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Moritz Pasch (8 November 1843, Breslau,
''Vorlesungen über neuere Geometrie''
Leipzig 1882;
''Einleitung in die Differential- und Integralrechnung''
Leipzig 1882
''Grundlagen der Analysis''
Leipzig, 1908
''Mathematik und Logik''
Leipzig, 1919
'' Die Begriffsswelt des Mathematikers in der Vorhalll der Geometrie''
Leipzig, 1922
Pasch.
{{DEFAULTSORT:Pasch, Moritz 1843 births 1930 deaths 19th-century German Jews 19th-century German mathematicians Geometers 20th-century German mathematicians
Moritz Pasch (8 November 1843, Breslau, Prussia
Prussia, , Old Prussian: ''Prūsa'' or ''Prūsija'' was a German state on the southeast coast of the Baltic Sea. It formed the German Empire under Prussian rule when it united the German states in 1871. It was ''de facto'' dissolved by an e ...
(now Wrocław
Wrocław (; german: Breslau, or . ; Silesian German: ''Brassel'') is a city in southwestern Poland and the largest city in the historical region of Silesia. It lies on the banks of the River Oder in the Silesian Lowlands of Central Europe, r ...
, Poland
Poland, officially the Republic of Poland, is a country in Central Europe. It is divided into 16 administrative provinces called voivodeships, covering an area of . Poland has a population of over 38 million and is the fifth-most populou ...
) – 20 September 1930, Bad Homburg
Bad Homburg vor der Höhe () is the district town of the Hochtaunuskreis, Hesse, on the southern slope of the Taunus mountains. Bad Homburg is part of the Frankfurt Rhein-Main urban area. The town's official name is ''Bad Homburg v.d.Höhe'', w ...
, Germany
Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populous country in Europe after Russia, and the most populous member state of the European Union. Germany is situated betwee ...
) 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, structure, space, models, and change.
History
On ...
of Jewish
Jews ( he, יְהוּדִים, , ) or Jewish people are an ethnoreligious group and nation originating from the Israelites Israelite origins and kingdom: "The first act in the long drama of Jewish history is the age of the Israelites""The ...
ancestry specializing in the foundations of geometry
Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, bu ...
. He completed his Ph.D. at the University of Breslau
A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States, th ...
at only 22 years of age. He taught at the University of Giessen
University of Giessen, official name Justus Liebig University Giessen (german: Justus-Liebig-Universität Gießen), is a large public research university in Giessen, Hesse, Germany. It is named after its most famous faculty member, Justus von ...
, 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 mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
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
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The sta ...
, Mario Pieri
Mario Pieri (22 June 1860 – 1 March 1913) was an Italian mathematician who is known for his work on foundations of geometry.
Biography
Pieri was born in Lucca, Italy, the son of Pellegrino Pieri and Ermina Luporini. Pellegrino was a lawyer. Pie ...
, Alessandro Padoa
Alessandro Padoa (14 October 1868 – 25 November 1937) was an Italian mathematician and logician, a contributor to the school of Giuseppe Peano. He is remembered for a method for deciding whether, given some formal theory, a new primitive noti ...
* Hilbert'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 mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
.
Pasch is perhaps best remembered for Pasch's axiom
In geometry, Pasch's axiom is a statement in plane geometry, used implicitly by Euclid, which cannot be derived from the postulates as Euclid gave them. Its essential role was discovered by Moritz Pasch in 1882.
Statement
The axiom states tha ...
:
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 edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
, that line must also cross one of the two remaining sides of the same triangle.
Selected publications
''Vorlesungen über neuere Geometrie''
Leipzig 1882;
''Einleitung in die Differential- und Integralrechnung''
Leipzig 1882
''Grundlagen der Analysis''
Leipzig, 1908
''Mathematik und Logik''
Leipzig, 1919
'' Die Begriffsswelt des Mathematikers in der Vorhalll der Geometrie''
Leipzig, 1922
Translations
*See also
*Pasch's theorem
In geometry, Pasch's theorem, stated in 1882 by the German mathematician Moritz Pasch, is a result in plane geometry which cannot be derived from Euclid's postulates.
Statement
The statement is as follows: ere, for example, (, , ) means tha ...
* Pasch hypergraph
*Ordered geometry Ordered geometry is a form of geometry featuring the concept of intermediacy (or "betweenness") but, like projective geometry, omitting the basic notion of measurement. Ordered geometry is a fundamental geometry forming a common framework for affi ...
References
External links
* * The Mathematics Genealogy ProjectPasch.
{{DEFAULTSORT:Pasch, Moritz 1843 births 1930 deaths 19th-century German Jews 19th-century German mathematicians Geometers 20th-century German mathematicians