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Traditional theories of how children learn the positive integers start from infants' abilities in detecting the quantity of physical objects. Our target article examined this view and found no plausible accounts of such development. Most of our commentators appear to agree that no adequate developmental theory is presently available, but they attempt to hold onto a role for early enumeration. Although some defend the traditional theories, others introduce new basic quantitative abilities, new methods of transformation, or new types of end states. A survey of these proposals, however, shows that they do not succeed in bridging the gap to knowledge of the integers. We suggest that a better theory depends on starting with primitives that are inherently structural and mathematical.
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 children's 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.
However much we may disparage deduction, it cannot be denied that the laws established by induction are not enough.
Frege (1884/1974, p. 23)
At the yearly proseminar for first-year graduate students at Northwestern, we presented some evidence that reasoning draws on separate cognitive systems for assessing deductive versus inductive arguments (Rips, 2001a, 2001b). In the experiment we described, separate groups of participants evaluated the same set of arguments for deductive validity or inductive strength. For example, one of the validity groups decided whether the conclusions of these arguments necessarily followed from the premises, while one of the strength groups decided how plausible the premises made the conclusions. The results of the study showed that the percentages of “yes” responses (“yes” the argument is deductively valid or “yes” the argument is inductively strong) were differently ordered for the validity and the strength judgments. In some cases, for example, the validity groups judged Argument A to be valid more often than Argument B, but the strength groups judged B inductively strong more often than A. Reversals of this sort suggest that people do not just see arguments as ranging along a single continuum of convincingness or probability but instead employ different methods when engaged in deductive versus inductive reasoning.
Earlier imaging evidence by Goel, Gold, Kapur, and Houle (1997) and by Osherson et al. (1998) had implicated separate brain regions when participants evaluated arguments for validity versus plausibility, and these earlier data had inspired our own experiment. All these results cast doubt on the view that there's a homogeneous “analytic” reasoning system responsible for correctly solving deductive and probabilistic problems.
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