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- Mathematics In Europe - Philosophy of mathematics: logicism

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 propositions and hence mathematical theorems Frege

Original URL path: http://mathematics-in-europe.eu/tr/43-information/history-philosophy/82-philosophy-of-mathematics-logicism (2013-11-18)

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

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 can be derived from

Original URL path: http://mathematics-in-europe.eu/tr/43-information/history-philosophy/84-philosophy-of-mathematics-formalism (2013-11-18)

Open archived version from archive - Mathematics In Europe - Anasayfa

mathematical axioms are linguistic conventions that implicitly define the meaning of the expressions that contain them And derived theorems represent merely the logical consequences of these linguistic conventions Philosophy of mathematics logicism Ayrıntılar Kategori History philosophy The basis for logicism whose originators were Gottlob Frege 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 propositions and hence mathematical theorems Frege adopted a Platonic point of view while that of Russell was empirical In the end the logicist program foundered on certain paradoxes antinomies that follow from the logicist principles While Russell eventually was able to develop methods that avoided these contradictions the principles that had to be employed were not of a logical nature In this sense the logicist program of reducing all of mathematics to logic could be viewed as having failed Philosophy of mathematics intuitionism Ayrıntılar Kategori History philosophy Alongside logicism and formalism intuitionism is the third front in the struggle to establish the foundations of mathematics Its philosophical roots can be found in the Kantian philosophy of mathematics The most important proponent of intuitionism was the Dutch mathematician L E J Brouwer L E J Brouwer Of central importance to intuitionism is the idea that mathematics is a creation of the human mind According to Brouwer we recognize the properties of mathematical objects by constructing them through introspection The mathematician does not discover facts that exist independently of the field rather one discovers so to speak mathematical formulas 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

Original URL path: http://mathematics-in-europe.eu/tr/43-information/history-philosophy (2013-11-18)

Open archived version from archive - Mathematics In Europe - 1913: Publication of The Concept of a Riemann Surface, by Hermann Weyl

up in regions where the function was already defined obtaining values however of the function that differ from those originally obtained at the same points There are two apparent ways to deal with the problem Either one gives up on the idea of uniqueness of function values or else one alters the domain of definition in such a way that it covers the complex plane several perhaps infinitely many times the individual layers being joined in a complicated way somewhat analogously to how the floors of a parking garage are linked Before the appearance of Weyl s book this concept was implemented intuitively in simple cases being exemplified by cardboard models with the multiple layers glued one to the next I well remember that the construction of such models was among the problems in complex analysis that I had to solve during my student years back in the 1950s The essence of Weyl s book was to describe such surfaces called Riemann surfaces after the first mathematician to describe them axiomatically as two dimensional manifolds and to study complex valued functions on them without reference to how they were originally constructed and without reference to their embedding in a three dimensional space which ultimately led to the characterization of surfaces by the functions that exist on them Closed surface of genus 3 Nonorientable closed surface Consequently the book consists of two parts called chapters by Weyl In the first part Riemann surfaces are defined and studied using topological concepts and methods where it is to be kept in mind that before 1913 much was not clearly understood in this area with most ideas resting on intuition Thus this chapter is of considerable independent interest even without reference to functions of a complex variable for the development of the topology of

Original URL path: http://mathematics-in-europe.eu/tr/bilgiler/tarih-felsefe/47-information/math-calendar/971-1913-publication-of-the-concept-of-a-riemann-surface-by-hermann-weyl (2013-11-18)

Open archived version from archive - Mathematics In Europe - 2013: 50th Anniversary of the Proof of the Independence of the Continuum Hypothesis

they therefore cannot be mapped bijectively to a set that is countable in the sense of the model only because these mappings are lacking in the model as sets of ordered pairs In 1938 Kurt Gödel constructed a countable minimal model of the Zermelo Fraenkel axiom system of set theory in which both the likewise contested axiom of choice and the generalized continuum hypothesis are satisfied that is in which there is no intermediate cardinality between an infinite set and its power set In 1963 Paul Joseph Cohen 1934 2007 brought this entire issue to a satisfactory conclusion by employing a new method of model construction which he called forcing to show that it follows from the assumption of the existence of a countable basic model of set theory in which even the axiom of choice is satisfied that there exist models in which there are arbitrarily many cardinalities intermediate between the cardinality of an infinite set and that of its power set The significance of this is that just as in the case of the validity of the axiom of choice we may freely choose whether or not to assume the existence of a cardinality between that of the natural numbers and that of the real numbers Therefore there is no uniquely determined continuum that exists independently of us rather our continuum has in many respects only the properties that we grant it based on our predilections Like Gödel s model Cohen s models are countable and consist essentially of strings of characters They are therefore almost tangible Paul Joseph Cohen Cohen s accomplishment was honoured at the 1966 International Congress of Mathematicians in Moscow with the award of the Fields Medal the highest honour that a mathematician can be granted and his method of model construction has since been used by many others to prove new theorems about the pseudo universe of unimaginably large sets a universe that has taken on quite a dubious aspect Those looking for information on Cohen s biography and the technique of forcing can find a great deal of information in the Internet From a purely formal point of view axiomatic set theory s relationship to the axiom of choice and the continuum hypothesis is analogous to elementary geometry s relationship to the parallel postulate However there is an important difference In the case of geometry in our everyday lives practical experience this refers to the inapplicability in principle of both Euclidean and hyperbolic geometry to the physics of elementary particles and to the physics of the cosmos causes us to decide in favour of Euclidean geometry In questions concerning uncountable sets we can neither experience nor experiment since such sets do not exist in a material sense To be sure a relatively small group of specialists in axiomatic set theory have since 1963 established a number of further astounding and admirable results on uncountable sets and even on inaccessibly large sets But the majority of mathematicians have not been affected by these

Original URL path: http://mathematics-in-europe.eu/tr/47-information/math-calendar/972-2013-50th-anniversary-of-the-proof-of-the-independence-of-the-continuum-hypothesis (2013-11-18)

Open archived version from archive - Mathematics In Europe - 1912: There Is a Winning Strategy for Every Strategic Two-Person Game

that has become accepted only since this transformation as proper mathematics including the notion of directed and undirected graphs possibly with labelled vertices or edges Thus from today s point of view every mathematical result is at bottom an application of set theory Zermelo interpreted the states or situations of a game as vertices of a directed graph and the game moves as directed edges that lead from one game state to another There is a start state and the player who makes the first move from this state is called the first player the other player is called the second player The players alternate moves It is assumed that there are only finitely many game states altogether and no possibility of returning to a game state that has already been achieved once or else there is a rule limiting the number of such returns from which it follows that there are only finitely many ways that the game can proceed from beginning to end and every game must end after finitely many moves In the resulting graph of the game which is essentially a tree each vertex with no edge leading out of it is according to the rules specified as a winning state for one of the two players Zermelo s idea was that beginning at these end situations one can determine recursively proceeding through the graph layer by layer whether the first or second player has a guaranteed winning strategy when the game is in a given state The player whose turn it is has a winning strategy if he or she has at least one possible move that leads to a winning situation If on the other hand every possible move of that player leads to a state that is a winning situation for the other player then the given state is a winning situation for this other player To be sure this process of determining winning states can be illustrated only on very simple games since in games with any complexity the number of situations that can arise and the number of moves that generally constitute a playing of the game are much to large to be drawn on a sheet of paper As an example let us consider a simple game of nim Beginning with nine tiles the players alternately remove one two or three of them The player forced to remove the last tile loses the game In the figure the first player s moves are indicated by a single line while those of the second player are shown by a double line where the direction of the game is from top to bottom in the graph and each game state is indicated as a winning state for the first player by a circle and for the second player by a square It turns out that the second player has at least one and indeed several how many winning strategies From this graphical representation one can count the number of ways that a

Original URL path: http://mathematics-in-europe.eu/tr/47-information/math-calendar/770-1912-there-is-a-winning-strategy-for-every-strategic-two-person-game (2013-11-18)

Open archived version from archive - Mathematics In Europe - 2012: 1612 The first book on recreational mathematics is published

years later he achieved what would make his name famous in the annals of mathematics the first Greek Latin edition of the Arithmetica of Diophantus 1621 It was in the margin of his copy of the second edition of this book that Fermat wrote down his famous conjecture that became known as Fermat s last theorem Here however our topic is the Problèmes plaisans The book begins with problems of the following type I am thinking of a number that Bachet presents many such problems some with unusual variants such as this one About a number less than 60 I will tell you only its remainders on division by 3 4 and 5 Or this I am thinking of a number and tell you whether thrice its square is even or odd If it is even I take half of it otherwise I add 1 and then take half then multiply the result by 3 and tell you the result following integer division by 9 discarding the remainder What number am I thinking of In addition there are problems on weighing objects of integral weight with a minimal number of weights with integer values where the cases are distinguished according to whether the weights can be used on one or both pans of a balance In the latter case one can for example weigh two units by placing a 1 weight on one pan and a 3 weight on the other There follow problems on measuring out a certain quantity of liquid by pouring back and forth among several pitchers of given volume and problems on Latin squares formulated in terms of playing cards that are to laid out in a certain pattern as well as a recipe for creating magic squares In a later edition one had to place

Original URL path: http://mathematics-in-europe.eu/tr/47-information/math-calendar/643-2012-1612-the-first-book-on-recreational-mathematics-is-published (2013-11-18)

Open archived version from archive - Mathematics In Europe - 2011: 75 Years of Turing Machines and Turing Computability

unit While Turing s first machine was developed for the purpose of producing the infinite sequences of the decimal digits of computable real numbers and hence it would never cease operating once set in motion Post envisaged mechanical transformation processes of character strings that would produce a result when a computation halted after finitely many steps This first clear definition of the computability of string functions that is functions whose arguments and values are character strings from a given finite alphabet includes a definition of the concepts of decidability and countability of a formal language Turing introduced the notion of a universal machine and showed how simply with the help of this concept one can define algorithmically undecidable problems From then on there commenced a pronounced bifurcation in the development of the subject In one direction there followed a host of results on the connections between computability countability and decidability as well as on the undecidability of special mathematical and logical problems such as the various word problems of group and semigroup theory All of this can be considered part of mathematics in the traditional sense In the other direction first came proofs of the equivalence of Turing computability with various other clarifications of the intuitive notion of computability for example Alonzo Church s l calculus or the recursive functions defined inductively by basis functions and generating principles which supported the thesis formulated by Church in 1936 and named after him Church s thesis on the equivalence of the precise and intuitive notions of computability This thesis cannot of course be proved in the mathematical sense since it equates a precise notion with an imprecise one Thereafter numerous variants of the original Turing machine were invented including those with two dimensional instead of linear storage and with apparently stronger or

Original URL path: http://mathematics-in-europe.eu/tr/47-information/math-calendar/615-2011-75-years-of-turing-machines-and-turing-computability (2013-11-18)

Open archived version from archive