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.

Introduction: The History of Modern Mathematics – Writing and Rewriting

2004 ◽  
Vol 17 (1-2) ◽  
pp. 1-21 ◽  
Author(s):  
Leo Corry
Keyword(s):  
Nineteenth Century ◽  
History Of Mathematics ◽  
Present Issue ◽  
Tel Aviv ◽  
History Of ◽  
New Ideas ◽  
Ancient Mathematics ◽  
Modern Mathematics ◽  
September Issue

The present issue of Science in Context comprises a collection of articles dealing with various, specific aspects of the history of mathematics during the last third of the nineteenth century and the first half of the twentieth. Like the September issue of 2003 of this journal (Vol. 16, no. 3), which was devoted to the history of ancient mathematics, this collection originated in the aftermath of a meeting held in Tel-Aviv and Jerusalem in May 2001, under the title: “History of Mathematics in the Last 25 Years: New Departures, New Questions, New Ideas.” Taken together, these two topical issues are meant as a token of appreciation for the work of Sabetai Unguru and his achievements in the history of mathematics.

scholarly journals The Assembly of a Criminal Self

2020 ◽  
Vol 24 (2) ◽  
pp. 43-61
Author(s):  
Seán Manning ◽  
Dave Nicholls
Keyword(s):  
Indigenous People ◽  
20Th Century ◽  
Natural Extension ◽  
The Self ◽  
High Rate ◽  
Western Society ◽  
Psychotherapeutic Intervention ◽  
Criminal Histories ◽  
History Of

Beginning with the experience of working with men in prison and others who have considerable prison experience, all of whom have long criminal histories, and considering Aotearoa’s relatively high rate of imprisonment, particularly of indigenous people, this paper attempts to describe a theory of self as a performative assembly, rather than as a developmental achievement, which is the dominant view in psychotherapy. In doing so, a brief history of the self from the beginning of the 20th century is presented, illustrating how the self changes, not just in an individual subjectivity, but between eras in the history of Western society. This perspective is used to understand how a “criminal self” might develop as a product of incarceration and as a natural extension of the self in the neoliberal era, and why it might prove resistant to psychotherapeutic intervention. Drawing on the work of Foucault, Rose, and Butler, among others, the concept of “intoxicating performativity” is introduced. The role of anger as an antidote to fragmentation is explored. Some thoughts are added about why indigenous people are overrepresented in prison compared to the population at large.

scholarly journals Proof of Fermat's Last Theorem By Infinite Descent

2020 ◽  
Author(s):  
Djamel Himane
Keyword(s):  
20Th Century ◽  
Elementary Function ◽  
Mathematical Problem ◽  
History Of Mathematics ◽  
Peano Arithmetic ◽  
Basic Principles ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
History Of ◽  
Infinite Descent

Fermat's last theorem, one of the most challenging theories in the history of mathematics, has been conjectured by French lawyer Pierre de Verma in 1637. Since then, it wasconsidered the most difficult and unsolvable mathematical problem. However, more than three centuries later, a first proof was proposed by the British mathematician Andrew Wiles in 1994, relying on 20th-century techniques. Wiles's proof is based on elliptic (oval) curves that were not available at the time when the theory was first proposed. Most mathematicians argued that it was impossible to prove Fermat's theorem according to basic principles of arithmetic, though Harvey Friedman's grand conjecture states that mathematical theorems, including Fermat's Last Theorem, can be solved in very weak systems such as the Elementary Function Arithmetic (EFA). Friedman's grand conjecture states that "every theorem published in the journal, Annals of Mathematics, whose statement involves only finitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in EFA, which is the weak fragment of Peano Arithmetic based on the usual quantifier free axioms for 0,1,+,x, exp, together with thescheme of induction for all formulas in the language all of whose quantifiers are bounded." *

scholarly journals Academician Jonas Kubilius: works dedicated to the history of Lithuanian mathematics

2021 ◽  
Vol 62 ◽  
pp. 13-21
Author(s):  
Juozas Banionis
Keyword(s):  
Probability Theory ◽  
20Th Century ◽  
History Of Mathematics ◽  
Book Series ◽  
Informal Group ◽  
Mathematical School ◽  
Mathematics History ◽  
History Of ◽  
The 19Th Century ◽  
Fourth Book

The rise of the Lithuanian mathematical school in the second half of the 20th century is associated with the development of probability theory and its application, and the foundations of that school were insightfully laid by the famous Lithuanian mathematician Jonas Kubilius. However, the academician also had a second vocation – the history of mathematics. At the end of the 20th century, he purposefully researched the mathematical legacy of the poet, bishop A. Baranauskas, recognizing him as the first Lithuanian mathematician researcher of the second half of the 19th century. At the beginning of the 21st century, J. Kubilius undertook a detailed implementation of the idea of a work in the history of Lithuanian mathematics. For this purpose, an informal group of specialists was convened, the content of the work was planned, and the research-based book series ``From the History of Lithuanian Mathematics'' was published. The fourth book in this series, Mathematics in Lithuanian Higher Education Institutions in 1921–1944, presents the research of an academic who reveals the situation of mathematics in universities in Kaunas and Vilnius. In addition, the memoirs of mathematics history by J. Kubilius, dedicated to mathematicians Z. Žemaitis, G. Žilinskas and V. Statulevičius, should be mentioned. The article, at the end of which fragments of the author's memories are presented, is dedicated to the centenary of the birth of Academician J. Kubilius.

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.

Mathematics, Epistemology, Ethics: Robert Musil, the Man Without Qualities

2018 ◽  
pp. 93-125
Author(s):  
Nina Engelhardt
Keyword(s):  
History Of Mathematics ◽  
Analytic Philosophy ◽  
Profound Change ◽  
Robert Musil ◽  
Double Function ◽  
History Of ◽  
Critical Questioning ◽  
Modern Mathematics ◽  
Literary Aesthetic

Chapter 3 argues that mathematics in Musil’s The Man without Qualities not only exemplifies the side of rationality but also encompasses mystical elements and transitional states between these opposites, tracing the identification of a non-rational element in mathematics to the debate between the logicist/formalist and the intuitionist schools. The chapter thus re-examines notions of the rational and non-rational and attempts at their synthesis from the perspective of the history of mathematics. It also demonstrates that, for Musil, mathematics answers to two major instances of modern crisis: the failing of reason and the loss of trust. Paradoxically combining critical questioning of its foundations and confidence in its usefulness, modern mathematics connects the approaches of analytic philosophy and outcome-focused pragmatism. The chapter thus argues that mathematics becomes a model not only of exactitude but also of vagueness and that in this paradoxical double-function it serves to inspire the critical trust needed to adjust epistemology, ethics and aesthetics in a time of profound change. Not least, in its own form The Man without Qualities translates the model of mathematics into literary aesthetic by reflecting simultaneous examination of its conditions and trust in the credit of fiction.

Reviews and Evaluations

1968 ◽  
Vol 61 (2) ◽  
pp. 190-194
Author(s):  
Harold Tinnappel
Keyword(s):  
History Of Mathematics ◽  
Limited Success ◽  
Mathematical Thought ◽  
History Of ◽  
New Ideas ◽  
Working Out ◽  
Modern Mathematics

A translation of the first venture into the history of mathematics by a professor of Greek at Lausanne University, this book seeks with only very limited success “to explain the birth of modern mathematics by describing the progress of mathematical thought at the time of Plato” and to convince the reader that “twenty-five years of thought and discussion in Plato's Academy sufficed to delimit the field of mathematics in all its breadth [and] saw the working-out of new ideas on which the whole edifice of modern mathematics rests.”

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