 # Philosophy of Mathematics First published Tue Sep 25, 2007; substantive revision Wed May 2, 2012 If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. 1. On the one hand, philosophy of mathematics is concerned with problems that are closely related to central problems of metaphysics and epistemology. On the other hand, it has turned out that to some extent it is possible to bring mathematical methods to bear on philosophical questions concerning mathematics. When professional mathematicians are concerned with the foundations of their subject, they are said to be engaged in foundational research. 2. The general philosophical and scientific outlook in the nineteenth century tended toward the empirical. 2.1 Logicism φ ∨ ¬φ, ¬¬φ → φ Related:  Philosophy Major

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

Table of mathematical symbols When reading the list, it is important to recognize that a mathematical concept is independent of the symbol chosen to represent it. For many of the symbols below, the symbol is usually synonymous with the corresponding concept (ultimately an arbitrary choice made as a result of the cumulative history of mathematics), but in some situations a different convention may be used. For example, depending on context, the triple bar "≡" may represent congruence or a definition. Further, in mathematical logic, numerical equality is sometimes represented by "≡" instead of "=", with the latter representing equality of well-formed formulas. Each symbol is shown both in HTML, whose display depends on the browser's access to an appropriate font installed on the particular device, and in TeX, as an image. Guide This list is organized by symbol type and is intended to facilitate finding an unfamiliar symbol by its visual appearance. Basic symbols Symbols based on equality sign

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.

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.

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.

Supremum A set A of real numbers (blue balls), a set of upper bounds of A (red diamond and balls), and the smallest such upper bound, that is, the supremum of A (red diamond). In mathematics, the supremum (sup) of a subset S of a totally or partially ordered set T is the least element of T that is greater than or equal to all elements of S. Consequently, the supremum is also referred to as the least upper bound (lub or LUB). If the supremum exists, it is unique, meaning that there will be only one supremum. If S contains a greatest element, then that element is the supremum; otherwise, the supremum does not belong to S (or does not exist). For instance, the negative real numbers do not have a greatest element, and their supremum is 0 (which is not a negative real number). Supremum of a set of real numbers Examples Simple The "supremum" or "least upper bound" of the set of numbers { 1, 2, 3 } is 3. Mathematically, this is: sup { 1, 2, 3 } = 3 Advanced One basic property of the supremum is S.

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.

Set Theory First published Thu Jul 11, 2002 Set Theory is the mathematical science of the infinite. It studies properties of sets, abstract objects that pervade the whole of modern mathematics. The language of set theory, in its simplicity, is sufficiently universal to formalize all mathematical concepts and thus set theory, along with Predicate Calculus, constitutes the true Foundations of Mathematics. 1. The objects of study of Set Theory are sets. The language of set theory is based on a single fundamental relation, called membership. Basic Set Theory for further discussion. When dealing with sets informally, such operations on sets are self-evident; with the axiomatic approach, it is postulated that such operations can be applied: for instance, one postulates that for any sets A and B, the set {A,B} exists. One of the basic principles of set theory is the existence of an infinite set. for further discussion. The fundamental concept in the theory of infinite sets is the cardinality of a set. 2.

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.

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