The connection between science and mathematics is often considered necessary and insoluble. Therefore, a relationship between mathematics and humanities or arts is deemed exceptional or sometimes unnatural. Nevertheless, on the basis of historical, ontological and epistemological researches it can be noted that it’s impossible to warrant the immediate identification between mathematics and sciences on a deeper level than the practical one. Given the instrumentality and then the unnecessity of this connection, the relationship between mathematics and not-scientific disciplines is undeniable, (...) even if the mathematics in the explicit formalisms which we know doesn’t appear in them. It’s possible to demonstrate this relationship not only with philosophical argumentations, but also whit empirical verifications, e.g. in the music and in particular in the music of J. S. Bach. Such an epistemological thought finally leads to the question on the possibility of knowledge in the art in comparison to the epistemological characteristics of the Galilean and Post-Galilean science. (shrink)

We have reached the peculiar situation where the advance of mainstream science has required us to dismiss as unreal our own existence as free, creative agents, the very condition of there being science at all. Efforts to free science from this dead-end and to give a place to creative becoming in the world have been hampered by unexamined assumptions about what science should be, assumptions which presuppose that if creative becoming is explained, it will be explained (...) away as an illusion. In this paper it is shown that this problem has permeated the whole of European civilization from the Ancient Greeks onwards, leading to a radical disjunction between cosmology which aims at a grasp of the universe through mathematics and history which aims to comprehend human action through stories. By going back to the Ancient Greeks and tracing the evolution of the denial of creative becoming, I trace the layers of assumptions that must in some way be transcended if we are to develop a truly post-Egyptian science consistent with the forms of understanding and explanation that have evolved within history. (shrink)

Du Châtelet holds that mathematical representations play an explanatory role in natural science. Moreover, she writes that things proceed in nature as they do in geometry. How should we square these assertions with Du Châtelet’s idealism about mathematical objects, on which they are ‘fictions’ dependent on acts of abstraction? The question is especially pressing because some of her important interlocutors (Wolff, Maupertuis, and Voltaire) denied that mathematics informs us about the properties of material things. After situating Du (...) Châtelet in this debate, this chapter argues, first, that her account of the origins of mathematical objects is less subjectivist than it might seem. Mathematical objects are non-arbitrary, public entities. While mathematical objects are partly mind-dependent, so are material things. Mathematical objects can approximate the material. Second, it is argued that this moderate metaphysical position underlies Du Châtelet’s persistent claims that mathematics is required for certain kinds of general knowledge, including in natural science. The chapter concludes with an illustrative example: an analysis of Du Châtelet’s argument that matter is continuous. A key premise in the argument is that mathematical representations and material nature correspond. (shrink)

In the last couple of years, a few seemingly independent debates on scientific explanation have emerged, with several key questions that take different forms in different areas. For example, the questions what makes an explanation distinctly mathematical and are there any non-causal explanations in sciences (i.e., explanations that don’t cite causes in the explanans) sometimes take a form of the question of what makes mathematical models explanatory, especially whether highly idealized models in science can be explanatory and (...) in virtue of what they are explanatory. These questions raise further issues about counterfactuals, modality, and explanatory asymmetries: i.e., do mathematical and non-causal explanations support counterfactuals, and how ought we to understand explanatory asymmetries in non-causal explanations? Even though these are very common issues in the philosophy of physics and mathematics, they can be found in different guises in the philosophy of biology where there is the statistical interpretation of the Modern Synthesis theory of evolution, according to which the post-Darwinian theory of natural selection explains evolutionary change by citing statistical properties of populations and not the causes of changes. These questions also arise in philosophy of ecology or neuroscience in regard to the nature of topological explanations. The question here is can the mathematical (or more precisely topological) properties in network models in biology, ecology, neuroscience, and computer science be explanatory of physical phenomena, or are they just different ways to represent causal structures. The aim of this special issue is to unify all these debates around several overlapping questions. These questions are: are there genuinely or distinctively mathematical and non-causal explanations?; are all distinctively mathematical explanations also non-causal; in virtue of what they are explanatory; does the instantiation, implementation, or in general, applicability of mathematical structures to a variety of phenomena and systems play any explanatory role? This special issue provides a platform for unifying the debates around several key issues and thus opens up avenues for better understanding of mathematical and non-causal explanations in general, but also, it will enable even better understanding of key issues within each of the debates. (shrink)

The present yearbook (which is the fourth in the series) is subtitled Trends & Cycles. It is devoted to cyclical and trend dynamics in society and nature; special attention is paid to economic and demographic aspects, in particular to the mathematical modeling of the Malthusian and post-Malthusian traps' dynamics. An increasingly important role is played by new directions in historical research that study long-term dynamic processes and quantitative changes. This kind of history can hardly develop without the application (...) of mathematical methods. There is a tendency to study history as a system of various processes, within which one can detect waves and cycles of different lengths – from a few years to several centuries, or even millennia. The contributions to this yearbook present a qualitative and quantitative analysis of global historical, political, economic and demographic processes, as well as their mathematical models. This issue of the yearbook consists of three main sections: (I) Long-Term Trends in Nature and Society; (II) Cyclical Processes in Pre-industrial Societies; (III) Contemporary History and Processes. We hope that this issue of the yearbook will be interesting and useful both for historians and mathematicians, as well as for all those dealing with various social and natural sciences. (shrink)

[EDIT: This book will be published open access. Production is taking longer than expected but I will post the link to the whole book, hopefully by October.] One central aim of science is to provide explanations of natural phenomena. What role(s) does mathematics play in achieving this aim? How does mathematics contribute to the explanatory power of science? Rules to Infinity defends the thesis, common though perhaps inchoate among many members of the Vienna Circle, that mathematics contributes (...) to the explanatory power of science by expressing conceptual rules, rules which allow the transformation of empirical descriptions. Mathematics should not be thought of as describing, in any substantive sense, an abstract realm of eternal mathematical objects, as traditional platonists have thought. In Rules to Infinity, this view, which I call mathematical normativism, is updated with contemporary philosophical tools, and it is argued that normativism is compatible with mainstream semantic theory. This allows the normativist to accept that there are mathematical truths, while resisting the platonistic idea that there exist abstract mathematical objects that explain such truths. Furthermore, Rules to Infinity defends a particular account of the distinction between scientific explanations that are in some sense distinctively mathematical – those that explain natural phenomena in some uniquely mathematical way – and those that are only standardly mathematical, and it lays out desiderata for any account of this distinction. Normativism is compared with other prominent views in the philosophy of mathematics such as neo-Fregeanism, fictionalism, conventionalism, and structuralism. Rules to Infinity serves as an entry point into debates at the forefront of philosophy of science and mathematics, and it defends novel positions in these debates. (shrink)

The study investigated the effects of computer-assisted instruction on mathematics achievement among secondary school students in Rivers State. Two research questions and two hypotheses guided the study. The design was quasi-experimental. The population of the study was 215 students in a senior secondary school Kpor in Gokana. The sample of the study was 35 students. The sample size was drawn using simple random sampling technique. The instrument used to collect data was multiple choice achievement test. The instrument was validated by (...) experts in computer science education and the researchers, while the reliability of the instrument was established at 0.81. Mean and standard deviation was used to analyze pre-test and post-test scores to answer research questions, while z-test was used to test the hypotheses at 0.05 significant level. The findings of the study showed that students taught with computer assisted instruction performed higher than those taught with traditional instructional methods. Findings also revealed that there is a significant difference between the pretest and post-test achievement scores of students taught Mathematics with computer assisted instructional method. The study recommended among others that government should integrate computer and educational software into the mainstream of mathematics curriculum from the junior secondary school level as this will help to prepare students for stellar performance in Mathematics at senior secondary level. (shrink)

This study investigated the impact of collaborative learning on mathematics achievement and attitudes in sixth-grade students, comparing it to traditional didactic teaching. A quasi-experimental research design was utilized in which sixth-grade students were randomly assigned to either control or experimental groups. Pre- and post-tests assessed mathematics achievement using curriculum-aligned tests. In addition, attitudes toward mathematics were measured using the ‘attitude towards mathematics’ inventory developed by Tapai and Marsh in 2004. Both groups exhibited similar pre-test levels. The experimental group received (...) collaborative learning, while the control group received traditional teaching. Post-tests after a 12-week intervention showed significant improvements in the experimental group’s mathematics achievement, regardless of initial achievement levels. Positive changes in attitudes toward mathematics were also observed in the experimental group, with some progress in the control group. Collaborative learning appears promising for enhancing mathematics achievement and nurturing positive attitudes in elementary students. (shrink)

In his book “Galileo’s Error”, Philip Goff lays out what he calls “foundations for a new science of consciousness”, which are decidedly anti-physicalist (panpsychist), motivated by a critique of Galileo’s distinction into knowable objective and unknowable subjective properties and Arthur Eddington’s argument for the limitation of purely structural (physical) knowledge. Here we outline an alternative theory, premised on the Interface Theory of Perception, that too subscribes to a “post-Galilean” research programme. However, interface theorists disagree along several lines. 1. (...) They note that Galileo’s distinction should be replaced by a truly non-dual account, referring to a difference of degree only. 2. They highly appreciate the role of mathematics, in particular when it comes to actually engage scientifically with consciousness. Some notable features of the interface theory are its skepticism towards our epistemic capacities and its rejection of the existence of a public, mind-independent reality. In addition, some interface theorists further employ a thin concept of “conscious agency” to ground their theory. The interface theory leaves open many of the problems of consciousness science (e.g. what is a “self”?) as questions for further (scientific, mathematical) research. (shrink)

This research article was championed as a way of providing discourses pertaining to the concept of "Critical Realism (CR)" approach, which is amongst many othe forms of competing postmodern philosophical concepts for the engagement of dialogical discourses in the area of established econonetric methodologies for effective policy prescription in the economic science discipline. On the the whole, there is no doubt surrounding the value of empirical endeavours in econometrics to address real world economic problems, but equally so, the heavy (...) weighted use and reliance on mathematical contents as a way of justifying its scientific base seemed to be loosing traction of the intended focus of economics when it comes to confronting real world problems in the domain of social interaction. In this vein, the construction of mixed methods discourse(s), which favour that of CR philosophy is hereby suggested in this article as a way forward in confronting with issues raised by critics of mainstream economics and other professionals in the postmodern era. (shrink)

Climatology is a paradigmatic complex systems science. Understanding the global climate involves tackling problems in physics, chemistry, economics, and many other disciplines. I argue that complex systems like the global climate are characterized by certain dynamical features that explain how those systems change over time. A complex system's dynamics are shaped by the interaction of many different components operating at many different temporal and spatial scales. Examining the multidisciplinary and holistic methods of climatology can help us better understand the (...) nature of complex systems in general. -/- Questions surrounding climate science can be divided into three rough categories: foundational, methodological, and evaluative questions. "How do we know that we can trust science?" is a paradigmatic foundational question (and a surprisingly difficult one to answer). Because the global climate is so complex, questions like "what makes a system complex?" also fall into this category. There are a number of existing definitions of `complexity,' and while all of them capture some aspects of what makes intuitively complex systems distinctive, none is entirely satisfactory. Most existing accounts of complexity have been developed to work with information-theoretic objects (signals, for instance) rather than the physical and social systems studied by scientists. -/- Dynamical complexity, a concept articulated in detail in the first third of the dissertation, is designed to bridge the gap between the mathematics of contemporary complexity theory (in particular the formalism of "effective complexity" developed by Gell-Mann and Lloyd [2003]) and a more general account of the structure of science generally. Dynamical complexity provides a physical interpretation of the formal tools of mathematical complexity theory, and thus can be used as a framework for thinking about general problems in the philosophy of science, including theories, explanation, and lawhood. -/- Methodological questions include questions about how climate science constructs its models, on what basis we trust those models, and how we might improve those models. In order to answer questions about climate modeling, it's important to understand what climate models look like and how they are constructed. Climate model families are significantly more diverse than are the model families of most other sciences (even sciences that study other complex systems). Existing climate models range from basic models that can be solved on paper to staggeringly complicated models that can only be analyzed using the most advanced supercomputers in the world. I introduce some of the central concepts in climatology by demonstrating how one of the most basic climate models might be constructed. I begin with the assumption that the Earth is a simple featureless blackbody which receives energy from the sun and releases it into space, and show how to model that assumption formally. I then gradually add other factors (e.g. albedo and the greenhouse effect) to the model, and show how each addition brings the model's prediction closer to agreement with observation. After constructing this basic model, I describe the so-called "complexity hierarchy" of the rest of climate models, and argue that the sense of "complexity" used in the climate modeling community is related to dynamical complexity. -/- With a clear understanding of the basics of climate modeling in hand, I then argue that foundational issues discussed early in the dissertation suggest that computation plays an irrevocably central role in climate modeling. "Science by simulation" is essential given the complexity of the global climate, but features of the climate system--the presence of non-linearities, feedback loops, and chaotic dynamics--put principled limits on the effectiveness of computational models. This tension is at the root of the staggering pluralism of the climate model hierarchy, and suggests that such pluralism is here to stay, rather than an artifact of our ignorance. Rather than attempting to converge on a single "best fit" climate model, we ought to embrace the diversity of climate models, and view each as a specialized tool designed to predict and explain a rather narrow range of phenomena. Understanding the climate system as a whole requires examining a number of different models, and correlating their outputs. This is the most significant methodological challenge of climatology. -/- Climatology's role contemporary political discourse raises an unusually high number of evaluative questions for a physical science. The two leading approaches to crafting policy surrounding climate change center on mitigation (i.e. stopping the changes from occurring) and adaptation (making post hoc changes to ameliorate the harm caused by those changes). Crafting an effective socio-political response to the threat of anthropogenic climate change, however, requires us to integrate multiple perspectives and values: the proper response will be just as diverse and pluralistic as the climate models themselves, and will incorporate aspects of both approaches. I conclude by offering some concrete recommendations about how to integrate this value pluralism into our socio-political decision making framework. (shrink)

The aim of this paper is to investigate the current debate on the state of economics from a methodological perspective. We claim that the majority of contributions criticizing modern economics are not based on clear methodological principles and thus many of them are not correct. We show this with respect to such issues as the problem of realisticness of models and their assumptions, the role of mathematics in economics, the way we conceptualize the relation between economics (theory) and economy (empiria), (...) as well as the general problem of comparing theories. In doing so we use the research apparatus taken form the philosophy of science and also we benefit a lot from recent developments in the philosophy of economics. Finally, we show one of the paradoxes of that debate, namely that many critics of economics accuse economists of using the wrong language (mathematics) while they do not use proper language themselves while criticizing economics, namely the apparatus taken from the philosophy of science. (shrink)

The latest draft (posted 05/14/22) of this short, concise work of proof, theory, and metatheory provides summary meta-proofs and verification of the work and results presented in the Theory and Metatheory of Atemporal Primacy and Riemann, Metatheory, and Proof. In this version, several new and revised definitions of terms were added to subsection SS.1; and many corrected equations, theorems, metatheorems, proofs, and explanations are included in the main text. The body of the text is approximately 18 pages, with 3 sections; (...) 1, an Introduction (with a 108 page listing of key terms & definitions), sect. 2, the Results, sect. 3, Discussion (commentary & predictions), and sect. 4, Works Cited. As much as possible, the style is intended for readability and understanding by very bright children (with some interest & knowledge of maths, etc.) and very interested nonprofessionals. The results of this project also enable upgrades of number theory, set theory, proof theory, metamathematics, the foundations of science, and quantum mechanics theory (etc.). (shrink)

Preface -/- 1. 'Anima' and 'res cogitans'. The Cartesian idea of nature and mind as a residual concept. The first chapter discusses the genesis of the concept of mind in Cartesian Philosophy; the claim is advanced that 'res cogitans' is a residual concept, defined on the basis of a previous definition of matter as 'res extensa'. As a consequence, a contradictory ontology of the mind is Descartes's poisoned bequest to the following tradition of 'scientific' psychology. -/- 2. The Mathematical (...) method in the theory of the mind and the passions. The second chapter discusses the first ambitious attempt in this direction, namely Spinoza's theory of the mind and the passions in Ethics II and III; the topics treated are Spinoza's program of a mathematical science of the mind and his transformation of the genre 'treatise of the passions' into a would-be mathematical science of the motions of the soul. -/- 3. The definition of affections and the impossibility of reducing inadequate to adequate ideas. The third chapter discusses a tension in Spinoza project of a theory of the passions, namely a need to reduce concepts to pure mathematical definitions and the opposite need to keep inadequate imaginative definitions of the phenomena under scrutiny, namely the passions, in order to avoid a radical reduction of the subject-matter to the only truly existing reality (the world order in itself) that would leave the theory without any subject-matter at all. -/- 4. 'Conatus', 'potentia', 'vis'. Concepts of force in the theory of passions and the impossibility of eliminating occult qualities. The fourth chapter discusses the troubles arising in the original project deriving from difficulties in the treatment of dynamic concepts, such as the notion of conatus, which were an unavoidable consequence of Cartesian assumptions. -/- 5. Overview and discussion. Chapter five tries to draw a general discussion, focusing on the tension between the Cartesian assumptions and the actual practice in the new post-Galilean sciences, including attempts at giving birth to psychology or science of man. -/- Appendices I. The mind-body problem in Spinoza. II. Concepts of force in Newton and Spinoza. III. Materialist readings of Newton and Spinoza. IV. On the opposition of materialism and spiritualism and Geisteswissenschaften and natural science. V. Conatus, the idea of liberation and social theory in Spinoza. (shrink)

J. N. (Jitendra Nath) Mohanty (1928–2023). -/- Professor J. N. Mohanty has characterized his life and philosophy as being both “inside” and “outside” East and West, i.e., inside and outside traditions of India and those of the West, living in both India and United States: geographically, culturally, and philosophically; while also traveling the world: Melbourne to Moscow. Most of his academic time was spent teaching at the University of Oklahoma, The New School Graduate Faculty, and finally Temple University. Yet his (...) preeminent work in Husserlian phenomenology developed alongside his eminent work in Indian philosophy: describing his interests as “a fusion of disparate horizons.” -/- J. N. Mohanty was born September 26, 1928, in Cuttack (Odisha, East India). After graduating from high school, he went on to study both Indian and Western philosophy in Calcutta, earning bachelor’s and master’s degrees. There he read Whitehead and Kant’s First Critique; although he wanted to include mathematics in his curriculum, he was led instead to include Indian philosophy and Sanskrit. On the shelves of his teacher, Ras Vihar Das, he came upon a copy of the English translation (Ideas I, 1931) of Edmund Husserl’s classic Ideen I (1913), which presented Husserl’s ground-breaking conception of phenomenology. In 1952–1954 he left India for the first time, reaching Göttingen to study mathematics and philosophy, which earned him a doctorate in mathematics and “philosophy of mathematical sciences” (in his own words). In Göttingen, Mohanty found the powerful mathematical world that Husserl himself had earlier interacted with, where several Husserl students had formed the Göttingen school of phenomenology. During these years Mohanty studied primarily mathematics, alongside Kant and also Vedic Sanskrit. From his friend Günter Patzig, interpreter of Aristotle and Frege, Mohanty was drawn to Frege in relation to mathematical logic. He attended lectures of Heidegger, intrigued by his ontological thinking. Yet, despite the Husserlian legacy, Mohanty was completely self-taught in his studies of Husserl (as he has reported). With a doctorate in mathematics, and ideas from Kant and Frege in his philosophical background, Mohanty set about crafting his own conception of philosophy grounded in phenomenology, drawing on Husserl’s extensive work, critically sifting through Husserl’s texts and their emerging concepts of intentionality, meaning, subject, intersubjectivity, and world. In between he wrote his first book-length study: on phenomenological insights in Nicolai Hartmann and A. N. Whitehead (1958). -/- Over many decades Mohanty formulated and argued, in analytical detail, for a conception of phenomenology and its place in philosophy, later presented in a clear and concise book titled Transcendental Phenomenology: An Analytic Account (1989). Over these very decades the same scholar explored classical and recent Indian philosophy, thinking through kindred ideas of consciousness, self, and knowledge drawn from the Indian philosophical contexts. While writing on Nyāya theory of truth, he also pondered whether the world and finite individual are illusory or real, and whether Marx, Arendt, Gandhi (whom he heard speak in Calcutta), or Vinoba Bhave (with whom he marched across India for the land-grant movement) could best navigate post-Independent India’s social and svarāj or self-rule reforms. The two Mohantys, thinking through a vision of self and world, turned out to be “non-different” or “non-dual” as they each practiced critical phenomenology from both inside and outside the respective philosophical and cultural traditions. Numerous students, fellows, and colleagues or collaborators have benefited immensely from this infusion and unified approach to diversity in philosophical thought. In the 2000s, moving into retirement, Mohanty wrote two long books devoted to his understanding of Husserl and phenomenology and the calling of philosophy itself. This two-volume study shows Mohanty himself thinking through Husserl, critically, in The Philosophy of Edmund Husserl: A Historical Introduction (2008), and Edmund Husserl’s Freiburg Years: 1916–1938. As Mohanty worked on Husserl, he carefully indicated where he agreed and where he rejected or changed ideas, all part of his practice in phenomenology of “description and interpretation.” It was the same pattern he used in addressing the thought of Husserl vis-à-vis Kant or Frege or even Quine. -/- As Mohanty developed his understanding of phenomenology over the years, he wrote books on theory of meaning and the concept of intentionality, developing a model of ideal meaning and its foundation in intentionality, drawing on Husserl’s results. He followed these with the book Husserl and Frege (1982), linking the thought of those foundational figures for the “continental” and “analytic” traditions, respectively, in twentieth-century Western philosophy. Over his long career Mohanty addressed both traditions in his clear and accessible writing style. While developing his views on phenomenology, Mohanty regularly looked to “Husserl and his others,” evaluating views in Heidegger, Sartre, Merleau-Ponty, Ricoeur, and then Derrida and others in the wake of Husserl. With his background in Kant, Mohanty also looked toward Heidegger and Hegel in relation to Husserl’s later work in the Crisis (1935–38). Similarly, he looked to contemporary analytic philosophers, adjudicating his own, oft-wise Husserlian views in relation to Frege, Nagel, and others. Amid his active scholarly career, Mohanty co-founded the journal Husserl Studies, and was editorial advisor to Philosophy & Phenomenological Research, Philosophy East & West, Journal of Indian Philosophy, Sophia, among others. -/- Yet all this while Mohanty was also thinking and writing about Indian philosophy and its relation to phenomenology. In his own retrospective, Indian philosophy is “the permanent background” of his Husserlian thinking, while Kant is the recurrent Western background of his Husserlian phenomenology. Mohanty’s form of “transcendental phenomenology” evolved, in his own perspective, against the background of his studies of Navya-Nyāya on logic and Vedānta on consciousness, in Indian philosophy, and against the background of Kant’s First Critique on the transcendental, in Western philosophy. Accordingly, Mohanty’s study of logical form and of the intentionality of consciousness seeks a fusion of East and West in the conception of transcendental phenomenology. (Cf. Mohanty’s apt response to critics in The Empirical and the Transcendental (2000).) Even in the context of North American Husserl scholarship, Mohanty has exercised an earnest fusion of East and West. For the so-called East Coast and West Coast interpretations of Husserl’s crucial notion of noema both find a sympathetic spirit in J. N. Mohanty’s careful and nuanced interpretation of Husserlian transcendental phenomenology. With Mohanty the two faces of the noema are the logical (Fregean) and the phenomenal (Kantian), and these views of intentional structure join in consciousness—in a way resonant with Indian thought. -/- - David Woodruff Smith (UC Irvine) and Purushottama Bilimoria (SFSU, San Francisco; University of Melbourne) -/- For photo of Prof Mohanty visit APA online memorial_minutes2023 Courtesy of APA Memorial Minutes (& Proceedings) 2023 . (shrink)

Defending Robert Rosen’s claim that in every confrontation between physics and biology it is physics that has always had to give ground, it is shown that many of the most important advances in mathematics and physics over the last two centuries have followed from Schelling’s demand for a new physics that could make the emergence of life intelligible. Consequently, while reductionism prevails in biology, many biophysicists are resolutely anti-reductionist. This history is used to identify and defend a fragmented but progressive (...) tradition of anti-reductionist biomathematics. It is shown that the mathematicoephysico echemical morphology research program, the biosemiotics movement, and the relational biology of Rosen, although they have developed independently of each other, are built on and advance this antireductionist tradition of thought. It is suggested that understanding this history and its relationship to the broader history of post-Newtonian science could provide guidance for and justify both the integration of these strands and radically new work in post-reductionist biomathematics. (shrink)

Wheeler proposed that repeated acts of observation give rise to the reality that we observe, but offered no detailed mechanism for this. Here this creative process is accounted for on the basis of the idea that nature has a deep technological aspect that evolves as a result of selection processes that act upon observers making use of the technologies. This leads to the conclusion that our universe is the product of agencies that use these evolved technologies to suit particular purposes (...) (MS based on lecture prepared for 2015 Lindau meeting, which the author was unable to attend). (shrink)

The phenomenon of rationalism and irrationalism, contextually related to the transformation methodology and the social function of modern (post-industrial) science – social verification, interpretation and knowledge, etc., are analyzes.

-/- FILOSOFIA: CRÍTICA À METAFÍSICA -/- PHILOSOPHY: CRITICISM TO METAPHYSICS -/- Por: Emanuel Isaque Cordeiro da Silva - UFRPE Alana Thaís Mayza da Silva - CAP-UFPE RESUMO: A Metafísica (do grego: Μεταφυσική) é uma área inerente à Filosofia, dito isto, é uma esfera que compreende o mundo e os seres humanos sob uma fundamentação suprassensível da realidade, bem como goza de fundamentação ontológica e teológica para explicação dos dilemas do nosso mundo. Logo, não goza da experiência e explicação científica com (...) base na matemática, ciências, observação, análise, etc. e sim da explicação apenas teorética sem a análise empírica. Por fim, filósofos de divergentes escolas e eras da história da filosofia fundamentaram suas críticas quanto a Metafísica; e este breve trabalho tem como objetivo analisar as críticas dos filósofos e trazer uma conclusão clara e objetiva para os leitores, sejam leigos ou não. Palavras-chave: Metafísica. Crítica. Teoria. Suprassensível. Empírico. Experiência. ABSTRACT Metaphysics is an area inherent to philosophy, that is, it is a sphere that comprises the world and human beings under a supersensitive foundation of reality, as well as enjoying ontological and theological foundations for explaining the dilemmas of our world. Therefore, it does not enjoy the scientific experience and explanation based on mathematics, science, observation, analysis, etc., but only the theoretical explanation without empirical analysis. Finally, philosophers from divergent schools and eras in the history of philosophy based their criticism of Metaphysics, and this brief work aims to analyze the criticism of philosophers and bring a clear and objective conclusion to readers, whether lay or not. Keywords: Metaphysics. Criticism. Theory. Supersensitive. Empirical. Experience. INTRODUÇÃO Para uma fundamentação alicerçada na observação, análise e conclusões pragmáticas, é imprescindível aclarar o conceito de Metafísica como esfera inerente da Filosofia, bem como disciplina primordial para o currículo do Filósofo formado nas Universidades globais. Para isso, discorro acerca do conceito de Metafísica desde a alcunha aristotélica de 'Filosofia primeira', passando à análise modernista, em especial à kantiana e seu criticismo (Crítica da Metafísica é a alma desse trabalho), sob análise de autores didáticos e, por fim, sob a elucidação da esfera Metafísica mediante os dicionários de Filosofia de Japiassú e Marcondes, e de Nicola Abbagnano. No conjunto de obras aristotélicas entituladas de Metafísica, Aristóteles buscou elucidar 'o ser enquanto ser', isto é, buscar a objetividade das coisas e do mundo mediante a subjetividade e a realidade suprassensível dessas coisas. Logo, a explicação dos fenômenos que cercavam a Grécia clássica eram explicados além do alicerce das Ciências tradicionais da época (Física, Química, Biologia, etc.). Todavia, para Aristóteles, a Metafísica consiste na 'ciência primeira' no que se refere o fornecimento de um fundamento único para todas as demais ciências, ou seja, dar-lhes o objeto ao qual elas se referem e os princípios dos quais todas elas dependem. Por fim, a Metafísica implica ser uma enciclopédia das ciências um inventário completo e exaustivo de todas as ciências, em suas relações de coordenação e subordinação, nas tarefas e nos limites atribuídos a cada uma, de modo definitivo. Na modernidade, a Metafísica perde a centralidade do mundo da Filosofia, quem é o precursor de tal declínio é Immanuel Kant e seu criticismo. Tal descendência é elucidada e aclarada posteriormente no presente trabalho. A Metafísica moderna ganha uma nova interpretação, sendo alcunhada e levada por analogia à Ontologia, expressada, segundo Kant, como conceito de gnosiologia. Para Kant, Metafísica é a fonte inerente do estudo e explicação das formas alicerçadas na razão, bem como fundamento de toda realidade suprassensível que se deve basear para elucidar e aclarar o mundo moderno. Ainda segundo o filósofo, a Metafísica é a fonte de todos os princípios reais para a explicação da realidade. Sendo ela, por fim um 'sistema filosófico' alicerçado sobre uma perspectiva ontológica, teológica e/ou suprassensível da realidade. No nosso finalismo, a inerência da Metafísica à filosofia transcende a explicação de todo o universo e sua totalidade (matéria e forma como alcunha Aristóteles). Logo, na Filosofia Clássica com Platão e, especialmente em Aristóteles, o Mito passou a ser ciência e essa ciência era determinada de Metafísica. Vale salientar que a alcunha 'Metafísica' foi, primeiramente, usada por Andrônico de Rodes . Por fim, a Metafísica primordial para fundamentar as explicações necessárias para os fenômenos que se decorrera na Grécia antiga, foi perdendo centralidade ao passo da história da filosofia e, na modernidade, com Hume e Kant passa por uma grande crise, discorrida agora no trabalho. -/- DA CRÍTICA À METAFÍSICA Platão criou um sistema metafísico de explicação da realidade e do mundo que influenciou muitos pensadores e religiosos. Boa parte da história da filosofia foi composta de sistemas metafísicos. Mas, se, por um lado, o pensamento metafísico tem uma vida longa e profícua, por outro, é alvo de críticas constantes, principalmente por filósofos modernos e contemporâneos, que questionam a validade das teses metafísicas, seja porque tratam de coisas que não podem ser conhecidas, como alma, Deus e formas inteligíveis, seja porque suas conclusões são consideradas enganosas. ARISTÓTELES E A FILOSOFIA PRIMEIRA As ciências teóricas ou teoréticas são aquelas que produzem um saber universal, válido em qualquer situação, e necessário, isto é, que não pode ser de outro modo. Essas ciências estão voltadas para a contemplação da verdade. O sábio que se dedica a elas encontra um fim em si mesmo, pois o resultado de sua investigação não gera nenhum objeto exterior, como uma construção ou uma escultura, mas beneficia sua própria alma. Elas estão organizadas em Metafísica, Matemática e Física, que inclui a Psicologia ou a ciência da alma. As principais obras teoréticas escritas por Aristóteles são Física, De anima e Metafísica. Filosofia Primeira, que chegou até nós com o nome de Metafísica, também chamada muitas vezes por Aristóteles de Teologia .com a ressalva de que Teologia aqui não tem o sentido que é dado a ela contemporaneamente), é a expressão que designava a mais elevada das ciências teoréticas, diferenciando-se da chamada Filosofia Segunda, ou Física. Aristóteles abre o Livro I da Metafísica com a célebre passagem: Todos os homens têm, por natureza, desejo de conhecer: uma prova disso é o prazer das sensações, pois, fora até da sua utilidade, elas nos agradam por si mesmas e, mais que todas as outras, as visuais [...l. (ARISTÓTELES, 1969). A busca pelo conhecimento encontra-se na própria natureza humana, desde o nascimento, mesmo que se manifeste em seu grau mais rudimentar. Prova disso é o conhecimento por meio das sensações ou dos sentidos, entre os quais se destaca o da visão. É o chamado conhecimento por empiria ou saber empírico, que não pode ser ensinado porque é adquirido imediata e concretamente quando percebemos as coisas sensíveis. Também há um saber que vem da técnica. O técnico é aquele que tem o conhecimento dos meios para chegar a um fim. Como ensina o historiador da Filosofia, Julián Marias, apesar de o saber técnico ser superior ao saber empírico, ambos são necessários em nossa vida: 1..l Portanto, a tékhne [técnica) é superior à empeiria; mas esta também é necessária, por exemplo, para curar, porque o médico não tem de curar o homem, e sim Sócrates, e o homem apenas de modo mediato. (MARIAS, J. 2015). O conhecimento metafísico é menos necessário em nossa vida cotidiana, mas nenhum outro lhe é superior. A Metafísica é o conhecimento pelas causas e, como vimos, o conhecimento é científico quando ele nos dá os princípios e as causas das coisas. Para Aristóteles, tudo o que existe é efeito de uma causa e ela é a responsável por esse algo ser necessariamente de um jeito e não de outro. De acordo com Aristóteles, conhecer algo é conhecer pela causa. Dessa forma, a explicação de algo ou o seu conhecimento deve sempre dizer por que algo é necessariamente de determinado modo. Sem esse tipo de explicaçào, não pode haver propriamente conhecimento. Daí se constata o caráter rigoroso da explicação científica: ela não apenas sabe que "algo é isto", mas é capaz de explicar por que algo é necessariamente isto. A definição aristotélica de Metafísica Chauí indaga em seu livro Iniciação à Filosofia a questão central do que Aristóteles entende por sua alcunha ou Metafísica. Chauí explana: Embora Aristóteles admita que para cada tipo de Ser e suas essências existe uma ciência teorética própria (física, biologia, psicologia, matemática, astronomia), ele também defende a necessidade de uma ciência geral, mais ampla, mais universal, anterior a todas essas, cujo objeto não seja esse ou aquele tipo de Ser, essa ou aquela modalidade de essência, mas o Ser em geral, a essência em geral. Essa ciência, para Aristóteles, é a Filosofia Primeira ou metafísica, que investiga o que é a essência e aquilo que faz com que haja essências particulares diferenciadas, que estuda o Ser enquanto Ser. (CHAUÍ, 2010). O objeto investigado pela Metafísica também é muito diferente do objeto conhecido pelo saber empírico ou pelo saber técnico. O saber empírico nos faz conhecer as coisas particulares, como Sócrates, o Oráculo de Delfos e a caneta esferográfica; o saber técnico nos faz conhecer as diversas técnicas, como os procedimentos médicos e a engenharia de construir templos. O que o saber metafísico, então, nos permite conhecer? Segundo Aristóteles, a Metafisica estuda o ser enquanto ser. Não considera o ser de modo particular como fazem as demais ciências — por exemplo, ao investigar Sócrates, os procedimentos médicos etc. mas busca o que é universal, o que envolve tanto Sócrates quanto os procedimentos médicos, tanto o Oráculo de Delfos quanto a arte de construir templos. Segundo Aristóteles, a Metafísica trata das causas primeiras, que são as quatro seguintes: • a causa material, que é a matéria de que é feita alguma coisa. Em uma escultura, por exemplo, a causa material pode ser o bronze ou o mármore; • a causa formal, que é a forma ou a essência das coisas. No exemplo da escultura, é a forma ou a aparência que possibilita que a reconheçamos como uma escultura (e não como um poste, por exemplo); • a causa eficiente, que é o agente que produz a coisa. Uma escultura é produzida por um artista; • a causa final, que é a razão ou a finalidade das coisas, A finalidade da escultura é o prazer estético. A Metafísica também se ocupa da substância. A substância responde pelos significados do ser. Mas o que é a substância? Trata-se de uma questão complexa no pensamento aristotélico. Aristóteles recusa-se a entender a substância como sendo a forma platónica. Ela nào é o suprassensível. Seria entao a substância o sensível individual? As coisas são matéria e forma; amatéria é o substrato da coisa (por exemplo, o mármore é a matéria da estátua). Sem sua forma, ela é apenas potencialidade, mas sem a matéria qualquer realidade sensível se desvaneceria. Assim, a matéria pode ser dita substância em sentido impróprio. A forma, por seu turno, é aquilo que determina a matéria, é a sua essência; ela é substancia em sentido próprio. Mas a matéria é também matéria enformada. As coisas sensíveis são, a um só tempo, matéria e forma. E esse composto também pode ser legitimamente chamado de substância. Aristóteles apresenta três gêneros de substâncias: • as substâncias sensíveis, que nascem, morrem e, por isso, passam por todo tipo de mudança; • as substancias sensíveis e incorruptíveis, que não passam por nenhum tipo de mudança, como os planetas e as estrelas; • a substância suprassensível, que é superior às outras duas: o Primeiro Motor Imóvel, o deus aristotélico, causa de todo movimento, mas sendo ele mesmo imóvel. (Ele precisa ser imóvel porque se ele também se movesse precisaria haver uma causa para esse movimento, o que levaria a uma espécie de regressão ao infinito). Assim como o demiurgo de Platão, o deus aristotélico nao é um deus criador. Não criou o Universo nem gerou o movimento dos corpos. Ao contrário, o deus de Aristóteles exerce uma atraçáo sobre o Universo, motivando sua existência, e sobre os corpos, gerando seu movimento, ou seja, atrai tudo para si como objeto de amor. Não é, portanto, causa eficiente do mundo e do movimento do mundo, mas causa final. O deus aristotélico nao cria o mundo do nada por um ato de vontade. Ele nao é causa eficiente do mundo, como indicamos ser o artista causa eficiente de sua escultura. A METAFÍSICA CLÁSSICA OU MODERNA Grandes revoluções no pensamento do século XVII deram estopim à uma grande crise na esfera metafísica. De modo resumido a Metafísica sofreu uma grande transformação e uma nova elaboração, sem a fundamentação embasada no pensamento platônico, aristotélico ou neoplatônico. Sendo assim a nova metafísica é caracterizada por: [...] afirmação da incompatibilidade entre fé e razão, acarretando a separação de ambas, de sorte que a religião e a filosofia possam seguir caminhos próprios, mesmo que a segunda não esteja publicamente autorizada a expor ideias que contradigam as verdades ou dogmas da fé [...] [...] redefinição do conceito de Ser ou substância. Os modernos conservam a definição tradicional da substância como o Ser que existe em si e por si mesmo, que subsiste em si e por si mesmo. Porém, em lugar de considerar que a substância se define por gênero e espécie, havendo tantos tipos de substâncias quantos gêneros e espécies houver, passa-se a definir a substância levando em consideração seus predicados essenciais ou seus atributos essenciais, isto é, aquelas propriedades ou atributos sem os quais uma substância não é o que ela é. Após o supracitado, os cartesianos e Descartes dirão que há somente três substâncias essenciais, divergindo totalmente do pensamento aristotélico, as substâncias cartesianas são a Alma, a matéria dos corpos e Deus. Para empiristas, só se é possível conhecer a substância corpórea, logo não se é possível elaborar uma metafísica, e sim uma geometria ou uma física que se ocupa do estudo dessas matérias corpóreas. Baruch Spinoza (1632-1677) explanou a primordialidade de se elaborar uma definição genérica e universal, comumente aceita por toda a comunidade de filósofos de Aristóteles a modernidade. Podemos reduzir o conceito proposto da seguinte maneira: “substância é aquilo que existe em si e por si e não depende de outros para existir”.Logo, diz Espinosa, que há apenas uma substância no Universo que não depende de outra para existir, tal substância é o próprio Deus ou a natureza. Nessa revolução do pensamento filosófico, em peculiar à metafísica, houve também a redefinição do conceito de causa e causalidade. Em Aristóteles haviam quatro tipos de causas, agora, na modernidade, se propunha apenas dois tipos: a eficiente e a final. Chauí explana que a Causa eficiente é aquela na qual uma ação anterior determina como consequência necessária a produção de um efeito, e que seu alcance é universal na natureza. Causa final é aquela que determina, para os seres pensantes, a escolha da realização ou não realização de uma ação, e que só opera na ação de Deus e nas ações humanas. Houve também, nessa revolução, uma quebra do paradigma metafísico, isto é, a metafísica não se dividia mais entre a teologia, a psicologia racional e a cosmologia racional. Agora, a metafísica ganha um novo rumo, um rumo próprio de estudo embasado em suas próprias fundamentações. Logo, a metafísica se apropria de três e apenas três ideias de substância. A substância infinita, ou o próprio Deus. A substância pensante que é a consciência como faculdade de reflexão e de representação da realidade alicerçada na razão. E, por fim, a substância extensa que é a natureza embasada nos princípios e leis regidas pela matemática e a mecânica. DAVID HUME E A CRISE DA METAFÍSICA O filósofo escocês David Hume (1711-1776) nos ajuda a entender em parte o teor das críticas ao pensamento metafísico. "Se tomarmos em nossas mãos um volume qualquer, de teologia ou metafísica escolástica, por exemplo, façamos a pergunta: contém ele qualquer raciocínio abstrato referente a números e quantidades? Não. Contém qualquer raciocínio experimental referente a questões de fato e de existência? Não. Às chamas com ele, então, pois não pode conter senão sofismas e ilusão." (HUME, 2004) Hume critica a teologia e a metafísica escolástica, que foi a metafísica praticada principalmente nas escolas cristãs durante a Idade Média, porque suas concepções não tratam nem de conceitos matemáticos, que são formais e podem levar a conclusões seguras - por exemplo, "2 + 2 = 4" -, nem de afirmações sobre fatos, que podem ser verificados empiricamente, isto é, pelos órgãos dos sentidos —como "a aceleração de um corpo em queda livre é de 9,8 metros por segundo", Assim, Hume aconselha o leitor a jogar as afirmações metafísicas na fogueira, pois são consideradas ilusões, fantasias ou produto de erros de raciocínio e nada têm a acrescentar ao conhecimento humano. A partir de Hume, a metafísica, tal como existira desde os gregos, tornara-se impossível. O CRITICISMO KANTIANO E O FIM DA METAFÍSICA CLÁSSICA Entre a Dissertação de 1770 e a Crítica da razão pura, mais de dez anos se passaram. Apesar de a obra ser volumosa, a redaçào da Crítica da razão pura não chegou a cinco meses de trabalho. De início, Kant pretendia apenas revisar sua dissertação, mas o trabalho acabou por levá-lo a figurar entre os grandes nomes do pensamento filosófico, como Aristóteles e Descartes. Kant fez um diagnóstico bastante negativo da situação da Filosofia de sua época, em particular da Metafísica, que, embora fosse considerada a Filosofia Primeira, patinava em questões impossíveis de serem resolvidas. Ao mesmo tempo, buscou inserir a Filosofia no rumo seguido pelas Ciências Naturais, pela Lógica e pela Matemática, que já tinham encontrado o caminho da Ciência. Em alguns pontos, o projeto kantiano se aproximou de algumas características do projeto cartesiano. Descartes duvidou de todo o saber conhecido, pois queria encontrar um fundamento seguro que sustentasse a edificação da Filosofia e, mais particularmente, da Ciência. Diferentemente de Descartes, contudo, Kant nào duvidou de todo o conhecimento que havia aprendido. Para ele, bastava estabelecer os limites do conhecimento racional, ou seja, o que a razão podia conhecer e o que era impossível de ser conhecido. No fim do século XVIII, de acordo com a proposta de Wolff, a Metafísica era dividida em duas partes: Metafísica geral e Metafísica especial. A primeira tratava da Ontologia (ciência do ser enquanto ser, como definido por Aristóteles), e a segunda se ocupava do ser humano (Psicologia), do mundo (Cosmologia) e de Deus (Teologia racional), Os principais temas in-vestigados eram a imortalidade da alma, a liberdade a finitude ou infinitude do mundo e a existência de Deus. No entanto, de acordo com Kant, as questões metafísicas clássicas extrapolavam as capacidades de conhecimento do ser humano, pois estavam tão afastadas da experiência que a razão só podia pensá-las, sem conhecê-las realmente. As questões postas pela Metafísica clássica extrapolavam o terreno de qualquer experiência possível e, no entanto, a razão, por sua própria natureza, apresenta questões que não pode evitar, mas que também náo pode responder por não terem nenhuma pedra de toque na experiência. Disso decorre a pergunta fundamental sobre a possibilidade de haver um conhecimento metafísico. Publicada em 1781, a Crítica da razão pura foi uma tentativa de responder a essa pergunta. A obra instituiu um "tribunal da razao" para examinar a própria razão, e não para tomar partido no conflito entre racionalistas, como Descartes e Leibniz, que elaboraram sistemas metafísicos, e empiristas, como Locke e Hume, que basearam o conhecimento unicamente nos sentidos. Criticar a razão é determinar as possibilidades, os limites e o alcance do próprio conhecimento. A transformação foi tao grande para a Filosofia que essa obra nao pôde ser ignorada. A situação da Metafísica No texto a seguir, retirado do célebre prefácio à primeira edição de Crítica da razão pura, podemos ver o diagnóstico de Kant sobre a situaçao da Metafísica no período em que vivia. A razão humana, num determinado dominio dos seus conhecimentos, possui o singular destino de se ver atormentada por questões, que náo pode evitar, pois lhe são impostas pela sua natureza, mas às quais também não pode dar resposta por ultrapassarem completamente as suas possibilidades. Não é por culpa sua que cai nessa perplexidade. Parte de princípios, cujo uso é inevitável no decorrer da experiência e, ao mesmo tempo, suficientemente garantido por esta. Ajudada por estes princípios eleva-se cada vez mais alto (como de resto lho consente a natureza) para condições mais remotas. Porém, logo se apercebe de que, desta maneira, a sua tarefa há de ficar sempre inacabada, porque as questões nunca se esgotam; vê-se obrigada, por conseguinte, a refugiar-se em princípios, que ultrapassam todo o uso possível da experiência e, não obstante, estão ao abrigo de qualquer suspeita, pois o senso comum está de acordo com eles. Assim, a razão humana cai em obscuridades e contradições, que a autorizam a concluir dever ter-se apoiado em erros, ocultos algures, sem contudo os poder descobrir. Na verdade, os princípios de que se serve, uma vez que ultrapassam os limites de toda experiência, já não reconhecem nesta qualquer pedra de toque. O teatro destas disputas infindáveis chama-se Metafísica. (KANT, I. 2010). O fim da Metafísica clássica Podemos agora retomar o questionamento sobre os motivos pelos quais a física newtoniana é possível como ciência e entender também por que a metafísica racionalista tradicional não é conhecimento científico, segundo a teoria kantiana. A física trata de regras e leis dos fenômenos, das representações conformadas pela sensibilidade e pelo entendimento, ou seja, refere-se a objetos do conhecimento humano. A metafísica clássica, por sua vez, trata de coisas que não fazem parte da experiência sensível. Deus e a alma, por exemplo, não são apreendidos pela sensibilidade, não são impressões conformadas ou fenômenos e, portanto, não podem ser conhecidos pelo ser humano. Ideias ou conceitos dessa espécie podem ser pensados pela razão, mas é impossível conhecê-los, pois estão além da capacidade humana de conhecimento. Por esse motivo, a física é possível como ciência e a metafísica tradicional não. Com base nisso, Kant provocou uma revolução na teoria do conhecimento, que ele mesmo designou como "virada copernicana na filosofia", Se Copémico mudou a forma de entender o mundo ao defender a ideia de que o Sol está no centro do sistema solar, Kant mudou a forma de o ser humano compreender o mundo e a si próprio ao estabelecer a ideia de que o ser humano, e não os objetos externos, é o centro do conhecimento, O ser humano não é um agente passivo, que só recebe informações, mas atua decisivamente na constituição do conhecimento e do que se compreende como realidade. Os elementos a priori que determinam essa constituição do conhecimento são o escopo da filosofia kantiana, também conhecida como transcendental. Logo, o pensamento kantiano alterou profundamente a Metafísica. A existência de Deus ou a imortalidade da alma, por exemplo, passaram a ser questões que já não podiam ser respondidas. Se percebemos apenas as coisas que ocorrem em um tempo específico (antes, agora, depois), como seremos capazes de perceber algo que está na eternidade (Deus)? Como afirmar que a alma é imortal se não conseguimos percebê-la com os nossos sentidos? O conhecimento a partir de Kant, então, é sempre conhecimento de objetos, isto é, de conteúdos elaborados e modificados pelo sujeito que conhece. O conhecimento passou a se ocupar, assim, de fenómenos, e não mais da própria coisa, como pretendiam os metafísicos clássicos. Com a crítica kantiana, tornou-se impossível conhecer efetivamente os temas da Metafísica clássica e emitir qualquer opinião segura sobre assuntos que não podem ser assentados na experiência. Podemos pensar sobre Deus, podemos querer que nossa a seja imortal, mas jamais teremos a possibilidade de verificar esses assuntos em nossa experiência ou mesmo ter sobre eles verdadeiro conhecimento. Nada nos impede de acreditar em Deus ou na imortalidade da alma, porém não podemos exigir que a Filosofia se pronuncie a respeito desses temas. Na verdade, a crítica kantiana não diz que Deus existe nem que não existe, mas apenas que não é capaz de saben Deus, não obstante sua importância para a Filosofia do próprio Kant, é uma questão de fé. AUGUSTE COMTE E A METAFÍSICA NO SÉCULO XIX O filósofo francês Auguste Comte (1798-1857), considerado o pai do positivismo, fundamenta sua filosofia também em oposição às concepções metafísicas. Em seu entendimento, a busca por princípios gerais ou essências, entidades que estão além do que podemos observar ou perceber pelos órgãos dos sentidos, é algo infrutífero. Comte explana: "[...] não é supérfluo assinalar agora, de modo direto, a preponderância contínua da observação sobre a imaginação, como o principal caráter lógico da sã filosofia moderna, dirigindo nossas pesquisas, não para causas essenciais, mas para leis efetivas, dos diversos fenômenos naturais. Sem ser doravante imediatamente contestado, permanece este princípio fundamental muitas vezes desconhecido nos trabalhos especiais. Embora as diferentes ordens de especulações concedam, sem dúvida, à imaginação uma alta participação, isto é, constan temente empregada para criar ou aperfeiçoar os meios da vinculação entre os fatos constatados, mas o ponto de partida e a sua direçào não lhe poderiam pertencer em nenhum caso. Ainda quando procedemos verdadeiramente a priori, é claro que as considerações gerais que nos guiam foram inicialmente fundadas, quer na ciência correspondente, quer em outra, na simples observação, única fonte de sua realidade e também de sua fecundidade. Ver para prever: tal é o caráter permanente da verdadeira ciência. Tudo prever sem ter nada visto constitui somente uma absurda utopia metafísica, ainda muito seguida." (COMTE, 1978). A observação é considerada a única fonte de realidade. Abandonando a observação e apoiando-se única e exclusivamente na imaginação, como seria característico das especulaçôes metafísicas, o pensamento pode levar não para o conhecimento das coisas, mas para a fantasia. A ciência, então, deve ter como base as coisas que podem ser observadas. Todas as especulações e as teorias científicas, em última instância, devem ter como fundamento a realidade observável. LUDWIG WITTGENSTEIN E A METAFÍSICA CONTEMPORÂNEA O filósofo austríaco Ludwig Wittgenstein (1889-1951) também se opõe às afirmações metafísicas por considerar que estas tratam de problemas (ou pseudoproblemas) que não podem ser formulados claramente e, portanto, não podem ser respondidos. “O método correto da filosofia seria o seguinte: só dizer o que pode ser dito, ou seja, as proposições das ciências naturais [...] e depois, quando alguém quisesse dizer algo metafísico, mostrar-lhe que nas suas proposições existem sinais aos quais não foi dada uma denotação.” (WITTGENSTEIN, 1995). Quer dizer, as sentenças ou os termos metafísicos se referem a coisas, objetos ou seres enigmáticos ou misteriosos, que não podem ser tratados com clareza, isto é, não se sabe exatamente o que são, o que impossibilitaria a formulação de problemas e soluções reais. "Deus", "alma" ou "vida após a morte", por exemplo, são termos que se referem especificamente a quê? Os "problemas" metafísicos estariam no campo do mistério e não no da investigação racional. Por isso, como afirma Wittgenstein, "acerca daquilo de que se não pode falar, tem que se ficar em silêncio". CONCLUSÃO (ÕES) Ora, a Metafísica na Grécia antiga foi imprescindível para o entendimento dos mais variados dilemas do mundo antigo, sejam perguntas simples, ou complexas. Iniciada como um sistema filosófico platônico, passando a ser filosofia primeira em Aristóteles, a metafísica incorporou elementos teológicos, psicológicos e cosmológicos durante o mundo antigo, e ainda mais durante o período medieval e a hegemônica Igreja Católica. No mundo moderno e embasado pelo “Luz da razão”, a metafísica começa a ser refutada pelos pensadores da época em questão, dentre eles o célebre cético Hume, e o emblemático Kant, além de Descartes e seus cartesianistas. Kant e seu criticismo incorporaram à Metafísica uma nova interpretação, esta que fez a mesma entrar em declínio e sair da centralidade do mundo filosófico. Kant, graças às indagações e refutações de Hume, pôde acordar do célebre “sono dogmático”. O que fez Kant se importa com a Metafísica. Kant trouxe à luz o fim da Metafísica clássica e deu legado ao início da metafísica contemporânea alcunhada de Ontologia. No decorrer da história da filosofia, a Metafísica teve intensa ascenção e vertiginosos declives, sendo seu maior declive o crítico Kant. Por fim, muitos filósofos se apropriaram de ser heterodoxos e enfrentaram de frente o mundo metafísica e a base do catolicismo sobre a realidade. Posta realidade que hoje, com o legado de Kant, da fenomenologia de Husserl, etc., originaram a nova escola filosófica chamada de Ontologia. -/- REFERÊNCIAS BIBLIOGRÁFICAS ABBAGNANO, N. Dicionário de filosofia. 5ª. ed. Trad. Ivone Castillo Benedetti. São Paulo: Martins Fontes, 2007. ARANHA, M. L. de A.; MARTINS, M. H. P. Filosofando: introdução à filosofia. 5ª. ed. São Paulo: Moderna, 2013. ARISTÓTELES. Metafísica. Trad. L. Vallandro. Porto Alegre: Globo, 1969. BELO, R. dos S. Filosofia. Vol. Único. 2ª. ed. São Paulo: FTD, 2016. (Coleção Filosofia). BENOIT, L. O. Augusto Comte: fundador da física social. 1ª. ed. São Paulo: Moderna, 2002. CHAUÍ, M. Iniciação à Filosofia. 1ª. ed. São Paulo: Ática, 2010. COMTE, A. Curso de filosofia positiva. In: FERNANDES, F.; FILHO, E. de M. Comte: sociologia. São Paulo: Ática, 1978. (Coleção Grandes Cientistas Sociais). COTRIM, G.; FERNANDES, M. Fundamentos de filosofia. 2ª. ed. São Paulo: Saraiva, 2013. FERREIRO, H. Siete ensayos sobre la muerte de la Metafísica: Una introducción al idealismo absoluto a partir de la ontología. 1ª. ed. Porto Alegre: Editora Fi, 2016. FIGUEIREDO, V. de. (Org.); et. al. Filosofia: Temas e Percursos. Vol. Único. 2ª. ed. São Paulo: Berlendis & Vertecchia, 2016. FILHO, J. S. Filosofia e filosofias: existência e sentidos. 1ª. ed. Belo Horizonte: Autêntica, 2016. HUME, D. Investigações sobre o entendimento humano e sobre os principios da moral. São Paulo: Unesp, 2004. ________. Tratado da natureza humana: uma tentativa de introduzir o método experimental de raciocínio nos assuntos morais. São Paulo: Unesp, 2001. JAPIASSÚ, H.; MARCONDES, D. Dicionário básico de filosofia. 3ª. ed. Rio de Janeiro: Zahar, 2001. KANT, I. Crítica da razão prática. Lisboa: Edições 70, 1989. [Textos filosóficos]. ______. Crítica da razão pura. São Paulo: Abril Cultural, 1979. (Col. Os Pensadores). MARIAS, J. História da filosofia. 2ª. ed. Trad. Claudia Berliner. São Paulo: Martins Fontes, 2015. MELANI, R. Diálogo: primeiros estudos em filosofia. Vol. Único. 2ª. ed. São Paulo: Moderna, 2016. OTERO, J. G. Compendio de filosofía: para uso de los jóvenes estudiosos. 1ª. ed. Bogotá: Imprensa de San Bernardo, 1919. SILVA, F. L. e. Descartes, a metafísica da modernidade. 1ª. ed. São Paulo: Moderna, 2006. WITTGENSTEIN, L. Tratado lógico-filosófico. 2ª. ed. Lisboa: Fundação Calouste Gulbenkian, 1995. -/- Emanuel Isaque Cordeiro da Silva – estudante, pesquisador, professor, agropecuarista, altruísta e defensor dos direitos humanos e dos animais. -/- Alana Thaís Mayza da Silva – estudante, potterhead, pesquisadora, projetista, musicista, filantropa, defensora dos direitos humanos e dos animais, LGBT. (shrink)

In Galileo's Error, Philip Goff sets out a manifesto for a post-Galilean science of consciousness. Article four of the manifesto reads: 'Anti-Dualism: Consciousness is not separate from the physical world; rather consciousness is located in the intrinsic nature of the physical world.' I argue that there is an important sense of ‘dualism’ in which Goff’s arguments are not only compatible with but entail dualism, and not only dualism but substance dualism. Substance dualism, in the sense I have in (...) mind, is the view that (i) there are two sides to reality, a fundamentally mental side and a fundamentally nonmental side; and (ii) the fundamentally mental side consists of mental substances resembling Cartesian souls. I do not suggest that this is bad news for post-Galilean science of consciousness. My aim is rather to push for a certain view about what a theory of consciousness motivated by Goff’s arguments must look like. (shrink)

This paper centers on the implicit metaphysics beyond the Theory of Relativity and the Principle of Indeterminacy – two revolutionary theories that have changed 20th Century Physics – using the perspective of Husserlian Transcedental Phenomenology. Albert Einstein (1879-1955) and Werner Heisenberg (1901-1976) abolished the theoretical framework of Classical (Galilean- Newtonian) physics that has been complemented, strengthened by Cartesian metaphysics. Rene Descartes (1596- 1850) introduced a separation between subject and object (as two different and self- enclosed substances) while Galileo and Newton (...) did the “mathematization” of the world. Newtonian physics, however, had an inexplicable postulate of absolute space and absolute time – a kind of geometrical framework, independent of all matter, for the explication of locality and acceleration. Thus, Cartesian modern metaphysics and Galilean- Newtonian physics go hand in hand, resulting to socio- ethical problems, materialism and environmental destruction. Einstein got rid of the Newtonian absolutes and was able to provide a new foundation for our notions of space and time: the four (4) dimensional space- time; simultaneity and the constancy of velocity of light, and the relativity of all systems of reference. Heisenberg, following the theory of quanta of Max Planck, told us of our inability to know sub- atomic phenomena and thus, blurring the line between the Cartesian separation of object and subject, hence, initiating the crisis of the foundations of Classical Physics. But the real crisis, according to Edmund Husserl (1859-1930) is that Modern (Classical) Science had “idealized” the world, severing nature from what he calls the Lebenswelt (life- world), the world that is simply there even before it has been reduced to mere mathematical- logical equations. Husserl thus, aims to establish a new science that returns to the “pre- scientific” and “non- mathematized” world of rich and complex phenomena: phenomena as they “appear to human consciousness”. To overcome the Cartesian equation of subject vs. object (man versus environment), Husserl brackets the external reality of Newtonian Science (epoché = to put in brackets, to suspend judgment) and emphasizes (1) the meaning of “world” different from the “world” of Classical Physics, (2) the intentionality of consciousness (L. in + tendere = to tend towards, to be essentially related to or connected to) which means that even before any scientific- logical description of the external reality, there is always a relation already between consciousness and an external reality. The world is the equiprimordial existence of consciousness and of external reality. My paper aims to look at this new science of the pre- idealized phenomena started by Husserl (a science of phenomena as they appear to conscious, human, lived experience, hence he calls it phenomenology), centering on the life- world and the intentionality of consciousness, as providing a new way of looking at ourselves and the world, in short, as providing a new metaphysics (as an antidote to Cartesian metaphysics) that grounds the revolutionary findings of Einstein and Heisenberg. The environmental destruction, technocracy, socio- ethical problems in the modern world are all rooted in this Galilean- Newtonian- Cartesian interpretation of the relationship between humans and the world after the crumbling of European Medieval Ages. Friedrich Nietzsche (1844-1900) comments that the modern world is going toward a nihilism (L. nihil = nothingness) at the turn of the century. Now, after two World Wars and the dropping of Atomic bomb, the capitalism and imperialism on the one hand, and on the other hand the poverty, hunger of the non- industrialized countries alongside destruction of nature (i.e., global warming), Nietzsche might be correct: unless humanity changes the way it looks at humanity and the kosmos. The works of Einstein, Heisenberg and Husserl seem to be pointing the way for us humans to escape nihilism by a “great existential transformation.” What these thinkers of post- modernity (after Cartesian/ Newtonian/ Galilean modernity) point to are: a) a new therapeutic way of looking at ourselves and our world (metaphysics) and b) a new and corrective notion of “rationality” (different from the objectivist, mathematico- logical way of thinking). This paper is divided into four parts: 1) A summary of Classical Physics and a short history of Quantum Theory 2) Einstein’s Special and General Relativity and Heisenberg’s Indeterminacy Principle 3) Husserl’s discussion of the Crisis of Europe, the life- world and intentionality of consciousness 4) A Metaphysics of Relativity and Indeterminacy and a Corrective notion of Rationality in Husserl’s Phenomenology . (shrink)

This volume celebrates the various facets of Alan Turing (1912–1954), the British mathematician and computing pioneer, widely considered as the father of computer science. It is aimed at the general reader, with additional notes and references for those who wish to explore the life and work of Turing more deeply. -/- The book is divided into eight parts, covering different aspects of Turing’s life and work. -/- Part I presents various biographical aspects of Turing, some from a personal point (...) of view. -/- Part II presents Turing’s universal machine (now known as a Turing machine), which provides a theoretical framework for reasoning about computation. His 1936 paper on this subject is widely seen as providing the starting point for the field of theoretical computer science. -/- Part III presents Turing’s working on codebreaking during World War II. While the War was a disastrous interlude for many, for Turing it provided a nationally important outlet for his creative genius. It is not an overstatement to say that without Turing, the War would probably have lasted longer, and may even have been lost by the Allies. The sensitive nature of Turning’s wartime work meant that much of this has been revealed only relatively recently. -/- Part IV presents Turing’s post-War work on computing, both at the National Physical Laboratory and at the University of Manchester. He made contributions to both hardware design, through the ACE computer at the NPL, and software, especially at Manchester. Part V covers Turing’s contribution to machine intelligence (now known as Artificial Intelligence or AI). Although Turing did not coin the term, he can be considered a founder of this field which is still active today, authoring a seminal paper in 1950. -/- Part VI covers morphogenesis, Turing’s last major scientific contribution, on the generation of seemingly random patterns in biology and on the mathematics behind such patterns. Interest in this area has increased rapidly in recent times in the field of bioinformatics, with Turing’s 1952 paper on this subject being frequently cited. -/- Part VII presents some of Turing’s mathematical influences and achievements. Turing was remarkably free of external influences, with few co-authors – Max Newman was an exception and acted as a mathematical mentor in both Cambridge and Manchester. -/- Part VIII considers Turing in a wider context, including his influence and legacy to science and in the public consciousness. -/- Reflecting Turing’s wide influence, the book includes contributions by authors from a wide variety of backgrounds. Contemporaries provide reminiscences, while there are perspectives by philosophers, mathematicians, computer scientists, historians of science, and museum curators. Some of the contributors gave presentations at Turing Centenary meetings in 2012 in Bletchley Park, King’s College Cambridge, and Oxford University, and several of the chapters in this volume are based on those presentations – some through transcription of the original talks, especially for Turing’s contemporaries, now aged in their 90s. Sadly, some contributors died before the publication of this book, hence its dedication to them. -/- For those interested in personal recollections, Chapters 2, 3, 11, 12, 16, 17, and 36 will be of interest. For philosophical aspects of Turing’s work, see Chapters 6, 7, 26–31, and 41. Mathematical perspectives can be found in Chapters 35 and 37–39. Historical perspectives can be found in Chapters 4, 8, 9, 10, 13–15, 18, 19, 21–25, 34, and 40. With respect to Turing’s body of work, the treatment in Parts II–VI is broadly chronological. We have attempted to be comprehensive with respect to all the important aspects of Turing’s achievements, and the book can be read cover to cover, or the chapters can be tackled individually if desired. There are cross-references between chapters where appropriate, and some chapters will inevitably overlap. -/- We hope that you enjoy this volume as part of your library and that you will dip into it whenever you wish to enter the multifaceted world of Alan Turing. (shrink)

Peirce and Dewey were generally more concerned with the process of scientific activity than purely mathematical work. However, their accounts of knowledge production afford some insights into the epistemology of mathematical postulates, especially definition and axioms. Their rejection of rationalist metaphysics and their emphasis on continuity in inquiry provides the pretext for the pragmatic a priori – hypothetical and operational assumptions whose justification relies on their fruitfulness in the long run. This paper focuses on the application of this (...) idea to the epistemology of definitions and an account of progress in mathematics, although it has broader implications for the study of conceptual change and the function and basis of presuppositions in the sciences. (shrink)

The sudden shift in the education system during the pandemic and its subsequent evolution during the post-pandemic era have been pivotal in fostering significant educational development and growth. However, this paradigm shift has not been without challenges. This paper aims to investigate the challenges faced by 68 mathematics teachers in four public elementary schools in the Philippines. The respondents were purposively selected to answer the study’s instrument. Using a descriptive survey research methodology, this study explored the five domains in (...) teaching mathematics, namely planning lessons, executing lessons, assessment, mathematical knowledge, and using technology tools. This paper revealed that novice, experienced, and highly experienced teachers alike had moderate difficulty enriching mathematical knowledge and utilizing technology in teaching the subject matter. Moreover, teachers with more than five years’ experience had moderate difficulty in the five domains of teaching mathematics in the post-pandemic era. This finding emphasizes the widespread impact of the education shift for teachers of all levels of experience. The paper further emphasizes the importance of supporting experienced teachers during this transformative phase of formulating innovative learning activities and adapting to reimagined pedagogical concepts. Consequently, this study underscores the need for additional training interventions to enhance their mathematical knowledge and use of technology as an integral teaching tool. (shrink)

The triumph of scientific materialism in the Seventeenth Century not only bifurcated nature into matter and mind and primary and secondary qualities, as Alfred North Whitehead pointed out in Science and the Modern World. It divided science and the humanities. The core of science is the effort to comprehend the cosmos through mathematics. The core of the humanities is the effort to comprehend history and human nature through narratives. The life sciences can be seen as the zone (...) in which the conflict between these two very different ways of comprehending the world collide. Evolutionary theory as defended by Schelling developed out of natural history, but efforts have been made to formulate neo-Darwinism through mathematical models. However, it is impossible to eliminate stories from biology. As Stuart Kauffman argued, mathematical models attempt to pre-state all possibilities, but in evolution there can be adjacent possibles that can be embraced by organisms but cannot be pre-stated. To account for such actions it is necessary to tell stories. Mathematics provides analytic precision allowing long chains of deductions, but tends to deny temporal becoming and cannot do justice to the openness of the future, while narratives focus on processes and events, but lack exactitude that would provide precise deductions and predictions. In advancing mathematics adequate to life, Robert Rosen argued that living beings as anticipatory systems must have models of themselves, and strove to develop a form of mathematics able to model life itself. It has been convincingly argued that narratives are central to human self-creation and they are lived out before being explicitly told. Their models of themselves are first and foremost, stories or narratives. If this is the case, might not living beings as biological entities be characterized by proto-stories or narratives in their models of themselves? Biosemiotics, largely inspired by C.S. Peirce, provides a bridge between mathematical and narrative comprehension, conceiving them as different forms of semiosis. The study of life through biosemiotics could reveal how mathematics and narratives can be understood in relation to each other. This could have implications for how we understand science and the humanities and their relationship to each other. In this paper I will examine work in theoretical biology that might advance these efforts. (shrink)

Attempts to ‘naturalize’ phenomenology challenge both traditional phenomenology and traditional approaches to cognitive science. They challenge Edmund Husserl’s rejection of naturalism and his attempt to establish phenomenology as a foundational transcendental discipline, and they challenge efforts to explain cognition through mainstream science. While appearing to be a retreat from the bold claims made for phenomenology, it is really its triumph. Naturalized phenomenology is spearheading a successful challenge to the heritage of Cartesian dualism. This converges with the reaction against (...) Cartesian thought within science itself. Descartes divided the universe between res cogitans, thinking substances, and res extensa, the mechanical world. The latter won with Newton and we have, in most of objective science since, literally lost our mind, hence our humanity. Despite Darwin, biologists remain children of Newton, and dream of a grand theory that is epistemologically complete and would allow lawful entailment of the evolution of the biosphere. This dream is no longer tenable. We now have to recognize that science and scientists are within and part of the world we are striving to comprehend, as proponents of endophysics have argued, and that physics, biology and mathematics have to be reconceived accordingly. Interpreting quantum mechanics from this perspective is shown to both illuminate conscious experience and reveal new paths for its further development. In biology we must now justify the use of the word “function”. As we shall see, we cannot prestate the ever new biological functions that arise and constitute the very phase space of evolution. Hence, we cannot mathematize the detailed becoming of the biosphere, nor write differential equations for functional variables we do not know ahead of time, nor integrate those equations, so no laws “entail” evolution. The dream of a grand theory fails. In place of entailing laws, a post-entailing law explanatory framework is proposed in which Actuals arise in evolution that constitute new boundary conditions that are enabling constraints that create new, typically unprestatable, Adjacent Possible opportunities for further evolution, in which new Actuals arise, in a persistent becoming. Evolution flows into a typically unprestatable succession of Adjacent Possibles. Given the concept of function, the concept of functional closure of an organism making a living in its world, becomes central. Implications for patterns in evolution include historical reconstruction, and statistical laws such as the distribution of extinction events, or species per genus, and the use of formal cause, not efficient cause, laws. (shrink)

Powers, capacities and dispositions (in what follows I will use these terms synonymously) have become prominent in recent debates in metaphysics, philosophy of science and other areas of philosophy. In this paper I will analyse in some detail a well-known argument from scientific practice to the existence of powers/capacities/dispositions. According to this argument the practice of extrapolating scientific knowledge from one kind of situation to a different kind of situation requires a specific interpretation of laws of nature, namely as (...) attributing dispositions to systems. My main interest will be to discuss what characteristics these dispositions need to have in order to account for the scientific practice in question. I will furthermore assess whether the introduction of dispositions in the context of the extrapolation argument can be described as a ‘revitalization’ or as a ‘return’ to those notions repudiated by early modern philosophers. More particularly I will argue for the following claims: I. In repudiating scholastic terminology, including substantial forms with their causal powers, post-cartesian philosophers focussed on a concept of causation that was much stronger than 21st century conceptions of causation. For this reason alone, whatever ‘causal’ is supposed to mean in today’s causal powers, embracing causal powers is not a simple return to a pre-cartesian notion. II. The dispositions presupposed in scientific practice need not (and should not) be construed in causal terms (whether strong or weak). III. While some early modern philosophers contrasted the characterisation of the natural world in terms of substantial forms (and their causal powers) on the one hand and a mathematical characterization on the other and suggested that these approaches are incompatible, the dispositions postulated by the extrapolation argument to account for scientific practice are themselves characterized in mathematical terms. More precisely: The behaviour the systems are disposed to display is – at least in physics – often characterized in mathematical terms. IV. The dispositions assumed in the law-statements in scientific practice are determinable rather than determinate properties. (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)

Speaking from our experience as department chairs in fields in which women are traditionally underrepresented, we offer reflections and advice on how one might move beyond the chilly climate and create a warmer environment for women students and faculty members.

There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and (...) compares these options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege’s widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized. (shrink)

A SECOND COMPLETELY REVISED EDITION OF THIS TEXTBOOK ON INTERDISCIPLINARY RESEARCH WAS PUBLISHED WITH AMSTERDAM UNIVERSITY PRESS IN 2022. Check out that version here and a PDF of its ToC and Introduction, as this first edition (AUP 2016) is no longer available. [This book (128 pp.) serves as an introduction and manual to guide students through the interdisciplinary research process. We are becoming increasingly aware that, as a result of technological developments and globalisation, problems are becoming so complex that they (...) can only be solved through cooperation between multiple disciplines. Healthcare, climate change, food security, energy, financial markets and quality of life are just a few examples of issues that require scientists and academics to work in a crossdisciplinary way. As a result of these developments, an interdisciplinary approach is becoming increasingly popular in higher education and must be considered an absolute necessity. Young academics are being called on to step beyond the boundaries of traditional disciplines in order to contribute to addressing fundamental problems and solving challenges facing society. There is a need for people who are not afraid to ask critical questions, who can work together well and can look beyond the boundaries of their own field. This means students need to learn more about how they can integrate and apply knowledge, methods and skills from different fields. Interdisciplinary research projects and practical training courses offer students more than one perspective on the same subject. Comparing and contrasting, connecting, adding and adapting concepts, theories and methodologies from different disciplines ultimately results in new insights and better answers to complex problems.]. (shrink)

A perspective is provided on how to move beyond postmodernism while struggling to do philosophy in the twenty-first century. The ontological structures of time, history, and mathematics are analyzed from the vantagepoint of the Heideggerian theory of nonspatial Fold.

This paper sets mathematics among the sciences, despite not being empirical, because it studies relations of various sorts, like the sciences. Each empirical science studies the relations among objects, which relations determining which science. The mathematical science studies relations as such, regardless of what those relations may be or be among, how relations themselves are related. This places it at the extreme among the sciences with no objects of its own (A Subject with no Object, by (...) J.P. Burgess and G. Rosen). Examples are discussed. The historical development of written mathematics from algorithms on clay tablets to theorems with proofs is said to show that application of algorithms to specific problems and theorems to scientific objects is like allegorical interpretation. Mathematics fits into the modern scientific context because the sciences, beginning with Galileo, have been constructed in imitation of mathematics. Viewing mathematics this way does not solve any ontological problems, but it does show how mathematics avoids them. Epistemological problems, insuperable for object realism, are simplified. For example, we have access to relations among any objects that we can consider objectively, first physical objects and then mathematical objects of greater and greater abstraction. (shrink)

This chapter elaborates and develops the thesis originally put forward by Mary Morgan (2005) that some mathematical models may surprise us, but that none of them can completely confound us, i.e. let us unable to produce an ex post theoretical understanding of the outcome of the model calculations. This chapter intends to object and demonstrate that what is certainly true of classical mathematical models is however not true of pluri-formalized simulations with multiple axiomatic bases. This chapter thus (...) proposes to show that - and why - some of these computational simulations that are now booming in the sciences not only surprise us but also confound us. To do so, it shows too that it is needed to elaborate and articulate with some new precision the concept of weak emergence initially due, for its part, to Mark A. Bedau (1997). (shrink)

Mathematics is often taken to play one of two roles in the empirical sciences: either it represents empirical phenomena or it explains these phenomena by imposing constraints on them. This article identifies a third and distinct role that has not been fully appreciated in the literature on applicability of mathematics and may be pervasive in scientific practice. I call this the “bridging” role of mathematics, according to which mathematics acts as a connecting scheme in our explanatory reasoning about why and (...) how two different descriptions of an empirical phenomenon relate to each other. I discuss two bridging roles appearing in biological and physical explanations. (shrink)

That the state-of-affairs of cognitive science is not good is brought into figural salience in "What happened to cognitive science?" (Núñez et al., 2019). We extend their objective description of 'what's wrong' to a prescription of 'how to correct'. Cognitive science, in its quest to elucidate 'how we know', embraces a long list of subjects, while ignoring Mathematics (Fig. 1a, Núñez et al., 2019). Mathematics is known for making the unknown to be known (cf. solving for unknowns). (...) This acknowledgement naturally raises the question: does mathematical knowing inform knowing in general? Here we show that drawing parallels to mathematical knowing can facilitate the advancement of cognitive science (Lawvere, 1994). (shrink)

The need to optimize student interactions in universities for enhanced academic performance has been a subject of debate and discussion in different academic fora. A number of studies have shown that students, both male and female, can assert themselves academically if they are provided with opportunities for active participation and interaction with their lecturers and peers for both the horizontal and the vertical sharing of knowledge. The purpose of this study, therefore, was to investigate the gender-based interaction practices of (...) class='Hi'>science, mathematics and technology university students, and how these interactive patterns influence their academic performance. Using a quantitative approach located in the post-positivist paradigm, the study employed a structured questionnaire to collect the data from a sample of 1285 students from three universities. The results of the study showed that institutional practices, lecturers, parents, peers, learning content and artifacts, as well as the classroom environment, have a significant influence on the gender-based interaction practices of university students. Furthermore,, the results showed that the levels of interaction have a significant influence on the academic performance of university students, according to gender. As a main recommendation, it was proposed that universities should come up with gender-equity policies that would guide how the universities and their stakeholders could cater for the issues of gender equity. (shrink)

This article aims to contribute to the economic science’s discourse analysis, bringing the concepts of Foucault’s post-structuralism to this debate. We seek to understand the epistemological change in the economy, started in the 80s, without having to resort to a split in social interpretation in abstract cultural spheres (postmodernism) and other material-economic spheres (neoliberalism). The field of rhetoric in economics has sparked an intense debate in the social sciences. The abandonment of Keynesian theses, empirically tested throughout the 20th (...) century, due to the absorption of neoliberal theses was interpreted as an expression of “postmodernity” in economic science. Arida and Paulani’s analysis trace the methodological criticism to the intense mathematization of the economy, to the resumption of premises based on ontologically questionable abstractions without, however, understanding the role of language in structuring the discourse of economic science. It is the search for an adequate ontology that guides the methodological critique of marginalist-neoclassical economic science. Based on Foucault’s concepts, we propose a new ontology for the understanding of the rhetoric field in economics. (shrink)

Logic has pride of place in mathematics and its 20th century offshoot, computer science. Modern symbolic logic was developed, in part, as a way to provide a formal framework for mathematics: Frege, Peano, Whitehead and Russell, as well as Hilbert developed systems of logic to formalize mathematics. These systems were meant to serve either as themselves foundational, or at least as formal analogs of mathematical reasoning amenable to mathematical study, e.g., in Hilbert’s consistency program. Similar efforts continue, (...) but have been expanded by the development of sophisticated methods to study the properties of such systems using proof and model theory. In parallel with this evolution of logical formalisms as tools for articulating mathematical theories (broadly speaking), much progress has been made in the quest for a mechanization of logical inference and the investigation of its theoretical limits, culminating recently in the development of new foundational frameworks for mathematics with sophisticated computer-assisted proof systems. In addition, logical formalisms developed by logicians in mathematical and philosophical contexts have proved immensely useful in describing theories and systems of interest to computer scientists, and to some degree, vice versa. Three examples of the influence of logic in computer science are automated reasoning, computer verification, and type systems for programming languages. (shrink)

The objective of this case study concentrated on examining the learning gap, going through some components of the transformation process, and coming up with some ways for aiding students who were experiencing lost learning. A qualitative research design was utilized by the researchers to understand and solve the cases related to Mathematics learning difficulties. Creswell (2008) asserts that qualitative research can be used to discover and comprehend the significance that certain people or groups assign to social or human issues. The (...) researchers used purposive sampling to determine 15 student participants to understand the cases of learning loss and prescribe better interventions to the students. To solve the problems associated with loss of learning experienced by the learners after remote learning, a case study will be utilized as the strategy of inquiry since it fits the situation that is to focus the said key cases, discuss some transforming process, and develop interventions. After analyzing the interviews, researchers identified three common themes across the three cases which serve to clarify the learners' experiences after the post-remote learning: the learners' learning experiences during remote learning, factors that caused learning loss in Mathematics, and learners' coping strategies. In a deeper analysis, the impact of distance learning on students during the pandemic goes beyond just the difficulties in learning and understanding lessons which resulted in difficulty in learning math during in-person classes. The lack of in-person interaction with classmates and teachers also contributes to a sense of isolation and boredom, negatively affecting students' overall well-being and mental health. (shrink)

Over the years, mathematics and statistics have become increasingly important in the social sciences1 . A look at history quickly confirms this claim. At the beginning of the 20th century most theories in the social sciences were formulated in qualitative terms while quantitative methods did not play a substantial role in their formulation and establishment. Moreover, many practitioners considered mathematical methods to be inappropriate and simply unsuited to foster our understanding of the social domain. Notably, the famous Methodenstreit also (...) concerned the role of mathematics in the social sciences. Here, mathematics was considered to be the method of the natural sciences from which the social sciences had to be separated during the period of maturation of these disciplines. All this changed by the end of the century. By then, mathematical, and especially statistical, methods were standardly used, and their value in the social sciences became relatively uncontested. The use of mathematical and statistical methods is now ubiquitous: Almost all social sciences rely on statistical methods to analyze data and form hypotheses, and almost all of them use (to a greater or lesser extent) a range of mathematical methods to help us understand the social world. Additional indication for the increasing importance of mathematical and statistical methods in the social sciences is the formation of new subdisciplines, and the establishment of specialized journals and societies. Indeed, subdisciplines such as Mathematical Psychology and Mathematical Sociology emerged, and corresponding journals such as The Journal of Mathematical Psychology (since 1964), The Journal of Mathematical Sociology (since 1976), Mathematical Social Sciences (since 1980) as well as the online journals Journal of Artificial Societies and Social Simulation (since 1998) and Mathematical Anthropology and Cultural Theory (since 2000) were established. What is more, societies such as the Society for Mathematical Psychology (since 1976) and the Mathematical Sociology Section of the American Sociological Association (since 1996) were founded. Similar developments can be observed in other countries. The mathematization of economics set in somewhat earlier (Vazquez 1995; Weintraub 2002). However, the use of mathematical methods in economics started booming only in the second half of the last century (Debreu 1991). Contemporary economics is dominated by the mathematical approach, although a certain style of doing economics became more and more under attack in the last decade or so. Recent developments in behavioral economics and experimental economics can also be understood as a reaction against the dominance (and limitations) of an overly mathematical approach to economics. There are similar debates in other social sciences. It is, however, important to stress that problems of one method (such as axiomatization or the use of set theory) can hardly be taken as a sign of bankruptcy of mathematical methods in the social sciences tout court. This chapter surveys mathematical and statistical methods used in the social sciences and discusses some of the philosophical questions they raise. It is divided into two parts. Sections 1 and 2 are devoted to mathematical methods, and Sections 3 to 7 to statistical methods. As several other chapters in this handbook provide detailed accounts of various mathematical methods, our remarks about the latter will be rather short and general. Statistical methods, on the other hand, will be discussed in-depth. (shrink)

The 'Science of Knowing: Mathematics' textbook is the first book to put forward and substantiate the thesis that the mathematical understanding of mathematics, as exemplified in F. William Lawvere's Functorial Semantics, constitutes the science of knowing i.e. cognitive science. -/- This is a textbook, i.e. teaching material.

We give an overview of Lakatos’ life, his philosophy of mathematics and science, as well as of this issue. Firstly, we briefly delineate Lakatos’ key contributions to philosophy: his anti-formalist philosophy of mathematics, and his methodology of scientific research programmes in the philosophy of science. Secondly, we outline the themes and structure of the masterclass Lakatos’ Undone Work – The Practical Turn and the Division of Philosophy of Mathematics and Philosophy of Science, which gave rise to this (...) special issue. Lastly, we provide a summary of the contributions to this issue. (shrink)

This paper deals with Natorp’s version of the Marburg mathematical philosophy of science characterized by the following three features: The core of Natorp’s mathematical philosophy of science is contained in his “knowledge equation” that may be considered as a mathematical model of the “transcendental method” conceived by Natorp as the essence of the Marburg Neo-Kantianism. For Natorp, the object of knowledge was an infinite task. This can be elucidated in two different ways: Carnap, in the (...) Aufbau, contended that this endeavor can be divided into two distinct parts, namely, a finite “constitution” of the object of knowledge and an infinite incompletable empirical description. In contrast, and more in the original spirit of Cohen and Natorp, the physicist and philosopher Margenau in The Nature of Physical Reality (Margenau. 1950) conceived the infinity of this “Aufgabe” as an infinite dialectical process, in which relative “data” and “conceptual constructs” determine each other. This dialectical process eliminates the dichotomy between Anschauung and Begriff that distinguished the Marburg Neo-Kantianism from Kantian orthodoxy, namely, the abandonment of the difference between intuition and concept. Finally, the paper deals with the non-Archimedean geometrical systems that played a central role in Natorp’s defence of Cohen’s “infinitesimal” metaphysics. (shrink)

Marr’s seminal distinction between computational, algorithmic, and implementational levels of analysis has inspired research in cognitive science for more than 30 years. According to a widely-used paradigm, the modelling of cognitive processes should mainly operate on the computational level and be targeted at the idealised competence, rather than the actual performance of cognisers in a specific domain. In this paper, we explore how this paradigm can be adopted and revised to understand mathematical problem solving. The computational-level approach applies (...) methods from computational complexity theory and focuses on optimal strategies for completing cognitive tasks. However, human cognitive capacities in mathematical problem solving are essentially characterised by processes that are computationally sub-optimal, because they initially add to the computational complexity of the solutions. Yet, these solutions can be optimal for human cognisers given the acquisition and enactment of mathematical practices. Here we present diagrams and the spatial manipulation of symbols as two examples of problem solving strategies that can be computationally sub-optimal but humanly optimal. These aspects need to be taken into account when analysing competence in mathematical problem solving. Empirically informed considerations on enculturation can help identify, explore, and model the cognitive processes involved in problem solving tasks. The enculturation account of mathematical problem solving strongly suggests that computational-level analyses need to be complemented by considerations on the algorithmic and implementational levels. The emerging research strategy can help develop algorithms that model what we call enculturated cognitive optimality in an empirically plausible and ecologically valid way. (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)

Ian James has carved a rigorous analysis of four philosophers—Jean-Luc Nancy, François Laruelle, Catherine Malabou and Bernard Stiegler—who not only engage with the limits of thought through variegated, albeit embedded, disciplinary tendencies but have also, arguably, spearheaded a critical reorientation of continental philosophy, slowly opening the doors for transcending the traditional terms of the analytic-continental divide by engaging with a pluralized understanding of the sciences. A parallel plexus of American naturalist philosophy accompanies James’ analysis, as he stakes the claim that (...) these four thinkers engage with pluralist ontologies and the limit-conditions of the real to stoke a proximal entanglement between philosophy and science. However, The Technique of Thought is by no means a synoptic account of Nancy, Laruelle, Malabou and Stiegler, as James surveys discourses in philosophy of mind, quantum gravity, causality and biosemiology to index various recent horizons of thought and their developments. The rigor and deft with which James approaches scientific-realist perspectives produce a rich picture of post-metaphysical thinking. (shrink)

The COVID-19 pandemic has had a significant impact on the education sector globally, with over one billion students being held out of school as a result of quarantine measures. In response, education systems had to quickly shift to online learning to ensure that students could continue their education. The sudden shift to online learning has resulted in educators having to adapt to the use of technology in education rapidly. The COVID-19 pandemic has highlighted the importance of digital literacy for educators (...) and the need for ongoing training in digital skills. The pandemic has served as an example of emerging, more creative methods of teaching methods, and technology use in education will continue to evolve. Hybrid learning will be critical to delivering realistic educational experiences, and teachers must adapt to become facilitators instead of teachers. The pandemic experience will affect educators’ attitudes towards integrating information and communication technology (ICT) in education, resulting in a rise in the proportion of people who have a favorable opinion of integrating ICT to education. Ongoing training in digital abilities is necessary to gain a viable education. This paper highlights the need for educators to embrace digital literacy and adapt to emerging technologies to provide a quality education in the post-vaccine world. (shrink)

Mathematics clearly plays an important role in scientific explanation. Debate continues, however, over the kind of role that mathematics plays. I argue that if pure mathematical explananda and physical explananda are unified under a common explanation within science, then we have good reason to believe that mathematics is explanatory in its own right. The argument motivates the search for a new kind of scientific case study, a case in which pure mathematical facts and physical facts are explanatorily (...) unified. I argue that it is possible for there to be such cases, and provide some toy examples to demonstrate this. I then identify a potential source of scientific case studies as a guide for future work. (shrink)