The meaning of 'most' can be described in many ways. We offer a framework for distinguishing semantic descriptions, interpreted as psychological hypotheses that go beyond claims about sentential truth conditions, and an experiment that tells against an attractive idea: 'most' is understood in terms of one-to-one correspondence. Adults evaluated 'Most of the dots are yellow', as true or false, on many trials in which yellow dots and blue dots were displayed for 200 ms. Displays manipulated the ease of using a (...) 'one-to-one with remainder' strategy, and a strategy of using the Approximate Number System to compare of (approximations of) cardinalities. Interpreting such data requires care in thinking about how meaning is related to verification. But the results suggest that 'most' is understood in terms of cardinality comparison, even when counting is impossible. (shrink)

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. (shrink)

We see no grounds for insisting that, because the concept natural number is abstract, its foundations must be innate. It is possible to specify domain general learning processes that feed into more abstract concepts of numerical infinity. By neglecting the messiness of children's slow acquisition of arithmetical concepts, Rips et al. present an idealized, unnecessarily insular, view of number development.

Neurosciences and cognitive sciences provide us with myriad empirical findings that shed light on hypothesised primitive numerical processes in the brain and in the mind. Yet, the hypotheses on which the experiments are based, and hence the results, depend strongly on sophisticated abstract models used to describe and explain neural data or cognitive representations that supposedly are the empirical roots of primary arithmetical activity. I will question the foundational role of such models. I will even cast doubt upon the search (...) for a general and unified philosophical foundation of an empirical science. First, it seems to me hard to draw a global and coherent view from the innumerable and piecemeal neuropsychological experiments and their variable, and sometimes uneasily compatible or fully divergent interpretations. Secondly, I think that the aim of empirical research is to describe dynamical processes, establishing correlations between different sets of data, without meaning to fix an origin or to point to a cause, let alone to a ground. From the very scientific and philosophical point of view it is essential to distinguish between explanations, which provide correlations or, at best, causal mechanisms, and grounding, which involves a claim to some form of determinism. (shrink)

Rips et al.'s analyses have boosted the plausibility of proposals that the human mind embodies some critical properties of natural numbers. I suggest that such proposals can be further evaluated by infant studies, neuropsychological data, and evolution-based considerations, and additionally, that Rips et al.'s model may need to be modified in order to more completely reflect infants' quantitative abilities.