The most recent special issue of* positions: asia critique,* “Benjamin’s Travel,” edited by Briankle G. Chang, is now available.

Walter Benjamin’s writings are popular among Chinese scholars, but variances of translation and interpretation have created an understanding of Benjamin that bears little resemblance to how Western scholars discuss and use Benjamin. This special issue uses that dissemblance as a starting point to explore what Benjamin’s writings have meant and continue to mean, bringing these multiple different versions of Benjamin into conversation. Contributors explore Benjamin’s fascination with the spiritual power of color, connect his youthful fascination with Chinese thought with his later writings, compare his ideas to the work of Chinese filmmaker Jia Zhangke and Vietnamese author Bùi Anh Tuấn, and analyze his experiments in imbuing book reviews with social commentary. This issue also includes a new translation of Benjamin’s essay “Chinese Paintings at the National Gallery.”

Browse the table of contents and read the introduction to the issue, now freely available.

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AMERICAN AND EUROPEAN CULTURE

ABSTRACT

The Western world (i.e. Europe, the Americas, Australia and New Zealand) could be considered as a single “Western civilisation”. ‘Westernness’ could be defined by people who are ethnically or culturally European, in other words people of European descent or speaking a European language as their mother-tongue.

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ABSTRACT

Joseph-Louis Lagrange is usually considered to be a French mathematician, but the Italian Encyclopaedia [40] refers to him as an Italian mathematician. They certainly have some justification in this claim since Lagrange was born in Turin and baptised in the name of Giuseppe Lodovico Lagrangia. Lagrange’s father was Giuseppe Francesco Lodovico Lagrangia who was Treasurer of the Office of Public Works and Fortifications in Turin, while his mother Teresa Grosso was the only daughter of a medical doctor from Cambiano near Turin. Lagrange was the eldest of their 11 children but one of only two to live to adulthood.

Turin had been the capital of the duchy of Savoy, but became the capital of the kingdom of Sardinia in 1720, sixteen years before Lagrange’s birth. Lagrange’s family had French connections on his father’s side, his great-grandfather being a French cavalry captain who left France to work for the Duke of Savoy. Lagrange always leant towards his French ancestry, for as a youth he would sign himself Lodovico LaGrange or Luigi Lagrange, using the French form of his family nam

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School children meet the number line in the early grades. By high school algebra and geometry, the real number line has become a central concept. But really, what is the real number line? Is it a figment of our imagination? How do we define it as something more concrete?

A child’s intuition of the real number line as a straight line in a plane or in space is derived from experience with straight line segments in real life, as the edge of a ruler, the border of a page of paper, the lines on graph paper, the edges of tables, or the lines where the walls meet the ceiling. But what if the line is extended into space, say to Jupiter, or beyond? What happens as the line approaches the outer reaches of space? Even the concept of space itself is based on a precise notion for number line.

And what are the individual real numbers? The child’s intuitive model for a real number corresponds to a dot made with pencil on paper. But each dot really corresponds to a multitude of points, a mound of graphite. Does the heap of graphite represent something other than vacuum? What really are “pi” and “the square root of 2”?

An intuitively appealing construction of the rational numbers is based upon Euclidean geometry. It runs as follows. One starts with a straight line, one marks a point and labels it 0, and one marks a different point and labels it 1. Then one constructs the other integers by marking off steps of equal length, and one constructs the rational numbers by dividing the segments between integers into equal parts. In this model, the real number line, stripped of its arithmetic, is taken as a primitive concept and subjected to the axioms of Euclidean geometry (say Hilbert’s axioms, which are studied in a course on the foundations of geometry; Euclid himself simply proceeded with blind faith that the constructions he performed did not stumble into any holes). And how do we know there is a model of Euclidean geometry? The canonical model for Euclidean geometry is the Cartesian plane consisting of ordered pairs of real numbers, and the verification of the axioms of Euclidean geometry depends on the properties of the real number line. If we follow this route to construct the real numbers from a Euclidean straight line, we find we have traveled in a logical circle.

The circular reasoning that appears in some high school algebra textbooks is not so subtle. In one of them, the rational numbers are defined as quotients of integers, the irrational numbers are defined as the real numbers that are not rational, and then the real numbers are defined as the aggregate of the rational and the irrational numbers.

The book Mathematics for High School Teachers, by Usiskin, Stanley, et al., treats the real numbers in Chapters 2 and 6. In Chapter 2, reference is made to various methods of constructing the real numbers from the rational numbers, without attempting to give a precise definition of the real numbers. Then the authors take a straight line, mark off 0 and 1, represent the rational numbers on the line, and go on to explore in some detail the decimal representation of real numbers. They return in Chapter 6 to the field axioms, and they establish the uniqueness of a complete ordered field. The question of existence is never completely nailed down. Yet they come close, when they say: “In school algebra, real numbers are commonly described as numbers that can be represented by finite or infinite decimals.”

EXERCISE: Suppose a persistent high school student asks you to explain exactly what real numbers are. What explanation would you give the student?

The goal of these notes is to bring you to a point where you can give the student a satisfactory answer to this question. Your answer might be brief, but you should feel confident that you can supply as much detail as the student might insist upon. In particular, you should understand in what sense the real numbers “are” the set of decimals.

the REAL Number LIne

Rather than specify concretely what a real number is, we will describe the real number line by listing its properties. This is done by defining an axiom system. The primitive concepts in the axiom system are points (real numbers), the operations of addition and multiplication, and an order relation. The list of axioms is quite long, but with one exception they are not difficult to understand. They are familiar properties of the rational numbers. The one exception is the “completeness axiom,” which says that there are no “holes” in the real number line. We refer to any model for the axiom system as “the real number line” or “the field of real numbers.” In other words, the real number line is a set with arithmetic and ordering that satisfies the “real number axioms.”

There are two important facts that justify our use of the expression “the real number line.” First, there is a model for the axiom system. Second, any two models for the axiom system are isomorphic, that is, they can be put in a one-to-one correspondence so that the arithmetic and the ordering correspond. In other words, the real number line exists, and it is unique. We may perform arithmetic operations on the set with confidence, without pausing to consider where the set comes from or where it is going. (The K-12 student is generally happy to perform arithmetic operations on real numbers, oblivious of the defining properties of the real numbers, confident that there is such an entity, and not the least concerned about whether such an entity is unique.)

So what are the real number axioms? The axioms come in three batches corresponding to arithmetic, ordering, and completeness. The axioms taken together assert that the real numbers form a “complete ordered field.”

The construction of the real numbers is usually carried out in a foundational upper division course in analysis (Math 131A at UCLA). The arithmetic axioms, in various combinations, are studied in more detail in upper division algebra courses (Math 110AB and Math 117 at UCLA). The arithmetic axioms assert that the real numbers form a field. The completeness axiom in the form of the Least Upper Bound Axiom is usually introduced in the first calculus course. Completeness is treated in more detail in the foundational analysis course or in a more advanced topology course (Math 121 at UCLA), in the context of metric spaces. The ordering and completeness axioms also appear in some form in Hilbert’s axiom system for Euclidean geometry, which is treated in a course on the foundations of geometry (Math 123 at UCLA).

The order axioms assert that there is a relation “ < ” defined between certain elements, which satisfies the following rules.

5. The trichotomy law asserts that exactly one of the relations x<y, y<x, or x=y holds between any two given x and y.

We write x <= y as shorthand for x x to mean x = x to mean x <= y.

6. The law of transitivity asserts that if x<y and y<z, then x<z.

7. The law of compatibility with addition asserts that if x < y, then x+z < y+z.

8. The law of compatibility with multiplication asserts that if x 0, then ax < ay.

A field with an ordering that satisfies these axioms is called an ordered field.

EXERCISE: Show from the axioms that -1 < 0 and 0 = 0 for any x in an ordered field. Deduce from this that the complex numbers cannot be ordered to become an ordered field.

EXERCISE: Show from the axioms that in an ordered field, the elements 1, 1+1, 1+1+1, 1+1+1+1, … are distinct.

If the elements 1, 1+1, 1+1+1, 1+1+1+1, … of a field are distinct, we say that the field has characteristic zero. If these elements are not distinct, there is a first positive integer p such that 1+1+…+1 [p summands] is 0. In this case, we say that the field has characteristic p.

EXERCISE: Show that the characteristic of a field is either 0 or a prime, that is, show that the number p above is a prime number.

There is some standard notation that is convenient. In any field, we write 1+1 = 2, 1+1+1 = 3, 1+1+1+1 = 4, and so on. As usual, -n denotes the additive inverse of n. If the field has characteristic zero, we identify these elements with the integers Z, and we regard Z as a subset of the field. Under this identification, addition and multiplication in Z are the same as addition and multiplication in the field. Further, the subfield generated by 0 and 1 (the smallest subfield containing 0 and 1) is isomorphic to the field of rational numbers. In other words, we can regard the rational numbers as being a subset of any field of characteristic zero, and in particular of any ordered field.

EXERCISE: Define the absolute value function by |x| = x if x>= 0, and |x| = -x if x < 0. Show from the axioms that |x+y| 0 and b > 0, there is an integer m>0 such that ma > b.

If the ordered field satisfies the Archimedean order axiom, we call it an Archimedean ordered field. By taking a=1 in the Archimedean ordering axiom we see that each b > 0 in the field is bounded above by some positive integer m. Let n be the first integer such that b = 0, and n <= b < n+1. The integer n is the leading entry in the decimal expansion of b. We return to decimal expansions later.

EXERCISE: In an ordered field, let (a,b) denote the open interval from a to b, that is, the set of x in the field satisfying a < x

0, there is a positive integer n such that x > 1/10n.THE COMPLETENESS AXIOM

The completeness axiom for the real numbers is the tersest, yet the most difficult to understand. To state it, we need some preliminary definitions. Let S be a subset of the ordered field. We say that b is an upper bound for S if x <= b for all elements x of S. We say that b is a least upper bound for S if b is an upper bound for S, and

b 0. Let S be the set of multiples a, 2a, 3a, … of a. Let c be an upper bound for S. Then (n+1)a <= c for all positive integers n, so that na <= c – a for all positive integers n. Thus c – a is also an upper bound for S, and further c – a < c. We conclude that S does not have a least upper bound. By the LUB axiom, S is not bounded above. Consequently for each b, there is some n such that b 0 and b > 0 such that ma <= b for all positive integers m.”

three models for the real number line

There are three methods that are often used to construct the real numbers. Each method has its advantages and its disadvantages. Each method leads to a model for the real numbers, that is, a set with addition, multiplication, and ordering that satisfy the axioms for complete ordered field. We shall refer to the three models respectively as the Weierstrass-Stolz model (decimal expansions, the most intuitive model), the Dedekind model (Dedekind cuts, the slickest model), and the Meray-Cantor model (completion of a metric space, the most far-reaching model).

Decimal expansions

It was Otto Stolz (1886) who pointed out that decimal expansions can be used to define the real numbers. In the Weierstrass-Stolz model, we define the real numbers to be the set of all decimal expansions a = a0.a1a2a3…, where a0 is an integer (positive or negative), and a1, a2, a3, … are integers between 0 and 9, except that we declare a decimal expansion that terminates in all nines to be the same real number as the (terminating) decimal expansion obtained by incrementing the last non-nine term by 1 and replacing the subsequent 9’s by 0’s. Thus for instance we regard 3.2599999… and 3.2600000… as the same real number. (This unfortunate complication is not an essential difficulty, but it does make the verification of the arithmetic axioms into a tedious exercise.)

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Camille Jordan’s father, Esprit-Alexandre Jordan (1800-1888), was an engineer who had been educated at the École Polytechnique. Camille’s mother, Joséphine Puvis de Chavannes, was the sister of the famous painter Pierre Puvis de Chavannes who was the foremost French mural painter of the second half of the 19th century. Camille’s father’s family were also quite well known; a grand-uncle also called Ennemond-Camille Jordan (1771-1821) achieved a high political position while a cousin Alexis Jordan (1814-1897) was a famous botanist.

Jordan studied at the Lycée de Lyon and at the Collège d’Oullins. He entered the École Polytechnique to study mathematics in 1855. This establishment provided training to be an engineer and Jordan, like many other French mathematicians of his time, qualified as an engineer and took up that profession. Cauchy in particular had been one to take this route and, like Cauchy, Jordan was able to work as an engineer and still devote considerable time to mathematical research. Jordan’s doctoral thesis was in two parts with the first part Sur le nombre des valeurs des fonctions Ⓣ being on algebra. The second part entitled Sur des periodes des fonctions inverses des intégrales des différentielles algebriques Ⓣ was on integrals of the form ∫ u dz where u is a function satisfying an algebraic equation f (u, z) = 0. Jordan was examined on 14 January 1861 by Duhamel, Serret and Puiseux. In fact the topic of the second part of Jordan’s thesis had been proposed by Puiseux and it was this second part which the examiners preferred. After the examination he continued to work as an engineer, first at Privas, then at Chalon-sur-Saône, and finally in Paris.

Jordan married Marie-Isabelle Munet, the daughter of the deputy mayor of Lyon, in 1862. They had eight children, two daughters and six sons.

From 1873 he was an examiner at the École Polytechnique where he became professor of analysis on 25 November 1876. He was also a professor at the Collège de France from 1883 although until 1885 he was at least theoretically still an engineer by profession. It is significant, however, that he found more time to undertake research when he was an engineer. Most of his original research dates from this period.

Jordan was a mathematician who worked in a wide variety of different areas essentially contributing to every mathematical topic which was studied at that time. The references [3], [4], [5], [6] are to the four volumes of his complete works and the range of topics is seen from the contents of these. Volumes 1 and 2 contain Jordan’s papers on finite groups, Volume 3 contains his papers on linear and multilinear algebra and on the theory of numbers, while Volume 4 contains papers on the topology of polyhedra, differential equations, and mechanics.

Topology (called analysis situs at that time) played a major role in some of his first publications which were a combinatorial approach to symmetries. He introduced important topological concepts in 1866 built on his knowledge of Riemann’s work in topology but not the work by Möbius for he was unaware of it. Jordan introduced the notion of homotopy of paths looking at the deformation of paths one into the other. He defined a homotopy group of a surface without explicitly using group terminology.

Jordan was particularly interested in the theory of finite groups. In fact this is not really an accurate statement, for it would be reasonable to argue that before Jordan began his research in this area there was no theory of finite groups. It was Jordan who was the first to develop a systematic approach to the topic. It was not until Liouville republished Galois’s original work in 1846 that its significance was noticed at all. Serret, Bertrand and Hermite had attended Liouville’s lectures on Galois theory and had begun to contribute to the topic but it was Jordan who was the first to formulate the direction the subject would take.

To Jordan a group was what we would call today a permutation group; the concept of an abstract group would only be studied later. To give an illustration of the way he tried to build up groups theory we will say a little about his contributions to finite soluble groups. The standard way to define such groups today would be to say that they are groups whose composition factors are abelian groups. Indeed Jordan introduced the concept of a composition series (a series of subgroups each normal in the preceding with the property that no further terms could be added to the series so that it retains that property). The composition factors of a group G are the groups obtained by computing the factor groups of adjacent groups in the composition series. Jordan proved the Jordan-Hölder theorem, namely that although groups can have different composition series, the set of composition factors is an invariant of the group.

Although the classification of finite abelian groups is straightforward, the classification of finite soluble groups is well beyond mathematicians today and for the foreseeable future. Jordan, however, clearly saw this as an aim of the subject, even if it was not one which might ever be solved. He made some remarkable contributions to how such a classification might proceed setting up a recursive method to determine all soluble groups of order n for a given n.

A second major piece of work on finite groups was the study of the general linear group over the field with p elements, p prime. He applied his work on classical groups to determine the structure of the Galois group of equations whose roots were chosen to be associated with certain geometrical configurations.

His work on group theory done between 1860 and 1870 was written up into a major text Traité des substitutions et des équations algebraique Ⓣ which he published in 1870. This treatise gave a comprehensive study of Galois theory as well as providing the first ever group theory book. For this work he was awarded the Poncelet Prize of the Académie des Sciences. The treatise contains the ‘Jordan normal form’ theorem for matrices, not over the complex numbers but over a finite field. He appears not to have known of earlier results of this type by Weierstrass. His book brought permutation groups into a central role in mathematics and, until Burnside wrote his famous group theory text nearly 30 years later, this work provided the foundation on which the whole subject was built. It would also be fair to say that group theory was one of the major areas of mathematical research for 100 years following Jordan’s fundamental publication. as observed prof dr mircea orasanu for Louis University

Jordan’s use of the group concept in geometry in 1869 was motivated by studies of crystal structure. He considered the classification of groups of Euclidean motions. His work had gained him a wide international reputation and both Sophus Lie and Felix Klein visited him in Paris in 1870 to study with him. Jordan’s interest in groups of Euclidean transformations in three dimensional space influenced Lie and Klein in their own theories of continuous and discontinuous groups. but here ene horia ,marcel chirita ,prof dr ioan rosca or FAC MATEM Bucharest or sorin radulescu wirh arhimede publication have discovered EQ GR !and II and also other but prof drd doc discovered stream line and free lines and potential function , same stefan zarea and some

The publication of Traité des substitutions et des équations algebraique Ⓣ did not mark the end of Jordan’s contribution to group theory. He went on over the next decade to produce further results of fundamental importance. He studied primitive permutation groups and proved a finiteness theorem. He defined the class of a subgroup of the symmetric group to be c > 1 if c was the smallest number such that the subgroup had an element moving c points. His finiteness theorem showed that for a given c there are only finitely many primitive groups with class c other than the symmetric and alternating groups.

Generalising a result of Fuchs on linear diff

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Mathematics culture loses a great teacher and thinker, for on the 19th of December 1998 professor Francesco Speranza left us. He was born in Milano the 4th October 1932. In 1954 he graduated in Mathematics from Pavia University and he was assistant lecturer in Geometry at the University of Bologna from 1954 to 1967. In 1967, as full professor of Differential Geometry he taught at the University of Messina. In 1969 he moved to the University of Parma where, from 1974, he had the chair of “Matematiche Complementari”. He was a member (until his death) of the scientific commission of the Italian Mathematics Union, of the Italian Commission for Mathematics teaching, and other important commissions. His extensive researches, motivated by a great curiosity and desire to investigate, goes from differential geometry to didactics, history and epistemology of mathematics to art in mathematics. In the eighties he found in Parma a Centre for Mathematics Education and a group of researchers and teachers for concerned with researches in didactics. He was persuaded of the need of providing an epistemological basis for didactics. Following Enriques, Gonseth and Lakatos he didn’t erect barriers between philosophy of mathematics and experimental sciences. He considered himself to be a neo-empiricist, a non-absolutist, and a fallibilist. He claimed that this position parallels developments in the philosophy of science based on the great scientific revolutions of the 19th century and the beginning of the 20th century, and Popper’s school. In the early 1990s he founded a National Group of Epistemology of Mathematics which organises conferences every year. His great variety of cultural and scientific interests led him to publish more than 200 paper.and thus prof dr mircea orasanu has been considered

The Group of Mathematical Education Research of Parma: Lucia Grugnetti, Carlo Marchini, Daniela Medici, Margherita Michelotti, Maria Gabriella Rinaldi, Paola Vighi.

THE EDGE

Recommended free Web Journal touching on matters intellectual and philosophical including philosophy of mathematics

http://www.edge.org/

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for many situations are observed certain situations considered prof dr mircea orasanu as we obsrved that in these cases are considered certain situations as observed prof dr mircea orasanu as followed,and thus ) can be answered by the text, rather than by generalizations or by copious external research (e.g., “Why did Shakespeare depict madness in the way that he did?”).

Tips to keep in mind:

• “How” and “why” questions generally require more analysis than “who/ what/when/where.”

• Good analytical questions can highlight patterns/connections, or contradictions/dilemmas/ problems.

• Good analytical questions can also ask about some implications or consequences of your analysis.

Thus the question should be answerable, given the available evidence, but not immediately, and not in the same way by all readers. Your thesis should give at least a provisional answer to the question, an answer that needs to be defended and developed. Your goal is to help readers understand why this question is worth answering, why this feature of the text is problematic, and to send them back to t• Barbara A. Crawford

and here prof dr mircea orasanu observed that this prof dr must reviews many and more situations so must contact me

CONTACT INFORMATION

• barbarac@uga.edu

• 706-542-0989 (office)

View/Download CV

OFFICE LOCATION

104 C Aderhold Hall

110 Carlton Street

Athens, Georgia 30602

LINKS

• The Fossil Finders Project

• Undergraduate Biology Education Research (UBER)

Education

DEGREE CONCENTRATION INSTITUTION YEAR

Ph. D. Science Education The University of Michigan 1996

M. S. Biology (Limnology) The University of Michigan

B. S. Microbiology The University of MIchigan

Awards and Accolades

YEAR AWARD FROM

2012 Elected Fellow of the American Association for the Advancement of Science (AAAS). AAAS

2000 Provosts’ Award for Collaboration, Science Education Group Pennsylvania State University

1996 Recipient of the James B. Edmondson Award, Outstanding Dissertation Award for Secondary Curriculum and Instruction, School of Education The University of Michigan

2016 Elected President-elect of NARST A worldwide organization NARST

AREAS OF EXPERTISE

• science teacher education

• science teaching and learning

RESEARCH INTERESTS

• inquiry-based science teaching and learning

• teacher professional development

• teacher

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