For the past hundred years, mathematics, for its own reasons, has been shifting away from the study of “mathematical objects” and towards the study of “structures”. One would have expected philosophers to jump onto the bandwagon, as in many other cases, to proclaim that this shift is no accident, since mathematics is “essentially” about structures, not objects. In fact, structuralism has not been a very popular philosophy of mathematics, probably because of the hostility of Frege and other influential logicists, and (...) quasi-logicists like Quine. Recently, however, structuralism has finally penetrated the philosophical community and a number of “structuralist” volumes in the philosophy of mathematics have seen the light of day. Michael Resnik is one of the most distinguished representatives of this school, along with Stewart Shapiro, who has also recently published a structuralist tract. (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)

Number concepts must support arithmetic inference. Using this principle, it can be argued that the integer concept of exactly ONE is a necessary part of the psychological foundations of number, as is the notion of the exact equality - that is, perfect substitutability. The inability to support reasoning involving exact equality is a shortcoming in current theories about the development of numerical reasoning. A simple innate basis for the natural number concepts can be proposed that embodies the arithmetic principle, supports (...) exact equality and also enables computational compatibility with real- or rational-valued mental magnitudes. (shrink)

Despite their importance in public discourse, numbers in the range of 1 million to 1 trillion are notoriously difficult to understand. We examine magnitude estimation by adult Americans when placing large numbers on a number line and when qualitatively evaluating descriptions of imaginary geopolitical scenarios. Prior theoretical conceptions predict a log-to-linear shift: People will either place numbers linearly or will place numbers according to a compressive logarithmic or power-shaped function (Barth & Paladino, ; Siegler & Opfer, ). While about half (...) of people did estimate numbers linearly over this range, nearly all the remaining participants placed 1 million approximately halfway between 1 thousand and 1 billion, but placed numbers linearly across each half, as though they believed that the number words “thousand, million, billion, trillion” constitute a uniformly spaced count list. Participants in this group also tended to be optimistic in evaluations of largely ineffective political strategies, relative to linear number-line placers. The results indicate that the surface structure of number words can heavily influence processes for dealing with numbers in this range, and it can amplify the possibility that analogous surface regularities are partially responsible for parallel phenomena in children. In addition, these results have direct implications for lawmakers and scientists hoping to communicate effectively with the public. (shrink)

The mapping of numbers onto space is fundamental to measurement and to mathematics. Is this mapping a cultural invention or a universal intuition shared by all humans regardless of culture and education? We probed number-space mappings in the Mundurucu, an Amazonian indigene group with a reduced numerical lexicon and little or no formal education. At all ages, the Mundurucu mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western adults used linear mapping with small or symbolic numbers and logarithmic (...) mapping when numbers were presented nonsymbolically under conditions that discouraged counting. This indicates that the mapping of numbers onto space is a universal intuition and that this initial intuition of number is logarithmic. The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education. (shrink)

Five-year-old children categorized as skilled versus unskilled counters were given verbal estimation and number word comprehension tasks with numerosities 20 – 120. Skilled counters showed a linear relation between number words and nonsymbolic numerosities. Unskilled counters showed the same linear relation for smaller numbers to which they could count, but not for larger number words. Further tasks indicated that unskilled counters failed even to correctly order large number words differing by a 2 : 1 ratio, whereas they performed well on (...) this task with smaller numbers, and performed well on a nonsymbolic ordering task with the same numerosities. These findings provide evidence that large, approximate numerosity representations become linked to number words around the time that children learn to count to those words reliably. (shrink)