Bootstrapping & the Origin of Concepts The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Carey, Susan E. 2004. Bootstrapping & the origin of concepts. Daedalus 133(1): 59-68. Published Version doi:10.1162/001152604772746701 Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:5109360 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA Susan Carey Bootstrapping & the origin of concepts All animals learn. But only human ysis and our capacity to reason about beings create scienti½c theories, mathe- the mental states of others. Each of matics, literature, moral systems, and these factors doubtless contributes complex technology. And only humans to our prodigious ability to learn. have the capacity to acquire such cultur- But in my view another factor is even ally constructed knowledge in the nor- more important: our uniquely human mal course of immersion in the adult ability to ‘bootstrap.’ Many psycholo- world. gists, historians, and philosophers of There are many reasons for the differ- science have appealed to the metaphor ences between the minds of humans of bootstrapping in order to explain and other animals. We have bigger learning of a particularly dif½cult sort– brains, and hence more powerful infor- those cases in which the endpoint of the mation processors; sometimes differ- process transcends in some qualitative ences in the power of a processor can way the starting point. The choice of create what look like qualitative differ- metaphor may seem puzzling–it is self- ences in kind. And of course human evidently impossible to pull oneself up beings also have language–the main by one’s own bootstrap. After all, the medium for the cultural transmission process I describe below is not impos- of acquired knowledge. Comparative sible, but I keep the term because of studies of humans and other primates its historical credentials and because it suggest that we differ from them as well seeks to explain cases of learning that in our substantive cognitive abilities– many have argued are impossible. for example, our capacity for causal anal- Sometimes learning requires the cre- ation of new representational resources Susan Carey, professor of psychology at Harvard that are more powerful than those pres- University, has played a leading role in transform- ent at the outset. Early in the cultural ing our understanding of cognitive development. history of mathematics, for instance, A Fellow of the American Academy since 2001, the concept of the number included only she is the author of numerous articles and essays positive integers: with subsequent de- and the book “Conceptual Change in Childhood” velopment the concept came to encom- (1985). pass zero, rational numbers (fractions), negative numbers, irrational numbers © 2004 by the American Academy of Arts like pi, and so on. & Sciences Dædalus Winter 2004 59 Susan Bootstrapping is the process that un- in solving it, I will examine how children Carey derlies the creation of such new con- acquire one speci½c set of concepts: the on learning cepts, and thus it is part of the answer to positive integers–i.e., concepts such as the question: What is the origin of con- one, two, three, nine, eighteen, etc. cepts? Individual concepts are the units of Before they acquire language, infants thought. They are constituents of larger form several different types of represen- mental structures–of beliefs that are tation with numerical content, at least formed out of them and of systems of two of which they share with other ver- representation such as intuitive theories. tebrate animals. Concepts are individuated on the basis One, described by Stanislas Dehaene of two kinds of considerations: their ref- in his delightful book The Number Sense, erence to different entities in the world uses mental symbols that are neural and their role in distinct mental systems magnitudes linearly related to the num- of inferential relations. ber of individuals in a set. Because the How do human beings acquire con- symbols get bigger as the represented cepts? Logic dictates three parts to any entity gets bigger, they are called analog explanation of the origin of concepts. magnitudes. Figure 1 gives an external First, we must specify the innate repre- analog magnitude representation of sentations that provide the building number, where the symbol is a line, and blocks of the target concepts of interest. length is the magnitude linearly related Second, we must describe how the target to number. Mental computations using concepts differ from these innate repre- these symbols include comparison, to sentations–that is, we must describe de- establish numerical difference or equal- velopmental change. And third, we must ity, and also addition and subtraction. characterize the learning mechanisms Mental analog magnitudes represent that enable the construction of new con- many dimensions of experience–for cepts out of the prior representations. example, brightness, loudness, and tem- Claims about all three parts of the ex- poral duration. In each case as the physi- planation of the origin of concepts are cal magnitudes get bigger, it becomes highly controversial. Many believe that increasingly harder to discriminate be- innate representations are either percep- tween pairs of values that are separated tual or sensory, while others (including by the same absolute difference. You can myself ) hold that humans and other ani- see in ½gure 1 that it is harder to tell that mals are endowed with some innate rep- the symbol for seven is different from resentations with rich conceptual con- (and smaller than) that for eight than it tent. Some researchers also debate the is to tell that the symbol for two is dif- existence, even the possibility, of quali- ferent from (and smaller than) that for tative changes to the child’s initial repre- three. Analog magnitude representa- sentations. One argument for the impos- tions follow Weber’s law, according to sibility of such radical changes in the which the discriminability of two values course of development is the putative is a function of their ratio. lack of learning mechanisms that could You can con½rm for yourself that you explain them. This is the gap that my have an analog magnitude system of rep- appeal to bootstrapping is meant to ½ll. resentation of number that conforms to To make clear both what the problem Weber’s law. Tap out as fast as you can is, and what role bootstrapping may play without counting (you can prevent your- 60 Dædalus Winter 2004 Figure 1 dots. After habituation they were pre- Bootstrapping Analog magnitude models sented with new displays containing ei- & the origin Number represented by a quantity linearly related ther the same number of dots to which of concepts to the cardinal value of the set they had been habituated or the other one: –– number. Xu and Spelke found that the infants recovered interest to the new two: –––– number, and so concluded that they are capable of representing number. Xu and three: –––––– Spelke also found evidence for Weber’s law: infants could discriminate eight seven: –––––––––––––– from sixteen and sixteen from thirty- two, but not eight from twelve or six- eight: ––––––––––––––– teen from twenty-four.1 Infants and animals can form analog self from counting by thinking ‘the’ with magnitude representations of fairly large each tap) the following numbers of taps: sets, but these representations are only 4, 15, 7, and 28. If you carried this out approximate. Analog magnitude repre- several times, you’d ½nd the mean num- sentations of number fall short of the ber of taps to be 4, 15, 7, and 28, with the representational power of integers; in range of variation very tight around 4 this system one cannot represent exactly (usually 4, occasionally 3 or 5) and very ½fteen, or ½fteen as opposed to fourteen. great around 28 (from 14 to 40 taps, for Nonetheless, analog magnitude repre- example). Discriminability is a function sentations clearly have numerical con- of the absolute numerical value, as dic- tent: they refer to numerical values, tated by Weber’s law. Since you were not and number-relevant computations counting, some other numerical repre- are de½ned over them. sentation must have been guiding your tapping performance–presumably ana- A second system of representations log magnitudes, as your adherence to with numerical content works very Weber’s law, again, would seem to indi- differently. Infants and nonhuman pri- cate. mates have the capacity to form sym- Space precludes my reviewing the ele- bols for individuals and to create men- gant evidence for analog magnitude rep- tal models of ongoing events in which resentations of number in animals and each individual is represented by a single human infants, but let me give just one symbol. Figure 2 shows how, in this sys- example. Fei Xu and Elizabeth Spelke tem, sets of one, two, or three boxes showed infants arrays of dots, one dot might be represented. The ½gure repre- array at a time, until the infants got sents three different possibilities for bored with looking at them. All other the format and content of the symbols. variables that could have been con- 1 For an overview of the evidence for analog founded with number (total array size, magnitude representations of number in both total volume of dots, density of dots, nonhuman animals and human adults, see and so on) were controlled in these stud- Stanislas Dehaene, The Number Sense (Oxford: Oxford University Press, 1997). For evidence ies, such that the only possible basis for in human infants, see Fei Xu and Elizabeth S.