Constructing a concept of number

Numbers are concepts whose content, structure, and organization are influenced by the material forms used to represent and manipulate them. Indeed, as argued here, it is the inclusion of multiple forms (distributed objects, fingers, single- and two-dimensional forms like pebbles and abaci, and written notations) that is the mechanism of numerical elaboration. Further, variety in employed forms explains at least part of the synchronic and diachronic variability that exists between and within cultural number systems. Material forms also impart characteristics like linearity that may persist in the form of knowledge and behaviors, ultimately yielding numerical concepts that are irreducible to and functionally independent of any particular form. Material devices used to represent and manipulate numbers also interact with language in ways that reinforce or contrast different aspects of numerical cognition. Not only does this interaction potentially explain some of the unique aspects of numerical language, it suggests that the two are complementary but ultimately distinct means of accessing numerical intuitions and insights. The potential inclusion of materiality in contemporary research in numerical cognition is advocated, both for its explanatory power, as well as its influence on psychological, behavioral, and linguistic aspects of numerical cognition

Toward a Kripkean Concept of Number

Saul Kripke once remarked to me that natural numbers cannot be posits inferred from their indispensability to science, since we’ve always had them. This left me wondering whether numbers are objects of Russellian acquaintance, or accessible by analysis, being implied by known general principles about how to reason correctly, or both. To answer this question, I discuss some recent (and not so recent) work on our concepts of number and of particular numbers, by leading psychologists and philosophers. Special attention is paid to Kripke’s theory that numbers possess structural features of the numerical systems that stand for them, and to the relation between his proposal about numbers and his doctrine that there are contingent truths known a priori. My own proposal, to which Kripke is sympathetic, is that numbers are properties of sets. I argue for this by showing the extent to which it can avoid the problems that plague the various views under discussion, including the problems raised by Kripke against Frege. I also argue that while the terms ‘the number of F’s’, ‘natural number’ and ‘0’, ‘1’, ‘2’ etc. are partially understood by the folk, they can only be fully understood by reflection and analysis, including reflection on how to reason correctly. In this last respect my thesis is a retreat position from logicism. I also show how it dovetails with an account of how numbers are actually grasped in practice, via numerical systems, and in virtue of a certain structural affinity between a geometric pattern that we grasp intuitively, and our fully analyzed concepts of numbers. I argue that none of this involves acquaintance with numbers

Concepts, Frames and Cascades in Semantics, Cognition and Ontology

This open access book presents novel theoretical, empirical and experimental work exploring the nature of mental representations that support natural language production and understanding, and other manifestations of cognition. One fundamental question raised in the text is whether requisite knowledge structures can be adequately modeled by means of a uniform representational format, and if so, what exactly is its nature. Frames are a key topic covered which have had a strong impact on the exploration of knowledge representations in artificial intelligence, psychology and linguistics; cascades are a novel development in frame theory. Other key subject areas explored are: concepts and categorization, the experimental investigation of mental representation, as well as cognitive analysis in semantics. This book is of interest to students, researchers, and professionals working on cognition in the fields of linguistics, philosophy, and psychology

Micro/nano devices for blood analysis

[Excerpt] The development of microdevices for blood analysis is an interdisciplinary subject that demandsan integration of several research fields such as biotechnology, medicine, chemistry, informatics, optics,electronics, mechanics, and micro/nanotechnologies.Over the last few decades, there has been a notably fast development in the miniaturization ofmechanical microdevices, later known as microelectromechanical systems (MEMS), which combineelectrical and mechanical components at a microscale level. The integration of microflow and opticalcomponents in MEMS microdevices, as well as the development of micropumps and microvalves,have promoted the interest of several research fields dealing with fluid flow and transport phenomenahappening at microscale devices. [...

The role of representation in Bayesian reasoning: Correcting common misconceptions

The terms nested sets, partitive frequencies, inside-outside view, and dual processes add little but confusion to our original analysis (Gigerenzer & Hoffrage 1995; 1999). The idea of nested set was introduced because of an oversight; it simply rephrases two of our equations. Representation in terms of chances, in contrast, is a novel contribution yet consistent with our computational analysis - it uses exactly the same numbers as natural frequencies. We show that non-Bayesian reasoning in children, laypeople, and physicians follows multiple rules rather than a general-purpose associative process in a vaguely specified "System 1.” It is unclear what the theory in "dual process theory” is: Unless the two processes are defined, this distinction can account post hoc for almost everything. In contrast, an ecological view of cognition helps to explain how insight is elicited from the outside (the external representation of information) and, more generally, how cognitive strategies match with environmental structure

A reappraisal and 3D characterisation of fracture systems within the Devonian Orcadian Basin and its underlying basement: an onshore analogue for the Clair Group

The Orcadian Basin is a Devonian (Old Red Sandstone) sedimentary basin formed as a result of extensional tectonics after the end of the Caledonian Orogeny in onshore-offshore northeast Scotland. The Clair oil field lies in a smaller basin with similar types of continental sedimentation and it represents the largest remaining oilfield in the UKCS. Oil is found within Devonian-Carboniferous red beds of the Clair Group directly overlying crystalline basement rocks of the Rona Ridge. Recent work has shown that Lewisian Complex exposed in NW Scotland is excellent for assessing the nature and importance ofthe Rona Ridge basement fracture network. Here, geological evidence suggests that the Devonian rocks of the Orcadian Basin are also a suitable analogue for the Clair cover sequences to some degree and that a reappraisal of the deformation history of the Orcadian Basin is necessary to enhance the understanding of spatial and temporal characterization of structures in the subsurface. Faults, fault rocks, associated mineralization and deformation allow differentiation of three groups of structures within the Devonian Orcadian Basin: i. N-S, NW-SE, WNW-ESE trending faults showing little or no carbonate mineralisation (Group 1); ii. Metre- to kilometre-scale N-S to NE-SW trending folds and thrusts related to a highly heterogeneous regional inversion event, recognized locally throughout the field area, but especially on Orkney (Group 2); iii. dextral oblique NE-SW trending faults and sinistral E-W trending faults with widespread syn-deformational carbonate mineralisation (± base metal sulphides and bitumen) both along faults and in associated mineral veins (Group 3). Localized folds are associated with Group 3 structures due to reactivation of pre-existing faults. Crucially, these later folds are synchronous with carbonate and associated mineralisation events. Re-Os model ages of syn-deformational pyrite in two faults in the Caithness area (Dounreay) are 268.4 ± 4.9 & 266.4 ± 5.2 Ma (Permian). This is consistent with the field observation that Group 3 deformation is synchronous with the emplacement of ENE-WSW-trending lamprophyres of the Orkney Dyke Swarm (ca. 250 Ma based on K-Ar dating). Thus we suggest that Group 3 structures are synchronous with Permian rifting (NW-SE extension) which new faults and locally reactivated earlier structures. It appears that the Devonian rocks of the Orcadian basin were taken through the oil window at this time during a thermal event associated with regional alkaline basic igneous activity in Northern Britain. Scalability studies from 1D transect analysis show that fracture attributes (length and aperture)in the Devonian in Caithness (Scotland) are well described by a power-law distribution over 8 and 4 orders of magnitude. Also, 2D fracture connectivity is highly variable in the system and appears to be mainly associated with corridor structures at a large scale and on longer structures at the mesoscale. The onshore dataset shows sub-vertical fault intersections (3D) suggesting that horizontal drill orientation is favoured to vertical if these rocks were to be drilled as a reservoir

European Research in Mathematics Education I, volume 2

International audienc

Combination of rainfall thresholds and susceptibility maps for dynamic landslide hazard assessment at regional scale

We propose a methodology to couple rainfall thresholds and susceptibility maps for dynamic landslide hazard assessment at regional scale. Both inputs are combined in a purposely-built hazard matrix to get a spatially and temporally variable definition of landslide hazard: while statistical rainfall thresholds are used to accomplish a temporal forecasting with very coarse spatial resolution, landslide susceptibility maps provide static spatial information about the probability of landslide occurrence at fine spatial resolution. The test site is the Northern part of Tuscany (Italy), where a recent landslide susceptibility map and a set of recently updated rainfall thresholds are available. These products were modified and updated to meet the requirements of the proposed procedure: the susceptibility map was reclassified and the threshold set was expanded defining additional thresholds. The hazard matrix combines three susceptibility classes (S1, low susceptibility; S2 medium susceptibility; S3 high susceptibility) and three rainfall rate classes (R1, R2, R3), defining five hazard classes, from H0 (null hazard) to H4 (high hazard). A key passage of the procedure is the appropriate calibration and validation of the matrix, letting the hazard classes have a precise meaning in terms of expected consequences and hazard management. The employ of the proposed procedure in a regional warning system brings two main advantages: (i) it is possible to better hypothesize when and where landslide are expected and with which hazard degree, thus fostering a more effective hazard and risk management (e.g., setting priorities of intervention); (ii) the spatial resolution of the regional scale warning system is markedly refined because from time to time the areas where landslides are expected represent only a fraction of the alert zone