Ground and Explanation in Mathematics

1997

Biological Theory, 2009

What is the nature of number systems and arithmetic that we use in science for quantification, analysis, and modeling? I argue that number concepts and arithmetic are neither hardwired in the brain, nor do they exist out there in the universe. Innate subitizing and early cognitive preconditions for numberwhich we share with many other species-cannot provide the foundations for the precision, richness, and range of number concepts and simple arithmetic, let alone that of more complex mathematical concepts. Numbers and arithmetic, and mathematics in general, have unique features-precision, objectivity, rigor, generalizability, stability, symbolizability, and applicability to the real world-that must be accounted for. They are sophisticated concepts that developed culturally only in recent human history. I suggest that numbers and arithmetic are realized through precise combinations of non-mathematical everyday cognitive mechanisms that make human imagination and abstraction possible. One such mechanism, conceptual metaphor, is a neurally instantiated inference-preserving cross-domain mapping that allows the conceptualization of abstract entities in terms of grounded bodily experience. I analyze how the inferential organization of the properties and "laws" of arithmetic emerge metaphorically from everyday meaningful actions. Numbers and arithmetic are thus-outside of natural selection-the product of the biologically constrained interaction of individuals with the appropriate cultural and historical phenotypic variation supported by language, writing systems, and education.

American Educator, 2002

he most familiar of objects, numbers are nonetheless surprisingly slippery, their sheer slipperiness interesting evidence that certain intellectual tools may be successfully used before they are successfully understood. Numbers tend to sort themselves out by clans or systems, with each new system arising as the result of a perceived infirmity in the one that precedes it. The natural numbers 1, 2, 3, 4,…, start briskly at 1 and then go on forever, although how we might explain what it means for anything to go on forever without in turn using the natural numbers is something of a mystery. In almost every respect, they are, those numbers, simply given to us, and they express a primitive and intimate part of our experience. Like so many gifts, they come covered with a cloud. Addition makes perfect sense within the natural numbers; so, too, multiplication. Any two natural numbers may be added, any two multiplied. But subtraction and division are curiously disabled operations. It is possible to subtract 5 from 10. The result is 5. What of 10 from 5? No answer is forthcoming from within the natural numbers. They start at 1. The integers represent an expansion, a studied enlargement, of the system of natural numbers, one motivated by obvious intellectual distress and one made possible by two fantastic inventions. The distress, I have just described. And those inventions? The first is the number 0, the creation of some nameless but commanding Indian mathematician.

The Journal of Philosophy, 2001

Complexity, 1998

Theoria, 2023

The field of numerical cognition provides a fairly clear picture of the processes through which we learn basic arithmetical facts. This scientific picture, however, is rarely taken as providing a response to a much-debated philosophical question, namely, the question of how we obtain number knowledge, since numbers are usually thought to be abstract entities located outside of space and time. In this paper, I take the scientific evidence on how we learn arithmetic as providing a response to the philosophical question of how we obtain number knowledge. I reject the view that numbers are abstract entities located outside of space and time and, alternatively, derive from the scientific evidence a novel account of the nature of numbers. In this account, numbers are reifications of the counting procedure and arithmetic statements are seen as describing the functioning of counting and calculation techniques.

This extended article uses several examples to illustrate the logic and process of mathematical thinking.

Different kinds of knowledge are acquired in different ways. In this chapter, I consider four kinds of knowledge: (1) Core knowledge, which is implicit, nonlinguistic and learned easily and automatically; (2) Explicit, linguistic knowledge that fits into an existing conceptual structure (i.e., a new word for an old concept); (3) Explicit, linguistic knowledge that does not fit into any prior conceptual structure (e.g., how to knit); and (4) Explicit, linguistic knowledge that is incompatible with some prior conceptual structure, making it very difficult to learn (e.g., knowing that the earth is a round ball floating in space.) It is this fourth type of learning that Carey (2009) calls ‘conceptual change.’ In this chapter I argue that mathematics includes all four types of knowledge, but that natural numbers are an example of the fourth and most interesting type—knowledge that is the result of conceptual change.