In the first part of this investigation we highlight two, seemingly irreconcilable, beliefs that suggest an impending crisis in the teaching, research, and practice of—primarily state-supported—mathematics: (a) the belief, with increasing, essentially faith-based, conviction and authority amongst academics that first-order Set Theory can be treated as the lingua franca of mathematics, since its theorems—even if unfalsifiable—can be treated as ‘knowledge’ because they are finite proof sequences which are entailed finitarily by self-evidently Justified True Beliefs; and (b) the slowly (...) emerging, but powerfully rooted in economic imperatives, belief that only those Justified True Beliefs can be treated as ‘knowledge’ which are, further, categorically communicable as Factually Grounded Beliefs—hence as comprehensible pre-formal ‘mathematical truths’—by any intelligence that lays claims to a mathematical lingua franca. We argue that this seeming irreconcilability merely reflects a widening chasm between an increasing under-estimation of the value to society of ‘semantic arithmetical truth’, and an increasing over-estimation of the value to society of ‘syntactic set-theoretical provability’; a chasm which must be narrowed and bridged explicitly. We thus proffer a Complementarity Thesis which seeks to recognize that mathematical ‘provability’ and mathematical ‘truth’ need to be interdependent and complementary, ‘evidence-based’, assignments-by-convention to the formulas of a formal mathematical language; where (a) the goal of mathematical ‘proofs’ may be viewed as seeking to ensure that any mathematical language intended to formally represent our pre-formal conceptual metaphors and their inter-relatedness is unambiguous, and free from contradiction; whilst (b) the goal of mathematical ‘truths’ must be viewed as seeking to ensure that any such representation does, indeed, uniquely identify and adequately represent such metaphors and their inter-relatedness. Our thesis is that, by universally ignoring the need to introduce goal (b) in our curriculum, the teaching of, and research in, mathematics at the post-graduate level cannnot justify its value to society beyond the mere intellectual stimulation of individual minds. In the second part we appeal to some recent—and hitherto unsuspected—unarguably constructive Tarskian interpretations, of the first-order Peano Arithmetic PA, which establish PA as both finitarily consistent, and categorical. Since we also show that the second order subsystem ACA0 of Peano Arithmetic and PA cannot both be assumed provably consistent, we conclude that there can be no mathematical, or meta-mathematical, proof of consistency for Set Theory. Hence the theorems of any Set Theory are not sufficient for distinguishing between (i) what we can believe to be a ‘mathematical truth’; (ii) what we can evidence as a ‘mathematical truth’; and (iii) what we ought not to believe is a ‘mathematical truth’; whilst the theorems of the first-order Peano Arithmetic PA are sufficient for distinguishing between (i), (ii) and (iii). We conclude from this that the holy grail of mathematics ought to be ‘arithmetical truth’, not ‘set-theoretical proof’. (shrink)

Adams and Aizawa (2010b) define cognitivism as the processing of representations with underived content. In this paper, I respond to their use of this stipulative definition of cognition. I look at the plausibility of Adams and Aizawa’s cognitivism, taking into account that they have no criteria for cognitive representation and no naturalistic theory of content determination. This is a glaring hole in their cognitivism—which requires both a theory of representation and underived content to be successful. I also explain why my (...) own position, cognitive integration, is not susceptible to the supposed causal-coupling fallacy. Finally, I look at the more interesting question of whether the distinction between derived and underived content is important for cognition. Given Adams and Aizawa’s concession that there is no difference in content between derived and underived representations (only a difference in how they get their content) I conclude that the distinction is not important and show that there is empirical research which does not respect the distinction. (shrink)

Pritchard calls his epistemological disjunctivism ‘the holy grail of epistemology’. What this metaphor means is that the acceptance of this thesis puts the internalism-externalism debate to an end, thanks to satisfaction of intuitions standing behind both competing views. Simultaneously, Pritchard strongly emphasizes that the endorsement of epistemological disjunctivism does not commit one to metaphysical disjunctivism. In this paper I analyze the formulations and motivations of epistemological disjunctivism presented by Pritchard and McDowell. Then I consider the most common argument (...) for the claim that epistemological disjunctivism can be held without the support of metaphysical disjunctivism. I conclude that the plausibility of epistemological disjunctivism depends on the plausibility of metaphysical disjunctivism. If the latter is false, the former postulates a set of conditions for epistemic justification that are impossible to be fulfilled. (shrink)

Pritchard argues that epistemological disjunctivism seems plainly false at first sight, but if it were right, it would represent the “holy grail of epistemology” (1), a view that allows us “to have our cake and eat it too” (3). This prospect motivates Pritchard to develop and defend an account that prima facie might seem simply false. It is disputable whether ED really seems plainly false at first sight or whether this intuition is based on a particular philosophical tradition. (...) However, in this paper I will not discuss whether ED is actually true. Rather, I will investigate whether, if true, it has the advantages over rival accounts that Pritchard claims. (shrink)

This book explores the idea that all of logic can be reduced to two very simple rules that are sensitive to logical polarity. The authors show that this idea has profound consequences for our understanding of the nature of human inferential capacities, and for some of the key issues in contemporary linguistics.

The development of the science of poverty has largely been driven by the need to define more precisely what poverty is, as well as to provide theoretical and empirical criteria for identifying those who suffer from it. This thesis focuses on a notable response to these and related questions: the conception and measure of poverty by the British sociologist Peter Townsend. Townsend defines poverty as relative deprivation caused by lack of resources. This conception, along with his corresponding cut-off measure, constitutes (...) some of the key components of his theory of poverty. The theory discusses what poverty is, its causes, and how it can be eradicated, and is detailed in Poverty in the United Kingdom: A Survey of Household Resources and Standards of Living (1979), his magnum opus. The primary aim of this thesis is to interpret the theory from Townsend’s adherence to the value-free ideal. He pursued this ideal in response to what he perceived as the pervasive and deleterious influence of moral and political values on measures, theories, and policies on poverty throughout its scientific and political history. I argue that to be free from such influence, Townsend aimed to be as guided by epistemic values as possible, which resulted in a systematic theory of poverty. -/- The thesis is structured as follows: In Chapter 1, I situate Townsend’s conception and measurement of poverty within the context of both the recent history of scientific poverty lines and the most important approaches to measuring the phenomenon. Chapter 2 discusses the systematic nature of the theory, including its role as a middle-range theory that bridges broad-range theories with empirical data. This connection is achieved through several elements: a conception of poverty; hypotheses and anthropological observations related to the elements of the conception; measures to test these hypotheses; an explanation of poverty; and reports on how the research was conducted. Furthermore, Townsend’s theory aligns with Bradburn et al.’s (2017) measurement theory, which asserts that a good measure must include a conception of the relevant phenomenon, a representation, and procedures to gather data. Additionally, all three components must be supported by arguments showing that they fit together properly. -/- In the remaining chapters, I present and discuss the main elements of the theory: his conception of poverty as relative deprivation caused by lack of resources, along with the related hypotheses and observations (Chapter 3); the representations of this conception and the procedures for data collection, processing, as well as ensuring their reliability (Chapter 4); and the explanation of poverty (Chapter 5). I conclude by presenting what Townsend considered the “policy prescriptions” of his theory. Despite its potential shortcomings, I also argue that Townsend’s theory fulfills an important epistemic value: fruitfulness. (shrink)

This paper introduces the "Status Loss Theory of Humor," as detailed in "How Humor Works" and "How Humor Works, Part II" , in a single page. This theory has the potential to fully, clearly, and naturally explain the human humor instinct, and has made predictions that are being confirmed by other studies.

Within this text is The Proof, beyond an ounce of doubt, by way of the language of the Universe: Mathematics, that The Triune God is The One True God and The Holy Bible is His Word.

The Theory of Relativity has been portrayed as a theory that redefined the way we look at the cosmos, enabling us to unlock the reality we live in. Its proponents are constantly reminding us of how Einstein managed to reveal the true nature of the universe with his groundbreaking theory, which has been proved multiple times until now. Yet, philosophy of science teaches us that no theory has any privileged connection with what we call reality per se. The role of (...) science is to formulate models of the cosmos we see and not to try to interpret or reveal reality. This paper tries to show how this holds true even for the famous relativity theory, by showing specific objections to the connection of the theory with the Holy Grail of philosophers. By analyzing various subjects related to the theory, from the twins’ paradox to the GPS satellites, this paper illustrates that relativity is much less connected to reality than what we would like to think. At the end, what Einstein’s theory provides is nothing more than a way to formalize the interactions of the world but without being able to make any claims whatsoever regarding the ‘reality’ of its conclusions. (shrink)

Summary: Throughout the history of science, indeed throughout the history of knowledge, unification has been touted as a central aim of intellectual inquiry. We’ve always wanted to discover not only numerous bare facts about the universe, but to show how such facts are linked and interrelated. Large amounts of time and effort have been spent trying to show diverse arrays of things can be seen as different manifestations of some common underlying entities or properties. Thales is said to have originated (...) philosophy and science with his declaration that everything was, at base, a form of water. Plato’s theory of the forms was thought to be a magnificent accomplishment because it gave a unified solution to the separate problems of the relation between knowledge and belief, the grounding of objective values, and how continuity is possible amid change. Pasteur made numerous medical advancements possible by demonstrating the interconnection between microorganisms and human disease symptoms. Many technological advances were aided by Maxwell’s showing that light is a kind of electromagnetic radiation. The attempt to unify the various known forces is often referred to as “The Holy Grail” of physics. Some philosophers have even suggested that providing explanations is itself just a sort of unifying of our knowledge. But while unification (like simplicity) has often been hailed as a tremendous virtue in science, the meaning of the term is not altogether clear. Scientists often don’t specify what, precisely, they mean by unification. And in cases where what they mean is clear, different thinkers plainly mean different things by the term. What are the various senses of unification, and why has unification been such an important aim in the history of inquiry? (shrink)

The Language of Thought program has a suicidal edge. Jerry Fodor, of all people, has argued that although LOT will likely succeed in explaining modular processes, it will fail to explain the central system, a subsystem in the brain in which information from the different sense modalities is integrated, conscious deliberation occurs, and behavior is planned. A fundamental characteristic of the central system is that it is “informationally unencapsulated” -- its operations can draw from information from any cognitive domain. The (...) domain general nature of the central system is key to human reasoning; our ability to connect apparently unrelated concepts enables the creativity and flexibility of human thought, as does our ability to integrate material across sensory divides. The central system is the holy grail of cognitive science: understanding higher cognitive function is crucial to grasping how humans reach their highest intellectual achievements. But according to Fodor, the founding father of the LOT program and the related Computational Theory of Mind (CTM), the holy grail is out of reach: the central system is likely to be non-computational (Fodor 1983, 2000, 2008). Cognitive scientists working on higher cognitive function should abandon their efforts. Research should be limited to the modules, which for Fodor rest at the sensory periphery (2000).1 Cognitive scientists who work in the symbol processing tradition outside of philosophy would reject this pessimism, but ironically, within philosophy itself, this pessimistic streak has been very influential, most likely because it comes from the most well-known proponent of LOT and CTM. Indeed, pessimism about centrality has become assimilated into the mainstream conception of LOT. (Herein, I refer to a LOT that appeals to pessimism about centrality as the “standard LOT”). I imagine this makes the standard LOT unattractive to those philosophers with a more optimistic approach to what cognitive science can achieve.. (shrink)

A lean hylomorphism stands as a metaphysical holy grail. An embarrassing feature of traditional hylomorphic ontologies is prime matter. Prime matter is both so basic that it cannot be examined (in principle) and its engagement with the other hylomorphic elements is far from clear. One particular problem posed by prime matter is how it is to be understood both as a principle of individuation for material substances and as pure potency. I present Thomas Aquinas’s way of squeezing some (...) intelligibility out of prime matter by modeling it on the idea of logical genus. Such a modeling provides insight into understanding prime matter as substratum, as maximally indeterminate, and as ontologically vague. One of the unusual but exciting things that fall out from this analysis of prime matter is the Entirety Thesis: “For any substance x, if x has prime matter then the prime matter of x is the same* as x,” where ‘same*’ is understood as “indeterminately identical.”. (shrink)

Terms Artificial General Intelligence (AGI) and Human-Level Artificial Intelligence (HLAI) have been used interchangeably to refer to the Holy Grail of Artificial Intelligence (AI) research, creation of a machine capable of achieving goals in a wide range of environments. However, widespread implicit assumption of equivalence between capabilities of AGI and HLAI appears to be unjustified, as humans are not general intelligences. In this paper, we will prove this distinction.

Theorising the interplay of structure and agency is the quintessential focus of sociological endeavour. This paper aims to be part of that continuing endeavour, arguing for a stratified social ontology, where structure and agency are held to be irreducible to each other and causally efficacious, yet necessarily interdependent. It thus aims not to be part of that on-going journey in search of the 'ontological holy grail'. Instead, it offers a way of linking structure and agency which enables the (...) practical education researcher concretely to examine their relative interplay over time. The methodological key to teasing out their relative interplay is held to be analytical dualism. It will be argued that such a methodological device is precluded by Giddens' structuration theory. (shrink)

The quest for a unified “Theory of Everything” that explains the fundamental nature of the universe has long been a holy grail for scientists and philosophers, dating back to the ancient Greeks’ search for Arche. -/- So far, the mainstream of research on A Theory of Everything primarily focuses on the lifeless phenomena and laws of physics while ignores the realm of biology. However, a fundamentally different approach to the ToE has been put forward, presenting a viable alternative (...) to address the challenge of a Theory of Everything. This approach does not seek the ultimate “building block” but rather aims to uncover the intangible rules that fundamentally govern everything in the universe, seeking their universality across the vast spectrum, from the minute subatomic world to the mega mass cosmic world and the magical biological world. -/- To address the challenge, a set of the fundamental interrelationships is introduced and represented by a model, the Fundamental Interrelationships Model. This model cohesively represents these fundamental interrelationships, including serial-parallel relationships, transition of state, critical point, continuation-discontinuation, convergence-divergence, contraction-expansion, singularity-plurality, commonality-difference, similarity, symmetry-asymmetry, periodicity, dynamics-stability, order-disorder, limitation-without limitation, hierarchical structure, and cohesive interconnectedness. These interrelationships, found in nature, can be interpreted as the fundamental laws of physics. -/- Built on the foundation of the Fundamental Interrelationships Model, a collection of the well-established laws of physics and theories are represented and unified, including Newton’s three laws of motion, the four laws of thermodynamics, Einstein’s space-time relationship, Heisenberg’s uncertainty principle, Noether’s theorem, Schrodinger’s wave function collapse, chaos theory and complexity theory. The list continues to grow… Based on this model, a groundbreaking research has, for the first time, put forward a new hypothesis that unifies the Big Bang theory with the evolutionary theory, published in a book in 2022. This research unequivocally demonstrates that the evolution of life also follows the fundamental laws of physics, marking a significant stride toward addressing the longstanding challenge of a comprehensive Theory of Everything. -/- Subsequent studies reveal that the evolution of life (including the evolution of multicellularity), embryonic development, evolution of society (civilisation), and the evolution of the universe all adhere to these fundamental interrelationships. These events are specific expressions of the underlying principles of the Fundamental Interrelationships Model. -/- Ultimately, as a truly all-encompassing theory, the Theory of Everything should not only incorporate non-biological events and the laws of physics but also extend to all facets of life. Thus, unlike most existing candidates, the Fundamental Interrelationships Model offers a comprehensive framework, encompassing both non-biological and living phenomena. Therefore, any hypothesis failing to integrate biology and sociology shouldn’t be considered a comprehensive Theory of Everything -/- This article and the subsequent ones are the abstracts derived from the book, Behind Civilisation/civilization (multiple editions), which represents the culmination of decades of research. The book offers a more comprehensive exploration of how fundamental interrelationships govern all aspect of the universe. -/- . (shrink)

An e-book devoted to 13 critical discussions of Thaddeus Metz's book "Meaning in Life: An Analytic Study", with a lengthy reply from the author. -/- Preface Masahiro Morioka i -/- Précis of Meaning in Life: An Analytic Study Thaddeus Metz ii-vi -/- Source and Bearer: Metz on the Pure Part-Life View of Meaning Hasko von Kriegstein 1-18 -/- Fundamentality and Extradimensional Final Value David Matheson 19-32 -/- Meaningful and More Meaningful: A Modest Measure Peter Baumann 33-49 -/- Is Meaning in (...) Life Comparable?: From the Viewpoint of ‘The Heart of Meaning in Life’ Masahiro Morioka 50-65 -/- Agreement and Sympathy: On Metz’s Meaning in Life Sho Yamaguchi 66-89 -/- Metz’s Quest for the Holy Grail James Tartaglia 90-111 -/- Meaning without Ego Christopher Ketcham 112-133 -/- Death and the Meaning of Life: A Critical Study of Metz’s Meaning in Life Fumitake Yoshizawa 134-149 -/- Metz’ Incoherence Objection: Some Epistemological Considerations Nicholas Waghorn 150-168 -/- Meaning in Consequences Mark Wells 169-179 -/- Defending the Purpose Theory of Meaning in Life Jason Poettcker 180-207 -/- Review of Thaddeus Metz’s Meaning in Life Minao Kukita 208-214 -/- A Psychological Model to Determine Meaning in Life and Meaning of Life Yu Urata 215-227 -/- Assessing Lives, Giving Supernaturalism Its Due, and Capturing Naturalism: Reply to 13 Critics of Meaning in Life Thaddeus Metz 228-278 . (shrink)

In "The Nature and Meaning of Numbers," Dedekind produces an original, quite remarkable proof for the holy grail in the foundations of elementary arithmetic, that there are an infinite number of things. It goes like this. [p, 64 in the Dover edition.] Consider the set S of things which can be objects of my thought. Define the function phi(s), which maps an element s of S to the thought that s can be an object of my thought. Then (...) phi is evidently one-to-one, and the image of phi is contained in S. Indeed, it is properly contained in S, because I myself can be an object of my thoughts and so belong to S, but I myself am not a mere thought. Thus S is infinite. (shrink)

The task of this brief presentation is to “establish a dialogue” with Streeck’s text, attempting to fill the hiatus between the answer and the original question that Habermas’ interpretation intended to pose to those wishing to simply dispose of economic and monetary union, ending up by dismantling the political and cultural integration project that inspired the founding fathers. Streeck complains about the “levity” with which many reviewers accepted “as a slogan” the “killer-argument” [Totschlagargument] of the “nostalgic option” provided by the (...) interpretation of his most recent publication and the excuse for his removal from the assembly of “guardians of the Holy Grail”. Now, while acknowledging that in the thesis of the author of Gekaufte Zeit [4] there is no trace of nationalist seeds, which – as Altiero Spinelli observed – “had caused the catastrophe of the two World Wars” [5], Habermas’ critique seizes the unrealistic thoughtlessness of an idea that ends up evoking the system of “European cooperation” between the now impotent national states. We can do without the “specialisation in Europeanism” that according to Streeck, would be required as a preliminary to discuss European policy in the German public arena, but not criticism that is constructive and not defeatist. (shrink)

Grasping the identity of hybrids, that is beings which cross the binarism of nature and technology (e.g. genetically-modified organisms (GMOs), syn-bio inventions, biomimetic projects), is problematic since it is still guided by self-evident dualistic categories, either as artefacts or as natural entities. To move beyond the limitations of such a one-sided understanding of hybrids, we suggest turning towards the categories of affordances and the juxtaposition of needs and patterns of proper use, as inspired by the Heideggerian version of phenomenology. Drawing (...) upon selected concepts by Heidegger, we argue that hybrids can be conceptualised as a regenerative design and use to serve the planet. We argue that the ideal type of non-exploitative account of hybrids consists of the adaptive approach to the environment, which does not, however, exclude the possibility of designing and constructing new beings. We also point out that hybrids undermine the divide of being destructive/regenerative which marks the boundaries of nature and technology. (shrink)

The transgressive ontological character of hybrids—entities crossing the ontological binarism of naturalness and artificiality, e.g., biomimetic projects—calls for pondering the question of their ethical status, since metaphysical and moral ideas are often inextricably linked. The example of it is the concept of “moral considerability” and related to it the idea of “intrinsic value” understood as a non-instrumentality of a being. Such an approach excludes hybrids from moral considerations due to their instrumental character. In the paper, we revisit the boundaries of (...) moral considerability by reexamining the legitimacy of identifying intrinsic value with a non-instrumental one. We offer the concept of “functional value,” which we define as a simultaneous contribution to the common good of the ecosystem and the possibility to disclose the full variety of aspects of a being’s identity. We argue that such a value of hybrids allows us to include them into the scope of moral considerability. (shrink)

Until recently, discussion of virtues in the philosophy of mathematics has been fleeting and fragmentary at best. But in the last few years this has begun to change. As virtue theory has grown ever more influential, not just in ethics where virtues may seem most at home, but particularly in epistemology and the philosophy of science, some philosophers have sought to push virtues out into unexpected areas, including mathematics and its philosophy. But there are some mathematicians already there, (...) ready to meet them, who have explicitly invoked virtues in discussing what is necessary for a mathematician to succeed. In both ethics and epistemology, virtue theory tends to emphasize character virtues, the acquired excellences of people. But people are not the only sort of thing whose excellences may be identified as virtues. Theoretical virtues have attracted attention in the philosophy of science as components of an account of theory choice. Within the philosophy of mathematics, and mathematics itself, attention to virtues has emerged from a variety of disparate sources. Theoretical virtues have been put forward both to analyse the practice of proof and to justify axioms; intellectual virtues have found multiple applications in the epistemology of mathematics; and ethical virtues have been offered as a basis for understanding the social utility of mathematical practice. Indeed, some authors have advocated virtue epistemology as the correct epistemology for mathematics (and perhaps even as the basis for progress in the metaphysics of mathematics). This topical collection brings together several of the researchers who have begun to study mathematical practices from a virtue perspective with the intention of consolidating and encouraging this trend. (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)

The physical foundations of mathematics in the theory of emergent space-time-matter were considered. It is shown that mathematics, including logic, is a consequence of equation which describes the fundamental field. If the most fundamental level were described not by mathematics, but something else, then instead of mathematics there would be consequences of this something else.

The human attempts to access, measure and organize physical phenomena have led to a manifold construction of mathematical and physical spaces. We will survey the evolution of geometries from Euclid to the Algebraic Geometry of the 20th century. The role of Persian/Arabic Algebra in this transition and its Western symbolic development is emphasized. In this relation, we will also discuss changes in the ontological attitudes toward mathematics and its applications. Historically, the encounter of geometric and algebraic perspectives enriched the (...) mathematical practices and their foundations. Yet, the collapse of Euclidean certitudes, of over 2300 years, and the crisis in the mathematical analysis of the 19th century, led to the exclusion of “geometric judgments” from the foundations of Mathematics. After the success and the limits of the logico-formal analysis, it is necessary to broaden our foundational tools and re-examine the interactions with natural sciences. In particular, the way the geometric and algebraic approaches organize knowledge is analyzed as a cross-disciplinary and cross-cultural issue and will be examined in Mathematical Physics and Biology. We finally discuss how the current notions of mathematical (phase) “space” should be revisited for the purposes of life sciences. (shrink)

The city of Kruja (Albania)contains three types of dwellings that date back to different periods of time: the historic ones, the socialist ones, the modern ones. This paper has to deal only with the historic building's typology. The questionnaire that is applied will be considered for the development of mathematical regression based on specific data for this category. Variation between the relevant variables of the questionnaire is fairly or inverse-linked with a certain percentage of influence. The aim of this study (...) is to have better insights into the important role of building occupancy, people's lifestyle, economic level, and the quality of life of the residents in the inner medieval citadel, the physics characteristics, and the energy efficiency of the historical dwellings. The variables such as the number of residents, quality of life, the surface of the dwellings, heating instruments and time spent in the dwelling, current living conditions, monthly bills, level of satisfaction, restoration, and building improvements, interact with each other with probabilities with different positive or negative percentages. (shrink)

This study delved into the perceptions of Mathematics’ student teachers regarding the implementation of gamification in secondary schools at Nasugbu, Batangas. This research investigates the rising global trend of implementing gamification in education, particularly in Mathematics teaching, to address contemporary learner needs by examining student teachers' use of gamified activities, their design factors, encountered challenges, and perceived benefits. Purposive sampling was utilized in a multiple-case study approach to select ten (10) secondary school Mathematics’ student teachers engaged in (...) practice teaching in Nasugbu, Batangas. Qualitative findings revealed common gamified activities employed by student teachers, emphasizing the importance of student-centered design, alignment with learning objectives, student engagement, assessment, and adaptation to the classroom context. Challenges faced by student teachers in integrating gamification included technical limitations, sustaining engagement, time management, and addressing diverse learning styles. Despite these challenges, integrating gamification showed benefits such as increased engagement, improved learning outcomes, reduced math anxiety, and enhanced attitudes towards Mathematics. In conclusion, gamified activities play a significant role in enhancing student engagement and learning outcomes in Mathematics classrooms. The findings suggest student teachers to explore and integrate gamified activities into Mathematics lessons to create engaging environments supporting student success. (shrink)

In this paper, I introduce the idea that some important parts of contemporary pure mathematics are moving away from what I call the extensional point of view. More specifically, these fields are based on criteria of identity that are not extensional. After presenting a few cases, I concentrate on homotopy theory where the situation is particularly clear. Moreover, homotopy types are arguably fundamental entities of geometry, thus of a large portion of mathematics, and potentially to all mathematics, (...) at least according to some speculative research programs. (shrink)

This essay offers a strategic reinterpretation of Kant’s philosophy of mathematics in Critique of Pure Reason via a broad, empirically based reconception of Kant’s conception of drawing. It begins with a general overview of Kant’s philosophy of mathematics, observing how he differentiates mathematics in the Critique from both the dynamical and the philosophical. Second, it examines how a recent wave of critical analyses of Kant’s constructivism takes up these issues, largely inspired by Hintikka’s unorthodox conception of Kantian (...) intuition. Third, it offers further analyses of three Kantian concepts vitally linked to that of drawing. It concludes with an etymologically based exploration of the seven clusters of meanings of the word drawing to gesture toward new possibilities for interpreting a Kantian philosophy of mathematics. (shrink)

I begin by outlining Du Châtelet’s ontology of mathematical objects: she is an idealist, and mathematical objects are fictions dependent on acts of abstraction. Next, I consider how this idealism can be reconciled with her endorsement of necessary truths in mathematics, which are grounded in essences that we do not create. Finally, I discuss how mathematics and physics relate within Du Châtelet’s idealism. Because the primary objects of physics are partly grounded in the same kinds of acts as (...) yield mathematical objects, she thinks we are sometimes licensed in drawing conclusions about physical things from mathematical premises. (shrink)

Abstract In the new millennium there have been important empirical developments in the philosophy of mathematics. One of these is the so-called “Empirical Philosophy of Mathematics”(EPM) of Buldt, Löwe, Müller and Müller-Hill, which aims to complement the methodology of the philosophy of mathematics with empirical work. Among other things, this includes surveys of mathematicians, which EPM believes to give philosophically important results. In this paper I take a critical look at the sociological part of EPM as a (...) case study of ... (shrink)

Mathematical objects are divided into (1) those which are autonomous, i.e., not dependent for their existence upon mathematicians’ conscious acts, and (2) intentional objects, which are so dependent. Platonist philosophy of mathematics argues that all objects belong to group (1), Brouwer’s intuitionism argues that all belong to group (2). Here we attempt to develop a dualist ontology of mathematics (implicit in the work of, e.g., Hilbert), exploiting the theories of Meinong, Husserl and Ingarden on the relations between autonomous (...) and intentional objects. In particular we develop a phenomenology of mathematical works, which has the stratified intentional structure discovered by Ingarden in his study of the literary work. (shrink)

Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for non-diagrammatic expressions. Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs. In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic proofs, one from topology and one from algebra, I show (...) that diagrams form genuine notational systems, and I argue that this explains why they can play a role in the inferential structure of proofs without undermining their reliability. I then consider whether diagrams can be essential to the proofs in which they appear. (shrink)

This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.

Mathematics is everywhere and yet its objects are nowhere. There may be five apples on the table but the number five itself is not to be found in, on, beside or anywhere near the apples. So if not in space and time, where are numbers and other mathematical objects such as perfect circles and functions? And how do we humans discover facts about them, be it Pythagoras’ Theorem or Fermat’s Last Theorem? The metaphysical question of what numbers are and (...) the epistemological question of how we know about them are central to the philosophy of mathematics. These and related philosophical questions are of particular interest because of mathematics’ unusual status. Mathematics is exceptional in that, on the one hand, it appears unhesitatingly true—no one doubts that 2 + 3 = 5—but on the other, as just noted, it is not about the physical world. This ambivalent status is what gives the philosophy of mathematics its special interest. The philosophy of mathematics is also one of the oldest academic fields, more or less coeval with philosophy itself. But contemporary philosophy of mathematics is rather different from its pre-twentieth-century antecedents, largely for three reasons. The first is that since the seventeenth century, mathematics has become integral to science. Science has over the past few centuries become increasingly mathematical, and indeed the fundamental science of nature, physics, is today recognised as a branch of applied mathematics. The second is that mathematics underwent a transformation in the course of nineteenth century: having started the century as a rather traditional-looking science of quantity it emerged a hundred years later a radically transformed abstract theory of structure. The final factor in the transformation of the philosophy of mathematics is the rise of modern logic. Developed by Frege, Cantor and others in the late nineteenth century, modern logic pervades contemporary mathematics, philosophy and computer science, and has had an immeasurable effect on the philosophy of mathematics. These volumes will collect the major works in this major field, with a focus on the last few decades. The anthology will include technical work, which interprets philosophically significant mathematical results or subfields of mathematics, as well as purely philosophical writing, aimed at those without advanced mathematics. The collection should be of interest to both philosophers and mathematicians, as well as to anyone who is susceptible to wondering what the main intellectual tool used in science, economics and finance, and indeed everyday life is ultimately about. (shrink)

Throughout his work, John Dewey seeks to emancipate philosophical reflection from the influence of the classical tradition he traces back to Plato and Aristotle. For Dewey, this tradition rests upon a conception of knowledge based on the separation between theory and practice, which is incompatible with the structure of scientific inquiry. Philosophical work can make progress only if it is freed from its traditional heritage, i.e. only if it undergoes reconstruction. In this study I show that implicit appeals to the (...) classical tradition shape prominent debates in philosophy of mathematics, and I initiate a project of reconstruction within this field. (shrink)

According to Steiner (1998), in contemporary physics new important discoveries are often obtained by means of strategies which rely on purely formal mathematical considerations. In such discoveries, mathematics seems to have a peculiar and controversial role, which apparently cannot be accounted for by means of standard methodological criteria. M. Gell-Mann and Y. Ne׳eman׳s prediction of the Ω− particle is usually considered a typical example of application of this kind of strategy. According to Bangu (2008), this prediction is apparently based (...) on the employment of a highly controversial principle—what he calls the “reification principle”. Bangu himself takes this principle to be methodologically unjustifiable, but still indispensable to make the prediction logically sound. In the present paper I will offer a new reconstruction of the reasoning that led to this prediction. By means of this reconstruction, I will show that we do not need to postulate any “reificatory” role of mathematics in contemporary physics and I will contextually clarify the representative and heuristic role of mathematics in science. (shrink)

Standard histories of mathematics and of analytic philosophy contend that work on the foundations of mathematics was motivated by a crisis such as the discovery of paradoxes in set theory or the discovery of non-Euclidean geometries. Recent scholarship, however, casts doubt on the standard histories, opening the way for consideration of an alternative motive for the study of the foundations of mathematics—unification. Work on foundations has shown that diverse mathematical practices could be integrated into a single framework (...) of axiomatic systems and that much of mathematics could be expressed in a single language. The new framework was the product of an interdisciplinary coalition whose ideas resemble those later adopted by the Vienna Circle and logical empiricists. (shrink)

This paper argues that scientific realism commits us to a metaphysical determination relation between the mathematical entities that are indispensible to scientific explanation and the modal structure of the empirical phenomena those entities explain. The argument presupposes that scientific realism commits us to the indispensability argument. The viewpresented here is that the indispensability of mathematics commits us not only to the existence of mathematical structures and entities but to a metaphysical determination relation between those entities and the modal structure (...) of the physical world. The no-miracles argument is the primary motivation for scientific realism. It is a presupposition of this argument that unobservable entities are explanatory only when they determine the empirical phenomena they explain. I argue that mathematical entities should also be seen as explanatory only when they determine the empirical facts they explain, namely, the modal structure of the physical world. Thus, scientific realism commits us to a metaphysical determination relation between mathematics and physical modality that has not been previously recognized. The requirement to account for the metaphysical dependence of modal physical structure on mathematics limits the class of acceptable solutions to the applicability problem that are available to the scientific realist. (shrink)

Fefermans argument is indeed convincing in a certain context, it can be dissolved entirely by modifying the context appropriately.

Analytical philosophy defines mathematics as an extension of logic. This research will restructure the progress in mathematical philosophy made by analytical thinkers like Wittgenstein, Russell, and Frege. We are setting up a new theory of mathematics and arithmetic’s familiar to Wittgenstein’s philosophy of language. The analytical theory proposed here proves that mathematics can be defined with non-logical terms, like numbers, theorems, and operators. We’ll explain the role of the arithmetical operators and geometrical theorems to be foundational in (...) mathematics. Our position states that mathematical operators and theorems are tautologies. Here numbers are arbitrary signs of magnitudes whose meaning is arbitrated by the mathematical operator. We will prove that set theory in mathematics doesn’t form a branch of logic and operators arbitrarily form sets into units. Herein are the foundations of numbers. (shrink)

Paying attention to the inner workings of mathematicians has led to a proliferation of new themes in the philosophy of mathematics. Several of these have to do with epistemology. Philosophers of mathematical practice, however, have not (yet) systematically engaged with general (analytic) epistemology. To be sure, there are some exceptions, but they are few and far between. In this chapter, I offer an explanation of why this might be the case and show how the situation could be remedied. I (...) contend that it is only by conceiving the knowing subject(s) as embodied, fallible, and embedded in a specific context (along the lines of what has been done within social and feminist epistemology) that we can pursue an epistemology of mathematics sensitive to actual mathematical practice. I further suggest that this reconception of the knowing subject(s) does not force us to abandon the traditional framework of epistemology in which knowledge requires justified true belief. It does, however, lead to a fallible conception of mathematical justification that, among other things, makes Gettier cases possible. This shows that topics considered to be far removed from the interests of philosophers of mathematical practice might reveal to be relevant to them. (shrink)

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)

It is undeniable Poincaré was a very famous and influential scientist. So, possibly because of it, it was relatively easy for him to participate in the heated discussions of the foundations of mathematics in the early 20th century. We can say it was “easy” because he didn't get involved in this subject by writing great treatises, or entire books about his own philosophy of mathematics (as other authors from the same period did). Poincaré contributed to the philosophy of (...) mathematics by writing short essays and letters. (shrink)

I defend a new position in philosophy of mathematics that I call mathematical inferentialism. It holds that a mathematical sentence can perform the function of facilitating deductive inferences from some concrete sentences to other concrete sentences, that a mathematical sentence is true if and only if all of its concrete consequences are true, that the abstract world does not exist, and that we acquire mathematical knowledge by confirming concrete sentences. Mathematical inferentialism has several advantages over mathematical realism and fictionalism.

Arnošt Kolman (1892–1979) was a Czech mathematician, philosopher and Communist official. In this paper, we would like to look at Kolman’s arguments against logical positivism which revolve around the notion of the fetishization of mathematics. Kolman derives his notion of fetishism from Marx’s conception of commodity fetishism. Kolman is aiming to show the fact that an entity (system, structure, logical construction) acquires besides its real existence another formal existence. Fetishism means the fantastic detachment of the physical characteristics of real (...) things or phenomena from these things. We identify Kolman’s two main arguments against logical positivism. In the first argument, Kolman applied Lenin’s arguments against Mach’s empiricism-criticism onto Russell’s neutral monism, i.e. mathematical fetishism is internally related to political conservativism. Kolman’s second main argument is that logical and mathematical fetishes are epistemologically deprived of any historical and dynamic dimension. In the final parts of our paper we place Kolman’s thinking into the context of his time, and furthermore we identify some tenets of mathematical fetishism appearing in Alain Badiou’s mathematical ontology today. (shrink)

Some have argued for a division of epistemic labor in which mathematicians supply truths and philosophers supply their necessity. We argue that this is wrong: mathematics is committed to its own necessity. Counterfactuals play a starring role.

We propose a way to explain the diversification of branches of mathematics, distinguishing the different approaches by which mathematical objects can be studied. In our philosophy of mathematics, there is a base object, which is the abstract multiplicity that comes from our empirical experience. However, due to our human condition, the analysis of such multiplicity is covered by other empirical cognitive attitudes (approaches), diversifying the ways in which it can be conceived, and consequently giving rise to different mathematical (...) disciplines. This diversity of approaches is founded on the manifold categories that we find in physical reality. We also propose, grounded on this idea, the use of Aristotelian categories as a first model for this division, generating from it a classification of mathematical branches. Finally we make a history review to show that there is consistency between our classification, and the historical appearance of the different branches of mathematics. (shrink)

Modern philosophy of mathematics has been dominated by Platonism and nominalism, to the neglect of the Aristotelian realist option. Aristotelianism holds that mathematics studies certain real properties of the world – mathematics is neither about a disembodied world of “abstract objects”, as Platonism holds, nor it is merely a language of science, as nominalism holds. Aristotle’s theory that mathematics is the “science of quantity” is a good account of at least elementary mathematics: the ratio of (...) two heights, for example, is a perceivable and measurable real relation between properties of physical things, a relation that can be shared by the ratio of two weights or two time intervals. Ratios are an example of continuous quantity; discrete quantities, such as whole numbers, are also realised as relations between a heap and a unit-making universal. For example, the relation between foliage and being-a-leaf is the number of leaves on a tree,a relation that may equal the relation between a heap of shoes and being-a-shoe. Modern higher mathematics, however, deals with some real properties that are not naturally seen as quantity, so that the “science of quantity” theory of mathematics needs supplementation. Symmetry, topology and similar structural properties are studied by mathematics, but are about pattern, structure or arrangement rather than quantity. (shrink)

Drawing mainly from the Tractatus Logico-Philosophicus and his middle period writings, strategic issues and problems arising from Wittgenstein’s philosophy of mathematics are discussed. Topics have been so chosen as to assist mediation between the perspective of philosophers and that of mathematicians on their developing discipline. There is consideration of rules within arithmetic and geometry and Wittgenstein’s distinctive approach to number systems whether elementary or transfinite. Examples are presented to illuminate the relation between the meaning of an arithmetical generalisation or (...) theorem and its proof. An attempt is made to meet directly some of Wittgenstein’s critical comments on the mathematical treatment of infinity and irrational numbers. (shrink)

A new hypothesis on the basic features characterising the Foundations of Mathematics is suggested. By means of them the entire historical development of Mathematics before the 20th Century is summarised through a table. Also the several programs, launched around the year 1900, on the Foundations of Mathematics are characterised by a corresponding table. The major difficulty that these programs met was to recognize an alternative to the basic feature of the deductive organization of a theory - more (...) precisely, to Hilbert’s main tenet. Ironically, already half a century before the births of these programs the alternative organization has been substantially represented by Lobachevsky's theory on parallel lines. Moreover, although each program’s founder recognised the basic features in a partial way only, all together these programs represented just the four possible foundational approaches. (shrink)