Felix Hausdorff was a German mathematician, considered to be one of the founders of modern topology and who contributed to set theory, descriptive set theory, measure theory, function theory, functional analysis. Life became difficult for Hausdorff and his family after Kristallnacht in 1938; the next year he initiated efforts to emigrate to the United States, but was unable to make arrangements to receive a research fellowship. On 26 January 1942, Felix Hausdorff, along with his wife and his sister-in-law, committed suicide by taking an overdose of veronal, rather than comply with German orders to move to the Endenich camp, there suffer the implications, about which he held no illusions. Hausdorff's father, the Jewish merchant Louis Hausdorff, moved in the autumn of 1870 with his young family to Leipzig and worked over time at various companies, including a linen-and cotton goods factory, he was an educated man and had become a Morenu at the age of 14. There are several treatises from his pen, including a long work on the Aramaic translations of the Bible from the perspective of Talmudic law.
Hausdorff's mother, referred to in various documents as Johanna, came from the Jewish Tietz family. From another branch of this family came Hermann Tietz, founder of the first department store, co-owner of the department store chain called "Hermann Tietz". During the period of Nazi dictatorship the name was "Aryanised" to Hertie. From 1878 to 1887 Felix Hausdorff attended the Nicolai School in Leipzig, a facility that had a reputation as a hotbed of humanistic education, he was an excellent student, class leader for many years and recited self-written Latin or German poems at school celebrations. In his graduation in 1887, he was the only one; the choice of subject was not easy for Hausdorff. Magda Dierkesmann, a guest in the home of Hausdorff as a student in Bonn in the years 1926–1932, reported in 1967 that: His versatile musical talent was so great that only the insistence of his father made him give up his plan to study music and become a composer; the decision was made to study the sciences in high school.
From summer term 1887 to summer semester 1891 Hausdorff studied mathematics and astronomy in his native city of Leipzig, interrupted by one semester in Freiburg and Berlin. The surviving testimony of other students show him as versatile interested young man, who, in addition to the mathematical and astronomical lectures, attended lectures in physics and geography, lectures on philosophy and history of philosophy as well as on issues of language and social sciences. In Leipzig he heard lectures on the history of music from musicologist Paul, his early love of music lasted a lifetime. As a student in Leipzig, he was an admirer and connoisseur of the music of Richard Wagner. In semesters of his studies, Hausdorff was close to Heinrich Bruns. Bruns was professor of director of the observatory at the University of Leipzig. Under him, Hausdorff graduated in 1891 with a work on the theory of astronomical refraction of light in the atmosphere. Two publications on the same subject followed, in 1895 his habilitation followed with a thesis on the absorbance of light in the atmosphere.
These early astronomical works of Hausdorff have—despite their excellent mathematical working through—not gained importance. Firstly, the underlying idea of Bruns has not proved viable. On the other hand, the progress in the direct measurement of atmospheric data has since made the painstaking accuracy of this data from refraction observations unnecessary. In the time between PhD and habilitation Hausdorff completed the yearlong-volunteer military requirement and worked for two years as a human computer at the observatory in Leipzig. With his habilitation, Hausdorff became a lecturer at the University of Leipzig and began an extensive teaching in a variety of mathematical areas. In addition to teaching and research in mathematics, he went with his literary and philosophical inclinations. A man of varied interests, educated sensitive and sophisticated in thinking and experiencing, he frequented in his Leipzig period with a number of famous writers and publishers such as Hermann Conradi, Richard Dehmel, Otto Erich Hartleben, Gustav Kirstein, Max Klinger, Max Reger and Frank Wedekind.
The years 1897 to about 1904 mark the high point of his literary and philosophical creativity, during which time 18 of his 22 pseudonymous works were published, including a book of poetry, a play, an epistemological book and a volume of aphorisms. Hausdorff married Charlotte Goldschmidt in daughter of Jewish doctor Siegismund Goldschmidt, her stepmother was preschool teacher Henriette Goldschmidt. Hausdorff's only child, daughter Lenore, was born in 1900. In December 1901 Hausdorff was appointed as adjunct associate professor at the University of Leipzig; the repeated assertion that Hausdorff got a call from Göttingen and rejected it cannot be verified and is wrong. When applying in Leipzig, Dean Kirchner had been led to positive vote of his colleagues, written by Hei
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of one object from each bin if the collection is infinite. Formally, it states that for every indexed family i ∈ I of nonempty sets there exists an indexed family i ∈ I of elements such that x i ∈ S i for every i ∈ I; the axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. In many cases, such a selection can be made without invoking the axiom of choice. An illustrative example is sets picked from the natural numbers. From such sets, one may always select the smallest number, e.g. in the smallest elements are. In this case, "select the smallest number" is a choice function. If infinitely many sets were collected from the natural numbers, it will always be possible to choose the smallest element from each set to produce a set.
That is, the choice function provides the set of chosen elements. However, no choice function is known for the collection of all non-empty subsets of the real numbers. In that case, the axiom of choice must be invoked. Bertrand Russell coined an analogy: for any collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection. For an infinite collection of pairs of socks, there is no obvious way to make a function that selects one sock from each pair, without invoking the axiom of choice. Although controversial, the axiom of choice is now used without reservation by most mathematicians, it is included in the standard form of axiomatic set theory, Zermelo–Fraenkel set theory with the axiom of choice. One motivation for this use is that a number of accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs. Contemporary set theorists study axioms that are not compatible with the axiom of choice, such as the axiom of determinacy.
The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced. A choice function is a function f, defined on a collection X of nonempty sets, such that for every set A in X, f is an element of A. With this concept, the axiom can be stated: Formally, this may be expressed as follows: ∀ X. Thus, the negation of the axiom of choice states that there exists a collection of nonempty sets that has no choice function; each choice function on a collection X of nonempty sets is an element of the Cartesian product of the sets in X. This is not the most general situation of a Cartesian product of a family of sets, where a given set can occur more than once as a factor; the axiom of choice asserts the existence of such elements. In this article and other discussions of the Axiom of Choice the following abbreviations are common: AC – the Axiom of Choice. ZF – Zermelo–Fraenkel set theory omitting the Axiom of Choice.
ZFC – Zermelo–Fraenkel set theory, extended to include the Axiom of Choice. There are many other equivalent statements of the axiom of choice; these are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it. One variation avoids the use of choice functions by, in effect, replacing each choice function with its range. Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains one element in common with each of the sets in X; this guarantees for any partition of a set X the existence of a subset C of X containing one element from each part of the partition. Another equivalent axiom only considers collections X that are powersets of other sets: For any set A, the power set of A has a choice function. Authors who use this formulation speak of the choice function on A, but be advised that this is a different notion of choice function, its domain is the powerset of A, and
John L. Kelley
John L. Kelley was an American mathematician at University of California, Berkeley who worked in general topology and functional analysis. Kelley's 1955 text, General Topology, which appeared in three editions and several translations, is a classic and cited graduate level introduction to topology. An appendix sets out a new approach to axiomatic set theory, now called Morse–Kelley set theory, that builds on Von Neumann–Bernays–Gödel set theory. After earning B. A. and M. A. degrees from the University of California, Los Angeles, he went to the University of Virginia, where he obtained his Ph. D. in 1940. Gordon Whyburn, a student of Robert Lee Moore, supervised his thesis, entitled A Study of Hyperspaces, he taught at the University of Notre Dame until the outbreak of World War II. From 1942 to 1945, he did mathematics for the war effort at the Aberdeen Proving Grounds, where his work unit included his future Berkeley colleagues Anthony Morse and Charles Morrey. After teaching at the University of Chicago, 1946–47, Kelley spent the rest of his career at Berkeley, from which he retired in 1985.
He chaired the Mathematics Department at Berkeley 1957-60 and 1975-80. He held visiting appointments at Cambridge University and the Indian Institute of Technology in Kanpur, India. An Indian Mathematician, Vashishtha Narayan Singh was among those mentored by Kelley. In 1950, Kelley was one of 29 tenured Berkeley faculty dismissed for refusing to sign a McCarthy-era loyalty oath mandated by the UC Board of Regents, he taught at Tulane University and the University of Kansas. He returned to Berkeley in 1953, after the California Supreme Court declared the oath unconstitutional and directed UC Berkeley to rehire the dismissed academics, he was an outspoken opponent of the Vietnam War. Kelley's interest in teaching extended well beyond the higher reaches of mathematics. In 1960, he took a leave of absence to serve as the National Teacher on NBC's Continental Classroom television program, he was an active member of the School Mathematics Study Group which played an important role in designing and promulgating the "new math" of that era.
In 1964, he led his department to introduce a new major called Mathematics for Teachers, taught one of its core courses. These endeavors culminated in Richert. In 1977-78, he was a member of the U. S. Commission on Mathematical Instruction, his doctoral students include Vashishtha Narayan Singh, James Michael Gardner Fell, J. M. G. Fell, Isaac Namioka, Reese Prosser. 1953. Exterior ballistics; the University of Denver Press 1955. General Topology. David Van Nostrand Company, link from Internet Archive. Reprinted by Springer Verlag. ISBN 0-387-90125-6 1960. Introduction to Modern Algebra. Van Nostrand. 1963. Linear Topological Spaces. Van Nostrand. 1970. Elementary Mathematics for Teachers. Autobiographical article and other memorials John L. Kelley from University of California, Berkeley. John L. Kelley from Department of Mathematics, UC Berkeley. John L. Kelley at the Mathematics Genealogy Project