1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
2.
Sequence
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In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members, the number of elements is called the length of the sequence. Unlike a set, order matters, and exactly the elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the numbers or the set of the first n natural numbers. The position of an element in a sequence is its rank or index and it depends on the context or of a specific convention, if the first element has index 0 or 1. For example, is a sequence of letters with the letter M first, also, the sequence, which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, the empty sequence is included in most notions of sequence, but may be excluded depending on the context. A sequence can be thought of as a list of elements with a particular order, Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations, Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers. There are a number of ways to denote a sequence, some of which are useful for specific types of sequences. One way to specify a sequence is to list the elements, for example, the first four odd numbers form the sequence. This notation can be used for sequences as well. For instance, the sequence of positive odd integers can be written. Listing is most useful for sequences with a pattern that can be easily discerned from the first few elements. Other ways to denote a sequence are discussed after the examples, the prime numbers are the natural numbers bigger than 1, that have no divisors but 1 and themselves. Taking these in their natural order gives the sequence, the prime numbers are widely used in mathematics and specifically in number theory. The Fibonacci numbers are the integer sequence whose elements are the sum of the two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is, for a large list of examples of integer sequences, see On-Line Encyclopedia of Integer Sequences
3.
Scalar (mathematics)
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A scalar is an element of a field which is used to define a vector space. A quantity described by multiple scalars, such as having both direction and magnitude, is called a vector, more generally, a vector space may be defined by using any field instead of real numbers, such as complex numbers. Then the scalars of that space will be the elements of the associated field. A scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, a vector space equipped with a scalar product is called an inner product space. The real component of a quaternion is also called its scalar part, the term is also sometimes used informally to mean a vector, matrix, tensor, or other usually compound value that is actually reduced to a single component. Thus, for example, the product of a 1×n matrix and an n×1 matrix, the term scalar matrix is used to denote a matrix of the form kI where k is a scalar and I is the identity matrix. The word scalar derives from the Latin word scalaris, a form of scala. The English word scale also comes from scala, according to a citation in the Oxford English Dictionary the first recorded usage of the term scalar in English came with W. R. A vector space is defined as a set of vectors, a set of scalars, and a multiplication operation that takes a scalar k. For example, in a space, the scalar multiplication k yields. In a function space, kƒ is the function x ↦ k, the scalars can be taken from any field, including the rational, algebraic, real, and complex numbers, as well as finite fields. According to a theorem of linear algebra, every vector space has a basis. It follows that every vector space over a scalar field K is isomorphic to a vector space where the coordinates are elements of K. For example, every vector space of dimension n is isomorphic to n-dimensional real space Rn. Alternatively, a vector space V can be equipped with a function that assigns to every vector v in V a scalar ||v||. By definition. If ||v|| is interpreted as the length of v, this operation can be described as scaling the length of v by k, a vector space equipped with a norm is called a normed vector space. The norm is defined to be an element of Vs scalar field K. Moreover, if V has dimension 2 or more, K must be closed under square root, as well as the four operations, thus the rational numbers Q are excluded
4.
Cartesian product
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In Set theory, a Cartesian product is a mathematical operation that returns a set from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs where a ∈ A and b ∈ B, products can be specified using set-builder notation, e. g. A table can be created by taking the Cartesian product of a set of rows, If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form. More generally, a Cartesian product of n sets, also known as an n-fold Cartesian product, can be represented by an array of n dimensions, an ordered pair is a 2-tuple or couple. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, an illustrative example is the standard 52-card deck. The standard playing card ranks form a 13-element set, the card suits form a four-element set. The Cartesian product of these sets returns a 52-element set consisting of 52 ordered pairs, Ranks × Suits returns a set of the form. Suits × Ranks returns a set of the form, both sets are distinct, even disjoint. The main historical example is the Cartesian plane in analytic geometry, usually, such a pairs first and second components are called its x and y coordinates, respectively, cf. picture. The set of all such pairs is thus assigned to the set of all points in the plane, a formal definition of the Cartesian product from set-theoretical principles follows from a definition of ordered pair. The most common definition of ordered pairs, the Kuratowski definition, is =, note that, under this definition, X × Y ⊆ P, where P represents the power set. Therefore, the existence of the Cartesian product of any two sets in ZFC follows from the axioms of pairing, union, power set, let A, B, C, and D be sets. × C ≠ A × If for example A =, then × A = ≠ = A ×, the Cartesian product behaves nicely with respect to intersections, cf. left picture. × = ∩ In most cases the above statement is not true if we replace intersection with union, cf. middle picture. Other properties related with subsets are, if A ⊆ B then A × C ⊆ B × C, the cardinality of a set is the number of elements of the set. For example, defining two sets, A = and B =, both set A and set B consist of two elements each. Their Cartesian product, written as A × B, results in a new set which has the following elements, each element of A is paired with each element of B. Each pair makes up one element of the output set, the number of values in each element of the resulting set is equal to the number of sets whose cartesian product is being taken,2 in this case
5.
Function (mathematics)
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In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that each real number x to its square x2. The output of a function f corresponding to a x is denoted by f. In this example, if the input is −3, then the output is 9, likewise, if the input is 3, then the output is also 9, and we may write f =9. The input variable are sometimes referred to as the argument of the function, Functions of various kinds are the central objects of investigation in most fields of modern mathematics. There are many ways to describe or represent a function, some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function, in science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation, sometimes the codomain is called the functions range, but more commonly the word range is used to mean, instead, specifically the set of outputs. For example, we could define a function using the rule f = x2 by saying that the domain and codomain are the numbers. The image of this function is the set of real numbers. In analogy with arithmetic, it is possible to define addition, subtraction, multiplication, another important operation defined on functions is function composition, where the output from one function becomes the input to another function. Linking each shape to its color is a function from X to Y, each shape is linked to a color, there is no shape that lacks a color and no shape that has more than one color. This function will be referred to as the color-of-the-shape function, the input to a function is called the argument and the output is called the value. The set of all permitted inputs to a function is called the domain of the function. Thus, the domain of the function is the set of the four shapes. The concept of a function does not require that every possible output is the value of some argument, a second example of a function is the following, the domain is chosen to be the set of natural numbers, and the codomain is the set of integers. The function associates to any number n the number 4−n. For example, to 1 it associates 3 and to 10 it associates −6, a third example of a function has the set of polygons as domain and the set of natural numbers as codomain
6.
Set (mathematics)
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In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. For example, the numbers 2,4, and 6 are distinct objects when considered separately, Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a part of mathematics. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, the German word Menge, rendered as set in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite. A set is a collection of distinct objects. The objects that make up a set can be anything, numbers, people, letters of the alphabet, other sets, Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements. Cantors definition turned out to be inadequate, instead, the notion of a set is taken as a notion in axiomatic set theory. There are two ways of describing, or specifying the members of, a set, one way is by intensional definition, using a rule or semantic description, A is the set whose members are the first four positive integers. B is the set of colors of the French flag, the second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets, one often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance, A = C and B = D, there are two important points to note about sets. First, in a definition, a set member can be listed two or more times, for example. However, per extensionality, two definitions of sets which differ only in one of the definitions lists set members multiple times, define, in fact. Hence, the set is identical to the set. The second important point is that the order in which the elements of a set are listed is irrelevant and we can illustrate these two important points with an example, = =. For sets with many elements, the enumeration of members can be abbreviated, for instance, the set of the first thousand positive integers may be specified extensionally as, where the ellipsis indicates that the list continues in the obvious way. Ellipses may also be used where sets have infinitely many members, thus the set of positive even numbers can be written as
7.
Subset
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In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is contained inside B, that is, all elements of A are also elements of B. The relationship of one set being a subset of another is called inclusion or sometimes containment, the subset relation defines a partial order on sets. The algebra of subsets forms a Boolean algebra in which the relation is called inclusion. For any set S, the inclusion relation ⊆ is an order on the set P of all subsets of S defined by A ≤ B ⟺ A ⊆ B. We may also partially order P by reverse set inclusion by defining A ≤ B ⟺ B ⊆ A, when quantified, A ⊆ B is represented as, ∀x. So for example, for authors, it is true of every set A that A ⊂ A. Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively and this usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may or may not equal y, but if x < y, then x definitely does not equal y, and is less than y. Similarly, using the convention that ⊂ is proper subset, if A ⊆ B, then A may or may not equal B, the set A = is a proper subset of B =, thus both expressions A ⊆ B and A ⊊ B are true. The set D = is a subset of E =, thus D ⊆ E is true, any set is a subset of itself, but not a proper subset. The empty set, denoted by ∅, is also a subset of any given set X and it is also always a proper subset of any set except itself. These are two examples in both the subset and the whole set are infinite, and the subset has the same cardinality as the whole. The set of numbers is a proper subset of the set of real numbers. In this example, both sets are infinite but the set has a larger cardinality than the former set. Another example in an Euler diagram, Inclusion is the partial order in the sense that every partially ordered set is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example—if each ordinal n is identified with the set of all ordinals less than or equal to n, then a ≤ b if and only if ⊆. For the power set P of a set S, the partial order is the Cartesian product of k = |S| copies of the partial order on for which 0 <1. This can be illustrated by enumerating S = and associating with each subset T ⊆ S the k-tuple from k of which the ith coordinate is 1 if and only if si is a member of T
8.
Real line
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In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the line is the set R of all real numbers, viewed as a geometric space. It can be thought of as a space, a metric space, a topological space. Just like the set of numbers, the real line is usually denoted by the symbol R. However. This article focuses on the aspects of R as a space in topology, geometry. The real numbers also play an important role in algebra as a field, for more information on R in all of its guises, see real number. The real line is a linear continuum under the standard < ordering, specifically, the real line is linearly ordered by <, and this ordering is dense and has the least-upper-bound property. In addition to the properties, the real line has no maximum or minimum element. It also has a dense subset, namely the set of rational numbers. It is a theorem that any linear continuum with a dense subset. The real line also satisfies the countable chain condition, every collection of mutually disjoint, in order theory, the famous Suslin problem asks whether every linear continuum satisfying the countable chain condition that has no maximum or minimum element is necessarily order-isomorphic to R. This statement has been shown to be independent of the axiomatic system of set theory known as ZFC. The real line forms a space, with the distance function given by absolute difference. The metric tensor is clearly the 1-dimensional Euclidean metric, since the n-dimensional Euclidean metric can be represented in matrix form as the n by n identity matrix, the metric on the real line is simply the 1 by 1 identity matrix, i. e.1. If p ∈ R and ε >0, then the ε-ball in R centered at p is simply the open interval. This real line has several important properties as a space, The real line is a complete metric space. The real line is path-connected, and is one of the simplest examples of a metric space The Hausdorff dimension of the real line is equal to one. The real line carries a standard topology which can be introduced in two different, equivalent ways, first, since the real numbers are totally ordered, they carry an order topology
9.
Nicolas Bourbaki
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With the goal of grounding all of mathematics on set theory, the group strove for rigour and generality. Their work led to the discovery of several concepts and terminologies still used, in 1934, young French mathematicians from various French universities felt the need to form a group to jointly produce textbooks that they could all use for teaching. André Weil organized the first meeting on 10 December 1934 in the basement of a Parisian grill room, Bourbakis main work is the Elements of Mathematics series. This series aims to be a completely self-contained treatment of the areas of modern mathematics. Assuming no special knowledge of mathematics, it takes up mathematics from the beginning, proceeds axiomatically. The volume on spectral theory from 1967 was for almost four decades the last new book to be added to the series, after that several new chapters to existing books as well as revised editions of existing chapters appeared until the publication of chapters 8-9 of Commutative Algebra in 1983. Then a long break in publishing activity occurred, leading many to suspect the end of the publishing project, however, chapter 10 of Commutative Algebra appeared in 1998, and after another long break a completely re-written and expanded chapter 8 of Algèbre was published in 2012. More importantly, the first four chapters of a new book on algebraic topology were published in 2016. Besides the Éléments de mathématique series, lectures from the Séminaire Bourbaki also have been published in monograph form since 1948. Notations introduced by Bourbaki include the symbol ∅ for the empty set and a dangerous bend symbol ☡, and the terms injective, surjective, and bijective. The emphasis on rigour may be seen as a reaction to the work of Henri Poincaré, the impact of Bourbakis work initially was great on many active research mathematicians world-wide. For example, Our time is witnessing the creation of a monumental work and it provoked some hostility, too, mostly on the side of classical analysts, they approved of rigour but not of high abstraction. This led to a gulf with the way theoretical physics was practiced, Bourbakis direct influence has decreased over time. This is partly because certain concepts which are now important, such as the machinery of category theory, are not covered in the treatise. It also mattered that, while especially algebraic structures can be defined in Bourbakis terms. On the other hand, the approach and rigour advocated by Bourbaki have permeated the current mathematical practices to such extent that the task undertaken was completed and this is particularly true for the less applied parts of mathematics. The Bourbaki seminar series founded in post-WWII Paris continues, it has going on since 1948. It is an important source of survey articles, with sketches of proofs, the topics range through all branches of mathematics, including sometimes theoretical physics
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Set theory
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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 often to objects that are relevant to mathematics, the language of set theory can be used in the definitions of nearly all mathematical objects. The modern study of set theory was initiated by Georg Cantor, Set theory is commonly employed as a foundational system for mathematics, particularly 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, 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 typically emerge and evolve through interactions among many researchers, Set theory, however, 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, especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1867–71, with Cantors work on number theory, an 1872 meeting between Cantor and Richard Dedekind influenced Cantors thinking and culminated in Cantors 1874 paper. Cantors work initially 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. This utility of set theory led to the article Mengenlehre contributed in 1898 by Arthur Schoenflies to Kleins encyclopedia, in 1899 Cantor had himself posed the question What is the cardinal number of the set of all sets. 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 William Henry Young and his wife Grace Chisholm Young, published by Cambridge University Press. The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment, the work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory. The work of such as Henri Lebesgue demonstrated the great mathematical utility of set theory. Set theory is used as a foundational system, although in some areas category theory is thought to be a preferred foundation. Set theory begins with a binary relation between an object o and a set A. If o is a member of A, the notation o ∈ A is used, since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, for example, is a subset of, and 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
11.
Norbert Wiener
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Norbert Wiener was an American mathematician and philosopher. He was a professor of mathematics at MIT, Wiener was born in Columbia, Missouri, the first child of Leo Wiener and Bertha Kahn, Jews of Polish and German origin, respectively. Leo had educated Norbert at home until 1903, employing teaching methods of his own invention, earning his living teaching German and Slavic languages, Leo read widely and accumulated a personal library from which the young Norbert benefited greatly. Leo also had ample ability in mathematics and tutored his son in the subject until he left home, in his autobiography, Norbert described his father as calm and patient, unless he failed to give a correct answer, at which his father would lose his temper. After graduating from Ayer High School in 1906 at 11 years of age and he was awarded a BA in mathematics in 1909 at the age of 14, whereupon he began graduate studies of zoology at Harvard. In 1910 he transferred to Cornell to study philosophy, the next year he returned to Harvard, while still continuing his philosophical studies. Back at Harvard, Wiener became influenced by Edward Vermilye Huntington, whose mathematical interests ranged from axiomatic foundations to engineering problems. Harvard awarded Wiener a Ph. D. in 1912, when he was merely 17 years old, for a dissertation on mathematical logic, supervised by Karl Schmidt, the essential results of which were published as Wiener. In that dissertation, he was the first to state publicly that ordered pairs can be defined in terms of set theory. Hence relations can be defined by set theory, thus the theory of relations does not require any axioms or primitive notions distinct from those of set theory. In 1921, Kazimierz Kuratowski proposed a simplification of Wieners definition of ordered pairs, and that simplification has been in common use ever since. In 1914, Wiener traveled to Europe, to be taught by Bertrand Russell and G. H. Hardy at Cambridge University, during 1915–16, he taught philosophy at Harvard, then was an engineer for General Electric and wrote for the Encyclopedia Americana. Although Wiener eventually became a staunch pacifist, he contributed to the war effort in World War I. In 1916, with Americas entry into the war drawing closer, Wiener attended a camp for potential military officers. One year later Wiener again tried to join the military, in the summer of 1918, Oswald Veblen invited Wiener to work on ballistics at the Aberdeen Proving Ground in Maryland. Living and working with other mathematicians strengthened his interest in mathematics, however, Wiener was still eager to serve in uniform, and decided to make one more attempt to enlist, this time as a common soldier. Wiener wrote in a letter to his parents, I should consider myself a pretty kind of a swine if I were willing to be an officer. This time the army accepted Wiener into its ranks and assigned him, by coincidence, to a unit stationed at Aberdeen, World War I ended just days after Wieners return to Aberdeen and Wiener was discharged from the military in February 1919
12.
Principia Mathematica
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The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910,1912, and 1913. In 1925–27, it appeared in an edition with an important Introduction to the Second Edition. PM was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this project is of great importance in the history of mathematics and philosophy. One of the inspirations and motivations for PM was the earlier work of Gottlob Frege on logic. PM sought to avoid this problem by ruling out the creation of arbitrary sets. This was achieved by replacing the notion of a set with the notion of a hierarchy of sets of different types. Contemporary mathematics, however, avoids paradoxes such as Russells in less unwieldy ways, PM is not to be confused with Russells 1903 The Principles of Mathematics. PM states, The present work was intended by us to be comprised in a second volume of Principles of Mathematics. PM has long known for its typographical complexity. Famously, several hundred pages are required in PM to prove the validity of the proposition 1+1=2, the Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century. The Principia covered only set theory, cardinal numbers, ordinal numbers and it was also clear how lengthy such a development would be. A fourth volume on the foundations of geometry had been planned, as noted in the criticism of the theory by Kurt Gödel, unlike a formalist theory, the logicistic theory of PM has no precise statement of the syntax of the formalism. A raw formalist theory would not provide the meaning of the symbols that form a primitive proposition—the symbols themselves could be absolutely arbitrary, the theory would specify only how the symbols behave based on the grammar of the theory. Then later, by assignment of values, a model would specify an interpretation of what the formulas are saying. Thus in the formal Kleene symbol set below, the interpretation of what the symbols commonly mean, but this is not a pure Formalist theory. The following formalist theory is offered as contrast to the theory of PM. A contemporary formal system would be constructed as follows, Symbols used, This set is the starting set, symbol strings, The theory will build strings of these symbols by concatenation
13.
Felix Hausdorff
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Life became difficult for Hausdorff and his family after Kristallnacht in 1938. The next year he initiated efforts to emigrate to the United States and he was an educated man and had become a Morenu at the age of 14. There are several treatises from his pen, including a work on the Aramaic translations of the Bible from the perspective of Talmudic law. Hausdorffs mother, Hedwig, who is referred to in various documents as Johanna. From another branch of this family came Hermann Tietz, founder of the first department store, during the period of Nazi dictatorship the name was Aryanised to Hertie. From 1878 to Felix Hausdorff attended the Nicolai School in Leipzig and he was an excellent student, class leader for many years and often recited self-written Latin or German poems at school celebrations. In his graduation in 1887, he was the one who reached the highest grade. The choice of subject was not easy for Hausdorff, the decision was made to study the sciences in high school. From summer term 1887 to summer semester 1891 Hausdorff studied mathematics and astronomy, mainly in his city of Leipzig, interrupted by one semester in Freiburg. In Leipzig he heard lectures on the history of music from musicologist Paul and his early love of music lasted a lifetime, in Hausdorffs house there were impressive musical evenings with the landlord at the piano, according to witness statements made by various participants. Even as a student in Leipzig, he was an admirer, in later semesters of his studies, Hausdorff was close to Heinrich Bruns. Bruns was professor of astronomy and director of the observatory at the University of Leipzig, under him, Hausdorff graduated in 1891 with a work on the theory of astronomical refraction of light in the atmosphere. Two publications on the subject followed, and in 1895 his habilitation also followed with a thesis on the absorbance of light in the atmosphere. These early astronomical works of Hausdorff have — despite their excellent mathematical working through — not gained importance, firstly, the underlying idea of Bruns has not proved viable. On the other hand, the progress in the measurement of atmospheric data has soon made the painstaking accuracy of this data from refraction observations unnecessary. In the time between PhD and habilitation Hausdorff completed the military requirement and worked for two years as a human computer at the observatory in Leipzig. With his habilitation, Hausdorff became a lecturer at the University of Leipzig, in addition to teaching and research in mathematics, he went with his literary and philosophical inclinations. Hausdorff married Charlotte Goldschmidt in 1899, daughter of Jewish doctor Siegismund Goldschmidt and her stepmother was the famous suffragist and preschool teacher Henriette Goldschmidt
14.
Kazimierz Kuratowski
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Kazimierz Kuratowski was a Polish mathematician and logician. He was one of the representatives of the Warsaw School of Mathematics. Kazimierz Kuratowski was born in Warsaw, Vistula Land, on 2 February 1896 and he was a son of Marek Kuratow, a barrister, and Róża Karzewska. He completed a Warsaw secondary school, which was named after general Paweł Chrzanowski, in 1913, he enrolled in an engineering course at the University of Glasgow in Scotland, in part because he did not wish to study in Russian, instruction in Polish was prohibited. He completed only one year of study when the outbreak of World War I precluded any further enrollment, in 1915, Russian forces withdrew from Warsaw and Warsaw University was reopened with Polish as the language of instruction. Kuratowski restarted his university education there the year, this time in mathematics. He obtained his Ph. D. in 1921, in newly independent Poland, in autumn 1921 Kuratowski was awarded the Ph. D. degree for his groundbreaking work. His thesis statement consisted of two parts, one was devoted to an axiomatic construction of topology via the closure axioms. This first part has been cited in hundreds of scientific articles, the second part of Kuratowskis thesis was devoted to continua irreducible between two points. This was the subject of a French doctoral thesis written by Zygmunt Janiszewski, since Janiszewski was deceased, Kuratowskis supervisor was Wacław Sierpiński. Kuratowskis thesis solved certain problems in set theory raised by a Belgian mathematician, Charles-Jean Étienne Gustave Nicolas, two years later, in 1923, Kuratowski was appointed deputy professor of mathematics at Warsaw University. He was then appointed a professor of mathematics at Lwów Polytechnic in Lwów. He was the head of the Mathematics department there until 1933, Kuratowski was also dean of the department twice. Kuratowski left Lwów for Warsaw in 1934, before the famous Scottish Book was begun and he did however, collaborate closely with Banach in solving important problems in measure theory. In 1934 he was appointed the professor at Warsaw University, a year later Kuratowski was nominated as the head of Mathematics Department there. From 1936 to 1939 he was secretary of the Mathematics Committee in The Council of Science, during World War II, he gave lectures at the underground university in Warsaw, since higher education for Poles was forbidden under German occupation. In February 1945, Kuratowski started to lecture at the reopened Warsaw University, in 1952 he became a member of the Polish Academy of Sciences, of which he was the vice-president from 1957 to 1968. After World War II, Kuratowski was actively involved in the rebuilding of scientific life in Poland and he helped to establish the State Mathematical Institute, which was incorporated into the Polish Academy of Sciences in 1952
15.
Intersection (set theory)
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In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B, but no other elements. For explanation of the used in this article, refer to the table of mathematical symbols. The intersection of A and B is written A ∩ B, formally, A ∩ B = that is x ∈ A ∩ B if and only if x ∈ A and x ∈ B. For example, The intersection of the sets and is, the number 9 is not in the intersection of the set of prime numbers and the set of odd numbers. More generally, one can take the intersection of sets at once. The intersection of A, B, C, and D, Intersection is an associative operation, thus, A ∩ = ∩ C. Additionally, intersection is commutative, thus A ∩ B = B ∩ A, inside a universe U one may define the complement Ac of A to be the set of all elements of U not in A. We say that A intersects B if A intersects B at some element, a intersects B if their intersection is inhabited. We say that A and B are disjoint if A does not intersect B, in plain language, they have no elements in common. A and B are disjoint if their intersection is empty, denoted A ∩ B = ∅, for example, the sets and are disjoint, the set of even numbers intersects the set of multiples of 3 at 0,6,12,18 and other numbers. The most general notion is the intersection of a nonempty collection of sets. If M is a nonempty set whose elements are themselves sets, then x is an element of the intersection of M if, the notation for this last concept can vary considerably. Set theorists will sometimes write ⋂M, while others will instead write ⋂A∈M A, the latter notation can be generalized to ⋂i∈I Ai, which refers to the intersection of the collection. Here I is a nonempty set, and Ai is a set for every i in I. In the case that the index set I is the set of numbers, notation analogous to that of an infinite series may be seen. When formatting is difficult, this can also be written A1 ∩ A2 ∩ A3 ∩, even though strictly speaking, A1 ∩ (A2 ∩ (A3 ∩. Finally, let us note that whenever the symbol ∩ is placed before other symbols instead of them, it should be of a larger size. Note that in the section we excluded the case where M was the empty set
16.
Union (set theory)
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In set theory, the union of a collection of sets is the set of all elements in the collection. It is one of the operations through which sets can be combined and related to each other. For explanation of the used in this article, refer to the table of mathematical symbols. The union of two sets A and B is the set of elements which are in A, in B, for example, if A = and B = then A ∪ B =. Sets cannot have duplicate elements, so the union of the sets and is, multiple occurrences of identical elements have no effect on the cardinality of a set or its contents. Binary union is an operation, that is, A ∪ = ∪ C. The operations can be performed in any order, and the parentheses may be omitted without ambiguity, similarly, union is commutative, so the sets can be written in any order. The empty set is an identity element for the operation of union and that is, A ∪ ∅ = A, for any set A. This follows from analogous facts about logical disjunction, since sets with unions and intersections form a Boolean algebra, intersection distributes over union A ∩ = ∪ and union distributes over intersection A ∪ = ∩. One can take the union of several sets simultaneously, for example, the union of three sets A, B, and C contains all elements of A, all elements of B, and all elements of C, and nothing else. Thus, x is an element of A ∪ B ∪ C if and only if x is in at least one of A, B, and C. In mathematics a finite union means any union carried out on a number of sets. The most general notion is the union of a collection of sets. If M is a set whose elements are themselves sets, then x is an element of the union of M if, in symbols, x ∈ ⋃ M ⟺ ∃ A ∈ M, x ∈ A. This idea subsumes the preceding sections, in that A ∪ B ∪ C is the union of the collection, also, if M is the empty collection, then the union of M is the empty set. The notation for the concept can vary considerably. For a finite union of sets S1, S2, S3, …, S n one often writes S1 ∪ S2 ∪ S3 ∪ ⋯ ∪ S n or ⋃ i =1 n S i. In the case that the index set I is the set of natural numbers, whenever the symbol ∪ is placed before other symbols instead of between them, it is of a larger size
17.
Braces (punctuation)
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A bracket is a tall punctuation mark typically used in matched pairs within text, to set apart or interject other text. The matched pair may be described as opening and closing, or left, forms include round, square, curly, and angle brackets, and various other pairs of symbols. Chevrons were the earliest type of bracket to appear in written English, desiderius Erasmus coined the term lunula to refer to the rounded parentheses, recalling the shape of the crescent moon. Some of the names are regional or contextual. Sometimes referred to as angle brackets, in cases as HTML markup. Occasionally known as broken brackets or brokets, ⸤ ⸥, ｢ ｣ – corner brackets ⟦ ⟧ – double square brackets, white square brackets Guillemets, ‹ › and « », are sometimes referred to as chevrons or angle brackets. The characters ‹ › and « », known as guillemets or angular quote brackets, are actually quotation mark glyphs used in several European languages, which one of each pair is the opening quote mark and which is the closing quote varies between languages. In English, typographers generally prefer to not set brackets in italics, however, in other languages like German, if brackets enclose text in italics, they are usually set in italics too. Parentheses /pəˈrɛnθᵻsiːz/ contain material that serves to clarify or is aside from the main point, a milder effect may be obtained by using a pair of commas as the delimiter, though if the sentence contains commas for other purposes, visual confusion may result. In American usage, parentheses are considered separate from other brackets. Parentheses may be used in writing to add supplementary information. They can also indicate shorthand for either singular or plural for nouns and it can also be used for gender neutral language, especially in languages with grammatical gender, e. g. he agreed with his physician. Parenthetical phrases have been used extensively in informal writing and stream of consciousness literature, examples include the southern American author William Faulkner as well as poet E. E. Cummings. Parentheses have historically been used where the dash is used in alternatives, such as parenthesis) is used to indicate an interval from a to c that is inclusive of a. That is, [5, 12) would be the set of all numbers between 5 and 12, including 5 but not 12. The numbers may come as close as they like to 12, including 11.999 and so forth, in some European countries, the notation [5, 12[ is also used for this. The endpoint adjoining the bracket is known as closed, whereas the endpoint adjoining the parenthesis is known as open, if both types of brackets are the same, the entire interval may be referred to as closed or open as appropriate. Whenever +∞ or −∞ is used as an endpoint, it is considered open
18.
Ordinal number
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In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another. Any finite collection of objects can be put in order just by the process of counting, labeling the objects with distinct whole numbers, Ordinal numbers are thus the labels needed to arrange collections of objects in order. An ordinal number is used to describe the type of a well ordered set. Whereas ordinals are useful for ordering the objects in a collection, they are distinct from cardinal numbers, although the distinction between ordinals and cardinals is not always apparent in finite sets, different infinite ordinals can describe the same cardinal. Like other kinds of numbers, ordinals can be added, multiplied, a natural number can be used for two purposes, to describe the size of a set, or to describe the position of an element in a sequence. When restricted to finite sets these two concepts coincide, there is one way to put a finite set into a linear sequence. This is because any set has only one size, there are many nonisomorphic well-orderings of any infinite set. Whereas the notion of number is associated with a set with no particular structure on it. A well-ordered set is an ordered set in which there is no infinite decreasing sequence, equivalently. Ordinals may be used to label the elements of any given well-ordered set and this length is called the order type of the set. Any ordinal is defined by the set of ordinals that precede it, in fact, the most common definition of ordinals identifies each ordinal as the set of ordinals that precede it. For example, the ordinal 42 is the type of the ordinals less than it, i. e. the ordinals from 0 to 41. Conversely, any set of ordinals that is downward-closed—meaning that for any ordinal α in S and any ordinal β < α, β is also in S—is an ordinal. There are infinite ordinals as well, the smallest infinite ordinal is ω, which is the type of the natural numbers. After all of these come ω·2, ω·2+1, ω·2+2, and so on, then ω·3, now the set of ordinals formed in this way must itself have an ordinal associated with it, and that is ω2. Further on, there will be ω3, then ω4, and so on, and ωω, then ωωω, then later ωωωω and this can be continued indefinitely far. The smallest uncountable ordinal is the set of all countable ordinals, in a well-ordered set, every non-empty subset contains a distinct smallest element. Given the axiom of dependent choice, this is equivalent to just saying that the set is ordered and there is no infinite decreasing sequence
19.
Mizar system
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The system is maintained and developed by the Mizar Project, formerly under the direction of its founder Andrzej Trybulec. In 2009 the Mizar Mathematical Library was the largest coherent body of formalized mathematics in existence. The Mizar Project was started around 1973 by Andrzej Trybulec as an attempt to reconstruct mathematical vernacular so it can be checked by a computer and this is in-line with the influential QED manifesto. Currently the project is developed and maintained by groups at Białystok University, Poland, the University of Alberta, Canada. While the Mizar proof checker remains proprietary, the Mizar Mathematical Library—the sizable body of formalized mathematics that it verified—is licensed open-source, papers related to the Mizar system regularly appear in the peer-reviewed journals of the mathematic formalization academic community. These include Studies in Logic, Grammar and Rhetoric, Intelligent Computer Mathematics, Interactive Theorem Proving, Journal of Automated Reasoning, the distinctive feature of the Mizar language is its readability. As is common in text, it relies on classical logic. Yet, the language enables the increased level of formality necessary for automated proof checking, for a proof to be admitted, all steps have to be justified either by elementary logical arguments or by citing previously verified proofs. This results in a level of rigor and detail than is customary in mathematical textbooks. Thus, a typical Mizar article is about four times as long as an equivalent paper written in ordinary style, formalization is relatively labor-intensive, but not impossibly difficult. Once one is versed in the system, it takes one week of full-time work to have a textbook page formally verified. This suggests that its benefits are now in the reach of applied fields such as probability theory, the Mizar Mathematical Library includes all theorems to which authors can refer in newly written articles. Once approved by the proof checker they are evaluated in a process of peer-review for appropriate contribution. If accepted they are published in the associated Journal of Formalized Mathematics, as of July 2012, the MML included 1150 articles written by 241 authors. In aggregate, these more than 10,000 formal definitions of mathematical objects. More than 180 named mathematical facts have so benefited from formal codification, some examples are the Hahn–Banach theorem, Königs lemma, Brouwer fixed point theorem, Gödels completeness theorem and Jordan curve theorem. This breadth of coverage has led some to suggest Mizar as one of the approximations to the QED utopia of encoding all core mathematics in computer verifiable form. All MML articles are available in PDF form as the papers of the Journal of Formalized Mathematics, the full text of the MML is distributed with the Mizar checker and can be freely downloaded from the Mizar website
20.
Willard Van Orman Quine
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Willard Van Orman Quine was an American philosopher and logician in the analytic tradition, recognized as one of the most influential philosophers of the twentieth century. He filled the Edgar Pierce Chair of Philosophy at Harvard from 1956 to 1978, a 2009 poll conducted among analytic philosophers named Quine as the fifth most important philosopher of the past two centuries. This led to his famous quip that philosophy of science is philosophy enough, in philosophy of mathematics, he and his Harvard colleague Hilary Putnam developed the Quine–Putnam indispensability thesis, an argument for the reality of mathematical entities. According to his autobiography, The Time of My Life, Quine grew up in Akron, Ohio and his father, Cloyd R. was a manufacturing entrepreneur and his mother, Harriett E. was a schoolteacher and later a housewife. He received his B. A. in mathematics from Oberlin College in 1930 and his thesis supervisor was Alfred North Whitehead. He was then appointed a Harvard Junior Fellow, which excused him from having to teach for four years. During the academic year 1932–33, he travelled in Europe thanks to a Sheldon fellowship, meeting Polish logicians and members of the Vienna Circle and it was through Quines good offices that Tarski was invited to attend the September 1939 Unity of Science Congress in Cambridge. To attend that Congress, Tarski sailed for the USA on the last ship to leave Danzig before the Third Reich invaded Poland, Tarski survived the war and worked another 44 years in the USA. For the academic year 1964–1965, Quine was a fellow on the faculty in the Center for Advanced Studies at Wesleyan University, in 1980 Quine received an honorary doctorate from the Faculty of Humanities at Uppsala University, Sweden. Quine was an atheist when he was a teenager and he had four children by two marriages. Guitarist Robert Quine was his nephew, Quine was politically conservative, but the bulk of his writing was in technical areas of philosophy removed from direct political issues. Quines Ph. D. thesis and early publications were on formal logic, only after World War II did he, by virtue of seminal papers on ontology, epistemology and language, emerge as a major philosopher. By the 1960s, he had worked out his naturalized epistemology whose aim was to answer all questions of knowledge and meaning using the methods. Quine roundly rejected the notion that there should be a first philosophy and these views are intrinsic to his naturalism. Quine could lecture in French, Spanish, Portuguese and German, like the logical positivists, Quine evinced little interest in the philosophical canon, only once did he teach a course in the history of philosophy, on Hume. This distinction was central to logical positivism, like other Analytic philosophers before him, Quine accepted the definition of analytic as true in virtue of meaning alone. Unlike them, however, he concluded that ultimately the definition was circular, in other words, Quine accepted that analytic statements are those that are true by definition, then argued that the notion of truth by definition was unsatisfactory. Quines chief objection to analyticity is with the notion of synonymy, the objection to synonymy hinges upon the problem of collateral information
21.
Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
22.
Image (mathematics)
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In mathematics, an image is the subset of a functions codomain which is the output of the function from a subset of its domain. Evaluating a function at each element of a subset X of the domain, the inverse image or preimage of a particular subset S of the codomain of a function is the set of all elements of the domain that map to the members of S. Image and inverse image may also be defined for binary relations. The word image is used in three related ways, in these definitions, f, X → Y is a function from the set X to the set Y. If x is a member of X, then f = y is the image of x under f, Y is alternatively known as the output of f for argument x. The image of a subset A ⊆ X under f is the subset f ⊆ Y defined by, f = When there is no risk of confusion and this convention is a common one, the intended meaning must be inferred from the context. This makes the image of f a function whose domain is the set of X. The image f of the entire domain X of f is called simply the image of f, let f be a function from X to Y. The set of all the fibers over the elements of Y is a family of sets indexed by Y, for example, for the function f = x2, the inverse image of would be. Again, if there is no risk of confusion, we may denote f −1 by f −1, the notation f −1 should not be confused with that for inverse function. The notation coincides with the one, though, for bijections. The traditional notations used in the section can be confusing. {\displaystyle f=\left\ The image of the set under f is f =, the image of the function f is. The preimage of a is f −1 =, the preimage of is the empty set. F, R → R defined by f = x2, the image of under f is f =, and the image of f is R+. The preimage of f is f −1 =. The preimage of set N = under f is the empty set, F, R2 → R defined by f = x2 + y2. The fibres f −1 are concentric circles about the origin, the origin itself, and the empty set, depending on whether a >0, a =0, or a <0, respectively
23.
Cardinality
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In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = contains 3 elements, there are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is called its size, when no confusion with other notions of size is possible. The cardinality of a set A is usually denoted | A |, with a bar on each side, this is the same notation as absolute value. Alternatively, the cardinality of a set A may be denoted by n, A, card, while the cardinality of a finite set is just the number of its elements, extending the notion to infinite sets usually starts with defining the notion of comparison of arbitrary sets. Two sets A and B have the same cardinality if there exists a bijection, that is, such sets are said to be equipotent, equipollent, or equinumerous. This relationship can also be denoted A≈B or A~B, for example, the set E = of non-negative even numbers has the same cardinality as the set N = of natural numbers, since the function f = 2n is a bijection from N to E. A has cardinality less than or equal to the cardinality of B if there exists a function from A into B. A has cardinality less than the cardinality of B if there is an injective function. If | A | ≤ | B | and | B | ≤ | A | then | A | = | B |, the axiom of choice is equivalent to the statement that | A | ≤ | B | or | B | ≤ | A | for every A, B. That is, the cardinality of a set was not defined as an object itself. However, such an object can be defined as follows, the relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets. The equivalence class of a set A under this relation then consists of all sets which have the same cardinality as A. There are two ways to define the cardinality of a set, The cardinality of a set A is defined as its class under equinumerosity. A representative set is designated for each equivalence class, the most common choice is the initial ordinal in that class. This is usually taken as the definition of number in axiomatic set theory. Assuming AC, the cardinalities of the sets are denoted ℵ0 < ℵ1 < ℵ2 < …. For each ordinal α, ℵ α +1 is the least cardinal number greater than ℵ α
24.
Anthony Morse
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Anthony Perry Morse was an American mathematician who worked in both analysis, especially measure theory, and in the foundations of mathematics. He is best known as the co-creator, together with John L. Kelley and this theory first appeared in print in Kelleys General Topology. Morses own version appeared later in A Theory of Sets and he is also known for his work on the Morse–Sard theorem and the Federer–Morse theorem. Anthony Morse should not be confused with Marston Morse, known for developing Morse Theory and he received his PhD in 1937 at Brown University with C. R. Adams as thesis advisor. After two years at the Institute for Advanced Study he joined the faculty at Berkeley where except for two interruptions he worked for the rest of his life on mathematics. In the first of these, from 1943 until the end of World War II, in 1950 his life was interrupted by the McCarthy Era loyalty oath controversy. He was one of 29 non-signers, but he was also one of 6 who took advantage of a 10-day grace period to sign, while continuing to refer to the remaining non-signers as patriots. His work is characterized by a degree of formality. His students include Herbert Federer and Woody Bledsoe, obituary A. P. Morse at the Mathematics Genealogy Project
25.
Product (category theory)
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Essentially, the product of a family of objects is the most general object which admits a morphism to each of the given objects. Let C be a category with some objects X1 and X2, the morphisms π1 and π2 are called the canonical projections or projection morphisms. Above we defined the binary product, instead of two objects we can take an arbitrary family of objects indexed by some set I. Then we obtain the definition of a product, × Xn and the product of morphisms is denoted < f1. Alternatively, the product may be defined through equations, so, for example, for the binary product, Existence of f is guaranteed by existence of the operation < -, - >. Commutativity of the diagrams above is guaranteed by the equality ∀f1, ∀f2, ∀i∈, πi o < f1, uniqueness of f is guaranteed by the equality ∀g, Y → X, < π1 o g, π2 o g > = g. The product is a case of a limit. This may be seen by using a category as the diagram required for the definition of the limit. The discrete objects will serve as the index of the components, if we regard this diagram as a functor, it is a functor from the index set I considered as a discrete category. The definition of the product then coincides with the definition of the limit, i being a cone, just as the limit is a special case of the universal construction, so is the product. Starting with the definition given for the property of limits. The diagonal functor Δ, C → C × C assigns to each object X the ordered pair, the product X1 × X2 in C is given by a universal morphism from the functor Δ to the object in C × C. This universal morphism consists of an object X of C and a morphism → which contains projections, in the category of sets, the product is the cartesian product. Given a family of sets Xi the product is defined as Πi∈I Xi, = with the canonical projections πj, Πi∈I Xi → Xj, πj, = xj. Given any set Y with a family of functions fi, Y → Xi, other examples, In the category of topological spaces, the product is the space whose underlying set is the cartesian product and which carries the product topology. The product topology is the coarsest topology for which all the projections are continuous, in the category of modules over some ring R, the product is the cartesian product with addition defined componentwise and distributive multiplication. In the category of groups, the product is the product of groups given by the cartesian product with multiplication defined componentwise. In the category of relations, the product is given by the disjoint union, in the category of algebraic varieties, the categorical product is given by the Segre embedding
26.
Universal property
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In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and this article gives a general treatment of universal properties. Before giving a definition of universal properties, we offer some motivation for studying such constructions. Proofs often become short and elegant if the property is used rather than the concrete details. For example, the algebra of a vector space is slightly painful to actually construct. Universal properties define objects uniquely up to a unique isomorphism, therefore, one strategy to prove that two objects are isomorphic is to show that they satisfy the same universal property. Universal constructions are functorial in nature, if one can carry out the construction for every object in a category C then one obtains a functor on C, furthermore, this functor is a right or left adjoint to the functor U used in the definition of the universal property. Universal properties occur everywhere in mathematics, by understanding their abstract properties, one obtains information about all these constructions and can avoid repeating the same analysis for each individual instance. Suppose that U, D → C is a functor from a category D to a category C, consider the following dual notions, An initial morphism from X to U is an initial object in the category of morphisms from X to U. In each definition, the existence of the morphism g intuitively expresses the fact that is general enough, since the notions of initial and terminal are dual, it is often enough to discuss only one of them, and simply reverse arrows in C for the dual discussion. Alternatively, the universal is often used in place of both words. Note, some authors may call one of these constructions a universal morphism. Which is which depends on the author, although in order to be consistent with the naming of limits and colimits the latter construction should be named universal and this article uses the unambiguous terminology of initial and terminal objects. Below are a few examples, to highlight the general idea, the reader can construct numerous other examples by consulting the articles mentioned in the introduction. Let C be the category of vector spaces K-Vect over a field K, let U, K-Alg → K-Vect be the forgetful functor which assigns to each algebra its underlying vector space. Given any vector space V over K we can construct the tensor algebra T of V, the tensor algebra is characterized by the fact, “Any linear map from V to an algebra A can be uniquely extended to an algebra homomorphism from T to A. Since this construction works for any vector space V, we conclude that T is a functor from K-Vect to K-Alg and this means that T is left adjoint to the forgetful functor U. A categorical product can be characterized by a terminal property, for concreteness, one may consider the Cartesian product in Set, the direct product in Grp, or the product topology in Top, where products exist
27.
Natural transformation
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In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a transformation can be considered to be a morphism of functors. Indeed, this intuition can be formalized to define so-called functor categories, natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications. If F and G are functors between the categories C and D, then a natural transformation η from F to G is a family of morphisms that satisfies two requirements. The natural transformation must associate to every object X in C a morphism ηX, the morphism ηX is called the component of η at X. If η is a transformation from F to G, we also write η, F → G or η, F ⇒ G. Two functors F and G are called naturally isomorphic or simply if there exists a natural isomorphism from F to G. An infranatural transformation η from F to G is simply a family of morphisms ηX, thus a natural transformation is an infranatural transformation for which ηY ∘ F = G ∘ ηX for every morphism f, X → Y. The naturalizer of η, nat, is the largest subcategory of C containing all the objects of C on which η restricts to a natural transformation, statements such as Every group is naturally isomorphic to its opposite group abound in modern mathematics. We will now give the meaning of this statement as well as its proof. Consider the category Grp of all groups with group homomorphisms as morphisms, if is a group, we define its opposite group as follows, Gop is the same set as G, and the operation ∗op is defined by a ∗op b = b ∗ a. All multiplications in Gop are thus turned around, forming the opposite group becomes a functor from Grp to Grp if we define fop = f for any group homomorphism f, G → H. Note that fop is indeed a group homomorphism from Gop to Hop, fop = f = f ∗ f = fop ∗op fop. The content of the statement is, The identity functor IdGrp, Grp → Grp is naturally isomorphic to the opposite functor op. To prove this, we need to provide isomorphisms ηG, G → Gop for every group G, the formulas −1 = b−1 a−1 and −1 = a show that ηG is a group homomorphism with inverse ηGop. To prove the naturality, we start with a homomorphism f, G → H and show ηH ∘ f = fop ∘ ηG. This is true since fop = f and every group homomorphism has the property −1 = f, given a group G, we can define its abelianization Gab = G/. Let πG, G → Gab denote the map onto the cosets of
28.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
29.
Word and Object
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The thought experiment of radical translation and the accompanying notion of indeterminacy of translation are original to Word and Object, which is Quines most famous book. Quine emphasizes his naturalism, the doctrine that philosophy should be pursued as part of natural science, central to Quine’s philosophy is his linguistic behaviorism. Quine has remarked that one may or may not choose to be a behaviorist in psychology and this influence can be seen in Word and Object. In chapter 2 a linguist has to translate a natives unknown language into English, what is so specifically behavioristic there is that the linguist has nothing to go on but verbal behavior from the native and the visible environment the native interacts with. The same view is displayed in chapter 3 where Quine describes how a baby learns its first words, in this chapter Quine also mentions B. F. Skinner, a well known behaviorist, as one of his influences. In the second chapter of Word and Object, Quine investigates the concept of meaning and he shows to what extent his own, empirical, notion of meaning can give an account for our intuitive concept of meaning, what a sentence shares with its translation. The linguist can thus only use empirical information, therefore, radical translation will tell us which part of our language can be accounted for by stimulus conditions, in the experiment, Quine assumes that functional Jungle equivalents of Yes and No are relatively easy to be found. This allows the linguist to actively query the utterances of the natives, by repeating words he has heard the native utter, for example, if the linguist sees a rabbit, and the native says ‘Gavagai’, the linguist may think that ‘Gavagai’ means ‘Rabbit’. He will then try the sentence ‘Gavagai’ in different situations caused by the stimulation of a rabbit, the native’s reaction is elicited by the linguist’s question and the prompting stimulation together. It is the stimulation that prompts the assent or dissent, not the object in the world, because an object in the world can be replaced by a replica, but then the stimulation stays the same. The class of all the stimulations that would prompt his assent. is the stimulus meaning of a certain sentence for a given speaker. Negative stimulus meaning is defined likewise, with assent and dissent interchanged, Quine calls these affirmative and negative stimulus meaning combined the stimulus meaning of the sentence. Once the stimulus meaning has been found, the linguist can then compare it to the meanings of sentences in English. The English sentence with identical meaning to Gavagai functions as a translation of Gavagai. After Quine has set out the concept of meaning, he continues by comparing it with our intuitive notion of meaning. For this, he distinguished two kinds of sentences, occasion sentences and standing sentences, occasion sentences are the sentences that are only affirmed or dissented after an appropriate stimulation, e. g. Look, a rabbit walks by. On the other hand, there are standing sentences, which do not rely on stimulation for assent or dissent, they can be prompted by stimulation, thus, the stimulus meaning is less useful to approximate the intuitive meaning of standing sentences. However, the difference between occasion and standing sentences is only a gradual difference and this difference depends on the modulus because ‘an occasion sentence modulo n seconds can be a standing sentence modulo n – 1’
![{\displaystyle \varphi (x):=\sigma [x]=\{\sigma (\alpha )|\alpha \in x\}=(x\setminus \mathbb {N} )\cup \{n+1:n\in (x\cap \mathbb {N} )\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c033c1194a787ecfe3644898aeaf94109bff6004)
![{\displaystyle \psi (x):=\sigma [x]\cup \{0\}=\varphi (x)\cup \{0\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3ca1d6cd2933c7de1458ca2adb006bdc27503cf)
![{\displaystyle (A,B):=\varphi [A]\cup \psi [B]=\{\varphi (a):a\in A\}\cup \{\varphi (b)\cup \{0\}:b\in B\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb6ca4663acb45c04927b9af820b2b60125ae381)
