Bootstrapping & the Origin of Concepts

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Citation Carey, Susan E. 2004. Bootstrapping & the origin of concepts. Daedalus 133(1): 59-68.

Published Version doi:10.1162/001152604772746701

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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. the infants’ discrimination was numeric. Spelke, “Large Number Discrimination in 6- Seven-month-old infants were habituat- Month-Old Infants,” Cognition 74 (2000): ed either to arrays of eight or sixteen B1–B11.

Dædalus Winter 2004 61 Susan There is one symbol for each box, so Figure 2 Carey Parallel individuation models on number is implicitly represented; the learning symbols in the model stand in one-one Number Image Abstract Speci½c correspondence with the objects in the of boxes world. To give you a feel for the evidence that infants indeed employ such models, dis- 1 box
obj box tinct from the analog magnitude repre- sentations sketched above, consider the 2 boxes

obj obj box box following experiment from my laborato-


ry. Ten- to fourteen-month-old infants 3 boxes obj obj obj box box box are shown a box into which they can reach to retrieve objects, but into which Note: one symbol for each individual; no symbols for they cannot see. If you show infants integers. three objects being placed, one at a time numerical comparison is a function or all at once, into this box, and then of the ratios of the numbers being com- allow them to reach in to retrieve them pared, and that the representations can one at a time, they show by their pattern handle sets of objects at least as big as of reaching that they expect to ½nd ex- thirty-two. But in this reaching task, actly three objects there. If the infant infants succeed at ratios of 2:1 and 3:2, has a mental representation of a set of but fail at 4:2 and even 4:1; as soon as two objects (e.g., object, object) that are the set exceeds three, infants cannot hidden from view, and the infant sees a hold a model of distinct items in their new object being added to the set, the short-term memory.2 infant creates a mental representation of a set of three (object, object, object). In sum, human infants (and other pri- Further, computations of one-one corre- mates) are endowed with at least two spondence carried out over these models distinct systems of representation with allow the child to establish numerical numerical content. Both take sets of in- equivalence and number order (e.g., dividuals as input. One creates a summa- Have I got all the objects out of the box ry analog representation that is a linear or are there more?) function of the number of individuals So far, this is just another demon- in the set. This process is noisy, and the stration that infants represent number. noise is itself a linear function of the set However, an exploration of the limits size, with the consequence that the rep- on infants’ performance of this task im- resentations are merely approximate. plicates a different system of representa- For several reasons, this system is too tion from the analog magnitude system weak to represent the positive integers. sketched above. For one, there is likely an upper bound to Performance breaks down at four the set sizes that can be represented by objects. If the infants see four objects analog magnitudes. More importantly, being placed into the box and are al- lowed to retrieve two of them, or even 2 For evidence of the set-size limits on in- just one of them, they do not reach per- fants’ representations of small numbers of objects, see Lisa Feigenson and Susan Carey, sistently for the remaining objects. Re- “Tracking Individuals Via Object Files: Evi- member that in the analog magnitude dence from Infants’ Manual Search,” Develop- system of representation, success at mental Science 6 (2003): 568–584.

62 Dædalus Winter 2004 animals and infants cannot discriminate symbols in the list refer to cardinal val- Bootstrapping adjacent integer values once the sets ues exactly one apart: 5 is 4 plus 1, 6 is 5 & the origin of concepts contain more than three or four individ- plus 1, and so on. uals; that is, they cannot represent exact- I have argued so far that the count-list ly ½fteen or twenty-½ve or forty-nine, or representation of number transcends any other large exact integer. Finally, the representational power of both of analog magnitude representations the representational systems with nu- obscure one of the foundational rela- merical content that are available to pre- tions among successive integers–that verbal infants, for these precursors lack each one is exactly one more than the the capacity to represent integers. If this one before. It is this relation, called the is so, it should be dif½cult for children to successor relation, that underlies how come to understand the numerical func- counting algorithms work and provides tion of counting. the mathematical foundation of integer And so, indeed, it is dif½cult for chil- concepts. Since discriminabilty of ana- dren to learn how counting represents log magnitudes is a function of the ratio number, and details about the partial between them, the relation between two understanding they achieve along the and three is not experienced as the same way constrain our theories of the learn- as that between twenty-four and twenty- ing process. In the United States (and ½ve; indeed, the latter two values cannot every other place where early counting really even be discriminated within this has been studied, including Western system of representation. Europe, Russia, China, and Japan) The second system–one symbol for children learn to recite the count list each individual–falls even shorter as a as young two-year-olds, and at this age representation of integers. There are no can even engage in the routine of count- symbols for number in this system at all; ing–touching objects in a set one by one the symbols in ½gure 2 each represent an as they recite the list. But it takes anoth- individual object, unlike those in ½gure er year and a half before they work out 1, which represent an approximate cardi- how counting represents number, and nal value. Furthermore, what can be rep- in every culture yet studied, children go resented in this system is limited in through similar stages in working out number to sets of one, two, and three. the meanings of the number words in the count list. The count list (‘one, two, three . . .’) is First, children learn what ‘one’ means a system of representation that has the and take all other words in the list to power to represent the positive integers, contrast with ‘one,’ meaning ‘more than so long as it contains a generative sys- one’ or ‘some.’ The behaviors that dem- tem for creating an in½nite list. When onstrate this are quite striking. If you deployed in counting, it provides a rep- present young two-year-olds with a pile resentation of exact integer values based of pennies and ask them to give you one on the successor function. That is, when penny, they comply. If you ask for two applied in order, in one-one correspon- pennies or three pennies or ½ve pennies, dence with the individuals in a set, the they grab a bunch, always more than ordinal position of the last number word one, and hand them over. They do in the count provides a representation not create a larger set for ‘½ve’ than for of the cardinal value of the set–of how ‘two.’ You might suppose that the plural many individuals it contains. Successive in ‘pennies’ is doing the work here, but

Dædalus Winter 2004 63 Susan the same phenomenon is observed in and then to ½x the set, counters invari- Carey on China and Japan, even though Chinese ably adjust the set in the right direction, learning and Japanese do not have a singular- taking an object away if the set is too plural distinction, and also in the large or adding one if it is too small. United States when the contrast is One-, two-, and three-knowers, in con- between ‘one ½sh,’ ‘two ½sh,’ and trast, almost always add more to the ‘½ve ½sh.’ set–even if they had counted to ½ve or Let us call children at this stage of six or seven when they were checking working out the meanings of number whether it had four–con½rming that words ‘one-knowers.’ Many other tasks they really do not understand how provide additional evidence that one- counting determines the meaning of knowers truly know only the meaning number words. of the word ‘one’ among all the words in These data suggest that the partial their count list. For example, if you ask a meanings of number words seem to be one-knower to tell you what’s on a series organized initially by the semantics of of cards that contain one, two, or three quanti½ers–the singular-plural distinc- ½sh (up to eight ½sh), they say ‘a ½sh’ or tion and the meanings of words like ‘one ½sh’ for the card with one, and ‘two ‘some’ and ‘a.’ If this is right, then we ½sh’ or ‘two ½shes’ or ‘two ½shies’ for might expect that children learning lan- all of the other cards. This again indi- guages with quanti½er systems that cates a single cut between the meaning mark numerical contrasts differently of ‘one,’ which they grasp, and words for from English would entertain different the number of individuals in larger sets, hypotheses concerning the partial mean- which they do not. ings of number words. They might break After having been one-knowers for into the system differently. And indeed about six to nine months, children learn they do. what ‘two’ means. At this point they can Consider ½rst classi½er languages correctly give you two objects if you ask such as Chinese and Japanese that do for ‘two,’ but they still just grab a bunch not mark the distinction between singu- (always greater than two), if you ask for lar and plural in nouns, verbs, or adjec- ‘three,’ ‘four,’ ‘½ve,’ or ‘six.’ After some tives. Two independent studies have months as two-knowers, they become found that although children in China three-knowers, and some months later and Japan learn the count list as young induce how counting works. as English-speaking children do, they The performance of children who have become one-knowers several months worked out how counting works is quali- later and are relatively delayed at each tatively different from that of the one-, stage of the process. Conversely, Russian two-, and three-knowers in a variety of has a complex plural system in which the ways that reflect the conceptual under- morphological markers for sets of two, standing of counting. three, and four differ from those for ½ve To give just one example, in the task in through ten. Two independent studies which children are asked to give the ex- have shown that even Russian one- and perimenter a certain number of items, two-knowers distinguish between the say four, one-, two-, and three-knowers meanings of the number words ‘two,’ usually give the wrong number, and the ‘three,’ and ‘four,’ on the one hand, and young counters also sometimes make an ‘½ve,’ ‘six,’ ‘seven,’ and ‘eight,’ on the error. When asked to check by counting other. Unlike the one- and two-knowers

64 Dædalus Winter 2004 described above, Russian children in the have the power to represent natural Bootstrapping early stages of working out how count- numbers either. & the origin of concepts ing works grab smaller sets when asked to give the experimenter ‘two,’ ‘three,’ The problem of the origin of the posi- or ‘four’ than when asked to give the ex- tive integers arises at two different time perimenter ‘½ve’ or more, and use larger scales–historical and ontogenetic. At numbers for larger sets in the what’s-on- the dawn of modern anthropology, this-card task.3 when colonial of½cers went out into These phenomena concerning young the French and English colonial worlds, children’s partial understanding of the they discovered many systems of explicit meanings of number words support number representation that fell short of three interrelated conclusions. First, a full representation of natural number. that it is so dif½cult for children to learn They described languages that marked what ‘two’ means, let alone what ‘½ve’ number on nouns, adjectives, and verbs, and ‘eight’ mean, lends support to the and which had quanti½ers like the Eng- claims that preverbal number represen- lish ‘one,’ ‘two,’ ‘many,’ ‘some,’ ‘each,’ tations are not representations of inte- ‘every,’ and ‘more,’ but which had no gers, at least not in the format of an inte- count list. In this vein, the psychologist ger list. Young children–for a full six to Peter Gordon has described the language nine months before they work out what of the Piraha, an isolated Amazonian ‘two’ means, and a full year and a half people. He has shown that in addition before they work out how the count to linguistic quanti½ers meaning ‘one,’ list represents integers–know how to ‘two,’ and ‘many,’ the Piraha also have count, know what ‘one’ means, and access to the nonverbal systems de- know that ‘two,’ ‘three,’ ‘four,’ ‘½ve,’ scribed above (parallel individuation ‘six,’ ‘seven,’ and ‘eight’ represent num- of small sets and analog magnitude rep- bers larger than ‘one.’ Second, coming resentations of large numerosities). Gor- to understand how the count list repre- don con½rms that they have no repre- sents numbers reflects a qualitative sentations of large exact numerical val- change in the child’s representational ues. capacities; I would argue that it does Anthropologists and archeologists nothing less than create a representa- have described intermediate systems of tion of the positive integers where none integer representation, short of integer was available before. Finally, a third pos- lists, and these intermediate systems sible developmental source of natural provide evidence for a process of cultur- number representations, in addition to al construction over generations and the preverbal systems described above, centuries of historical time.4 Here I con- may be the representations of numbers centrate on ontogenetic time. How do within natural language quanti½er se- three-year-olds do it? How do they cre- mantics. Of course, natural language ate a representational system with more quanti½ers, other than the number 4 I would recommend the linguist James Hur- words in the count list itself, do not ford’s review of this literature to any reader in- terested in this process. James Hurford, Lan- 3 For a characterization of the early stages of guage and Number (Oxford: Basil Blackwell, counting in English, see Karen Wynn, “Chil- 1987). For work on the Piraha, see Peter Gor- dren’s Acquisition of the Number Words and don, “The Role of Language in Numerical Cog- the Counting System,” Cognitive Psychology 24 nition: Evidence from Amazonia” (under re- (2) (1992): 220–257. view).

Dædalus Winter 2004 65 Susan power than any on which it is built?5 As described above, the child learns Carey on In answering this question, I would the meanings of the ½rst number words learning appeal to bootstrapping processes. as natural language quanti½ers. Children Bootstrapping processes make essen- learn the meaning of ‘one’ just as they tial use of the human capacity for creat- learn the meaning of the singular deter- ing and using external symbols such as miner ‘a’ (indeed, in many languages, words and icons. Bootstrapping capital- such as French, they are the same lexi- izes on our ability to learn sets of sym- cal item). bols and the relations among them di- Some months later, ‘two’ is learned, rectly, independently of any meaning just as dual markers are in languages that assigned to them in terms of anteced- have singular/dual/plural morphology. ently interpreted mental representa- Languages with dual markers have a dif- tions. These external symbols then ferent plural af½x for sets of two than the serve as placeholders, to be ½lled in af½x for sets greater than two. It is as if with richer and richer meanings. The English nouns were declined ‘box’ (sin- processes that ½ll the placeholders gular), ‘boxesh’ (dual), ‘boxeesh’ (plu- create mappings between previously ral). In this system, the suf½x ‘esh’ separate systems of representation, would apply just when the set referred to drawing on the human capacity for contained exactly two items. By hypoth- analogical reasoning and inductive in- esis, children would learn the meaning ference. The power of the resulting sys- of the word ‘two’ just as they would tem of concepts derives from the com- learn the morphological marker ‘esh’– bination and integration of previously if English plural markers worked that distinct representational systems. way. By extension, some months later, Let’s see how this might work in the ‘three’ is learned just as trial markers present case. We must allow the child are in the rare languages that have sin- one more prenumerical capacity–that gular/dual/trial/plural morphology. of representing serial order. This is no In the early stages of being a one-, problem–young children learn a vari- two-, or three-knower, the child repre- ety of meaningless ordered lists, such sents other number words as quanti½ers, as ‘eeny, meeny, miney, mo.’ meaning ‘many,’ where ‘many’ is more We seek to explain how the child than any known number word. As I will learns the meanings of the number argue below, it is likely that the nonver- words–what ‘two’ means, what ‘seven’ bal number representations that support means–and how the child learns how the meanings of the known words is the the list itself represents number–that system of parallel individuation (½gure the cardinal value of a set enumerated 2), with natural language quanti½cation by counting is determined by the order articulation in terms of notions like ‘set’ on the list, and that successive numbers and ‘individual.’ on the list are related by the arithmetic Meanwhile, the child has learned the successor relation. count list, which initially has no seman- tic content other than its order. The 5 In a forthcoming book I argue that the same child knows one must recite ‘one, two, bootstrapping process underlying this marvel- ous feat in childhood also accounts for the de- three, four, ½ve,’ not ‘two, three, one, velopment in historical time, but that argument ½ve, four,’ just as one must say ‘a, b, c, d, is beyond the scope of this brief paper. e,’ not ‘c, a, e, d, b.’

66 Dædalus Winter 2004 The stage is now set for a series of would be empirically correct, at least for Bootstrapping mappings between representations. some children. We do know that chil- & the origin of concepts Children may here make a wild analo- dren come to integrate their integer list gy–that between the order of a particu- with analog magnitudes, such that ‘½ve’ lar quantity within an ordered list, and comes to mean both ‘one more than that between this quantity’s order in a four, which is one more than three. . .’ series of sets related by additional indi- and ‘––––––––––,’ the analog magnitude viduals. These are two quite different symbol for the cardinality of a set of ½ve bases of ordering–but if the child recog- individuals. This integration is undoubt- nizes this analogy, she is in the position edly very important; bootstrapping pro- to make the crucial induction: For any vides richer representations precisely word on the list whose quanti½cational through integration of previously dis- meaning is known, the next word on the tinct systems of representation. list refers to a set with another individ- As important as the integration of the ual added. Since the quanti½er for single integer list representation with analog individuals is ‘one,’ this is the equivalent magnitude representations may be, to the following induction: If number there is good reason to believe that this word X refers to a set with cardinal value integration is not part of the bootstrap- n, the next number word in the list re- ping process through which the concept fers to a set with cardinal value n + 1. of positive integers is ½rst understood. This bootstrapping story provides dif- Research suggests that it is not until after ferent answers for how the child learns children have worked out how the count the meaning of the word ‘two’ than for list represents number–in fact some six how she learns the meaning of ‘½ve.’ months later–that they know which According to the proposal, the child analog magnitudes correspond to which ascertains the meaning of ‘two’ from numbers above ½ve in their count list. the resources that underlie natural lan- That ½nding–along with the fact that guage quanti½ers, and from the system the precise meanings of number words of parallel individuation, whereas she are learned in the order ‘one,’ then ‘two,’ comes to know the meaning of ‘½ve’ then ‘three,’ followed by the induction through the bootstrapping process– of how the count list works–leads me to i.e., that ‘½ve’ means ‘one more than favor the bootstrapping proposal above. four, which is one more than three . . .’ I doubt that anybody would deny that –by integrating representations of natu- language helps us occupy the distinctive ral language quanti½ers with the exter- cognitive niche that we human beings nal serial ordered count list. enjoy. It is obvious that culturally con- structed knowledge is encoded in lan- I began by sketching two systems of guage and can then be passed on to new preverbal representation with numerical generations through verbal communica- content: the analog magnitude system tion–you can tell your children some- and the system of parallel individuation. thing, saving them from having to dis- You may have noticed that the analog cover it themselves. Still, this account magnitude system played no role in my misses the equally obvious point that bootstrapping story. It would be quite children are often unable to understand possible to imagine a role for this system what we tell them, because they lack the in a slightly different bootstrapping pro- concepts that underlie our words. The posal, and it may be that such a proposal problem then becomes accounting for

Dædalus Winter 2004 67 Susan how they acquire the relevant concepts Carey on they need to understand what we are learning telling them. I have argued that bootstrapping mechanisms provide part of the solution to this problem. In thinking about how bootstrapping might work, we are led to a fuller appreciation of the role of language in supporting the cultural transmission of knowledge. We cannot just teach our children to count and ex- pect that they will then know what ‘two’ or ‘½ve’ means. Learning such words, even without fully understanding them, creates a new structure, a structure that can then be ½lled in by mapping rela- tions between these novel words and other, familiar concepts. And so eventu- ally our children do know what ‘½ve’ means: through the medium of language and the bootstrapping process sketched here they have acquired a new concept.

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