ICM 2002 · Vol. III · 1–3
arXiv:math/0305022v1 [math.HO] 1 May 2003
“Algebraic Truths”
vs
“Geometric Fantasies”:
Weierstrass’ Response to Riemann
U. Bottazzini∗
Abstract
In the 1850s Weierstrass succeeded in solving the Jacobi inversion prob-
lem for the hyper-elliptic case, and claimed he was able to solve the general
problem. At about the same time Riemann successfully applied the geometric
methods that he set up in his thesis (1851) to the study of Abelian inte-
grals, and the solution of Jacobi inversion problem. In response to Riemann’s
achievements, by the early 1860s Weierstrass began to build the theory of
analytic functions in a systematic way on arithmetical foundations, and to
present it in his lectures. According to Weierstrass, this theory provided the
foundations of the whole of both elliptic and Abelian function theory, the lat-
ter being the ultimate goal of his mathematical work. Riemann’s theory of
complex functions seems to have been the background of Weierstrass’s work
and lectures. Weierstrass’ unpublished correspondence with his former stu-
dent Schwarz provides strong evidence of this. Many of Weierstrass’ results,
including his example of a continuous non-differentiable function as well as
his counter-example to Dirichlet principle, were motivated by his criticism of
Riemann’s methods, and his distrust in Riemann’s “geometric fantasies”. In-
stead, he chose the power series approach because of his conviction that the
theory of analytic functions had to be founded on simple “algebraic truths”.
Even though Weierstrass failed to build a satisfactory theory of functions of
several complex variables, the contradiction between his and Riemann’s geo-
metric approach remained effective until the early decades of the 20th century.
2000 Mathematics Subject Classification: 01A55, 30-03.
Keywords and Phrases: Abelian integrals, Complex function theory, Ja-
cobi inversion problem, Riemann, Weierstrass.
Introduction
∗ Dipartimento di matematica, Università di Palermo, via Archirafi 34, 90123 Palermo, Italy.
E-mail:
, 924 U. Bottazzini
In 1854 Crelle’s Journal published a paper on Abelian functions by an un-
known school teacher. This paper announced the entry in the mathematical world
of a major figure, Karl Weierstrass (1815-1897), who was to dominate the scene
for the next forty years to come. His paper presented a solution of Jacobi inver-
sion problem in the hyper-elliptic case. In analogy with the inversion of elliptic
integrals of the first kind, Jacobi unsuccessfully attempted a direct inversion of a
hyper-elliptic integral of the first kind. This led him to consider multi-valued, “un-
reasonable” functions having a “strong multiplicity” of periods, including periods
of arbitrarily small (non-zero) absolute value. Jacobi confessed he was “almost in
despair” about the possibility of the inversion when he realized “by divination”
that Abel’s theorem provided him with the key for resurrecting the analogy with
the inversion of elliptic integrals by considering the sum of a suitable number of
(linearly independent) hyper-elliptic integrals instead of a single integral. In his
memoir submitted to the Paris Academy in 1826 (and published only in 1841) Abel
had stated a theorem which extended Euler’s addition R theorem for elliptic integrals
to more general (Abelian) integrals of the form R(x, y)dx in which R(x, y) is a
rational function and y = y(x) is an algebraic function defined by a (irreducible)
polynomial equation f (x, y) = 0. According to Abel’s theorem, the sum of any
number of such integrals reduces to the sum of a number p of linearly independent
integrals and of an algebraic-logarithmic expression (p was later called by Clebsch
the genus of the algebraic curve f (x, y) = 0). In 1828 Abel published an excerpt of
his Paris memoir dealing with the particular (hyper-elliptic) case of the theorem,
when f (x, y) = y 2 − P (x), P is a polynomial of degree n > 4 having no multiple
roots. In this case p = [(n − 1)/2], and for hyper-elliptic integrals of the first kind
R Q(x)dx
√ (Q is a polynomial of degree ≤ p − 1) the algebraic-logarithmic expression
P (x)
vanishes ([1], vol. 1, 444-456).
On the basis of Abel’s theorem in 1832 Jacobi formulated the problem of
investigating the inversion of a system of p hyper-elliptic integrals
p−1
X Z xj xk dx
uk = p (0 ≤ k ≤ p − 1) (degP = 2p + 1 or 2p + 2)
j=0 0 P (x)
by studying x0 , x1 , · · · , xp−1 as functions of the variables u0 , u1 , · · · , up−1 . These
functions xi = λi (u0 , u1 , · · · , up−1 ) generalized the elliptic functions to 2p-periodic
functions of p variables. Jacobi’s “general theorem” claimed that x0 , x1 , · · · , xp−1
were the roots of an algebraic equation of degree p whose coefficient were single-
valued, 2p-periodic functions of u0 , u1 , · · · , up−1 . Therefore, the elementary sym-
metric functions of x0 , x1 , · · · , xp−1 could be expressed by means of single-valued
functions in C p . In particular, Jacobi considered the case p = 2 ([4], vol. 2,
7-16). His ideas were successfully developed by A. Göpel in 1847 (and, indepen-
dently of him, J. G. Rosenhain in 1851). The required 4-fold periodic functions of
two complex variables were expressed as the ratio of two θ-series of two complex
variables obtained by a direct and cumbersome computation. This involved an im-
pressive amount of calculations and could hardly be extended to the case p > 2.
arXiv:math/0305022v1 [math.HO] 1 May 2003
“Algebraic Truths”
vs
“Geometric Fantasies”:
Weierstrass’ Response to Riemann
U. Bottazzini∗
Abstract
In the 1850s Weierstrass succeeded in solving the Jacobi inversion prob-
lem for the hyper-elliptic case, and claimed he was able to solve the general
problem. At about the same time Riemann successfully applied the geometric
methods that he set up in his thesis (1851) to the study of Abelian inte-
grals, and the solution of Jacobi inversion problem. In response to Riemann’s
achievements, by the early 1860s Weierstrass began to build the theory of
analytic functions in a systematic way on arithmetical foundations, and to
present it in his lectures. According to Weierstrass, this theory provided the
foundations of the whole of both elliptic and Abelian function theory, the lat-
ter being the ultimate goal of his mathematical work. Riemann’s theory of
complex functions seems to have been the background of Weierstrass’s work
and lectures. Weierstrass’ unpublished correspondence with his former stu-
dent Schwarz provides strong evidence of this. Many of Weierstrass’ results,
including his example of a continuous non-differentiable function as well as
his counter-example to Dirichlet principle, were motivated by his criticism of
Riemann’s methods, and his distrust in Riemann’s “geometric fantasies”. In-
stead, he chose the power series approach because of his conviction that the
theory of analytic functions had to be founded on simple “algebraic truths”.
Even though Weierstrass failed to build a satisfactory theory of functions of
several complex variables, the contradiction between his and Riemann’s geo-
metric approach remained effective until the early decades of the 20th century.
2000 Mathematics Subject Classification: 01A55, 30-03.
Keywords and Phrases: Abelian integrals, Complex function theory, Ja-
cobi inversion problem, Riemann, Weierstrass.
Introduction
∗ Dipartimento di matematica, Università di Palermo, via Archirafi 34, 90123 Palermo, Italy.
E-mail:
, 924 U. Bottazzini
In 1854 Crelle’s Journal published a paper on Abelian functions by an un-
known school teacher. This paper announced the entry in the mathematical world
of a major figure, Karl Weierstrass (1815-1897), who was to dominate the scene
for the next forty years to come. His paper presented a solution of Jacobi inver-
sion problem in the hyper-elliptic case. In analogy with the inversion of elliptic
integrals of the first kind, Jacobi unsuccessfully attempted a direct inversion of a
hyper-elliptic integral of the first kind. This led him to consider multi-valued, “un-
reasonable” functions having a “strong multiplicity” of periods, including periods
of arbitrarily small (non-zero) absolute value. Jacobi confessed he was “almost in
despair” about the possibility of the inversion when he realized “by divination”
that Abel’s theorem provided him with the key for resurrecting the analogy with
the inversion of elliptic integrals by considering the sum of a suitable number of
(linearly independent) hyper-elliptic integrals instead of a single integral. In his
memoir submitted to the Paris Academy in 1826 (and published only in 1841) Abel
had stated a theorem which extended Euler’s addition R theorem for elliptic integrals
to more general (Abelian) integrals of the form R(x, y)dx in which R(x, y) is a
rational function and y = y(x) is an algebraic function defined by a (irreducible)
polynomial equation f (x, y) = 0. According to Abel’s theorem, the sum of any
number of such integrals reduces to the sum of a number p of linearly independent
integrals and of an algebraic-logarithmic expression (p was later called by Clebsch
the genus of the algebraic curve f (x, y) = 0). In 1828 Abel published an excerpt of
his Paris memoir dealing with the particular (hyper-elliptic) case of the theorem,
when f (x, y) = y 2 − P (x), P is a polynomial of degree n > 4 having no multiple
roots. In this case p = [(n − 1)/2], and for hyper-elliptic integrals of the first kind
R Q(x)dx
√ (Q is a polynomial of degree ≤ p − 1) the algebraic-logarithmic expression
P (x)
vanishes ([1], vol. 1, 444-456).
On the basis of Abel’s theorem in 1832 Jacobi formulated the problem of
investigating the inversion of a system of p hyper-elliptic integrals
p−1
X Z xj xk dx
uk = p (0 ≤ k ≤ p − 1) (degP = 2p + 1 or 2p + 2)
j=0 0 P (x)
by studying x0 , x1 , · · · , xp−1 as functions of the variables u0 , u1 , · · · , up−1 . These
functions xi = λi (u0 , u1 , · · · , up−1 ) generalized the elliptic functions to 2p-periodic
functions of p variables. Jacobi’s “general theorem” claimed that x0 , x1 , · · · , xp−1
were the roots of an algebraic equation of degree p whose coefficient were single-
valued, 2p-periodic functions of u0 , u1 , · · · , up−1 . Therefore, the elementary sym-
metric functions of x0 , x1 , · · · , xp−1 could be expressed by means of single-valued
functions in C p . In particular, Jacobi considered the case p = 2 ([4], vol. 2,
7-16). His ideas were successfully developed by A. Göpel in 1847 (and, indepen-
dently of him, J. G. Rosenhain in 1851). The required 4-fold periodic functions of
two complex variables were expressed as the ratio of two θ-series of two complex
variables obtained by a direct and cumbersome computation. This involved an im-
pressive amount of calculations and could hardly be extended to the case p > 2.
