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  • Mathematics In Europe - Set theory and logic
    ducks are not white one cannot find out that they are equivalent by means of propositional logic In predicate logic statements P x containing a variable x like the mentioned x is even become propositions if either a value is assigned to the variable e g put x 5 to get 5 is even and now one can determine that it is a false proposition or by using one of the two quantifiers for all universal quantifier and exists existential quantifier where x P x means for all x in the universe P x and x P x means there exists a x in the universe P x In both cases the truth depends on what one takes as the universe For example x such that x is even is true if the universe is that of integers but false if the universe is the set of all odd integers Proof theory is a branch of mathematical logic in which proofs are considered as mathematical objects It is of a syntactic nature meaning that it deals with formal rules without consideration of their interpretation In contrast model theory the study of mathematical structures using the tools of mathematical logic is of a semantic nature it is interested in interpreations Other subdisciplines of mathematical logic include type theory computability and recursion theory algebraic logic Consider two sets of points in the coordinate plane the yellow rectangle A and the red dots B as shown above the third picture in the row above shows the relative position of the two sets The orange region on the left is the intersection and the blue on the right the union of A and B Figure 1 Illustration of two basic operations from algebra of sets intersection and union Many school maths curricula include the description of some topics from naive set thery sets and a few basic operations with sets union intersection difference But rare nonprofessionals know that set theory is not about finding an intersection of a set of groceries with a set of fruits from a market stand or the intersection union and diference of sets shown in figure 1 above Set theory provides an exact way to deal with the infinite as well as answering fundamental questions like what a positive integer is Cantor s proof from 1873 showing that there are various levels of infinity not all infinities are equally infinite is usually taken as the birthmark of set theory Cantor found a way to compare the sizes of infinite sets The idea is simple if you want to know if two finite sets have the same number of elements the cardinality all one has to do is to check if each of the elements of one set can be paired with one element of the other set For example if you want to check if the number of seats in the classroom is enough for all the students you can instead of counting both the seats and the students

    Original URL path: http://mathematics-in-europe.eu/tr/bilgiler/avrupa-da-matematik?id=91 (2013-11-18)
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  • Mathematics In Europe - Information
    there is probably no one answer to which all mathematicians would subscribe We have collected a number of partial answers There is great variation in the ambitions of the authors and the demands placed by them on the reader The mathematical landscape Here is information on the mathematics profession Which areas are of greatest interest What important concepts are at work What of value can be said about particular numbers and formulas Research This is an enormous topic We present an overview of where in Europe outside of the universities special mathematics research institutes have been set up There are also articles on the Millennium Problems each of which carries a million dollar prize for its solution In addition you can learn about the impact of current mathematical problems on our daily lives History and philosophy of mathematics In many instances the development of mathematics was motivated by concrete problems in the natural sciences economics and technology The current state of the mathematical sciences has many historical precedents which we shall illustrate with examples It is by no means clear how well founded the methods of mathematics actually are This issue has been repeatedly addressed by philosophers of mathematics over

    Original URL path: http://mathematics-in-europe.eu/da/information (2013-11-18)
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  • Mathematics In Europe - Perustietoa
    probably no one answer to which all mathematicians would subscribe We have collected a number of partial answers There is great variation in the ambitions of the authors and the demands placed by them on the reader The mathematical landscape Here is information on the mathematics profession Which areas are of greatest interest What important concepts are at work What of value can be said about particular numbers and formulas Research This is an enormous topic We present an overview of where in Europe outside of the universities special mathematics research institutes have been set up There are also articles on the Millennium Problems each of which carries a million dollar prize for its solution In addition you can learn about the impact of current mathematical problems on our daily lives History and philosophy of mathematics In many instances the development of mathematics was motivated by concrete problems in the natural sciences economics and technology The current state of the mathematical sciences has many historical precedents which we shall illustrate with examples It is by no means clear how well founded the methods of mathematics actually are This issue has been repeatedly addressed by philosophers of mathematics over the course

    Original URL path: http://mathematics-in-europe.eu/fi/perustietoa (2013-11-18)
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  • Mathematics In Europe - Contenuti
    very difficult question and there is probably no one answer to which all mathematicians would subscribe We have collected a number of partial answers There is great variation in the ambitions of the authors and the demands placed by them on the reader The mathematical landscape Here is information on the mathematics profession Which areas are of greatest interest What important concepts are at work What of value can be said about particular numbers and formulas Research This is an enormous topic We present an overview of where in Europe outside of the universities special mathematics research institutes have been set up There are also articles on the Millennium Problems each of which carries a million dollar prize for its solution In addition you can learn about the impact of current mathematical problems on our daily lives History and philosophy of mathematics In many instances the development of mathematics was motivated by concrete problems in the natural sciences economics and technology The current state of the mathematical sciences has many historical precedents which we shall illustrate with examples It is by no means clear how well founded the methods of mathematics actually are This issue has been repeatedly addressed by

    Original URL path: http://mathematics-in-europe.eu/it/contenuti (2013-11-18)
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  • Mathematics In Europe - Informasjon
    is probably no one answer to which all mathematicians would subscribe We have collected a number of partial answers There is great variation in the ambitions of the authors and the demands placed by them on the reader The mathematical landscape Here is information on the mathematics profession Which areas are of greatest interest What important concepts are at work What of value can be said about particular numbers and formulas Research This is an enormous topic We present an overview of where in Europe outside of the universities special mathematics research institutes have been set up There are also articles on the Millennium Problems each of which carries a million dollar prize for its solution In addition you can learn about the impact of current mathematical problems on our daily lives History and philosophy of mathematics In many instances the development of mathematics was motivated by concrete problems in the natural sciences economics and technology The current state of the mathematical sciences has many historical precedents which we shall illustrate with examples It is by no means clear how well founded the methods of mathematics actually are This issue has been repeatedly addressed by philosophers of mathematics over the

    Original URL path: http://mathematics-in-europe.eu/nb/informasjon (2013-11-18)
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  • Mathematics In Europe - Informacje
    probably no one answer to which all mathematicians would subscribe We have collected a number of partial answers There is great variation in the ambitions of the authors and the demands placed by them on the reader The mathematical landscape Here is information on the mathematics profession Which areas are of greatest interest What important concepts are at work What of value can be said about particular numbers and formulas Research This is an enormous topic We present an overview of where in Europe outside of the universities special mathematics research institutes have been set up There are also articles on the Millennium Problems each of which carries a million dollar prize for its solution In addition you can learn about the impact of current mathematical problems on our daily lives History and philosophy of mathematics In many instances the development of mathematics was motivated by concrete problems in the natural sciences economics and technology The current state of the mathematical sciences has many historical precedents which we shall illustrate with examples It is by no means clear how well founded the methods of mathematics actually are This issue has been repeatedly addressed by philosophers of mathematics over the course

    Original URL path: http://mathematics-in-europe.eu/pl/informacje (2013-11-18)
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  • Mathematics In Europe - Informação
    and there is probably no one answer to which all mathematicians would subscribe We have collected a number of partial answers There is great variation in the ambitions of the authors and the demands placed by them on the reader The mathematical landscape Here is information on the mathematics profession Which areas are of greatest interest What important concepts are at work What of value can be said about particular numbers and formulas Research This is an enormous topic We present an overview of where in Europe outside of the universities special mathematics research institutes have been set up There are also articles on the Millennium Problems each of which carries a million dollar prize for its solution In addition you can learn about the impact of current mathematical problems on our daily lives History and philosophy of mathematics In many instances the development of mathematics was motivated by concrete problems in the natural sciences economics and technology The current state of the mathematical sciences has many historical precedents which we shall illustrate with examples It is by no means clear how well founded the methods of mathematics actually are This issue has been repeatedly addressed by philosophers of mathematics

    Original URL path: http://mathematics-in-europe.eu/po/informacao (2013-11-18)
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  • Mathematics In Europe - Информация
    is probably no one answer to which all mathematicians would subscribe We have collected a number of partial answers There is great variation in the ambitions of the authors and the demands placed by them on the reader The mathematical landscape Here is information on the mathematics profession Which areas are of greatest interest What important concepts are at work What of value can be said about particular numbers and formulas Research This is an enormous topic We present an overview of where in Europe outside of the universities special mathematics research institutes have been set up There are also articles on the Millennium Problems each of which carries a million dollar prize for its solution In addition you can learn about the impact of current mathematical problems on our daily lives History and philosophy of mathematics In many instances the development of mathematics was motivated by concrete problems in the natural sciences economics and technology The current state of the mathematical sciences has many historical precedents which we shall illustrate with examples It is by no means clear how well founded the methods of mathematics actually are This issue has been repeatedly addressed by philosophers of mathematics over the

    Original URL path: http://mathematics-in-europe.eu/ru/2013-03-25-18-15-24 (2013-11-18)
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