Matrix (mathematics) Each element of a matrix is often denoted by a variable with two subscripts. For instance, a2,1 represents the element at the second row and first column of a matrix A. Applications of matrices are found in most scientific fields. The numbers, symbols or expressions in the matrix are called its entries or its elements. The size of a matrix is defined by the number of rows and columns that it contains. Matrices are commonly written in box brackets: An alternative notation uses large parentheses instead of box brackets: The entry in the i-th row and j-th column of a matrix A is sometimes referred to as the i,j, (i,j), or (i,j)th entry of the matrix, and most commonly denoted as ai,j, or aij. Sometimes, the entries of a matrix can be defined by a formula such as ai,j = f(i, j). In this case, the matrix itself is sometimes defined by that formula, within square brackets or double parenthesis. Schematic depiction of the matrix product AB of two matrices A and B. whereas Ax = b
Hilbert's Program First published Thu Jul 31, 2003 In the early 1920s, the German mathematician David Hilbert (1862-1943) put forward a new proposal for the foundation of classical mathematics which has come to be known as Hilbert's Program. It calls for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is consistent. The consistency proof itself was to be carried out using only what Hilbert called “finitary” methods. The special epistemological character of finitary reasoning then yields the required justification of classical mathematics. Although Hilbert proposed his program in this form only in 1921, various facets of it are rooted in foundational work of his going back until around 1900, when he first pointed out the necessity of giving a direct consistency proof of analysis. 1. 1.1 Early work on foundations 1.2 The influence of Principia Mathematica 1.3 Finitism and the quest for consistency proofs 2.
Dave's short course in trigonometry Table of Contents Who should take this course? Trigonometry for you Your background How to learn trigonometry Applications of trigonometry Astronomy and geography Engineering and physics Mathematics and its applications What is trigonometry? Trigonometry as computational geometry Angle measurement and tables Background on geometry The Pythagorean theorem An explanation of the Pythagorean theorem Similar triangles Angle measurement The concept of angle Radians and arc length Exercises, hints, and answers About digits of accuracy Chords What is a chord? Ptolemy’s sum and difference formulas Ptolemy’s theorem The sum formula for sines The other sum and difference formulas Summary of trigonometric formulas Formulas for arcs and sectors of circles Formulas for right triangles Formulas for oblique triangles Formulas for areas of triangles Summary of trigonometric identities More important identities Less important identities Truly obscure identities About the Java applet.
The Problem of Dirty Hands First published Wed Apr 29, 2009; substantive revision Mon Jan 27, 2014 Should political leaders violate the deepest constraints of morality in order to achieve great goods or avoid disasters for their communities? This question poses what has become known amongst philosophers as the problem of dirty hands. There are many different strands to the philosophical debate about this topic, and they echo many of the complexities in more popular thinking about politics and morality. All, however, involve the idea that correct political action must sometimes conflict with profound moral norms. In the course of addressing these issues, the dirty hands challenge is also distinguished from that of political realism, with which it has some affinities, and the resort to role morality to render dirty hands coherent is discussed, as is the issue of the desirability of shaming or punishing dirty hands agents. 1. 2. 3. Let us examine these in turn.
Non-Euclidean Geometry In three dimensions, there are three classes of constant curvature geometries. All are based on the first four of Euclid's postulates, but each uses its own version of the parallel postulate. The "flat" geometry of everyday intuition is called Euclidean geometry (or parabolic geometry), and the non-Euclidean geometries are called hyperbolic geometry (or Lobachevsky-Bolyai-Gauss geometry) and elliptic geometry (or Riemannian geometry). Spherical geometry is a non-Euclidean two-dimensional geometry. It was not until 1868 that Beltrami proved that non-Euclidean geometries were as logically consistent as Euclidean geometry. Aesthetic Judgment First published Fri Feb 28, 2003; substantive revision Thu Jul 22, 2010 Beauty is an important part of our lives. Ugliness too. It is no surprise then that philosophers since antiquity have been interested in our experiences of and judgments about beauty and ugliness. 1. What is a judgment of taste? 1.1 Subjectivity The first necessary condition of a judgment of taste is that it is essentially subjective. This subjectivist thesis would be over-strict if it were interpreted in an “atomistic” fashion, so that some subjective response corresponds to every judgment of taste, and vice versa. However, it is not obvious what to make of the subjectivity of the judgment of taste. Beyond a certain point, this issue cannot be pursued independently of metaphysical issues about realism, for the metaphysics we favor is bound to affect our view of the nature of the pleasure we take in beauty. This is all important as far as it goes, but it is all negative. 1.2 Normativity 1.3 Recasting Normativity 2.
Non-Euclidean geometry Behavior of lines with a common perpendicular in each of the three types of geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is set aside. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry. Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line: History Early history Terminology
Intute: Encouraging Critical Thinking Online Encouraging Critical Thinking Online is a set of free teaching resources designed to develop students' analytic abilities, using the Web as source material. Two units are currently available, each consisting of a series of exercises for classroom or seminar use. Students are invited to explore the Web and find a number of sites which address the selected topic, and then, in a teacher-led group discussion, to share and discuss their findings. The exercises are designed so that they may be used either consecutively to form a short course, or individually. The resources encourage students to think carefully and critically about the information sources they use. A comprehensive Teacher's Guide provides an overview of the course, lesson/seminar outlines, suggestions of illustrative websites, and points for discussion. Teacher's Guide (Units 1 and 2) Printable version (PDF) Resources for Unit 1: Checking Facts and Gathering Opinions Resources for Unit 1 Resources for Unit 2
Non-Euclidean geometry Version for printing In about 300 BC Euclid wrote The Elements, a book which was to become one of the most famous books ever written. Euclid stated five postulates on which he based all his theorems: To draw a straight line from any point to any other. Proclus (410-485) wrote a commentary on The Elements where he comments on attempted proofs to deduce the fifth postulate from the other four, in particular he notes that Ptolemy had produced a false 'proof'. Playfair's Axiom:- Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line. Although known from the time of Proclus, this became known as Playfair's Axiom after John Playfair wrote a famous commentary on Euclid in 1795 in which he proposed replacing Euclid's fifth postulate by this axiom. Many attempts were made to prove the fifth postulate from the other four, many of them being accepted as proofs for long periods of time until the mistake was found.