In this paper I argue that constructivism in mathematics faces a dilemma. In particular, I maintain that constructivism is unable to explain (i) the application of mathematics to nature and (ii) the intersubjectivity of mathematics unless (iii) it is conjoined with two theses that reduce it to a form of mathematical Platonism. The paper is divided into five sections. In the first section of the paper, I explain the difference between mathematical constructivism and mathematical (...) Platonism and I outline my argument. In the second, I argue that the best explanation of how mathematics applies to nature for a constructivist is a thesis I call Copernicanism. In the third, I argue that the best explanation of how mathematics can be intersubjective for a constructivist is a thesis I call Ideality. In the fourth, I argue that once constructivism is conjoined with these two theses, it collapses into a form of mathematical Platonism. In the fifth, I confront some objections. (shrink)

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

Indigenous communities have a rich cultural heritage encompassing diverse ways of knowing, learning, and understanding the world around them. This mixed methods study utilized the explanatory sequential design to determine the level and relationship of the IP mathematics teachers' beliefs and teaching practices and gain a deeper insight into these beliefs and attitudes. There are 115 respondents for the quantitative phase, while 10 participants in the qualitative phase. Data were collected through survey and key informant interviews and were analyzed through (...) mean, standard deviation, Pearson product moment correlation, and thematic analysis. Results showed that the IP mathematics teachers possess high levels for both beliefs and teaching practices, and a significant moderate positive relationship exists between the two variables. Qualitative data analysis revealed that teachers' beliefs include the ideal qualities of an effective mathematics teacher, respect for students' culture and background, and a constructivist approach to teaching. Teaching practices include discussing student performance and intervention during meetings, implementing culturally relevant approaches, technology, varied activities, and peer mentoring and coaching. Further, it is concluded that teachers’ belief, which influences teaching practices, is shaped by different environmental factors. It is recommended that IP mathematics teachers continue to participate in personal and professional development activities and strengthen collaboration and reflection. (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)

Since the publication of the Remarks on the Foundations of Mathematics, Wittgenstein’s interpreters have endeavored to reconcile his general constructivist/anti-realist attitude towards mathematics with his confessed anti-revisionary philosophy. In this article, the author revisits the issue and presents a solution. The basic idea consists in exploring the fact that the so-called “non-constructive results” could be interpreted so that they do not appear non-constructive at all. The author substantiates this solution by showing how the translation of mathematical results, given by (...) the possibility of translation between logics, can be seen as a tool for partially implementing the solution Wittgenstein had in mind. (shrink)

In this paper, I study the kind of questions we can ask about the existence of numbers. In addition to asking whether numbers exist, and how, I argue that there is also a third relevant question: why numbers exist. In platonist and nominalist accounts this question may not make sense, but in the psychologist account I develop, it is as well-placed as the other two questions. In fact, there are two such why-questions: the causal why-question asks what causes numbers to (...) exist and the teleological why-question asks for what purpose numbers exist. I argue that in a psychologist constructivist account, in which numbers are understood to exist as referents of a particular type of culturally shared concepts, both why-questions can get plausible answers. (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)

What is so special and mysterious about the Continuum, this ancient, always topical, and alongside the concept of integers, most intuitively transparent and omnipresent conceptual and formal medium for mathematical constructions and the battle field of mathematical inquiries ? And why it resists the century long siege by best mathematical minds of all times committed to penetrate once and for all its set-theoretical enigma ? -/- The double-edged purpose of the present study is to save from the (...) transfinite deadlock of higher set theory the jewel of mathematical Continuum -- this genuine, even if mostly forgotten today raison d'etre of all set-theoretical enterprises to Infinity and beyond, from Georg Cantor to W. Hugh Woodin to Buzz Lightyear, by simultaneously exhibiting the limits and pitfalls of all old and new reductionist foundational approaches to mathematical truth: be it Cantor's or post-Cantorian Idealism, Brouwer's or post-Brouwerian Constructivism, Hilbert's or post-Hilbertian Formalism, Goedel's or post-Goedelian Platonism. -/- In the spirit of Zeno's paradoxes, but with the enormous historical advantage of hindsight, we claim that Cantor's set-theoretical methodology, powerful and reach in proof-theoretic and similar applications as it might be, is inherently limited by its epistemological framework of transfinite local causality, and neither can be held accountable for the properties of the Continuum already acquired through geometrical, analytical, and arithmetical studies, nor can it be used for an adequate, conceptually sensible, operationally workable, and axiomatically sustainable re-creation of the Continuum. -/- From a strictly mathematical point of view, this intrinsic limitation of the constative and explicative power of higher set theory finds its explanation in the identified in this study ultimate phenomenological obstacle to Cantor's transfinite construction, similar to topological obstacles in homotopy theory and theoretical physics: the entanglement capacity of the mathematical Continuum. (shrink)

The current literature suggests that the use of Husserl’s and Heidegger’s approaches to phenomenology is still practiced. However, a clear gap exists on how these approaches are viewed in the context of constructivism, particularly with non-traditional female students’ study of mathematics. The dissertation attempts to clarify the constructivist role of phenomenology within a transcendental framework from the first-hand meanings associated with the expression of the relevancy as expressed by interviews of six nontraditional female students who have studied undergraduate mathematics. (...) Comparisons also illustrate how the views associated with Husserl’s stance on phenomenology inadvertently relate to the stances of the participants interviewed as part of the study. The research questions focus on the emotional association with studying mathematics and how pre-conceived opinions regarding the study of mathematics may have influenced the essences of the experiences of the participants who have studied collegiate-level mathematics. The essences of the experiences of the participants are analyzed using bracketing and epoché to ensure personal biases of the researcher do not affect the interpretation of the expressed essences of the participants. Data collection is accomplished through two series of qualitative interviews seeking the participants’ firsthand impressions of how they view the way instructional design is oriented with regard to mathematics. Additional questions seek to illuminate the participants’ point of view regarding their emotional association with mathematics as well as their opinions and theoretical perspectives on the study of mathematics. (shrink)

The paper aims to establish if Grassmann’s notion of an extensive form involved an epistemological change in the understanding of geometry and of mathematical knowledge. Firstly, it will examine if an ontological shift in geometry is determined by the vectorial representation of extended magnitudes. Giving up homogeneity, and considering geometry as an application of extension theory, Grassmann developed a different notion of a geometrical object, based on abstract constraints concerning the construction of forms rather than on the homogeneity conditions (...) required by the modern version of the theory of proportions. Secondly, Grassmann’s conception of mathematical knowledge will be investigated. Parting from the traditional definition of mathematics as a science of magnitudes, Grassmann considered mathematical forms as particulars rather than universals: the classification of the branches of mathematics was thus based on different operational rules, rather than on empirical criteria of abstraction or on the distinction of different species belonging to a common genus. It will be argued that a different notion of generalization is thus involved, and that the knowledge of mathematical forms relies on the understanding of the rules of generation of the forms themselves. Finally, the paper will analyse if Grassmann’s approach in the first edition of the Ausdehnungslehre should be explained in terms of the notion of purity of method, and if it clashes with Grassmann’s later conventionalism. Although in the second edition the features of the operations are chosen by convention, as it is the case for the anti-commutative property of the multiplication, the choice is oriented by our understanding of the resulting forms: a simplification in the algebraic calculus need not correspond to a simplification in the ‘dimensional’ interpretation of the result of the multiplicative operation. (shrink)

A new form of skepticism is described, which holds that objectivity and understanding are incompossible ideals of modern science. This is attributed to Weyl, hence its name: Weylean skepticism. Two general defeat strategies are then proposed, one of which is rejected.

The quest to understand the natural and the mathematical as well as philosophical principles of dynamics of life forms are ancient in the human history of science. In ancient times, Pythagoras and Plato, and later, Copernicus and Galileo, correctly observed that the grand book of nature is written in the language of mathematics. Platonism, Aristotelian logism, neo-realism, monadism of Leibniz, Hegelian idealism and others have made efforts to understand reasons of existence of life forms in nature and the underlying (...) principles through the lenses of philosophy and mathematics. In this paper, an approach is made to treat the similar question about nature and existential life forms in view of mathematical philosophy. The approach follows constructivism to formulate an abstract model to understand existential life forms in nature and its dynamics by selectively combining the elements of various schools of thoughts. The formalisms of predicate logic, probabilistic inference and homotopy theory of algebraic topology are employed to construct a structure in local time-scale horizon and in cosmological time-scale horizon. It aims to resolve the relative and apparent conflicts present in various thoughts in the process, and it has made an effort to establish a logically coherent interpretation. (shrink)

Heinz Von Forester characterizes the objects “known” by an autopoietic system as eigen-solutions, that is, as discrete, separable, stable and composable states of the interaction of the system with its environment. Previous articles have presented the FBST, Full Bayesian Significance Test, as a mathematical formalism specifically designed to access the support for sharp statistical hypotheses, and have shown that these hypotheses correspond, from a constructivist perspective, to systemic eigen-solutions in the practice of science. In this article several issues related (...) to the role played by language in the emergence of eigen-solutions are analyzed. The last sections also explore possible connections with the semiotic theory of Charles Sanders Peirce. (shrink)

This article explores some open questions related to the problem of verification of theories in the context of empirical sciences by contrasting three epistemological frameworks. Each of these epistemological frameworks is based on a corresponding central metaphor, namely: (a) Neo-empiricism and the gambling metaphor; (b) Popperian falsificationism and the scientific tribunal metaphor; (c) Cognitive constructivism and the object as eigen-solution metaphor. Each of one of these epistemological frameworks has also historically co-evolved with a certain statistical theory and method for (...) testing scientific hypotheses, respectively: (a) Decision theoretic Bayesian statistics and Bayes factors; (b) Frequentist statistics and p-values; (c) Constructive Bayesian statistics and e-values. This article examines with special care the Zero Probability Paradox (ZPP), related to the verification of sharp or precise hypotheses. Finally, this article makes some remarks on Lakatos’ view of mathematics as a quasi-empirical science. (shrink)

Open peer commentary on the article “Constructivist Model Building: Empirical Examples From Mathematics Education” by Catherine Ulrich, Erik S. Tillema, Amy J. Hackenberg & Anderson Norton. Upshot: I argue that radical constructivism poses a series of deep methodological constraints on educational research. We focus on the work of Ulrich et al. to illustrate the practical implications of these constraints.

These remarks take up the reflexive problematics of Being and Nothingness and related texts from a metalogical perspective. A mutually illuminating translation is posited between, on the one hand, Sartre’s theory of pure reflection, the linchpin of the works of Sartre’s early period and the site of their greatest difficulties, and, on the other hand, the quasi-formalism of diagonalization, the engine of the classical theorems of Cantor, Gödel, Tarski, Turing, etc. Surprisingly, the dialectic of mathematical logic from its inception (...) through the discovery of the diagonal theorems can be recognized as a particularly clear instance of the drama of reflection according to Sartre, especially in the positing and overcoming of its proper value-ideal, viz. the synthesis of consistency and completeness. Conversely, this translation solves a number of systematic problems about pure reflection’s relations to accessory reflection, phenomenological reflection, pre-reflective self-consciousness, conversion, and value. Negative foundations, the metaphysical position emerging from this translation between existential philosophy and metalogic, concurs by different paths with Badiou’s Being and Event in rejecting both ontotheological foundationalisms and constructivist antifoundationalisms. (shrink)

Nietzsche was persistently concerned with what an object is and how different views of objects lead to different views of facts, causality, personhood, substance, truth, mathematics and logic, and even nihilism. Yet his treatment of objects is incredibly puzzling. In many passages he assumes that objects such as trees and leaves, tables and chairs, and dogs and cats are just ordinary entities of experience. In other places he reports that objects do not exist. Elsewhere he claims that objects exist, but (...) as mere bundles of forces. And sometimes he proposes that we bring all objects into existence. Nietzsche’s writings, then, appear to support various secondary readings, which are jointly inconsistent. My chief aim is to present and defend the reading that Nietzsche embraces constructivism about objects, the neo-Kantian view that all objects are socially constructed. I first explain this view and argue that all non-constructivist readings are not supported by Nietzsche’s texts. I then present Nietzsche’s object constructivism, reconstruct his argument for the position, and defend it from internal objections. I finish by suggesting that Nietzsche might have embraced such a radical conception of objects because it plays a crucial role in overcoming nihilism. (shrink)

Western science claims to provide unique, objective information about the world. This is supported by the observation that peoples across cultures will agree upon a common description of the physical world. Further, the use of scientific instruments and mathematics is claimed to enable the objectification of science. In this work, carried out by reviewing the scientific literature, the above claims are disputed systematically by evaluating the definition of physical reality and the scientific method, showing that empiricism relies ultimately upon the (...) human senses for the evaluation of scientific theories and that measuring instruments cannot replace the human sensory system. Nativist and constructivist theories of human sensory development are reviewed, and it is shown that nativist claims of core conceptual knowledge cannot be supported by the findings in the literature, which shows that perception does not simply arise from a process of maturation. Instead, sensory function requires a long process of learning through interactions with the environment. To more rigorously define physical reality and systematically evaluate the stability of perception, and thus the basis of empiricism, the development of the method of dimension analysis is reviewed. It is shown that this methodology, relied upon for the mathematical analysis of physical quantities, is itself based upon empiricism, and that all of physical reality can be described in terms of the three fundamental dimensions of mass, length and time. Hereafter the sensory modalities that inform us about these three dimensions are systematically evaluated. The following careful analysis of neuronal plasticity in these modalities shows that all the relevant senses acquire from the environment the capacity to apprehend physical reality. It is concluded that physical reality is acquired rather than given innately, and leads to the position that science cannot provide unique results. Rather, those it can provide are sufficient for a particular environmental setting. (shrink)

According to the iterative conception of set, each set is a collection of sets formed prior to it. The notion of priority here plays an essential role in explanations of why contradiction-inducing sets, such as the Russell set, do not exist. Consequently, these explanations are successful only to the extent that a satisfactory priority relation is made out. I argue that attempts to do this have fallen short: understanding priority in a straightforwardly constructivist sense threatens the coherence of the empty (...) set and raises serious epistemological concerns; but the leading realist interpretations---ontological and modal interpretations of priority---are deeply problematic as well. I conclude that the purported explanatory virtues of the iterative conception are, at present, unfounded. (shrink)

Abstract We examine the construction of electromagnetism in its current form, and in an alternative form, from a point of view that combines a minimal realism with strict rational demands. We begin by discussing the requests of reason when constructing a theory and next, we follow the historical development as presented in the record of original publications, the underlying epistemology (often explained by the authors) and the mathematical constructions. The historical construction develops along socio-political disputes (mainly, the reunification of (...) Germany and the second industrial revolution), epistemic disputes (at least two demarcations of science in conflict) and several theories of electromagnetism. Such disputes resulted in the militant adoption of the ether by some, a position that expanded in parallel with the expansion of Prussia. This way of thinking was facilitated by the earlier adoption of a standpoint that required, as a condition for understanding, the use of physical hypothesis in the form of analogies; an attitude that is antithetic to Newton's “hypotheses non fingo”. While the material ether was finally abandoned, the epistemology survived in the form of “substantialism” and a metaphysical ether: the space. The militants of the ether attributed certainties regarding the ether to Faraday and Maxwell, when they only expressed doubts and curiosity. Thus, the official story is not the real history. This was achieved by the operation of detaching Maxwell's electromagnetism from its construction and introducing a new game of formulae and interpretations. Large and important parts of Maxwell work are today not known, as for example, the rules for the transformation of the electromagnetic potentials between moving systems. When experiments showed that all the theories based in the material ether were incorrect, a new interpretation was offered: Special Relativity (SR). At the end of the transformation period a pragmatic view of science, well adapted to the industrial society, had emerged, as well as a new protagonist: the theoretical physicist. The rival theory of delayed action at distance initiated under the influence of Gauss was forgotten in the midst of the intellectual warfare. The theory is indistinguishable in formulae from Maxwell's and its earlier versions are the departing point of Maxwell for the construction of his equations. We show in a mathematical appendix that such (relational) theory can incorporate Lorentz' contributions as well as Maxwell's transformations and C. Neumann's action, without resource to the ether. Demarcation criteria was further changed at the end of the period making room for habits and intuitions. When these intuited criteria are examined by critical reason (seeking for the fundaments) they can be sharpened with the use of the Non Arbitrariness Principle, which throws light over the arbitrariness in the construction of SR. Under a fully rational view SR is not acceptable, it requires to adopt a less demanding epistemology that detaches the concept from the conception, such as Einstein's own view in this respect, inherited from Hertz. In conclusion: we have shown in this relevant exercise how the reality we accept depends on earlier, irrational, decisions that are not offered for examination but rather are inherited from the culture. (shrink)

This article explores the metaphor of Science as provider of sharp images of our environment, using the epistemological framework of Objective Cognitive Constructivism. These sharp images are conveyed by precise scientific hypotheses that, in turn, are encoded by mathematical equations. Furthermore, this article describes how such knowledge is pro-duced by a cyclic and recursive development, perfection and reinforcement process, leading to the emergence of eigen-solutions characterized by the four essential properties of precision, stability, separability and composability. Finally, this (...) article discusses the role played by ontology and metaphysics in the scientific production process, and in which sense the resulting knowledge can be considered objective. (shrink)

The purpose of this paper is to contextualize the study of metaphors within constructivist-informed research, in the hope that this process will orient cognitive scientists to the usefulness of implementing qualitative research methodologies, especially to using the person of the researcher as the primary research instrument. First, I explore some of the differences between Johnson and Lakoff’s Contemporary Metaphor Theory (CMT) and approaches evolving from it on one hand, and the clinical approach to metaphor based on a constructivist therapy model, (...) on the other. CMT has been one of the most significant forces that helped shift cognitive science toward an embodied approach to cognition. While it has succeeded to place physical experience back where it belongs in reason and meaning, CMT has, however, also fallen into some positivist traps which lead to problems such as a dualism, a split between the knower and the known, and with that, to a distrust of introspective, first-person accounts. In the process of finding conceptual metaphors — generalizations that govern metaphorical expressions — CMT often deletes the idiosyncratic characteristics and presuppositions implicit in linguistic metaphors; it divorces them from people’s sensory experiences, the “here and now” and the intent of their communications. The constructivist approach to metaphor that I present here accepts as a priori assumptions much of what workers of CMT are out to prove. In particular, it takes the correlation of conceptual metaphors and physical experience, as well as the unity of language and thought as pragmatic givens. Emulating the constructivist therapist’s approach to metaphors, I show how it is possible to deconstruct conceptual metaphors into minute sensory distinctions, using one’s own person as the main tool, for the purpose of helping people change their experiences in desired ways, at will. I illustrate this process by numerous examples from a wide field of applications, including mathematics education and psychotherapy. (shrink)

Context: The infinite has long been an area of philosophical and mathematical investigation. There are many puzzles and paradoxes that involve the infinite. Problem: The goal of this paper is to answer the question: Which objects are the infinite numbers (when order is taken into account)? Though not currently considered a problem, I believe that it is of primary importance to identify properly the infinite numbers. Method: The main method that I employ is conceptual analysis. In particular, I argue (...) that the infinite numbers should be as much like the finite numbers as possible. Results: Using finite numbers as our guide to the infinite numbers, it follows that infinite numbers are of the structure w + (w* + w) a + w*. This same structure also arises when a large finite number is under investigation. Implications: A first implication of the paper is that infinite numbers may be large finite numbers that have not been investigated fully. A second implication is that there is no number of finite numbers. Third, a number of paradoxes of the infinite are resolved. One change that should occur as a result of these findings is that “infinitely many” should refer to structures of the form w + (w* + w) a + w*; in contrast, there are “indefinitely many” natural numbers. Constructivist content: The constructivist perspective of the paper is a form of strict finitism. (shrink)

This paper considers non-standard analysis and a recently introduced computational methodology based on the notion of ①. The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations requiring these notions. Non-standard analysis is a classical purely symbolic technique that works with ultrafilters, external and internal sets, standard and non-standard numbers, etc. In its turn, the ①-based methodology does not use any of these (...) notions and proposes a more physical treatment of mathematical objects separating the objects from tools used to study them. It both offers a possibility to create new numerical methods using infinities and infinitesimals in floating-point computations and allows one to study certain mathematical objects dealing with infinity more accurately than it is done traditionally. In these notes, we explain that even though both methodologies deal with infinities and infinitesimals, they are independent and represent two different philosophies of Mathematics that are not in a conflict. It is proved that texts :539–555, 2017; Gutman and Kutateladze in Sib Math J 49:835–841, 2008; Kutateladze in J Appl Ind Math 5:73–75, 2011) asserting that the ①-based methodology is a part of non-standard analysis unfortunately contain several logical fallacies. Their attempt to show that the ①-based methodology can be formalized within non-standard analysis is similar to trying to show that constructivism can be reduced to the traditional Mathematics. (shrink)

This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we do not want to (...) equate falsity with the non-existence of a proof since this would render a statement such as “pi is transcendental” false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for statements for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorov’s logic of problems. (shrink)

The first aim of this paper is to elucidate Russell’s construction of spatial points, which is to be <br>considered as a paradigmatic case of the "logical constructions" that played a central role in his epistemology and theory of science. Comparing it with parallel endeavours carried out by Carnap and Stone it is argued that Russell’s construction is best understood as a structural representation. It is shown that Russell’s and Carnap’s representational constructions may be considered as incomplete and sketchy harbingers of (...) Stone’s representation theorems. The representational program inaugurated by Stone’s theorems was one of the success stories of 20th century’s mathematics. This suggests that representational constructions à la Stone could also be important for epistemology and philosophy of science. More specifically it is argued that the issues proposed by Russellian definite descriptions, logical constructions, and structural representations still have a place on the agenda of contemporary epistemology and philosophy of science. Finally, the representational interpretation of Russell’s logical constructivism is used to shed some new light on the recently vigorously discussed topic of his structural realism. (shrink)

In this paper we analyze methodological and philosophical implications of algorithmic aspects of unconventional computation. At first, we describe how the classical algorithmic universe developed and analyze why it became closed in the conventional approach to computation. Then we explain how new models of algorithms turned the classical closed algorithmic universe into the open world of algorithmic constellations, allowing higher flexibility and expressive power, supporting constructivism and creativity in mathematical modeling. As Goedels undecidability theorems demonstrate, the closed algorithmic (...) universe restricts essential forms of mathematical cognition. In contrast, the open algorithmic universe, and even more the open world of algorithmic constellations, remove such restrictions and enable new, richer understanding of computation. (shrink)

Open peer commentary on the article “Ethics: A Radical-constructivist Approach” by Andreas Quale. Upshot: The first of my two main goals in this commentary is to establish thinking of ethics as concepts rather than as non-cognitive knowledge. The second is to argue that establishing models of individuals’ ethical concepts is a scientific enterprise that is quite similar to establishing models of individuals’ mathematical concepts. To accomplish these two primary goals, I draw from my experience of working scientifically with von (...) Glasersfeld for 25 years while he was developing radical constructivism as a coherent model of knowing, and appeal to several of his basic insights to establish constructing models of ethical concepts as a scientific enterprise. (shrink)

Kant's arguments for the synthetic a priori status of geometry are generally taken to have been refuted by the development of non-Euclidean geometries. Recently, however, some philosophers have argued that, on the contrary, the development of non-Euclidean geometry has confirmed Kant's views, for since a demonstration of the consistency of non-Euclidean geometry depends on a demonstration of its equi-consistency with Euclidean geometry, one need only show that the axioms of Euclidean geometry have 'intuitive content' in order to show that both (...) Euclidean and non-Euclidean geometry are bodies of synthetic a priori truths. Michael Friedman has argued that this defence presumes a polyadic conception of logic that was foreign to Kant. According to Friedman, Kant held that geometrical reasoning itself relies essentially on intuition, and that this precludes the very possibility of non-Euclidean geometry. While Friedman's characterization of Kant's views on geometrical reasoning is correct, I argue that Friedman's conclusion that non-Euclidean geometries are logically impossible for Kant is not. I argue that Kant is best understood as a proto-constructivist and that modern constructive axiomatizations (unlike Hilbert-style axiomatizations) of both Euclidean and non-Euclidean geometry capture Kant's views on the essentially constructive nature of geometrical reasoning well. (shrink)

Lakatos constructed his major contribution to the philosophy of science, the methodology of scientific research programmes (MSRP), in the late sixties and early seventies in England, after he had already become estranged from the Popperian philosophy of science. In this paper, we attempt to show that the MSRP was motivated by his philosophical and political ideas from the forties and fifties in Hungary, when he was imbued with the communist ideology and was influenced by the philosophy of Georg Lukács. From (...) this point of view, the MSRP can be considered to be a special representation of Lakatos' earlier political values and practice in the field of history and philosophy of science. (shrink)

Nền kinh tế nước ta đang chuyển biến mạnh mẽ từ nền kinh tế bao cấp sang kinh tế thị trường, nhất là từ khi nước ta gia nhập WTO. Đảng và chính phủ đã đề ra rất nhiều các chính sách để cải tiến các thể chế quản lý nền kinh tế và tài chính. Thị trường chứng khoán Việt Nam đã ra đời và đang đóng một vai trò quan trọng trong việc huy động vốn phục vụ cho (...) việc phát triển nền kinh tế quốc dân. Việc phân tích tác động của các chính sách đến nền kinh tế vĩ mô và nền tài chính, những đối sách của chính phủ ta và chính phủ các nước đối với cuộc khủng khoảng tài chính toàn cầu là các vấn đề rất thời sự đang thu hút sự chú ý của nhiều nhà kinh tế và toán học. (shrink)

In this paper epistemological, ontological and sociological questions concerning the statistical significance of sharp hypotheses in scientific research are investigated within the framework provided by Cognitive Constructivism and the FBST (Full Bayesian Significance Test). The constructivist framework is contrasted with the traditional epistemological settings for orthodox Bayesian and frequentist statistics provided by Decision Theory and Falsificationism.

I describe a new problem for metaethical constructivism. The problem arises when agents make conflicting judgments, so that the constructivist is implausibly committed to denying they have any reason for any of the available options. The problem is illustrated primarily with reference to Sharon Street’s version of constructivism. Several possible solutions to the problem are explained and rejected.

Given constructivism’s enduring popularity and appeal, it is perhaps something of a surprise that there remains considerable uncertainty among many philosophers about what constructivism is even supposed to be. My aim in this article is to make some progress on the question of how constructivism should be understood. I begin by saying something about what kind of theory constructivism is supposed to be. Next, I consider and reject both the standard proceduralist characterization of constructivism and (...) also Sharon Street’s ingenious standpoint characterization. I then suggest an alternative characterization according to which what is central is the role played by certain standards of correct reasoning. I conclude by saying something about the implications of this account for evaluating the success of constructivism. I suggest that certain challenges that have been raised against constructivist theories are based on dubious understandings of constructivism, whereas other challenges only properly come into focus once a proper understanding is achieved. (shrink)

The study assessed the conceptual understanding and attitude toward fractions of teacher education students in a socio-constructivist learning environment. Specifically, it determined the students’ level of conceptual understanding before and after instruction; verified the types of conceptual changes that occurred; and ascertained the attitude of students toward fractions before and after instruction and its relationship to their levels of understanding. Descriptive-correlational research method was used. Socio-constructivist context-based teaching method was employed to introduce the concept of fractions. Achievement tests and interviews (...) were administered to determine the students’ level of conceptual understanding. Conceptual analysis based on Jensen and Finley’s (1995) method with Tiberghien’s (1994) classification of changes was utilized to describe students’ conceptual understanding and conceptual changes. In order to determine their attitude on fractions, students were asked to answer the socio-constructivist attitude questionnaire. The level of conceptual understanding of teacher education students in fraction was functional misconception and partial understanding before and after instruction, respectively. The type of conceptual change that occurred among teacher education students was change for the better. Socio-constructivist learning more likely to improve students’ attitudes toward fractions; promoted prosocial behavior among students; and tend to increase students’ activeness in the classroom activities as evidenced. (shrink)

Are there such things as moral truths? How do we know what we should do? And does it matter? Constructivism states that moral truths are neither invented nor discovered, but rather are constructed by rational agents in order to solve practical problems. While constructivism has become the focus of many philosophical debates in normative ethics, meta-ethics and action theory, its importance is still to be fully appreciated. These new essays written by leading scholars define and assess this new (...) approach in ethics, addressing such questions as the nature of constructivism, how constructivism improves our understanding of moral obligations, how it accounts for the development of normative practices, whether moral truths change over time, and many other topics. The volume will be valuable for advanced students and scholars of ethics and all who are interested in questions about the foundation of morality. (shrink)

Constructivism is frequently met with objections, criticism and often equated with nihilism or relativism. Sometimes even blamed for what some would randomly picture as unwanted side effects of radicalism or of a progressivist era: such misconceptions are not only due to an imprecise grasp of the premises shared by the constructivist family of systems. The structure of media, political systems, and economic models, still up today impel societal understandings of knowledge on neo-positivistic grounds. The first part of this essay (...) outlines such pressures while sketching how language and worldviews play critical roles in our knowledge construction. Focusing on recent mediatic events, this work advances displaying some essential limits regarding the construction of human knowledge. Though unavoidable, some of the distinguishing aspects regarding the nature of our narratives are then critically reviewed. Later, it is shown how a special kind of self-denial that certain sub-stories implicitly hold about their own narrative nature leaves us with clashing worldviews that eventually collide onto crisis. Finally, it’ll be argued that it’s precisely in this scenario where a constructivist depiction of social discourses may move us away from any adolescent intent of elucidating absolutes within mere heuristics, to the pragmatic need of arriving on satisfactory agreements between parties (DOI: 10.4236/ojpp.2022). (shrink)

In this paper I introduce a version of constructivism that relies on a theory of practical wisdom. Wise judgment constructivism is a type of constructivism because it takes correct judgments about what we have “all-in” reason to do to be the result of a process we can follow, where our interest in the results of this process stems from our practical concerns. To fully defend the theory would require a comprehensive account of wisdom, which is not available. (...) Instead, I describe a constructivist methodology for defending an account of wisdom and outline its main features. This gives us enough to see what wise judgment constructivism would look like, why it might be an attractive theory, and how it is different from other versions of constructivism. (shrink)

Many neo-Aristotelians argue that practical identities are normative, that is, they provide us with reasons for action and create binding obligations. Kantian constructivists agree with this insight but argue that contemporary Aristotelians fail to fully justify it. Practical identities are normative, Kantian constructivists contend, but their normativity necessarily derives from the normativity of humanity. In this paper, I shed light on this underexplored similarity between neo-Aristotelian and Kantian constructivist accounts of the normativity of practical identities, and argue that both ultimately (...) fail. I end by suggesting an alternative justification of the claim that practical identities are normative. (shrink)

There is a major debate as to whether there are non-causal mathematical explanations of physical facts that show how the facts under question arise from a degree of mathematical necessity considered stronger than that of contingent causal laws. We focus on Marc Lange’s account of distinctively mathematical explanations to argue that purported mathematical explanations are essentially causal explanations in disguise and are no different from ordinary applications of mathematics. This is because these explanations work not by (...) appealing to what the world must be like as a matter of mathematical necessity but by appealing to various contingent causal facts. (shrink)

This paper presents a comparative evaluation of constructivist and error theoretic accounts of moral claims. It is argued that constructivism has distinct advantages over error theory.

There are at least two politically salient senses of “representation”—acting-for-others and portraying-something-as-something. The difference is not just semantic but also logical: relations of representative agency are dyadic (x represents y), while portrayals are triadic (x represents y as z). I exploit this insight to disambiguate constructivism and to improve our theoretical vocabulary for analyzing political representation. I amend Saward’s claims-based approach on three points, introducing the “characterization” to correctly identify the elements of representational claims; explaining the “referent” in pragmatic, (...) not metaphysical terms; and differentiating multiple forms of representational activity. This enables me to clarify how the represented can be both prior to representation and constituted by it, and to recover Pitkin’s idea that representatives ought to be “responsive” to the represented. These points are pertinent to debates about the role of representatives, the nature of representative democracy, and the dynamics of revolutionary movements. (shrink)

The paper responds to a common charge against constructivism about objects, the view that all objects are essentially socially constructed. The objection is that constructivism is false because there must exist unconstructed objects for there to be constructed objects. I contend that the worry is unsound because whatever exists fully independently of our activities cannot be an object.

Sharon Street defines her constructivism about practical reasons as the view that whether something is a reason to do a certain thing for a given agent depends on that agent’s normative point of view. However, Street has also maintained that there is a judgment about practical reasons which is true relative to every possible normative point of view, namely constructivism itself. I show that the latter thesis is inconsistent with Street’s own constructivism about epistemic reasons and discuss (...) some consequences of this incompatibility. (shrink)

While it is uncontroversial that Kantian constructivism has implications for normative ethics, its status as a metaethical view has been contested. In this article, I provide a characterisation of metaethical Kantian constructivism that withstands these criticisms. I start by offering a partial defence of Sharon Street’s practical standpoint characterisation. However, I argue that this characterisation, as presented by Street, is ultimately incomplete because it fails to demonstrate that the claims of Kantian constructivism constitute a distinctive contribution to (...) metaethics. I then try to complete the practical standpoint characterisation by elaborating on Christine Korsgaard’s suggestion that metaethical Kantian constructivism takes up a position on the source of morality’s normativity. (shrink)

Modes of teaching and learning have had to rapidly shift amid the COVID-19 pandemic. As an emergency response, students from Philippine public schools were provided learning modules based on a minimized list of essential learning competencies in Biology. Using a cross-sectional survey method, we investigated students’ perceptions of the Biology self-learning modules (BSLM) that were designed in print and digitized formats according to a constructivist learning approach. Senior high school STEM students from grades 11 (n = 117) and 12 (n (...) = 104) participated in a survey using a 3-point Likert-scale questionnaire uploaded online through Google Forms. The survey results indicate that majority of the students perceived the modules positively, suggesting that aspects of the modules that were salient to students corresponded to essential elements of constructivist pedagogies. However, during interviews, students reported several difficulties in learning with BSLM as it was constrained by, to name a few, the use of unfamiliar words, lack of access to supporting resources, slow internet connection, and time constraints. To address these problems, teachers reported that they gave deadline extensions, complemented modules with other channels of support, and used online and offline platforms for reaching out to students to answer their queries and plan out their schedule for the week. The findings across the data sources point to the complex demands of emergency distance education that teachers, as curriculum designers and enactors, need to bear in mind in order to craft productive pedagogies, constructivist or otherwise, during this unprecedented time. (shrink)

The dissertation defends that the often-assumed link between constructivism and universalism builds on non-constructivist, perfectionist grounds. To this end, I argue that an exemplary form of universalist constructivism – i.e., O’Neill’s Kantian constructivism – can defend its universalist commitments against an influential particularist form of constructivism – i.e., political liberalism as advanced by Rawls, Macedo, and Larmore – only if it invokes a perfectionist view of the good. (En route, I show why political liberalism is a (...) form of particularism and reconstruct the role of its conception of public justification, its implied contextualism about justified belief, its conceptions of toleration, neutrality, good reasons, and legitimacy, and, not least, its justification-constitutive conception of reasonableness.) Contrary to what is often assumed, then, at the level of a vindication of the very project of a universalist constructivism, universalist constructivists should construe perfectionists not as their opponents, but as partial, though uneasy, allies. (shrink)

The Polish philosophy of mathematics in the 19th century is not a well-researched topic. For this period, only five philosophers are usually mentioned, namely Jan Śniadecki, Józef Maria Hoene-Wroński, Henryk Struve, Samuel Dickstein, and Edward Stamm. This limited and incomplete perspective does not allow us to develop a well-balanced picture of the Polish philosophy of mathematics and gauge its influence on 19th- and 20th-century Polish philosophy in general. To somewhat complete our picture of the history of the Polish philosophy of (...) mathematics in those times, we here present the profiles of some lesser-known Polish Romantic philosophers of the 19th century, namely Karol Libelt, Bronisław Trentowski, and Józef Kremer. We discuss their contributions to the philosophy of mathematics and their metaphysical perspectives, and we also show how their metaphysical ideas have found some continuity in the studies of some Catholic philosophers. (shrink)

The conventional wisdom regarding the aims and shortcomings of Kantian constructivism is mistaken. The aim of metaethical constructivism is not to provide a naturalistic account of the objectivity of normative facts by deriving substantive morality from a conception of agency so thin as to be uncontroversial (a task at which it is generally regarded to have failed). Its aim is to explain the “grip” that normative facts have on us—to avoid what I call the problem of normative alienation. (...) So understood, Kantian constructivism faces two problems: that determinate normative facts cannot be derived from agency and that its individualistic conception of agency cannot account for the sociality of morality. I propose and elaborate a social conception of agency that is better able to address the latter problem while still avoiding normative alienation, and evaluate two different strategies for responding to the former problem. (shrink)

An inquiry on the training needs in Mathematics was conducted to Laura Vicuña Center - Palawan (LVC-P) learners. Specifically, this aimed to determine their level of performance in numbers, measurement, geometry, algebra, and statistics, identify the difficulties they encountered in solving word problems and enumerate topics where they needed coaching. -/- To identify specific training needs, the study employed a descriptive research design where 36 participants were sampled purposively. The data were gathered through a problem set test and focus group (...) discussion. Findings revealed that LVC-P learners had an unsatisfactory performance in numbers, measurement, and statistics while alarmingly poor in geometry and algebra. They also faced difficulties in remembering, understanding, applying, and analyzing mathematical concepts when solving problems. Further, the learners were able to name certain topics subjected to tutorials. -/- The above facts and observations suggest that learners of LVC-P have urgent training needs in Mathematics. It is recommended that the Western Philippines University - College of Education RDE Unit continue its research, development, and extension services to LVC-P. More importantly, the results of this inquiry regarding the training needs of the learners will serve as bases for conducting an extension project and development program for LVC-P. Series of tutorial sessions is deemed necessary to address the needs and difficulties of the learners. (shrink)