|
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] |
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 axiomatization of arithmetic provided by Peano axioms is commonly called Peano arithmetic. The importance of formalizing 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 operation and induction. In 1881, Charles Sanders Peirce provided an 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 ... [...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 the 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''),Peano, Giuseppe (1903). De Latino Sine Flexione. Lingua Auxiliare Internationale', ''Revista de Mathematica'' (''Revue de Mathématiques''), Tomo VIII, pp. 74-83. Fratres Bocca Editores: Torino. 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 int ... [...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 () 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 that its Gauss map has a W ... [...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 ith step constructs a set S_i of squares, and a sequence P_i 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_ is partitioned into nine smaller equal squares, and its center point c is replaced by a contiguous subsequence of the centers of these ni ... [...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 ... [...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'(t) = f\left(t, y(t)\right) a continuous, explicit first-order differential equation defined on ''D'', then every initial value problem y\left(t_0\right) = y_0 for ''f'' with (t_0, y_0) \in D has a local solution z\colon I \to \mathbb where I is a neighbourhood of t_0 in \mathbb, such that z'(t) = f\left(t, z(t)\right) for all t \in I . The solution need not be unique: one and the same initial value (t_0, y_0) may give ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to th ... [...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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called scalar (mathematics), ''scalars''. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: real numbers and complex numbers. Scalars can also be, more generally, elements of any field (mathematics), field. Vector spaces generalize Euclidean vectors, which allow modeling of Physical quantity, physical quantities (such as forces and velocity) that have not only a Magnitude (mathematics), magnitude, but also a Orientation (geometry), direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix (mathematics), matrices, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maria Gramegna
Maria Paola Gramegna (1887–1915) was an Italian mathematician and a student of Giuseppe Peano. Her work with Peano on systems of linear differential equations has been cited as an important early milestone in the history of functional analysis and its transition from working with concrete matrices to more abstract basis-independent formulations of linear algebra. After becoming a schoolteacher, she died in the 1915 Avezzano earthquake. Early life and education Gramegna was born on 11 May 1887 in Tortona, the youngest of five children of a pasta factory owner. After studying mathematics in high school in Voghera with Giuseppe Vitali, she was admitted to the University of Turin in 1906, where she became a student of Peano. She completed her mathematics degree in 1910, with the thesis ''Serie di equazioni differenziali lineari ed equazioni integro-differenziali'' 'Series of linear differential equations and integro-differential equations'' supervised by Peano. Peano also presented ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiomatization
In mathematics and logic, an axiomatic system is a set of formal statements (i.e. axioms) used to logically derive other statements such as lemmas or theorems. A proof within an axiom system is a sequence of deductive steps that establishes a new statement as a consequence of the axioms. An axiom system is called complete with respect to a property if every formula with the property can be derived using the axioms. The more general term theory is at times used to refer to an axiomatic system and all its derived theorems. In its pure form, an axiom system is effectively a syntactic construct and does not by itself refer to (or depend on) a formal structure, although axioms are often defined for that purpose. The more modern field of model theory refers to mathematical structures. The relationship between an axiom systems and the models that correspond to it is often a major issue of interest. Properties Four typical properties of an axiom system are consistency, relativ ... [...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 f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of ''naive set theory''. After the discovery of Paradoxes of set theory, paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
