Summary In the attempt to understand science, definition is not much of a help. Due to the ever growing complexity of science, no unique and universally valid set of defining criteria is available. Hence, certain distinctions are resorted to: science is both a body of tested knowledge and a method of inquiry. The latter, in turn, consists of two phases: discovery and confirmation. The writer argues that it is mainly in respect of its method that science can best be characterized (...) and distinguished from other types of inquiry. (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)

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)

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)

A proof of ‘correctness’ for a mathematical algorithm cannot be relevant to executions of a program based on that algorithm because both the algorithm and the proof are based on assumptions that do not hold for computations carried out by real-world computers. Thus, proving the ‘correctness’ of an algorithm cannot establish the trustworthiness of programs based on that algorithm. Despite the (deceptive) sameness of the notations used to represent them, the transformation of an algorithm into an executable program is a (...) wrenching metamorphosis that changes a mathematical abstraction into a prescription for concrete actions to be taken by real computers. Therefore, it is verification of program executions (processes) that is needed, not of program texts that are merely the scripts for those processes. In this view, verification is the empirical investigation of: (a) the behavior that programs invoke in a computer system and (b) the larger context in which that behavior occurs. Here, deduction can play no more, and no less, a role than it does in the empirical sciences. (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)

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)

We explain the general significance of integer-based descriptors for natural shapes and show that the evolution of two such descriptors, called mechanical descriptors (the number _N_(_t_) of static balance points and the Morse–Smale graph associated with the scalar distance function measured from the center of mass) appear to capture (unlike classical geophysical shape descriptors) one of our most fundamental intuitions about natural abrasion: shapes get monotonically _simplified_ in this process. Thus mechanical descriptors help to establish a correlation between subjective and (...) objective descriptors of perceived objects. (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)

If true, this is one the the most important papers in the history of mathematics. the natural numbers are defined and one to one correspondence between the natural numbers is made precise. the paper deals with the very fundamentals of arithmetic and the logical principles differ quite markedly from those used by georg cantor.

People can be taught to manipulate symbols according to formal mathematical and logical rules. Cognitive scientists have traditionally viewed this capacity—the capacity for symbolic reasoning—as grounded in the ability to internally represent numbers, logical relationships, and mathematical rules in an abstract, amodal fashion. We present an alternative view, portraying symbolic reasoning as a special kind of embodied reasoning in which arithmetic and logical formulae, externally represented as notations, serve as targets for powerful perceptual and sensorimotor systems. Although symbolic reasoning often (...) conforms to abstract mathematical principles, it is typically implemented by perceptual and sensorimotor engagement with concrete environmental structures. (shrink)