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...reasons the differences between the minds of humans and other animals. We have bigger brains, and hence more powerful information processors; sometimes differences in the power of a processor can create what look like qualitative differences in kind. And of course human beings also have language--the main medium for the cultural transmission of acquired knowledge. Comparative studies of humans and other primates suggest that we differ from them as well in our substantive cognitive abilities--for example, our capacity for causal analysis and our capacity to reason about the mental states of others. Each of these factors doubtless contributes to our prodigious ability to learn.
But in my view another factor is even more important: our uniquely human ability to 'bootstrap.' Many psychologists, historians, and philosophers of science have appealed to the metaphor of bootstrapping in order to explain learning of a particularly difficult sort--those cases in which the endpoint of the process transcends in some qualitative way the starting point. The choice of metaphor may seem puzzling--it is self-evidently impossible to pull oneself up by one's own bootstrap. After all, the process I describe below is not impossible, but I keep the term because of its historical credentials and because it seeks to explain cases of learning that many have argued are impossible.
Sometimes learning requires the creation of new representational resources that are more powerful than those present at the outset. Early in the cultural history of mathematics, for instance, the concept of the number included only positive integers: with subsequent development the concept came to encompass zero, rational numbers (fractions), negative numbers, irrational numbers like pi, and so on.
Bootstrapping is the process that underlies the creation of such new concepts, and thus it is part of the answer to the question: What is the origin of concepts?
Individual concepts are the units of thought. They are constituents of larger mental structures--of beliefs that are formed out of them and of systems of representation such as intuitive theories. Concepts are individuated on the basis of two kinds of considerations: their reference to different entities in the world and their role in distinct mental systems of inferential relations.
How do human beings acquire concepts? Logic dictates three parts to any explanation of the origin of concepts. First, we must specify the innate representations that provide the building blocks of the target concepts of interest. Second, we must describe how the target concepts differ from these innate representations--that is, we must describe developmental change. And third, we must characterize the learning mechanisms that enable the construction of new concepts out of the prior representations.
Claims about all three parts of the explanation of the origin of concepts are highly controversial. Many believe that innate representations are either perceptual or sensory, while others (including myself) hold that humans and other animals are endowed with some innate representations with rich conceptual content. Some researchers also debate the existence, even the possibility, of qualitative changes to the child's initial representations. One argument for the impossibility of such radical changes in the course of development is the putative lack of learning mechanisms that could explain them. This is the gap that my appeal to bootstrapping is meant to fill.
To make clear both what the problem is, and what role bootstrapping may play in solving it, I will examine how children acquire one specific set of concepts: the positive integers--i.e., concepts such as one, two, three, nine, eighteen, etc.
Before they acquire language, infants form several different types of representation with numerical content, at least two of which they share with other vertebrate animals.
One, described by Stanislas Dehaene in his delightful book The Number Sense, uses mental symbols that are neural magnitudes linearly related to the number of individuals in a set. Because the symbols get bigger as the represented entity gets bigger, they are called analog magnitudes. Figure 1 gives an external analog magnitude representation of number, where the symbol is a line, and length is the magnitude linearly related to number. Mental computations using these symbols include comparison, to establish numerical difference or equality, and also addition and subtraction.
Mental analog magnitudes represent many dimensions of experience--for example, brightness, loudness, and temporal duration. In each case...
NOTE: All illustrations and photos
have been removed from this article.
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