The Concept of Number: Multiplicity and Succession between Cardinality and Ordinality

Abstract

This article sets out to analyse some of the most basic elements of our number concept - of our awareness of the one and the many in their coherence with multiplicity, succession and equinumerosity. On the basis of the definition given by Cantor and the set theoretical definition of cardinal numbers and ordinal numbers provided by Ebbinghaus, a critical appraisal is given of Frege’s objection that abstraction and noticing (or disregarding) differences between entities do not produce the concept of number. By introducing the notion of subject functions, an account is advanced of the (nominalistic) reason why Frege accepted physical, kinematic and spatial properties (subject functions) of entities, but denied the ontic status of their quantitative properties (their quantitative subject function). With reference to intuitionistic mathematics (Brouwer, Weyl, Troelstra, Kreisel, Van Dalen) the primitive meaning of succession is acknowledged and connected to an analysis of what is entailed in the term ‘Gleichzahligkeit’ (‘equinumerosity’). This expression enables an analysis of the connections between ordinality and cardinality on the one hand and succession and wholeness (totality) on the other. The conceptions of mathematicians such as Frege, Cantor, Dedekind, Zermelo, Brouwer, Skolem, Fraenkel, Von Neumann, Hilbert, Bernays and Weyl, as well as the views of the philosopher Cassirer, are discussed in order to arrive at an assessment of the relation between ordinality and cardinality (also taking into account the relation between logic and arithmetic) - and on the basis of this evaluation, attention is briefly given to the definition of an ordered pair in axiomatic set theory (with reference to the set theory of Zermelo-Fraenkel) and to the defmition of an ordered pair advanced by Wiener and Kuratowski.     

Improved set‐size labeling mediates the effect of a counting intervention on children's understanding of cardinality

How does improving children's ability to label set sizes without counting affect the development of understanding of the cardinality principle? It may accelerate development by facilitating subsequent alignment and comparison of the cardinal label for a given set and the last word counted when counting that set (Mix et al., 2012). Alternatively, it may delay development by decreasing the need for a comprehensive abstract principle to understand and label exact numerosities (Piantadosi et al., 2012). In this study, preschoolers (N = 106, M_age = 4;8) were randomly assigned to one of three conditions: (a) count‐and‐label, wherein children spent 6 weeks both counting and labeling sets arranged in canonical patterns like pips on a die; (b) label‐first,wherein children spent the first 3 weeks learning to label the set sizes without counting before spending 3 weeks identical to the count‐and‐label condition; (c) print referencing control. Both counting conditions improved understanding of cardinality through increases in children's ability to label set sizes without counting. In addition to this indirect effect, there was a direct effect of the count‐and‐label condition on progress toward understanding of cardinality. Results highlight the roles of set labeling and equifinality in the development of children's understanding of number concepts.     

Fregean Abstraction, Referential Indeterminacy and the Logical Foundations of Arithmetic

In Die Grundlagen der Arithmetik, Frege attempted to introduce cardinalnumbers as logical objects by means of a second-order abstraction principlewhich is now widely known as ``Hume's Principle' (HP): The number of Fsis identical with the number of Gs if and only if F and G are equinumerous.The attempt miscarried, because in its role as a contextual definition HP fails tofix uniquely the reference of the cardinality operator ``the number of Fs'. Thisproblem of referential indeterminacy is usually called ``the Julius Caesar problem'.In this paper, Frege's treatment of the problem in Grundlagen is critically assessed. In particular, I try to shed new light on it by paying special attention to the framework of his logicism in which it appears embedded. I argue, among other things, that the Caesar problem, which is supposed to stem from Frege's tentative inductive definition of the natural numbers, is only spurious, not genuine; that the genuine Caesar problem deriving from HP is a purely semantic one and that the prospects of removing it by explicitly defining cardinal numbers as objects which are not classes are presumably poor for Frege. I conclude by rejecting two closely connected theses concerning Caesar put forward by Richard Heck: (i) that Frege could not abandon Axiom V because he could not solve the Julius Caesar problem without it; (ii) that (by his own lights) his logicist programme in Grundgesetze der Arithmetik failed because he could not overcome that problem.     

Partial knowledge in the development of number word understanding

A common measure of number word understanding is the give‐N task. Traditionally, to receive credit for understanding a number, N, children must understand that N does not apply to other set sizes (e.g. a child who gives three when asked for ‘three’ but also when asked for ‘four’ would not be credited with knowing ‘three’). However, it is possible that children who correctly provide the set size directly above their knower level but also provide that number for other number words (‘N + 1 givers’) may be in a partial, transitional knowledge state. In an integrative analysis including 191 preschoolers, subset knowers who correctly gave N + 1 at pretest performed better at posttest than did those who did not correctly give N + 1. This performance was not reflective of ‘full’ knowledge of N + 1, as N + 1 givers performed worse than traditionally coded knowers of that set size on separate measures of number word understanding within a given timepoint. Results support the idea of graded representations (Munakata, Trends in Cognitive Sciences, 5, 309–315, 2001.) in number word development and suggest traditional approaches to coding the give‐N task may not completely capture children's knowledge.     

Frege's Cardinals Do Not Always Obey Hume's Principle

Hume's Principle, dear to neo-Logicists, maintains that equinumerosity is both necessary and sufficient for sameness of cardinal number. All the same, Whitehead demonstrated in Principia Mathematica's logic of relations (where non-homogeneous relations are allowed) that Cantor's power-class theorem entails that Hume's Principle admits of exceptions. Of course, Hume's Principle concerns cardinals and in Principia's ‘no-classes’ theory cardinals are not objects in Frege's sense. But this paper shows that the result applies as well to the theory of cardinal numbers as objects set out in Frege's Grundgesetze. Though Frege did not realize it, Cantor's power-theorem entails that Frege's cardinals as objects do not always obey Hume's Principle.     

Attaching meaning to the number words: contributions of the object tracking and approximate number systems

Children's understanding of the quantities represented by number words (i.e., cardinality) is a surprisingly protracted but foundational step in their learning of formal mathematics. The development of cardinal knowledge is related to one or two core, inherent systems – the approximate number system (ANS) and the object tracking system (OTS) – but whether these systems act alone, in concert, or antagonistically is debated. Longitudinal assessments of 198 preschool children on OTS, ANS, and cardinality tasks enabled testing of two single‐mechanism (ANS‐only and OTS‐only) and two dual‐mechanism models, controlling for intelligence, executive functions, preliteracy skills, and demographic factors. Measures of both OTS and ANS predicted cardinal knowledge in concert early in the school year, inconsistent with single‐mechanism models. The ANS but not the OTS predicted cardinal knowledge later in the school year as well the acquisition of the cardinal principle, a critical shift in cardinal understanding. The results support a Merge model, whereby both systems initially contribute to children's early mapping of number words to cardinal value, but the role of the OTS diminishes over time while that of the ANS continues to support cardinal knowledge as children come to understand the counting principles.     

Is there really a link between exact‐number knowledge and approximate number system acuity in young children?

Although everyone perceives approximate numerosities, some people make more accurate estimates than others. The accuracy of this estimation is called approximate number system (ANS) acuity. Recently, several studies have reported that individual differences in young children's ANS acuity are correlated with their knowledge of exact numbers such as the word ‘six’ (Mussolin et al., 2012, Trends Neurosci. Educ., 1, 21; Shusterman et al., 2011, Connecting early number word knowledge and approximate number system acuity; Wagner & Johnson, 2011, Cognition, 119, 10; see also Abreu‐Mendoza et al., 2013, Front. Psychol., 4, 1). This study argues that this correlation should not be trusted. It seems to be an artefact of the procedure used to assess ANS acuity in children. The correlation arises because (1) some experimental designs inadvertently allow children to answer correctly based on the size (rather than the number) of dots in the display and/or (2) young children with little exact‐number knowledge may not understand the phrase ‘more dots’ to mean numerically more. When the task is modified to make sure that children respond on the basis of numerosity, the correlation between ANS acuity and exact‐number knowledge in normally developing children disappears.     

Greater response cardinality indirectly reduces confidence

Koriat's Self-Consistency Model of subjective confidence proposes that the consistency and accessibility with which a response is retrieved are cues to confidence in the accuracy of that response. The Self-Consistency Model, however, has only been assessed using two-alternative forced-choice questions. Consideration of open-ended questions suggested that response cardinality—the number of unique response options that might come to mind—should directly reduce response consistency, and thus indirectly reduce confidence. Australian undergraduate students (N?=?389) completed a 20-item open-ended general-knowledge test with confidence ratings attached to each answer. Replicating previous Self-Consistency Model findings, but in an open-ended format for the first time, consistency and accessibility were independent positive predictors of confidence, though the prediction of accessibility did not reach statistical significance. Extending the model, the number of unique responses given by participants, a measure of response cardinality, was found to be a strong negative predictor of consistency and indirect negative predictor of confidence. Implications of the results therefore include the use of simple strategies such as prompting or providing varying numbers of response options as potentially effective for manipulating cardinality and thus confidence.     

Neural correlates of belief- and desire-reasoning in 7- and 8-year-old children: an event-related potential study

Theory of mind requires belief‐ and desire‐understanding. Event‐related brain potential (ERP) research on belief‐ and desire‐reasoning in adults found mid‐frontal activations for both desires and beliefs, and selective right‐posterior activations only for beliefs. Developmentally, children understand desires before beliefs; thus, a critical question concerns whether neural specialization for belief‐reasoning exists in childhood or develops later. Neural activity was recorded as 7‐ and 8‐year‐olds (N = 18) performed the same diverse‐desires, diverse‐beliefs, and physical control tasks used in a previous adult ERP study. Like adults, mid‐frontal scalp activations were found for belief‐ and desire‐reasoning. Moreover, analyses using correct trials alone yielded selective right‐posterior activations for belief‐reasoning. Results suggest developmental links between increasingly accurate understanding of complex mental states and neural specialization supporting this understanding.     

Older (but not younger) preschoolers understand that knowledge differs between people and across time

We examined 3‐ to 5‐year‐olds' understanding of general knowledge (e.g., knowing that clocks tell time) by investigating whether (1) they recognize that their own general knowledge has changed over time (i.e., they knew less as babies than they know now), and (2) such intraindividual knowledge differences are easier/harder to understand than interindividual differences (i.e., Do preschoolers understand that a baby knows less than they do?). Forty‐eight 3‐ to 5‐year‐olds answered questions about their current general knowledge (‘self‐now’), the general knowledge of a 6‐month‐old (‘baby‐now’), and their own general knowledge at 6 months (‘self‐past’). All age groups were significantly above chance on the self‐now questions, but only 5‐year‐olds were significantly above chance on the self‐past and baby‐now questions. Moreover, children's performance on the baby‐now and self‐past questions did not differ. Our findings suggest that younger preschoolers do not fully appreciate that their past knowledge differs from their current knowledge, and that others may have less knowledge than they do. We situate these findings within the research on knowledge understanding, more specifically, and cognitive development, more broadly.     

Children with autism are impaired in the understanding of teaching

Children learn novel information using various methods, and one of the most common is human pedagogical communication or teaching – the purposeful imparting of information from one person to another. Neuro‐typically developing (TD) children gain the ability to recognize and understand teaching as a core method for acquiring knowledge from others. However, it is not known when children with Autism Spectrum Disorder (ASD) acquire the ability to recognize and understand teaching. This study (total N = 70) examined whether children with ASD recognize the two central elements that define teaching: (1) that teaching is an intentional activity; and (2) that teaching requires a knowledge difference between teacher and learner. Theory of mind understanding was also tested. Compared to individually matched TD children, high cognitively functioning children with ASD were impaired in their comprehension of both components of teaching understanding, and their performance was correlated with theory of mind understanding. These findings could have broad implications for explaining learning in children with autism, and could help in designing more effective interventions, which could ultimately lead to improved learning outcomes for everyday life skills, school performance, health, and overall well‐being.     

Find the picture of eight turtles: a link between children's counting and their knowledge of number word semantics

An essential part of understanding number words (e.g., eight) is understanding that all number words refer to the dimension of experience we call numerosity. Knowledge of this general principle may be separable from knowledge of individual number word meanings. That is, children may learn the meanings of at least a few individual number words before realizing that all number words refer to numerosity. Alternatively, knowledge of this general principle may form relatively early and proceed to guide and constrain the acquisition of individual number word meanings. The current article describes two experiments in which 116 children (2½- to 4-year-olds) were given a Word Extension task as well as a standard Give-N task. Results show that only children who understood the cardinality principle of counting successfully extended number words from one set to another based on numerosity—with evidence that a developing understanding of this concept emerges as children approach the cardinality principle induction. These findings support the view that children do not use a broad understanding of number words to initially connect number words to numerosity but rather make this connection around the time that they figure out the cardinality principle of counting.     

K‐means clustering: A half‐century synthesis

This paper synthesizes the results, methodology, and research conducted concerning the K‐means clustering method over the last fifty years. The K‐means method is first introduced, various formulations of the minimum variance loss function and alternative loss functions within the same class are outlined, and different methods of choosing the number of clusters and initialization, variable preprocessing, and data reduction schemes are discussed. Theoretic statistical results are provided and various extensions of K‐means using different metrics or modifications of the original algorithm are given, leading to a unifying treatment of K‐means and some of its extensions. Finally, several future studies are outlined that could enhance the understanding of numerous subtleties affecting the performance of the K‐means method.     

IPSEITY,ALTERITY, AND COMMUNITY: THE TRI‐UNITY OF MAYA THERAPEUTIC HEALING

Taking K'iche’ Maya therapeutic consultations in Guatemala as its focus, this essay explores some astonishing indigenous accounts of “healing‐at‐a‐distance” and “pain passing” between healers and wellness‐seekers. Rather than exoticizing or dismissing such reports, we attempt to understand what it means to conceive of the bodily boundaries of healers and wellness‐seekers (self and other) as sympathetically defiable and transgressable in healing. Within the moral space of K'iche’ healing, when one cares to feel, if one dares to feel with another or others, the experiential space between healer and wellness‐seeker is transformed as the alterity (otherness) of what is felt and who feels becomes (through a sympathy in ipseity) but one thing. I argue that Maya therapeutic healing may be seen as a tri‐unity, involving a movement from an enfolded illness experience (alterity) to an unfolding sickness experience (ipseity), passing through empathy until participants together arrive at sympathy (community) to experience healing.     

What You Know When You Know an Answer to a Question

A significant argument for the claim that knowing‐wh is knowing‐that, implicit in much of the literature, including Stanley and Williamson (2001), is spelt out and challenged. The argument includes the assumption that a subject's state of knowing‐wh is constituted by their involvement in a relation with an answer to a question. And it involves the assumption that answers to questions are propositions or facts. One of Lawrence Powers’ counterexamples to the conjunction of these two assumptions is developed, responses to it are rebutted, and the possibility of rejecting the second rather than the first of these assumptions is explored briefly.     

Random word generation reveals spatial encoding of syllabic word length

Existing random number generation studies demonstrate the presence of an embodied attentional bias in spontaneous number production corresponding to the horizontal Mental Number Line: Larger numbers are produced on right-hand turns and smaller numbers on left-hand turns (Loetscher et al.,2008, Curr. Biol., 18, R60). Furthermore, other concepts were also shown to rely on horizontal attentional displacement (Di Bono and Zorzi, 2013, Quart. J. Exp. Psychol., 66, 2348). In two experiments, we used a novel random word generation paradigm combined with two different ways to orient attention in horizontal space: Participants randomly generated words on left and right head turns (Experiment 1) or following left and right key presses (Experiment 2). In both studies, syllabically longer words were generated on right-hand head turns and following right key strokes. Importantly, variables related to semantic magnitude or cardinality (whether the generated words were plural-marked, referred to uncountable concepts, or were associated with largeness) were not affected by lateral manipulations. We discuss our data in terms of the ATOM (Walsh, 2015, The Oxford handbook of numerical cognition, 552) which suggests a general magnitude mechanism shared by different conceptual domains.