Otto Stolz (3 July 1842 – 23 November 1905) was an

Austrian Austrian may refer to: * Austrians, someone from Austria or of Austrian descent ** Someone who is considered an Austrian citizen, see Austrian nationality law * Austrian German dialect * Something associated with the country Austria, for example: ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

noted for his work on

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...

and

infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...

s. Born in

Hall in Tirol Hall in Tyrol is a town in the Innsbruck-Land district of Tyrol, Austria. Located at an altitude of 574 m, about 5 km (3 mi) east of the state's capital Innsbruck in the Inn valley, it has a population of about 13,000 (Jan 2013). Histo ...

, he studied in

Innsbruck Innsbruck (; bar, Innschbruck, label=Austro-Bavarian ) is the capital of Tyrol and the fifth-largest city in Austria. On the River Inn, at its junction with the Wipp Valley, which provides access to the Brenner Pass to the south, it had a p ...

from 1860 and in

Vienna en, Viennese , iso_code = AT-9 , registration_plate = W , postal_code_type = Postal code , postal_code = , timezone = CET , utc_offset = +1 , timezone_DST ...

from 1863, receiving his habilitation there in 1867. Two years later he studied in

Berlin Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constitu ...

under

Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...

Ernst Kummer Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned ...

and

Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers, ...

, and in 1871 heard lectures in

Göttingen Göttingen (, , ; nds, Chöttingen) is a university city in Lower Saxony, central Germany, the capital of the eponymous district. The River Leine runs through it. At the end of 2019, the population was 118,911. General information The ori ...

Alfred Clebsch Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) 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. ...

and

Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...

(with whom he would later correspond), before returning to Innsbruck permanently as a professor of mathematics. His work began with

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

(on which he wrote his thesis) but after the influence of Weierstrass it shifted to

real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...

, and many small useful theorems are credited to him. For example, he proved that a

continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...

''f'' on a

closed interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...

'a'', ''b''with midpoint convexity, i.e.,

f\left(\frac2\right) \leq \frac

, has left and right

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

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 In mathematics, the Stolz–Cesàro theorem is a criterion for proving the convergence of a sequence. The theorem is named after mathematicians Otto Stolz and Ernesto Cesàro, who stated and proved it for the first time. The Stolz–Cesàro theo ...

Work on non-Archimedean systems

Stolz published a number of papers containing constructions of non-Archimedean extensions of the

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s, as detailed by Ehrlich (2006). His work, as well as that of Paul du Bois-Reymond, was sharply criticized by

Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...

as an "abomination". Cantor published a "proof-sketch" of the inconsistency of infinitesimals. The errors in Cantor's proof are analyzed by Ehrlich (2006).

Notes

Bibliography

*Philip Ehrlich (2006). "The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes",

Archive for History of Exact Sciences ''Archive for History of Exact Sciences'' is a peer-reviewed academic journal currently published bimonthly by Springer Science+Business Media, covering the history of mathematics and of astronomy observations and techniques, epistemology of scienc ...

60, no. 1, pp. 1–121.

External links

Almanach
for 1906, containing obituary *

containing Stolz's photograph

Haus Der Mathematik 1842 births 1905 deaths 19th-century Austrian mathematicians Mathematical analysts Austro-Hungarian mathematicians {{europe-mathematician-stub