A Dozen Episodes from the Mathematics of the Nineteenth Century ...

A Dozen Episodes from the Mathematics of the
Nineteenth Century
Lecture notes and weekly reports
HENRIK KRAGH SØRENSEN
History of Science Department, University of Aarhus
Spring 2001
http://home.imf.au.dk/hkragh/hom/episoder/
Compilation date: 15th May 2001

2 Contents
Contents
Week 5: Welcome 3
The dozen episodes in schematic form 4
Week 7: Institutional and national aspects of nineteenth century mathematics 9
Week 8: Complex numbers throughout the nineteenth century 12
Week 9: Algebraic solution of equations 14
Week 10: Algebra and number theory 16
Week 11: Rigor in analysis – continuity and differentiability 18
Week 12: Rigor in analysis – infinite series 20
Week 13: Fourier series and integration 21
Week 14: Theory of complex functions 23
Week 16: Non-Euclidean geometry 25
Week 17: Differential geometry and models of non-Euclidean geometry 27
Week 18: Projective geometry 29
Week 19: Axiomatization of set theory 31
Complete list of references 34
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Week 5: Welcome 3
Week 5: Welcome
At the introductory lecture and welcome meeting, the course was briefly presented and
the schedule was made. Unfortunately, it was impossible to accomodate all wishes but we
managed to agree upon the following schedule:
The schedule for the course was laid out as follows:
Lectures on Fridays 14–16. A room will be announced.
Exercises on Tuesdays 12–14 at IVH Aud.
However, to accomodate everybody, please email me if you are able to attend
exercises on Thursdays 16-18.
The first lecture and exercises will be held in week 7, ie. if no further notice is given,
First exercises on Tuesday 13th February 12.15 in IVH Aud.
For the exercises in week 7, please read the texts 1–3 and study the questions posed on
the weekly report for week 7.
I would strongly encourage two participants to volunteer for the exercises in week 8,
ie. texts 4–5. Further arrangements will be made on Tuesday 13th.
All material handed out will be placed on Semesterhylden at the rear of the IVH
library. Later in the term, participants will be asked to pay for the material handed out for
the exercises.
Besides the material handed out, background literature for the lectures will draw upon
the 2nd and 3rd volume of M. KLINE’s Mathematical thought from ancient to modern
times (?). The episodes will deal with material covered by chapters 26, 27, 31, 32 (volume
2) and 34, 35, 36, 37, 38, 40, 41, 42, 43, 49 (volume 3). Each weekly report will include
references to the relevant chapters. Another good supplement is I. GRATTAN-GUINNESS’
book The Fontana history of mathematical sciences (?).
For the lectures and exercises in week 7, the most relevant part of (?) is chapter 26.
The material handed out for the exercises also constitutes part of the background literature
of the lectures.
For the lectures and exercises in week 8, the background literature consists of (?,
chapters 27.3, 32).
Useful links
• http://home.imf.au.dk/hkragh/hom/episoder/: Homepage of the
course
• http://www.ams.org/mathscinet/: Mathematical Reviews online (by AMS)
Henrik Kragh Sørensen
hkragh@imf.au.dk
Office H1.19, phone 3508
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4 The dozen episodes in schematic form
The dozen episodes in schematic form
Week 7: Institutional and national aspects of nineteenth century mathematics
Exercises: On the social and institutional setting of mathematics in the nineteenth century:
(Struik 1954, 201–227), (Grattan-Guinness 1994b), and (Schubring 1994)
Presentation of the episodes and historiographical remarks: temporal, disciplinary, and
national.
The social and political background of the French Revolution and the Napoleonic
Wars. The institutionalizing of mathematics in France and Germany; the emergence of
a professional class of mathematicians. The change of mathematical interaction; the
creation of the first professional journals, in particular those by GERGONNE 1810 and
CRELLE 1826. Mathematics in Britain and in Scandinavia.
Week 8: Complex numbers throughout the nineteenth century
Biography: WESSEL
Exercises: On HAMILTON’s construction of complex numbers as pairs of time steps:
(Hamilton 1837, 76–84) and (Hankins 1980, 268–275)
Geometric representation of complex numbers with WESSEL and ARGAND. The low
and high priority questions concerning complex numbers among mathematicians. Symbolic
approach to complex numbers leading to quaternions and hypercomplex numbers;
CAUCHY and HAMILTON.
Week 9: Algebraic solution of equations
Biography: GALOIS
Exercises: A selection from Gauss’ division of the circle: (Gauss 1986, 407–428)
The continuity from the algebraic investigations of LAGRANGE. The works of RUFINI and
ABEL concerning the quintic equation. Solubility by other means. Problem of algebraic
solubility. GALOIS’ theory. JORDAN’s assimilation of Galois-theory. The emergence of
the theory of groups.
Week 10: Algebra and number theory, division problems, reciprocity, and FERMAT’s
Last Theorem
Biography: GAUSS
Exercises: HÖLDER on the concept of quotient groups and the JORDAN-HÖLDER Theorem:
(Hölder 1889, 26–38)
GAUSS’ Disquisitiones arithmeticae; the division of the circle with GAUSS, the division
of the lemniscate with ABEL, and Abelian equations. Quadratic (and higher) reciprocity.
The importance of FERMAT’s Last Theorem for the evolution of abstract algebra.
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The dozen episodes in schematic form 5
Week 11: Rigor in analysis
Biography: WEIERSTRASS
Exercises: BOLZANO (Russ 1980, 159–168, 171–172, 174–176) and CAUCHY (Grabiner 1981,
167–168) on the definition of continuity and the intermediate value theorem
Focus on limits and inequalities. CAUCHY’s introduction of continuity and differentiability
in Cours d’analyse and Resumé des leçons. CAUCHY’s innovation founded in the
previous generation, primarily LAGRANGE. The intermediate value theorem, importance
and proofs. BOLZANO. Real numbers, DEDEKIND and CANTOR. WEIERSTRASS. The
WEIERSTRASS Monster.
BJARNE AAGAARD will lecture on the biography of WEIERSTRASS, the philosophical
theory of LAKATOS, and the importance of WEIERSTRASS’ Monster for the development
of analysis in the 19th century.
Week 12: Theory of series and uniform convergence
Biography: ABEL
Exercises: SEIDEL’s proof analysis (Seidel 1847, 35–45) and LAKATOS’ philosophy
(Lakatos 1976, 127–141)
CAUCHY’s numerical concept of equality based on limits as opposed to EULER’s formal
equality. CAUCHY’s ideas about rigorous, critical revision of the theory of infinite series:
definition and criteria of convergence. ABEL’s reading of CAUCHY and the exception.
Towards uniform convergence.
Week 13: Fourier series and the concept of integrals
Biography: FOURIER
Exercises: RIEMANN and the integral concept (Riemann 1854, 227–244)
FOURIER and the emergence of Fourier series from the theory of heat. CAUCHY’s concept
of the integral and his integral calculus. Convergence of Fourier series, CAUCHY
and DIRICHLET, the Dirichlet conditions. Are all trigonometric series Fourier series?
Do Fourier series represent the functions? Is Fourier series representation unique? RIE-
MANN’s Habilitationsaufsatz, 1854. The Fundamental Theorem of the Calculus. CAN-
TOR and the theory of point sets. Uniqueness of representation. Replacement condition:
JORDAN and bounded variation.
Week 14: Theory of complex functions
Biography: CAUCHY
Exercises: WEIERSTRASS on power series and analytic functions (Weierstrass 1988, 63–
70,93–97,127–130)
Logarithms of negative numbers. GAUSS on integration in the complex plane. CAUCHY
on integration along the sides of a rectangle. CAUCHY’s theory of complex functions. Elliptic
functions, the inversion by ABEL. RIEMANN’s theory of complex functions. Briefly
on WEIERSTRASS’ theory. Differences between the theories. Unification.
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6 The dozen episodes in schematic form
Week 16: Non-Euclidean geometry
Biography:
Exercises: LOBACHEVSKY’s redefinition of parallels (Lobachevski 1955, 11–23)
EUCLID’s geometry. The problematic fifth postulate (P5). Attemts at proving P5. The
philosophical aspect: is geometry “true”? Correspondance. PLAYFAIR’s axiom and
hyperbolic geometry. On the geometries of BOLYAI and LOBACHEVSKY. Video with
JEREMY GRAY on the discovery of hyperbolic geometry.
Week 17: Differential geometry
Biography: RIEMANN
Exercises: GAUSS’ intrinsic geometry and the concept of curvature (Gauss 1902, )
Week 18: Projective geometry and the Erlanger programme
Biography: PONCELET
Exercises: HILBERT’s axiomatization of geometry
Week 19: Mathematics of the nineteenth century in hindsight
Biography: HILBERT
Exercises:
Week 20: The theory of sets (TERESE M. O. NIELSEN)
Biography: CANTOR
Exercises: ZERMELO
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The dozen episodes in schematic form 7
References
Gauss, C. F. (1986). Disquisitiones Arithmeticae. English Edition. New York, Berlin,
Heidelberg, Tokyo: Springer-Verlag. Edited by A. A. Clarke and W. C. Waterhouse.
Gauss, K. F. (1902). General Investigations of Curved Surfaces of 1827 and 1825.
Princeton (NJ): Princeton University Library. Translated and edited by J. C. Morehead
and A. M. Hiltebeitel.
Grabiner, J. V. (1981). The Origins of Cauchy’s Rigorous Calculus. Cambridge
(Mass.): MIT Press.
Grattan-Guinness, I. (Ed.) (1994a). Companion encyclopedia of the history and philosophy
of the mathematical sciences. London: Routledge. 2 vols.
Grattan-Guinness, I. (1994b). France. In volume 2 of Grattan-Guinness 1994a, Chapter
11.1, pp. 1430–1441.
Hamilton, W. R. (1837). Theory of conjugate functions, or algebraic couples; with a
preliminary and elementary essay on algebra as the science of pure time. In The
mathematical papers of Sir William Rowan Hamilton, Volume 3, pp. 3–96. Cambridge:
at the University Press. First published Trans. Roy. Irish Acad. vol. XVIII
(1837), pp. 293–422. Read on 4 November 1833 and 1 June 1835.
Hankins, T. L. (1980). Sir William Rowan Hamilton. Baltimore and London: The Johns
Hopkins University Press.
Hölder, O. (1889). Zurückführung einer beliebigen algebraischen Gleichung auf eine
Kette von Gleichungen. Mathematische Annalen 34, 26–56.
Lakatos, I. (1976). Proofs and Refutations. The Logic of Mathematical Discovery.
Cambridge: Cambridge University Press.
Lobachevski, N. (1955). Geometrical researches on the theory of parallels. In
R. Bonola (Ed.), Non-Euclidean Geometry. New York: Dover Publications, Inc.
Translated by G. B. Halsted. German orignal published 1840.
Riemann, B. (1854). Ueber die Darstellbarkeit einer Function durch eine
trigonometrische Reihe. In The Collected Works of Bernard Riemann (Gesammelte
mathematische Werke und wissenschaftlicher Nachlass), pp. 227–264. New York:
Dover Publications. First published Abhandlungen der Königlichen Gesellschaft
der Wissenschaften zu Göttingen, volume 13.
Russ, S. B. (1980). A Translation of Bolzano’s Paper on the Intermediate Value Theorem.
Historia Mathematica 7(2), 156–185.
Schubring, G. (1994). Germany to 1933. In volume 2 of Grattan-Guinness 1994a,
Chapter 11.2, pp. 1442–1456.
Seidel, P. L. (1847). Note über eine Eigenschaft der Reihen, welche discontinuirliche
Functionen darstellen. In H. Liebmann (Ed.), Die Darstellung ganz willkürlicher
Functionen durch Sinus- und Cosinusreihen von Lejeune Dirichlet (1837) und Note
über eine Eigenschaft der Reihen, welche discontinuirliche Functionen darstellen
von Philipp Ludwig Seidel (1847). Leipzig: Verlag von Wilhelm Engelmann. First
published Abhandl. der Math. Phys. Klasse der Kgl. Bayerischen Akademie der
Wissenschaften, V (1847), 381–394.
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8 The dozen episodes in schematic form
Struik, D. J. (1954). A concise history of mathematics. London: G. Bell and Sons ltd.
First published 1948.
Weierstrass, K. (1988). Einleitung in die Theorie der analytischen Funktionen. Vorlesung
Berlin 1878, Volume 4 of Dokumente zur Geschichte der Mathematik.
Braunsweig: Deutsche Mathematiker-Vereinigung, Vieweg & Sohn. Notes taken
by A. Hurwitz. Edited by P. Ullrich.
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Week 7: Institutional and national aspects of nineteenth century mathematics 9
Week 7: Institutional and national aspects of nineteenth
century mathematics
Lectures
The lectures in week 7 will serve a three-fold purpose:
1. Introductions of the course, the lecturer, and the participants will be given.
2. Based on the selection of episodes, some historiographical remarks will be given
concerning central choices.
3. The main core of the lectures and the exercises will deal with the external — institutional
and social — background of mathematics in the nineteenth century.
The background literature will consist of (Kline 1972, chapter 26).
Apart from the literature handed out, (Struik 1981) gives a general introduction to the
mathematics of the early nineteenth century and its close links with the dramatic changes
that took place on the political scene. Many other general histories of mathematics dealing
with the nineteenth century can be named, eg. (Klein 1967, Kline 1972). Of course,
expert detailed studies also exist of almost all the episodes covered in this course. References
will be given at the appropriate places.
The impact of the French Revolution on mathematics education has been described in
eg. (Grattan-Guinness 1990).
The evolution of mathematics in Germany and its relation to the reform of mathematics
education has been described in (Biermann 1988, Jahnke 1990, Schubring 1993).
The British mathematical community were largely centered in Cambridge and in
Dublin. The scene in Cambridge has recently been studied in a speciale at our department,
(Nielsen 2001).
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10 Week 7: Institutional and national aspects of nineteenth century mathematics
Exercises
For the exercises in week 7, we read three secondary sources describing aspects of the general
development of mathematics in the early nineteenth century as well as some of the important
social and institutional changes that took place in the period: (Grattan-Guinness 1998,
chapters 7,11; 347–363,479–492), (Grattan-Guinness 1994b), and (Schubring 1994).
Since these exercises are the first in the course, no students were appointed to present
questions. Instead, besides questions concerning the textual contents of the texts, we can
discuss some the questions raised below. When we have learned more about the contents
of nineteenth century mathematics, we may pause to consider some of the questions again.
You may be unfamiliar with many of the names and mathematical topics mentioned in
the texts. Please, try to read on and get to the general contents of the arguments presented.
1. What is your general impression from reading GRATTAN-GUINNESS’ presentation
(text 1)? Was it easy to read? How much of the (sparse) mathematics did you
recognize? How about the texts from the Companion?
2. Discuss the various motivations for education reforms in mathematics in the early
nineteenth century in France and Germany.
3. Based on the limited information contained in the texts, discuss the creation of
mathematical schools around eg. WEIERSTRASS and KLEIN.
4. Describe similarities and differences between French and German mathematics in
the nineteenth century as these are expressed in the three sources.
5. How do the authors describe the interaction between “pure” and “applied” mathematics
in the nineteenth century? Any national differences?
6. Based on the texts, how is your reaction to the following statement: “Mathematics
as we know it — as an independent discipline of university research — came into
being only in the nineteenth century”?
7. Characterize the purposes of education in mathematics that were involved in the
reforms of the nineteenth century. Compare these to the modern discussion of the
objectives of mathematical education in Denmark and elsewhere.
8. Discuss the importance of the expressed purposes of mathematics for the contents
of mathematical research and education, in particular in the French polytechnical
tradition.
9. Can you come up with any (other) strong interactions between mathematics and the
surrounding society or non-adjacent disciplines?
10. Can you problematize the idea of national styles and differences in mathematics?
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Week 7: Institutional and national aspects of nineteenth century mathematics 11
References
Biermann, K.-R. (1988). Die Mathematik und ihre Dozenten an der Berliner Universität
1810–1933. Stationen auf dem Wege eines mathematischen Zentrums von
Weltgeltung. Berlin: Akademie-Verlag Berlin.
Grattan-Guinness, I. (1990). Convolutions in French Mathematics 1800–1840, Volume
2–4 of Science Networks — Historical Studies. Basel, Boston, Berlin: Birkhäuser
Verlag.
Grattan-Guinness, I. (Ed.) (1994a). Companion encyclopedia of the history and philosophy
of the mathematical sciences. London: Routledge. 2 vols.
Grattan-Guinness, I. (1994b). France. In volume 2 of Grattan-Guinness 1994a, Chapter
11.1, pp. 1430–1441.
Grattan-Guinness, I. (1998). The Fontana history of the mathematical sciences. London:
FontanaPress.
Jahnke, H. N. (1990). Mathematik und Bildung in der Humboldtschen Reform. Göttingen:
Vandenhoeck & Ruprecht.
Klein, F. (1967). Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert.
New York: Chelsea Publishing Company. First published in three volumes
1926–27.
Kline, M. (1972). Mathematical Thought from Ancient to Modern Times. Oxford: Oxford
University Press. Reprinted in three volumes 1990.
Nielsen, R. R. (2001, January). Robert Woodhouse og The Analytical Society. Opgøret
med fluxionerne i Cambridge i første halvdel af 1800-tallet. Speciale, Institut for
videnskabshistorie, Aarhus Universitet, Aarhus.
Schubring, G. (1993). The German mathematical community. In J. Fauvel, R. Flood,
and R. Wilson (Eds.), Möbius and his band. Mathematics and Astronomy in
Nineteenth-century Germany, Chapter 2, pp. 21–33. Oxford: Oxford University
Press.
Schubring, G. (1994). Germany to 1933. In volume 2 of Grattan-Guinness 1994a,
Chapter 11.2, pp. 1442–1456.
Struik, D. J. (1981). Mathematics in the Early Part of the Nineteenth Century. In
H. Mehrtens, H. Bos, and I. Schneider (Eds.), Social History of Nineteenth Century
Mathematics, pp. 6–20. Boston, Basel, Stuttgart: Birkhäuser.
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12 Week 8: Complex numbers throughout the nineteenth century
Week 8: Complex numbers throughout the nineteenth
century
Lectures
The lectures in week 8 will treat the conceptions of complex and hypercomplex numbers
throughout the 19th century. Starting with the geometrical interpretations of complex
numbers advocated by WESSEL and ARGAND, both on the mathematical periphery
(Andersen 1999, Wessel 1799), we will turn to the proofs of the Fundamental Theorem
of Algebra (FTA) by GAUSS, and the symbolic interpretation of complex numbers with
CAUCHY and HAMILTON. Finally, the further development of generalized complex numbers,
leading to hypercomplex numbers and algebras, will be traced, (van der Waerden 1985,
Hoffmann 2000).
The background literature will consist of (Kline 1972, chapters 27.3, 32).
Exercises
For the exercises, we will read part of WILLIAM ROWAN HAMILTON’s Theory of conjugate
functions or algebraic couples (text 4) as well as a short passage from THOMAS
HANKINS’ biography of HAMILTON (text 5).
JOHN and ERIK have prepared the following guiding questions for the exercises:
1. What does HAMILTON actually introduce in the text?
2. What is the general outline of the selection? What are HAMILTON’s intentions?
3. What are the differences (from a modern perspective) between step and momentcouple?
4. What do pure primary and pure imaginary translate into in modern terminlogy?
5. Where are the differences between HAMILTON and modern mathematics the most/the
least apparent? The focus is here on the presentation of mathematical material.
6. Based on the introduction, would HAMILTON be classified as a pure or as an applied
mathematician?
7. Compare HAMILTON’s way of presenting mathematics to the modern tradition. Are
they different? How?
8. At a central point, HAMILTON introduced two constants, γ1 and γ2. What are their
roles? How could they be interpreted from a modern perspective (think geometrically).
9. Importantly, HAMILTON fixed γ1 = −1, γ2 = 0. Why? What did he achieve? What
were his arguments?
10. Based on the texts, discuss the Kantian foundation of HAMILTON’s philosophy of
numbers.
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Week 8: Complex numbers throughout the nineteenth century 13
References
Andersen, K. (1999). Wessel’s work on complex numbers and its place in history. In
B. Branner and J. Lützen (Eds.), Caspar Wessel. On the Analytic Representation
of Direction. An Attempt Applied Chiefly to Solving Plane and Spherical Polygons.
Copenhagen: C. A. Reitzel.
Gericke, H. (1996). Talbegrebets historie. ˚Arhus: Matematiklærerforeningen og Institut
for de Eksakte Videnskabers Historie, Aarhus Universitet. Danish edition translated
and edited by Kirsti Andersen and Kate Larsen.
Guld, T. L. (1997). Hamilton og kvaternionerne. Speciale, Institut for de Eksakte Videnskabers
Historie, Aarhus Universitet, Aarhus.
Hamilton, W. R. (1837). Theory of conjugate functions, or algebraic couples; with a
preliminary and elementary essay on algebra as the science of pure time. In The
mathematical papers of Sir William Rowan Hamilton, Volume 3, pp. 3–96. Cambridge:
at the University Press. First published Trans. Roy. Irish Acad. vol. XVIII
(1837), pp. 293–422. Read on 4 November 1833 and 1 June 1835.
Hoffmann, M. (2000, June). Fra tal til algebraer. Den britiske begyndelse i 1830’erne
og aspekter af den tyske fortsættelse op til 1890. Speciale, Institut for Videnskabs–
historie, Aarhus Universitet, Aarhus.
Kline, M. (1972). Mathematical Thought from Ancient to Modern Times. Oxford: Oxford
University Press. Reprinted in three volumes 1990.
van der Waerden, B. L. (1985). A History of Algebra. From al-Khwārizmī to Emmy
Noether. Berlin, Heidelberg, New York, Tokyo: Springer-Verlag.
Wessel, C. (1799). Om Directionens analytiske Betegning, et Forsøg, anvendt fornemmelig
til plane og sphæriske Polygoners Opløsning. Nye Samling af det Kongelige
Danske Videnskabernes Selskabs Skrifter 5, 469–518.
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14 Week 9: Algebraic solution of equations
Lectures
Week 9: Algebraic solution of equations
The lectures in week 9 (27th February) will be concerned with the theory of equations and
its impact on the formation of the concept of groups. A brief introduction to the theory
of equations in the 18th century will be given before the theory of algebraic solvability
is treated in some detail. The continuity from LAGRANGE to ABEL and GALOIS will be
outlined, and the background of the insolubility proofs for the quintic will be sketched.
The assimilation of Galois theory will be described. Together, the lectures and exercises
of weeks 9 and 10 will illustrate the introduction of the concept of an abstract group.
The background literature will be (Kline 1972, chapters 25.2, 31, 49.1, 49.2). Further
important literature on the subject includes (Wussing 1969, Kiernan 1971–72).
Exercises
For the exercises (2nd March), we will read a selection (text 6) from GAUSS’ Disquisitiones
arithmeticae (Gauss 1986, 407–428) which should facilitate a discussion of GAUSS’
working with an implicit group-like structure.
The text is at places rather technical. Therefore, the reader should focus her/his attention
on the paragraphs §335–344 and §352. The mathematical details of the other paragraphs
can also be treated in the exercises, and the overall purpose and scope of GAUSS’
book will be discussed in lectures.
While reading the text, you can have the following questions prepared by PETER and
MICHAEL in mind:
1. Was the text difficult to understand? Why/why not? What did GAUSS think of its
level of difficulty?
2. What is GAUSS’ concern in the text (§335)? Which theories are listed by GAUSS
as being important? Which roles do these theories play today?
3. What do you make of GAUSS’ introduction to section 7 (§335)?
4. What does GAUSS explain in §337? What do you think of his way of doing so?
5. What is the equation which GAUSS wants to solve (end of §337)? Does he succeed
in solving this equation in the text?
6. Study the contents of §339–340 so that you can make out what is going on and
comment on GAUSS’ arguments.
7. In §341, GAUSS proves the irreducibility of the equation X = 0. How does he do
that?
8. How to you understand GAUSS general program as set forth in §342? It is further
developped in §352.
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Week 9: Algebraic solution of equations 15
9. Compare the structure and notation of GAUSS’ text with the ones common today.
E.g. What is a ‘complex’ etc.?
10. Do you recognize any ‘modern’ concepts in the text? Groups?
11. Starting in §343, GAUSS introduced a concept of periods. Can you (do you) understand
the concept and his results concerning it? It might help to revise GAUSS’
notation.
References
Gauss, C. F. (1986). Disquisitiones Arithmeticae. English Edition. New York, Berlin,
Heidelberg, Tokyo: Springer-Verlag. Edited by A. A. Clarke and W. C. Waterhouse.
Kiernan, B. M. (1971–72). The Development of Galois Theory from Lagrange to Artin.
Archive for History of Exact Sciences 8, 40–154.
Kline, M. (1972). Mathematical Thought from Ancient to Modern Times. Oxford: Oxford
University Press. Reprinted in three volumes 1990.
Wussing, H. (1969). Die Genesis des abstrakten Gruppenbegriffes. Ein Beitrag zur
Entstehungsgeschichte der abstrakten Gruppentheorie. Berlin: VEB Deutscher
Verlag der Wissenschaften.
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16 Week 10: Algebra and number theory
Lectures
Week 10: Algebra and number theory
The lectures in week 10 will summarize and analyze the introduction of the concept of abstract
groups. Through a short recap of the previous lecture on Galois theory and examples
drawn from number theory, the introduction of and focus on abstract structures in algebra
will be illustrated. Among the examples, we will deal with the law of quadratic reciprocity,
composition of quadratic forms, the understanding of quotient groups, SYLOW’s
theorems, and the introduction of ideal numbers to prove FERMAT’s Last Theorem.
Background literature will consist of (Kline 1972, chapters 31, 34). The interested
student may also consult (Wussing 1984), (Scholz 1990).
Exercises
For the exercises on Friday March 9, we will read an article by OTTO HÖLDER included
in the material as text 7 (Hölder 1889, 26–38). KIM has prepared the following questions
which you can have in mind while reading the source. Page numbers refer to HÖLDER’s
pagination.
1. Compare the definition of a group (page 29) with the one you are accustomed to.
Describe the differences.
2. Why do you think HÖLDER distinguishes between a group theoretical and an algebraic
section? What does he mean by algebraic?
3. What does HÖLDER mean in §2, page 29 when he speaks of ausgezeichnete or
invariante subgroups? What is the modern equivalent of this concept?
4. What does HÖLDER mean by holoedrish isomorphe groups?
5. What do you make of his definition of quotient groups in §4?
6. What would we call a meroedische Isomorphie?
7. What does he mean when he says (bottom of page 32) that G is “split in the factors
G|H and H”?
8. Does the direct product employed in §8 mean the same as it does today?
9. Compare the version of the JORDAN-HÖLDER-Theorem contained in the article
with the one you (perhaps) know. How do they differ?
References
Hölder, O. (1889). Zurückführung einer beliebigen algebraischen Gleichung auf eine
Kette von Gleichungen. Mathematische Annalen 34, 26–56.
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Week 10: Algebra and number theory 17
Kline, M. (1972). Mathematical Thought from Ancient to Modern Times. Oxford: Oxford
University Press. Reprinted in three volumes 1990.
Scholz, E. (Ed.) (1990). Geschichte der Algebra: eine Einfürung, Volume 16 of
Lehrbücher und Monographien zur Didaktik der Mathematik. Mannheim, Wien,
Zürich: Bibliographisches Institut, Wissenschaftsverlag.
Wussing, H. (1984). The genesis of the abstract group concept. Cambridge (Mass.):
MIT Press. A contribution to the history of the origin of abstract group theory,
Translated from the German by Abe Shenitzer and Hardy Grant.
A Dozen Episodes from the Mathematics of the Nineteenth Century HKS, Spring 2001

18 Week 11: Rigor in analysis – continuity and differentiability
Lectures
Week 11: Rigor in analysis – continuity and
differentiability
The lectures in week 11 will give an introduction to the ‘rigorization movement’ of the
19th century. The background and intentions of this ‘movement’ will be discussed. In
subsequent lectures, the impact of rigorization on various theories (e.g. infinite series
and integration) will be discussed. In week 11, however, we focus on the concepts of
continuity and differentiability. Among the issues to be discussed, the concept of ‘proof’
will take a central position.
Cand. scient. BJARNE AAGAARD has been invited to present a biography of WEIER-
STRASS, a presentation of LAKATOS’ philosophy of mathematics, and discuss the final
distinction between continuity and differentiability which only occurred in the 1870s.
BJARNE has worked with these questions in his speciale, (Aagaard 2001).
Background literature for the lectures will be (Kline 1972, chapters 26, 40, 41).
The primary literature dealt with in the lectures in week 11 includes (Cauchy 1821,
Bolzano 1817). One of the very best references for the history of analysis in the nineteenth
century is (Bottazzini 1986). For the relation between CAUCHY’s new rigor and the studies
of the foundation of analysis carried out by his predecessors, see (Grabiner 1981). For
the history of the concept of real numbers, the reader may start by consulting (Gericke 1996).
Exercises
The exercises for March 16 consist of two proofs concerning the intermediate value theorem.
Text 8 is an excerpt from a proof given by the Czech mathematician BERNARD
BOLZANO in 1817 and text 9 is the ‘technical’ proof which AUGUSTIN-LOUIS CAUCHY
gave in his textbook Cours d’analyse in 1821. Both texts are given in English translations.
HENRIK and FRANK have prepared the following questions, which you can keep in
mind while reading the texts. You should read the proofs in sufficient detail to be able to
participate in the discussion of question 9.
1. What were BOLZANO’s intentions with the text (text 8)?
2. What are the differences between BOLZANO’s proof and the previous proofs?
3. What is the difference between confirmation and justification? Compare to your
own notions.
4. What are basic truths?
5. Where do examples and metaphorical phrases (e.g. time, motion, and space) belong
in ‘pure’ science? Compare to a modern view.
6. Discuss BOLZANO’s presentation of earlier “proofs” and their flaws.
7. Look at the two propositions mentioned in the text (intermediate value theorem and
divisibility of functions). Which is the more general? Why?
A Dozen Episodes from the Mathematics of the Nineteenth Century HKS, Spring 2001

Week 11: Rigor in analysis – continuity and differentiability 19
8. What is the difference between “the truth” and “the more general truth” mentioned
at the beginning of page 69.
9. Discuss the proof given by BOLZANO in some detail.
10. Compare the two proofs given by BOLZANO and CAUCHY.
11. New scientific results can be published in two ways. Which one did BOLZANO
prefer? Why? Compare to modern ways of publishing results.
References
Aagaard, B. (2001, January). Patologiske funktioners historie og Lakatos. Speciale,
Institut for Videnskabshistorie, Aarhus Universitet, Aarhus.
Bolzano, B. (1817). Rein analytischer Beweis des Lehrsatzes, daß zwischen je zwey
Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle
Wurzel der Gleichung liege. Prag: Gottlieb Haase.
Bottazzini, U. (1986). The Higher Calculus: A History of Real and Complex Analysis
from Euler to Weierstrass. New York: Springer-Verlag.
Cauchy, A.-L. (1821). Cours d’analyse de l’École Royale Polytechnique. Premier partie.
Analyse algébrique. Paris: de l’Imprimerie Royale.
Gericke, H. (1996). Talbegrebets historie. ˚Arhus: Matematiklærerforeningen og Institut
for de Eksakte Videnskabers Historie, Aarhus Universitet. Danish edition translated
and edited by Kirsti Andersen and Kate Larsen.
Grabiner, J. V. (1981). The Origins of Cauchy’s Rigorous Calculus. Cambridge
(Mass.): MIT Press.
Kline, M. (1972). Mathematical Thought from Ancient to Modern Times. Oxford: Oxford
University Press. Reprinted in three volumes 1990.
A Dozen Episodes from the Mathematics of the Nineteenth Century HKS, Spring 2001

20 Week 12: Rigor in analysis – infinite series
Lectures
Week 12: Rigor in analysis – infinite series
The lectures on March 20 deals with the rigorization of the theory of infinite series. The
radical position of CAUCHY to ban divergent series from analysis is seen in the tradition of
the 18th century. Contributions and a counter example of ABEL are presented which shed
light on the evolution of a concept of uniform convergence. This theme will be taken up
again in the exercises on March 23. The importance of criteria for testing convergence is
emphasized and illustrated. The development of the theory of Fourier series in connection
with the theory of integration will be taken up in more detail in week 13.
Background literature for the lectures will be (Kline 1972, chapters 40, 47).
Exercises
For the exercises on March 23, we will read selections from SEIDEL (Seidel 1847, 35–45)
and LAKATOS (Lakatos 1976, 127–141) (texts 10 and 11). Remember, that although the
text might seem difficult, you are asked to read the SEIDEL-text to such a level of detail
that you can participate in a discussion of the proof.
MICHAEL and PETER have prepared the following questions for the discussion.
1. What were SEIDEL’s motivations for the paper?
2. What does the theorem on page 37 (original pagination) say? What is meant by
“beliebig langsam convergirt’? How did SEIDEL prove it? What do you think of
the proof?
3. LAKATOS states on page 136 (original pagination) that SEIDEL discovered uniform
convergence. Do you agree with LAKATOS? Why/why not? In your opinion, who
should be credited with the introduction of uniform convergence?
4. Place CAUCHY, ABEL, and SEIDEL in LAKATOS’ theory of Proof and Refutations.
Discuss!
References
Kline, M. (1972). Mathematical Thought from Ancient to Modern Times. Oxford: Oxford
University Press. Reprinted in three volumes 1990.
Lakatos, I. (1976). Proofs and Refutations. The Logic of Mathematical Discovery.
Cambridge: Cambridge University Press.
Seidel, P. L. (1847). Note über eine Eigenschaft der Reihen, welche discontinuirliche
Functionen darstellen. In H. Liebmann (Ed.), Die Darstellung ganz willkürlicher
Functionen durch Sinus- und Cosinusreihen von Lejeune Dirichlet (1837) und Note
über eine Eigenschaft der Reihen, welche discontinuirliche Functionen darstellen
von Philipp Ludwig Seidel (1847). Leipzig: Verlag von Wilhelm Engelmann. First
published Abhandl. der Math. Phys. Klasse der Kgl. Bayerischen Akademie der
Wissenschaften, V (1847), 381–394.
A Dozen Episodes from the Mathematics of the Nineteenth Century HKS, Spring 2001

Week 13: Fourier series and integration 21
Week 13: Fourier series and integration
Due to illness, the lectures were postponed from March 27 to March 30. Rescheduling
of exercises will be announced.
Lectures
The lectures on March 30 deal with Fourier series and conceptions of integrals. The
interaction between important questions concerning Fourier series, the new concept of
uniform convergence, and the extension of the concept of integrals is emphasized.
Background literature for the lectures will be (Kline 1972, chapters 20, (22.2), 28.2,
40.4, (44)). More information on the prehistory of LEBESGUE’s theory of integration
can be found in (Hawkins 1975). A good source of information on the development of
Fourier theory is the speciale (Ramskov 1992). DIRICHLET’s papers (Dirichlet 1829,
Dirichlet 1837) make quite interesting and not-too-difficult reading. For comprehensive
presentations of the rigorization of analysis, you may also consult (Bottazzini 1986) or
(Grabiner 1981). (Mejlbro 1989) is an interesting list of pathologocial counter examples.
Exercises
For the exercises of week 13 (which have been postponed to April 3), we will read an
excerpt from RIEMANN’s Habilitationsaufsatz (Riemann 1854, 227–244) (text 12). It
consists mainly of a historical introduction and a single mathematical idea. You can keep
the following questions, which have been prepared by ULRIK, in mind while reading the
text. Special emphasis should be placed upon the introduction of the Riemann integral.
1. What do you think of RIEMANN’s historical introduction? Is it ‘neutral’? Does it
serve a purpose?
2. How did RIEMANN present the state of affairs 1854?
3. What were RIEMANN’s motivations for generalizing DIRICHLET’s work?
4. In §4, RIEMANN introduced his definition of the definite integral. Compare it to
CAUCHY’s integral. Also compare with to the modern introduction of the integral
(Mat11).
5. Describe RIEMANN’s criterion of integrability as given in §5.
6. How did RIEMANN employ the first counter example presented in §6? Is it illuminating
to you? Hard: Can you see a connection with BJARNE’s guest lecture on
non-differentiable functions?
References
Bottazzini, U. (1986). The Higher Calculus: A History of Real and Complex Analysis
from Euler to Weierstrass. New York: Springer-Verlag.
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22 Week 13: Fourier series and integration
Dirichlet, G. L. (Werke). G. Lejeune Dirichlet’s Werke. Berlin: Georg Reimer. 1889–
1897. 2 vols. Edited by L. Kronecker and L. Fuchs.
Dirichlet, L. (1829). Sur la convergence des séries trigonométriques qui servent
représenter une fonction arbitraire entre des limites données. Journal für die reine
und angewandte Mathematik 4(2), 157–169. Reproduced (Dirichlet Werke, I, 117–
132).
Dirichlet, L. (1837). Ueber die Darstellung ganz willkürlicher Functionen durch Sinusund
Cosinusreihen. In H. Liebmann (Ed.), Die Darstellung ganz willkürlicher
Functionen durch Sinus- und Cosinusreihen von Lejeune Dirichlet (1837) und Note
über eine Eigenschaft der Reihen, welche discontinuirliche Functionen darstellen
von Philipp Ludwig Seidel (1847), pp. 3–34. Leipzig: Verlag von Wilhelm Engelmann.
First published Repertorium der Physik (Dove und Moser), I, p. 152–174,
1837. Reproduced (Dirichlet Werke, I, 133–160).
Grabiner, J. V. (1981). The Origins of Cauchy’s Rigorous Calculus. Cambridge
(Mass.): MIT Press.
Hawkins, T. (1975). Lebesgue’s Theory of Integration. Its Origins and Development
(2nd ed.). The Bronx, New York: Chelsea Publishing Company.
Kline, M. (1972). Mathematical Thought from Ancient to Modern Times. Oxford: Oxford
University Press. Reprinted in three volumes 1990.
Mejlbro, L. (1989). Mit Rædselskabinet. Matematisk Institut DTH.
Ramskov, K. (1992, April). Aspekter af de trigonometriske rækkers historie. De
trigonometriske rækkers udvikling indtil slutningen af 1800-tallet med hovedvægten
lagt p˚a perioden fra Riemann til du Bois-Reymond. Specialeopgave, Institut
for de Eksakte Videnskabers Historie, ˚Arhus Universitet.
Riemann, B. (1854). Ueber die Darstellbarkeit einer Function durch eine
trigonometrische Reihe. In The Collected Works of Bernard Riemann (Gesammelte
mathematische Werke und wissenschaftlicher Nachlass), pp. 227–264. New York:
Dover Publications. First published Abhandlungen der Königlichen Gesellschaft
der Wissenschaften zu Göttingen, volume 13.
A Dozen Episodes from the Mathematics of the Nineteenth Century HKS, Spring 2001

Week 14: Theory of complex functions 23
Lectures
Week 14: Theory of complex functions
The lectures on April 6 treat the origins and development of the theory of complex functions.
The interaction with problems arising in the theory of elliptic functions and generalizations
thereof is also described.
Background literature for the lectures will be (Kline 1972, chapters 19.3, (19.4),
27). More information can be found in (Bottazzini 1986). On CAUCHY’s theory of complex
functions, one may also consult (Smithies 1997). The interaction between elliptic
functions and complex analysis is the topic of the speciale (Nørgaard 1990).
Exercises
For the exercises on April 10, we will read part of the HURWITZ’ transcript from WEIER-
STRASS’ introductory lectures on analytic functions (text 13) (Weierstrass 1988, 63–
70,93–97,127–130). The text is rather lengthy, but focus your attention on the contents of
chapters 7 and 10. You need not go into all details but you should not find the mathematics
too strange...
ANN METTE and JAKOB have prepared the following useful questions which you can
consider while reading the text. The last three questions pertain to chapter 13.
1. First of all, try to relate WEIERSTRASS’ work to the previous works on the subject.
2. WEIERSTRASS commences his treatment of power series in section 7.1 with a
“Fundamentalsatz über ihre Convergenz”. What is the statement of the theorem?
What did he actually prove? Is the proof “modern”?
3. Do you recognize other theorems in section 7.1?
4. What does it mean, according to WEIERSTRASS, to be an internal, external or limit
point? What does he say of the convergence at these points?
5. In section 7.2, WEIERSTRASS wants to show when a sum of infinitely many power
series is unconditionally convergent. What do you think of his proof?
6. What does the Identity Theorem say? What can it be used for?
7. What is it that WEIERSTRASS introduces in section 10.1? What du you think of
the proof in this section? What does WEIERSTRASS mean when he says that one
element is a continuation of another one?
8. What is, according to WEIERSTRASS, a function element?
9. How does WEIERSTRASS (in section 10.2) define an analytic function? Discuss the
use (and usefulness) of this definition?
10. What are, again according to WEIERSTRASS, single-valued and multiple-valued
functions?
A Dozen Episodes from the Mathematics of the Nineteenth Century HKS, Spring 2001

24 Week 14: Theory of complex functions
11. Which types of isolated singularities arise acconding to WEIERSTRASS?
12. What is a meromorphic function (term introduced by the editor, ULLRICH, who is
responsible for all the section headings)?
13. Discuss the representation of meromorphic functions.
References
Bottazzini, U. (1986). The Higher Calculus: A History of Real and Complex Analysis
from Euler to Weierstrass. New York: Springer-Verlag.
Kline, M. (1972). Mathematical Thought from Ancient to Modern Times. Oxford: Oxford
University Press. Reprinted in three volumes 1990.
Nørgaard, S. (1990). Elliptiske funktioner og kompleks funktionsteori 1825-1860. Aspekter
af inversionsproblemet for elliptiske integraler set i relation til udviklingen
af en kompleks funktionsteori i Frankrig. Speciale, Institut for de Eksakte Videnskabers
Historie, Aarhus Universitet.
Smithies, F. (1997). Cauchy and the Creation of Complex Function Theory. Cambridge:
Cambridge University Press.
Weierstrass, K. (1988). Einleitung in die Theorie der analytischen Funktionen. Vorlesung
Berlin 1878, Volume 4 of Dokumente zur Geschichte der Mathematik.
Braunsweig: Deutsche Mathematiker-Vereinigung, Vieweg & Sohn. Notes taken
by A. Hurwitz. Edited by P. Ullrich.
A Dozen Episodes from the Mathematics of the Nineteenth Century HKS, Spring 2001

Week 16: Non-Euclidean geometry 25
Lectures
Week 16: Non-Euclidean geometry
The lectures on Friday April 20 will treat the discovery and development of non-Euclidean
geometry (NEG). Emphasis will be placed upon the philosophical implications of NEG
and the impact of models of NEG. At the end of the lecture, a video with JEREMY GRAY
on NEG will be shown.
Background literature for the lectures will be (Kline 1972, chapters 36, (38.2, 38.4,
38.6)). The development of non-Euclidean geometry is described in many textbooks on
the history of mathematics, eg. (Stillwell 1989, Katz 1998). The primary sources of
BOLYAI and LOBACHEVKSY are translated into English in (Bonola 1955). For the history
of the parallel postulate, see (Rosenfeld 1988, Jeppesen 1998).
Exercises
For the exercises on Tuesday April 24, we will read the introduction to LOBACHEVSKY’s
geometrical researches. The selection contains LOBACHEVSKY’s redefinition of parallel
lines. TUURE has prepared the following questions:
1. Discuss LOBACHEVSKY’s introduction and his style of presenting his arguments.
2. LOBACHEVSKY states some initial assumptions (theorems) 1–15 from which, according
to him, the other theorems follow. Are these initial theorems as trivial as
LOBACHEVSKY claims? Do you recognize all terms and objects mentioned? How
do these theorems follow from the Euclidean axioms without the parallel postulate?
3. When are two lines called parallel in LOBACHEVSKY’s sense? How does he come
to this definition? In how many ways can two lines be related to each other? What
do you think of this division?
4. What are the contents of theorems 17 and 18?
5. How are the parallelism of two lines and the angle sum of triangles related? The
sum of the angles in a triangle cannot exceed π. Why?
6. Discuss paragraph 22. What is Π(p)?
7. LOBACHEVSKY has given the name “imaginary geometry” to his new geometry;
can you see the reason for this name?
8. In Euclidean geometry, lines are either parallel or intersecting; do you think that
the discovery of other possibilities could have affected the birth of different logics
(intuitionistic, three-valued, ...)? Do you see other relations to other branches of
mathematics or to other sciences?
A Dozen Episodes from the Mathematics of the Nineteenth Century HKS, Spring 2001

26 Week 16: Non-Euclidean geometry
References
Bonola, R. (1955). Non-Euclidean Geometry. New York: Dover Publications, Inc.
Translated by H. S. Cartan.
Jeppesen, J. L. (1998, September). Euklids femte postulat. 2. delsopgave, Institut for
de Eksakte Videnskabers Historie, Aarhus Universitet, Aarhus.
Katz, V. J. (1998). A History of Mathematics. An Introduction (2nd ed.). Reading
(Mass.) etc.: Addison-Wesley.
Kline, M. (1972). Mathematical Thought from Ancient to Modern Times. Oxford: Oxford
University Press. Reprinted in three volumes 1990.
Rosenfeld, B. A. (1988). A History of non-Euclidean Geometry. Evolution of the Concept
of a Geometric Space. Number 12 in Studies in the History of Mathematics
and Physical Sciences. New York: Springer-Verlag.
Stillwell, J. (1989). Mathematics and Its History. New York: Springer-Verlag.
A Dozen Episodes from the Mathematics of the Nineteenth Century HKS, Spring 2001

Week 17: Differential geometry and models of non-Euclidean geometry 27
Lectures
Week 17: Differential geometry and models of
non-Euclidean geometry
The lectures on April 27 will deal with the differential geometry of GAUSS and RIEMANN.
Furthermore, the construction and implications of models of non-Euclidean geometry will
be discussed.
Background literature for the lectures will be (Kline 1972, chapters 36.8, 37, (38.4,
38.6)).
Exercises
For the exercises in week 17, we will read an excerpt from the English translation of
GAUSS’ Disquisitioens generales circa superficies curvas (text 15). The text is a points
quite technical, and special attention should be given to the results and methods of paragraphs
7 and 11-13.
JOHN and ERIK have prepared the following questions and problems for discussion
of the text. At points you are asked to compare GAUSS’ arguments and presentation with
modern textbooks of which we shall take DO CARMO’s book (Do Carmo 1976) as an
example.
1. Compare GAUSS’ way of presenting mathematics in the present text to that of
LOBACHEVSKY (text 14) and GAUSS’ own 1801 Disquisitiones arithmeticae (text
6). Relate differences to eg. time, disciplinary and national traditions.
2. Which concept is GAUSS discussing on page 10 (about the center)?
3. Comment on GAUSS’ implicit and explicit use of charts throughout the text. Where
are they explicitly used?
4. Compare the deduction in §7 to that in a modern textbook, eg. (Do Carmo 1976,
163). Comment on similaries and differences.
5. Comment on §11. What was achieved?
6. Comment on GAUSS’ use of the word developed in §12. Which modern formulation
would be used? Compare to the assumptions made in (Do Carmo 1976).
7. To what extent does GAUSS’ presentation differ from that of modern textbooks?
8. Comment on GAUSS’ use of infinitesimals and the rigor of his arguments. Compare
with other sources read in the course.
9. Comment on GAUSS’ concept of a surface.
A Dozen Episodes from the Mathematics of the Nineteenth Century HKS, Spring 2001

28 Week 17: Differential geometry and models of non-Euclidean geometry
References
Do Carmo, M. P. (1976). Differential Geometry of Curves and Surfaces. Prentice-Hall.
Kline, M. (1972). Mathematical Thought from Ancient to Modern Times. Oxford: Oxford
University Press. Reprinted in three volumes 1990.
A Dozen Episodes from the Mathematics of the Nineteenth Century HKS, Spring 2001

Week 18: Projective geometry 29
Lectures
Week 18: Projective geometry
The lectures on May 4 deals with the renewed attention paid to projective geometry in the
19th century. The attempt by PONCELET to enhance the deductive strength of synthetic
geometry will be discussed, as will the attempts by German mathematicians (in particular
VON STAUDT) of introducing coordinates into projective geometry by purely projective
means. The fusion of geometry with the theory of groups will be discussed. As time
permits, something will be said about HILBERT’s re-axiomatization of geometry.
Background literature for the lectures will be (Kline 1972, chapters 35, 38.5).
For a technical description of PONCELET’s principles and results, see (Jeppesen 2000).
On HILBERT’s Grundlagen der Geometry, see eg. (Freudenthal 1957, Freudenthal 1960).
Exercises
For the exercises on May 8, we will read an excerpt from the English translation of
HILBERT’s Grundlagen der Geometrie (text 16).
PATRICK has prepared the following questions which could guide your reading and
our discussion.
1. Compare the beginning of the text with other texts read in the course.
2. Discuss the objects introduced in §1.
3. Do you find the choice of axioms obvious? More specifically: is it obvious what
should be included as axioms and what should named theorems?
4. Is the text (its presentation) modern in the language and style? Compare with eg.
LOBACHEVSKY.
5. Are the presented ideas intuitively clear? Is this important?
6. Discuss the definition of “betweenness” in §3.
7. Ad the definition in §5. What is HILBERT really talking about here? (Algebra 1)?
Please note the keywords “symmetry” and “transitivity”.
8. Comment on HILBERT’s definition of angle in §5, p. 11.
9. What are your comments on the few proofs in the presentation? Do you follow
them easily?
10. At the beginning of §9, what is meant by a “field”? Is it the same thing today?
11. What do you think of theorem 32 (§8, p. 27)? And of the proof?
12. Now, considering the entire text, do you recognize ideas from other branches of
mathematics? If so, what branches and where? And why?
13. What are the purposes and contents of §9–10?
A Dozen Episodes from the Mathematics of the Nineteenth Century HKS, Spring 2001

30 Week 18: Projective geometry
References
Freudenthal, H. (1957). Zur Geschichte der Grundlagen der Geometrie. zugleich eine
Besprechung der 8. Aufl. von Hilberts “Grundlagen der Geometrie”. Nieuw Archief
voor Wiskunde 5(4), 105–142.
Freudenthal, H. (1960). Die Grundlagen der Geometrie um die Wende des 19. Jahrhunderts.
Mathematische-Physikalische Semesterberichte 7(1), 2–25.
Jeppesen, J. L. (2000, September). Poncelet og den projektive geometri. Poncelets
Traité des propriétés projectives des figures og dens plads i geometriens historie.
Speciale, Institut for Videnskabshistorie, Aarhus Universitet, Aarhus.
Kline, M. (1972). Mathematical Thought from Ancient to Modern Times. Oxford: Oxford
University Press. Reprinted in three volumes 1990.
A Dozen Episodes from the Mathematics of the Nineteenth Century HKS, Spring 2001

Week 19: Axiomatization of set theory 31
Lectures
Week 19: Axiomatization of set theory
A replacement lecture had been scheduled for Monday May 14. On this occasion, a brief
presentation of the axiomatic method as it stood around 1900 was given. Furthermore,
some of the Hilbert Problems were introduced in a non-technical way to allow for a discussion
of motives and methods in axiomatization.
The proper lecture of the week is Tuesday May 15 where cand. scient. TERESE M.
O. NIELSEN lectures on the development and axiomatization of set theory into the 20th
century. TERESE is a PhD-student at IVH specializing in the philosophy of mathematics.
Background literature for both the lectures will be (Kline 1972, chapters 41, 51).
TERESE has produced the following background information for the lectures and the
exercises. The book (Moore 1982) is recommendable. You can find more information
about ZERMELO on the following URL and its links:
www-history.mcs.st-and.ac.uk/history/Mathematicians/Zermelo.html
Background: Set Theory was created by Georg Cantor in the last quarter of the 19th
century. It originated from Cantor’s distinguishing between two ’sizes’ of infinite sets,
namely what we today call countable and uncountable sets. In an 1874 paper Cantor
proved that there are as many algebraic numbers as there are natural numbers, but that
there are more reals than natural numbers. These results led Cantor to state the Continuum
Hypothesis (CH) in 1878: Every infinite subset of the continuum (R) is either countable or
can be put into 1-1 correspondance with the continuum itself. In his famous 1900 lecture,
Hilbert made the proof or disproof of CH the first on his list of mathematical problems.
Furthermore, Hilbert suggested that the solution of this problem was best approached by
first proving another of Cantor’s conjectures, namely that any set can be well-ordered.
Thus, Zermelo set to work on the latter question and in 1904 he sent Hilbert a letter
(Zermelo 1904), containing his proof of the Well-ordering Theorem. However, Zermelo’s
result was far from accepted by his colleagues; on the contrary, it was massively criticised.
Zermelo’s (1908b) contains a new proof for his Well-ordering Theorem and a discussion
of the criticisms directed against the 1904 proof. Furthermore, in 1903 Russell’s paradox
was published, and this brought set theory in its entirety into doubt. Hence, Zermelo
wanted to put set theory on a secure footing by providing an axiomatization and proving
it consistent. Even though he did not succeed in the latter, he published his axiomatization
as Zermelo 1908a.
Biography: Ernst Zermelo was born on July 27th 1871 in Berlin and died on May 21st
1953 in Freiburg im Breisgau. He was the son of a college professor and received his education
from three different German universities, namely Berlin, Halle and Freiburg. He
received his doctorate in 1894 on the basis of a dissertation on the calculus of variations,
following Weierstrass’ approach. He then began work on his habilitation as an assistant
to Planck in Berlin in the field of hydrodynamics. He moved to Göttingen in 1897 and
completed his dissertation two years later and was immediately appointed lecturer.
When Zermelo heard of Cantor’s set theory, presumably from Hilbert who was also
in Göttingen, he changed the direction of his research. He published his first set theoret-
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32 Week 19: Axiomatization of set theory
ical paper in 1902, and in 1904 he proved the Well-ordering Theorem from what was to
become his Axiom of Choice (Axiom der Auswahl). The Theorem immediately became
the subject of much controversy, which Zermelo sort to dispel by publishing a new proof
of his theorem and an axiomatization for set theory in 1908. The use of axiomatization as
a method for clarification may also be inspired by Hilbert.
In 1910 Zermelo left Göttingen for the chair of mathematics at the Zürich University.
He resigned in 1916, due to bad health. He was appointed an honorary chair in Freiburg im
Breisgau in 1926, which he held until 1935 where he resigned it because of his disapproval
of the Hitler regime. In 1946 he was however reinstated to the post.
Exercises
LARS BO has prepared guiding questions for the texts. These questions have been supplemented
by questions suggested by TERESE. As a result, the list has grown quite long
and we will have to focus the discussion on Friday.
Introductory questions
1. What is the axiom of choice? Pay attention to how Zermelo employs it.
Questions on “Proof that every set can be well ordered” (Zermelo 1904)
2. How does Zermelo use the cardinalities of M and M ′ ?
3. How does Zermelo prove the existence of his ordering? What does a < b mean?
Questions on “A new proof of the possibility of a well-ordering” (Zermelo 1908b)
4. What were Zermelo’s motivations for presenting a new proof?
5. What does ‘an element of [a] subset [of M] is associated by some law as a “distinquished
element”’ (italics added) mean? Is a law necessarily an explicit rule, an
arbitrary function, or something else?
6. What are the “γ-sets” and “Θ-chains” introduced? How are these concepts used?
7. What is the argument for the validity of the axiom of choice? What does “necessary
for science” mean in this context (bottom of p. 187)?
8. Is Zermelo aware that the well-ordering theorem implies the axiom of choice?
9. What are the differences between Zermelo’s and Peano’s views on mathematics?
10. What are the differences between Zermelo’s and Poincaré’s views on mathematics?
What are “impredicative definitions”? Give an account of Zermelo’s defence
of impredicative definitions. How does Zermelo treat Poincaré’s criticism? Is it
convincing?
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Week 19: Axiomatization of set theory 33
Questions on “Investigations in the foundations of set theory” (Zermelo 1908a)
11. What were Zermelo’s purposes with this text? What role does axiomatization play?
Is he satisfied with his achievement?
12. What does a “set” mean? What is an “individual”? Whare does “aɛb” mean?
13. How does Zermelo justify his axioms?
14. Why does Zermelo describe the null set (and not other sets) as “fictious”?
15. Consider the general structure and wording of the axioms. Any surprises?
16. What is the purpose of Axiom III (axiom of separation)? Is it easily applied? What
is achieved in the theorem following this axiom?
17. How does Zermelo avoid the antinomies of set theory?
18. What is the purpose of Axiom VII, the axiom of infinity? Is it plausible?
General questions
19. Discuss similarities and differences between the 1908 and the 1904 proof, both
mathematically and structurally. Note especially the differences in the definitions
of a well-ordering.
20. Compare Zermelo’s axiom of choice and his proof of the well-ordering theorem
with a modern approach, eg. (Pedersen 1989, 2–5).
21. Compare the postulates I, II, III from “A new proof” with the axioms stated in
“Investigations”.
22. What is your opinion on the axiom of choice and the well-ordering theorem?
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34 Complete list of references
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A Dozen Episodes from the Mathematics of the Nineteenth Century HKS, Spring 2001