Visual Thinking in Mathematics

Philosophy ◽  
2019 ◽  
Author(s):  
Jessica Carter
Keyword(s):  
20Th Century ◽  
Euclidean Geometry ◽  
18Th Century ◽  
History Of Mathematics ◽  
Mathematical Practice ◽  
Visual Representations ◽  
Visual Thinking ◽  
General Acceptance ◽  
History Of ◽  
Euclid’S Elements

In contemporary philosophy, “visual thinking in mathematics” refers to studies of the kinds and roles of visual representations in mathematics. Visual representations include both external representations (i.e., diagrams) and mental visualization. Currently, three main areas and questions are being investigated. The first concerns the roles of diagrams, or the diagram-based reasoning, found in Euclid’s Elements. Second is the epistemic role of diagrams: the question of whether reasoning based on diagrams can be rigorous. This debate includes the question of whether beliefs based on visual input can be justified, and whether visual perception may lead to mathematical knowledge. The third observes that diagrams abound in (contemporary) mathematical practice, and so tries to understand the role they play, going beyond the traditional debates on the legitimacy of using diagrams in mathematical proofs. Looking at the history of mathematics, one will find that it is only recently that diagrammatic proofs have become discredited. For about 2,000 years, Euclid’s Elements was conceived as the paradigm of (mathematical) rigorous reasoning, and so until the 18th century, Euclidean geometry served as the foundation of many areas of mathematics. One includes the early history of analysis, where the study of curves draws on results from (Euclidean) geometry. During the 18th and 19th centuries, however, diagrams gradually disappear from mathematical texts, and around 1900 one finds the famous statements of Pasch and Hilbert claiming that proofs must not rely on figures. The development of formal logic during the 20th century further contributed to a general acceptance of a view that the only value of figures, or diagrams, is heuristic, and that they have no place in mathematical rigorous proofs. A proof, according to this view, consists of a discrete sequence of sentences and is a symbolic object. In the latter half of the 20th century, philosophers, sensitive to the practice of mathematics, started to object to this view, leading to the emergence of the study of visual thinking in mathematics.

LITERARY ROMANTICISM AS A THEORETICAL ISSUE

2020 ◽  
pp. 19-27
Author(s):  
Vera V. Serdechnaia ◽  
Keyword(s):  
20Th Century ◽  
Conceptual Analysis ◽  
18Th Century ◽  
Theoretical Concept ◽  
Historical Research ◽  
Theoretical Issue ◽  
Main Research ◽  
Literary Process ◽  
History Of ◽  
The Subject

The article is devoted to the analysis of the concept of literary romanticism. The research aims at a refinement of the “romanticism” concept in relation to the history of the literary process. The main research methods include conceptual analysis, textual analysis, comparative historical research. The author analyzes the semantic genesis of the term “romanticism”, various interpretations of the concept, compares the definitions of different periods and cultures. The main results of the study are as follows. The history of the term “romanticism” shows a change in a number of definitions for the same concept in relation to the same literary phenomena. By the end of the 20th century, realizing the existence of significant contradictions in the content of the term “romanticism”, researchers often come to abandon it. At the same time, the steady use of the term “romanticism” testifies to the subject-conceptual component that exists in it, which does not lose its relevance, but just needs a theoretical refinement. Conclusion: one have to revise an approach to romanticism as a theoretical concept, based on the change in the concept of an individual in Europe at the end of the 18th century. It is the newly discovered freedom of an individual predetermines the rethinking for the image of the author as a creator and determines the artistic features of literary romanticism.

scholarly journals Um Passeio pelo Labirinto da Lógica Matemática em companhia de Malba Tahan

2018 ◽  
Vol 15 (19) ◽  
pp. 247-264
Author(s):  
Inocêncio Fernandes Balieiro Filho
Keyword(s):  
Mathematical Logic ◽  
Euclidean Geometry ◽  
History Of Mathematics ◽  
Logical Structure ◽  
Deductive System ◽  
Axiomatic Method ◽  
Concept Definition ◽  
History Of ◽  
Contemporary Approach

O presente artigo tem por objetivo discutir numa perspectiva contemporânea os conteúdos de Lógica, Matemática, Filosofia da Matemática e História da Matemática presentes no livro A Lógica na Matemática, escrito por Malba Tahan. Para isso, mediante o uso da historiografia, foram selecionados temas concernentes com os assuntos da pesquisa. Foram tratados os seguintes temas: a base lógica da Matemática, a definição de conceito, os princípios para se definir um objeto, as definições e a natureza dos axiomas em Matemática, o método axiomático e as diversas axiomáticas para a geometria euclidiana, a estrutura lógica de um sistema dedutivo, os métodos de demonstração em Matemática, a indução, analogia e dedução em Matemática.   Palavras-chave: Lógica Matemática; História da Matemática; Filosofia da Matemática.   A TOUR BY THE LABYRINTH OF MATHEMATICAL LOGIC IN THE COMPANY OF MALBA TAHAN   Abstract   In this paper we discuss the Mathematics, the Logic of Mathematics, the Philosophy and History of Mathematics that presents in the book A Lógica na Matemática of the Malba Tahan, in a contemporary approach. For that, we use the historiography to select matters in adherence with the research. Are treated this topics: the basis of the Logic of Mathematics; the concept definition; principles to define an object; definitions and nature of the axioms in Mathematics; the axiomatic method and the diverse axiomatic to the Euclidean Geometry; the logical structure of a deductive system; demonstration methods in mathematics; the induction, analogy and deduction in mathematics.  

scholarly journals Considerations to some forms of the verb a fi: sum and sunt

Diacronia ◽  
2018 ◽  
Author(s):  
Gheorghe Chivu
Keyword(s):  
20Th Century ◽  
19Th Century ◽  
18Th Century ◽  
History Of ◽  
Literary Writing ◽  
The 19Th Century

The history of the verbal forms sum and sunt, introduced into the literary writing by the Transylvanian Latinist School, reveals a winding process in the elaboration of certain cultured norms proper to the modern literary Romanian. Not at all linear, this process was concurrently influenced by two, often divergent, tendencies that were active from the end of the 18th century up to the beginning of the 20th century: the use of some cultured forms, borrowed from Latin or created according to Latin patterns; and the revitalization of certain linguistic forms with regional diffusion. Initially proposed as literary pronunciations, the two verbal forms were soon adopted and used as etymological graphic forms that corresponded to sîm and suntu from certain conservative patois. During the second half of the 19th century (sum), and during the first decades of the 20th century (sunt), the two graphic forms became orthoepic norms as well, due to the phonological tradition of the Romanian writing.

scholarly journals A short remark on Gödel incompleteness theorem and its self-referential paradox from Neutrosophic Logic perspective

2019 ◽  
pp. 21-26
Author(s):  
V. Christianto ◽  
◽  
◽  
F. Smarandache
Keyword(s):  
20Th Century ◽  
History Of Mathematics ◽  
The Self ◽  
Incompleteness Theorem ◽  
Aristotelian Logic ◽  
Binary Logic ◽  
Neutrosophic Logic ◽  
History Of ◽  
Incompleteness Theorems ◽  
Modern Mathematics

It is known from history of mathematics, that Gödel submitted his two incompleteness theorems, which can be considered as one of hallmarks of modern mathematics in 20th century. Here we argue that Gödel incompleteness theorem and its self-referential paradox have not only put Hilbert’s axiomatic program into question, but he also opened up the problem deep inside the then popular Aristotelian Logic. Although there were some attempts to go beyond Aristotelian binary logic, including by Lukasiewicz’s three-valued logic, here we argue that the problem of self-referential paradox can be seen as reconcilable and solvable from Neutrosophic Logic perspective. Motivation of this paper: These authors are motivated to re-describe the self-referential paradox inherent in Godel incompleteness theorem. Contribution: This paper will show how Neutrosophic Logic offers a unique perspective and solution to Godel incompleteness theorem.

scholarly journals Protestant cemeteries in Czechia: a necrogeographical perspective

Geografie ◽  
2016 ◽  
Vol 121 (1) ◽  
pp. 187-207
Author(s):  
Zdeněk R. Nešpor
Keyword(s):  
20Th Century ◽  
18Th Century ◽  
Regional Distribution ◽  
Social Geography ◽  
Data Sources ◽  
Important Research ◽  
History Of ◽  
Research Questions ◽  
Czech Lands ◽  
The Impact

The article introduces the field of necrogeography to Czech social geography and provides information on existing (nationwide) data sources. The author takes the issue of Protestant confessional cemeteries as an example, briefly outlines the history of these special types of burial fields (established principally from the end of the 18th century until approximately the mid-20th century), and provides a historical geographical analysis of their regional distribution in the Czech Lands. The article proves the impact of religious and geographical factors on the emergence (and eventual demise) of non-Catholic Christian confessional cemeteries and, at the same time, the research unveiled a number of important research questions to be addressed by Czech necrogeography in the future.

Scientific novelties implemented into teaching mathematics in secondary schools on the Polish territories in the 19th century. The case of descriptive geometry

2020 ◽  
pp. 205-218
Author(s):  
Karolina Karpińska
Keyword(s):  
Mathematics Education ◽  
19Th Century ◽  
18Th Century ◽  
History Of Mathematics ◽  
Descriptive Geometry ◽  
Teaching Mathematics ◽  
Education History ◽  
Secondary School Education ◽  
History Of ◽  
The 19Th Century

This article is dedicated to discussing the implementation of the descriptive geometry, i.e. the scientific novelty from the end of the 18th century, in secondary school education on the Polish territories in the 19th century. At that time, Polish lands were under the occupation of three empires: Prussia, Austria, and Russia. Over the time, the policy of the partition empires toward the Poles was changing in intensity. As a consequence, in the 19th century, there were schools on the Polish territories with Polish, Prussian, Austrian and Russian curricula and relevant lecture languages. The article analyses the implementation of descriptive geometry into teaching mathematics in schools located in all three partitions. Keywords: descriptive geometry, history of mathematics education, history of mathematics

Declining fires in Larix-dominated forests in northern Irkutsk district

2011 ◽  
Vol 20 (2) ◽  
pp. 248 ◽  
Author(s):  
Tuomo Wallenius ◽  
Markku Larjavaara ◽  
Juha Heikkinen ◽  
Olga Shibistova
Keyword(s):  
20Th Century ◽  
Forest Fires ◽  
Fire History ◽  
18Th Century ◽  
Fire Suppression ◽  
Tree Ring Chronology ◽  
Fire Cycle ◽  
History Of ◽  
Historical Accounts ◽  
Additional Explanation

To study the poorly known fire history of Larix-dominated forest in central Siberia, we collected samples from 200 trees in 46 systematically located study plots. Our study area stretches ~90 km from north to south along the River Nizhnyaya Tunguska in northern Irkustk district. Cross-dated tree-ring chronology for all samples combined extended from the year 1360 AD to the present and included 76 fire years and 88 separate fire events. Average fire cycle gradually lengthened from 52 years in the 18th century to 164 years in the 20th century. During the same time, the number of recorded fires decreased even more steeply, i.e. by more than 85%. Fires were more numerous but smaller in the past. Contrary to expectations, climate change in the 20th century has not resulted in increased forest fires in this region. Fire suppression may have contributed to the scarcity of fires since the 1950s. However, a significant decline in fires was evident earlier; therefore an additional explanation is required, a reduction in human-caused ignitions being likely in the light of historical accounts.

The History of Christian Missions to Africa

2019 ◽  
Author(s):  
Norman Etherington
Keyword(s):  
20Th Century ◽  
French Revolution ◽  
18Th Century ◽  
Sub Saharan Africa ◽  
Foreign Born ◽  
13Th Century ◽  
Christian Missions ◽  
Late 20Th Century ◽  
History Of ◽  
Sub Saharan

Christianity came very early to Africa, as attested by the Gospels. The agencies by which it spread across North Africa and into the Kingdom of Aksum remain largely unknown. Even after the rise of Islam cut communications between sub-Saharan Africa and the churches of Rome and Constantinople, it survived in the eastern Sudan kingdom of Nubia until the 15th century and never died in Ethiopia. The documentary history of organized missions begins with the Roman Catholic monastic orders founded in the 13th century. Their evangelical work in Africa was closely bound up with Portuguese colonialism, which both helped and hindered their operations. Organized European Protestant missions date from the 18th-century evangelical awakening and were much less creatures of states. Africa was a particular object of attention for Evangelicals opposed to slavery and the slave trade. Paradoxically this gave an impetus to colonizing ventures aimed at undercutting the moral and economic foundations of slavery in Africa. Disease proved to be a deadly obstacle to European- and American-born missionaries in tropical Africa, thus spurring projects for enrolling local agents who had acquired childhood immunity. Southern Africa below the Zambezi River attracted missionaries from many parts of Europe and North America because of the absence of the most fearsome diseases. However the turbulent politics of the region complicated their work by restricting their access to organized African kingdoms and chieftaincies. The prevalent mission model until the late 19th century was a station under the direction of a single European family whose religious and educational endeavors were directed at a small number of African residents. Catholic missions acquired new energy following the French Revolution, the old Portuguese system of partnership with the state was displaced by enthusiasm for independent operations under the authority of the Pope in Rome. Several new missionary orders were founded with a particular focus on Africa. Mission publications of the 19th and 20th centuries can convey a misleading impression that the key agents in the spread of African Christianity were foreign-born white males. Not only does this neglect the work of women as wives and teachers, but it diverts attention from the Africans who were everywhere the dominant force in the spread of modern Christianity. By the turn of the 20th century, evangelism had escaped the bounds of mission stations driven by African initiative and the appearance of so-called “faith missions” based on a model of itinerant preaching. African prophets and independent evangelists developed new forms of Christianity. Once dismissed as heretical or syncretic, they gradually came to be recognized as legitimate variants of the sort that have always accompanied the acculturation of religion in new environments. Decolonization caught most foreign mission operations unawares and required major changes, most notably in the recruitment of African clergy to the upper echelons of church hierarchies. By the late 20th century Africans emerged as an independent force in Christian missions, sending agents to other continents.

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