Titles and Abstracts

 

Emergence of Number Conference (June 27-29, OSU): Titles and Abstracts

 

Jessica Cantlon (Psych, Rochester)

Title: Math, Monkeys, and the Developing Brain

Abstract: In this talk I discuss the fundamental and universal perception of number in humans, its evolutionary homolog in non-human primates, the primitive logic of mathematics, and its neural origin in humans. These studies from multiple populations and methods are important for identifying the interacting constraints on human thought from evolution, development, and the structure of the human brain.

 

David Geary (Psych, Missouri)

Title: Early Development of Number Knowledge

Abstract: An overview will be provided of a preschool through first-grade longitudinal study of children’s quantitative development, with a focus on their understanding of the cardinal value of number words. The factors that influence the early emergence of this core mathematical concept will be discussed, as will the relation between the age at which children achieve this conceptual insight and their subsequent number development and mathematics achievement.

 

Michael Glanzberg (Phil, Northwestern)

Title: The Cognitive Roots of Adjectival Meaning

Abstract: In this paper, I illustrate a way that work in cognitive psychology can fruitfully interact with truth-conditional semantics. A widely held view takes the meanings of gradable adjectives to be measure functions, which map objects to degrees on a scale. Scales come equipped with dimensions that fix what the degrees are. Following Bartsch and Vennemann, I observe that this allows dimensions to play the role of lexical roots, that provide the distinctive contents for each lexical entry. I review evidence that the grammar provides a limited range of scale structures, presumably dense linear orderings with a limited range of topological properties. I then turn to how the content of the root can be fixed. In the verbal domain, there is evidence suggesting roots are linked to concepts. In many cases for adjectives, it is not concepts but approximate magnitude representation systems that fix root contents. However, these magnitude representation systems are approximate or analog, and do not provide precise values. I argue that the roots of adjectives like these provide a weak, discrimination-based constraint on a grammatically fixed scale structure. Other adjectives can find concepts to fix roots, which can support a well-known equivalence class construction which can fix precise values on a scale. I conclude that though adjectives have a uniform truth-conditional semantics, they show substantial differences in the cognitive sources of their root meanings. This shows that there are (at least) two sub-classes of adjectives, with roots fixed by different mechanisms and with different degrees of precision, and showing very different cognitive properties.

 

 

Chris Kennedy (Ling, Chicago) with Kristen Syrett (Ling, Rutgers)

Title: Numerals denote degree quantifiers: Evidence from child language

Abstract: A large body of work in both the theoretical and experimental literature suggests that upper bound implications in simple sentences with bare numerals are entailments arising from the semantics of the numeral, rather than scalar implicatures of the sort seen with other scalar terms. However, not all semantic analyses of numerals designed to introduce upper bounding entailments in simple sentences make the same predictions about upper bounding implications in other sentences.  Here we focus on the case of upper bounded construals of numerals embedded under existential root modals, which are derived as entailments by a semantic analysis of numerals as generalized quantifiers over degrees, but can only be derived as scalar implicatures by other semantic analyses. We provide experimental evidence from child language that upper bounding implications in such cases are in fact entailments and not implicatures, thereby providing support for the degree quantifier analysis. We conclude with a discussion of the emergence of numerals in child language.

 

Øystein Linnebo (Phil, Oslo)

Title: An Ordinal Conception of the Natural Numbers

Abstract: According to the ordinal conception, the natural numbers are individuated by their position in the number sequence. A version of this conception of developed and defended—along broadly abstractionist lines. This conception is argued to be superior to the competing cardinal conception, which currently enjoys more support among proponents of an abstractionist approach to mathematics.

 

John Opfer (Psychology, Ohio State)

Title: The Predicament of Quantity:  Estimating the limits of the Analog Magnitude System.

Abstract: To follow…

 

Jessica Rett (Ling, UCLA)

Title: Individual/Degree Polysemy

Abstract: There are a lot of constructions in which determiner phrases (DPs) can denote quantities, not just individuals: this polysemy has been observed in relative clauses (so-called ‘amount relatives’); in pseudopartitives (e.g. the bottle of wine); measure phrases (e.g. two cups of wine); and how-many questions (e.g. How many books must Jane read?). Existing accounts of this polysemy are construction-specific; I’ve argued (Rett 2014) that we need a more general account for two reasons: 1) Any DP can denote an individual or a degree (and the interpretation conditions agreement): compare Many guests are drunk to Many guests is more than Bill had anticipated. And 2) The degree interpretation isn’t restricted to quantities, but to dimensions of measurement that are monotonic on the part-whole structure of the denoted individual (Schwarzschild 2006). My semantic account of this individual/degree polysemy involves a standard null measurement operator, which effects a homomorphism between the denoted individual and its salient dimension of measurement; this homomorphism is only informative if the dimension is monotonic. I also present recent processing work (Grant et al. to appear) suggesting that this polysemy carries the sort of processing cost that we’d expect from other types of polysemy, but that it differs in strength and direction across types of DPs.

 

Lance Rips (Psych, Northwestern)

Title: Possible Number Systems

Abstract: Number systems—such as the natural numbers and the reals—play a foundational role in mathematics, but these systems present hurtles for students. The first part of this talk looks at the claim that children in kindergarten and the early grades have a concept of the integers 1-100 that is dominated by an approximate number system. According to this theory, children’s understanding of the integers is systematically distorted because the approximate number system’s measure is nonlinearly (e.g., logarithmically) related to the numbers’ true magnitude. I’ll argue that although children may have an approximate number system, it does not exhaust their knowledge of the positive integers’ structure: They believe that successive integers are linearly related. If the right math structures are in place, though, why do students have trouble learning new number systems, such as the rational numbers and the complex numbers? One possibility is that they have incorrect preconceptions about what a number system can be, based on earlier knowledge. The second part of the talk presents evidence that even college students, who have a generally correct view of number systems, may retain some incorrect ideas about their properties.

 

Susan Rothstein (Linguistics, Bar Illan)

Title: Counting, Plurality and Portions

Abstract: To follow…

 

Richard Samuels, Stewart Shapiro, and Eric Snyder (Phil, Ohio State)

Title: On the Acquisition of number concepts: A new puzzle

Abstract: In this talk, we sketch a new puzzle arising from an apparent tension in number concept acquisition and the meanings of number expressions, which we call ‘The Acquisition Puzzle’. It runs roughly as follows. It is widely accepted among developmental psychologists that children acquire cardinality concepts, i.e. the sorts of concepts employed when counting, well before acquiring basic arithmetic concepts, i.e. the sorts of concepts employed when doing basic arithmetic. On the other hand, it is widely accepted by semanticists that the meanings of number expressions employed in cardinality statements at least implicitly reference numbers, and that numbers are the sorts of things referenced by numerals in basic arithmetic statements. This seems to suggest that acquiring cardinality concepts requires prior possession of arithmetic concepts, just as e.g. acquiring the concept BACHELOR seemingly requires prior possession of the concept MALE. Thus, the Puzzle is this: How can children acquire cardinality concepts before basic arithmetic concepts if possessing the former presupposes prior possession of the latter? The purpose of the talk is to motivate the assumptions underlying The Acquisition Puzzle, as well as to consider certain seemingly plausible ways out. These include adopting (i) a non-referential semantics for numerals, (ii) a referential semantics for numerals according to which numbers are derived (via type-shifting) from the meanings of cardinals, (iii) nativism with respect to the Dedekind-Peano Axioms or the Approximate Number System, or (iv) a certain structuralist-inspired account on which children acquire both cardinality and basic arithmetic concepts via abstraction on the basis of “intransitive counting” (reciting the numerals in their usual order). Ultimately, though we would like option (iv) to work, none of these is without significant challenges.

 

Barbara Sarnecka (Psyc, UC Irvine)

Title: Numbers, with and without language

Abstract: After about 20 years of thinking about language and number, and I’ve come to believe three, mutually contradictory things. I believe that the mental representation of numbers is (1) unrelated to language; (2) related to particular languages; and (3) related to language in general. In my talk, I will explain how I think all of these statements are true, depending on what we mean by ‘numbers.’

 

Greg Scontras (Ling, UC Irvine)

Title: On the semantics of number morphology

Abstract: In this talk, I develop a semantic account of morphological number in the presence of numerals. In addition to accounting for number morphology on basic nouns like apple in English, the approach extends to cover data from two seemingly disparate domains: 1) number marking on measure terms like kilo, which is determined by the numeral co-occurring with these terms: one kilo of apples vs. two kilos of apples; and 2) cross-linguistic variation in patterns of number marking: numerals other than ‘one’ obligatorily combining with plural-marked nouns (e.g., English), all numerals obligatorily combining with singular (i.e., unmarked) nouns (e.g., Turkish, Hungarian), and numerals optionally combining with either singular or plural nouns (e.g., Western Armenian). Building on the presuppositional approach to morphological number from Sauerland (2003), all of the data considered receive an account once we assume variation in the selection of the measure relevant to the one-ness presupposition of the morphological singular form. Different classes of words and different languages employ diverging criteria in the determination of whether or not something measures 1 for the purpose of morphological singularity.