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- Mathematics In Europe - Philosophy of mathematics: intuitionism

constructing them The purpose of a mathematical theorem then is to express the implementation of such a mental construction And while formalism emphasized the importance of mathematical symbols intuitionism assigns essential significance to verbal expression in any formal system The verbal formulation of mathematical results serves exclusively communicative purposes and represents the essentials of mathematics the mental constructions only in imperfect approximation According to intuitionism mathematical axioms need not be reduced to logic or proven to be consistent rather it is their intuitive rightness that somehow determines their validity To such an extent intuitionism is a purely philosophical position It becomes a foundational program for mathematics only through its attack on traditional logic led above all by Brouwer In a number of papers and lectures Brouwer advocated a restriction on the application of the law of the excluded middle and the method of indirect proof based on that law The law of the excluded middle asserts that every assertion is either true or false and a proof is said to be indirect if it proves the validity of an assertion by showing that its negation leads to an absurdity that is leads to some sort of contradiction In the case of universally quantified statements that is statements of the form all x are or among all x there exists one such that then according to Brouwer the two principles are valid only if applied to finite sets represented by the variable x In that case universally quantified propositions can always be verified in principle by checking each individual case and according to Brouwer it is precisely this condition the ability to verify the truth of every statement in principle that expresses the law of the excluded middle In contrast when we are dealing with infinite sets in which the possibility

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Open archived version from archive - Mathematics In Europe - Philosophy of mathematics: formalism

axioms whose truth could be grasped intuitively such as the truth of the logical axioms The conception of the formalists on the other hand is that while mathematics is to be given an axiomatic form questions about the truth of the axioms do not fall within the purview of mathematics In the formalists view the mathematician is to deal not with content oriented theorems but initially with uninterpreted symbols Axioms therefore are not bearers of truth rather they determine the formal properties of the mathematical symbols And the task of the mathematician consists in discovering derivability relationships among these symbols by operating on them according to certain rules For the formalists mathematics is a game played with meaningless symbols comparable to the rule governed pushing around of figures on a chessboard Questions about truth and falsity come into play only in applications of the game that is when the mathematical symbols are to be interpreted in a particular way Thus as Hilbert emphasized the concepts of Euclidean geometry are open to quite different interpretations not merely say as points and lines in space but for example as pairs of real numbers In this sense formalistically conceived mathematics possesses a conditional character It maintains nothing with regard to the validity of particular axioms or axiomatic systems Instead if a particular interpretation of mathematical symbols is given and the axioms thus interpreted are valid then the theorems derived from them are true as well The possibility of an application that is an interpretation of the mathematical symbols in such a way that the axioms formed from them are true depends according to the ideas of the formalists entirely on the consistency of the chosen system of axioms A mathematical axiom system is therefore to be accepted if and only if no contradictions

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Open archived version from archive - Mathematics In Europe - Algebra

independent on the order in which we perform the operation when calculating 2 3 4 it is unimportant if we calculate this as 2 3 4 5 4 9 or as 2 3 4 2 7 9 In both cases there is a special element called the neutral element in the set such that the operation performed on him and any other element doesn t change the other element if you add 0 to any integer you don t change it and if you multiply 1 by any positive real you don t change it Finally for every element of the set we can find another such that the performance of the operation on these two elements results in the neutral element if you add n to any integer n you get 0 if you multiply any positive real number x by 1 x you get 1 When a set and the operation on it have all the above mentioned properties we call it a commutative group the sets of integers with addition and positive reals with multiplications are examples of commutative groups The group D 2 described above is also a commutative group but contrary to the group of natural numbers with addition and positive real numbers with multiplication it is finite it has only four elements Not all groups are commutative For example if you consider the set S 3 the possible rearrangements of three objects A B and C and take the composition of rearrangements as the operation of the group you will notice that the order of rearrangements matters see Figure 31 below the three squares start in order shown in the first row We consider the following two rearrangements switching the first two squares on the left rearrangement 1 and switching the first and the last one rearrangement 2 The second row of Figure 3 shows what happens if we first perform rearrangement 1 and then rearrangement 2 and the third row shows what happens if we first perform rearrangement 2 and then rearrangement 1 You can see that we end up with two different arrangements if we change the order of the rearrangements Thus S 3 with the operation of composition of rearrangements is not commutative but it is a group i e all of the other properties mentioned for integers with addition hold Figure 3 Rearrangements of three squares illustrating the noncommutativity of the group S 3 Besides groups there are many other kinds of algebraic structures rings fields vector spaces modules The study of algebraic structures has its origins in the study of solutions to algebraic equations linear quadratic cubic quartic After the formulas for solutions of cubic and quartic equations were discovered in the reneissance mathematicians worked on equations of higher order In the beginning of the 19th century N H Abel and E Galois proved that it is not possible to describe the solutions of higher degree equations by radicals i e by formulas giving the rule how to calculate

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Open archived version from archive - Mathematics In Europe - Combinatorics

things cannot be done both at the same time there are m n ways to do one of the things the product rule if there are m ways to do one thing and n ways to do another there are mn ways to do both things and the pigeonhole principle if n things are to be put into m boxes and m n then at least one box has to contain more than one thing Figure 2 a The sum and product rules the objects to choose from If you are given three cylinders four balls and five cubes and want to choose two objects of different kinds in how many ways can you do it You can choose one cylinder and one ball in 3 4 ways one cylinder and one cube in 3 5 ways and one ball and one cube in 4 5 ways product rule Consequently you can choose two objects of different kinds in 3 4 3 5 4 5 47 ways sum rule The 47 choices are shown below Figure 2 b The 47 possible choices of two objects of different kinds There are also other more advanced counting techniques For example the cardinalities of sequences of sets are often arranged into generating functions Generating functions are formal power series with coefficients that contain information about a sequence They are analysed with techniques of analysis For example the famous Fibonacci numbers are defined as the sequence of numbers formed from the starting numbers 0 and 1 in such a way that each next Fibonacci number is the sum of the previous two Thus one obtains the Fibonacci sequence 0 1 1 2 3 5 8 13 21 34 The easiest way to obtain a formula for the n th Fibonacci number f n is by using a generating function We define it as the sum F x of all terms of the form f n x n for n 0 1 2 This sum does not converge for any x this means that no matter what value one inserts for x there is no reasonable way to assign a number to the infinite sum Still but by performing formal computations one can still obtain many insights in the sequence of coefficients Since by definition f 0 0 f 1 1 f n f n 1 f n 2 the sum of all f n x n can be written as x plus the sum of all terms of the form f n 1 f n 2 x n for n 1 After rearrangements one gets that F x x xF x x 2 F x so F x x 1 x x 2 The last fraction can be written as 1 1 Φ x 1 1 φ x 5 where Φ 1 5 2 and φ 1 5 2 As 1 1 ax can be written in analysis only if x a because otherwise the series does not converge but here one performs the

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Open archived version from archive - Mathematics In Europe - Probability and statistics

assign a number say 0 to heads and another say 1 to tails In this way we define a real valued function X on Ω Such a function is called a random variable a function X defined on the sample space is a random variable if it assigns unique numerical values to elementary events the function must also meet some other requirements and there are also random variable taking values in other sets besides R If one is interested in how many people come to concerts in a city and is therefore recording the numbers for as many concerts as possible then each counting of the number of visitors at a particular concert can be considered as an experiment As the results will be integers but there is no explicit upper bound you cannot identify the largest possible number of visitors the range of the corresponding random variable is best taken to be the set of all possible nonnegative integers Another example of a random variable would be the following if one measures the pH of samples of various solutions the corresponding random variable would assign to each experiment i e sample solution the value of its pH Compared to the previous two examples one can note while in the first three the possible values are clearly separated in the last experiment the possible values cannot be separated pH can take any real value usually between 0 and 14 The first three cases are examples of discrete random variables while the last one is an example of a continuous one Consider now the following sequence of experiments A person is repeatedly throwing a die and keeping track how many times a 6 rolls This is an example of a stochastic process a sequence of random variables their values are usually called states Even more it is an example of a Markov chain a discrete stochastic process with the property that the next state does not depend on previous ones except possibly on the current one For a given number n of rolls we consider the random variable X that describes the number of sixes rolled The probability of rolling a 6 in one roll can now be denoted by p 1 6 and the probability of not rolling a six is q 1 p 5 6 As each roll is independent on the previous one it is not hard to see that the probability of rolling exactly k sixes in n rolls is B n k p k q n k where B n k is the binomial coefficient equal to n n k k One can now define a function F from R into 0 1 by setting F x equal to the probability of of X having value at most x i e of having rolled at most x sixes in n rolls Such functions F R 0 1 with the property that F x equal to the probability of of X having value at most x are

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Open archived version from archive - Mathematics In Europe - The landscape: formulas

Sponsor Munich RE The landscape formulas Ayrıntılar Kategori Landscape Some important formulas from mathematics and physics Pythagoras formula pdf Newton s universal law of gravity pdf Euler s most beautiful formula of mathematics pdf The prime number theorem of Gauß pdf Maxwell s equations of electrodynamics pdf Clausius equation of thermodynamics pdf Einstein I How changes time for a moving object pdf Einstein II E mc 2 pdf Einstein III

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Open archived version from archive - Mathematics In Europe - Number theory

1 7 and the remainder is 0 and we conclude that the greatest common divisor of 247 and 16 is 1 Two integers are congruent to each other modulo an integer n if they give the same remainder when divided by n or equivalently if their difference is divisible by n For example congruences modulo 12 classify integers into 12 classes depending on their remainder modulo 12 Chinese remainder theorem known in 13th century China states that if you take a number of pairwise coprime integers n m and the same number of any integers a b there exist integers x that are at the same time congruent to a modulo n to b modulo m and so on all of these x are congruent to each other modulo the product n m Examples of more recent results in number theory are the Fermat s little theorem Lagrange s four square theorem and the prime number theorem Fermat s little theorem states that if a is an integer and p is a prime then a p a is divisible by p or equivalently if a is not divisible by a prime p then the remainder of the division of a p 1 by p is 1 For example as 127 is a prime number and 250 is not divisible by 127 without the actual calculation we know that 250 126 divided by 127 has a remainder 1 The Lagrange s four square theorem states that any positive integer can be written as the sum of four square numbers some can be zero For example 239 14 2 5 2 3 2 3 2 The prime number theorem gives an estimate for the number π x of primes up to a number x π x is approximately equal to x ln x and the approximation is the more correct the larger x is For example there are π 100 25 primes not greater than 100 and 100 ln 100 21 71 and there are π 1 000 000 78 498 primes not greater than 1 000 000 and 1 000 000 ln 1 000 000 72382 414 Many of the most famous open problems in mathematics are number theoretical The Goldbach s conjecture is one of the oldest stated by Goldbach in 1742 unsolved problems in mathematics it states that every even integer can be expressed as the sum of two primes It was checked by computers for all even integers up to the order of 10 18 but a general proof is still unknown The twin prime conjecture states that there is an infinite number of pairs of primes with difference 2 Diophantine equations are also a typical number theoretical topic These are indeterminate polynomial equations that are to be solved in integers only For example Pell s equation is x 2 n y 2 1 where n is a nonsquare integer e g 2 It was studied as early as 400 BC in India and Lagrange proved that it

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Open archived version from archive - Mathematics In Europe - Mathematics in Europe

Anasayfa Haberler Bilgiler Halka ulaşma faaliyetleri Ulusal faaliyetler Halka açılma makaleler Yarışmalar Matematik Yardım Meslek olarak matematik Karışık Misyon Hoşgeldin Mesajları EMS Destekçiler Diller Iletişim Yasal Bilgi arama The European Mathematical Society Our Sponsor Munich RE Mathematics in Europe Ayrıntılar Kategori math in europe Mathematicians are being educated in all the countries of Europe The significance for the development of the subject and the intensity with which research is being

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Open archived version from archive