Neurosciences and cognitive sciences provide us with myriad empirical findings that shed light on hypothesised primitive numerical processes in the brain and in the mind. Yet, the hypotheses on which the experiments are based, and hence the results, depend strongly on sophisticated abstract models used to describe and explain neural data or cognitive representations that supposedly are the empirical roots of primary arithmetical activity. I will question the foundational role of such models. I will even cast doubt upon the search (...) for a general and unified philosophical foundation of an empirical science. First, it seems to me hard to draw a global and coherent view from the innumerable and piecemeal neuropsychological experiments and their variable, and sometimes uneasily compatible or fully divergent interpretations. Secondly, I think that the aim of empirical research is to describe dynamical processes, establishing correlations between different sets of data, without meaning to fix an origin or to point to a cause, let alone to a ground. From the very scientific and philosophical point of view it is essential to distinguish between explanations, which provide correlations or, at best, causal mechanisms, and grounding, which involves a claim to some form of determinism. (shrink)

Academic anxiety interferes with achievement and performance, as well as social and psychological development among children and adults. Although the writings of Michel Foucault do not address anxiety directly, his themes of knowledge and power have been applied to education and describe relationships that are likely to create anxiety among some participants. This article utilizes concepts from the work of Michel Foucault, as well as the experimental psychology and educational psychology literatures, to understand the causes and consequences of academic anxiety (...) in school and learning environments. In addition to general anxiety that is associated with academic performance and achievement, extensive theoretical and empirical work has been done in two specific areas: test anxiety and mathematics anxiety. Consequently, relevant literature related to these two specific forms of academic anxiety will be incorporated when appropriate. (shrink)

This paper explores some of the constructive dimensions and specifics of human theoretic cognition, combining perspectives from (Husserlian) genetic phenomenology and distributed cognition approaches. I further consult recent psychological research concerning spatial and numerical cognition. The focus is on the nexus between the theoretic development of abstract, idealized geometrical and mathematical notions of space and the development and effective use of environmental cognitive support systems. In my discussion, I show that the evolution of the theoretic cognition of space apparently follows (...) two opposing, but in truth, intrinsically aligned trajectories. On the epistemic plane, which is the main focus of Husserl’s genetic phenomenological investigations, theoretic conceptions of space are progressively constituted by way of an idealizing emancipation of spatial cognition from the concrete, embodied intentionality underlying the human organism’s perception of space. As a result of this emancipation, it ultimately becomes possible for the human mind to theoretically conceive of and posit space as an ideal entity that is universally geometrical and mathematical. At the same time, by synthesizing a range of literature on spatial and mathematical cognition, I illustrate that for the theoretic mind to undertake precisely this emancipating process successfully, and further, for an ideal and objective notion of geometrical and mathematical space to first of all become fully scientifically operative, the cognitive support provided by a range of specific symbolic technologies is central. These include lettered diagrams, notation systems, and more generally, the technique of formalization and require for their functioning various cognitively efficacious types of embodiment. Ultimately, this paper endeavors to understand the specific symbolic-technological dimensions that have been instrumental to major shifts in the development of idealized, scientific conceptions of space. The epistemic characteristics of these shifts have been previously discussed in genetic phenomenology, but without devoting sufficient attention to the constructive role of symbolic technologies. At the same time, this paper identifies some of the irreducible phenomenological and epistemic dimensions that characterize the functioning of the historically situated, embodied and distributed theoretic mind. (shrink)

In recent work, Sören Stenlund (2015) contextualizes Wittgenstein’s philosophy of mathematics as aligned with the tradition of symbolic mathematics. In the early modern era, mathematicians began using purely formal methods disconnected from any obvious empirical applications, transforming their subject into a symbolic discipline. With this, Stenlund argues, they were freeing themselves of ancient ontological presuppositions and discovering the ultimately autonomous nature of mathematical symbolism, which eventually formed the basis for Wittgenstein’s thinking. A crucial premise of Wittgenstein’s philosophy of mathematics, on (...) this view, is that the development of mathematical concepts is independent of any ontological implications and occurs in principle without normative connections to empirical applicability. This paper critically examines this narrative and arrives at the conclusion that Stenlund’s view of mathematical progress is in stark contrast to the later Wittgenstein’s writing, which emphasizes links between symbolisms and their applications. (shrink)

Following the recent recognition that humans are an active force in nature that gave rise to a new geological epoch, this article explores the implications of the shift to the Anthropocene for social theory. The argument assumes that the emerging conditions compel an expansion and deepening of the timescale of the social-theoretical perspective and that such an enhancement has serious repercussions for the concept of human agency. First, the Anthropocene is conceptualized as a nascent cognitively structured cultural model rather than (...) simply a geological epoch. Second, the vast and deep timescale, in the light of which the new time unit and its generative agency alone make sense, is analysed along with the human world's objective, sociocultural and subjective axes. Finally, the elements of the concept of agency are recomposed in their temporal and relational contexts. At the reflexive level throughout, the need for social theory to develop a cognitive-theoretical approach in conjunction with a weak naturalistic ontology is suggested. (shrink)

Since the discovery of the paradoxes of Zeno, the problem of infinity was dominated by the meaning of endlessness—a view also adhered to by Herman Dooyeweerd. Since Aristotle, philosophers and mathematicians distinguished between the potential infinite and the actual infinite. The main aim of this article is to highlight the strengths and limitations of Dooyeweerd’s philosophy for an understanding of the foundations of mathematics, including Dooyeweerd’s quasi-substantial view of the natural numbers and his view of the other types of numbers (...) as functions of natural numbers. Dooyeweerd’s rejection of the actual infinite is turned upside down by the exploring of an alternative perspective on the interrelations between number and space in support of the idea of infinite totalities, or infinite wholes. No other trend has succeeded in justifying the mathematical use of the actual infinite on the basis of an analysis of the intermodal coherence between number and space. (shrink)

This chapter aims to make Introduction to Objectivist Epistemology (ITOE) more accessible both to students of epistemology without a background in Objectivism and to students of Objectivism without a background in epistemology. It begins with a discussion of some figures and issues in the history of philosophy that helps to appreciate what Ayn Rand meant by the advocacy of reason and why she saw the issue of concepts as central to epistemology. The chapter then considers Rand view of consciousness and (...) sense perception, and also discusses her view of the process by which concepts and a whole system of conceptual knowledge are built upon a perceptual foundation. Next, it looks more closely at Rand's conception of objectivity and how it relates to her theory of concepts. Finally, the chapter considers the standards that follow from Rand's theory for the proper formation and use of concepts. (shrink)

The paper aims to put certain basic mathematical elements and operations into an empirical perspective, evaluate the empirical status of various analytic operations widely used within psychology and suggest alternatives to procedures criticized as inadequate. Experimentation shows the "manyness" of items to be a perceptual quality for both young children and animals and that natural operations are performed by naive children analogous to those performed by persons tutored in arithmetic. Number, counting, arithmetic operations therefore can make distinctions that are not (...) inevitably arbitrary, and conceptual operations can obviously have a status as natural events with psychology. If the elements and conceptual operations involved in mathematical systems were not inherent in physiological process, various primitive discriminations could not be possible. Also, since some calculi have a natural status in a given empirical circumstance, the axioms of others can not be satisfied. Therefore the psychologist when acting as an empirical scientist seeks a calculus having a structure whose elements are isomorphic with natural units of stimulus and response and whose operations are isomorphic with whatever natural processes are involved. Measurement poses a special problem for the empirical scientist. It concerns but a single class of natural qualities and this only in a limited way. The concept of quantity has a natural counterpart but quantity and measurement are not wholly analogous. Measurement is defined, as H. S. Leonard suggests, as a theoretical activity. Measurement theory has a formal structure but empirical end. Measurement hypothesizes about the position of a particular quality within a definite range of qualities. It therefore is beholden to definite empirical restrictions. Some hypotheses-making systems use terms and relations per se as the context and starting point for dealing with discriminable events. Such procedures are 'transcendent." In empirical science, questions are part of problem-solving activity and their reference is naturally restricted. In providing description and explanation, psychological researchers frequently use calculi in a transcendent way. This results in theories that are only quasi-empirical and "half" true. The roles measurement plays in psychology are discussed. Of particular concern are those cases in which the results of measuring or a theory of measurement are used to define the "real" units, or the "real" relations involved in problematic psychological events, and thence to describe and explain behaviors of interest. Metaphysical or ontological usages of measurement sometimes occur. The implication of these arguments with regard to a view of empirical science is discussed. (shrink)

The first part of this article makes the case that human cognition is an intergenerational project enabled by the inheritance and bequeathal of cognitive technology (Sects. 2–4). The final two sections of the article (Sects. 5 and 6) explore the normative significance of this claim. My case for the intergenerational claim draws results from multiple disciplines: philosophy (Sect. 2), cultural evolutionary approaches in cognitive science (Sect. 3), and developmental psychology and neuroscience (Sect. 4). In Sect. 5, I propose that cognitive (...) technology should be given to future generations in accordance with principles of pluralism and transparency. In the final main section of the article, Sect. 6, I apply these principles to topics such as the preservation of information, environmental offloading of cognition, and thinking itself. (shrink)

Human adults are thought to possess two dissociable systems to represent numbers: an approximate quantity system akin to a mental number line, and a verbal system capable of representing numbers exactly. Here, we study the interface between these two systems using an estimation task. Observers were asked to estimate the approximate numerosity of dot arrays. We show that, in the absence of calibration, estimates are largely inaccurate: responses increase monotonically with numerosity, but underestimate the actual numerosity. However, insertion of a (...) few inducer trials, in which participants are explicitly (and sometimes misleadingly) told that a given display contains 30 dots, is sufficient to calibrate their estimates on the whole range of stimuli. Based on these empirical results, we develop a model of the mapping between the numerical symbols and the representations of numerosity on the number line. (shrink)

My paper concentrates on Peirce’s late essay, “Issues of Pragmaticism,” which identifies “critical common-sensism” and Scotistic realism as the two primary products of pragmaticism. I argue that the doctrines of Peirce’s critical common-sensism provide a host of commendable curricular objectives for democratic Bildung. The second half of my paper explores Peirce’s Scotistic realism. I argue that Peirce eventually returned to Aristotelian intuitions that led him to a more robust realism. I focus on the development of signs from the vague and (...) indeterminate to the determinate and universal. The primary example will be the evolution of the very idea of number. I believe we will never arrive at the end of number history because we can never fully contain creativity. I draw similar conclusions for the idea of curriculum. Whether or not there is an end to the evolution of signs in Peirce is a matter of debate. I incline toward the opinion there is not, though I am unsure. I conclude by arguing that rationality itself is but the form and structure of poetic creation and that we should embrace paradox and even contradiction rather that become caught in totalizing and totalitarian end of history stories. (shrink)

The ability of predicting upcoming events or conditions in advance offers substantial selective advantage to living beings. The most successful systematic tool for fairly reliable prognoses is the use of dynamical causal models in combination with memorised experience. Surprisingly, causality is a fundamental but rather controversially disputed concept. For both models and memory, symbol processing is requisite. Symbols are a necessary and sufficient attribute of life from its very beginning; the process of their evolutionary emergence was discovered by Julian Huxley (...) a century ago. In behavioural biology, this universal symmetry-breaking kinetic phase transition became known as ritualisation. Symbol use for predicting future dynamical processes has culminated in the unprecedented complexity of mental models used in science and technology, coining the historical ascent of modern humans. Observation and measurement transform structural information of physical exchange processes into symbolic information from which state quantities are derived by means of mental models. However, phylogenetically inherited models such as naïve realism do not necessarily explain the sophisticated insights revealed by modern experiments with, say, entangled quantum states. It is suggested to carefully distinguish observed exchange quantities from predicted unobservable state quantities, and physical reality from mental models thereof. (shrink)

This paper deals with Parmenides of Elea’s way of inquiry about reality and the opposition emerging from it. In more detail, it analyses how Parmenides’ concepts of logos and doxa present some analogies with Bergson’s thoughts about duration and Time and how these theories influenced the understanding of visual media, especially the cinematographic camera. This survey will allow us to demonstrate that some scientific theories about space that accompanied the development of the cinematographic camera progressively allowed for the birth of (...) a new understanding of this device. In the last section of this study, we will then focus on the way through which the film camera - understood as an intelligent device - passes from the sphere of doxa to the sphere of logos. (shrink)

Experimental studies indicate that nonhuman animals and infants represent numerosities above three or four approximately and that their mental number line is logarithmic rather than linear. In contrast, human children from most cultures gradually acquire the capacity to denote exact cardinal values. To explain this difference, I take an extended mind perspective, arguing that the distinctly human ability to use external representations as a complement for internal cognitive operations enables us to represent natural numbers. Reviewing neuroscientific, developmental, and anthropological evidence, (...) I argue that the use of external media that represent natural numbers (like number words, body parts, tokens or numerals) influences the functional architecture of the brain, which suggests a two-way traffic between the brain and cultural public representations. (shrink)

EXPANDED EDITION (eBook): -/- Infinity Is Not What It Seems...Infinity is commonly assumed to be a logical concept, reliable for conducting mathematics, describing the Universe, and understanding the divine. Most of us are educated to take for granted that there exist infinite sets of numbers, that lines contain an infinite number of points, that space is infinite in expanse, that time has an infinite succession of events, that possibilities are infinite in quantity, and over half of the world’s population believes (...) in a divine Creator infinite in knowledge, power, and benevolence. -/- According to this treatise, such assumptions are mistaken. In reality, to be is to be finite. The implications of this assessment are profound: the Universe and even God must necessarily be finite. -/- The author makes a compelling case against infinity, refuting its most prominent advocates. Any defense of the infinite will find it challenging to answer the arguments laid out in this book. But regardless of the reader’s position, FOREVER FINITE offers plenty of thought-provoking material for anyone interested in the subject of infinity from the perspectives of philosophy, mathematics, science, and theology. -/- This electronic edition of FOREVER FINITE is offered free online for all and is expanded with content that does not appear in the published edition due to print production page limitations. From the Introduction to the Conclusion, Chapters 1–27 are identical to the published print edition, including the page numbering for the chapters. This electronic edition adds an appendix and associated content—specifically, updates to the book’s back matter (68 new endnotes, 17 new bibliographic references, 3 new figure credits, 14 new glossary terms) and front matter (an updated table of contents and this preface note). (shrink)