• Journal for History of Mathematics
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• v.24 no.4
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• pp.45-59
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• 2011

Constructionists believe that mathematical knowledge is obtained by a series of purely mental constructions, with all mathematical objects existing only in the mind of the mathematician. But constructivism runs the risk of rejecting the classical laws of logic, especially the principle of bivalence and L. E. M.(Law of the Excluded Middle). This philosophy of mathematics also does not take into account the external world, and when it is taken to extremes it can mean that there is no possibility of communication from one mind to another. Two constructionists, Brouwer and Dummett, are common in rejecting the L. E. M. as a basic law of logic. As indicated by Dummett, those who first realized that rejecting realism entailed rejecting classical logic were the intuitionists of the school of Brouwer. However for Dummett, the debate between realists and antirealists is in fact a debate about semantics - about how language gets its meaning. This difference of initial viewpoints between the two constructionists makes Brouwer the intuitionist and Dummettthe the semantic anti-realist. This paper is confined to show that Dummett's proposal in favor of intuitionism differs from that of Brouwer. Brouwer's intuitionism maintained that the meaning of a mathematical sentence is essentially private and incommunicable. In contrast, Dummett's semantic anti-realism argument stresses the public and communicable character of the meaning of mathematical sentences.

• Journal for History of Mathematics
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• v.24 no.4
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• pp.1-6
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• 2011

We investigate a method to find the least common multiples of numbers in the mathematics book GuIlJib(구일집(九一集), 1724) written by the greatest mathematician Hong Jung Ha(홍정하(洪正夏), 1684~?) in Chosun dynasty and then show his achievement on Number Theory. He first noticed that for the greatest common divisor d and the least common multiple l of two natural numbers a, b, l = $a\frac{b}{d}$ = $b\frac{a}{d}$ and $\frac{a}{d}$, $\frac{b}{d}$ are relatively prime and then obtained that for natural numbers $a_1,\;a_2,{\ldots},a_n$, their greatest common divisor D and least common multiple L, $\frac{ai}{D}$($1{\leq}i{\leq}n$) are relatively prime and there are relatively prime numbers $c_i(1{\leq}i{\leq}n)$ with L = $a_ic_i(1{\leq}i{\leq}n)$. The result is one of the most prominent mathematical results Number Theory in Chosun dynasty. The purpose of this paper is to show a process for Hong Jung Ha to capture and reveal a mathematical structure in the theory.

• Journal for History of Mathematics
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• v.20 no.3
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• pp.127-146
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• 2007

From 1876 to 1883, British mathematician James Joseph Sylvester worked as the founding head of Mathematics Department at the Johns Hopkins University which has been known as America's first school of mathematical research. Sylvester established the American Journal of Mathematics, the first sustained mathematics research journal in the United States. It is natural that we think this is the most exciting and important period in American mathematics. But we found out that the International Congress of Mathematicians held at the World's Columbian Exposition in Chicago, August 21-26, 1893 was the real turning point in American's dedication to mathematical research. The University of Chicago was founded in 1890 by the American Baptist Education Society and John D. Rockefeller. The founding head of mathematics department Eliakim Hastings Moore was the one who produced many excellent American mathematics Ph.D's in early stage. Many of Moore's students contributed to build up real American mathematics research power in early 20 century The University also has a well-deserved reputation as the "teacher of teachers". Beginning with Sylvester, we analyze what E.H. Moore had done as a teacher and a head of the new department that produced many mathematical talents such as L.E. Dickson(1896), H. Slaught(1898), O. Veblen(1903), R.L. Moore(1905), G.D. Birkhoff(1907), T.H. Hilderbrants(1910), E.W. Chittenden(1912) who made the history of American mathematics. In this article, we study how Moore's vision, new system and new way of teaching influenced American mathematical society at early stage of the top class mathematical research. and the meaning that early University of Chicago case gave.

• The Journal of Art Theory & Practice
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• no.7
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• pp.165-188
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• 2009

In the development of linear perspective, Brook Taylor's theory has achieved a special position. With his method described in Linear Perspective(1715) and New Principles of Linear Perspective(1719), the subject of linear perspective became a generalized and abstract theory rather than a practical method for painters. He is known to be the first who used the term 'vanishing point'. Although a similar concept has been used form the early stage of Renaissance linear perspective, he developed a new method of British perspective technique of measure points based on the concept of 'vanishing points'. In the 15th and 16th century linear perspective, pictorial space is considered as independent space detached from the outer world. Albertian method of linear perspective is to construct a pavement on the picture in accordance with the centric point where the centric ray of the visual pyramid strikes the picture plane. Comparison to this traditional method, Taylor established the concent of a vanishing point (and a vanishing line), namely, the point (and the line) where a line (and a plane) through the eye point parallel to the considered line (and the plane) meets the picture plane. In the traditional situation like in Albertian method, the picture plane was assumed to be vertical and the center of the picture usually corresponded with the vanishing point. On the other hand, Taylor emphasized the role of vanishing points, and as a result, his method entered the domain of projective geometry rather than Euclidean geometry. For Taylor's theory was highly abstract and difficult to apply for the practitioners, there appeared many perspective treatises based on his theory in England since 1740s. Joshua Kirby's Dr. Brook Taylor's Method of Perspective Made Easy, Both in Theory and Practice(1754) was one of the most popular treatises among these posterior writings. As a well-known painter of the 18th century English society and perspective professor of the St. Martin's Lane Academy, Kirby tried to bridge the gap between the practice of the artists and the mathematical theory of Taylor. Trying to ease the common readers into Taylor's method, Kirby somehow abbreviated and even omitted several crucial parts of Taylor's ideas, especially concerning to the inverse problems of perspective projection. Taylor's theory and Kirby's handbook reveal us that the development of linear perspective in European society entered a transitional phase in the 18th century. In the European tradition, linear perspective means a representational system to indicated the three-dimensional nature of space and the image of objects on the two-dimensional surface, using the central projection method. However, Taylor and following scholars converted linear perspective as a complete mathematical and abstract theory. Such a development was also due to concern and interest of contemporary artists toward new visions of infinite space and kaleidoscopic phenomena of visual perception.

• School Mathematics
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• v.7 no.4
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• pp.391-402
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• 2005

The purpose of this study is to resolve misunderstanding for centroid of a triangle and to clarify several logical problems in finding the centroid of a Polygon. The conclusions are the followings. For a triangle, the misunderstanding that the centroid of a figure is the intersection of two lines that divide the area of the figure into two equal part is more easily accepted caused by the misinterpretation of a median. Concerning the equilibrium of a triangle, the median of it has the meaning that it makes the torques of both regions it divides to be equal, not the areas. The errors in students' strategies aiming for finding the centroid of a polygon fundamentally lie in the lack of their understanding of the mathematical investigation of physical phenomena. To investigate physical phenomena mathematically, we should abstract some mathematical principals from the phenomena which can provide the appropriate explanations for then. This abstraction is crucial because the development of mathematical theories for physical phenomena begins with those principals. However, the students weren't conscious of this process. Generally, we use the law of lever, the reciprocal proportionality of mass and distance, to explain the equilibrium of an object. But some self-evident principles in symmetry may also be logically sufficient to fix the centroid of a polygon. One of the studies by Archimedes, the famous ancient Greek mathematician, gives a solution to this rather awkward situation. He had developed the general theory of a centroid from a few axioms which concerns symmetry. But it should be noticed that these axioms are achieved from the abstraction of physical phenomena as well.

• East Asian mathematical journal
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• v.28 no.4
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• pp.383-401
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• 2012

The purpose of mathematics education includes two important areas; cognitive area that emphasizes mathematical knowledge and understanding and affective area that stresses mathematical interest and attitude. The purpose of mathematics education is not only in acquiring the contents and knowledge but also rousing up interest and attention toward mathematics. Therefore, effort to accomplish this affective purpose has to be made. Introducing history of mathematics to teaching can be a important method for the students to arouse interest and attention toward mathematics. History of mathematics can help the students who are familiar to only manipulation of the symbols to develop a new way of thinking and mathematical thoughts arousing reflective thinking. According to the survey, although the effect of using mathematics history has been recognized, the mathematics history has neither been developed as teaching materials nor reflected in the courses of study. The purpose of this research is to develop the reading materials into suit for the mathematics curriculum to extract contents of the mathematics valuable in using in elementary mathematics teaching, and to investigate the effect of reading materials using the history of mathematics on learning attitude in elementary school. The way of developing materials in this study is as follows. First, to select the interesting and instructive subject for the elementary students such as the story and life of a mathematician, developmental stages of mathematical theory and calculation currently used and finding the patterns of the rules that requires mathematical thoughts. Second, to classify the selected items according to mathematics curriculum. Third, to reorganize the classified items of the appropriate grade with the reading materials of dialogue pattern in order to draw attention and interest from the students I developed 18 kinds materials in accordance with the above procedure and applied 5 materials among them to one class in 4th grade. Analysing the student's responses, First, using history of mathematics helps the students to arouse interest and confidence on mathematical learning attitude. And the students became better attitude of studying by oneself and attention on class. Second, as know by opinions after lesson, most students have a chance refresh one's thinking of mathematics, want to know the other content of history of mathematics and responded to study hard in mathematics. As a result, the reading materials on the basis of the history of mathematics motivates students for mathematics and helps them become confident in mathematics. If the materials are complemented properly, they will be useful and effective for students and teachers.

• School Mathematics
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• v.18 no.3
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• pp.589-609
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• 2016

In this study, researchers have modernly reinterpreted geometric solving of cubic equations presented by an arabic mathematician, Omar Khayyam in medieval age, and have considered the pedagogical significance of geometric solving of the cubic equations using two conic sections in terms of analytic geometry. These efforts allow to analyze educational application of mathematics instruction and provide useful pedagogical implications in school mathematics such as 'connecting algebra-geometry', 'induction-generalization' and 'connecting analogous problems via analogy' for the geometric approaches of cubic equations: $x^3+4x=32$, $x^3+ax=b$, $x^3=4x+32$ and $x^3=ax+b$. It could be possible to reciprocally convert between algebraic representations of cubic equations and geometric representations of conic sections, while geometrically approaching the cubic equations from a perspective of connecting algebra and geometry. Also, it could be treated how to generalize solution of cubic equation containing variables from geometric solution in which coefficients and constant terms are given under a perspective of induction-generalization. Finally, it could enable to provide students with some opportunities to adapt similar solving procedures or methods into the newly-given cubic equation with a perspective of connecting analogous problems via analogy.

• Journal of Gifted/Talented Education
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• v.23 no.6
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• pp.1019-1034
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• 2013

The purpose of this study is to investigate the characteristics of convergence gifted students through the their perception of the differences about scientists between Science class and Convergence class in gifted science education center. Consequently, this article reports that there are differences in the perception about scientists was distinction between applicants of Convergence and Science class. Science class applicants mainly recognized scientists as pure scientists, but Convergence class applicants recognized scientist were including mathematician, artist, architect, etc. Also Convergence class applicants thought that affective domain including 'effort', 'patience', 'interest' was more important that Science class applicants. On the other hand, when they described the scientists, Science class applicants knew their achievements as scientists more specifically than Convergence class. And to conclude, the characteristics were different between Convergence and Science class applicants in gifted science education center. Based on the result of this study, this paper suggests the following: Firstly, conceptual study is urgent about convergence gifted students in their definition and characteristics. Secondly, information regarding the criteria to select student for convergence class in gifted science education center. Finally, when teaching convergence gifted students, attention should be paid to their characteristics such affective domain.

• Journal of Broadcast Engineering
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• v.24 no.3
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• pp.411-419
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• 2019

Nowadays we are not able to consider and imagine anything without taking into account what is called Artificial Intelligence. Even broadcasting media technologies could not be thought of outside this newly emerging technology of A.I.. Since the last part of 20th century, this technology seemingly is accelerating it's development thanks to an unbelievably enormous computational capacity of data information treatments. In conjunction with the firmly established worldwide platform companies like GAFA(Google, Amazon, Facebook, Apple), the key cutting edge technologies dubbed NBIC(Nanotech, Biotech, Information Technology, Cognitive science) converge to change the map of the current civilization by affecting the human relationship with the world and hence modifying what is essential in humans. Under the sign of the converging technologies, the relatively recently coined concepts such as 'trans(post)humanism' are emerging in the academic sphere in the North American and Major European regions. Even though the so-called trans(post)human movements are prevailing in the major technological spots, we have to say that these terms do not yet reach an unanimous acceptation among many experts coming from diverse fields. Indeed trans(post)humanism as a sort of obscure term has been a largely controversial trend. Because there have been many different opinions depending on scientific, philosophical, medical, engineering scholars like Peter Sloterdijk, K. N. Hayles, Neil Badington, Raymond Kurzweil, Hans Moravec, Laurent Alexandre, Gilbert Hottois just to name a few. However, considering the highly dazzling development of artificial intelligence technology basically functioning in conjunction with the cybernetic communication system firstly conceived by Nobert Wiener, MIT mathematician, we can not avoid questioning what A. I. signifies and how it will affect the current media communication environment.

• Journal of Ginseng Culture
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• v.1
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• pp.28-42
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• 2019

What is unknown about Leibniz (Gottfried Wilhelm Leibniz, 1646~1716), a great philosopher and mathematician, is that he inquired about ginseng. Why Leibniz, one of the leading figures of the Enlightenment, became interested in ginseng? This paper excavates Leibniz's references on ginseng in his vast amount of correspondences and traces the path of his personal life and cultural context where the question about ginseng arose. From the sixteenth century, Europe saw a notable growth of medical botany, due to the rediscovery of such Greek-texts as Materia Medica and the introduction of a variety of new plants from the New World. In the same context, ginseng, the renowned panacea of the Old World began to appear in a number of European travelogues. As an important part of mercantilistic projects, major scientific academies in Europe embarked on the researches of valuable foreign plants including ginseng. Leibniz visited such scientific academies as the Royal Society in London and $Acad{\acute{e}}mie$ royale des sciences in Paris, and envisioned to establish such scientific society in Germany. When Leibniz visited Rome, he began to form a close relationship with Jesuit missionaries. That opportunity amplified his intellectual curiosity about China and China's famous medicine, ginseng. He inquired about the properties of ginseng to Grimaldi and Bouvet who were the main figures in Jesuit China mission. This article demonstrates ginseng, the unnoticed subject in the Enlightenment, could be an important clue that interweaves the academic landscape, the interactions among the intellectuals, and the mercantilistic expansion of Europe in the late 17th century.