Jennifer Asmuth

From numerical concepts to concepts of number

Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for childrens understanding of number terms do not necessarily tap these concepts. (2) True concepts of number do appear, however, when children are able to understand generalizations over all numbers; for example, the principle of additive commutativity (a+b=b+a). Theories of how children learn such principles usually rely on a process of mapping from physical object groupings. But both experimental results and theoretical considerations imply that direct mapping is insufficient for acquiring these principles. We suggest instead that children may arrive at natural numbers and arithmetic in a more top-down way, by constructing mathematical schemas.

Relational categories are more mutable than entity categories

Across three experiments, we explore differences between relational categories—whose members share common relational patterns—and entity categories, whose members share common intrinsic properties. Specifically, we test the claim that relational concepts are more semantically mutable in context, and therefore less stable in memory, than entity concepts. We compared memory for entity nouns and relational nouns, tested either in the same context as at encoding or in a different context. We found that (a) participants show better recognition accuracy for entity nouns than for relational nouns, and (b) recognition of relational nouns is more impaired by a change in context than is recognition of entity nouns. We replicated these findings even when controlling for factors highly correlated with relationality, such as abstractness–concreteness. This suggests that the contextual mutability of relational concepts is due to the core semantic property of conveying relational structure and not simply to accompanying characteristics such as abstractness. We note parallels with the distinction between nouns and verbs and suggest implications for lexical and conceptual structure. Finally, we relate these patterns to proposals that a deep distinction exists between words with an essentially referential function and those with a predicate function.

Inductive Reasoning: Mathematical Induction and Induction in Mathematics

property of the natural number system that is responsible for the correctness of math induction. It is still possible, of course, that math induction picks up quasi-inductive support from its fruitfulness in proving further theorems, but in this respect it doesn’t differ from any other math axiom. Although we think that math induction doesn’t threaten the distinction between deductive and inductive reasoning, there is a related issue about generalization in math that might. Math proofs often proceed by selecting an “arbitrary instance” from a domain, showing that some property is true of this instance, and then generalizing to all the domain’s members. For this universal generalization to work, the instance in question must be an abstraction or stand-in (an “arbitrary name” or variable) for all relevant individuals, and there is no real concern that such a strategy is not properly deductive. However, there’s psychological evidence that students don’t always recognize the difference between such an abstraction and an arbitrarily selected exemplar. Sometimes, in fact, students use exemplars in their proofs (and evaluate positively proofs that contain exemplars) that don’t even look arbitrary but are simply convenient, perhaps because the exemplars lend themselves to concrete arguments that are easy to understand. In these cases, students are using an inductive strategy, since the exemplar can at most increase their confidence in the to-be-proved proposition. It’s no news, of course, that people make mistakes in math. And it’s also no news that ordinary induction has a role to play in math, especially in the context of discovery. The question here is whether these inductive intrusions provide evidence that the deduction/induction split is psychologically untenable. Does the use of arbitrary and not-so-arbitrary instances show that people have a single type of reasoning mechanism that delivers conclusions that are quantitatively stronger or weaker, but not qualitatively inductive versus deductive? We’ve considered one way in which this might be the case. Perhaps people look for counterexamples, continuing their search through increasingly arbitrary (i.e., haphazard or atypical) cases until they’ve found such a counterexample or have run out of steam. The longer the search, the more secure the conclusion. We’ve seen, however, that this procedure doesn’t Mathematical Induction and Induction in Mathematics / 25 extend to deductively valid proofs; no matter how obscure the instance, it will still have an infinite number of properties that prohibit you from generalizing from it. It is possible to contend that this procedure is nevertheless all that people have at their disposal—that they can never ascend from their search for examples and counterexamples to a deductively adequate method. But although the evidence on proof evaluation paints a fairly bleak picture of students’ ability to recognize genuine math proofs, the existence of such proofs shows they are not completely out of reach. Mathematical Induction and Induction in Mathematics / 26 References Dedekind, R. (1963). The nature and meaning of numbers (W. W. Beman, Trans.). In Essays on the theory of numbers (pp. 31-115). New York: Dover. (Original work published 1888) Eliaser, N. M. (2000). What constitutes a mathematical proof? Dissertation Abstracts International, 60 (12), 6390B. (UMI No. AAT 9953274) Evans, J. St. B. T., Newstead, S. E., & Byrne, R. M. J. (1993). Human reasoning. Hillsdale, NJ: Erlbaum. Fallis, D. (1997). The epistemic status of probabilistic proofs. 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Metaphoric extension, relational categories, and abstraction

ABSTRACT We propose that concepts exist along a continuum of abstraction, from highly concrete to highly abstract, and we explore a critical kind of abstract category: relational abstractions. We argue that these relational categories emerge gradually from concrete concepts through a process of progressive analogical abstraction that renders their common structure more salient. This account is supported by recent findings in historical linguistics, language acquisition and neuroscience. We suggest that analogical abstraction provides a major route for the development of abstractions in language and cognition.