1.
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

2.
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

3.
Philosophy of science
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Philosophy of science is a branch of philosophy concerned with the foundations, methods, and implications of science. The central questions of this study concern what qualifies as science, the reliability of theories. This discipline overlaps with metaphysics, ontology, and epistemology, for example, in addition to these general questions about science as a whole, philosophers of science consider problems that apply to particular sciences. Some philosophers of science also use contemporary results in science to reach conclusions about philosophy itself, Karl Popper and Charles Sanders Pierce moved on from positivism to establish a modern set of standards for scientific methodology. Subsequently, the coherentist approach to science, in which a theory is validated if it makes sense of observations as part of a coherent whole, became prominent due to W. V. Quine and others. Some thinkers such as Stephen Jay Gould seek to ground science in axiomatic assumptions, another approach to thinking about science involves studying how knowledge is created from a sociological perspective, an approach represented by scholars like David Bloor and Barry Barnes. Finally, a tradition in continental philosophy approaches science from the perspective of an analysis of human experience. Philosophies of the particular sciences range from questions about the nature of time raised by Einsteins general relativity, a central theme is whether one scientific discipline can be reduced to the terms of another. That is, can chemistry be reduced to physics, or can sociology be reduced to individual psychology, the general questions of philosophy of science also arise with greater specificity in some particular sciences. For instance, the question of the validity of scientific reasoning is seen in a different guise in the foundations of statistics, the question of what counts as science and what should be excluded arises as a life-or-death matter in the philosophy of medicine. Distinguishing between science and non-science is referred to as the demarcation problem, for example, should psychoanalysis be considered science. How about so-called creation science, the multiverse hypothesis, or macroeconomics. Karl Popper called this the question in the philosophy of science. However, no unified account of the problem has won acceptance among philosophers, Martin Gardner has argued for the use of a Potter Stewart standard for recognizing pseudoscience. Early attempts by the logical positivists grounded science in observation while non-science was non-observational, Popper argued that the central property of science is falsifiability. That is, every genuinely scientific claim is capable of being proven false, a closely related question is what counts as a good scientific explanation. In addition to providing predictions about events, society often takes scientific theories to provide explanations for events that occur regularly or have already occurred. One early and influential theory of scientific explanation is the deductive-nomological model and it says that a successful scientific explanation must deduce the occurrence of the phenomena in question from a scientific law

4.
Germany
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Germany, officially the Federal Republic of Germany, is a federal parliamentary republic in central-western Europe. It includes 16 constituent states, covers an area of 357,021 square kilometres, with about 82 million inhabitants, Germany is the most populous member state of the European Union. After the United States, it is the second most popular destination in the world. Germanys capital and largest metropolis is Berlin, while its largest conurbation is the Ruhr, other major cities include Hamburg, Munich, Cologne, Frankfurt, Stuttgart, Düsseldorf and Leipzig. Various Germanic tribes have inhabited the northern parts of modern Germany since classical antiquity, a region named Germania was documented before 100 AD. During the Migration Period the Germanic tribes expanded southward, beginning in the 10th century, German territories formed a central part of the Holy Roman Empire. During the 16th century, northern German regions became the centre of the Protestant Reformation, in 1871, Germany became a nation state when most of the German states unified into the Prussian-dominated German Empire. After World War I and the German Revolution of 1918–1919, the Empire was replaced by the parliamentary Weimar Republic, the establishment of the national socialist dictatorship in 1933 led to World War II and the Holocaust. After a period of Allied occupation, two German states were founded, the Federal Republic of Germany and the German Democratic Republic, in 1990, the country was reunified. In the 21st century, Germany is a power and has the worlds fourth-largest economy by nominal GDP. As a global leader in industrial and technological sectors, it is both the worlds third-largest exporter and importer of goods. Germany is a country with a very high standard of living sustained by a skilled. It upholds a social security and universal health system, environmental protection. Germany was a member of the European Economic Community in 1957. It is part of the Schengen Area, and became a co-founder of the Eurozone in 1999, Germany is a member of the United Nations, NATO, the G8, the G20, and the OECD. The national military expenditure is the 9th highest in the world, the English word Germany derives from the Latin Germania, which came into use after Julius Caesar adopted it for the peoples east of the Rhine. This in turn descends from Proto-Germanic *þiudiskaz popular, derived from *þeudō, descended from Proto-Indo-European *tewtéh₂- people, the discovery of the Mauer 1 mandible shows that ancient humans were present in Germany at least 600,000 years ago. The oldest complete hunting weapons found anywhere in the world were discovered in a mine in Schöningen where three 380, 000-year-old wooden javelins were unearthed

5.
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