Moritz Pasch (8 November 1843,
Breslau,
Prussia
Prussia (; ; Old Prussian: ''Prūsija'') was a Germans, German state centred on the North European Plain that originated from the 1525 secularization of the Prussia (region), Prussian part of the State of the Teutonic Order. For centuries, ...
(now
Wrocław
Wrocław is a city in southwestern Poland, and the capital of the Lower Silesian Voivodeship. It is the largest city and historical capital of the region of Silesia. It lies on the banks of the Oder River in the Silesian Lowlands of Central Eu ...
,
Poland
Poland, officially the Republic of Poland, is a country in Central Europe. It extends from the Baltic Sea in the north to the Sudetes and Carpathian Mountains in the south, bordered by Lithuania and Russia to the northeast, Belarus and Ukrai ...
) – 20 September 1930,
Bad Homburg,
Germany
Germany, officially the Federal Republic of Germany, is a country in Central Europe. It lies between the Baltic Sea and the North Sea to the north and the Alps to the south. Its sixteen States of Germany, constituent states have a total popu ...
) 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 with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
of
Jewish
Jews (, , ), or the Jewish people, are an ethnoreligious group and nation, originating from the Israelites of History of ancient Israel and Judah, ancient Israel and Judah. They also traditionally adhere to Judaism. Jewish ethnicity, rel ...
ancestry specializing in the
foundations of geometry. He completed his Ph.D. at the
University of Breslau at only 22 years of age. He taught at the
University of Giessen, where he is known to have supervised 30 doctorates.
In 1882, Pasch published a book, ''Vorlesungen über neuere Geometrie'', calling for the grounding of
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
in more precise
primitive notions and
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
s, and for greater care in the deductive methods employed to develop the subject. He drew attention to a number of heretofore unnoted tacit assumptions in Euclid's ''
Elements''. He then argued that mathematical reasoning should not invoke the physical interpretation of the primitive terms, but should instead rely solely on formal manipulations justified by axioms. This book is the point of departure for:
*Similarly concerned Italians:
Peano,
Mario Pieri,
Alessandro Padoa
*
Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosophy of mathematics, philosopher of mathematics and one of the most influential mathematicians of his time.
Hilbert discovered and developed a broad ...
's work on geometry and mathematical axiomatics in general;
*All modern thinking about the foundations of
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
.
Pasch is perhaps best remembered for
Pasch's axiom:
Given three noncollinear points ''a, b, c'' and a line ''X'' not containing any of these points, if ''X'' includes a point between ''a'' and ''b'', then ''X'' also includes one and only one of the following: a point between ''a'' and ''c'', or a point between ''b'' and ''c''.
In other words, if a line crosses one side of a
triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
, that line must also cross one of the two remaining sides of the same triangle. Pasch's axiom is not to be confused with
Pasch's theorem.
Selected publications
''Vorlesungen über neuere Geometrie'' Leipzig 1882;
''Einleitung in die Differential- und Integralrechnung'' Leipzig 1882
''Grundlagen der Analysis'' Leipzig, 1908
Leipzig, 1919
'' Die Begriffsswelt des Mathematikers in der Vorhalll der Geometrie'' Leipzig, 1922
Translations
*
See also
*
Pasch's theorem
*
Pasch's axiom
*
Pasch configuration
*
Pasch hypergraph
*
Ordered geometry
References
External links
*
* The Mathematics Genealogy Project
Pasch.
{{DEFAULTSORT:Pasch, Moritz
1843 births
1930 deaths
19th-century German Jews
19th-century German mathematicians
German geometers
20th-century German mathematicians
Mathematicians from the Kingdom of Prussia