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- Mathematics In Europe - How far?

Maciejowice to Ryczywól I have to cross the Vistula so I need to find a bridge The closest bridge is in Deblin and the road there is about 60 kilometres So by distance we need do not necessarily mean the shortest segment joining two points By the way we regularly think about distances in different ways sometimes not even realising what we are doing How do we answer the question about the distance between Berlin and Sydney Considering a straight line Not at all We would travel through the centre of the planet We investigate instead the length of an arc on the surface of the Earth These examples show that there are distances of various kinds No wonder then that we may decide consider a general definition of distance Such a procedure in which a number of examples provide the foundation for a general definition is quite frequent in mathematics We may introduce the notion of distance in any nonempty set We require that the distance between every two points contained in our set should be defined the distance between the point and itself should be equal to 0 the distance between two different points should be greater than 0 the distance is symmetric i e the distance between A and B is the same as the distance between B and A the triangle inequality holds i e the sum of distances between A and C and between C and B is greater or equal to the distance between A and B that is if we go from A to B and want visit C in our way then we cannot shorten the route by visiting C on the way Mathematicians call a distance with these properties a metric on the set and the set is called a metric

Original URL path: http://mathematics-in-europe.eu/tr/bilgiler?id=78 (2013-11-18)

Open archived version from archive - Mathematics In Europe - Stefan Banach

very surprising applications of mathematics He had very original ideas and knew a lot He was a man of great culture and deep knowledge of literature His aphorisms remarks and thoughts are famous to this day Here are a few that are unfortunately most of them including the best are not translatable It is easy to remove God from his place in the universe But such good positions don t remain vacant for long Strip tease should be strictly forbidden This is the only way of keeping this beautiful and useful custom alive It is easy to go from the house of reality to the forest of mathematics but only few know how to go back Once when somebody was decorated with a medal Steinhaus said Now I know what to do in order to be awarded a medal Nothing but for a very long time After the Second World War Steinhaus was elected a member of sone scientific committee Among the members of this committee there were many poor scientists who were the members of it because of political reasons Once Steinhaus did not come for the meeting of the committee and he was asked to explain the reason of his absence He answered I am not going to give any reason of my absence until some others will give reasons for their presence here He used to say that a computer is an extremely efficient idiot He was an accomplished populariser of mathematics His book Mathematical Snapshots first published in 1938 was translated into many languages Strangely enough it was not reissued in Poland between 1957 and 1990 Stanislaw Mazur 1905 1981 a mathematician who offered a goose as a reward for solving one problem Banach s pupil and friend was also an excellent mathematician Like Banach he did not publish many of his results but for another reason Banach had too many ideas and results whereas Mazur did not like publishing For Mazur only two things were interesting mathematics and communism Before the war it was not known that he was a member of the Communist Party Mazur was 13 years younger than Banach He started his study when Banach was already a professor Nevertheless Banach treated Mazur as a partner They worked together Mazur frequently judged Banach s ideas gave details of proofs Other Banach s excellent pupils working in Lvov were Juliusz Pawel Schauder and Wladyslaw Orlicz In the thirties Kazimierz Kuratowski a Warsaw mathematician came for some years to Lvov One of Kuratowski s pupils was Ulam Stanislaw Ulam known later throughout the world as Stan Ulam was 5 born in 1909 in Lvov where he studied and initially worked Already as a first year student he obtained original mathematical results which were soon published Following an invitation by John von Neumann one of the greatest mathematicians of the first half of the 20th century he went to the United States in 1935 and settled there Among other things Ulam is most famous for his research in nuclear physics performed at Los Alamos for 25 years 1943 1967 He was one of the discoverers of the theoretical foundations of the construction of the hydrogen bomb He had broke scientific interests and obtained important results in various areas of mathematics set theory topology measure theory group theory functional analysis ergodic theory probability and game theory as well as in a number of sciences technology computer science physics astronomy and biology He developed original methods of propulsion of vessels moving above the earth s atmosphere Ulam died in 1984 His life and work are described in the book From Cardinals to Chaos published posthumously at Los Alamos and in his autobiography Adventures of a Mathematician Another young mathematician who moved abroad from Lvov before the war was Mark Kac well known because of his achievements in probability theory and statistics The names of many mathematicians from the Lvov Mathematical School are until now very well known in the world but for sure the most frequently mentioned is Banach s name It turned out that Banach is the mathematician from the whole world who is mentioned most frequently in the titles of mathematical scientific papers in the 20th century Banach s name is associated with many important by now classical theorems the Hahn Banach Theorem the Banach Steinhaus Theorem the Banach Open Mapping Theorem the Banach Alaoglu Theorem the Banach Closed Graph Theorem and the Banach Fixed Point Theorem And above all there is the fundamental mathematical concept of a Banach space What is it We learn in school about straight lines planes and three dimensional space We can describe these geometric objects by means of numbers Specifically we can identify the points on a straight line with single real numbers the points on a plane with pairs of real numbers and the points in space with triples of real numbers This idea can be extended to the study of finite sequences of numbers These sequences can be added and multiplied by numbers very much like vectors In this way we create so called finitedimensional spaces We can and do go further We add and multiply by constants numerical valued functions by defining the sum of two such 6 functions at a point to be the sum of their values at the point in question In this we are no longer dealing with a finite dimensional spaces It turned out for a variety of reasons that function spaces are very useful in many investigations and applications To a large extent modern mathematics is concerned with the study of general structures specific models of which have been known for a long time One advantage of studying a general structure is the economy of thought a theorem proved for the general structure need not be re proved for its different models Moreover the general proof makes it easier to identify the properties utilized in the course of the proof and thus makes it more transparent It is paradoxical but

Original URL path: http://mathematics-in-europe.eu/tr/bilgiler?id=79 (2013-11-18)

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

from the notion that mathematical terms and concepts denote abstract mathematical objects that exist in a separate mathematical world that is independent of the work of mathematicians According to the Platonists the task of the mathematician consists in determining the properties of and relationships among mathematical objects and thereby increasing our knowledge of mathematical reality Empiricism The philosophy of empiricism whose origins are to be found in the Anglo Saxon philosophy of the seventeenth century is determined by an epistemological idea Empiricists maintain that to perceive is the same as to experience in the sense that all knowledge comes through experience and only through experience This idea is opposed above all to the philosophy of rationalism which denies to the senses any contribution to knowledge and declares rational understanding as the only means of obtaining truth The expression of mathematics represents a problem for traditional empiricism because here we are dealing with an area of knowledge that is commonly seen as independent of experience The empiricists countered this by asserting that mathematical theorems describe quite general features of empirical reality In this sense they agree most closely with the assertions of physics differentiating themselves from these only in their greater generality Kant s Philosophy of Mathematics Kant s philosophy is based on the idea that all knowledge of empirical reality is a product of intuition on the one hand and certain active products of reason on the other According to Kant reason arranges the material of experience according to certain principles such as space and time which are not gathered from experience but precede it to the extent that they make experience possible in the first place We become aware of such principles of reasoning among which Kant includes the theorems of pure mathematics by reflecting on the conditions for

Original URL path: http://mathematics-in-europe.eu/tr/bilgiler?id=81 (2013-11-18)

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

and Bertrand Russell was the idea that mathematics should be given a secure foundation that would protect it from fallacies and misunderstandings of a logico mathematical or metaphysical nature This was to be accomplished by the definition of foundational mathematical concepts indeed those of arithmetic for the essential trouble spot seemed to lie in the supposed undefinability of such concepts Gottlob Frege Bertrand Russell Now to define a concept means to explain it in terms of an equivalent expression that is already understood In the case of arithmetic according to logicism logical symbols should play the role of the already known expressions Thus Frege and Russell gave definitions of the idea of number and the elementary calculational operations that contain exclusively logical constants such as and or there exists an x such that and allow for the proof of elementary arithmetic statements such as 2 2 4 by purely logical means What is remarkable here is that in spite of the collaborative nature of their program the philosophical positions of Russell and Frege exhibited clear differences Thus while the two were in agreement that logic should serve as the foundation of mathematics when it came to the nature of logical

Original URL path: http://mathematics-in-europe.eu/tr/bilgiler?id=82 (2013-11-18)

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

by 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

Original URL path: http://mathematics-in-europe.eu/tr/bilgiler?id=83 (2013-11-18)

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

theorems 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

Original URL path: http://mathematics-in-europe.eu/tr/bilgiler?id=84 (2013-11-18)

Open archived version from archive - Mathematics In Europe - Algebra

is 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

Original URL path: http://mathematics-in-europe.eu/tr/bilgiler?id=85 (2013-11-18)

Open archived version from archive - Mathematics In Europe - Combinatorics

two 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

Original URL path: http://mathematics-in-europe.eu/tr/bilgiler?id=90 (2013-11-18)

Open archived version from archive