Moritz Pasch
Encyclopedia
Moritz Pasch was a German
mathematician
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 in more precise primitive notion
s and axiom
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:
Pasch is perhaps best remembered for Pasch's axiom
:
In other words, if a line crosses one side of a triangle, that line must also cross one of the two remaining sides of the same triangle.
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
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....
specializing in the foundations of 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 ....
. He completed his Ph.D. at the University of Breslau at only 22 years of age. He taught at the University of Giessen
University of Giessen
The University of Giessen is officially called the Justus Liebig University Giessen after its most famous faculty member, Justus von Liebig, the founder of modern agricultural chemistry and inventor of artificial fertiliser.-History:The University of Gießen is among the oldest institutions of...
, 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 in more precise primitive notion
Primitive notion
In mathematics, logic, and formal systems, a primitive notion is an undefined concept. In particular, a primitive notion is not defined in terms of previously defined concepts, but is only motivated informally, usually by an appeal to intuition and everyday experience. In an axiomatic theory or...
s and axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
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
Euclid's Elements
Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
. 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 PieriMario PieriMario Pieri was an Italian mathematician who is known for his work on foundations of geometry.Pieri was born in Lucca, Italy, the son of Pellegrino Pieri and Ermina Luporini. Pellegrino was a lawyer. Pieri began his higher education at University of Bologna where he drew the attention of Salvatore...
, Alessandro PadoaAlessandro PadoaAlessandro Padoa 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 notion is truly independent of the other primitive notions... - HilbertDavid HilbertDavid Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
's work on geometry and mathematical axiomatics in general; - All modern thinking about the foundations of Euclidean geometryEuclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
.
Pasch is perhaps best remembered for Pasch's axiom
Pasch's axiom
In geometry, Pasch's axiom is a result of plane geometry used by Euclid, but yet which cannot be derived from Euclid's postulates. Its axiomatic role was discovered by Moritz Pasch.The axiom states that, in the plane,...
:
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, that line must also cross one of the two remaining sides of the same triangle.
External links
- MacTutor biography: Moritz Pasch.
- The Mathematics Genealogy Project: Pasch.