Indonesia has recently faced problems in various aspects of life. The results of a social media survey in Indonesia in early 2021 that the biggest threat to the Pancasila ideology is communism and other western ideologies. Communism has a dark history in the life of the Indonesian people. It shows the problem of thinking and philosophical views of the Indonesian people. This research is textbook research that aims to analyze philosophy books, namely mathematics education philosophy textbooks (...) written with a Western cultural background by Paul Ernest. This book was chosen as the object of analysis because this book is the essential reference textbook for the philosophy of mathematics education in universities providing prospective mathematics teachers in Indonesia. The research results show that this book contributes to developing Western thoughts that are contrary to Pancasila. It needs to eliminate the most critical thing in this book. It is the atheistic and individualistic philosophy underlying the ideology of mathematics education because this is not in line with the theistic collectivistic Pancasila. Therefore, this research recommends developing a textbook on the philosophy of mathematics education. It should be under the Pancasila ideology or written based on the Pancasila ideology. (shrink)

A basic thesis of Neokantian epistemology and philosophy of science contends that the knowing subject and the object to be known are only abstractions. What really exists, is the relation between both. For the elucidation of this “knowledge relation ("Erkenntnisrelation") the Neokantians of the Marburg school used a variety of mathematical metaphors. In this con-tribution I reconsider some of these metaphors proposed by Paul Natorp, who was one of the leading members of the Marburg school. It is shown (...) that Natorp's metaphors are not unrelated to those used in some currents of contemporary epistemology and philosophy of science. (shrink)

This paper surveys Bolzano's Beyträge zu einer begründeteren Darstellung der Mathematik (Contributions to a better-grounded presentation of mathematics) on the 200th anniversary of its publication. The first and only published issue presents a definition of mathematics, a classification of its subdisciplines, and an essay on mathematical method, or logic. Though underdeveloped in some areas (including,somewhat surprisingly, in logic), it is nonetheless a radically innovative work, where Bolzano presents a remarkably modern account of axiomatics and the epistemology of the (...) formal sciences. We also discuss the second, unfinished and unpublished issue, where Bolzano develops his views on universal mathematics. Here we find the beginnings of his theory of collections, for him the most fundamental of the mathematical disciplines. Though not exactly the same as the later Cantorian set theory, Bolzano's theory of collections was used in very similar ways in mathematics, notably in analysis. In retrospect, Bolzano's debut in philosophy was a remarkably successful one, though its fruits would only become generally known much later. (shrink)

This article examines the possibility of philosophizing about mathematics with children. It aims at outlining the nature of the practice of philosophy of mathematics with children in a mainly theoretical and exploratory way. First, an attempt at a definition is proposed. Second, I suggest some reasons that might motivate such a practice. My thesis is that one can identify an intrinsic as well as two extrinsic goals of philosophizing about mathematics with children. The intrinsic goal is (...) related to a presumed inherent importance of presenting children with some philosophical questions about mathematics. The extrinsic goals consist of first the positive effects such a practice can have on mathematical learning and abilities and second the fostering of children's understanding of philosophical method of inquiry and thinking and therefore of their philosophical thinking competences. Third, some examples found in the literature of previously developed ways of practising philosophy of mathematics with children are presented. This article aims at giving a general outlining picture of the issues surrounding the practice of philosophy of mathematics with children and should therefore be read as an encouragement to further development and studies. (shrink)

In this paper, I present and discuss critically the main elements of Mario Bunge’s philosophy of mathematics. In particular, I explore how mathematical knowledge is accounted for in Bunge’s systemic emergent materialism.

Creativity pervades human life. It is the mark of individuality, the vehicle of self-expression, and the engine of progress in every human endeavor. It also raises a wealth of neglected and yet evocative philosophical questions: What is the role of consciousness in the creative process? How does the audience for a work for art influence its creation? How can creativity emerge through childhood pretending? Do great works of literature give us insight into human nature? Can a computer program really be (...) creative? How do we define creativity in the first place? Is it a virtue? What is the difference between creativity in science and art? Can creativity be taught? The new essays that comprise The Philosophy of Creativity take up these and other key questions and, in doing so, illustrate the value of interdisciplinary exchange. Written by leading philosophers and psychologists involved in studying creativity, the essays integrate philosophical insights with empirical research. CONTENTS I. Introduction Introducing The Philosophy of Creativity Elliot Samuel Paul and Scott Barry Kaufman II. The Concept of Creativity 1. An Experiential Account of Creativity Bence Nanay III. Aesthetics & Philosophy of Art 2. Creativity and Insight Gregory Currie 3. The Creative Audience: Some Ways in which Readers, Viewers and/or Listeners Use their Imaginations to Engage Fictional Artworks Noël Carroll 4. The Products of Musical Creativity Christopher Peacocke IV. Ethics & Value Theory 5. Performing Oneself Owen Flanagan 6. Creativity as a Virtue of Character Matthew Kieran V. Philosophy of Mind & Cognitive Science 7. Creativity and Not So Dumb Luck Simon Blackburn 8. The Role of Imagination in Creativity Dustin Stokes 9. Creativity, Consciousness, and Free Will: Evidence from Psychology Experiments Roy F. Baumeister, Brandon J. Schmeichel, and C. Nathan DeWall 10. The Origins of Creativity Elizabeth Picciuto and Peter Carruthers 11. Creativity and Artificial Intelligence: a Contradiction in Terms? Margaret Boden VI. Philosophy of Science 12. Hierarchies of Creative Domains: Disciplinary Constraints on Blind-Variation and Selective-Retention Dean Keith Simonton VII. Philosophy of Education (& Education of Philosophy) 13. Educating for Creativity Berys Gaut 14. Philosophical Heuristics Alan Hájek. (shrink)

In this paper, we aim to get clear about why renewal is needed in philosophy of religion and how to achieve it. We begin with a fundamental distinction between someone’s perspective in the field and the perspective of the field, arguing that any philosopher of religion is responsible to both. Then we identify eight problems that should prevent the status quo in philosophy from appearing acceptable to anyone who takes the perspective of the field, as well as seven (...) practical suggestions which, if implemented, would help to solve these problems. The problems include such things as tribalism, ignorance about religion, and viewing issues exclusively in terms of what is religiously familiar; the mitigations such things as allowing non-Christians to join the Society of Christian Philosophers, broadening the education of philosophers of religion, and paying more attention to the deep history of religion. (shrink)

An important aspect of moral expertise is moral sensitivity, which is the ability to be sensitive to the presence of morally salient features in a context. This requires being able to see and acquire the morally relevant information, as well as organise and interpret it, so that you can undertake the related work of moral judgement, focus (or motivation) and action. As a distinct but interrelated component of ethical expertise, moral sensitivity can and must be trained and educated. However, despite (...) its importance to moral education, there has been comparatively little discussion about the role of ethical sensitivity and the ways that it can be trained within the context of Kant’s, and Kantian, ethics. This paper seeks to address this gap. While Kant does not explicitly focus in detail on moral sensitivity, by breaking sensitivity down into seven distinct aspects through drawing on the Four Component Model, we are able to identify a wealth of resources in Kant’s work from which we can con- struct an account of moral sensitivity education that draws on his underlying moral theory. (shrink)

In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the interpretation. (...) The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines. (shrink)

This collection of newly commissioned essays, edited by NYU philosophers Paul Boghossian and Christopher Peacocke, resumes the current surge of interest in the proper explication of the notion of a priori. The authors discuss the relations of the a priori to the notions of definition, meaning, justification, and ontology, explore how the concept figured historically in the philosophies of Leibniz, Kant, Frege, and Wittgenstein, and address its role in the contemporary philosophies of logic, mathematics, mind, and science. The (...) editors’ Introduction familiarizes the reader with the issues that are to be explored in detail in later parts of the anthology. (shrink)

In this article, modern standards of early years mathematics education are criticized and a proposal for change is presented. Today's early years mathematics education standards rest on a view of mathematics that became obsolete already at the end of the 19th century while the spirit of children's mathematics is precisely the spirit of modern mathematics. The proposal for change is not a return to the “new mathematics” movement, but something different.

It is argued that logic, and in particular mathematical logic, should play a key role in the undergraduate curriculum for students in the computing fields, which include electrical engineering (EE), computer engineering (CE), and computer science (CS). This is based on 1) the history of the field of computing and its close ties with logic, 2) empirical results showing that students with better logical thinking skills perform better in tasks such as programming and mathematics, and 3) the skills students (...) are expected to have in the job market. Further, the authors believe teaching logic to students explicitly will improve student retention, especially involving underrepresented minorities. Though this work focuses specifically on the computing fields, these results demonstrate the importance of logic education to STEM (science, technology, engineering, and mathematics) as a whole. (shrink)

For Kant, we cannot understand how to approach moral education without confronting the radical evil of humanity. But if we start out, as Kant thinks we do, from a morally corrupt state, how...

There are different narratives on mathematics as part of our world, some of which are more appropriate than others. Such narratives might be of the form ‘Mathematics is useful’, ‘Mathematics is beautiful’, or ‘Mathematicians aim at theorem-credit’. These narratives play a crucial role in mathematics education and in society as they are influencing people’s willingness to engage with the subject or the way they interpret mathematical results in relation to real-world questions; the latter yielding important normative (...) considerations. Our strategy is to frame current narratives of mathematics from a virtue-theoretic perspective. We identify the practice of mathematizing, put forward by Freudenthal’s ‘Realistic mathematics education’, as virtuous and use it to evaluate different narratives. We show that this can help to render the narratives more adequately, and to provide implications for societal organization. (shrink)

In this paper I argue that Kant develops, in a number of texts, a detailed three stage theory of moral development which resembles the contemporary accounts of moral development defended by Lawrence Kohlberg and John Rawls. The first stage in this process is that of physical education and disciplining, followed by cultivating and civilising, with a third and final stage of moralising. The outcome of this process of moral development is a fully autonomous person. However, Kant’s account of moral development (...) appears to be in tension with other elements of his moral philosophy. I identify two such tensions, which I call the knowledge and revolution tensions, and show why these tensions are illusory. As such, a proper understanding of Kant’s theory of moral development, far from exposing genuine tensions, helps rather to deepen our understanding of Kant’s moral philosophy. (shrink)

The purpose of this paper is to critically analyse and discuss the views of constructivism, on the teaching and learning of mathematics. I provide a background to the learning of mathematics as constructing and reconstructing knowledge in the form of new conceptual networks; the nature, role and possibilities of constructivism as a learning theoretical framework in Mathematics Education. I look at the major criticisms and conclude that it passes the test of a learning theoretical framework but there (...) is still a gap between theory and mathematics classroom practice. (shrink)

Kant begins his Lectures on Pedagogy by stating, “[t]he human being is the only creature that must be educated” (Kant, 2007, 9:441), and he argues that it is through education that we can transform our initial “animal nature into human nature” (ibid. 2007, 9:441). Kant understands education as involving an ordered process of care, discipline, instruction and formation through enculturating, civilizing and moralizing (Formosa 2011). Further, Kant envisages that we should pursue as a species the “moral perfection” that is the (...) “final destiny of the human race” through education (Collins, 1997, 27:470; see Dean, 2014). However, to engage in this pursuit Kant believes that, through education and social change, we have to regulate our “animal nature” and counter the moral corruption of our species, which he calls the “radical innate evil in human nature (not any the less brought upon us by ourselves)” (Kant, 1998, 6:32). If humanity is to pursue its final destiny of moral perfection, then education will need to respond responsibly to the propensity to evil that is deeply rooted in us as finite and imperfect rational beings living in imperfect and at times even in morally corrupted social conditions. This paper outlines some of the relevant issues. (shrink)

In spite of Ernst Cassirer’s criticisms of psychologism throughout Substance and Function, in the final chapter he issues a demand for a “psychology of relations” that can do justice to the subjective dimensions of mathematics and natural science. Although these remarks remain somewhat promissory, the fact that this is how Cassirer chooses to conclude Substance and Function recommends it as a topic worthy of serious consideration. In this paper, I argue that in order to work out the details of (...) Cassirer’s psychology of relations in Substance and Function, we need to situate it within two broader frameworks. First, I position Cassirer’s view in relation to the view of psychology and logic endorsed by his Marburg Neo-Kantian predecessors, Hermann Cohen and Paul Natorp. Second, I augment Cassirer’s early account of the psychology of relations in Substance and Function with the more mature view of psychology that he presents in The Philosophy of Symbolic Forms. By placing Cassirer’s account within these contexts, I claim that we gain insight into his psychology of relations, not just as it pertains to mathematics and natural science, but to culture as a whole. Moreover, I maintain that pursuing this strategy helps shed light on one of the most controversial features of his philosophy of mathematics and natural science, viz., his theory of the a priori. (shrink)

While Philosophy for/with Children (P4wC) provides a better alternative to the usual ‘banking’ model of education, questions have been raised regarding its applicability in non-western contexts. Despite its adherence to the ideals of democratic dialogue, not all members of a Community of Inquiry (COI) will be disposed to participate in the inquiry, not because they are incapable of doing so, but because they are positioned inferiorly within the group thereby affecting their efforts to speak out on topics that are (...) meaningful to them. In this article, I claim that it is essential to integrate positionality in P4wC research/practice. Aside from its role in helping a practitioner/researcher choose the appropriate method and materials that match the unique contexts of children, it also increases one’s awareness of the subtle forms of epistemic injustice that could leak in the COI, as well as the other subtle ways in which children are marginalized. In this regard, a P4wC researcher/practitioner must have a higher degree of sensitivity towards her positionality as this inevitably gets entangled with the positionality of children. I present some ‘areas’ in which the importance of positionality in the COI manifests, namely, restructuring classroom power relations, navigating a multi-ethnic classroom, facilitating meaning-making, and modeling reflective thinking. (shrink)

Ludwig Wittgenstein, one of the most influential philosophers of the 20th century, made significant contributions to the philosophy of language. His work is often divided into two periods: early and later Wittgenstein. The concept of the logic of language is central to both, though his understanding of it evolved significantly over time. Wittgenstein's exploration of the logic of language fundamentally reshaped our understanding of how language relates to the world. His early work provided a foundation for logical positivism and (...) analytic philosophy, while his later work opened new avenues for understanding the complexities of human communication. Together, these contributions offer a rich and nuanced perspective on the nature of language, meaning, and human interaction. In his early work, the "Tractatus Logico-Philosophicus," Wittgenstein aimed to establish a clear relationship between language and reality. He believed that language mirrors the world through logical structures, which he referred to as "logical form." According to early Wittgenstein, the world consists of facts, not things, and language's primary function is to represent these facts. Wittgenstein’s "Philosophical Investigations" presents a radical shift from his earlier work and offers a deep and nuanced understanding of language and its role in human life. By focusing on the practical uses of language and the various contexts in which it operates, Wittgenstein dissolves many traditional philosophical problems and opens up new ways of thinking about meaning, communication, and the nature of reality. His work remains a cornerstone of contemporary philosophy and continues to inspire debate and exploration. (shrink)

The paper brings a critique of Freire’s notions in regard to education, philosophy, and politics. It explains how exploitation and oppression affect humanization. It also criticizes the “Banking model” of education and recommends the dialogical approach which can transform individuals.

Cosmopolis A Review of Cosmopolitics -/- 2015/3-4 -/- Editorial Dominique de Courcelles & Paul Ghils -/- This issue addresses the general concept of “spirituality” as it appears in various cultural contexts and timeframes, through contrasting ideological views. Without necessarily going back to artistic and religious remains of primitive men, which unquestionably show pursuits beyond the biophysical dimension and illustrate practices seeking to unveil the hidden significance of life and death, the following papers deal with a number of interpretations covering (...) a wide field extending from belief to theory, from emotions to concepts, from the wisdom of personal experience to the most sophisticated doctrines. Spirit and spirituality are indeed many-faceted notions. They may refer to the intricate world of the interacting spirits which inhabit living beings in animistic traditions, without excluding a “grand force” linking human beings within a dynamic whole on which their very existence rely . They also bear upon more atomistic and either/or approaches of Western philosophy, which have become embodied in Cartesian dualism against a monotheist background, to the point of freezing the essence of individuals and culminating in the extreme individualism that characterizes our contemporaries. However, this equally refers to the opposite conception of materialism, across times and cultures, from ancient India and Greece (Cārvāka, originally known as Lokāyata, or some Buddhist doctrines for the former, Democritus or Lucretius for the latter) to more contemporary materialistic schools, whether modern or postmodern. -/- The following papers look at the contrasting forms of the philosophy and spirit of the human factor set into a whole, with no artificial disjoint between the psychical and the physical levels, as Wittgenstein put it: “And how can a body have a soul?”. This approach is not unrelated to the notion of anthropocene examined in a recent issue of Cosmopolis, with provides another comprehensive framework open to a spiritual life emerging from the very environment that generated it. -/- *** The first section of this issue was edited by Dominique de Courcelles, director at the National French Research Centre (CNRS), whom we wish to thank for collecting relevant studies relating to the religious and political questions, with a view to focusing on the war of ideas inevitably waged behind images, concepts and perceptions, taking an asymmetrical approach. To the extent that they are mindful of global/local interactions and include representations, opinions and beliefs, such disciplines as philosophy, philology, history and social sciences can provide useful studies accounting for new practices in geopolitics and a fair diplomacy. -/- In her introduction, Dominique de Courcelles first poses the question of how the religious and political spheres interrelate, with their corresponding religious demands and humanistic values. She then suggests that the right question today may be breaking with the philosophy of human rights concerned with the defense of human beings against the hazards of arbitrary politics or the instrumental use of religion, in favour of a fair philosophy of humankind, a new humanism. This would consist in recognizing a common loyalty of all towards one interhuman, not only interstate community, to protect it from both the autonomy demanded by individuals and the instrumental use of minorities. -/- Considering the fact of diversity, so important today in terms of both politics and religion, Abdelhai Azarkan looks at the conditions under which tolerance could obtain the double status of right and duty. He revisits to two philosophers, John Locke and Voltaire, who thought about it from the historical reality of religious wars. The former made tolerance into a right, basing his analysis on the political-legal level, while the latter saw tolerance as a duty, from an analysis based on ethical-political criteria. -/- Mathieu Guidère examines what he calls semantic denominationalism, a term which implies religious attributes and identities, whichever national loyalties or personal belonging they may have at the same time. Since the early 2000s, thie phenomenon has expanded tremendously, compounded by the “war on terror” and the over media-oriented terrorist actions. Denominational expressions act as formal names for ordinary and high-profile players in domestic and foreign policies of democratic states. These systems reveal a receding secularization, while the powerful comeback of religious identities signals the failure of nation-states and the weakening of the humanist spirit. -/- Barbara De Poli retraces the history of a contemporary jihadism claiming its Islamic essence and asserting the truth of genuine coranic principles via the war on infidels, with a view to restoring the Caliphate. After defining the term jihad, she shows that even if this contemporary jiadism is spreading in the Muslim world, it radically departs from Islamic law and the received use of the term jihad, in so far as it is rooted in the early radical thinking of Islamic ideologues in the 20th century, starting with with Hassan al-Banna, the founder of the Muslim Brotherhood. This current has been fueled by by international conflicts since the outbreak of the war in Afghanistan, in which the so-called Western countries bear a major responsibility. -/- Abderrazak Sayadi starts from the Tunisian experience to ask the question of humanist values and democracy within the relationships between the religious and political spheres. As a historian of religion, he is brought to demystify certain islamic principles and to paying attention to the reform of law, seeing the separation of religion and politics as a precondition to a successful democratic gamble and the establishment of a renewed humanism. -/- Dominique de Courcelles reminds us that getting a better knowledge of narrations and words makes it possible to better understand how logical and rhetorical thinking works for those who wage an asymetriccal war, re-enchanting and mystifying the world to better take control. As soon as 1932, an exchange of letters between Einstein and Freud made it clear that, in order to free man from fatality and war, education understood as culture was fundamental. Such illustrations as the exécution of Oussama ben Laden and the Caliph’s speech in Mossoul show that a premiminary analysis of images and words is essential to a fair diplomacy conducted by people from civil society, whose culture and wisdom allow justice and force to speak together and better resist war. -/- Marcel Boisard thinks that on the day the guns fall silent, exhausted by war, we will not return to the state borders that have prevailed for a century as an outcome of the Sykes-Picot agreement of 1916. It is time to prepare the “day after”, which will be a huge challenge. To this end, a summit of Middle-East nations is urgently needed to globally decide the fate of those peoples. On the condition that we know who the enemy is and accept to name it, to understand the history of the countries, groups and alliances, and to question any false or self-interested sense of certainty. -/- *** In the second section, Paul Schafer provides the author’s experiences to explain how culture, from the artistic to the biological, has the power to to open the doors to spirituality, from the inner self to the global environment. He asks himself whether a relative permanence of spirituality can arise from the specific moments that characterize it. Laurent Ledoux synthesizes the conclusion of a symposium held on 22 January 2015 on the links between philosophy and management, on the basis of the spiritual dimension conceived as “natural” and the answers it may suggest to the issues that face the organizations in a “contemporaniversal” world. Jacques Rifflet makes that question in a secular perspective, based on the wellsprings of personal commitment before it can be caught by any religious creed or scientific theory. In this sense spirituality, in alliance with reason, both inspires human consciousness and illuminates its destiny. Sami Aldeeb asks himself whether Islam can be reconciled with human rights. Caught between the belief in an absolute and final Word descended from the sky, and evidence showing that any religion is the creation of a given culture and a society situated in time and space, the Makkan and … contexts et médinois call for differentiated, if not opposite answers and exegeses. Bernard Carmona provides the outline of a dialogical framework, which is known to be a feature of debates between the various philosophical schools of classical India, exemplified here by the transdisciplinary perspective of debates within a Buddhist context. *** The articles not focused on the previous topic include a study by Landry Signé on China’s strategy, competing with the United States to control African resources. The author deals with the specific case of China’s rapprochement with Southern Sudan since Sudan was broken up. In the last paper, Goran Fejić and Rada Iveković, return to the essential role that women should play, and comments upon the role of some international legal instruments related in particular to the elimination of all forms of discrimination. The perspective is transnational and transethnic and is based on secular criteria, as regards nation-building and more generally society-building. Considering the persistence of widespread violence, whether in times of armed conflict or in times of peace, the question remains whether it is possible to fully implement rights and justice instruments. (shrink)

The concept of discrete space can be termed as “the ex­ternal mathematical reality hypothesis”. The concept was already known among the ancient Greek philosophers (≈ 500 BC). Unfortunately the phenomenological point of view has dominated science during more than 2000 years and it is only recently that the concept of discrete space gets “tangible” attention again in philosophy and theoretical physics. Although the model de­scribes the existence of the universal conservation laws, constants and principles in a convincing way, the (...) re­lation between phenomenological reality and the geo­metrical description of discrete space is difficult to ima­gine for everyone who is only familiar with phenomeno­logical reality. The purpose of this paper is to describe some easy to imagine properties of discrete space. DOI: 10.5281/zenodo.5498120. (shrink)

What is analytical philosophy of education (APE)? And what has been its place in the history of the subject over the last fifty years? In a recent essay in Ethics and Education (Vol 2, No 2 October 2007) on ‘Rival conceptions of the philosophy of education’, Paul Standish described a number of features of APE. Relying on both historical and philosophical argument, the present paper critically assesses these eight points, as well as another five points delineating APE (...) in the Introduction to The Blackwell Guide to the Philosophy of Education. (shrink)

Zeno’s Arrow and Nāgārjuna’s Fundamental Wisdom of the Middle Way Chapter 2 contain paradoxical, dialectic arguments thought to indicate that there is no valid explanation of motion, hence there is no physical or generic motion. There are, however, diverse interpretations of the latter text, and I argue they apply to Zeno’s Arrow as well. I also find that many of the interpretations are dependent on a mathematical analysis of material motion through space and time. However, with modern philosophy and (...) physics we find that the link from no explanation to no phenomena is invalid and that there is a valid explanation and understanding of physical motion. Hence, those arguments are both invalid and false, which banishes the MMK/2 and The Arrow under this and derivative interpretations to merely the history of philosophy. However, a view that maintains their relevance is that each is used as a koan or sequence of koans designed to assist students in spiritual meditation practice. This view is partly justified by the realization that both Nāgārjuna and Zeno were likely meditation masters in addition to being logicians. The works are, therefore, not works that should be assessed as having valid arguments and true conclusions by the standards of modern analytic philosophy—contrary to some of the literature—but rather are therapeutic and perhaps more appropriately considered as part of an experientially focused philosophy such as existentialism, phenomenology or religion. (shrink)

The Computational Theory of Mind (CTM) holds that cognitive processes are essentially computational, and hence computation provides the scientific key to explaining mentality. The Representational Theory of Mind (RTM) holds that representational content is the key feature in distinguishing mental from non-mental systems. I argue that there is a deep incompatibility between these two theoretical frameworks, and that the acceptance of CTM provides strong grounds for rejecting RTM. The focal point of the incompatibility is the fact that representational content is (...) extrinsic to formal procedures as such, and the intended interpretation of syntax makes no difference to the execution of an algorithm. So the unique 'content' postulated by RTM is superfluous to the formal procedures of CTM. And once these procedures are implemented in a physical mechanism, it is exclusively the causal properties of the physical mechanism that are responsible for all aspects of the system's behaviour. So once again, postulated content is rendered superfluous. To the extent that semantic content may appear to play a role in behaviour, it must be syntactically encoded within the system, and just as in a standard computational artefact, so too with the human mind/brain - it's pure syntax all the way down to the level of physical implementation. Hence 'content' is at most a convenient meta-level gloss, projected from the outside by human theorists, which itself can play no role in cognitive processing. (shrink)

Since 2004, it has been mandated by law that all Danish undergraduate university programmes have to include a compulsory course on the philosophy of science for that particular program. At the Faculty of Science and Technology, Aarhus University, the responsibility for designing and running such courses were given to the Centre for Science Studies, where a series of courses were developed aiming at the various bachelor educations of the Faculty. Since 2005, the Centre has been running a dozen different (...) courses ranging from mathematics, computer science, physics, chemistry over medical chemistry, biology, molecular biology to sports science, geology, molecular medicine, nano science, and engineering. -/- We have adopted a teaching philosophy of using historical and contemporary case studies to anchor broader philosophical discussions in the particular subject discipline under consideration. Thus, the courses are tailored to the interests of the students of the particular programme whilst aiming for broader and important philosophical themes as well as addressing the specific mandated requirements to integrate philosophy, some introductory ethics, and some institutional history. These are multiple and diverse purposes which cannot be met except by compromise. -/- In this short presentation, we discuss our ambitions for using case studies to discuss philosophical issues and the relation between the specific philosophical discussions in the disciplines and the broader themes of philosophy of science. We give examples of the cases chosen to discuss various issues of scientific knowledge, the role of experiments, the relations between mathematics and science, and the issues of responsibility and trust in scientific results. Finally, we address the issue of how and why science students can be interested in and benefit from mandatory courses in the philosophy of their subject. (shrink)

This chapter examines the three projects that constitute contemporary African epistemology and suggests various ways in which they can be put on a firmer footing, and by so doing advance the epistemic goal of the discipline. These three projects include ethno-epistemology, analytic African epistemology and what one might call ameliorative African epistemology. Ethno-epistemology is the study of the phenomenon of knowledge from the perspective of particular African communities as revealed in their cultural heritage, proverbs, folklores, traditions, and practices. Analytic African (...) epistemology involves the philosophical study of concepts such as ‘knowledge,’ ‘justification,’ ‘belief’ and ‘truth’ from the African perspective using the methods of analysis, criticism, arguments, ordinary language philosophy, and so on. And finally, ameliorative African epistemology addresses the predicament of African knowledge systems and voices in the global knowledge and credibility economy within the broader context of the problem of epistemic injustice suffered by historically marginalized groups. (shrink)

In 1949, the Department of Philosophy at the University of Manchester organized a symposium “Mind and Machine” with Michael Polanyi, the mathematicians Alan Turing and Max Newman, the neurologists Geoff rey Jeff erson and J. Z. Young, and others as participants. Th is event is known among Turing scholars, because it laid the seed for Turing’s famous paper on “Computing Machinery and Intelligence”, but it is scarcely documented. Here, the transcript of this event, together with Polanyi’s original statement and (...) his notes taken at a lecture by Jeff erson, are edited and commented for the fi rst time. Th e originals are in the Regenstein Library of the University of Chicago. Th e introduction highlights elements of the debate that included neurophysiology, mathematics, the mind-body-machine problem, and consciousness and shows that Turing’s approach, as documented here, does not lend itself to reductionism. (shrink)

Einstein's theory of Relativity describes variances of the volume of our universe in a model that is known as "spacetime". Quantum field theory describes the volume of our universe as a composition of a limited number of basic quantum fields. Both models exclude each other.

The investigation into logical form and structure of natural sciences and mathematics covers a significant part of contemporary philosophy. In contrast to this, the metatheory of normative theories is a slowly developing research area in spite of its great predecessors, such as Aristotle, who discovered the sui generis character of practical logic, or Hume, who posed the “is-ought” problem. The intrinsic reason for this situation lies in the complex nature of practical logic. The metatheory of normative educational (...) class='Hi'>philosophy and theory inherits all the difficulties inherent in the general metatheory but has also significantly contributed to its advancement. In particular, the discussion on its mixed normative-descriptive character and complex composition has remained an important part of research in educational philosophy and theory. The two points seem to be indisputable. First, the content of educational philosophy and theory is a complex one, connecting different disciplines. Second, these disciplines are integrated within the logical form of practical inference or means-end reasoning. On the other hand, the character of consequence relation in this field, although generally recognized as specific, represents an unresolved prob- lem, a solution of which requires a sophisticated logical theory and promises to influence the self- understanding of educational philosophy and theory. (shrink)

THE GREAT GUIDE What David Hume can teach us about being human and living well 328pp. Princeton University Press. £20 (US $24.95). Julian Baggini "... The most successful aspect of The Great Guide is the “Hop-On Hop-Off” intellectual tour that it offers. The reader is taken around the various locations where Hume’s life and ideas developed, moving from country to country, city to city, and stopping off at a few stately homes en route. This tour begins with Hume’s birthplace and (...) early education in Edinburgh and the Scottish Borders, moves on to La Flèche in France (where Hume composed much of his Treatise of Human Nature), then back to London and England, and carries on from there. As we travel around with Hume, Baggini provides his readers with a steady commentary and description of his subject’s various friendships and controversies, along with brief sketches of Hume’s core ideas and contributions. All this is lively and engaging. But there are limitations to Baggini’s method and approach. ... Perhaps the most unsatisfactory aspect of Baggini’s book is his effort to turn Hume’s life and philosophy into a self-help manual. It is true that there is much about both Hume’s character and philosophy to admire, but it is not obvious that this is usefully packaged into a set of 145 'maxims' guiding us towards 'the good life'. ... It does Hume little service to present his thought like this.". (shrink)

This is not a mathematics book, but a book about mathematics, which addresses both student and teacher, with a goal as practical as possible, namely to initiate and smooth the way toward the student’s full understanding of the mathematics taught in school. The customary procedural-formal approach to teaching mathematics has resulted in students’ distorted vision of mathematics as a merely formal, instrumental, and computational discipline. Without the conceptual base of mathematics, students develop over time (...) a “mathematical anxiety” and abandon any effort to understand mathematics, which becomes their “traditional enemy” in school. This work materializes the results of the inter- and trans-disciplinary research aimed toward the understanding of mathematics, which concluded that the fields with the potential to contribute to mathematics education in this respect, by unifying the procedural and conceptual approaches, are epistemology and philosophy of mathematics and science, as well as fundamentals and history of mathematics. These results argue that students’ fear of mathematics can be annulled through a conceptual approach, and a student with a good conceptual understanding will be a better problem solver. The author has identified those zones and concepts from the above disciplines that can be adapted and processed for familiarizing the student with this type of knowledge, which should accompany the traditional content of school mathematics. The work was organized so as to create for the reader a unificatory image of the complex nature of mathematics, as well as a conceptual perspective ultimately necessary to the holistic understanding of school mathematics. The author talks about mathematics to convince readers that to understand mathematics means first to understand it as a whole, but also as part of a whole. The nature of mathematics, its primary concepts (like numbers and sets), its structures, language, methods, roles, and applicability, are all presented in their essential content, and the explanation of non-mathematical concepts is done in an accessible language and with many relevant examples. (shrink)

Abstract The values of the healthcare sector are fairly ubiquitous across the globe, focusing on caring and respect, patient health, excellence in care delivery, and multi-stakeholder collaboration. Many individual pharmacists embrace these core values. But their ability to honor these values is significantly determined by the nature of the system they work in. -/- The paper starts with a model of the prevailing pharmacist workforce model in Scotland, in which core roles are predominantly separated into hierarchically disaggregated jobs focused on (...) one professional ‘pillar’: Clinician /Practice Provider; Educator; Leader/Manager; and Researcher. This is the ‘Atomistic’ Model. This skills-segregation yields a workforce of individuals working in isolation rather than collaborating together, and lacking a shared information flow, purpose and identity. Key strategic flaws include suboptimal responsiveness to population and subpopulation needs, inconsistency and inequity of care, an erosion of professional agency, and lower job satisfaction. It is conjectured that this results from a lack of congruence between values, professional ethos, and organizational structure. ‘Atomism’ culminates in a syndrome of widespread professional-level cognitive dissonance. -/- The paper contrasts this with a new emerging workforce vision, The Collaborative Care Model. This new model defines a systems-first-approach, built on the principle that all jobs must include all four professional ‘pillars’. Vertical skills integration, involving education and task sharing, will support sustainability and succession planning. Horizontal skills integration (across practice, leadership and research) is included to improve responsiveness to population need and individual professional agency. The working conditions, supportive ethos, and career structure needed to make the model work are described. Moral theory and workforce theory are used to justify why the model may be more effective for population health, delivering greater job satisfaction for individuals and ultimately helping systematically realize and honor healthcare values. Finally, the paper sketches the first steps needed to implement the model at the national level, starting with the operationalization of new multi-pillar professional curricula across the career spectrum. Potential pitfalls and challenges are also discussed. -/- Co-Authors: 1. Paul Forsyth (PF) Lead Pharmacist Clinical Cardiology, Pharmacy, NHS Greater Glasgow & Clyde. Contribution: Conceptualization; Model Curation; Model Theory; Model Visualization; Writing - original draft (lead author), Writing - review & editing (lead author) 2. Andrew Radley (AR), Consultant in Public Health Pharmacy, NHS Tayside. Contribution: Conceptualization; Model curation; Model Theory; Writing - review & editing 3. Gordon Rushworth (GR), MPharm MSc FFRPS FRPharmS (Consultant). Programme Director, Highland Pharmacy Education & Research Centre, NHS Highland, Inverness. Contribution: Model curation; Writing - review & editing 4. Fiona Marra (FM) National Lead Clinician Scottish Infection and Immunology Network (SPAIIN) / Advanced Pharmacist HCV / HIV, NHS Greater Glasgow & Clyde. Contribution: Model curation; Writing - review & editing 5. Susan Roberts (SR) Associate Postgraduate Pharmacy Dean, NHS Education for Scotland. Contribution: Model curation; Writing - review & editing 6. Roisin O’Hare (RO) Lead Teacher Practitioner Pharmacist, Northern Ireland University Network, Southern Health and Social Care Trust. Contribution: Model curation; Writing - review & editing 7. Catherine Duggan (CD) Chief Executive Officer, International Pharmaceutical Federation (FIP). Contribution: Model curation; Writing - review & editing 8. Barry Maguire (BM) Senior Lecturer, School of Philosophy, Psychology and Life Sciences, The University of Edinburgh. Contribution: Conceptualization; Model Curation; Model Theory; Model Visualization; Supervision; Writing - original draft (senior academic supervisor), Writing - review & editing (senior academic supervisor) -/- . (shrink)

The emergence of the COVID-19 pandemic initiated major disruptions to higher education systems. Physical spaces that previously supported interpersonal interaction and community were abruptly inactivated, and faculty largely took on the responsibility of accommodating classroom structures in rapidly changing situations. This study employed interviews to examine how undergraduate Science, Technology, Engineering, and Mathematics (STEM) instructors adapted instruction to accommodate the mandated transition to virtual learning and how these accommodations supported or hindered community and belonging during the onset of the (...) pandemic. Interviews with 25 STEM faculty at an undergraduate Hispanic Serving Institution revealed a wide range of accommodations they made to their courses and how they managed communication with students. Faculty strived to support student belonging with responses ranging from caring to crisis management, though some faculty expressed feelings of powerlessness when unable to accommodate certain challenges. The case of a responsive and flexible instructor is presented to highlight a productive response to a crisis. These retrospective findings point to strategies to support faculty teaching in virtual learning environments in the future; increasing opportunities for student–student and student–faculty interaction, supporting faculty in learning technologies that support these interactions and addressing faculty’s feelings of powerlessness. (shrink)

In this paper, I describe some strategies for teaching an introductory philosophy of science course to Science, Technology, Engineering, and Mathematics (STEM) students, with reference to my own experience teaching a philosophy of science course in the Fall of 2020. The most important strategy that I advocate is what I call the “Second Philosophy” approach, according to which instructors ought to emphasize that the problems that concern philosophers of science are not manufactured and imposed by philosophers (...) from the outside, but rather are ones that arise internally, during the practice of science itself. To justify this approach, I appeal to considerations from both educational research and the epistemology of testimony. In addition, I defend some distinctive learning goals that philosophy of science instructors ought to adopt when teaching STEM students, which include rectifying empirically well-documented shortcomings in students’ conceptions of the “scientific method” and the “nature of science.” Although my primary focus will be on teaching philosophy of science to STEM students, much of what I propose can be applied to non-philosophy majors generally. Ultimately, as I argue, a successful philosophy of science course for non-philosophy majors must be one that advances a student’s science education. The strategies that I describe and defend here are aimed at precisely that end. (shrink)

Randomness, a core concept of gambling, is seen in problem gambling as responsible for the formation of the math-related cognitive distortions, especially the Gambler’s Fallacy. In problem-gambling research, the concept of randomness was traditionally referred to as having a mathematical nature and categorized and approached as such. Randomness is not a mathematical concept, and I argue that its weak mathematical dimension is not decisive at all for the randomness-related issues in gambling and problem gambling, including the correction of the misconceptions (...) and fallacies about probability and statistical concepts applied in gambling. I distinguish between mathematical and nonmathematical dimensions of randomness (the epistemic, the theoretical-methodological, the functional, and the ethical) falling within the general concept, and I argue that both the studies having as object the math-related cognitive distortions among gamblers and the educational programs aiming at correcting them should employ this distinction in their design and content. (shrink)

Several areas of welfare economics seek to evaluate states of affairs as a function of interpersonally comparable individual utilities. The aim is to map each state of affairs onto a vector of individual utilities, and then to produce an ordering of these vectors that can be represented by a mathematical function assigning a real number to each. When this approach is used in intertemporal contexts, a central theoretical question concerns the evaluative weight to be applied to utility coming at different (...) times. This question concerns the rate of pure time preference, which is one key determinant of the social discount rate. This article argues that the standard philosophical account of pure time preference is mistaken, because it ascribes to economists a methodological commitment they need not, and often do not, accept. This in turn undercuts the most common philosophical objection to pure time preference, which traces at least to Rawls’s A Theory of Justice. The article then evaluates three further objections to pure time preference, concluding that it might still be defensible under certain circumstances. The article closes by articulating a final argument that is suggested by the “Social, Economic and Ethical Concepts and Methods” chapter of the most recent Intergovernmental Panel on Climate Change report. If this further argument is sound, it would constitute a decisive objection to pure time preference as it currently figures in much intertemporal welfare economics. (shrink)

James A. Harris's biography of David Hume is the first such study to appear since Ernest Mossner's The Life of David Hume (1954). Unlike Mossner, Harris aims to write a specifically "intellectual biography", one that gives "a complete picture of Hume's ideas" and "relates Hume's works to the circumstances in which they were conceived and written" (vii). Harris's study turns on four central theses or claims about the character of Hume's thought and how it is structured and developed. The claims (...) are: (1) Hume's Treatise has no claim to any sort of "privileged" status in relation to Hume's other major works -- which include his Essays and his History of England. -/- (2) There is no system, doctrine, or fundamental aims or ambitions that serve to unify Hume's thought. -/- (3) Irreligious aims and interests are not of any particular or unique importance for Hume's thought. Moreover, whatever irreligious aims and objectives Hume may have had, they reflect his moderate, neutral and detached attitude to this and all other subjects -- there is no underlying animus or hostility against religion that motivates his contributions to this subject. (4) Hume's thought should be understood and explained not in terms of his concern with some specific subject matter or body of doctrine but rather in terms of his style and his identity as a "philosophical man of letters". -/- The general picture of Hume that emerges from this study, as constructed around these four core theses, is, as I will explain, incomplete, unconvincing and, in important respects, seriously flawed and misleading. -/- . (shrink)

Curriculum integration is frequently promoted as a means of enabling deeper and more authentic learning, with Mathematics often considered a suitable subject for doing so. This review investigates which elements contribute to the effectiveness of Mathematics integration in secondary education. Teacher factors include attitudes towards integrative practices and knowledge of both disciplinary content and curriculum integration theory. Pedagogy factors concern utilising activities that best synthesise concepts from multiple subjects to enhance learning, especially projects. Institutional factors relate to curriculum (...) and assessment requirements as well as school-level variables such as access to planning time and professional development. Limitations in the research literature include a narrow focus on connecting Mathematics with scientific disciplines, a reliance on qualitative methods and small sample sizes, and unknown generalisability to education systems other than those in which studies were conducted. Even so, attempting to navigate the factors involved in effective curriculum integration with Mathematics appears worthwhile. (shrink)

Some authors have begun to appeal directly to studies of argumentation in their analyses of mathematical practice. These include researchers from an impressively diverse range of disciplines: not only philosophy of mathematics and argumentation theory, but also psychology, education, and computer science. This introduction provides some background to their work.

This paper analyzes Paul of Venice’s theory of measurement of natural properties and changes. The main sections of the paper correspond to Paul’s analysis of the three types of accidental changes, for which the Augustinian philosopher sought to provide rules of measurement. It appears that Paul achieved an original synthesis borrowing from both Parisian and Oxfordian sources. It is also argued that, on top of this theoretical synthesis, Paul managed to elaborate a quite original theory of (...) intensive properties that marks him out not only from the nominalist framework of his Parisian sources but also from the usual realist treatments of the problem. Finally, it is shown that, to a certain extent, Paul undertook to apply the mathematical and logical tools inherited from the Calculatores tradition to empirical problems of natural philosophy, leading to reevaluate his role in the evolution of scientific thought in early 15th-century Italy. (shrink)

An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, (...) are parts of the physical world and are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge. (shrink)

Holding students to high standards and assessing, measuring and evaluating their learning with challenging, authentic problems in the midterm and final exams is the goal of the professors who teach core signal processing concepts. However, the heavy reliance of these subjects on mathematics makes it difficult for students to genuinely grasp the concepts and relate to a conceptual framework. Specifically, analyzing the signals and the functionality of systems in Fourier domain; separating the system level analysis from signal level analysis; (...) and understanding how they are related in time domain and frequency domain are among the most challenging concepts. Students’ lower grades observed over past years in the introductory signal processing course exposed a potential disconnect between the actual level of learning and the high expectations set by the professors. In this paper, we present the active learning techniques that we implemented in one of the summer session offerings of this course in our department. The research explored Peer Instruction, pre-class reading quizzes and post-lecture quizzes. In addition to the mid and end of the quarter survey results, the comparison analysis of the grades students achieved in the active learning integrated course in the second summer session and the standard course offered in first summer session is discussed. According to our results, the developed techniques helped students in the active classroom perform significantly better than their peers participating in standard lectures when tested by challenging questions in their exams. (shrink)

Abstract -/- Aim: The present study aims of this study is to evaluate the Defense University's education policy by examining how the use of libraries in supporting scientific literacy. -/- Methodology: This research method is qualitative by using phenomenology approach. It means that in understanding something has an objective and subjective side, unlike the positivism of separating between the two (subject-object), for science to be objective. -/- Result: Apparently there is a process of transformation of values and symbolism of the (...) "Defense of Country / DoC" campus from the instructional education system to the science literacy that is pursued in accordance with the Ministry of Higher Education program but in line with the mission of the Defense Ministry. At present it has stood four Faculty of Defense Strategy, Defense Management, National Security and Defense Technology. The research concludes that the Indonesia Defense University (IDU) builds a distinctive character for both military and civilian needs to enhance the role of libraries as a center for the development of the Defense of Country (DoC) characters in semi-special model. In addition to fostering national character based on Pancasila can also follow the development of scientific literature supported by advanced information technology. -/- Conclution: The decrease of visits to the Library cannot indicate a decrease in student interest in science literature because in the same year all students load their rich works in the journals of each Study Program. Vision IDU Library itself is to be one of the best Library of Higher Education in Indonesia, . (shrink)

There has been little overt discussion of the experimental philosophy of logic or mathematics. So it may be tempting to assume that application of the methods of experimental philosophy to these areas is impractical or unavailing. This assumption is undercut by three trends in recent research: a renewed interest in historical antecedents of experimental philosophy in philosophical logic; a “practice turn” in the philosophies of mathematics and logic; and philosophical interest in a substantial body of (...) work in adjacent disciplines, such as the psychology of reasoning and mathematics education. This introduction offers a snapshot of each trend and addresses how they intersect with some of the standard criticisms of experimental philosophy. It also briefly summarizes the specific contribution of the other chapters of this book. (shrink)

What were the reasons of the Copernican Revolution ? How did modern science (created by a bunch of ambitious intellectuals) manage to force out the old one created by Aristotle and Ptolemy, rooted in millennial traditions and strongly supported by the Church? What deep internal causes and strong social movements took part in the genesis, development and victory of modern science? The author comes to a new picture of Copernican Revolution on the basis of the elaborated model of scientific revolutions (...) that takes into account some recent advances in philosophy, sociology and history of science. The model was initially invented to describe Einstein’s Revolution of the XX century beginning. The model considers the growth of knowledge as interaction, interpenetration and unification of the research programmes, springing out of different cultural traditions. Thus, Copernican Revolution appears as a result of revealation and (partial) resolution of the dualism , of the gap between Ptolemy’s mathematical astronomy and Aristotelian qualitative physics. The works of Copernicus, Galileo, Kepler and Newton were all the stages of mathematics descendance from skies to earth and reciprocal extrapolation of earth physics on skies. The model elaborated enables to reassess the role of some social factors crucial for the scientific revolution. It is argued that initially modern science was a result of the development of Christian Weltanschaugung . Later the main support came from the absolute monarchies. In the long run the creators of modern science appeared to be the “apparatchics” of the “regime of truth” built-in state machine. Natural science became a part of ideological state apparatus providing not only scientific education but the internalization of values crucial for the functioning of state. -/- . (shrink)

Many instances have presented mathematics as a difficult subject for students, and some of these difficulties have been perceived as an injustice to those who suffer from it. Meanwhile, no child is to be left behind in building a just society if all were to experience the collective good of mathematics. For this reason, it became necessary to explore pre-service teachers’ perceptions of mathematics as social justice in their lived experiences. In doing this, the qualitative interpretive approach (...) was adopted, where qualitative data was collected, transcribed, and analysed using the case study design. A sample of eleven pre-service teachers were involved in the study. The findings indicated that the pre-service teachers had experienced severe issues of mathematics as social justice in their previous schooling that were unfair to them. However, they now felt mathematics could be interesting if delivered in a manner that let their voices be heard in the delivery process. (shrink)

Some people believe that critical thinking is not a modern science, but its roots are old and deeply rooted in the history of philosophy. Its roots date back to Aristotle, the inventor of logic and who was called the first teacher by virtue of this invention. Aristotle was impressed by the language of mathematics and wanted to invent a language to logic similar to the language of Mathematics. What encouraged Aristotle to do so is that Math language (...) is quite different from daily life language. For example , when you say one in Mathematics language, what is this one you mean or intend? That is what is one? Is it a specific man or an animal or something abstract or any creature of the creatures of the natural world? The one we are talking about is neither this nor that and you can use it to talk about one human being or one animal or one creature. Therefore, the concept of the one is something abstract and it is so far from the creatures of the exterior world. Geometry has the same case. Triangle, quarter and rectangle as such are mere geometrical shapes and are far away from real life language. But when we use them in our daily life language we give them specific life meanings. -/- As I said, Aristotle was impressed by the language of Mathematics and wanted to build a similar language for logic .The rationale behind this is to get away from the language of ambiguity caused by the use of daily life language .One word or utterance in a language has many usages in one language or when translated into other languages , and therefore it causes many problems in usage instead of simplifying it and this is what Aristotle tried to avoid. Aristotle chose letters instead of words, the letter is part of the word, and the letter is meaningless unless it is a part of the word. The meaning of a letter depends on its occurrence in the word. If you compose a word from a group of letters and use it , it becomes ambiguous because it becomes part of the used language. -/- However, Aristotle did not combine letters in a word but he used them as abstract letters and gave them meanings in the language of logic where there is no divergence about its use. Aristotle used them in syllogism and they mean the limits of reason . Until recently critical thinking was part of formal logic and learning it means that you have logical background or at least you are familiar with the rules and fundamentals of logic. But critical thinking theorists soon took the initiative to separate it from logic because the latter started to suffer from its rules which are difficult to understand and which require some awareness and thoughts on the part of the scholar to understand. Hence , critical thinking is not supposed to possess a background of logic and it adopts a specific questioning approach where it suggests a number of questions the practitioner of critical thinking has to successively answer since he has to make sure that the opinion he reaches does not rely on any subjective considerations. This is how the skills that are acquired on a continuum of related critical questions are established ,this is not to mention that the concept of critical thinking itself is usually defined in this context as the awareness of such kinds of questions and the ability to arouse and answer them at appropriate times. My paper explains, How to be able to implement these skills in high education and what are the benefits of these skills? The paper includes; Critical Thinking and Logic Socrates `s Approach The impact of critical thinking on teaching methods Questioning How questions are organized in the light of critical thinking? Using Critical Thinking in Problem-Solving Use of Non-Oral language in Critical Thinking. (shrink)