RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

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tion that they are genuine themes, and I have sought to develop and document this position in this dissertation and in my subsequent research. The main theme, which I have idenfied and used to structure and analyse ABEL’S mathematics is one of a transition from a predominantly formula based approach to mathematics towards a more concept based one. The formula based approach to mathematics can be found most clearly in the works of such mathematicians of the eighteenth century as L. EULER (1707–1783) or A.-M. LEGENDRE (1752–1833). These mathematicians approached analysis (including the theory of equations) in a formula based way, in which formulae were the central objects of mathematics. Their mathematics — to a large extend — consisted of manip- ulations of formulae which produced new formulae as results. As a counterpart to the formula based approach, I introduce concept based approach of the nineteenth century. This way of thinking about and doing mathematics was championed by such mathematicians as G. P. L. DIRICHLET (1805–1859) and G. F. B. RIEMANN (1826–1866). They thought mainly in terms of concepts, and their mathematics consisted of researching the relations between concepts, including representations of concepts, extensions of concepts, and the precise delineation of concepts. This framework of formula based and concept based mathematics has helped me to organise my study of ABEL’S mathematics because it explains several particular aspects and provides an organising scheme. Thus, in the second part — on the theory of algebraic solubility — the framework provides an explanation and structuring of what I term the delineation problem: finding the precise extension of the quality of solv- able equations. In the third part (rigorisation of analysis), the framework suggests that mathematicians were struggling with changing conceptions about their objects, not just the ways of manipulating them. This led to the necessity for critical revision which is also better understood within a framework of transitions. And in the fourth part (elliptic and higher transcendentals), the framework finally suggests that habituation — coming to know new objects — was a prominent problem which played a part in the shaping of concept based mathematics. This extremely brief outline of the suggestive and explanatory powers of the framework is meant to justify and explain the organising principle of the dissertation. In my subsequent research, I have worked to expand more on the framework and its analyt- ical powers. Recent literature Since the dissertation was submitted in 2002, new literature on ABEL and his mathematics has emerged. In particular, ABEL’S Paris memoir was located — first partially, then completely; see (Del Centina, 2002; Del Centina, 2003). I am grateful to NILS VOJE JOHANSEN for providing me with a xerox copy of the famous treatise. Later, xviii
in connection with the Abel centennial in Oslo 2002, the proceedings were published (Laudal and Piene, 2004). These contain a mathematical introduction by CHRISTIAN HOUZEL which elaborates on the previous publications of this scholar. Besides these, a number of other publications deal with ABEL’S mathematics and his topics; these include (Radloff, 2002) which sheds interesting light on ABEL’S theory of solubility and (Pesic, 2003). In my future research and publishing on ABEL’S mathematics, these will provide good possibilities for discussions. Although this list is not complete and no use of these publications has been made in the present work, they are cited here for the convenience of the reader. xix
Page 1 and 2: RePoSS: Research Publications on Sc
Page 3 and 4: The Mathematics of NIELS HENRIK ABE
Page 5 and 6: The Mathematics of NIELS HENRIK ABE
Page 7 and 8: Contents Contents i List of Tables
Page 9 and 10: 8.3 Refocusing on the equation . .
Page 11: V ABEL’s mathematics and the rise
Page 15 and 16: List of Figures 2.1 NIELS HENRIK AB
Page 17: List of Boxes 1 The algebraic reduc
Page 21 and 22: Summary The present PhD dissertatio
Page 23: Preface to the 2004 edition For thi
Page 28 and 29: items published in the same year ar
Page 30 and 31: the Mittag-Leffler archives in Djur
Page 33 and 34: Chapter 1 Introduction In the after
Page 35 and 36: 1.2. The mathematical topics involv
Page 37 and 38: 1.3. Themes from early nineteenth-c
Page 39 and 40: 1.4. Reflections on methodology 9 l
Page 41 and 42: 1.4. Reflections on methodology 11
Page 43 and 44: 1.4. Reflections on methodology 13
Page 45: 1.4. Reflections on methodology 15
Page 48 and 49: 18 Chapter 2. Biography of NIELS HE
Page 69 and 70: Chapter 3 Historical background The
Page 71 and 72: 3.2. ABEL’s position in mathemati
Page 73 and 74: 3.3. The state of mathematics 43 tr
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3.4. ABEL’s legacy 45 As can be s
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Chapter 4 The position and role of
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4.1. Outline of ABEL’s results an
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4.2. Mathematical change as a histo
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4.2. Mathematical change as a histo
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58 Chapter 5. Towards unsolvable eq
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98 Chapter 6. Algebraic insolubilit
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100 Chapter 6. Algebraic insolubili
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Chapter 7 Particular classes of equ
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7.1. Solubility of Abelian equation
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7.2. Elliptic functions 153 Figure
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7.2. Elliptic functions 155 7.2.1 T
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7.3. The concept of irreducibility
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7.3. The concept of irreducibility
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7.4. Enlarging the class of solvabl
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164 Chapter 8. A grand theory in sp
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Part III Interlude: ABEL and the
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192 Chapter 9. The nineteenth-centu
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194 Chapter 10. Toward rigorization
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208 Chapter 11. CAUCHY’s new foun
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Chapter 12 ABEL’s reading of CAUC
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12.1. ABEL’s critical attitude 22
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12.3. Convergence 229 12.3 Converge
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12.3. Convergence 231 and {εm} was
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12.4. Continuity 233 it followed th
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12.4. Continuity 235 Actually, if t
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12.4. Continuity 237 DIRICHLET intr
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12.5. ABEL’s “exception” 239
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12.6. A curious reaction: Lehrsatz
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12.7. From power series to absolute
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12.7. From power series to absolute
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12.8. Product theorems of infinite
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12.8. Product theorems of infinite
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12.9. ABEL’s proof of the binomia
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12.10. Aspects of ABEL’s binomial
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12.10. Aspects of ABEL’s binomial
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266 Chapter 13. ABEL and OLIVIER on
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Chapter 14 Reception of ABEL’s co
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14.1. Reception of ABEL’s rigoriz
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14.2. Conclusion 281 of basic notio
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Part IV Elliptic functions and the
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286 Chapter 15. Elliptic integrals
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300 Chapter 16. The idea of inverti
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Chapter 17 Steps in the process of
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17.1. Infinite representations 323
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17.2. Elliptic functions as ratios
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17.2. Elliptic functions as ratios
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17.3. Characterization of ABEL’s
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Chapter 18 Tools in ABEL’s resear
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18.1. Transformation theory 333 The
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18.1. Transformation theory 335 Obv
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18.1. Transformation theory 337 Sum
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18.2. Integration in logarithmic te
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18.3. Conclusion 345 Summary. ABEL
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Chapter 19 The Paris memoir N. H. A
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19.1. ABEL’s approach to the Pari
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19.2. The contents of ABEL’s Pari
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19.3. Additional, tentative remarks
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19.3. Additional, tentative remarks
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19.4. The fate of the Paris memoir
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19.6. Conclusion 377 hyperelliptic
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Chapter 20 General approaches to el
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20.2. Other ways of introducing ell
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20.3. Conclusion 383 All these four
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Chapter 21 ABEL’s mathematics and
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21.1. From formulae to concepts 389
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21.1. From formulae to concepts 391
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21.2. Concepts and classes enter ma
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21.3. The role of counter examples
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21.4. Conclusion 401 It is beyond t
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404 Appendix A. ABEL’s correspond
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Appendix B ABEL’s manuscripts Man
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1. Précis d’une théorie des fon
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Bibliography Abel (MS:351:A). “M
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Bibliography 413 in: Journal für d
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Bibliography 415 — (1990). “Geo
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Bibliography 417 — (1830). “Mé
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Bibliography 419 — (1768). “Ins
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Bibliography 421 Grattan-Guinness,
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Bibliography 423 Jahnke, H. N. (198
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Bibliography 425 Legendre, A. M. (1
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Bibliography 427 Rosen, M. (1981).
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Bibliography 429 Toti Rigatelli, L.
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432 Index of names Frobenius, Georg
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Index École Normale, 40, 187 Écol
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Index 437 Lagrange interpolation, 3
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