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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 wrote 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, 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, gradu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peano Axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete. The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor function, successor operation and mathematical induction, induction. In 1881, Charles Sanders Peirce provided an Axiomatic system#Axiomatization, axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his book, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Latino Sine Flexione
Latino sine flexione ("Latin without inflections"), Interlingua de Academia pro Interlingua (IL de ApI) or Peano's Interlingua (abbreviated as IL), is an international auxiliary language compiled by the Academia pro Interlingua under chairmanship of the Italian mathematician Giuseppe Peano (1858–1932) from 1887 until 1914. It is a simplified version of Latin, and retains its vocabulary. Interlingua-IL was published in the journal ''Revue de Mathématiques'' in an article of 1903 entitled ''De Latino Sine Flexione, Lingua Auxiliare Internationale'' (meaning ''On Latin Without Inflection, International Auxiliary Language''), which explained the reason for its creation. The article argued that other auxiliary languages were unnecessary, since Latin was already established as the world's international language. The article was written in classical Latin, but it gradually dropped its inflections until there were none. Language codes ISO 639: ISO 639-2 and -1 were requested on 23 Ju ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peano Surface
In mathematics, the Peano surface is the graph of the two-variable function :f(x,y)=(2x^2-y)(y-x^2). It was proposed by Giuseppe Peano in 1899 as a counterexample to a conjectured criterion for the existence of maxima and minima of functions of two variables. The surface was named the Peano surface (german: Peanosche Fläche) by Georg Scheffers in his 1920 book ''Lehrbuch der darstellenden Geometrie''. It has also been called the Peano saddle. See especially section "Peano Saddle", pp. 562–563. Properties The function f(x,y)=(2x^2-y)(y-x^2) whose graph is the surface takes positive values between the two parabolas y=x^2 and y=2x^2, and negative values elsewhere (see diagram). At the origin, the three-dimensional point (0,0,0) on the surface that corresponds to the intersection point of the two parabolas, the surface has a saddle point. The surface itself has positive Gaussian curvature in some parts and negative curvature in others, separated by another parabola, implying th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peano Curve
In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890. Peano's curve is a surjective, continuous function from the unit interval onto the unit square, however it is not injective. Peano was motivated by an earlier result of Georg Cantor that these two sets have the same cardinality. Because of this example, some authors use the phrase "Peano curve" to refer more generally to any space-filling curve. Construction Peano's curve may be constructed by a sequence of steps, where the ''i''th step constructs a set ''Si'' of squares, and a sequence ''Pi'' of the centers of the squares, from the set and sequence constructed in the previous step. As a base case, ''S''0 consists of the single unit square, and ''P''0 is the one-element sequence consisting of its center point. In step ''i'', each square ''s'' of ''S''''i'' − 1 is partitioned into nine smaller equal squares, and its center point ''c'' is rep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logicism
In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that — for some coherent meaning of 'logic' — mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic. Bertrand Russell and Alfred North Whitehead championed this programme, initiated by Gottlob Frege and subsequently developed by Richard Dedekind and Giuseppe Peano. Overview Dedekind's path to logicism had a turning point when he was able to construct a model satisfying the axioms characterizing the real numbers using certain sets of rational numbers. This and related ideas convinced him that arithmetic, algebra and analysis were reducible to the natural numbers plus a "logic" of classes. Furthermore by 1872 he had concluded that the naturals themselves were reducible to sets and mappings. It is likely that other logicists, most importantly Frege, were also guided by the new theories of the r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peano Existence Theorem
In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees the existence of solutions to certain initial value problems. History Peano first published the theorem in 1886 with an incorrect proof. In 1890 he published a new correct proof using successive approximations. Theorem Let D be an open subset of \mathbb\times\mathbb with f\colon D \to \mathbb a continuous function and y'(x) = f\left(x,y(x)\right) a continuous, explicit first-order differential equation defined on ''D'', then every initial value problem y\left(x_0\right) = y_0 for ''f'' with (x_0, y_0) \in D has a local solution z\colon I \to \mathbb where I is a neighbourhood of x_0 in \mathbb, such that z'(x) = f\left(x,z(x)\right) for all x \in I . The solution need not be unique: one and the same initial value (x_0,y_0) may give ris ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peano Kernel Theorem
In numerical analysis, the Peano kernel theorem is a general result on error bounds for a wide class of numerical approximations (such as numerical quadratures), defined in terms of linear functionals. It is attributed to Giuseppe Peano. Statement Let \mathcal ,b/math> be the space of all functions f that are differentiable on (a,b) that are of bounded variation on ,b/math>, and let L be a linear functional on \mathcal ,b/math>. Assume that that L ''annihilates'' all polynomials of degree \leq \nu, i.e.Lp=0,\qquad \forall p\in\mathbb_\nu Suppose further that for any bivariate function g(x,\theta) with g(x,\cdot),\,g(\cdot,\theta)\in C^ ,b/math>, the following is valid:L\int_a^bg(x,\theta)\,d\theta=\int_a^bLg(x,\theta)\,d\theta,and define the Peano kernel of L ask(\theta)=L x-\theta)^\nu_+\qquad\theta\in ,busing the notation(x-\theta)^\nu_+ = \begin (x-\theta)^\nu, & x\geq\theta, \\ 0, & x\leq\theta. \endThe ''Peano kernel theorem'' states that, if k\in\mathcal ,b/math>, then ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, artificial intelligence, cognitive science, computer science and various areas of analytic philosophy, especially philosophy of mathematics, philosophy of language, epistemology, and metaphysics.Stanford Encyclopedia of Philosophy"Bertrand Russell" 1 May 2003. He was one of the early 20th century's most prominent logicians, and a founder of analytic philosophy, along with his predecessor Gottlob Frege, his friend and colleague G. E. Moore and his student and protégé Ludwig Wittgenstein. Russell with Moore led the British "revolt against idealism". Together with his former teacher A. N. Whitehead, Russell wrote ''Principia Mathematica'', a milestone in the development of classical logic, and a major attempt to reduce the whole ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jordan Measure
In mathematics, the Peano–Jordan measure (also known as the Jordan content) is an extension of the notion of size (length, area, volume) to shapes more complicated than, for example, a triangle, disk, or parallelepiped. It turns out that for a set to have Jordan measure it should be well-behaved in a certain restrictive sense. For this reason, it is now more common to work with the Lebesgue measure, which is an extension of the Jordan measure to a larger class of sets. Historically speaking, the Jordan measure came first, towards the end of the nineteenth century. For historical reasons, the term ''Jordan measure'' is now well-established, despite the fact that it is not a true measure in its modern definition, since Jordan-measurable sets do not form a σ-algebra. For example, singleton sets \_in \mathbb each have a Jordan measure of 0, while \mathbb\cap ,1/math>, a countable union of them, is not Jordan-measurable. For this reason, some authors prefer to use the term ''Jord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giovanni Vailati
Giovanni Vailati (24 April 1863 – 14 May 1909) was an Italian proto-analytic philosopher, historian of science, and mathematician. Life Vailati was born in Crema, Lombardy, and studied engineering at the University of Turin. He went on to lecture in the history of mechanics there from 1896 to 1899, after working as assistant to Giuseppe Peano and Vito Volterra. He resigned his university post in 1899 so that he could pursue his independent studies, making a living from high-school mathematics teaching. During his lifetime he became internationally known, his writings having been translated into English, French, and Polish, though he was largely forgotten after his death in Rome. He was rediscovered in the late 1950s. He did not publish any complete books, but left about 200 essays and reviews across a range of academic disciplines. Philosophy Vailati's view of philosophy was that it provided a preparation and the tools for scientific work. For that reason, and because phil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peano–Russell Notation
In mathematical logic, Peano–Russell notation was Bertrand Russell's application of Giuseppe Peano's logical notation to the logical notions of Frege and was used in the writing of ''Principia Mathematica'' in collaboration with Alfred North Whitehead: "The notation adopted in the present work is based upon that of Peano, and the following explanations are to some extent modelled on those which he prefixes to his ''Formulario Mathematico''." (Chapter I: Preliminary Explanations of Ideas and Notations, page 4) Variables In the notation, variables are ambiguous in denotation, preserve a recognizable identity appearing in various places in logical statements within a given context, and have a range of possible determination between any two variables which is the same or different. When the possible determination is the same for both variables, then one implies the other; otherwise, the possible determination of one given to the other produces a meaningless phrase. The alphabetic symbo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
