14,143 research outputs found
Mathematical symbols as epistemic actions
Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols are not only used to express mathematical conceptsâthey are constitutive of the mathematical concepts themselves. Mathematical symbols are epistemic actions, because they enable us to represent concepts that are literally unthinkable with our bare brains. Using case-studies from the history of mathematics and from educational psychology, we argue for an intimate relationship between mathematical symbols and mathematical cognition. This paper is the draft prior to peer review. The definitive version can be found in the Online first edition of Synthese at the followin
Why 'scaffolding' is the wrong metaphor : the cognitive usefulness of mathematical representations.
The metaphor of scaffolding has become current in discussions of the cognitive help we get from artefacts, environmental affordances and each other. Consideration of mathematical tools and representations indicates that in these cases at least (and plausibly for others), scaffolding is the wrong picture, because scaffolding in good order is immobile, temporary and crude. Mathematical representations can be manipulated, are not temporary structures to aid development, and are refined. Reflection on examples from elementary algebra indicates that Menary is on the right track with his âenculturationâ view of mathematical cognition. Moreover, these examples allow us to elaborate his remarks on the uniqueness of mathematical representations and their role in the emergence of new thoughts.Peer reviewe
Tools for Thought: The Case of Mathematics
The objective of this article is to take into account the functioning of representational cognitive tools, and in particular of notations and visualizations in mathematics. In order to explain their functioning, formulas in algebra and logic and diagrams in topology will be presented as case studies and the notion of manipulative imagination as proposed in previous work will be discussed. To better characterize the analysis, the notions of material anchor and representational affordance will be introduced
An Inquiry into the Practice of Proving in Low-Dimensional Topology
The aim of this article is to investigate speciïŹc aspects connected with visualization in the practice of a mathematical subïŹeld: low-dimensional topology. Through a case study, it will be established that visualization can play an epistemic role. The background assumption is that the consideration of the actual practice of mathematics is relevant to address epistemological issues. It will be shown that in low-dimensional topology, justiïŹcations can be based on sequences of pictures. Three theses will be defended. First, the representations used in the practice are an integral part of the mathematical reasoning. As a matter of fact, they convey in a material form the relevant transitions and thus allow experts to draw inferential connections. Second, in low-dimensional topology experts exploit a particular type of manipulative imagination which is connected to intuition of two- and three-dimensional space and motor agency. This imagination allows recognizing the transformations which connect diïŹerent pictures in an argument. Third, the epistemicâand inferentialâactions performed are permissible only within a speciïŹc practice: this form of reasoning is subject-matter dependent. Local criteria of validity are established to assure the soundness of representationally heterogeneous arguments in low-dimensional topology
Nonlocal quantum information transfer without superluminal signalling and communication
It is a frequent assumption that - via superluminal information transfers -
superluminal signals capable of enabling communication are necessarily
exchanged in any quantum theory that posits hidden superluminal influences.
However, does the presence of hidden superluminal influences automatically
imply superluminal signalling and communication? The non-signalling theorem
mediates the apparent conflict between quantum mechanics and the theory of
special relativity. However, as a 'no-go' theorem there exist two opposing
interpretations of the non-signalling constraint: foundational and operational.
Concerning Bell's theorem, we argue that Bell employed both interpretations at
different times. Bell finally pursued an explicitly operational position on
non-signalling which is often associated with ontological quantum theory, e.g.,
de Broglie-Bohm theory. This position we refer to as "effective
non-signalling". By contrast, associated with orthodox quantum mechanics is the
foundational position referred to here as "axiomatic non-signalling". In search
of a decisive communication-theoretic criterion for differentiating between
"axiomatic" and "effective" non-signalling, we employ the operational framework
offered by Shannon's mathematical theory of communication. We find that an
effective non-signalling theorem represents two sub-theorems, which we call (1)
non-transfer-control (NTC) theorem, and (2) non-signification-control (NSC)
theorem. Employing NTC and NSC theorems, we report that effective, instead of
axiomatic, non-signalling is entirely sufficient for prohibiting nonlocal
communication. An effective non-signalling theorem allows for nonlocal quantum
information transfer yet - at the same time - effectively denies superluminal
signalling and communication.Comment: 21 pages, 5 figures; The article is published with open acces in
Foundations of Physics (2016
Modeling of Phenomena and Dynamic Logic of Phenomena
Modeling of complex phenomena such as the mind presents tremendous
computational complexity challenges. Modeling field theory (MFT) addresses
these challenges in a non-traditional way. The main idea behind MFT is to match
levels of uncertainty of the model (also, problem or theory) with levels of
uncertainty of the evaluation criterion used to identify that model. When a
model becomes more certain, then the evaluation criterion is adjusted
dynamically to match that change to the model. This process is called the
Dynamic Logic of Phenomena (DLP) for model construction and it mimics processes
of the mind and natural evolution. This paper provides a formal description of
DLP by specifying its syntax, semantics, and reasoning system. We also outline
links between DLP and other logical approaches. Computational complexity issues
that motivate this work are presented using an example of polynomial models
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Reinventing discovery learning: a field-wide research program
© 2017, Springer Science+Business Media B.V., part of Springer Nature. Whereas some educational designers believe that students should learn new concepts through explorative problem solving within dedicated environments that constrain key parameters of their search and then support their progressive appropriation of empowering disciplinary forms, others are critical of the ultimate efficacy of this discovery-based pedagogical philosophy, citing an inherent structural challenge of students constructing historically achieved conceptual structures from their ingenuous notions. This special issue presents six educational research projects that, while adhering to principles of discovery-based learning, are motivated by complementary philosophical stances and theoretical constructs. The editorial introduction frames the set of projects as collectively exemplifying the viability and breadth of discovery-based learning, even as these projects: (a) put to work a span of design heuristics, such as productive failure, surfacing implicit know-how, playing epistemic games, problem posing, or participatory simulation activities; (b) vary in their target content and skills, including building electric circuits, solving algebra problems, driving safely in traffic jams, and performing martial-arts maneuvers; and (c) employ different media, such as interactive computer-based modules for constructing models of scientific phenomena or mathematical problem situations, networked classroom collective âvideo games,â and intercorporeal masterâstudent training practices. The authors of these papers consider the potential generativity of their design heuristics across domains and contexts
Apperceptive patterning: Artefaction, extensional beliefs and cognitive scaffolding
In âPsychopower and Ordinary Madnessâ my ambition, as it relates to Bernard Stieglerâs recent literature, was twofold: 1) critiquing Stieglerâs work on exosomatization and artefactual posthumanismâor, more specifically, nonhumanismâto problematize approaches to media archaeology that rely upon technical exteriorization; 2) challenging how Stiegler engages with Giuseppe Longo and Francis Baillyâs conception of negative entropy. These efforts were directed by a prevalent techno-cultural qualifier: the rise of Synthetic Intelligence (including neural nets, deep learning, predictive processing and Bayesian models of cognition). This paper continues this project but first directs a critical analytic lens at the Derridean practice of the ontologization of grammatization from which Stiegler emerges while also distinguishing how metalanguages operate in relation to object-oriented environmental interaction by way of inferentialism. Stalking continental (Kapp, Simondon, Leroi-Gourhan, etc.) and analytic traditions (e.g., Carnap, Chalmers, Clark, Sutton, Novaes, etc.), we move from artefacts to AI and Predictive Processing so as to link theories related to technicity with philosophy of mind. Simultaneously drawing forth Robert Brandomâs conceptualization of the roles that commitments play in retrospectively reconstructing the social experiences that lead to our endorsement(s) of norms, we compliment this account with Reza Negarestaniâs deprivatized account of intelligence while analyzing the equipollent role between language and media (both digital and analog)
Mathematical models of games of chance: Epistemological taxonomy and potential in problem-gambling research
Games of chance are developed in their physical consumer-ready form on the basis of mathematical models, which stand as the premises of their existence and represent their physical processes. There is a prevalence of statistical and probabilistic models in the interest of all parties involved in the study of gambling â researchers, game producers and operators, and players â while functional models are of interest more to math-inclined players than problem-gambling researchers. In this paper I present a structural analysis of the knowledge attached to mathematical models of games of chance and the act of modeling, arguing that such knowledge holds potential in the prevention and cognitive treatment of excessive gambling, and I propose further research in this direction
Embodiment and embodied design
Picture this. A preverbal infant straddles the center of a seesaw. She gently tilts her weight back and forth from one side to the other, sensing as each side tips downward and then back up again. This child cannot articulate her observations in simple words, let alone in scientific jargon. Can she learn anything from this experience? If so, what is she learning, and what role might such learning play in her future interactions in the world? Of course, this is a nonverbal bodily experience, and any learning that occurs must be bodily, physical learning. But does this nonverbal bodily experience have anything to do with the sort of learning that takes place in schools - learning verbal and abstract concepts? In this chapter, we argue that the body has everything to do with learning, even learning of abstract concepts. Take mathematics, for example. Mathematical practice is thought to be about producing and manipulating arbitrary symbolic inscriptions that bear abstract, universal truisms untainted by human corporeality. Mathematics is thought to epitomize our speciesâ collective historical achievement of transcending and, perhaps, escaping the mundane, material condition of having a body governed by haphazard terrestrial circumstance. Surely mathematics is disembodied
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