1.
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
2.
David Hilbert
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David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th, Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis, Hilbert adopted and warmly defended Georg Cantors set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed significantly to establishing rigor and developed important tools used in mathematical physics. Hilbert is known as one of the founders of theory and mathematical logic. In late 1872, Hilbert entered the Friedrichskolleg Gymnasium, but, after a period, he transferred to. Upon graduation, in autumn 1880, Hilbert enrolled at the University of Königsberg, in early 1882, Hermann Minkowski, returned to Königsberg and entered the university. Hilbert knew his luck when he saw it, in spite of his fathers disapproval, he soon became friends with the shy, gifted Minkowski. In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius, Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen. Hilbert remained at the University of Königsberg as a Privatdozent from 1886 to 1895, in 1895, as a result of intervention on his behalf by Felix Klein, he obtained the position of Professor of Mathematics at the University of Göttingen. During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world and he remained there for the rest of his life. Among Hilberts students were Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, john von Neumann was his assistant. At the University of Göttingen, Hilbert was surrounded by a circle of some of the most important mathematicians of the 20th century, such as Emmy Noether. Between 1902 and 1939 Hilbert was editor of the Mathematische Annalen, good, he did not have enough imagination to become a mathematician. Hilbert lived to see the Nazis purge many of the prominent faculty members at University of Göttingen in 1933 and those forced out included Hermann Weyl, Emmy Noether and Edmund Landau. One who had to leave Germany, Paul Bernays, had collaborated with Hilbert in mathematical logic and this was a sequel to the Hilbert-Ackermann book Principles of Mathematical Logic from 1928. Hermann Weyls successor was Helmut Hasse, about a year later, Hilbert attended a banquet and was seated next to the new Minister of Education, Bernhard Rust
3.
Logic
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Logic, originally meaning the word or what is spoken, is generally held to consist of the systematic study of the form of arguments. A valid argument is one where there is a relation of logical support between the assumptions of the argument and its conclusion. Historically, logic has been studied in philosophy and mathematics, and recently logic has been studied in science, linguistics, psychology. The concept of form is central to logic. The validity of an argument is determined by its logical form, traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logic. Informal logic is the study of natural language arguments, the study of fallacies is an important branch of informal logic. Since much informal argument is not strictly speaking deductive, on some conceptions of logic, formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as an application of a wholly abstract rule, that is. The works of Aristotle contain the earliest known study of logic. Modern formal logic follows and expands on Aristotle, in many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuances of natural language, Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. Symbolic logic is divided into two main branches, propositional logic and predicate logic. Mathematical logic is an extension of logic into other areas, in particular to the study of model theory, proof theory, set theory. Logic is generally considered formal when it analyzes and represents the form of any valid argument type, the form of an argument is displayed by representing its sentences in the formal grammar and symbolism of a logical language to make its content usable in formal inference. Simply put, formalising simply means translating English sentences into the language of logic and this is called showing the logical form of the argument. It is necessary because indicative sentences of ordinary language show a variety of form. Second, certain parts of the sentence must be replaced with schematic letters, thus, for example, the expression all Ps are Qs shows the logical form common to the sentences all men are mortals, all cats are carnivores, all Greeks are philosophers, and so on. The schema can further be condensed into the formula A, where the letter A indicates the judgement all - are -, the importance of form was recognised from ancient times
4.
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
5.
Mathematical theorem
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In mathematics, a theorem is a statement that has been proved on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a consequence of the axioms. The proof of a theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises, however, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol. Although they can be written in a symbolic form, for example, within the propositional calculus. In some cases, a picture alone may be sufficient to prove a theorem, because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being trivial, or difficult, or deep and these subjective judgments vary not only from person to person, but also with time, for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a theorem may be simply stated. Fermats Last Theorem is a particularly well-known example of such a theorem, logically, many theorems are of the form of an indicative conditional, if A, then B. Such a theorem does not assert B, only that B is a consequence of A. In this case A is called the hypothesis of the theorem and B the conclusion. The theorem If n is an natural number then n/2 is a natural number is a typical example in which the hypothesis is n is an even natural number. To be proved, a theorem must be expressible as a precise, nevertheless, theorems are usually expressed in natural language rather than in a completely symbolic form, with the intention that the reader can produce a formal statement from the informal one. It is common in mathematics to choose a number of hypotheses within a given language and these hypotheses form the foundational basis of the theory and are called axioms or postulates. The field of known as proof theory studies formal languages, axioms. Some theorems are trivial, in the sense that they follow from definitions, axioms, a theorem might be simple to state and yet be deep
6.
Richard's paradox
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In logic, Richards paradox is a semantical antinomy of set theory and natural language first described by the French mathematician Jules Richard in 1905. The paradox is used to motivate the importance of distinguishing carefully between mathematics and metamathematics. The paradox was also a motivation of the development of predicative mathematics, the original statement of the paradox, due to Richard, is strongly related to Cantors diagonal argument on the uncountability of the set of real numbers. The paradox begins with the observation that certain expressions of natural language define real numbers unambiguously, while other expressions of natural language do not. For example, The real number the integer part of which is 17, = 1693/99, while the phrase the capital of England does not define a real number. Thus there is an infinite list of English phrases that define real numbers unambiguously, arrange this list by length and then order lexicographically and this yields an infinite list of the corresponding real numbers, r1, r2. Now define a new real number r as follows, the preceding two paragraphs are an expression in English that unambiguously defines a real number r. Thus r must be one of the numbers rn, however, r was constructed so that it cannot equal any of the rn. Richards paradox results in a contradiction, which must be analyzed to find an error. The proposed definition of the new real number r clearly includes a sequence of characters. However, the definition refers to definability-in-English itself, if it were possible to determine which English expressions actually do define a real number, and which do not, then the paradox would go through. Thus the resolution of Richards paradox is that there is not any way to determine exactly which English sentences are definitions of real numbers. That is, there is not any way to describe in a number of words how to tell whether an arbitrary English expression is a definition of a real number. A similar phenomenon occurs in formalized theories that are able to refer to their own syntax, say that a formula φ defines a real number if there is exactly one real number r such that φ holds. Then it is not possible to define, by ZFC, the set of all formulas that define real numbers. For, if it were possible to define this set, it would be possible to diagonalize over it to produce a new definition of a real number, following the outline of Richards paradox above. Note that the set of formulas that define real numbers may exist, as a set F and this is related to Tarskis indefinability theorem. The example of ZFC illustrates the importance of distinguishing the metamathematics of a system from the statements of the formal system itself
7.
Russell's paradox
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According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. Symbolically, Let R =, then R ∈ R ⟺ R ∉ R In 1908, two ways of avoiding the paradox were proposed, Russells type theory and the Zermelo set theory, the first constructed axiomatic set theory. Zermelos axioms went well beyond Gottlob Freges axioms of extensionality and unlimited set abstraction, Let us call a set abnormal if it is a member of itself, and normal otherwise. For example, take the set of all squares in the plane and that set is not itself a square in the plane, and therefore is not a member of the set of all squares in the plane. On the other hand, if we take the set that contains all non-. Now we consider the set of all sets, R. This leads to the conclusion that R is neither normal nor abnormal, then by existential instantiation and universal instantiation we have y ∈ y ⟺ y ∉ y a contradiction. Modifications to this axiomatic theory proposed in the 1920s by Abraham Fraenkel, Thoralf Skolem and this theory became widely accepted once Zermelos axiom of choice ceased to be controversial, and ZFC has remained the canonical axiomatic set theory down to the present day. ZFC does not assume that, for property, there is a set of all things satisfying that property. Rather, it asserts that any set X, any subset of X definable using first-order logic exists. The object R discussed above cannot be constructed in this fashion, in some extensions of ZFC, objects like R are called proper classes. ZFC is silent about types, although the hierarchy has a notion of layers that resemble types. Zermelo himself never accepted Skolems formulation of ZFC using the language of first-order logic and this 2nd order ZFC preferred by Zermelo, including axiom of foundation, allowed a rich cumulative hierarchy. Ferreirós writes that Zermelos layers are essentially the same as the types in the versions of simple TT offered by Gödel. One can describe the cumulative hierarchy into which Zermelo developed his models as the universe of a cumulative TT in which types are allowed. Thus, simple TT and ZFC could now be regarded as systems that talk essentially about the same intended objects, the main difference is that TT relies on a strong higher-order logic, while Zermelo employed second-order logic, and ZFC can also be given a first-order formulation. The first-order description of the hierarchy is much weaker, as is shown by the existence of denumerable models
8.
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
9.
Gottlob Frege
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Friedrich Ludwig Gottlob Frege was a German philosopher, logician, and mathematician. Considered a major figure in mathematics, he is responsible for the development of modern logic and he is also understood by many to be the father of analytic philosophy, where he concentrated on the philosophy of language and mathematics. Though largely ignored during his lifetime, Giuseppe Peano and Bertrand Russell introduced his work to generations of logicians. Frege was born in 1848 in Wismar, Mecklenburg-Schwerin and his father Carl Alexander Frege was the co-founder and headmaster of a girls high school until his death. In childhood, Frege encountered philosophies that would guide his future scientific career, Frege studied at a gymnasium in Wismar and graduated in 1869. His teacher Gustav Adolf Leo Sachse, who was a poet, played the most important role in determining Freges future scientific career, Frege matriculated at the University of Jena in the spring of 1869 as a citizen of the North German Confederation. In the four semesters of his studies he attended approximately twenty courses of lectures and his most important teacher was Ernst Karl Abbe. Abbe was more than a teacher to Frege, he was a trusted friend, after Freges graduation, they came into closer correspondence. His other notable university teachers were Christian Philipp Karl Snell, Hermann Karl Julius Traugott Schaeffer, Frege married Margarete Katharina Sophia Anna Lieseberg on 14 March 1887. Though his education and early work focused primarily on geometry. His Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle a/S, the Begriffsschrift broke new ground, including a rigorous treatment of the ideas of functions and variables. Previous logic had dealt with the constants and, or. Freges conceptual notation however can represent such inferences, one of Freges stated purposes was to isolate genuinely logical principles of inference, so that in the proper representation of mathematical proof, one would at no point appeal to intuition. If there was an element, it was to be isolated and represented separately as an axiom, from there on. Already in the 1879 Begriffsschrift important preliminary theorems, for example a generalized form of law of trichotomy, were derived within what Frege understood to be pure logic and this idea was formulated in non-symbolic terms in his The Foundations of Arithmetic. Later, in his Basic Laws of Arithmetic, Frege attempted to derive, by use of his symbolism, most of these axioms were carried over from his Begriffsschrift, though not without some significant changes. The one truly new principle was one he called the Basic Law V, the crucial case of the law may be formulated in modern notation as follows. Let denote the extension of the predicate Fx, i. e. the set of all Fs, then Basic Law V says that the predicates Fx and Gx have the same extension iff ∀x
10.
Begriffsschrift
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Begriffsschrift is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book. Begriffsschrift is usually translated as concept writing or concept notation, the title of the book identifies it as a formula language, modeled on that of arithmetic. Freges motivation for developing his formal approach to logic resembled Leibnizs motivation for his calculus ratiocinator, Frege went on to employ his logical calculus in his research on the foundations of mathematics, carried out over the next quarter century. The calculus contains the first appearance of quantified variables, and is essentially classical bivalent second-order logic with identity and it is bivalent in that sentences or formulas denote either True or False, second order because it includes relation variables in addition to object variables and allows quantification over both. The modifier with identity specifies that the language includes the identity relation, Frege presents his calculus using idiosyncratic two-dimensional notation, connectives and quantifiers are written using lines connecting formulas, rather than the symbols ¬, ∧, and ∀ in use today. For example, that judgement B materially implies judgement A, i. e, let signify that the third of those possibilities does not obtain, but one of the three others does. So if we negate, that means the third possibility is valid, i. e. we negate A, Frege declared nine of his propositions to be axioms, and justified them by arguing informally that, given their intended meanings, they express self-evident truths. – govern material implication, – negation, and identity, expresses Leibnizs indiscernibility of identicals, and asserts that identity is a reflexive relation. This rule is much harder to articulate precisely than the two preceding rules, and Frege invokes it in ways that are not obviously legitimate. The main results of the chapter, titled Parts from a general series theory. Frege applied the results from the Begriffsschrifft, including those on the ancestral of a relation, thus, if we take xRy to be the relation y = x +1, then 0R*y is the predicate y is a natural number. Says that if x, y, and z are natural numbers, then one of the following must hold, x < y, x = y and this is the so-called law of trichotomy. For a careful recent study of how the Begriffsschrift was reviewed in the German mathematical literature, some reviewers, especially Ernst Schröder, were on the whole favorable. Some vestige of Freges notation survives in the turnstile symbol ⊢ derived from his Urteilsstrich │, Frege used these symbols in the Begriffsschrift in the unified form ├─ for declaring that a proposition is true. In his later Grundgesetze he revises slightly his interpretation of the ├─ symbol, in Begriffsschrift the Definitionsdoppelstrich │├─ indicates that a proposition is a definition. Furthermore, the negation sign ¬ can be read as a combination of the horizontal Inhaltsstrich with a vertical negation stroke and this negation symbol was reintroduced by Arend Heyting in 1930 to distinguish intuitionistic from classical negation. It also appears in Gerhard Gentzens doctoral dissertation, in the Tractatus Logico Philosophicus, Ludwig Wittgenstein pays homage to Frege by employing the term Begriffsschrift as a synonym for logical formalism. Freges 1892 essay, Sense and Reference, recants some of the conclusions of the Begriffsschrifft about identity, ancestral relation Freges propositional calculus Gottlob Frege
11.
Bertrand Russell
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Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, writer, social critic, political activist and Nobel laureate. At various points in his life he considered himself a liberal, a socialist, and a pacifist and he was born in Monmouthshire into one of the most prominent aristocratic families in the United Kingdom. In the early 20th century, Russell led the British revolt against idealism and he is considered one of the founders of analytic philosophy along with his predecessor Gottlob Frege, colleague G. E. Moore, and protégé Ludwig Wittgenstein. He is widely held to be one of the 20th centurys premier logicians, with A. N. Whitehead he wrote Principia Mathematica, an attempt to create a logical basis for mathematics. His philosophical essay On Denoting has been considered a paradigm of philosophy, Russell mostly was a prominent anti-war activist, he championed anti-imperialism. Occasionally, he advocated preventive nuclear war, before the opportunity provided by the monopoly is gone. He went to prison for his pacifism during World War I, in 1950 Russell was awarded the Nobel Prize in Literature in recognition of his varied and significant writings in which he champions humanitarian ideals and freedom of thought. Bertrand Russell was born on 18 May 1872 at Ravenscroft, Trellech, Monmouthshire and his parents, Viscount and Viscountess Amberley, were radical for their times. Lord Amberley consented to his wifes affair with their childrens tutor, both were early advocates of birth control at a time when this was considered scandalous. Lord Amberley was an atheist and his atheism was evident when he asked the philosopher John Stuart Mill to act as Russells secular godfather, Mill died the year after Russells birth, but his writings had a great effect on Russells life. His paternal grandfather, the Earl Russell, had asked twice by Queen Victoria to form a government. The Russells had been prominent in England for several centuries before this, coming to power, Lady Amberley was the daughter of Lord and Lady Stanley of Alderley. Russell often feared the ridicule of his grandmother, one of the campaigners for education of women. Russell had two siblings, brother Frank, and sister Rachel, in June 1874 Russells mother died of diphtheria, followed shortly by Rachels death. In January 1876, his father died of bronchitis following a period of depression. Frank and Bertrand were placed in the care of their staunchly Victorian paternal grandparents and his grandfather, former Prime Minister Earl Russell, died in 1878, and was remembered by Russell as a kindly old man in a wheelchair. His grandmother, the Countess Russell, was the dominant family figure for the rest of Russells childhood, the countess was from a Scottish Presbyterian family, and successfully petitioned the Court of Chancery to set aside a provision in Amberleys will requiring the children to be raised as agnostics. Her favourite Bible verse, Thou shalt not follow a multitude to do evil, the atmosphere at Pembroke Lodge was one of frequent prayer, emotional repression, and formality, Frank reacted to this with open rebellion, but the young Bertrand learned to hide his feelings
12.
Thoralf Skolem
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Thoralf Albert Skolem was a Norwegian mathematician who worked in mathematical logic and set theory. Although Skolems father was a school teacher, most of his extended family were farmers. Skolem attended secondary school in Kristiania, passing the university examinations in 1905. He then entered Det Kongelige Frederiks Universitet to study mathematics, also taking courses in physics, chemistry, zoology, in 1913, Skolem passed the state examinations with distinction, and completed a dissertation titled Investigations on the Algebra of Logic. He also traveled with Birkeland to the Sudan to observe the zodiacal light, in 1916 he was appointed a research fellow at Det Kongelige Frederiks Universitet. In 1918, he became a Docent in Mathematics and was elected to the Norwegian Academy of Science, Skolem did not at first formally enroll as a Ph. D. candidate, believing that the Ph. D. was unnecessary in Norway. He later changed his mind and submitted a thesis in 1926, titled Some theorems about integral solutions to certain algebraic equations and his notional thesis advisor was Axel Thue, even though Thue had died in 1922. In 1927, he married Edith Wilhelmine Hasvold, Skolem continued to teach at Det kongelige Frederiks Universitet until 1930 when he became a Research Associate in Chr. This senior post allowed Skolem to conduct research free of administrative, in 1938, he returned to Oslo to assume the Professorship of Mathematics at the university. There he taught the courses in algebra and number theory. Skolems Ph. D. student Øystein Ore went on to a career in the USA, Skolem served as president of the Norwegian Mathematical Society, and edited the Norsk Matematisk Tidsskrift for many years. He was also the editor of Mathematica Scandinavica. After his 1957 retirement, he made trips to the United States. He remained intellectually active until his sudden and unexpected death, for more on Skolems academic life, see Fenstad. Skolem published around 180 papers on Diophantine equations, group theory, lattice theory and he mostly published in Norwegian journals with limited international circulation, so that his results were occasionally rediscovered by others. An example is the Skolem–Noether theorem, characterizing the automorphisms of simple algebras, Skolem published a proof in 1927, but Emmy Noether independently rediscovered it a few years later. Skolem was among the first to write on lattices, in 1912, he was the first to describe a free distributive lattice generated by n elements. In 1919, he showed that every lattice is distributive and, as a partial converse
13.
Emil Leon Post
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Emil Leon Post was a Polish-born American mathematician and logician. He is best known for his work in the field eventually became known as computability theory. Post was born in Augustów, Suwałki Governorate, Russian Empire into a Polish-Jewish family that immigrated to New York City in May 1904 and his parents were Arnold and Pearl Post. Post had been interested in astronomy, but at the age of twelve lost his arm in a car accident. This loss was a significant obstacle to being a professional astronomer and he decided to pursue mathematics, rather than astronomy. Post attended the Townsend Harris High School and continued on to graduate from City College of New York in 1917 with a B. S. in Mathematics. After completing his Ph. D. in mathematics at Columbia University, supervised by Cassius Jackson Keyser, Post then became a high school mathematics teacher in New York City. Post married Gertrude Singer in 1929, with whom he had a daughter, Post spent at most three hours a day on research on the advice of his doctor in order to avoid manic attacks, which he had been experiencing since his year at Princeton. In 1936, he was appointed to the department at the City College of New York. He died in 1954 of an attack following electroshock treatment for depression. Post also devised truth tables independently of Wittgenstein and C. S. Peirce, jean Van Heijenoorts well-known source book on mathematical logic reprinted Posts classic article setting out these results. While at Princeton, Post came very close to discovering the incompleteness of Principia Mathematica, Post initially failed to publish his ideas as he believed he needed a complete analysis for them to be accepted. In 1936, Post developed, independently of Alan Turing, a model of computation that was essentially equivalent to the Turing machine model. Intending this as the first of a series of models of equivalent power but increasing complexity, Post devised a method of auxiliary symbols by which he could canonically represent any Post-generative language, and indeed any computable function or set at all. The unsolvability of his Post correspondence problem turned out to be exactly what was needed to obtain unsolvability results in the theory of formal languages and this question, which became known as Posts problem, stimulated much research. It was solved in the affirmative in the 1950s by the introduction of the priority method in recursion theory. Post made a fundamental and still-influential contribution to the theory of polyadic, or n-ary and his major theorem showed that a polyadic group is the iterated multiplication of elements of a normal subgroup of a group, such that the quotient group is cyclic of order n −1. He also demonstrated that a group operation on a set can be expressed in terms of a group operation on the same set
14.
Alonzo Church
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Alonzo Church was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the calculus, Church–Turing thesis, proving the undecidability of the Entscheidungsproblem, Frege–Church ontology. Alonzo Church was born on June 14,1903, in Washington, D. C. where his father, the family later moved to Virginia after his father lost this position because of failing eyesight. With help from his uncle, also named Alonzo Church, he was able to attend the Ridgefield School for Boys in Ridgefield and he stayed at Princeton, earning a Ph. D. in mathematics in three years under Oswald Veblen. He married Mary Julia Kuczinski in 1925 and the couple had three children, Alonzo Church, Jr. Mary Ann and Mildred, after receiving his Ph. D. he taught briefly as an instructor at the University of Chicago and then received a two-year National Research Fellowship. This allowed him to attend Harvard University in 1927–1928 and then both University of Göttingen and University of Amsterdam the following year and he taught philosophy and mathematics at Princeton, 1929–1967, and at the University of California, Los Angeles, 1967–1990. He was a Plenary Speaker at the ICM in 1962 in Stockholm, a deeply religious person, he was a lifelong member of the Presbyterian church. He died in 1995 and was buried in Princeton Cemetery and this is known as Churchs theorem. His proof that Peano arithmetic is undecidable and his articulation of what has come to be known as the Church–Turing thesis. He was the editor of the Journal of Symbolic Logic. His creation of the lambda calculus, the lambda calculus emerged in his 1936 paper showing the unsolvability of the Entscheidungsproblem. This result preceded Alan Turings work on the problem, which also demonstrated the existence of a problem unsolvable by mechanical means. This resulted in the Church–Turing thesis, the lambda calculus influenced the design of the LISP programming language and functional programming languages in general. The Church encoding is named in his honor, many of Churchs doctoral students have led distinguished careers, including C. Anthony Anderson, Peter B. Andrews, George A. Barnard, David Berlinski, William W. Boone, Martin Davis, Alfred L. Foster, Leon Henkin, John G. Kemeny, Stephen C. Kochen, Maurice LAbbé, Isaac Malitz, Gary R. Mar, Michael O. Rabin, Nicholas Rescher, Hartley Rogers, Jr. J. Barkley Rosser, Dana Scott, Raymond Smullyan, a more complete list of Churchs students is available via Mathematics Genealogy Project. Alonzo Church, Introduction to Mathematical Logic Alonzo Church, The Calculi of Lambda-Conversion Alonzo Church, A Bibliography of Symbolic Logic, Introduction to the Collected Works of Alonzo Church, MIT Press, not yet published. In memoriam, Alonzo Church, The Bulletin of Symbolic Logic, wade, Nicholas, Alonzo Church,92, Theorist of the Limits of Mathematics, The New York Times, September 5,1995, p. B6
15.
Stephen Cole Kleene
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Stephen Cole Kleene /ˈkleɪniː/ KLAY-nee was an American mathematician. Kleenes work grounds the study of functions are computable. A number of concepts are named after him, Kleene hierarchy, Kleene algebra, the Kleene star, Kleenes recursion theorem. He also invented regular expressions, and made significant contributions to the foundations of mathematical intuitionism, although his last name is commonly pronounced /ˈkliːniː/ KLEE-nee or /ˈkliːn/ kleen, Kleene himself pronounced it /ˈkleɪniː/ KLAY-nee. His son, Ken Kleene, wrote, As far as I am aware this pronunciation is incorrect in all known languages, I believe that this novel pronunciation was invented by my father. Kleene was awarded the BA degree from Amherst College in 1930 and he was awarded the Ph. D. in mathematics from Princeton University in 1934. His thesis, entitled A Theory of Positive Integers in Formal Logic, was supervised by Alonzo Church, in the 1930s, he did important work on Churchs lambda calculus. In 1935, he joined the department at the University of Wisconsin–Madison. After two years as an instructor, he was appointed assistant professor in 1937, while a visiting scholar at the Institute for Advanced Study in Princeton, 1939–40, he laid the foundation for recursion theory, an area that would be his lifelong research interest. In 1941, he returned to Amherst College, where he spent one year as a professor of mathematics. During World War II, Kleene was a lieutenant commander in the United States Navy. He was an instructor of navigation at the U. S. Naval Reserves Midshipmens School in New York, in 1946, Kleene returned to Wisconsin, becoming a full professor in 1948 and the Cyrus C. MacDuffee professor of mathematics in 1964 and he was chair of the Department of Mathematics and Computer Science, 1962–63, and Dean of the College of Letters and Science from 1969 to 1974. The latter appointment he took on despite the considerable student unrest of the day and he retired from the University of Wisconsin in 1979. In 1999 the mathematics library at the University of Wisconsin was renamed in his honor, Kleenes teaching at Wisconsin resulted in three texts in mathematical logic, Kleene and Kleene and Vesley, often cited and still in print. Kleene wrote alternative proofs to the Gödels incompleteness theorems that enhanced their status and made them easier to teach. Kleene and Vesley is the classic American introduction to intuitionist logic, Kleene served as president of the Association for Symbolic Logic, 1956–58, and of the International Union of History and Philosophy of Science,1961. In 1990, he was awarded the National Medal of Science, the importance of Kleenes work led to the saying that Kleeneness is next to Gödelness
16.
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
17.
Hilary Putnam
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Hilary Whitehall Putnam was an American philosopher, mathematician, and computer scientist, and a major figure in analytic philosophy in the second half of the 20th century. He made significant contributions to philosophy of mind, philosophy of language, philosophy of mathematics, at the time of his death, Putnam was Cogan University Professor Emeritus at Harvard University. As a result, he acquired a reputation for changing his own position. In the field of epistemology, he is known for his critique of the well known brain in a vat thought experiment and this thought experiment appears to provide a powerful argument for epistemological skepticism, but Putnam challenges its coherence. In the philosophy of perception Putnam came to endorse direct realism, in the past, he further held that there are no mental representations, sense data, or other intermediaries that stand between the mind and the world. Such transactions can further involve qualia, in his later work, Putnam became increasingly interested in American pragmatism, Jewish philosophy, and ethics, thus engaging with a wider array of philosophical traditions. He also displayed an interest in metaphilosophy, seeking to renew philosophy from what he identifies as narrow, outside philosophy, Putnam contributed to mathematics and computer science. Together with Martin Davis he developed the Davis–Putnam algorithm for the Boolean satisfiability problem and he was at times a politically controversial figure, especially for his involvement with the Progressive Labor Party in the late 1960s and early 1970s. Putnam was born in Chicago, Illinois, in 1926 and his father, Samuel Putnam, was a scholar of Romance languages, columnist, and translator who wrote for the Daily Worker, a publication of the American Communist Party, from 1936 to 1946. As a result of his fathers commitment to communism, Putnam had an upbringing, although his mother. The family lived in France until 1934, when returned to the United States. Putnam attended Central High School, there he met Noam Chomsky, the two had been friends—and often intellectual opponents—ever since. After teaching at Northwestern, Princeton, and MIT, he moved to Harvard in 1965 and his wife, Ruth Anna Jacobs took a teaching position in philosophy at Wellesley College. Hilary and Ruth Anna were married in 1962, the Putnams, rebelling against the antisemitism that they had experienced during their youth, decided to establish a traditional Jewish home for their children. Since they had no experience with the rituals of Judaism, they sought out invitations to other Jews homes for Seder and they had no idea how to do it, in the words of Ruth Anna. They therefore began to study Jewish ritual and Hebrew, and became more Jewishly interested, identified, in 1994, Hilary Putnam celebrated a belated Bar Mitzvah service. His wife had a Bat Mitzvah service four years later, Hilary was a popular teacher at Harvard. In keeping with the tradition, he was politically active
18.
Alfred Tarski
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Alfred Tarski was a renowned Polish logician, mathematician and philosopher. Tarski taught and carried out research in mathematics at the University of California, Alfred Tarski was born Alfred Teitelbaum, to parents who were Polish Jews in comfortable circumstances relative to other Jews in the overall region. He first manifested his mathematical abilities while in school, at Warsaws Szkoła Mazowiecka. Nevertheless, he entered the University of Warsaw in 1918 intending to study biology, Leśniewski recognized Tarskis potential as a mathematician and encouraged him to abandon biology. Tarski and Leśniewski soon grew cool to each other, however, in later life, Tarski reserved his warmest praise for Kotarbiński, as was mutual. In 1923, Alfred Teitelbaum and his brother Wacław changed their surname to Tarski, the Tarski brothers also converted to Roman Catholicism, Polands dominant religion. Alfred did so even though he was an avowed atheist, Tarski was a Polish nationalist who saw himself as a Pole and wished to be fully accepted as such — later, in America, he spoke Polish at home. In 1929 Tarski married fellow teacher Maria Witkowska, a Pole of Catholic background and she had worked as a courier for the army in the Polish–Soviet War. They had two children, a son Jan who became a physicist, and a daughter Ina who married the mathematician Andrzej Ehrenfeucht, Tarski applied for a chair of philosophy at Lwów University, but on Bertrand Russells recommendation it was awarded to Leon Chwistek. In 1930, Tarski visited the University of Vienna, lectured to Karl Mengers colloquium, thanks to a fellowship, he was able to return to Vienna during the first half of 1935 to work with Mengers research group. From Vienna he traveled to Paris to present his ideas on truth at the first meeting of the Unity of Science movement, in 1937, Tarski applied for a chair at Poznań University but the chair was abolished. Tarskis ties to the Unity of Science movement likely saved his life, thus he left Poland in August 1939, on the last ship to sail from Poland for the United States before the German and Soviet invasion of Poland and the outbreak of World War II. Tarski left reluctantly, because Leśniewski had died a few months before, oblivious to the Nazi threat, he left his wife and children in Warsaw. He did not see again until 1946. During the war, nearly all his Jewish extended family were murdered at the hands of the German occupying authorities, in 1942, Tarski joined the Mathematics Department at the University of California, Berkeley, where he spent the rest of his career. Tarski became an American citizen in 1945, although emeritus from 1968, he taught until 1973 and supervised Ph. D. candidates until his death. At Berkeley, Tarski acquired a reputation as an awesome and demanding teacher, Tarski was extroverted, quick-witted, strong-willed, energetic, and sharp-tongued. He preferred his research to be collaborative — sometimes working all night with a colleague — and was very fastidious about priority, some students were frightened away, but a circle of disciples remained, many of whom became world-renowned leaders in the field
19.
Non-Euclidean geometry
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In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. In the latter case one obtains hyperbolic geometry and elliptic geometry, 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. The essential difference between the geometries is the nature of parallel lines. In hyperbolic geometry, by contrast, there are many lines through A not intersecting ℓ, while in elliptic geometry. In elliptic geometry the lines curve toward each other and intersect, the debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclids work Elements was written. In the Elements, Euclid began with a number of assumptions. Other mathematicians have devised simpler forms of this property, regardless of the form of the postulate, however, it consistently appears to be more complicated than Euclids other postulates,1. To draw a line from any point to any point. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. For at least a thousand years, geometers were troubled by the complexity of the fifth postulate. Many attempted to find a proof by contradiction, including Ibn al-Haytham, Omar Khayyám, Nasīr al-Dīn al-Tūsī and these theorems along with their alternative postulates, such as Playfairs axiom, played an important role in the later development of non-Euclidean geometry. These early attempts did, however, provide some early properties of the hyperbolic and elliptic geometries. Another example is al-Tusis son, Sadr al-Din, who wrote a book on the subject in 1298, based on al-Tusis later thoughts and he essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements. His work was published in Rome in 1594 and was studied by European geometers and he finally reached a point where he believed that his results demonstrated the impossibility of hyperbolic geometry. His claim seems to have based on Euclidean presuppositions, because no logical contradiction was present. In this attempt to prove Euclidean geometry he instead unintentionally discovered a new viable geometry, in 1766 Johann Lambert wrote, but did not publish, Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure that today we call a Lambert quadrilateral and he quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyam, and then proceeded to prove many theorems under the assumption of an acute angle
20.
Hyperbolic geometry
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In mathematics, hyperbolic geometry is a non-Euclidean geometry. Hyperbolic plane geometry is also the geometry of saddle surface or pseudospherical surfaces, surfaces with a constant negative Gaussian curvature, a modern use of hyperbolic geometry is in the theory of special relativity, particularly Minkowski spacetime and gyrovector space. In Russia it is commonly called Lobachevskian geometry, named one of its discoverers. This page is mainly about the 2-dimensional hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry, Hyperbolic geometry can be extended to three and more dimensions, see hyperbolic space for more on the three and higher dimensional cases. Hyperbolic geometry is closely related to Euclidean geometry than it seems. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry, there are two kinds of absolute geometry, Euclidean and hyperbolic. All theorems of geometry, including the first 28 propositions of book one of Euclids Elements, are valid in Euclidean. Propositions 27 and 28 of Book One of Euclids Elements prove the existence of parallel/non-intersecting lines and this difference also has many consequences, concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry, new concepts need to be introduced. Further, because of the angle of parallelism hyperbolic geometry has an absolute scale, single lines in hyperbolic geometry have exactly the same properties as single straight lines in Euclidean geometry. For example, two points define a line, and lines can be infinitely extended. Two intersecting lines have the properties as two intersecting lines in Euclidean geometry. For example, two lines can intersect in no more than one point, intersecting lines have equal opposite angles, when we add a third line then there are properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given 2 intersecting lines there are many lines that do not intersect either of the given lines. While in some models lines look different they do have these properties, non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in Euclidean geometry, For any line R and any point P which does not lie on R. In the plane containing line R and point P there are at least two lines through P that do not intersect R. This implies that there are through P an infinite number of lines that do not intersect R. All other non-intersecting lines have a point of distance and diverge from both sides of that point, and are called ultraparallel, diverging parallel or sometimes non-intersecting. Some geometers simply use parallel lines instead of limiting parallel lines and these limiting parallels make an angle θ with PB, this angle depends only on the Gaussian curvature of the plane and the distance PB and is called the angle of parallelism
21.
Philosophy
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Philosophy is the study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language. The term was coined by Pythagoras. Philosophical methods include questioning, critical discussion, rational argument and systematic presentation, classic philosophical questions include, Is it possible to know anything and to prove it. However, philosophers might also pose more practical and concrete questions such as, is it better to be just or unjust. Historically, philosophy encompassed any body of knowledge, from the time of Ancient Greek philosopher Aristotle to the 19th century, natural philosophy encompassed astronomy, medicine and physics. For example, Newtons 1687 Mathematical Principles of Natural Philosophy later became classified as a book of physics, in the 19th century, the growth of modern research universities led academic philosophy and other disciplines to professionalize and specialize. In the modern era, some investigations that were part of philosophy became separate academic disciplines, including psychology, sociology. Other investigations closely related to art, science, politics, or other pursuits remained part of philosophy, for example, is beauty objective or subjective. Are there many scientific methods or just one, is political utopia a hopeful dream or hopeless fantasy. Major sub-fields of academic philosophy include metaphysics, epistemology, ethics, aesthetics, political philosophy, logic, philosophy of science, since the 20th century, professional philosophers contribute to society primarily as professors, researchers and writers. Traditionally, the term referred to any body of knowledge. In this sense, philosophy is related to religion, mathematics, natural science, education. This division is not obsolete but has changed, Natural philosophy has split into the various natural sciences, especially astronomy, physics, chemistry, biology and cosmology. Moral philosophy has birthed the social sciences, but still includes value theory, metaphysical philosophy has birthed formal sciences such as logic, mathematics and philosophy of science, but still includes epistemology, cosmology and others. Many philosophical debates that began in ancient times are still debated today, colin McGinn and others claim that no philosophical progress has occurred during that interval. Chalmers and others, by contrast, see progress in philosophy similar to that in science, in one general sense, philosophy is associated with wisdom, intellectual culture and a search for knowledge. In that sense, all cultures and literate societies ask philosophical questions such as how are we to live, a broad and impartial conception of philosophy then, finds a reasoned inquiry into such matters as reality, morality and life in all world civilizations. Socrates was an influential philosopher, who insisted that he possessed no wisdom but was a pursuer of wisdom
22.
Carl Friedrich Gauss
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Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick, in the Duchy of Brunswick-Wolfenbüttel, as the son of poor working-class parents. Gauss later solved this puzzle about his birthdate in the context of finding the date of Easter and he was christened and confirmed in a church near the school he attended as a child. A contested story relates that, when he was eight, he figured out how to add up all the numbers from 1 to 100, there are many other anecdotes about his precocity while a toddler, and he made his first ground-breaking mathematical discoveries while still a teenager. He completed Disquisitiones Arithmeticae, his opus, in 1798 at the age of 21. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day, while at university, Gauss independently rediscovered several important theorems. Gauss was so pleased by this result that he requested that a regular heptadecagon be inscribed on his tombstone, the stonemason declined, stating that the difficult construction would essentially look like a circle. The year 1796 was most productive for both Gauss and number theory and he discovered a construction of the heptadecagon on 30 March. He further advanced modular arithmetic, greatly simplifying manipulations in number theory, on 8 April he became the first to prove the quadratic reciprocity law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic, the prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers. Gauss also discovered that every integer is representable as a sum of at most three triangular numbers on 10 July and then jotted down in his diary the note, ΕΥΡΗΚΑ. On October 1 he published a result on the number of solutions of polynomials with coefficients in finite fields, in 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in magnetism and the discovery of Kirchhoffs circuit laws in electricity. It was during this time that he formulated his namesake law and they constructed the first electromechanical telegraph in 1833, which connected the observatory with the institute for physics in Göttingen. In 1840, Gauss published his influential Dioptrische Untersuchungen, in which he gave the first systematic analysis on the formation of images under a paraxial approximation. Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points and he derived the Gaussian lens formula. In 1845, he became associated member of the Royal Institute of the Netherlands, in 1854, Gauss selected the topic for Bernhard Riemanns Habilitationvortrag, Über die Hypothesen, welche der Geometrie zu Grunde liegen. On the way home from Riemanns lecture, Weber reported that Gauss was full of praise, Gauss died in Göttingen, on 23 February 1855 and is interred in the Albani Cemetery there. Two individuals gave eulogies at his funeral, Gausss son-in-law Heinrich Ewald and Wolfgang Sartorius von Waltershausen and his brain was preserved and was studied by Rudolf Wagner who found its mass to be 1,492 grams and the cerebral area equal to 219,588 square millimeters. Highly developed convolutions were also found, which in the early 20th century were suggested as the explanation of his genius, Gauss was a Lutheran Protestant, a member of the St. Albans Evangelical Lutheran church in Göttingen
23.
Boeotia
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Boeotia, sometimes alternatively Latinised as Beotia, is one of the regional units of Greece. It is part of the region of Central Greece and it was also a region of ancient Greece. Its capital is Livadeia, and its largest city is Thebes, Boeotia lies to the north of the eastern part of the Gulf of Corinth. It also has a coastline on the Gulf of Euboea. It bordered on Megaris in the south, Attica in the southeast, Euboea in the northeast, Opuntian Locris in the north and Phocis in the west. The main mountain ranges of Boeotia are Mount Parnassus in the west, Mount Helicon in the southwest, Cithaeron in the south and its longest river, the Cephissus, flows in the central part, where most of the low-lying areas of Boeotia are found. Lake Copais was a lake in the center of Boeotia. It was drained in the 19th century, lake Yliki is a large lake near Thebes. The earliest inhabitants of Boeotia, associated with the city of Orchomenus, were called Minyans, pausanias mentions that Minyans established the maritime Ionian city of Teos, and occupied the islands of Lemnos and Thera. The Argonauts were sometimes referred to as Minyans, also, according to legend the citizens of Thebes paid an annual tribute to their king Erginus. The early wealth and power of Boeotia is shown by the reputation and visible Mycenean remains of several of its cities, especially Orchomenus, the origin of the name Boeotians may lie in the mountain Boeon in Epirus. Some toponyms and the common Aeolic dialect indicate that the Boeotians were related to the Thessalians and they moved south and settled in another rich plain, while others filtered across the Aegean and settled on Lesbos and in Aeolis in Asia Minor. Others are said to have stayed in Thessaly, withdrawing into the hill country, many ancient Greek legends originated or are set in this region. The older myths took their form during the Mycenean age when the Mycenean Greeks established themselves in Boeotia. Many of them are related to the myths of Argos, and others indicate connections with Phoenicia, Boeotia was also notable for the ancient oracular shrine of Trophonius at Lebadea. Graea, an ancient city in Boeotia, is thought to be the origin of the Latin word Graecus, from which English derives the words Greece. The major poets Hesiod and Pindar were Boeotians, on the other hand, the lack of good harbours hindered its maritime development. The importance of the legendary Minyae has been confirmed by archaeological remains, the Boeotian population entered the land from the north possibly before the Dorian invasion
24.
Analytic philosophy
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Analytic philosophy is a style of philosophy that became dominant in English-speaking countries at the beginning of the 20th century. In the United Kingdom, United States, Canada, Australia, New Zealand, and Scandinavia, as a historical development, analytical philosophy refers to certain developments in early 20th-century philosophy that were the historical antecedents of the current practice. Central figures in historical development are Bertrand Russell, Ludwig Wittgenstein, G. E. Moore, Gottlob Frege. This may be contrasted with the traditional foundationalism, which considers philosophy to be a science that investigates the fundamental reasons. Consequently, many philosophers have considered their inquiries as continuous with, or subordinate to. This is an attitude that begins with John Locke, who described his work as that of an underlabourer to the achievements of scientists such as Newton. During the twentieth century, the most influential advocate of the continuity of philosophy with science was Willard Van Orman Quine, the principle that the logical clarification of thoughts can be achieved only by analysis of the logical form of philosophical propositions. The logical form of a proposition is a way of representing it, to reduce it to simpler components if necessary, however, analytic philosophers disagree widely about the correct logical form of ordinary language. The neglect of generalized philosophical systems in favour of more restricted inquiries stated rigorously and it is thus able, in regard to certain problems, to achieve definite answers, which have the quality of science rather than of philosophy. Its methods, in respect, resemble those of science. Analytic philosophy is often understood in contrast to other traditions, most notably continental philosophies such as existentialism and phenomenology. British idealism, as taught by such as F. H. Bradley and Thomas Hill Green. With reference to this basis the initiators of analytic philosophy, G. E. Moore and Bertrand Russell. Inspired by developments in logic, the early Russell claimed that the problems of philosophy can be solved by showing the simple constituents of complex notions. An important aspect of British idealism was logical holism — the opinion that the aspects of the world cannot be wholly without also knowing the whole world. This is closely related to the opinion that relations between items are actually internal relations, that is, properties internal to the nature of those items. Russell, along with Wittgenstein, in response promulgated logical atomism, Frege was also influential as a philosopher of mathematics in Germany at the beginning of the 20th century. Like Frege, Russell attempted to show that mathematics is reducible to logical fundamentals in The Principles of Mathematics, later, his book written with Whitehead, Principia Mathematica, encouraged many philosophers to renew their interest in the development of symbolic logic
25.
Formula
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In science, a formula is a concise way of expressing information symbolically as in a mathematical or chemical formula. The informal use of the formula in science refers to the general construct of a relationship between given quantities. The plural of formula can be spelled either as formulas or formulae, in mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language. Note that the volume V and the radius r are expressed as single instead of words or phrases. This convention, while important in a relatively simple formula, means that mathematicians can more quickly manipulate larger. Mathematical formulas are often algebraic, closed form, and/or analytical, for example, H2O is the chemical formula for water, specifying that each molecule consists of two hydrogen atoms and one oxygen atom. Similarly, O−3 denotes an ozone molecule consisting of three atoms and having a net negative charge. In a general context, formulas are applied to provide a solution for real world problems. Some may be general, F = ma, which is one expression of Newtons second law, is applicable to a range of physical situations. Other formulas may be created to solve a particular problem, for example. In all cases, however, formulas form the basis for calculations, expressions are distinct from formulas in that they cannot contain an equals sign. Whereas formulas are comparable to sentences, expressions are more like phrases, a chemical formula identifies each constituent element by its chemical symbol and indicates the proportionate number of atoms of each element. In empirical formulas, these begin with a key element and then assign numbers of atoms of the other elements in the compound. For molecular compounds, these numbers can all be expressed as whole numbers. For example, the formula of ethanol may be written C2H6O because the molecules of ethanol all contain two carbon atoms, six hydrogen atoms, and one oxygen atom. Some types of compounds, however, cannot be written with entirely whole-number empirical formulas. An example is boron carbide, whose formula of CBn is a variable non-whole number ratio with n ranging from over 4 to more than 6.5. When the chemical compound of the consists of simple molecules
26.
Language
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Language is the ability to acquire and use complex systems of communication, particularly the human ability to do so, and a language is any specific example of such a system. The scientific study of language is called linguistics, questions concerning the philosophy of language, such as whether words can represent experience, have been debated since Gorgias and Plato in Ancient Greece. Thinkers such as Rousseau have argued that language originated from emotions while others like Kant have held that it originated from rational and logical thought, 20th-century philosophers such as Wittgenstein argued that philosophy is really the study of language. Major figures in linguistics include Ferdinand de Saussure and Noam Chomsky, estimates of the number of languages in the world vary between 5,000 and 7,000. However, any precise estimate depends on an arbitrary distinction between languages and dialects. Natural languages are spoken or signed, but any language can be encoded into secondary media using auditory, visual, or tactile stimuli – for example, in whistling, signed and this is because human language is modality-independent. All languages rely on the process of semiosis to relate signs to particular meanings, human language has the properties of productivity and displacement, and relies entirely on social convention and learning. Its complex structure affords a wider range of expressions than any known system of animal communication. Language is processed in different locations in the human brain. Humans acquire language through interaction in early childhood, and children generally speak fluently when they are approximately three years old. The use of language is deeply entrenched in human culture, a group of languages that descend from a common ancestor is known as a language family. The languages of the Dravidian family that are mostly in Southern India include Tamil. Academic consensus holds that between 50% and 90% of languages spoken at the beginning of the 21st century will probably have become extinct by the year 2100. The English word language derives ultimately from Proto-Indo-European *dn̥ǵʰwéh₂s tongue, speech, language through Latin lingua, language, tongue, and Old French language. The word is used to refer to codes, ciphers. Unlike conventional human languages, a language in this sense is a system of signs for encoding and decoding information. This article specifically concerns the properties of human language as it is studied in the discipline of linguistics. As an object of study, language has two primary meanings, an abstract concept, and a specific linguistic system, e. g. French
27.
Arithmetic
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Arithmetic is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations between them—addition, subtraction, multiplication and division. Arithmetic is an part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory and are still used to refer to a wider part of number theory. The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations as early as 2000 BC and these artifacts do not always reveal the specific process used for solving problems, but the characteristics of the particular numeral system strongly influence the complexity of the methods. The hieroglyphic system for Egyptian numerals, like the later Roman numerals, in both cases, this origin resulted in values that used a decimal base but did not include positional notation. Complex calculations with Roman numerals required the assistance of a board or the Roman abacus to obtain the results. Early number systems that included positional notation were not decimal, including the system for Babylonian numerals. Because of this concept, the ability to reuse the same digits for different values contributed to simpler. The continuous historical development of modern arithmetic starts with the Hellenistic civilization of ancient Greece, prior to the works of Euclid around 300 BC, Greek studies in mathematics overlapped with philosophical and mystical beliefs. For example, Nicomachus summarized the viewpoint of the earlier Pythagorean approach to numbers, Greek numerals were used by Archimedes, Diophantus and others in a positional notation not very different from ours. Because the ancient Greeks lacked a symbol for zero, they used three separate sets of symbols, one set for the units place, one for the tens place, and one for the hundreds. Then for the place they would reuse the symbols for the units place. Their addition algorithm was identical to ours, and their multiplication algorithm was very slightly different. Their long division algorithm was the same, and the square root algorithm that was taught in school was known to Archimedes. He preferred it to Heros method of successive approximation because, once computed, a digit doesnt change, and the square roots of perfect squares, such as 7485696, terminate immediately as 2736. For numbers with a part, such as 546.934. The ancient Chinese used a positional notation. Because they also lacked a symbol for zero, they had one set of symbols for the place
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Thought
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Thought refers to ideas or arrangements of ideas that are the result of the process of thinking. Though thinking is an activity considered essential to humanity, there is no consensus as to how we define or understand it. Thinking allows humans to sense of, interpret, represent or model the world they experience. It is therefore helpful to an organism with needs, objectives, the word thought comes from Old English þoht, or geþoht, from stem of þencan to conceive of in the mind, consider. Definitions of thought may also be derived directly or indirectly from theories of thought, the notion of the fundamental role of non-cognitive understanding in rendering possible thematic consciousness informed the discussion surrounding Artificial Intelligence during the 1970s and 1980s. Phenomenology, however, is not the approach to thinking in modern Western philosophy. The mind-body problem concerns the explanation of the relationship exists between minds, or mental processes, and bodily states or processes. The main aim of working in this area is to determine the nature of the mind and mental states/processes. Someones desire for a slice of pizza, for example, will tend to cause that person to move his or her body in a specific manner and in a specific direction to obtain what he or she wants. The question, then, is how it can be possible for conscious experiences to arise out of a lump of gray matter endowed with nothing but electrochemical properties. A related problem is to explain how propositional attitudes can cause that individuals neurons to fire. These comprise some of the puzzles that have confronted epistemologists and philosophers of mind from at least the time of René Descartes, the above reflects a classical, functional description of how we work as cognitive, thinking systems. Therefore, functional analysis of the mind alone will always leave us with the problem which cannot be solved. A neuron is a cell in the nervous system that processes. Neurons are the components of the brain, the vertebrate spinal cord, the invertebrate ventral nerve cord. Motor neurons receive signals from the brain and spinal cord and cause muscle contractions, interneurons connect neurons to other neurons within the brain and spinal cord. Neurons respond to stimuli, and communicate the presence of stimuli to the nervous system. Neurons do not go through mitosis, and usually cannot be replaced after being destroyed, psychologists have concentrated on thinking as an intellectual exertion aimed at finding an answer to a question or the solution of a practical problem
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Gottfried Wilhelm Leibniz
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Leibnizs notation has been widely used ever since it was published. It was only in the 20th century that his Law of Continuity and he became one of the most prolific inventors in the field of mechanical calculators. He also refined the number system, which is the foundation of virtually all digital computers. Leibniz, along with René Descartes and Baruch Spinoza, was one of the three great 17th-century advocates of rationalism and he wrote works on philosophy, politics, law, ethics, theology, history, and philology. Leibnizs contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters and he wrote in several languages, but primarily in Latin, French, and German. There is no complete gathering of the writings of Leibniz in English, Gottfried Leibniz was born on July 1,1646, toward the end of the Thirty Years War, in Leipzig, Saxony, to Friedrich Leibniz and Catharina Schmuck. Friedrich noted in his journal,21. Juny am Sontag 1646 Ist mein Sohn Gottfried Wilhelm, post sextam vespertinam 1/4 uff 7 uhr abents zur welt gebohren, in English, On Sunday 21 June 1646, my son Gottfried Wilhelm is born into the world a quarter after six in the evening, in Aquarius. Leibniz was baptized on July 3 of that year at St. Nicholas Church, Leipzig and his father died when he was six and a half years old, and from that point on he was raised by his mother. Her teachings influenced Leibnizs philosophical thoughts in his later life, Leibnizs father had been a Professor of Moral Philosophy at the University of Leipzig, and the boy later inherited his fathers personal library. He was given access to it from the age of seven. Access to his fathers library, largely written in Latin, also led to his proficiency in the Latin language and he also composed 300 hexameters of Latin verse, in a single morning, for a special event at school at the age of 13. In April 1661 he enrolled in his fathers former university at age 15 and he defended his Disputatio Metaphysica de Principio Individui, which addressed the principle of individuation, on June 9,1663. Leibniz earned his masters degree in Philosophy on February 7,1664, after one year of legal studies, he was awarded his bachelors degree in Law on September 28,1665. His dissertation was titled De conditionibus, in early 1666, at age 19, Leibniz wrote his first book, De Arte Combinatoria, the first part of which was also his habilitation thesis in Philosophy, which he defended in March 1666. His next goal was to earn his license and Doctorate in Law, in 1666, the University of Leipzig turned down Leibnizs doctoral application and refused to grant him a Doctorate in Law, most likely due to his relative youth. Leibniz then enrolled in the University of Altdorf and quickly submitted a thesis, the title of his thesis was Disputatio Inauguralis de Casibus Perplexis in Jure. Leibniz earned his license to practice law and his Doctorate in Law in November 1666 and he next declined the offer of an academic appointment at Altdorf, saying that my thoughts were turned in an entirely different direction
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Calculus ratiocinator
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There are two contrasting points of view on what Leibniz meant by calculus ratiocinator. The first is associated with software, the second is associated with computer hardware. The received point of view in philosophy and formal logic, is that the calculus ratiocinator anticipates mathematical logic—an algebra of logic. The analytic point of view understands that the calculus ratiocinator is an inference engine or computer program. That logic began with Freges 1879 Begriffsschrift and C. S. Peirces writings on logic in the 1880s, frege intended his concept script to be a calculus ratiocinator as well as a lingua characteristica. That part of formal logic relevant to the calculus comes under the heading of proof theory, from this perspective the calculus ratiocinator is only a part of the universal characteristic, and a complete universal characteristic includes a logical calculus. A contrasting point of view stems from synthetic philosophy and fields such as cybernetics, electronic engineering and it is little appreciated in analytic philosophy. The synthetic view understands the calculus ratiocinator as referring to a calculating machine, as a computing machine, the ideal calculus ratiocinator would perform Leibnizs integral and differential calculus. In this way the meaning of the word, ratiocinator is clarified, hence the characteristic, calculus ratiocinator, and encyclopedia form three pillars of Leibnizs project. Gottfried Leibniz Characteristica universalis Louis Couturat,1901, donald Rutherfords English translation of some chapters. Hartley Rogers, Jr.1963, An Example in Mathematical Logic, The American Mathematical Monthly, Vol.70, norbert Wiener,1948, Time, communication, and the nervous system, Teleological mechanisms. --1965, Cybernetics, Second Edition, or the Control and Communication in the Animal, hermann Grassmann and the Prehistory of Universal Algebra, The American Mathematical Monthly, Vol.89, No. Language as Calculus versus Language as Universal Medium
