Otto Stolz
Encyclopedia
Otto Stolz was an Austrian
mathematician
noted for his work on mathematical analysis
and infinitesimal
s. Born in Hall in Tirol
, he studied in Innsbruck
from 1860 and in Vienna
from 1863, receiving his habilitation there in 1867. Two years later he studied in Berlin
under Karl Weierstrass
, Ernst Kummer
and Leopold Kronecker
, and in 1871 heard lectures in Göttingen
by Alfred Clebsch
and Felix Klein
(with whom he would later correspond), before returning to Innsbruck permanently as a professor of mathematics.
His work began with geometry
(on which he wrote his thesis) but after the influence of Weierstrass it shifted to real analysis
, and many small useful theorems are credited to him. For example, he proved that a continuous function
f on a closed interval [a, b] with the property
has left and right derivative
s at each point in (a, b).
He died in 1905 shortly after finishing work on Einleitung in die Funktionentheorie. His name lives on in the Stolz-Cesàro theorem.
as an "abomination". Cantor published a "proof-sketch" of the inconsistency of infinitesimals. The errors in Cantor's proof are analyzed by Ehrlich (2006).
Austrians
Austrians are a nation and ethnic group, consisting of the population of the Republic of Austria and its historical predecessor states who share a common Austrian culture and Austrian descent....
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
noted for his work on mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
and infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
s. Born in Hall in Tirol
Hall in Tirol
Hall in Tirol is a town in the Innsbruck-Land district of Tyrol, Austria. Located at an altitude of 574 m, about 5 km east of the state's capital Innsbruck in the Inn valley, it has a population of about 12,700 .-History:...
, he studied in Innsbruck
Innsbruck
- Main sights :- Buildings :*Golden Roof*Kaiserliche Hofburg *Hofkirche with the cenotaph of Maximilian I, Holy Roman Emperor*Altes Landhaus...
from 1860 and in Vienna
Vienna
Vienna is the capital and largest city of the Republic of Austria and one of the nine states of Austria. Vienna is Austria's primary city, with a population of about 1.723 million , and is by far the largest city in Austria, as well as its cultural, economic, and political centre...
from 1863, receiving his habilitation there in 1867. Two years later he studied in Berlin
Berlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...
under Karl Weierstrass
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....
, Ernst Kummer
Ernst Kummer
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.-Life:Kummer...
and Leopold Kronecker
Leopold Kronecker
Leopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...
, and in 1871 heard lectures in Göttingen
Göttingen
Göttingen is a university town in Lower Saxony, Germany. It is the capital of the district of Göttingen. The Leine river runs through the town. In 2006 the population was 129,686.-General information:...
by Alfred Clebsch
Alfred Clebsch
Rudolf Friedrich Alfred Clebsch was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Berlin. He subsequently taught in Berlin and Karlsruhe...
and Felix Klein
Felix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
(with whom he would later correspond), before returning to Innsbruck permanently as a professor of mathematics.
His work began with geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
(on which he wrote his thesis) but after the influence of Weierstrass it shifted to real analysis
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
, and many small useful theorems are credited to him. For example, he proved that a continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
f on a closed interval [a, b] with the property
has left and right derivativeDerivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
s at each point in (a, b).
He died in 1905 shortly after finishing work on Einleitung in die Funktionentheorie. His name lives on in the Stolz-Cesàro theorem.
Work on non-Archimedean systems
Stolz published a number of papers containing constructions of non-Archimedean extensions of the real numbers, as detailed by Ehrlich (2006). His work, as well as that of Paul du Bois-Reymond, was sharply criticized by Georg CantorGeorg Cantor
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...
as an "abomination". Cantor published a "proof-sketch" of the inconsistency of infinitesimals. The errors in Cantor's proof are analyzed by Ehrlich (2006).
External links
- Almanach for 1906, containing obituary
- Österreich Lexikon, containing Stolz's photograph
- http://www.hausdermathematik.at/dokumentationszentrum/oesterreichischemathematiker/stolzotto.html Haus Der Mathematik