Does the number sense represent number?

On a now orthodox view, humans and many other animals are endowed with a “number sense”, or approximate number system (ANS), that represents number. Recently, this orthodox view has been subject to numerous critiques, with critics maintaining either that numerical content is absent altogether, or else that some primitive analog of number (‘numerosity’) is represented as opposed to number itself. We distinguish three arguments for these claims – the arguments from congruency, confounds, and imprecision – and show that none succeed. We then highlight positive reasons for thinking that the ANS genuinely represents numbers. The upshot is that proponents of the orthodox view should not feel troubled by recent critiques of their position

Engaged learning

Engaged learning commits itself to community development and an active pedagogy as the state continues to withdraw funding from communities. Community engagement is a teaching, research and change mechanism for students who use ethnographic methods and experiential learning to practice citizenship and rehearse their roles in professional settings. Engaged learning provides the opportunity to explore “alternative moral frames for academic work”, for pedagogical innovation and for resistance against neoliberalism.Engaged learning commits itself to community development and an active pedagogy as the state continues to withdraw funding from communities. Community engagement is a teaching, research and change mechanism for students who use ethnographic methods and experiential learning to practice citizenship and rehearse their roles in professional settings. Engaged learning provides the opportunity to explore “alternative moral frames for academic work”, for pedagogical innovation and for resistance against neoliberalism

Grid graphs, Gorenstein polytopes, and domino stackings

We examine domino tilings of rectangular boards, which are in natural bijection with perfect matchings of grid graphs. This leads to the study of their associated perfect matching polytopes, and we present some of their properties, in particular, when these polytopes are Gorenstein. We also introduce the notion of domino stackings and present some results and several open questions. Our techniques use results from graph theory, polyhedral geometry, and enumerative combinatorics.Comment: 14 pages, 6 figures, uses graphs packag

The number sense represents (rational) numbers

On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system, that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique – the arguments from congruency, confounds, and imprecision – and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds or exotic substitutes for number, such as “numerosities” or “quanticals,” as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind of number being represented. In response, we propose that the ANS represents not only natural numbers, but also non-natural rational numbers. It does not represent irrational numbers, however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research

Canine Gastric Carcinomas: A Histopathological and Immunohistochemical Study and Similarities with the Human Counterpart

Canine gastric carcinoma (CGC) affects both sexes in relatively equal proportions, with a mean age of nine years, and the highest frequency in Staffordshire bull terriers. The most common histological subtype in 149 CGC cases was the undifferentiated carcinoma. CGCs were associated with increased chronic inflammation parameters and a greater chronic inflammatory score when Helicobacter spp. were present. Understanding the molecular pathways of gastric carcinoma is challenging. All markers showed variable expression for each subtype. Expression of the cell cycle regulator 14-3-3σ was positive in undifferentiated, tubular and papillary carcinomas. This demonstrates that 14-3-3σ could serve as an immunohistochemical marker in routine diagnosis and that mucinous, papillary and signet-ring cell (SRC) carcinomas follow a 14-3-3σ independent pathway. p16, another cell cycle regulator, showed increased expression in mucinous and SRC carcinomas. Expression of the adhesion molecules E-cadherin and CD44 appear context-dependent, with switching within tumor emboli potentially playing an important role in tumor cell survival, during invasion and metastasis. Within neoplastic emboli, acinar structures lacked expression of all markers, suggesting an independent molecular pathway that requires further investigation. These findings demonstrate similarities and differences between dogs and humans, albeit further clinicopathological data and molecular analysis are required

On positivity of Ehrhart polynomials

Ehrhart discovered that the function that counts the number of lattice points in dilations of an integral polytope is a polynomial. We call the coefficients of this polynomial Ehrhart coefficients, and say a polytope is Ehrhart positive if all Ehrhart coefficients are positive (which is not true for all integral polytopes). The main purpose of this article is to survey interesting families of polytopes that are known to be Ehrhart positive and discuss the reasons from which their Ehrhart positivity follows. We also include examples of polytopes that have negative Ehrhart coefficients and polytopes that are conjectured to be Ehrhart positive, as well as pose a few relevant questions.Comment: 40 pages, 7 figures. To appear in in Recent Trends in Algebraic Combinatorics, a volume of the Association for Women in Mathematics Series, Springer International Publishin