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Moritz Pasch
Moritz Pasch (8 November 1843, Breslau, Prussia (now Wrocław, Poland) – 20 September 1930, Bad Homburg, Germany) was a German mathematician of Jewish 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 in more precise primitive notions and axioms, 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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moritz Pasch (mathematician)
Moritz Pasch (8 November 1843, Breslau, Prussia (now Wrocław, Poland) – 20 September 1930, Bad Homburg, Germany) was a German mathematician of Jewish 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 in more precise primitive notions and axioms, 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, Mar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 standard axiomatization of the natural numbers is named the Peano axioms in his honor. As part of this effort, he made key contributions to the modern rigorous and systematic treatment of the method of mathematical induction. He spent most of his career teaching mathematics at the University of Turin. He also created an international auxiliary language, Latino sine flexione ("Latin without inflections"), which is a simplified version of Classical Latin. Most of his books and papers are in Latino sine flexione, while others are in Italian. Biography Peano was born and raised on a farm at Spinetta, a hamlet now belonging to Cuneo, Piedmont, Italy. He attended the Liceo classico Cavour in Turin, and enrolled at the University of Turin in 1876, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
19th-century German Mathematicians
The 19th century began on 1 January 1801 (represented by the Roman numerals MDCCCI), and ended on 31 December 1900 (MCM). It was the 9th century of the 2nd millennium. It was characterized by vast social upheaval. Slavery was abolished in much of Europe and the Americas. The First Industrial Revolution, though it began in the late 18th century, expanded beyond its British homeland for the first time during the 19th century, particularly remaking the economies and societies of the Low Countries, France, the Rhineland, Northern Italy, and the Northeastern United States. A few decades later, the Second Industrial Revolution led to ever more massive urbanization and much higher levels of productivity, profit, and prosperity, a pattern that continued into the 20th century. The Catholic Church, in response to the growing influence and power of modernism, secularism and materialism, formed the First Vatican Council in the late 19th century to deal with such problems and confirm ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1930 Deaths
Events January * January 15 – The Moon moves into its nearest point to Earth, called perigee, at the same time as its fullest phase of the Lunar Cycle. This is the closest moon distance at in recent history, and the next one will be on January 1, 2257, at . * January 26 – The Indian National Congress declares this date as Independence Day, or as the day for Purna Swaraj (Complete Independence). * January 28 – The first patent for a field-effect transistor is granted in the United States, to Julius Edgar Lilienfeld. * January 30 – Pavel Molchanov launches a radiosonde from Pavlovsk, Saint Petersburg, Slutsk in the Soviet Union. February * February 10 – The Việt Nam Quốc Dân Đảng launch the Yên Bái mutiny in the hope of ending French Indochina, French colonial rule in Vietnam. * February 18 – While studying photographs taken in January, Clyde Tombaugh confirms the existence of Pluto, a celestial body considered a planet until redefined as a dwarf planet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1843 Births
Events January–March * January 3 – The '' Illustrated Treatise on the Maritime Kingdoms'' (海國圖志, ''Hǎiguó Túzhì'') compiled by Wei Yuan and others, the first significant Chinese work on the West, is published in China. * January 6 – Antarctic explorer James Clark Ross discovers Snow Hill Island. * January 20 – Honório Hermeto Carneiro Leão, Marquis of Paraná is appointed by the Emperor, Dom Pedro, as the leader of the Brazilian Council of Ministers, although the office of Prime Minister of Brazil will not be officially created until 1847. * January ** Serial publication of Charles Dickens's novel ''Martin Chuzzlewit'' begins in London; in the July chapters, he lands his hero in the United States. ** Edgar Allan Poe's short story " The Tell-Tale Heart" is published in ''The Pioneer'', a Boston magazine. ** The Quaker magazine '' The Friend'' is first published in London. * February 3 – Uruguayan Civil War: Argentina supports Oribe of Uruguay, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 affine, Euclidean, absolute, and hyperbolic geometry (but not for projective geometry). History Moritz Pasch first defined a geometry without reference to measurement in 1882. His axioms were improved upon by Peano (1889), Hilbert (1899), and Veblen (1904). Euclid anticipated Pasch's approach in definition 4 of ''The Elements'': "a straight line is a line which lies evenly with the points on itself". Primitive concepts The only primitive notions in ordered geometry are points ''A'', ''B'', ''C'', ... and the ternary relation of intermediacy 'ABC''which can be read as "''B'' is between ''A'' and ''C''". Definitions The ''segment'' ''AB'' is the set of points ''P'' such that 'APB'' The ''interval'' ''AB'' is the segment ''AB'' and it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pasch Hypergraph
In geometry, a truncated projective plane (TPP), also known as a dual affine plane, is a special kind of a hypergraph or geometric configuration that is constructed in the following way. * Take a finite projective plane. * Remove one of the points (vertices) in the plane. * Remove all lines (edges) containing that point. These objects have been studied in many different settings, often independent of one another, and so, many terminologies have been developed. Also, different areas tend to ask different types of questions about these objects and are interested in different aspects of the same objects. Example: the Pasch hypergraph Consider the Fano plane, which is the projective plane of order 2. It has 7 vertices and 7 edges . It can be truncated e.g. by removing the vertex 7 and the edges containing it. The remaining hypergraph is the TPP of order 2. It has 6 vertices and 4 edges . It is a tripartite hypergraph with sides ,, (which are exactly the neighbors of the removed v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pasch Configuration
In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four Point (geometry), points in a Plane (geometry), plane, no three of which are Collinearity, on a common line, and of the six Line (geometry), lines connecting the six pairs of points. Duality (projective geometry), Dually, a ''complete quadrilateral'' is a system of four lines, no three of which pass through the same point, and the six points of Line–line intersection, intersection of these lines. The complete quadrangle was called a tetrastigm by , and the complete quadrilateral was called a tetragram; those terms are occasionally still used. The complete quadrilateral has also been called a Pasch configuration, especially in the context of Steiner triple systems. Diagonals The six lines of a complete quadrangle meet in pairs to form three additional points called the ''diagonal points'' of the quadrangle. Si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 that point lies between points and . Hilbert's use of Pasch's theorem David Hilbert originally included Pasch's theorem as an axiom in his modern treatment of Euclidean geometry in ''The Foundations of Geometry'' (1899). However, it was found by E. H. Moore in 1902 that the axiom is redundant, and revised editions now list it as a theorem. Thus Pasch's theorem is also known as Hilbert's discarded axiom. Pasch's axiom, a separate statement, is also included and remains an axiom in Hilbert's treatment. See also *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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent edges; the sum of angles of a triangle always equals a straight angle (180 degrees or π radians). The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the ''base'', in which case the opposite vertex is called the ''apex''; the shortest segment between the base and apex is the ''height''. The area of a triangle equals one-half the product of height and base length. In Euclidean geometry, any two points determine a unique line segment situated within a unique straight line, and any three points that do not all lie on the same straight line determine a unique triangle situated w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
