Hofweber (
Ontology and the ambitions of metaphysics
, Oxford University Press, 2016) argues for a thesis he calls “internalism” with respect to natural number discourse: no expressions purporting to ...refer to natural numbers in fact refer, and no apparent quantification over natural numbers actually involves quantification over natural numbers as objects. He argues that while internalism leaves open the question of whether other kinds of abstracta exist, it precludes the existence of natural numbers, thus establishing what he calls “restricted nominalism” about natural numbers. We argue that Hofweber’s internalism fails to establish restricted nominalism. Not only is his primary argument for restricted nominalism invalid, the analysis of quantification proposed threatens to collapse internalism into either a traditional form of error theory or realism.
Abstract
This research concerns the estimation of latent linear or polychoric correlations from fuzzy frequency tables. Fuzzy counts are of particular interest to many disciplines including social ...and behavioral sciences and are especially relevant when observed data are classified using fuzzy categories—as for socioeconomic studies, clinical evaluations, content analysis, inter-rater reliability analysis—or when imprecise observations are classified into either precise or imprecise categories—as for the analysis of ratings data or fuzzy-coded variables. In these cases, the space of count matrices is no longer defined over naturals and, consequently, the polychoric estimator cannot be used to accurately estimate latent linear correlations. The aim of this contribution is twofold. First, we illustrate a computational procedure based on generalized natural numbers for computing fuzzy frequencies. Second, we reformulate the problem of estimating latent linear correlations from fuzzy counts in the context of expectation–maximization-based maximum likelihood estimation. A simulation study and two applications are used to investigate the characteristics of the proposed method. Overall, the results show that the fuzzy EM-based polychoric estimator is more efficient to deal with imprecise count data as opposed to standard polychoric estimators that may be used in this context.
For a set of nonnegative integers S let RS(n) denote the number of unordered representations of the integer n as the sum of two different terms from S. In this paper we focus on partitions of the ...natural numbers into two sets affording identical representation functions. We solve a recent problem of Chen and Lev.
Research on rational numbers suggests that adults experience more difficulties in understanding the numerical magnitude of rational than natural numbers. Within rational numbers, the numerical ...magnitude of fractions has been found to be more difficult to understand than that of decimals. Using a number line estimation (NLE) task, the current study investigated two sources of difficulty in adults’ numerical magnitude understanding: number type (natural vs rational) and structure of the notation system (place-value-based vs non-place-value-based). This within-subjects design led to four conditions: natural numbers (natural/place-value-based), decimals (rational/place-value-based), fractions (rational/non-place-value-based), and separated fractions (natural/non-place-value-based). In addition to percentage absolute error (PAE) and response times, we collected eye-tracking data. Results showed that participants estimated natural and place-value-based notations more accurately than rational and non-place-value-based notations, respectively. Participants were also slower to respond to fractions compared with the three other notations. Consistent with the response time data, eye-tracking data showed that participants spent more time encoding fractions and re-visited them more often than the other notations. Moreover, in general, participants spent more time positioning non-place-value-based than place-value-based notations on the number line. Overall, the present study contends that when both sources of difficulty are present in a notation (i.e., both rational and non-place-value-based), adults understand its numerical magnitude less well than when there is only one source of difficulty present (i.e., either rational or non-place-value-based). When no sources of difficulty are present in a notation (i.e., both natural and place-value-based), adults have the strongest understanding of its numerical magnitude.
This article is an excerpt from an academic master's research that adopts the notion of didactic transposition by Yves Chevallard and aims to analyze aspects related to the internal didactic ...transposition process of the natural numbers division operation in elementary school. To support the analysis of the mathematics classes of three teachers, a study of the legal documents of Brazil and the State of Pernambuco was carried out, thus being analysis of textbooks for the 4th, 5th and 6th years and interviews with the teachers who work in those school years. Within the classes, the focus of analysis was the different meanings of the division, the calculation procedures, the treatment given to the rest and the interpretation of the results obtained, addressed or not into the observed classes. When comparing the knowledge expected to be taught and the knowledge effectively taught, changes, deformations and gaps were found, which could cause difficulties in the process of learning the operation.
For a very general principally polarized complex abelian 3-fold, the Chow
group of algebraic cycles is infinite modulo every prime number. In
particular, this gives the first examples of complex ...varieties with
infinite Chow groups modulo 2.
Topological representations Kwela, Adam; Sabok, Marcin
Journal of mathematical analysis and applications,
02/2015, Letnik:
422, Številka:
2
Journal Article
Recenzirano
Odprti dostop
This paper studies the combinatorics of ideals which recently appeared in ergodicity results for analytic equivalence relations. The ideals have the following topological representation. There are a ...separable metrizable space X, a σ-ideal I on X and a dense countable subset D of X such that the ideal consists of those subsets of D whose closure belongs to I. It turns out that this definition is independent of the choice of D. We show that an ideal is of this form if and only if it is dense and countably separated. The latter is a variation of a notion introduced by Todorčević for gaps. As a corollary, we get that this class is invariant under the Rudin–Blass equivalence. This also implies that the space X can be always chosen to be compact so that I is a σ-ideal of compact sets. We compute the possible descriptive complexities of such ideals and conclude that all analytic equivalence relations induced by such ideals are Π30. We also prove that a coanalytic ideal is an intersection of ideals of this form if and only if it is weakly selective.
We prove a new upper bound for diagonal two-colour Ramsey numbers, showing that there exists a constant C such that $r(k+1,k+1)\leq k^{-C\ \text{log}\ k/\text{log log}\ k}\left( \matrix 2k \\ k ...\endmatrix \right)$ .
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