 9.5 Denumerable (Countable) Infinity����������������������������������������������������������������������������������� 75 75 76 77 78 78 Contents ix 9.6 Paradoxes Regained�������������������������������������������������������������������������������������������������������������� 9.7 Arithmetic��������������������������������������������������������������������������������������������������������������������������������� 9.8 Two Major Problems ������������������������������������������������������������������������������������������������������������ 9.9 Conclusion�������������������������������������������������������������������������������������������������������������������������������� Problems and Projects ������������������������������������������������������������������������������������������������������� References �������������������������������������������������������������������������������������������������������������������������������� Further Reading���������������������������������������������������������������������������������������������������������������������� 79 80 80 81 82 83 83 Philosophy of Mathematics: From Hilbert to Gödel���������������������������������� 10 10.1 Introduction ���������������������������������������������������������������������������������������������������������������������������� 10.2 Logicism ������������������������������������������������������������������������������������������������������������������������������������ 10.3 Formalism���������������������������������������������������������������������������������������������������������������������������������� 10.4 Gödel’s Incompleteness Theorems �������������������������������������������������������������������������������� 10.5 Mathematics and Faith�������������������������������������������������������������������������������������������������������� 10.6 Intuitionism������������������������������������������������������������������������������������������������������������������������������ 10.7 Nonconstructive Proofs ������������������������������������������������������������������������������������������������������ 10.8 Conclusion�������������������������������������������������������������������������������������������������������������������������������� Problems and Projects �������������������������������������������������������������������������������������������������������� References �������������������������������������������������������������������������������������������������������������������������������� Further Reading���������������������������������������������������������������������������������������������������������������������� 85 85 86 87 88 89 89 90 91 92 92 93 11 Some Further Turning Points ������������������������������������������������������������������������������������ 95 11.1 Notation: From Rhetorical to Symbolic ������������������������������������������������������������������������ 95 11.2 Space Dimensions: From 3 to n (n >3)���������������������������������������������������������������������������� 96 11.3 Pathological Functions: From Calculus to Analysis�������������������������������������������������� 97 11.4 The Nature of Proof: From AxiomBased to ComputerAssisted������������������������ 99 11.5 Experimental Mathematics: From Humans to Machines �������������������������������������� 100 References �������������������������������������������������������������������������������������������������������������������������������� 101 Further Reading��������������������������������������������������������������������������������������������������������������������� 102 Index������������������������������������������������������������������������������������������������������������������������������������������ 105 1 1 Axiomatics—Euclid’s and Hilbert’s: From Material to Formal H. Grant, I. Kleiner, Turning Points in the History of Mathematics, Compact Textbooks in Mathematics, DOI 10.1007/9781493932641_1, © Springer Science+Business Media, LLC 2015 1.1 Euclid’s Elements The axiomatic method is, without doubt, the single most important contribution of ancient Greece to mathematics. The explicit recognition that mathematics deals with abstractions, and that proof by deductive reasoning from explicitly stated postulates offers a foundation for mathematics, was indeed an extraordinary development. When, how, and why this came about is open to conjecture. Various reasons—both internal and external to mathematics (Raymond Wilder calls them “hereditary” and “environmental” stresses, respectively [14])—have been advanced, with various degrees of certainty, for the emergence of the axiomatic method in ancient Greece. Among the suggested causes are: a. The nature of Greek society. Of course people have always—if often unconsciously— used “axioms” in conversations—shared presumptions from which one participant urges conclusions on the others. Two characteristic features of Greek experience, both dating from the fifth century BC, spurred reflection on these everyday occurrences. The courts of law and the citizen assemblies created by the (limited) democracy conferred high value on the techniques and strategies of skilful persuasion. A science of “rhetoric” developed, in which argument proceeded in specific modes and stages, analogous to the successive steps in a euclidean proof (see [10], 7 Chapter 4). b. The predisposition of the Greeks to the kind of philosophical inquiry in which answers to ultimate questions are of prime concern. For example, the attempt by Parmenides (ca. 515−450 BC), the founder of the “Eleatic” school of philosophy, to show that all of ultimate reality is an unchanging unity is generally taken by modern scholars to be the oldest deductive argument that has come down to us [10, p. 102]. Parmenides used the indirect method of proof, assuming the denial of his intended conclusion and reaching an untenable outcome. The famous paradoxes of Parmenides’ pupil Zeno, which claim to prove that motion is impossible, are (of course) also deductive arguments. Zeno also used the indirect method of proof (see [11], but also [9], in which an alternative thesis is proposed). Both of these thinkers may have consciously worked from explicit assumptions— in effect, axioms—though no hint of these survives. In this connection it is interesting to note the view of the eighteenthcentury mathematician and scientist A.C. Clairaut regarding Euclid’s proofs of obvious propositions [8, pp. 618 − 619]: 2 1 Chapter 1 • Axiomatics—Euclid’s and Hilbert’s: From Material to Formal »» It is not surprising that Euclid goes to the trouble of demonstrating that two circles which cut one another do not have a common centre, that the sum of the sides of a triangle which is enclosed within another is smaller than the sum of the sides of the enclosing triangle. This geometer had to convince obstinate sophists who glory in rejecting the most evident truths; so that geometry must, like logic, rely on formal reasoning in order to rebut the quibblers. c. The desire to decide among contradictory results bequeathed to the Greeks by earlier civilizations [12, p. 89]. For example, the Babylonians used the formula 3r2 for the area of 2 8 a circle, the Egyptians × 2r . (There is evidence that the Babylonians also used 3 1 as 9 8 an estimate for π [8, p. 11].) This encouraged the notion of mathematical demonstration, which in time evolved into the deductive method. d. The need to resolve the “crisis” engendered in the fifth century BC by proof of the incommensurability of the diagonal and side of the square [3]. A fundamental tenet of the Pythagoreans was that all phenomena can be described by numbers, which to them meant positive integers. They developed important parts of geometry with the aid of this principle. In particular, the principle implied that any two line segments a and b are commensurable (have a common measure), that is, that there exists a line segment t such that a = mt, b = nt, with m and n positive integers. But around 430 BC they proved that the side and diagonal of a unit square are not commensurable. (In a modern formulation we would say that 2 is irrational.) [7]. This must have been a great shock to them, as it went counter to their philosophy and their mathematics. And it might have provided an important impetus for a critical reevaluation of the logical foundations of mathematics [3(a)]. e. The need to teach. This may have forced the Greek mathematicians to consider the basic principles underlying their subject. It is noteworthy that the pedagogical motive in the formal organization of mathematics is also present in the works of later mathematicians, notably Lagrange, Cauchy, Weierstrass, and Dedekind [8]. Euclid’s great merit was to have collected, and arranged brilliantly in a grand axiomatic edifice called Elements, much of the mathematics of the previous three centuries (with notable exceptions, such as conic sections). His opus comprises over 450 propositions (theorems), deduced from five (!) postulates (axioms), and arranged in thirteen “Books” (chapters). The postulates are: 1. A straight line may be drawn between any two points. 2. A straight line segment may be produced indefinitely. 3. A circle may be drawn with any given point as centre and any given radius. 4. All right angles are equal. 5. If a straight line intersects two other straight lines lying in a plane, and if the sum of the interior angles thus obtained on one side of the intersecting line is less than two right angles, then the straight lines will eventually meet, if extended sufficiently, on the side on which the sum of the angles is less than two right angles. For over two thousand years, to teach elementary geometry meant to teach it essentially as Euclid had presented it. His masterpiece first appeared in print in 1482 (the printing press originated in ca. 1450). More than a thousand editions have appeared since, a profusion superseded 1.2 • Hilbert’s Foundations of Geometry 3 1 Euclid (fl. ca. 300 BC) only by the Bible. The Elements also inspired Newton to present his masterpiece of physics and cosmology, the Principia, axiomatically, and it inspired Spinoza to write his philosophical chef d’oeuvre, the Ethics, in the same style. 1.2 Hilbert’s Foundations of Geometry But despite this influence of the Elements, the practice of mathematics in the euclidean manner is a rather rare phenomenon in the five thousandyear history of mathematics. It was consciously undertaken for around two hundred years in ancient Greece and was resumed in the nineteenth century. Both of these periods were preceded by centuries of mathematical activity that was often vigorous but rarely rigorous. But for over two millennia there was only one geometry—Euclid’s. Its axioms, with the exception of the fifth, were taken to describe an idealization of physical space and were therefore viewed as “selfevident truths”, not open to critique. Its theorems, which were logical consequences of the axioms, were therefore also viewed as truths (see 7 Chapter 7). The nineteenth century brought a revolution in geometry, both in scope and in depth. New varieties emerged: projective geometry (Girard Desargues’ work in 1639 on the subject came to light only in 1845), hyperbolic geometry, elliptic geometry, Riemannian geometry, differential geometry, and algebraic geometry. Jean Victor Poncelet founded synthetic projective geometry in the early 1820s as an independent subject, but lamented its lack of general principles, and the validity of his “principle of duality”—that the truth of theorems is preserved by interchange of “point” and “line”—was questioned. The consistency of noneuclidean (hyperbolic) geometry and the relationship of axioms to the physical world were also in debate. And the relative merits of geometric methods were contested: the metric versus the projective, the synthetic versus the analytic. Important new ideas entered geometry: points and lines at infinity, use of complex numbers (cf. complex projective space), use of calculus, extension of geometry to n dimensions, Hermann Grassmann’s “calculus of extension”, invariants—for example, the “invariant theory of forms” developed by Cayley and Sylvester, and groups—for example, groups of the regular solids. An important development was Felix Klein’s proof (1871) that euclidean, hyperbolic, and elliptic geometries are subgeometries of projective geometry. For a time it was said that projective geometry was all of geometry [5, p. 239]. This period of profound changes left many mathematicians uneasy. The historian of mathematics Jeremy Gray (1947–) claimed that “signs 4 1 Chapter 1 • Axiomatics—Euclid’s and Hilbert’s: From Material to Formal of anxiety about the nature of geometry run like fissures through late 19thcentury mathematics” [5, p. 247]. Euclidean geometry did not escape scrutiny. Although Euclid was the paragon of rigor for more than two thousand years, logical shortcomings were now recognized in his masterpiece Elements. For example, his very first proposition in Book I, which presents the construction of an equilateral triangle, has a faulty proof: while Euclid assumed implicitly that two circles, each of which passes through the center of the other, intersect, this observation requires an axiom of continuity, supplied two millennia later by David Hilbert. Gauss pointed out that such concepts as “between”, used freely and intuitively by Euclid, must be given an axiomatic formulation. These challenges were taken up during the last two decades of the nineteenth century by a number of mathematicians. They provided, for projective, euclidean, and noneuclidean geometries, axioms free of the types of blemishes that appear in Euclid’s presentation. The first to do this was Moritz Pasch, who wrote an extensive work in 1882 on the foundations of projective geometry. Pasch set out clearly a crucial aspect of modern axiomatics, which departs radically from Euclid’s procedure [8, p. 1008]: »» If geometry is to become a genuine deductive science, it is essential that the way in which inferences are made should be altogether independent of the meaning of the geometrical concepts, and also of the diagrams; all that need be considered are the relationships between the geometrical concepts asserted by the propositions and definitions. The most influential work in this genre was Hilbert’s Foundations of Geometry of 1899. His aim was “to present a complete and simplest possible system [Hilbert’s italics] of axioms [for euclidean geometry], and to derive from these the most important geometrical theorems” [1, p. 344]. To avoid the pitfalls in Euclid’s Elements—reliance on intuitive arguments, often based on diagrams—Hilbert required twenty postulates; Euclid, recall, had five. Hilbert listed his axioms under five headings: I. axioms of connection, II. axioms of order, III. axiom of parallels (Euclid’s fifth axiom), IV. axioms of congruence, and V. axiom of continuity (Archimedes’ axiom). Crucial was the use, as urged by Pasch, of undefined terms, socalled primitive terms. Why are they needed? Because just as one cannot prove everything, hence the need for axioms, so one cannot define everything, hence the need for undefined terms. They are not uniquely determined; among Hilbert’s choices are “point”, “(straight) line”, and “plane”. Euclid defined all three terms, for example, a “point” as “that which has no part”—which is not very informative. Euclid considered his axioms to be selfevident truths, but Hilbert’s are neither selfevident nor true. They are simply the starting points, the basic building blocks, of the theory—assumptions about the relations among the primitive terms of the axiomatic system. The primitive terms are said to be “implicitly” defined by the axioms. As early as 1891 Hilbert highlighted the observation about the arbitrary nature of the primitive terms in the now classic remark that “It must be possible to replace in all geometric statements the words point, line, plane by table, chair, mug” [13, p. 14]. It follows that the axioms, hence also the theorems, are devoid of meaning. It is therefore not inappropriate to call Euclid’s system “material axiomatics” and Hilbert’s system “formal axiomatics” [3(a), p. 63 and 3(b), p. 171]. Hilbert’s Foundations of Geometry went through ten editions (in the original German), seven in Hilbert’s lifetime. It served as a model of what an axiom system should be like, and more broadly, it “demonstrated brilliantly the vitality of the new axiomatic approach to geometry” oundations that [1, p. 361]. Garrett Birkhoff and Mary Katherine Bennett wrote (1987) of the F 1.3 • The Modern Axiomatic Method 5 1 David Hilbert (1862–1943) it was “the most influential book on geometry written in the [nineteenth] century” [1, p. 343], and E. T. Bell claimed that it “inaugurated the abstract mathematics of the 20th century” [1, p. 343]. 1.3 The Modern Axiomatic Method In the wake of Hilbert’s Foundations one could define a “geometry” by picking a set of primitive terms—which, since it is to be a geometry, we might as well call “point”, “line” …—and a consistent set of axioms, and logically deducing consequences from the axioms, which are then theorems of the geometry. Considerations like these gave rise in the nineteenth and early twentieth centuries to such geometries as desarguesian, nondesarguesian, finite, neutral, nonarchimedean, and inversive. Soon one began to describe (define) in Hilbert’s manner mathematical structures other than geometries. Thus Giuseppe Peano defined (characterized) the positive integers in 1889 by means of the now classic Peano axioms, and Hilbert in 1900 gave a characterization of the real numbers as a complete ordered archimedean field. These accomplishments were in line with the spirit of rigor, generalization, abstraction, and axiomatization prevailing in late nineteenth and early twentiethcentury mathematics. Among early exponents of this approach were Dedekind, Peano, and especially Hilbert himself. Yet another approach to axiomatics was begun in algebra and resulted in the now familiar algebraic structures of groups, rings, fields, vector spaces, modules, and ideals. These structures arose mainly from mathematicians’ inability to solve old problems by old means, which necessitated the introduction of new structures. For the story of the rise of the group concept see [15]. Topological spaces, normed rings, Hilbert spaces, and lattices are among many other examples of mathematical structures defined by axiom systems. These structures, unlike (say) euclidean geometry, the natural numbers, or the real numbers, do not characterize a unique mathematical entity, but rather subsume many (usually infinitely many) different objects under the roof of a single set of axioms. The rise of modern axiomatics—one of the most distinctive features of modern mathematics—was gradual and slow, lasting for much of the nineteenth century and the early decades of the twentieth. In the 1920s the axiomatic method became well established in a number of major areas of mathematics, including algebra, analysis, geometry, and topology, and it flourished 6 1 Chapter 1 • Axiomatics—Euclid’s and Hilbert’s: From Material to Formal during the following three decades. Nicolas Bourbaki, among its most able practitioners and promoters, gave an eloquent description of the essence of the axiomatic method at what was perhaps the height of its power, in 1950 [2, p. 223]: »» What the axiomatic method sets as its essential aim, is exactly that which logical formalism by itself cannot supply, namely the profound intelligibility of mathematics. Just as the experimental method starts from the a priori belief in the permanence of natural laws, so the axiomatic method has its cornerstone in the conviction that, not only is mathematics not a randomly developing concatenation of syllogisms, but neither is it a collection of more or less “astute” tricks, arrived at by lucky combinations, in which purely technical cleverness wins the day. Where the superficial observer sees only two, or several, quite distinct theories, lending one another “unexpected support” through the intervention of a mathematician of genius, the axiomatic method teaches us to look for the deeplying reasons for such a discovery, to find the common ideas of these theories, buried under the accumulation of details properly belonging to each of them, to bring these ideas forward and to put them in their proper light. In this article Bourbaki presents a panoramic view of mathematics organized around what he calls “mother structures”—algebraic, ordered, and topological, and various substructures and crossfertilizing structures. This must have been an alluring, even bewitching, perspective to those growing up mathematically during this period. 1.4 Ancient vs. Modern Axiomatics There are significant differences between Euclid’s axiomatics and its modern incarnation in the nineteenth and twentieth centuries. Comparing Euclid’s Elements with Hilbert’s Foundations of Geometry makes starkly clear how standards of rigor have evolved. Moreover, while the chief role played by the axiomatic method in ancient Greece was (probably) that of providing a sure foundation, it became in the first half of the twentieth century a tool of research. Note, for example, the rich and deep theory of groups, which comprises the logical consequences of a “simple” set of four axioms. The modern axiomatic method was also indispensable in clarifying the status of various mathematical methods and results, such as the axiom of choice and the continuum hypothesis, to which mathematicians’ intuition provided little guide. And it played an essential role in the discovery of certain concepts, results, and theories. For example, the desarguesian and nondesarguesian geometries “could never have been discovered without [the axiomatic] method” [4, p. 182]. Thus the sometimes opposed activities of discovery and demonstration coexisted within the axiomatic method. The modern axiomatic method was however not universally endorsed. Although some, notably Hilbert, claimed that it is the central method of mathematical thought, others, for instance Klein, argued that as a method of discovery it tends to stifle creativity. And it has its limitations as a method of demonstration. The following quotation from Hermann Weyl (1885–1955) puts the issue in a broader perspective [13, p. 38; his italics]: »» Large parts of modern mathematical research are based on a dexterous blending of axiomatic and constructive procedures. References 7 1 And finally, a comment from Bourbaki—a masterful practitioner and strong advocate of the axiomatic method [2, p. 231]: »» The unity which [the axiomatic method] gives to mathematics is not the armor of formal logic, the unity of a lifeless skeleton; it is the nutritive fluid of an organism at the height of its development, the supple and fertile research instrument to which all the great mathematical thinkers since Gauss have contributed, all those who, in the words of LejeuneDirichlet, have always labored to “substitute ideas for calculations”. Problems and Projects 1. Write a brief biography of Hilbert. 2. Describe Hilbert’s characterization of the real numbers as a complete, ordered, archimedean field. What geometric purpose was it intended to serve? 3. Discuss several propositions in Euclid’s Elements dealing with number theory. 4. Discuss several propositions in Book II of the Elements which correspond to algebraic results. 5. Skech a “proof ”, using axioms in Euclid’s Elements, that every triangle is equilateral. Where is the error to be found? How is it to be corrected? 6. What was the fate of Euclid’s five postulates in Hilbert’s Foundations of Geometry? See for example [6]. 7. Discuss Hilbert’s “Axioms of Betweenness” and some of the uses he made of them. 8. Discuss Pasch’s axioms for projective geometry. 9. Describe how the (algebraic) concept of a “ring” arose in the late nineteenth and the early twentieth centuries. See [7, 8]. References Birkhoff, G., Bennett, M.K.: Hilbert’s Grundlagen der Geometrie. Rend. Circ. Mat. Palermo. Ser. II(36), 343–389 (1987) 2. Bourbaki, N.: The architecture of mathematics. Am. Math. Mon. 57, 221–232 (1950) 3. Eves, H.: Great Moments in Mathematics, 2 vols.; (a) before 1650, and (b) after 1650. Mathematical Association of America, (1983) 4. Gray, J.: Review of Uber die Enstehung von David Hilberts ‘Grundlagen der Geometrie’. Hist. Math. 15, 181–183 (1988) 5. Gray, J.: Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century. Springer, Berlin (2007) 6. Greenberg, M.J.: Euclidean and Noneuclidean Geometries: Development and History, 22nd edn. W. H. Freeman, New York (1980) 7. Katz, V.: A History of Mathematics: An Introduction, 3rd. edn.. AddisonWesley, Boston (2009) 8. Kline, M.: Mathematical Thought from Ancient to Modern Times. Oxford University Press, Oxford (1972) 9. Knorr, W.R.: On the early history of axiomatics: the interaction of mathematics and philosophy in Greek antiquity. In: Hintikka, J. et al. (eds.) Theory Change, Ancient Axiomatics and Galileo’s Methodology, pp. 145–186. D. Reidel, Dordrecht (1980) 10. Lloyd, G.E.R.: Magic, Reason and Experience: Studies in the Origin and Development of Greek Science. Cambridge University Press, Cambridge (1979) 11. Szabo, A.: The Beginnings of Greek Mathematics. D. Reidel, Dordrecht (1978) 12. van der Waerden, B.L.: Science Awakening. Noordhoff, The Netherlands (1954) 13. Weyl, H.: Axiomatic versus constructive procedures in mathematics. Math. Intell. 7(4), 10–17, 38 (1985) 1. 8 1 Chapter 1 • Axiomatics—Euclid’s and Hilbert’s: From Material to Formal 14. Wilder, R.L.: Evolution of Mathematical Concepts. Wiley, Hoboken (1968) 15. Wussing, H.: The Genesis of the Abstract Group Concept. MIT Press, Cambridge (1984). (Translated from the German by A. Shenitzer, with the editorial assistance of H. Grant) Further Reading 16. Artmann, B.: Euclid: The Creation of Mathematics. Springer, Berlin (1999) 17. Cuomo, S.: Ancient Mathematics. Routledge, London (2001) 18. Davis, P.J., Hersh, R.: The Mathematical Experience. Birkhäuser, Boston (1981) (Revised Study Edition, with E. Marchisotto, published in 1995) 19. Mueller, I.: Philosophy of Mathematics and Deductive Structure in Euclid’s Elements. MIT Press, Cambridge (1981) 20. Netz, R.: The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History. Cambridge University Press, Cambridge (1999) 21. von Neumann, J.: The mathematician. In: Newman, J.R. (ed.) The World of Mathematics, Vol. 4, pp. 2053–2063. Simon and Schuster, New York (1956) 22. von Neumann, J.: The role of mathematics in the sciences and in society. In: Taub, A.H. (ed.) Collected Works, Vol. 6, pp. 477–490. Macmillan, London (1963) 23. Weyl, H.: Mathematics and logic. Am. Math. Mon. 53, 2–13 (1946) 24. Wilder, R.L.: The role of the axiomatic method. Am. Math. Mon. 74, 115–127 (1967) 9 2 Solution by Radicals of the Cubic: From Equations to Groups and from Real to Complex Numbers H. Grant, I. Kleiner, Turning Points in the History of Mathematics, Compact Textbooks in Mathematics, DOI 10.1007/9781493932641_2, © Springer Science+Business Media, LLC 2015 2.1 Introduction Several ancient civilizations—the Babylonian, Egyptian, Chinese, and Indian—dealt with the solution of polynomial equations, mainly linear. The Babylonians (ca. 1600 BC ff.) were particularly proficient “algebraists”. They were able to solve quadratic equations, as well as equations that lead to quadratics—for example, x + y = a and x2 + y2 = b—by methods similar to ours. The equations were given in the form of “word problems”, and were often expressed in geometric language. Here is a typical example [7, p. 24]: »» I summed the area and twothirds of my squareside and it was 0;35 [35/60 in sexagesimal notation]. [What is the side of my square?] In modern notation, the problem is to solve the equation x2 + (2/3)x = 35/60. See [7, p. 24] for the Babylonians’ solution of this equation. The Chinese (ca. 200 BC ff.) and the Indians (ca. 600 BC ff.)—in each case the dates are very rough—made considerable advances in algebra. For example, both allowed negative coefficients in their equations—though not negative roots—and admitted two roots for a quadratic equation. They also described procedures for manipulating equations. The Chinese had methods for approximating roots of polynomial equations of any degree, and they solved systems of linear equations using “matrices” (rectangular arrays of numbers) well before such techniques were developed in Western Europe. The mathematics of the ancient Greeks, in particular their geometry and number theory, was relatively advanced, but their algebra was rather weak. (Note however that Diophantus (fl. ca. 250 AD), in his great numbertheoretic work Arithmetica, introduced various algebraic symbols [1].) Book II of Euclid’s remarkable work Elements (ca. 300 BC) presents, in geometric language, results which are familiar to us as algebraic, but most modern scholars believe that the Greeks of this period were not thinking algebraically. Islamic mathematicians made important contributions in algebra between the ninth and fifteenth centuries. Among the foremost was Muhammad ibnMūsā alKhwārizmī, dubbed by some “the Euclid of algebra” because he systematized the subject as it then existed and made it into an independent field of study. He did this in his book aljabr w almuqabalah. “Aljabr”, from which stems our word “algebra”, denotes the moving of a negative term of an equation to the other side so as to make it positive, and “almuqabalah” refers to cancelling equal (positive) 10 2 Chapter 2 • Solution by Radicals of the Cubic terms on the two sides of an equation. These are of course basic procedures for solving polynomial equations. alKhwārizmī, from whose name is derived the word “algorithm”, applied these procedures to the solution of quadratic equations, which he classified into five types: ax2 = bx, ax2 = b, ax2 + bx = c, ax2 + c = bx, and ax2 = bx + c. This categorization was necessary since alKhwārizmī did not admit negative coefficients or zero into the number system. He also had no algebraic notation, so that his problems and solutions were expressed rhetorically (in words). He did however offer (geometric) justification for his solution procedures. 2.2 Cubic and Quartic Equations The Babylonians (as we mentioned) were solving quadratic equations by about 1600 BC, using essentially an equivalent of our “quadratic formula”. A natural question was therefore whether cubic equations could be solved using “similar” formulas; three thousand years would pass before the answer was discovered. It was a great event in algebra when mathematicians of the sixteenth century succeeded in solving—by radicals—not only cubic but also quartic equations. This accomplishment was very much in character with the mood of the Renaissance—which wanted not only to absorb the classic works of the ancients but to strike out in new directions. Indeed, the solution of the cubic unquestionably proved a farreaching departure. A “solution by radicals” of a polynomial equation is a formula giving the roots of the equation in terms of its coefficients. The only permissible operations to be applied to the coefficients are the four algebraic operations (addition, subtraction, multiplication, and division) and the extraction of roots (square roots, cube roots, and so on, that is, “radicals”). For example, the 2 quadratic formula x = −b ± b − 4ac is a solution by radicals of the equation ax2 + bx + c = 0. 2a A solution by radicals of the cubic was first published in 1545 by Girolamo Cardano, in his Ars Magna (The Great Art, referring to algebra); it was discovered earlier by Scipione del Ferro and by Niccolò Tartaglia. The latter had passed on his method to Cardano, who had promised that he would not publish it; but he did. That is one version of events, which involved considerable drama and passion. A blowbyblow account is given by Oysten Ore [12, pp. 53–107]. Here is Cardano’s own rendition [7, p. 63]: »» Scipio Ferro of Bologna wellnigh thirty years ago [i.e., ca. 1515] discovered this rule and handed it on to Antonio Maria Fior of Venice, whose contest with Nicolò Tartaglia of Brescia gave Nicolò occasion to discover it. He [Tartaglia] gave it to me in response to my entreaties, though withholding the demonstration. Armed with this assistance, I sought out its demonstration in [various] forms. This was very difficult. What came to be known as “Cardano’s formula” for the solution of the cubic x3 = ax + b is given by x= 3 b 2 + ( b2 ) − ( a3 ) 2 3 + 3 b2 − ( b2 ) − ( a3 ) 2 3 . 2.3 • Beyond the Quartic: Lagrange 11 2 Girolamo Cardano (1501–1576) See for example [1, 2, 5]. Several comments are in order: i. Cardano used essentially no symbols, so his “formula” giving the solution of the cubic was expressed rhetorically. ii. He was usually content with determining a single root of a cubic. But in fact, if a proper choice is made of the cube roots involved, all three roots of the equation can be determined from his formula. iii. The coefficients and roots of the cubics he considered were specific positive numbers, so that he viewed (say) x3 = ax + b and x3 + ax = b as distinct. He devoted a chapter to the solution of each, and gave geometric justifications [13, p. 63 ff.]. iv. Negative numbers are found occasionally in his work, but he mistrusted them, and called them “fictitious”. Irrational numbers were admitted as roots. The solution by radicals of polynomial equations of the fourth degree—quartics—soon followed. The key idea was to reduce the solution of a quartic to that of a cubic. Ludovico Ferrari was the first to solve such equations, and his work was included in Cardano’s The Great Art [4]. It should be pointed out that cubic equations had arisen—in geometric guise—already in ancient Greece (ca. 400 BC), in connection with the problem of trisecting an angle, and that methods for finding approximate roots of cubics and quartics were known, for example by Chinese and Moslem mathematicians, well before such equations were solved by radicals. The latter solutions, though exact, were of little practical value. But the ramifications of these “impractical” ideas were very significant, and will now be briefly sketched. 2.3 Beyond the Quartic: Lagrange Having solved the cubic and quartic by radicals, mathematicians turned to finding a solution by radicals of the quintic (degreefive polynomial)—a quest that would take nearly 300 years. Some of the most distinguished mathematicians of the seventeenth and eighteenth centuries, among them François Viète, René Descartes, Gottfried Wilhelm Leibniz, Leonhard Euler, and Étienne Bezout, tackled the problem. The strategy was to seek new approaches to the solutions of the cubic and quartic, in the hope that at least one of them would generalize to the quintic. 12 Chapter 2 • Solution by Radicals of the Cubic Joseph Louis Lagrange (1736–1830) 2 But to no avail: although new ideas for solving the cubic and quartic were found, they did not yield the desired extensions. One approach, however, undertaken by Joseph Louis Lagrange in a paper of 1770 entitled Reflections on the Algebraic Solution of Equations, proved promising. Lagrange analyzed the various methods devised by his predecessors for solving cubic and quartic equations, and saw that—since those methods did not work when applied to the quintic—a deeper scrutiny was required. In his own words [14, p. 127]: »» I propose in this memoir to examine the various methods found so far for the algebraic so lution of equations, to reduce them to general principles, and to let us see a priori why these methods succeed for the third and fourth degree, and fail for higher degrees. Here are some of the key ideas of Lagrange’s approach. With each polynomial equation of arbitrary degree n he associated a “resolvent equation”, as follows: let f(x) be the original equation, with roots x1, x2, x3, …, xn. (As is the usual practice, we denote by “f(x)” both the polynomial and the polynomial equation.) Pick a rational function R(x1, x2, x3, …, xn) of the roots and coefficients of f(x). (Lagrange described a method for doing this.) Consider the different values which R(x1, x2, x3, …, xn) assumes under all the n! permutations of the roots x1, x2, x3, …, xn of f(x). If these values are denoted by y1, y2, y3, …, yk, the “resolvent equation” is (x − y1)(x − y2)… (x − yk). Lagrange showed that k divides n!—the source of what we call “Lagrange’s theorem” in group theory. For example, if f(x) is a quartic with roots x1, x2, x3, x4, then R(x1, x2, x3, x4) may be taken to be x1 x2 + x3x4, and this function assumes three distinct values under the 24 permutations of x1, x2, x3, and x4. Thus, the resolvent equation of a quartic is a cubic. However, in carrying over this analysis to the quintic, Lagrange found that the resolvent equation is of degree six, rather than the hopedfor degree four. Although Lagrange did not succeed in settling the problem of the solvability of the quintic by radicals, his work was a milestone. It was the first time that an association was made between the solutions of a polynomial equation and the permutations of its roots. In fact, Lagrange speculated that the study of the permutations of the roots of an equation was the cornerstone of the theory of algebraic equations—“the genuine principles of the solution of equations”, as he put it [14, p. 146]. He was of course vindicated in this by Evariste Galois. 13 2.5 • Complex Numbers: Birth 2 Evariste Galois (1811–1832) 2.4 Ruffini, Abel, Galois Paolo Ruffini and NielsHenrik Abel proved (in 1799 and 1826, respectively) the unsolvability by radicals of the “general quintic”. In fact, they proved the unsolvability of the “general equation” of degree n for every n > 4. They did this by building on Lagrange’s pioneering ideas on resolvents. Lagrange had shown that a necessary condition for the solvability of the general polynomial equation of degree n is the existence of a resolvent of degree less than n. (A “general equation” is an equation with arbitrary literal coefficients.) Ruffini and Abel showed that such resolvents do not exist for any n > 4. (Abel proved this result without knowing of Ruffini’s work; in any case, Ruffini’s work had a significant gap.) Although the general polynomial equation of degree > 4 is unsolvable by radicals, some specific equations of this form are solvable; for example, xn − 1 = 0 is solvable by radicals for every n > 4. Galois characterized those equations that are solvable by radicals in terms of group theory: A polynomial is solvable by radicals if and only if its “Galois group” is “solvable”. To prove this result Galois founded the elements of permutation group theory and introduced in it various important concepts, such as Galois group, normal subgroup, and solvable group. Thus ended, in the early 1830s, the great saga—beginning with Cardano in 1545—of solvability by radicals of equations of degrees greater than 2. 2.5 Complex Numbers: Birth A hugely important development arising from the solution of the cubic by radicals was the introduction of complex numbers. Recall that Cardano’s solution of the cubic x 3 = ax + b is given by x= 3 b 2 + ( b2 ) − ( a3 ) 2 3 + 3 b2 − ( b2 ) − ( a3 ) 2 3 . 14 Chapter 2 • Solution by Radicals of the Cubic Consider the cubic x 3 = 9 x + 2 . Its solution, using the above formula, is 2 x= 3 2 2 + ( 22 ) 2 − ( 93 )3 + 3 22 − ( 22 ) 2 − ( 93 )3 = 3 1 + −26 + 3 1 − −26 . What is one to make of this solution? Since Cardano was suspicious of negative numbers— calling them “fictitious” [10, p. 40]—he certainly had no taste for their square roots, which he named “sophistic negatives” [10, p. 40]. He therefore regarded his formula as inapplicable to equations such as x3 = 9x + 2. Judged by past experience, this was not an unreasonable attitude. For example, to preRenaissance mathematicians the quadratic formula could not be applied to x2 + 1 = 0. All this changed when the Italian Rafael Bombelli came on the scene. In his important book Algebra (1572) he applied Cardano’s formula to the equation x3 = 15x + 4 and obtained x = 3 2 + −121 + 3 2 − −121 . But he could not dismiss this solution, unpalatable as it would have been to Cardano, for he noted—by inspection—that x = 4 is also a root of this equation; its other two roots, −2 ± 3, are also real numbers. Here was a paradox: while all three roots of the cubic x3 = 15x + 4 are real, the formula used to obtain them involved square roots of negative numbers—meaningless at the time. How was one to resolve the paradox? Bombelli had a “wild thought”: since the radicands 2 + −121 and 2 − −121 differ only in sign, the same might be true of their cube roots. He thus let 3 2 + −121 = a + b −1, 3 2 − −121 = a − b −1, and proceeded to solve for a and b by manipulating these expressions according to the established rules for real variables. He deduced that a = 2 and b = 1 and thereby showed that, indeed, x = 3 2 + −121 + 3 2 − −121 = (2 + −1) + (2 − −1) = 4. Bombelli had given meaning to the “meaningless”. He put it thus [11, p. 19]: »» It was a wild thought in the judgment of many; and I too for a long time was of the same opinion. The whole matter seemed to rest on sophistry rather than on truth. Yet I sought so long, until I actually proved this to be the case. Moreover, Bombelli developed a “calculus” for complex numbers, stating such rules as ( + −1 ) ( + −1 ) = −1 and ( + −1 ) ( − −1 ) = 1, and defined addition and multiplication of some of these numbers. These innovations signaled the birth of complex numbers. But note that this is a retrospective view of what Bombelli had done. He did not postulate the existence of a system of numbers—called complex numbers—containing the real numbers and satisfying basic properties of numbers. To him, the expressions he worked with were just that; they were important because they “explained” hitherto inexplicable phenomena. Square roots of negative numbers could be manipulated in a meaningful way to yield significant results. This was a bold idea indeed. See [8, 10]. 2.6 • Growth 15 2 Rafael Bombelli (1526–1572) The equation x3 = 15x + 4 considered above is an example of an “irreducible cubic”, one with rational coefficients, irreducible over the rationals, all of whose three roots are real and distinct. It was shown in the nineteenth century that any solution by radicals of such a cubic—not just Cardano’s—must involve complex numbers [5, 10]. Thus complex numbers are unavoidable when determining solutions by radicals of irreducible cubics. It is for this reason that they arose in connection with the solution of cubic rather than (as seems much more plausible) quadratic equations. Note that the nonexistence of a solution of the quadratic x2 + 1 = 0 was accepted for centuries. 2.6 Growth Here are several examples of the penetration of complex numbers into mathematics in the centuries after Bombelli. As early as 1620, Albert Girard suggested that an equation of degree n may have n roots. Such statements of the “Fundamental Theorem of Algebra” were however vague and unclear. For example Descartes, who coined the unfortunate term “imaginary” for the new numbers (Gauss called them “complex”), stated that although one can imagine that every equation has as many roots as is indicated by its degree, no (real) numbers correspond to some of these imagined roots. Leibniz, who spent considerable time and effort on the question of the meaning of complex numbers and the possibility of deriving reliable results by applying the ordinary laws of algebra to them, thought of complex roots as “an elegant and wonderful resource of divine intellect, an unnatural birth in the realm of thought, almost an amphibium between being and nonbeing” [11, p. 159]. Complex numbers were used by Johann Heinrich Lambert for map projection, by Jean le Rond d’Alembert in hydrodynamics, and by Euler, d’Alembert, and Lagrange in (incorrect) proofs of the Fundamental Theorem of Algebra. Euler made important use of complex numbers in, for example, number theory and analysis; he also linked the exponential and trigonometric functions and, arguably, the five most important numbers in mathematics in, respectively, the following two famous formulas: eix = cos x + i sin x and eπi + 1 = 0 . (Euler was the first to designate −1 by “i”.) Yet he said of them [9, p. 594]: 16 Chapter 2 • Solution by Radicals of the Cubic »» Because all conceivable numbers are either greater than zero, less than zero or equal to zero, then it is clear that the square roots of negative numbers cannot be included among the possible numbers. Consequently we must say that these are impossible numbers. And this circumstance leads us to the concept of such numbers, which by their nature are impossible, and ordinarily are called imaginary or fancied numbers, because they exist only in the imagination. 2 Even the great Gauss, who in his doctoral thesis of 1797 gave the first essentially correct proof of the Fundamental Theorem of Algebra, claimed as late as 1825 that “the true metaphysics of −1 is elusive” [9, p. 631]. But by 1831 Gauss had overcome these metaphysical scruples and, in connection with a work on number theory, published his scheme for representing them geometrically, as points in the plane. Similar representations by Caspar Wessel in 1797 and by Jean Robert Argand in 1806 had gone largely unnoticed; but when given Gauss’ stamp of approval the geometric representation dispelled much of the mystery surrounding complex numbers. Doubts concerning the meaning and legitimacy of complex numbers persisted for two and a half centuries following Bombelli’s work. Yet during that same period these numbers were used extensively. How can inexplicable things be so useful? This is a recurrent theme in the history of mathematics. Bombelli’s resolution of the paradox dealing with the solution of the cubic x3 = 15x + 4 is an excellent example of this phenomenon. 2.7 Maturity In the next two decades further developments took place. In 1833 William Rowan Hamilton gave an essentially rigorous algebraic definition of complex numbers as pairs of real numbers, and in 1847 AugustinLouis Cauchy gave a completely rigorous definition in terms of congruence classes of real polynomials modulo x2 + 1. In this he modelled himself on Gauss’ definition of “congruences” for the integers. By the latter part of the nineteenth century most vestiges of mystery and distrust around complex numbers could be said to have disappeared [6]. But this is far from the end of their story. Various developments in mathematics in the nineteenth century gave us deeper insight into the role of complex numbers in mathematics and in other areas. These numbers offer just the right setting for dealing with many problems in mathematics in such diverse areas as algebra, analysis, geometry, and number theory. They have a symmetry and completeness that is often lacking in the real numbers. The following three quotations, by Gauss in 1811, Riemann in 1851, and Jacques Hadamard in the 1890s, respectively, say it well: »» Analysis … would lose immensely in beauty and balance and would be forced to add very hampering restrictions to truths which would hold generally otherwise, if … imaginary quantities were to be neglected [3, p. 31]. The original purpose and immediate objective in introducing complex numbers into mathematics is to express laws of dependence between variables by simpler operations on the quantities involved. If one applies these laws of dependence in an extended context, by giving the variables to which they relate complex values, there emerges a regularity and harmony which would otherwise have remained concealed [6, p. 64]. The shortest path between two truths in the real domain passes through the complex domain [9, p. 626]. 17 2.7 • Maturity 2 We give brief indications of what is involved in welcoming complex numbers into mathematics. In algebra, their introduction gave us the celebrated “Fundamental Theorem of Algebra”: every equation with complex coefficients has a complex root. The complex numbers offer an example of an “algebraically closed field”, relative to which many problems in linear algebra and other areas of abstract algebra have their “natural” formulation and solution. In analysis, the nineteenth century saw the development of a powerful and beautiful branch of mathematics: “complex function theory”. One indication of its efficacy is that a function in the complex domain is infinitely differentiable if once differentiable—which of course is false for functions of a real variable. In geometry, the complex numbers lend symmetry and generality to the formulation and description of its various branches, including euclidean, inversive, and noneuclidean geometry. For a specific example we mention Gauss’ use of complex numbers to show that the regular polygon of seventeen sides is constructible with straightedge and compass. In number theory, certain diophantine equations can be solved using complex numbers. For example, the domain consisting of the set of elements of the form a + b 2 i, with a and b integers, has unique factorization, and in it the Bachet equation x2 + 2 = y3 factors as ( x + 2 i) ( x − 2 i) = y3. This greatly facilitates its solution (in integers). An elementary illustration of Hadamard’s dictum that “the shortest path between two truths in the real domain passes through the complex domain” is supplied by the following proof that the product of sums of two squares of integers is again a sum of two squares of integers; that is, given integers a, b, c, and d, there exist integers u and v such that (a 2 + b 2 )(c 2 + d 2 ) = u 2 + v 2 . For, (a 2 )( ) + b 2 c 2 + d 2 = ( a + bi )( a − bi )( c + di )( c − di ) = ( a + bi )( c + di ) ( a − bi )( c − di ) = ( u + vi )( u − vi ) = u 2 + v 2 for some integers u and v. Try to prove this result without the use of complex numbers and without being given the u and v in terms of a, b, c, and d. In addition to their fundamental uses in mathematics, complex numbers have become indispensible in science and technology. For example, they are used in quantum mechanics and in electric circuitry. The “impossible” has become not only possible but essential [6]. Problems and Projects 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Discuss the solution of the quartic by radicals. Research the lives and work of two mathematicians discussed in this chapter. Show how to trisect an angle using trigonometric functions. Discuss the Italian Renaissance, including some of its accomplishments in mathematics (those not discussed in this chapter). Describe the “geometric algebra” of the ancient Greeks. Discuss the algebra of alKhwārizmī. Show how to solve an elementary problem in euclidean geometry using complex numbers. Discuss the meaning of the logarithms of negative and complex numbers. The “quaternions” (also known as “hypercomplex numbers”) contain the complex numbers. Discuss some of their properties that are like those of the complex numbers and some that differ. Show how to resolve the paradox of the irreducible cubic x3 = 15x + 4. 18 Chapter 2 • Solution by Radicals of the Cubic References 1. 2 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Bashmakova, I.G., Smirnova, G.S.: The beginnings and evolution of algebra. Mathematical Association of America (2000). (Translated from the Russian by A. Shenitzer) Berlinghoff, W.P., Gouvêa, F.Q.: Math through the ages: A gentle history for teachers and others, expanded edition. Mathematical Association of America (2004) Birkhoff, G. (ed.): A Source Book in Classical Analysis. Harvard University Press, Cambridge (1973) Cardano, G.: The Great Art (Ars Magna). Dover, New York (1993) Cooke, R.: Classical Algebra: Its Nature, Origins, and Uses. Wiley, New Jersey (2008) Ebbinghaus, H.D., et al.: Numbers. Springer, Berlin (1990) Katz, V.: A History of Mathematics, 3rd edn. AddisonWesley, Boston (2009) Katz, V., Parshall, K.H.: Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century. Princeton University Press, Princeton (2014) Kline, M.: Mathematical Thought from Ancient to Modern Times Oxford University Press, Oxford (1972) Mazur, B.: Imagining Numbers. Farrar, Straus and Giroux, New York (2003) Nahin, P.G.: An Imaginary Tale: The Story of −1 . Princeton University Press, Princeton (1998) Ore, O.: Cardano, The Gambling Scholar. Dover, New York (1965) Struik, D.J. (ed.): A Source Book in Mathematics, 1200–1800. Harvard University Press, Cambridge (1969) Tignol, J.P.: Galois Theory of Algebraic Equations. World Scientific Publ., River Edge (2001) Further Reading 15. Dobbs, D.E., Hanks, R.: A Modern Course on the Theory of Equations. Polygonal Publishing House, New Jersey (1980) 16. Pesic, P.: Abel’s Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability. MIT Press, Cambridge (2003) 17. Sesiano, J.: An introduction to the history of algebra: Solving equations from Mesopotamian times to the Renaissance. Am. Math. Soc. (2009) 18. Turnbull, H.W.: Theory of Equations. Oliver and Boyd, Edinburgh (1957) 19. van der Waerden, B.L.: Geometry and Algebra in Ancient Civilizations. Springer, Berlin (1983) 20. van der Waerden, B.L.: A History of Algebra, from alKhwārizmī to Emmy Noether. Springer, Berlin 1985 21. Wussing, H.: The Genesis of the Abstract Group Concept. MIT Press, Cambridge (1984). (Translated from the German by A. Shenitzer) 19 3 Analytic Geometry: From the Marriage of Two Fields to the Birth of a Third H. Grant, I. Kleiner, Turning Points in the History of Mathematics, Compact Textbooks in Mathematics, DOI 10.1007/9781493932641_3, © Springer Science+Business Media, LLC 2015 3.1 Introduction Analytic geometry was invented independently by René Descartes and Pierre de Fermat in the first half of the seventeenth century. (The term “analytic geometry” was coined by Sylvestre François Lacroix in 1792.) The independent and more or less simultaneous invention (discovery) of concepts, results, or even major theories is not uncommon in mathematics; two other outstanding instances are calculus and noneuclidean geometry (see 7 Chapters 5 and 7, respectively). As the mathematician Wolfgang Bolyai put it [1, p. 263]: »» Mathematical discoveries, like springtime violets in the woods, have their season which no human can hasten or retard. The analytic geometry of Fermat and Descartes did not originate in a vacuum. It drew inspiration from the Greek geometric tradition and, such as that was, the Renaissance algebraic tradition. The use of coordinates to designate position is possibly prehistoric, and appears in ancient Greek astronomy. Apollonius’ great work Conics (ca. 250 BC) contains what are essentially the equations of these curves with respect to a fixed coordinate system, although not, of course, in modern notation. Furthermore, graphical representation of physical laws goes back to the scholar Nicole Oresme (ca. 1320–1382). What then was left for Descartes and Fermat to do? It was (a) to recognize the following as a basic principle of analytic geometry: that to each geometric plane curve corresponds an equation in two unknowns, and, conversely, that to each equation involving two unknowns corresponds a curve; and (b) to develop this principle into an algorithmic procedure, showing how it can be used to solve problems in geometry. 3.2 Descartes Descartes wanted to devise a systematic method for the solution of geometric problems, especially those dealing with curves. Such a study received impetus from scientific developments in the early seventeenth century: Kepler’s use of conic sections in his study of planetary motion, Galileo’s use of parabolas to describe the motion of projectiles, and the use of curved lenses in the newly invented telescope and microscope. 20 Chapter 3 • Analytic Geometry: From the Marriage of Two Fields to the Birth of a Third René Descartes (1596–1650) 3 But the main motivation for the creation of analytic geometry came not from practical problems but from a desire to systematize the ancients’ problemsolving tools. Descartes noted that many constructions and proofs in euclidean geometry called for new, inventive, and ad hoc approaches. He therefore undertook to exploit the power of algebra to provide a broad methodology for solving geometric problems. Descartes was arguably the first great “modern” philosopher, as well as a firstrate mathematician and scientist. According to the distinguished historian of mathematics Henk Bos (1940–): »» Descartes’ mathematics was a philosopher’s mathematics. From the earliest documented phase in his intellectual career, mathematics was a source of inspiration and an example for his philosophy, and, conversely, his philosophical concerns strongly influenced his style and program in mathematics [2, p. 228]. His great mathematical work—Geometry (La Géométrie)—appeared as one of the appendices to his philosophical treatise Discourse on the Method of Reasoning Well and Seeking Truth in the Sciences. In the latter work he sought a way to establish truths in all fields of endeavor. Geometry was identified as one of the three disciplines exhibiting that general method; the other two were meteorology and optics. The essence of Descartes’ method in geometry is given in several places in his book; here is one [3, p. 90]: »» All points of a geometric curve [as defined by motions] must have a definite relation expressed by an equation. In his analysis of geometric problems Descartes admitted only certain types of curves, namely those defined by “motions” or by loci [9, p. 483]. As an application of this method, he singled out for special attention the socalled Problem of Pappus: given four straight lines, to find the locus of a point that moves so that the product of its distances from two of the lines is in a fixed ratio to the product of its distances from the other two lines [11, p. 128]. Descartes 3.3 • Fermat 21 3 Pierre de Fermat (16011665) showed that the locus is a conic section [7, p. 87]. This result was already known to the Greeks [2, 7 Chapter 23]. To exhibit the substantial power of his method, Descartes generalized the Problem of Pappus to the case of 2n lines and derived the equation of the locus for small values of n; for n = 3, he showed that the equation is of degree 3. But he showed little interest in the shapes of the curves given by such equations. While the salient idea for the subsequent development of mathematics was the association of equation and curve, for Descartes the idea was just a means to an end—the solution of geometric problems. 3.3 Fermat By the beginning of the seventeenth century the extant Greek mathematical works had been restored and had elicited great interest. Fermat introduced his new method in geometry after a careful study of the geometric works of Apollonius (ca. 225 BC) and Pappus (ca. 300 AD) and of the algebraic work of Viète. He noted that although the Greeks studied loci, they must have found them difficult, since some of the problems were not stated in full generality. He proceeded to rectify this in a twentypage work titled Introduction to Plane and Solid Loci. The basic principle of analytic geometry is stated at the outset [3, p. 75]: »» Whenever in a final equation two unknown quantities are found, we have a locus, the extremity of one of these describing a line, straight or curved. The historian of mathematics Carl Boyer (1906–1976) referred to this sentence as “one of the most significant statements in the history of mathematics”, for it introduced “not only analytic geometry, but also the immensely useful idea of an algebraic variable” [3, p. 75]. Fermat’s work had considerable influence on, among others, Newton and Leibniz. 22 3 Chapter 3 • Analytic Geometry: From the Marriage of Two Fields to the Birth of a Third The work of Fermat and Descartes had different emphases. While Descartes stressed the fact that curves can be represented by equations, Fermat’s point of departure was that indeterminate equations give rise to curves. He showed that equations of the first degree with two variables describe straight lines, and he carefully analyzed equations of the second degree with two variables, showing that they represent various conic sections. Although he did not consider equations of degree higher than two, he clearly recognized the potential of the subject he was dealing with to produce new curves, as is evident from his statement that “the species of curves are indefinite in number: circle, parabola, hyperbola, ellipse, etc.” [3, p. 79]. 3.4 Descartes’ and Fermat’s Works from a Modern Perspective Although the analytic geometry of Descartes and Fermat was groundbreaking, it was not in the form now familiar to us. In particular: (a) Remarkably, a rectangular coordinate system and formulas for distance and slope are missing. In fact, coordinate axes are not explicitly set forth. Only the horizontal axis appears explicitly in drawings, while the implicit vertical axis is usually oblique. (b) The unknowns x and y which appear in the equation of a curve were considered to be line segments rather than numbers. It was not until a century or more later that coordinates began to be viewed as numbers. The notion of a one–one correspondence between points in a plane and ordered pairs of real numbers, nowadays the basis of our formulation of analytic geometry, was foreign to Fermat and Descartes. (c) Descartes considered only curves whose equations are “algebraic” (that is, polynomials in x and y). Transcendental curves, such as y = log x, y = sin x, and y = ex, did not come under the scope of his general method. Fermat, as we noted, confined himself essentially to polynomial equations of degree two in x and y. (In another work, Fermat also considered the socalled higher parabolas and hyperbolas, y = xn and y = x−n, respectively.) (d) Curvesketching in the sense familiar to us was not a central aspect of the analytic geometry of Fermat and Descartes. Fermat emphasized the study of equations in x and y not via their graphical representation but via their properties as derived by the methods of calculus. Descartes (we recall) did not regard the equation of a curve as an adequate definition of the curve. (e) Both Descartes and Fermat used only positive coordinates, and such curves as were sketched appeared only in the first quadrant. Negative numbers were not a commonly acceptable part of the number system. Moreover, since Descartes’ objective, and to a large extent Fermat’s, was to solve geometric problems, the need for negative coordinates did not arise. At first the geometry of Descartes and Fermat was accessible only to a very small circle of the ablest mathematicians. The latter did not take kindly and quickly to the idea of algebra, conceived as a collection of formulas and rules of manipulation, playing the dominant role in the rigorous, axiomatic, venerable field of geometry. It is only with Gaspard Monge and Lacroix in the latter part of the eighteenth century that we find analytic geometry essentially as it appears in today’s textbooks. In the intervening years, analytic geometry was developed by, among others, Leibniz, who introduced transcendental curves into the study of geometry; Newton, who used negative coordinates freely, sketched curves from their equations, and introduced various 3.5 • The Significance of Analytic Geometry 23 3 Leonhard Euler (1707–1783) c oordinate systems, among them the polar; and Euler, who developed threedimensional analytic geometry (already hinted at by Descartes and Fermat), and who did much to systematize the subject in his outstanding book of 1748, Introductio in Analysin Infinitorum. 3.5 The Significance of Analytic Geometry Descartes’ and Fermat’s founding of the subject was revolutionary, although initially it might not have been viewed as such. Several decades after its creation, the new subject/method laid the mathematical groundwork for calculus and Newtonian physics. More specifically: (a) Fermat and Descartes were the first to highlight the very important notion of a (continuous) variable—indispensable in the development of calculus. (b) The use of equations to define curves opened up the possibility of introducing an unlimited number of new curves, beyond the conception of the synthetic method. Such curves, in turn, called for the invention of algorithmic techniques for their systematic investigation—an important factor in the creation of calculus. (c) Undoubtedly the study of the physical world calls for geometric knowledge: objects in space are geometric figures, paths of moving bodies are curves. Analytic geometry made possible the expression of shapes and paths in algebraic form, from which quantitative knowledge can be derived. Analytic geometry produced a most important coupling of algebra and geometry—a relationship that proved very fruitful for subsequent developments in mathematics. Lagrange expressed it as follows [11, p. 322]: »» As long as algebra and geometry travelled separate paths, their advance was slow and their application limited. But when these two sciences joined company, they drew from each other fresh vitality and thenceforward marched at a rapid pace toward perfection. The mathematician Keith Kendig (1938–) echoed these remarks [10, p. 161]: 24 Chapter 3 • Analytic Geometry: From the Marriage of Two Fields to the Birth of a Third »» [Analytic geometry] gave our imagination ‘two ends’—an algebraic one and a geometric one; geometric insight could often be translated into an algebraic one, and vice versa. Morris Hirsch (1933–), another prominent mathematician, was more specific [8, p. 604]: 3 »» If geometry lets us see what we are thinking about, algebra enables us to talk precisely about what we see, and above all to calculate. Moreover, it tends to organize our calculations and to conceptualize them; this, in turn, can lead to further geometrical construction and algebraic calculation. Linear algebra is another excellent example of the interplay of algebra and geometry. For instance, the algebraic formulation of dimension makes natural the extension to dimensions higher than three. On the other hand, speaking about “lines” and “planes” in dimensions higher than three makes the subject more intuitive, suggestive, and comprehensible. Analytic geometry—a bridge between algebra and geometry—also provides bridges between shape and quantity, number and form, the analytic and the synthetic, the discrete and the continuous. For, as was shown in the nineteenth century, the real numbers can be built up rigorously from the integers, and since the one–one correspondence between the real numbers and the points on a line is at the root of analytic geometry, this establishes a bridge between the continuous and the discrete. This correspondence—this tension—has been most fruitful in the development of mathematics. Hermann Weyl, one of the foremost mathematicians of the first half of the twentieth century, noted that it “represents a remarkable link between something which is given by our spatial intuition and something that is constructed in a purely logicoconceptual way” [5, p. 159]. In the twentieth century such bridgebuilding became enormously important, offering powerful tools to mathematicians. As examples, consider the following disciplines, which by merging two fields lent strength to each: analytic number theory, differential topology, geometric number theory, algebraic topology, algebraic number theory, differential geometry, and algebraic geometry. A grand synthesis—the Langlands Program—relating several areas of mathematics, in particular number theory, algebra, and analysis, was proposed by Robert Langlands (1936–) in the 1960s in a series of deep and farreaching conjectures, some by now established [6]. Problems and Projects 1. Discuss the thesis, advocated by some historians, that the Greeks invented analytic geometry. Consult [2, 3, 9, 11]. 2. What is the origin of the words “ellipse”, “hyperbola”, and “parabola”? See [2, 3, 9, 11]. 3. How did Descartes solve the fourline locus problem of Pappus? See [2, 7, 9]. 4. Discuss the coordinate systems (such as they were) of Descartes and Fermat. See [2, 3, 4, 9, 11, 12]. 5. Descartes’ geometry contains much on the theory of equations, especially in the third of the book’s three chapters. Describe it. See [2, 7, 9, 11]. 6. Write a brief biography of either Descartes or Fermat. 7. Describe the principles, as outlined in Descartes’ major work in philosophy, Discourse on Method, which were based on his general method of acquiring knowledge. See [2, 3, 7, 9]. 8. Discuss some contributions to analytic geometry of the successors of Fermat and Descartes. See [2, 3, 9, 11]. References 25 3 References 1. 2. Bell, E.T.: The Development of Mathematics, 2nd edn. McGrawHill, New York (1945) Bos, H.J.M.: Redefining Geometrical Exactness: Descartes’ Transformation of the Early Modern Concept of Construction. Springer, New York (2001) 3. Boyer, C.B.: History of Analytic Geometry. Scripta Mathematica, New York (1956) 4. Eves, H.: Great Moments in Mathematics (Before 1650). Mathematical Association of America, Washington DC (1980) 5. Gardiner, A.: Infinite Processes: Background to Analysis. SpringerVerlag, Berlin (1982) 6. Gelbart, S.: An elementary introduction to the Langlands Program. Bull. Am. Math. Soc. 10, 177–219 (1984) 7. Grabiner, J.: Descartes and problem solving. Math. Mag. 68, 83–97 (1995) 8. Hirsch, M.: Review of Linear Algebra Through Geometry, by T. Banchoff and J. Wermer. Am. Math. Mon. 92, 603–605 (1985) 9. Katz, V.: A History of Mathematics: An Introduction, 3rd edn. AddisonWesley, Boston (2009) 10. Kendig, K.M.: Algebra, geometry, and algebraic geometry. Am. Math. Mon. 90, 161–174 (1983) 11. Kline, M.: Mathematical Thought from Ancient to Modern Times. Oxford University Press, Oxford (1972) 12. Mahoney, M.: The Mathematical Career of Pierre de Fermat, 2nd edn. Princeton University Press, Princeton (1994) Further Reading 13. Descartes, R.: The Geometry of René Descartes, with a Facsimile of the First Edition. Dover, New York (1954) 14. Scott, J.F.: The Scientific Work of René Descartes (1596–1650). Taylor & Francis, New York (1952) 27 4 Probability: From Games of Chance to an Abstract Theory H. Grant, I. Kleiner, Turning Points in the History of Mathematics, Compact Textbooks in Mathematics, DOI 10.1007/9781493932641_4, © Springer Science+Business Media, LLC 2015 4.1 The Pascal–Fermat Correspondence Probability, like various other mathematical concepts and theories, emerged from the desire to solve realworld problems—in this case, to provide a mathematical framework for games of chance and for gambling. One must of course distinguish between “probability” as a concept and “probability” as a subject. We have occasionally used “probability theory” for the latter term. Normally “probability” is used for both concept and theory, the context making clear which is intended. Gambling is a longstanding activity, going back over three thousand years and engaged in by all civilizations. But the mathematical analysis of gambling, leading to the advent of probability, is of relatively recent origin. It began with Blaise Pascal and Pierre de Fermat. Around 1653 Pascal was approached by Antoine Gombauld, Chevalier de Méré (1607–1684)—a man of letters with a considerable knowledge of mathematics—to help him solve two gaming problems. Many writers say the request was made to improve de Méré’s gambling chances [3, p. 84]; Oysten Ore disputes that claim [14]. The two problems came to be known as the “Dice Problem” and the “Division Problem”, the latter also known as the “Problem of Points”. The Dice Problem: How many throws of two dice are needed in order to have a betterthaneven chance of getting two sixes? The Division Problem: What is a fair distribution of stakes in a game interrupted before its conclusion? The Dice Problem is much the simpler of the two. It was solved in the midsixteenth century by Girolamo Cardano, among others, using plausibility arguments. Cardano, called “The Gambling Scholar” by Ore [13], was a colorful figure. A physician and mathematician by profession, he was also a practicing astrologer and an inveterate gambler, who composed “a learned book on games and ways to win in gambling” [13, p. viii], entitled The Book on Games of Chance [13]. His most influential work, The Great Art, dealing with the solution of the cubic and quartic equations by radicals, was a fundamental contribution to mathematics (see 7 Chapter 2). The Division Problem presents a much greater challenge. Among the first to introduce it was the Italian mathematician Luca Pacioli, in his 1494 book Everything About Arithmetic, Geometry, and Proportions, better known as Suma. Here is his version of the problem [10, p. 489]: »» Two players are playing a fair game [the players are equally capable] that was to continue until one player had won six rounds. The game stops when the first player has won five rounds and the second player three. How should the stakes be divided between the two players? 28 Chapter 4 • Probability: From Games of Chance to an Abstract Theory Blaise Pascal (1623–1662) 4 Pacioli claimed that the stakes should be split in the ratio of 5:3, which is incorrect. Unsuccessful attempts to solve the problem were also made by several Italian mathematicians of the sixteenth century, including Cardano and Niccolò Tartaglia. Tartaglia ventured that the stakes should be divided in the ratio 2:1. And so we come to Pascal and the Chevalier de Méré. Pascal was intrigued by the Problem of Points proposed by de Méré and agreed to study it. Before long he had a solution. The problem was challenging and subtle, and so he wrote (in July 1654) to Fermat, the leading mathematician of France, asking if he would read his (Pascal’s) solution. Fermat obliged. Thus began the now famous Pascal–Fermat correspondence, lasting several months (July–November 1654), and resulting in the emergence of what turned out to be a most important mathematical discipline—probability. Seven letters of that correspondence are extant, though it is not known how many were exchanged; in particular, Pascal’s initial letter to Fermat is lost. The renowned mathematician and master probabilist Alfréd Rényi reconstructed four of Pascal’s letters to Fermat [15]. What did the letters in the Pascal–Fermat correspondence contain? First, what they did not contain: there are no formal definitions, nor proofs of theorems; even the word “probability” does not appear (it will first show up about a century later). What we have are solutions of a problem—the Division Problem (the Problem of Points), its specializations and extensions (for example, to three players), different ways of looking at it, giveandtake between two brilliant mathematicians, the emergence of approaches to the solution of the problem, some combinatorics, and crucial ideas such as a fair die, equally likely events, and “favorable” events—although these last two important ideas are already present in the solutions of gaming problems by Cardano and by Galileo. The thrust of the Pascal–Fermat correspondence was that it put in motion what came to be known as “probability theory”—a new branch of mathematics. Indeed, contemporary mathematicians recognized that Fermat and Pascal had done precisely that. Why was there no definition of probability, and why were there no theorems and proofs in the Pascal–Fermat correspondence—a work that initiated a new subject? First, such matters are not to be expected in a correspondence dealing with the solution of problems. Beyond that, it is almost always the case that the formal development of a subject comes at the end of an evolutionary process. Calculus is an excellent example of this phenomenon (see 7 Chapter 5). 4.2 • Huygens: The First Book on Probability 29 4 Pierre de Fermat (1601–1665) Probability theory followed a similar route. Mathematicians knew very well what “probability” meant without having to define the concept, and they put the ideas of probability theory to excellent use in the eighteenth and nineteenth centuries without having a formal structure of the subject, which was introduced in the early twentieth century. 4.2 Huygens: The First Book on Probability Christiaan Huygens was a firstrate Dutch mathematician, physicist, astronomer, and inventor. (Most mathematicians in the seventeenth and eighteenth centuries were also scientists.) He studied mathematics and law at the University of Leiden. On a visit to Paris in 1655 he became acquainted with the Problem of Points, though not with its solution. Taken with the problem, he promptly solved it. But he realized that one was dealing here with important ideas beyond the solution of problems. So he decided to write a book which would give expression to this broader point of view [9, p. 65]: »» I would like to believe that if someone studies these things a little more closely, then he will almost certainly come to the conclusion that it is not just a game which has been treated here, but that the principles and the foundations are laid of a very nice and very deep speculation. The result of these speculations was a sixteenpage treatise titled On Reckoning at Games of Chance, published in 1657. Here is how a historian of the subject, Florence Nightingale David (1909–1993), saw this work [3, p. 110] (but see [5, p. 138 ff.] for a contrary view): 30 Chapter 4 • Probability: From Games of Chance to an Abstract Theory »» The scientist who first put forward in a systematic way the new propositions evoked by the problems sent to Pascal and Fermat, who gave the rules and who first made definitive the idea of mathematical expectation, was Christianus Huygens. 4 Huygens’ book contained fourteen propositions, which were detailed solutions of problems dealing with games of chance. For example, two of the propositions were the Dice and Division Problems, presented by de Méré to Pascal. The ninth proposition discusses the Problem of Points involving an arbitrary number of players, and the twelfth proposition asks for the number of dice a player must use so that at least two sixes show up in a single throw. The solutions of all the problems were carefully justified. The justifications were based on the fundamental notion of mathematical “expectation” (expected gain)—which Huygens was the first to define and highlight—rather than on the concept of probability, which is not mentioned. (One can define expectation in terms of probability or probability in terms of expectation [1, p. 165].) Huygens concluded his book with five challenging problems, which came to be named after him, and which enticed prominent mathematicians, including Jakob Bernoulli and Abraham De Moivre, to work on their solution. Here is the second problem [16, p. 25]: »» Three players, A, B, C, take twelve balls, eight of which are black and four white. They play on the following conditions: they are to draw blindfold, and the first who draws a white ball wins. A is to have the first turn, B the next, C the next; then A again, and so on. Determine the chances of the players. Bernoulli solved the problem under several interpretations—for example, drawing the balls with or without replacement. See [9, p. 75] for Huygens’ own solution of the division problem. Huygens’ book served as a text in probability—the only one available for the next fifty years, when it was incorporated, with commentary, as Part I of Jakob Bernoulli’s Ars Conjectandi [2]. See the section below, as well as [3, 9] for details. 4.3 Jakob Bernoulli’s Ars Conjectandi (The Art of Conjecturing) The Bernoullis, a distinguished Swiss family, produced eight members who made significant contributions to mathematics. Most prominent among them were Jakob (1654–1705), his younger brother Johann (1667–1748), Johann’s son Daniel (1700–1782), and Jakob’s nephew Nikolaus (1687–1759). Jakob Bernoulli’s important and influential book Ars Conjectandi, published posthumously in 1713, may be said to have completed the first period in the evolution of probability and given a thrust to the second [2]. The modern philosopher Ian Hacking gives details [8, p. 143]: »» Jacques Bernoulli’s Ars conjectandi presents the most decisive conceptual innovations in the early history of probability. …[Upon its publication] probability came before the public with a brilliant portent of all the things we know about it now: its mathematical profundity, its unbounded practical applications…and its constant invitation for philosophizing. Probability had fully emerged. The Ars Conjectandi comprises four parts. The first is a reprint of Huygens’ On Reckoning at Games of Chance, with extensions and elaborations of his solutions. (Bernoulli was greatly 4.3 • Jakob Bernoulli’s Ars Conjectandi (The Art of Conjecturing) 31 4 Jakob Bernoulli (1654–1705) influenced by Huygens’ book.) There are also solutions of the five problems which Huygens left as exercises. Part II contains a systematic account of “the doctrine of permutations and combinations”, as Bernoulli called it, including what came to be known as the Bernoulli numbers [5]; and Part III applies the previous work to solve a series of games more challenging than those considered in Huygens’ treatise. It was in Part IV, however, that Ber