American Mathematical Monthly
The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Francis for the Mathematical Association of America; the American Mathematical Monthly is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the American Mathematical Monthly fulfills a different role from that of typical mathematical research journals; the American Mathematical Monthly is the most read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997-2010 are available online; the MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the American Mathematical Monthly. 2017-: Susan Colley 2012-2016: Scott T. Chapman 2007-2011: Daniel J. Velleman 2002-2006: Bruce Palka 1997-2001: Roger A.
Horn 1992-1996: John H. Ewing 1987-1991: Herbert S. Wilf 1982-1986: Paul Richard Halmos 1978-1981: Ralph Philip Boas, Jr. 1977-1978: Alex Rosenberg and Ralph Philip Boas Jr. 1974-1976: Alex Rosenberg 1969-1973: Harley Flanders 1967-1968: Robert Abraham Rosenbaum 1962-1966: Frederick Arthur Ficken 1957-1961: Ralph Duncan James 1952-1956: Carl Barnett Allendoerfer 1947-1951: Carroll Vincent Newsom 1942-1946: Lester Randolph Ford 1937-1941: Elton James Moulton 1932-1936: Walter Buckingham Carver 1927-1931: William Henry Bussey 1923-1926: Walter Burton Ford 1922: Albert Arnold Bennett 1919-1921: Raymond Clare Archibald 1918: Robert Daniel Carmichael 1916-1917: Herbert Ellsworth Slaught 1914-1915: Board of editors: C. H. Ashton, R. P. Baker, W. C. Brenke, W. H. Bussey, W. DeW. Cairns, Florian Cajori, R. D. Carmichael, D. R. Curtiss, I. M. DeLong, B. F. Finkel, E. R. Hedrick, L. C. Karpinski, G. A. Miller, W. H. Roever, H. E. Slaught 1913: Herbert Ellsworth Slaught 1909-1912: Benjamin Franklin Finkel, Herbert Ellsworth Slaught, George Abram Miller 1907-1908: Benjamin Franklin Finkel, Herbert Ellsworth Slaught 1905-1906: Benjamin Franklin Finkel, Leonard Eugene Dickson, Oliver Edmunds Glenn 1904: Benjamin Franklin Finkel, Leonard Eugene Dickson, Saul Epsteen 1903: Benjamin Franklin Finkel, Leonard Eugene Dickson 1894-1902: Benjamin Franklin Finkel, John Marvin Colaw Mathematics Magazine Notices of the American Mathematical Society, another "most read mathematics journal in the world" American Mathematical Monthly homepage Archive of tables of contents with article summaries Mathematical Association of America American Mathematical Monthly on JSTOR The American mathematical monthly, hathitrust
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most to objects that are relevant to mathematics; the language of set theory can be used to define nearly all mathematical objects. The modern study of set theory was initiated by Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, such as Russell's paradox, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with or without the axiom of choice, are the best-known. Set theory is employed as a foundational system for mathematics in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
Mathematical topics emerge and evolve through interactions among many researchers. Set theory, was founded by a single paper in 1874 by Georg Cantor: "On a Property of the Collection of All Real Algebraic Numbers". Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity. Notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1870–1874 and was motivated by Cantor's work in real analysis. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking and culminated in Cantor's 1874 paper. Cantor's work polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. Cantorian set theory became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, the "infinity of infinities" resulting from the power set operation.
This utility of set theory led to the article "Mengenlehre" contributed in 1898 by Arthur Schoenflies to Klein's encyclopedia. The next wave of excitement in set theory came around 1900, when it was discovered that some interpretations of Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes. Bertrand Russell and Ernst Zermelo independently found the simplest and best known paradox, now called Russell's paradox: consider "the set of all sets that are not members of themselves", which leads to a contradiction since it must be a member of itself and not a member of itself. In 1899 Cantor had himself posed the question "What is the cardinal number of the set of all sets?", obtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics. In 1906 English readers gained the book Theory of Sets of Points by husband and wife William Henry Young and Grace Chisholm Young, published by Cambridge University Press.
The momentum of set theory was such. The work of Zermelo in 1908 and the work of Abraham Fraenkel and Thoralf Skolem in 1922 resulted in the set of axioms ZFC, which became the most used set of axioms for set theory; the work of analysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory, which has since become woven into the fabric of modern mathematics. Set theory is used as a foundational system, although in some areas—such as algebraic geometry and algebraic topology—category theory is thought to be a preferred foundation. Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member of A, the notation o. Since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the subset relation called set inclusion. If all the members of set A are members of set B A is a subset of B, denoted A ⊆ B. For example, is a subset of, so is but is not; as insinuated from this definition, a set is a subset of itself.
For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined. A is called a proper subset of B if and only if A is a subset of B, but A is not equal to B. Note that 1, 2, 3 are members of the set but are not subsets of it. Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. The: Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both; the union of and is the set. Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B; the intersection of and is the set. Set difference of U and A, denoted U \ A, is the set of all members of U that are not members of A; the set difference \ is, conversely, the set difference \ is. When A is a subset of U, the set difference U \ A is called the complement of A in U. In this case, if the choice of U is clear from the context, the notation Ac is sometimes used instead of U \ A if U is a universal set as in the study of Venn diagrams.
Symmetric difference of sets A and B, denoted A △ B or A ⊖ B, is
Goodreads is a "social cataloging" website that allows individuals to search its database of books and reviews. Users can register books to generate library catalogs and reading lists, they can create their own groups of book suggestions, polls and discussions. The website's offices are located in San Francisco; the company is owned by the online retailer Amazon. Goodreads was founded in December 2006 and launched in January 2007 by Otis Chandler, a software engineer and entrepreneur, Elizabeth Khuri; the website grew in popularity after being launched. In December 2007, the site over 10,000,000 books had been added. By July 2012, the site reported 10 million members, 20 million monthly visits, 30 employees. On July 23, 2013, it was announced on their website that the user base had grown to 20 million members, having doubled in close to 11 months. On March 28, 2013, Amazon announced its acquisition of Goodreads; the Chandlers created Goodreads in 2006. Goodreads' stated mission is "to help people find and share books they love... to improve the process of reading and learning throughout the world."
Goodreads addressed "what publishers call the'discoverability' problem" by guiding consumers in the digital age to find books they might want to read. During its first year of business, the company was run without any formal funding. In December 2007, the site received; this funding lasted Goodreads until 2009, when Goodreads received two million dollars from True Ventures. In October 2010 the company opened its application programming interface, which enabled developers to access its ratings and titles. Goodreads received a small commission when a user clicks over from its site to an online bookseller and makes a purchase. In 2011, Goodreads acquired Discovereads, a book recommendation engine that employs "machine learning algorithms to analyze which books people might like, based on books they've liked in the past and books that people with similar tastes have liked." After a user has rated 20 books on its five-star scale, the site will begin making recommendations. Otis Chandler believed this rating system would be superior to Amazon's, as Amazon's includes books a user has browsed or purchased as gifts when determining its recommendations.
That year, Goodreads introduced an algorithm to suggest books to registered users and had over five million members. The New Yorker's Macy Halford noted that the algorithm wasn't perfect, as the number of books needed to create a perfect recommendation system is so large that "by the time I'd got halfway there, my reading preferences would have changed and I'd have to start over again."In October 2012, Goodreads announced it had grown to 11 million members with 395 million books catalogued and over 20,000 book clubs created by its users. A month in November 2012, Goodreads had surpassed 12 million members, with the member base having doubled in one year. In March 2013, Amazon made an agreement to acquire Goodreads in the second quarter of 2013 for an undisclosed sum. In September 2013, Goodreads announced it would delete, without warning, reviews that mention the behavior of an author or threats against an author. In January 2016, Amazon announced that it would shut down Shelfari in favor of Goodreads effective March 16, 2016.
Users were offered the ability to migrate accounts. In April 2016, Goodreads announced. On the Goodreads website, users can add books to their personal bookshelves and review books, see what their friends and authors are reading, participate in discussion boards and groups on a variety of topics, get suggestions for future reading choices based on their reviews of read books. Once users have added friends to their profile, they will see their friends' shelves and reviews and can comment on friends' pages. Goodreads features a rating system of one to five stars, with the option of accompanying the rating with a written review; the site provides default bookshelves—read, currently-reading, to-read—and the opportunity to create customized shelves to categorize a user's books. Goodreads users can read or listen to a preview of a book on the website using Kindle Cloud Reader and Audible. Goodreads offers quizzes and trivia, book lists, free giveaways. Members can receive the regular newsletter featuring new books, author interviews, poetry.
If a user has written a work, the work can be linked on the author's profile page, which includes an author's blog. Goodreads organizes offline opportunities as well, such as IRL book exchanges and "literary pub crawls"; the website facilitates reader interactions with authors through the interviews, authors' blogs, profile information. There is a special section for authors with suggestions for promoting their works on Goodreads.com, aimed at helping them reach their target audience. By 2011, "seventeen thousand authors, including James Patterson and Margaret Atwood" used Goodreads to advertise. Additionally, Goodreads has a presence on Facebook, Pinterest and other social networking sites. Linking a Goodreads account with a social networking account like Facebook enables the ability to import contacts from the social networking account to Goodreads, expanding one's Goodreads "Friends" list. There are settings available, as well, to allow Goodreads to post straight to a social networking account, which informs, e.g. Facebook friends, what one is reading or how one rated a book.
This constant linkage from Goodreads to other social networking sites keeps information flowing and connectivity continuous. The Amazon Kindle Paperw
Paul Richard Halmos was a Hungarian-born American mathematician who made fundamental advances in the areas of mathematical logic, probability theory, operator theory, ergodic theory, functional analysis. He was recognized as a great mathematical expositor, he has been described as one of The Martians. Born in Hungary into a Jewish family, Halmos arrived in the U. S. at 13 years of age. He obtained his B. A. from the University of Illinois, majoring in mathematics, but fulfilling the requirements for both a math and philosophy degree. He took only three years to obtain the degree, was only 19 when he graduated, he began a Ph. D. in philosophy, still at the Champaign-Urbana campus. Joseph L. Doob supervised his dissertation, titled Invariants of Certain Stochastic Transformations: The Mathematical Theory of Gambling Systems. Shortly after his graduation, Halmos left for the Institute for Advanced Study, lacking both job and grant money. Six months he was working under John von Neumann, which proved a decisive experience.
While at the Institute, Halmos wrote his first book, Finite Dimensional Vector Spaces, which established his reputation as a fine expositor of mathematics. Halmos taught at Syracuse University, the University of Chicago, the University of Michigan, the University of California at Santa Barbara, the University of Hawaii, Indiana University. From his 1985 retirement from Indiana until his death, he was affiliated with the Mathematics department at Santa Clara University. In a series of papers reprinted in his 1962 Algebraic Logic, Halmos devised polyadic algebras, an algebraic version of first-order logic differing from the better known cylindric algebras of Alfred Tarski and his students. An elementary version of polyadic algebra is described in monadic Boolean algebra. In addition to his original contributions to mathematics, Halmos was an unusually clear and engaging expositor of university mathematics, he won the Lester R. Ford Award in 1971 and again in 1977. Halmos chaired the American Mathematical Society committee that wrote the AMS style guide for academic mathematics, published in 1973.
In 1983, he received the AMS's Steele Prize for exposition. In the American Scientist 56: 375–389, Halmos argued that mathematics is a creative art, that mathematicians should be seen as artists, not number crunchers, he discussed the division of the field into mathology and mathophysics, further arguing that mathematicians and painters think and work in related ways. Halmos's 1985 "automathography" I Want to Be a Mathematician is an account of what it was like to be an academic mathematician in 20th century America, he called the book "automathography" rather than "autobiography", because its focus is entirely on his life as a mathematician, not his personal life. The book contains the following quote on Halmos' view of what doing mathematics means: Don't just read it. Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?
In these memoirs, Halmos claims to have invented the "iff" notation for the words "if and only if" and to have been the first to use the "tombstone" notation to signify the end of a proof, this is agreed to be the case. The tombstone symbol ∎ is sometimes called a halmos. In 2005, Halmos and his wife Virginia funded the Euler Book Prize, an annual award given by the Mathematical Association of America for a book, to improve the view of mathematics among the public; the first prize was given in 2007, the 300th anniversary of Leonhard Euler's birth, to John Derbyshire for his book about Bernhard Riemann and the Riemann hypothesis: Prime Obsession. 1942. Finite-Dimensional Vector Spaces. Springer-Verlag. 1950. Measure Theory. Springer Verlag. 1951. Introduction to Hilbert Space and the Theory of Spectral Multiplicity. Chelsea. 1956. Lectures on Ergodic Theory. Chelsea. 1960. Naive Set Theory. Springer Verlag. 1962. Algebraic Logic. Chelsea. 1963. Lectures on Boolean Algebras. Van Nostrand. 1967. A Hilbert Space Problem Book.
Springer-Verlag. 1973.. How to Write Mathematics. American Mathematical Society. 1978.. Bounded Integral Operators on L² Spaces. Springer Verlag 1985. I Want to Be a Mathematician. Springer-Verlag. 1987. I Have a Photographic Memory. Mathematical Association of America. 1991. Problems for Mathematicians and Old, Dolciani Mathematical Expositions, Mathematical Association of America. 1996. Linear Algebra Problem Book, Dolciani Mathematical Expositions, Mathematical Association of America. 1998.. Logic as Algebra, Dolciani Mathematical Expositions No. 21, Mathematical Association of America. 2009. Introduction to Boolean Algebras, Springer. Criticism of non-standard analysis The Martians J. H. Ewing. Paul Halmos: Celebrating 50 Years of Mathematics. Springer-Verlag. ISBN 0-387-97509-8. OCLC 22859036. Includes a bibliography of Halmos's writings through 1991. John Ewing. "Paul Halmos: In His Own Words". Notices of the American Mathematical Society. 54: 1136–1144. Retrieved 2008-01-15. Paul Halmos. I want to be a Mathematician: An Automathography.
Springer-Verlag. ISBN 0-387-96470-3. OCLC 230812318. O'Connor, John J..
Naive set theory
Naïve set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naïve set theory is defined informally, in natural language, it describes the aspects of mathematical sets familiar in discrete mathematics, suffices for the everyday use of set theory concepts in contemporary mathematics. Sets are of great importance in mathematics. Naïve set theory suffices for many purposes, while serving as a stepping-stone towards more formal treatments. A naïve theory in the sense of "naïve set theory" is a non-formalized theory, that is, a theory that uses a natural language to describe sets and operations on sets; the words and, or, if... not, for some, for every are treated as in ordinary mathematics. As a matter of convenience, use of naïve set theory and its formalism prevails in higher mathematics – including in more formal settings of set theory itself; the first development of set theory was a naïve set theory.
It was created at the end of the 19th century by Georg Cantor as part of his study of infinite sets and developed by Gottlob Frege in his Begriffsschrift. Naïve set theory may refer to several distinct notions, it may refer to Informal presentation of an axiomatic set theory, e.g. as in Naïve Set Theory by Paul Halmos. Early or versions of Georg Cantor's theory and other informal systems. Decidedly inconsistent theories, such as a theory of Gottlob Frege that yielded Russell's paradox, theories of Giuseppe Peano and Richard Dedekind; the assumption that any property may be used to form a set, without restriction, leads to paradoxes. One common example is Russell's paradox: there is no set consisting of "all sets that do not contain themselves", thus consistent systems of naïve set theory must include some limitations on the principles which can be used to form sets. Some believe that Georg Cantor's set theory was not implicated in the set-theoretic paradoxes. One difficulty in determining this with certainty is that Cantor did not provide an axiomatization of his system.
By 1899, Cantor was aware of some of the paradoxes following from unrestricted interpretation of his theory, for instance Cantor's paradox and the Burali-Forti paradox, did not believe that they discredited his theory. Cantor's paradox can be derived from the above assumption—that any property P may be used to form a set—using for P "x is a cardinal number". Frege explicitly axiomatized a theory in which a formalized version of naïve set theory can be interpreted, it is this formal theory which Bertrand Russell addressed when he presented his paradox, not a theory Cantor, who, as mentioned, was aware of several paradoxes had in mind. Axiomatic set theory was developed in response to these early attempts to understand sets, with the goal of determining what operations were allowed and when. A naïve set theory is not inconsistent, if it specifies the sets allowed to be considered; this can be done by the means of definitions. It is possible to state all the axioms explicitly, as in the case of Halmos' Naïve Set Theory, an informal presentation of the usual axiomatic Zermelo–Fraenkel set theory.
It is "naïve" in that the language and notations are those of ordinary informal mathematics, in that it doesn't deal with consistency or completeness of the axiom system. An axiomatic set theory is not consistent: not free of paradoxes, it follows from Gödel's incompleteness theorems that a sufficiently complicated first order logic system cannot be proved consistent from within the theory itself – if it is consistent. However, the common axiomatic systems are believed to be consistent. Based on Gödel's theorem, it is just not known – and never can be – if there are no paradoxes at all in these theories or in any first-order set theory; the term naïve set theory is still today used in some literature to refer to the set theories studied by Frege and Cantor, rather than to the informal counterparts of modern axiomatic set theory. The choice between an axiomatic approach and other approaches is a matter of convenience. In everyday mathematics the best choice may be informal use of axiomatic set theory.
References to particular axioms then occur only when demanded by tradition, e.g. the axiom of choice is mentioned when used. Formal proofs occur only when warranted by exceptional circumstances; this informal usage of axiomatic set theory can have the appearance of naïve set theory as outlined below. It is easier to read and write and is less error-prone than a formal approach. In naïve set theory, a set is described as a well-defined collection of objects; these objects are called the members of the set. Objects can be anything: numbers, other sets, etc. For instance, 4 is a member of the set of all integers; the set of numbers is infinitely large. The definition of sets goes back to Georg Cantor, he wrote 1915 in his article Beiträge zur Begründung der transfiniten Mengenlehre: “Unter einer'Menge' verstehen wir jede Zusammenfassung M von bestimmten wohlunterschi
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