UNIVERSITY OF PENNSYLVANIA INSTITUTE FOR RESEARCH IN COGNITIVE SCIENCE th
16 Annual Pinkel Endowed Lecture The Origin of Concepts Friday March 28, 2014 16th Annual Pinkel Endowed Lecture DAVID: Good afternoon. Welcome. Welcome to the 16 th Annual Benjamin & Anne Pinkel Lecture. This is an annual lecture. We have it once a year through the Institute for Research and Cognitive Science. It was endowed by a generous gift from the daughter of Benjamin & Anne Pinkel, Sheila Pinkel, in honor of her parents. Benjamin Pinkel got an engineering degree from Penn in 1930 and his career was in engineering. But he became interested in matters of cognitive science and, in fact, published a short monograph. I have a copy here, which we'll give to our speaker, Consciousness, Matter and Energy, concerned with topics in cognitive science. The lecture series is intended to promote discussion and thought in the general areas that were of interest to him. We've had now 15 previous Pinkel lectures. Each lecture has been wonderful and exciting and I'm very confident today's will be as well. There will be a reception after the lecture out back. Is that right? Out front. Every year it's in a different place. So at the end of the lecture we welcome everybody to join us for some, I guess, whatever the goodies are, but the goodies will be there. I want to mention that the same fund that endows the lecture now in fact also endows support for undergraduate research in cognitive science. It's an extra feature we've been able to add through the generosity of the Pinkel family just this year. With that introduction to the lecture series, I will turn it over to John Trueswell who will introduce today's speaker, Susan Carey. MR. JOHN TRUESWELL: Okay. Thanks David. So it's a true pleasure to introduce Professor Susan Carey. Professor Carey is currently the Henry and Elizabeth Morss Professor of Psychology at Harvard where she is part of the Laboratory for Developmental Studies. Professor Carey's research has focused on one of the most important questions within the field of cognitive science: How do infants and toddlers come to think about things, events, actions, and states of affairs in the world, all from just looking around and gathering information from their senses? This question is usually described as the question of conceptual development, and it has puzzled philosophers and psychologists really from the beginning of written records. What kinds of perceptions and conceptions of the world come for free? That is, evolutionarily defined, just by virtue of being a human being? How do humans use these basic building blocks to learn about new things in the world? These are the kinds of questions that Susan has focused on in her career. Over her impressive research career, Susan Carey has demonstrated that she can be 1
counted among the major figures in advancing our understanding of human and conceptual development. Through clever thinking and clever experimentation, which I'm sure you're going to be hearing about today, she's given us new insights into how children learn the meanings of words, develop theories of causation, reason about objects as they come and go from view, and develop the notion of precise number, the topic that she'll be talking about today. Before I hand over the stage to Professor Carey though, let me mention some precise and imprecise numbers that help give a picture of Carey's impressive academic career. So first, the number two: she has two degrees, Radcliffe College for her B.A. and Harvard for her Ph.D., where she now has returned. Four: she has four books, two of which are very well known on the origins of concepts and conceptual change throughout childhood. Greater than 120: that's the number of articles that she has published over the years and it's like watching the credits at the end of a movie here. We'll just sort of have to wait for a while. And it broke my animation too because there are even more there. Also the number 12, greater than 12: that's the number of honors and fellowships that she has been awarded over the years from the Academy of Arts and Sciences, the American Philosophical Society, the National Academy of Sciences. She has been recognized in every possible related discipline that is interested in cognition and cognitive development. Finally, I'd like to give you the one and the only Professor Susan Carey. [APPLAUSE] PROFESSOR SUSAN CAREY: My project, as John said, is: how do we account for the human capacity to think the thoughts we can think? We're the only animals that can ponder the causes and cures of global warming or pancreatic cancer. Yet our brains and minds are like other primates in many, many, many respects. Rightly, people addressing this question have concerned what makes it possible. But I'd like to add a second question, which is: Why is it sometimes so easy, and why is it sometimes so hard? "Easy," I mean, sometimes I'm first encountering a novel object or first hearing a novel word. We can form a concept that covers that object and that is expressed by that word, hold on to it forever. Other aspects of conceptual development, like learning the concept of force and Newtonian mechanics, takes years and most students who study Newtonian mechanics fail to do so. I'm going to argue today that considering the two questions together – what makes it easy in some cases and what makes it hard in the other – actually illuminates the issue of what makes it possible. As a matter of logic, a theory of conceptual development is going to have the following parts. We're going to have to be able to characterize the innate 2
representations, the innate primitives, in terms of which the animal or the baby looks out and interprets the world, because these are the input to any learning mechanisms, if you accept a representational‐computational theory of mind. Even the most radical empiricists agree with this. They thought the innate primitives were sensory primitives. But there had to be some, right? You need to characterize the adult conceptual repertoire. You have to ask how different those are. Is the adult conceptual repertoire just an elaboration of the beginning one? Or are there discontinuities? And the reason you need to do that is that you need to know how hard the learning problem is that's going to get you from here to there. This isn't meant to be controversial, but within that framework, it leaves room for all the controversy in the world. In one of those books that John mentioned, The Origin of Concepts, I argue that the innate primitives, contrary to the empiricists, are rich conceptual innate primitives for which we have good evidence. But nonetheless, there are important discontinuities. So we have a hard learning problem to account for. I'm going to illustrate those theses today. Just some more throat clearing. I don't mean anything fancy by the notion of concept. Concepts are a subset of mental representations and just laying my cards on the table I think, like most cognitive scientists, I take the notion of mental representation as a theoretical term in our field and I take it dead literally. This is not a metaphor. We are studying mental representations. What are mental representations? They are mental symbols, symbols in the mind. And like all symbols, in principle you can ask questions: What's their format? What are they like? Are they like pictures? Are they like words? Are they strings of zeros and ones? What's the format of the representation? What's their computational role? How do they figure in thought and inference in guiding behavior? They're symbols so what are their referents? What's in their extension and how do the symbols determine those referents? So all I'm saying here is that we as cognitive scientists, if you work at a representational‐computational theory, are committed to there being entities about which these questions can be asked. I take the consensus view in the field on where the human conceptual repertoire comes from to be the following: that our human adult conceptual repertoire is a function over the set of development computational primitives, and of the commentarial processes through which new concepts are composed from these primitives. The primitives plus the computational machinery determines a space of possible concepts and learning processes are hypothesis testing and confirmation over the space. That is, I think, a consensus view about how to think about learning and cognitive science. It's certainly the consensus view that my friend and teacher Jerry Fodor holds, and he famously says that accepting that view of the origin of 3
concepts leads us to a few challenges to cognitive science, conclusions that we might not want to accept. One of them is: learning cannot, as a matter of logic, increase the expressive power of the representational system. Another challenge related is mad dog nativism. So what I want to do in this talk is both illustrate the picture I have about how cognitive development works, argue against the consensus view, and get us out of Fodor's challenges. That's what I'm trying to do today. So what's the argument that expressive power cannot be increased by learning? Well the argument is a one‐liner, and this is it: here's what expressive power is. The expressive power of the mind or a machine is a function of the primitives and what can be constructed from them using the commentatorial machinery of language of thought. That's just what expressive power means. It's a logical semantic notion. But learning on this picture is characterized by the set of innate primitives and the logical resources through which one builds new representations from those primitives. If you accept both of those, then obviously learning can't increase expressive power, it's the same notion. Here's the argument for mad dog nativism. Premise one: all learning is a hypothesis confirmation. Premise two: logical construction. One can learn new concepts only by creating and confirming logical constructions from antecedently available concepts. Premise three, atomism. Most concepts, the ones that can be lexicalized by single morphings like dog or atom or wrench or carburetor cannot be analyzed as logical construction of other concepts, primitives or otherwise. So the conclusion from those three premises is innateness. In order to acquire a new monomomythic concept one would have to confirm hypotheses already containing it because all concept learning is hypothesis confirmation already containing the concept to be learned. That's because, by hypothesis they can't be constructed out of anything else, they're primitives. Therefore, no monomomythic concept can be learned. Now that really is a conclusion that we shouldn't want to accept because this says all 500,000 concepts that are lexicalized by words and the Oxford English Dictionary were innate. Quirk is an innate concept. Really we don't want that consequence. But the question is: how do we get out of this? Now, most cognitive scientists try to get out of it by denying premise three, denying atomism. Saying, "Look, Fodor has underestimated the possibility of definitions." I accept, actually, his arguments for atomism. I want to get out of it by denying premises one and two. I want to deny the whole consensus view of how concept learning works. So the resolution is: you can learn new primitives. Now that seems odd. That's because there's an ambiguity in the notion "primitive." Primitive means computational primitive. A computational primitive is simply the bottom unit that you can decompose your thoughts into. That's a 4
computational primitive. Remember, we're taking representations dead literally. But that's different from a developmental primitive. I mean it's logically a different notion. The claim here is that not all computational primitives are developmental primitives. And that requires that you have to be able to learn new primitives by some process other than constructing them out of biological combination, by other antecedently available concepts, because if you could do that, they wouldn't be computational primitives. I'm going to show you why that's not only actual, but happens all the time. Another thing to keep your eye on, as I go through the examples of why I reject premises one and two, is: I think that the analogy that you find in cognitive science of learning as navigating the space of possible concepts fixed by Bayesian hypothesis testing and enumerative induction is a terrible metaphor. I mean, it's not like there isn't that kind of learning, but it's not what all learning looks like by any means. Let me start by denying premises one and two. Premise one is: all learning is hypothesis testing and confirmation. Well, we have to agree, first of all, what we mean by "learning," and what I take learning to be is any computational process that yields new representations in response to representational inputs that can be conceptualized as providing relevant information. The information in there just says that these are representations. Now sometimes, this is explicit and implicit hypothesis testing. The organism evaluates the information with respect to its evidential status. In all forms of Bayesian learning, it's that. Association is learning connectionist back propagation and lots of learning mechanisms satisfy the Fodorian view. But also there are learning mechanisms which are domain‐specific adaptations that respond to information by simply causing a representational change. There's no hypothesis testing or logical construction involved. I'm going to give you an example so you can know what I'm talking about. The easy routes to concept learning are always cases like this. I'm going to give you an example and then I'll characterize them. Or there are also learning mechanisms that are bootstrapping mechanisms, which also are different from this classical view. Let's do the easy case first. In the easy case of learning new primitives, there's an innate conceptual role and then there are innate learning mechanisms that create representations that are needed for the computations that realize that conceptual role. The primitives that are being formed are primitive representations that are needed to carry out an innately supported conceptual role. So here's an example: the Indigo bunting navigational system. Indigo buntings, like many navigating birds, fly 3500 miles every spring and fall. They start in Canada in the fall, they end up in Middle America in the spring, and they go the other way around. That's 3500 miles. How do they manage this feat? That is, 5
how do they know how to get there? They probably don't have a map with landmarks that they're following for that 3500 miles. And in fact, we know they don't because they fly at night. So they fly at night; they're probably using the stars. We know they're using the stars because if you prevent them from seeing the stars, they don't know what direction to take off. Also, you can put them in a planetarium and orient the night sky arbitrarily with respect to the actual sky and they take off as specified by the sky. So they're using the stars and only the stars. How are they using the stars? There's two ways you could use the stars. You could take any arbitrary constellation, which sweeps through the sky at night. And if you know what time it is, then you can use its position in the sky the same way you would use the position of the sun in the sky to navigate if you were using the sun during the day and many animals do that. So maybe that's how they're using the sky. But the night sky also gives you something else. It gives you the North Star. To use the North Star, the North Star is always north. You don't need to know what time it is to navigate by the North Star. If they had a way of picking out Polaris, then they could use the night sky to navigate. They have a way of picking out Polaris, and you know that because if you reset their clock, it doesn't affect their navigation, whereas, if they were navigating by the sun or a particular other constellation, it would. So there's no doubt Indigo Buntings know where Polaris is, and a representation of Polaris in the sky is a primitive that allows them to compute north. They set off with respect to it and they monitor it while they're flying, so they're constantly adjusting their flight by monitoring north. Now here's the learning problem: How the hell do they know where Polaris is in the night sky? Now it could be innate, but that's highly unlikely because the earth tilts on its axis. Right now Polaris is the North Star. 30,000 years ago it was Vega. If you tried to navigate by Vega today, you'd end up in the middle of the ocean and you would die if you were an Indigo Bunting. So it's an empirical question because maybe what evolutionary mechanism is that allows a bird to know innately what Polaris is, maybe it can change. That seems unlikely. Anyway, it's an empirical question that's been answered. This is all Steven Emlen's work. There is a way you could identify Polaris, if you had a mechanism that allowed you to compute the center of rotation in the sky, because the earth rotates on a north/south axis. That center of rotation is roughly Polaris, and 30,000 years ago it was roughly Vega. So if the bird has a computational capacity to carry out that analysis and then print, make an iconic representation of what the part of the sky that is at that center is, they could then use that newly formed computational primitive to play the computational role that gives it the meaning "north," which is innately specified. And that's exactly how they do it and you can show that with planetarium studies by making the night sky rotate around any arbitrary star and then see where the bird goes off and the bird flies off as specified by north so specified. 6
So let's dwell on this case. Is this learning? For sure. Is it hypothesis testing? Not at all. There's never any confirmation. The bird just computes north and uses it, and if it's wrong he dies. He doesn't get any feedback. Is there an alternative space of hypotheses with associated likelihoods and priors? No. The bird never considers any other part of the sky other than the one that's the output of this mechanism. Can we think of it as specifying a space of possible north? Sure. Any part of the sky is a possible output of this mechanism. But that doesn't make it a space of hypotheses. It's a space of possibilities, but there's never any hypothesis testing over any other hypothesis. Does it involve logical combination? Certainly not. It doesn't have any of the properties that all learning is supposed to have according to Fodor. It's useful to explore the space of possible outputs. It allows you to further characterize the actual mechanism. This wouldn't work if the stars were packed too closely together. It wouldn't work if the stars had large‐scale repetitions of pattern. This mechanism could evolve, given the properties of the stars. But to think of this is as a hypothesis space is just wildly misleading. There's no hypothesis testing going on over here. This is what all the easy cases look like. There are innate conceptual roles. I'm not claim that the concept "north" is learned by this process. That's given innately. And it's given by the computational role and its content is given entirely in terms of its computational role in navigation. The primitive that's learned is the iconic representation of what north in the night sky looks that's needed to actually fulfill that computational role. Easy cases are all like that. There are innate conceptual roles and innate domain‐specific learning mechanisms that allow you compute other representations you need to fulfill those conceptual roles. Now this example is not conceptual learning, and I don't have time to give you examples, but there are plenty of examples of concept learning that have these properties as well. I want to talk about the hard cases though. The hard cases are cases where there are conceptual discontinuities. This is a case of learning new primitives when there isn't already a conceptual role for that representation to fill. New computational primitives. So there's co‐construction here, of conceptual roles and new conceptual primitives. The case study I'm gong to do very quickly, because it's in my book, is the case study of natural number. I like to do this because you can do it quickly. There are many other examples in my book as well too. And natural number is particularly fun to do because when you think about it, this seems very unpromising as a case of learning a new primitive, since after all if there's any concepts that have actual definitions, it's concepts like "seven." 7
That is, there are many axiomatic systems in which you can derive all the natural numbers. Peano's axiom tells us how to do that in terms of primitives of order and the successor function. Frega has a provably equivalent proof but based on different primitives. One primitive is the principle that one‐to‐one correspondence guarantees cardinal equality and then you need the resources of second or illogic and you can get all the arithmetic out of that as well. But those proofs are relevant to the question of the actual origin of representations of natural number only if the primitives out of which they build natural number are available to children in a form that would support the same construction, and children actually undergo the same construction to get to natural number. And nobody in their right mind thinks that that's possible. Nobody thinks that children construct natural number the way Frega did in an 80‐page proof. Really. It's really a non‐starter when you look at how the proof works. So how do children do it? There's very good evidence for three systems of core cognition. Remember, I said, thesis one in my picture of development is there are rich innate representations with conceptual content. There are actually three that have numerical content. One is Dehaene's number sense, – I'm going to characterize these in a second – one is a system of parallel representation of small sets of individuals, and then the third one is language itself and especially the number representations in natural language quantification where you need sort of the set base quantification that underlies quantifiers. I'm going to give the child all of these three innately. So let me characterize what they're like. Because remember, if you're taking mental representations seriously, you have to say, "What are they like? Not just what they can represent, how do they represent them?" Here's how one of the core systems of representation is. I'm running behind already, so I'd better do this quickly. Usually I have people do this out loud. But I'm going to flash up a bunch of dots. Just quickly, without counting them, say to yourself how many dots there are. Okay ready? The reason I have people say them out loud is because for this one which the answer is sixteen, you will hear anything from 12 to 20. This, where it's 8, you hear 7, 8, and 9. This one, which is 32, you hear everything from 15 to 60 and 24, something in between. Now if you did that and do this properly the average things that people would have yelled out would have been 16, 8, 32, 24. That's already remarkable. That says you have some way of computing representations of sets of dots without counting them and you've even mapped them to English, but that's not the important part of this. It's that you actually can recover the number in that way. 8
But the other thing is, it's very noisy. There's a little variance around 7, a huge variance around 32. That's just the signature of analog magnitude representations of all quantities. That's how analog magnitude representations of length, area, volume, brightness and number work. So we have a representational system that takes a quantity in the world and changes it into an analog symbol of it, that is, a linear or logarithmic function of that symbol, and the operations on those symbols are subject to Weber's law. That simply says that the standard deviation is a function of the value. Here are some external analog magnitude representations of one, two, three, seven, and eight. Again, this is just showing you that in such a representational system, we can easily tell the difference between two if the ratio is two to one, or three to one, or even three to two, but it's hard to tell the difference between seven and eight. That's just Weber's law applied to length. The idea here is, is that there are neural symbols that are analogs of the quantities in the same way length is an analog of number in that case. This is a system of number representation that we share with cockroaches. It's all throughout the animal kingdom, and it's found in neonates in every species that's been looked at, including humans. It's an innate representational system. And the experiments that show that, here's just an example – I'm trying to race through this – you can habituate babies through examples of eight‐ness or to examples of sixteen‐ness, and then you give them new stimuli that are both eight and sixteen. If they were habituated to eight‐ness, they'd generalize their habituation to the new eight and they recover interest to the sixteen and vice versa. If they are habituated to sixteen, they recover to the eight. These experiments control for every other variable that could be compounded with number. It's clearly number that they're responding to here. And it's also clearly the analog magnitude system, because the only thing that determines discriminability is ratio. That's the signature of analog magnitudes. At six months it has to be a ratio of two to one. At neonates it has to be three to one. At nine months it's already improved to two to three and adults it's seven to eight. But otherwise, all the other properties of this system are the same. So this is one of the systems of number representation that is innate. The characteristics of this system are there is evolutionary and ontogenetic continuity. It's amodal. You can enumerate any kind of set, sets of claps, sets of jumps, sets of dots, and any set of individuals. Any individuals you represent, you can enumerate with this system. It's an analog format, probably, given Weber's Law. These wouldn’t be number representations except for two things. They're sensitive to the number of elements in a set and not other properties of the set, but they also support numerical computations. You can use these representations to add, subtract, do numerical comparison, calculate ratios, divide by two, and multiply by two. At least all of those have been shown in babies. 9
I'm going very quickly here. So if you want details of the experiments that back up any of these claims, ask me it during the discussion or at the reception or read my book. The second system of numerical representation, that works really different from this, is actually the first one that the field studied. We didn't start asking whether babies represented big numbers, we started by asking whether they represented small numbers. You have two sets, you have two buckets. And the task is I bring out a cracker, I put one cracker in this bucket, a second cracker in this bucket, a third cracker in this bucket, and then a cracker in this bucket, and a cracker in this bucket. Then you let the baby crawl. What you find is that babies at ten to twelve months, if it's one versus two, 80% of the time, which is essentially always for a baby, I mean that's a very high percentage, go to the one that's more. So do twelve‐month olds. Two versus three, they also reliably go the one where it's more. Three versus four they're at chance. This suggests sensitivity to number but this could be analog magnitudes because I just told you that analog magnitudes, what determines the limits is their ratio. I also told you that at nine months they can do a two to three. But they can't do three to four. So that's just what you exactly would expect at these ages. But it's the next parts that tell you it's not analog magnitudes. Three versus six, which is the same ratio as one versus two, they're completely at chance. Two versus four, they're completely at chance. One versus four they're completely at chance. Okay? Dwell on this: one ‐ two, one ‐ two ‐ three, 80% correct. One, one ‐ two ‐ three ‐ four, chance. Okay. That's really bizarre because there's five crackers in both cases. So both events overall are taking the same amount of time. Furthermore, you've drawn much more attention to this one, it took longer…there's every reason that they could have succeeded at this without representing number. But instead they're failing. So what is happening here? What is happening here is that this particular situation, when you bring out a cracker, which they really like, they focus on that cracker and you put the cracker in there. They made a mental model, there's a cracker in that, and they've got a mental model of that cracker in that bucket. You then add a second cracker, they update that model and now they have a mental model of a cracker and a cracker in that bucket. This is just like your short‐term memory models for individuals. You add a third one, they've got a cracker and a cracker and a cracker. When you try to add a fourth one, they can't make a working memory model of a cracker, a cracker, a cracker, a cracker. Neither can I. There's a u‐
shaped curve here. All of you can do four, but you would fall apart at five. This is another continuous system. But it's the working memory system, which is sharply limited to the number of individuals that you can put into it. 10
So what these models are like, here is the representation of one box as I believe an iconic – and for reasons, again, no time to go through – an iconic symbol for a box. A representation of two boxes are iconic symbols of the actual two boxes that you're representing. Three boxes are the iconic symbol of three boxes. That's as high as it can go because you can't make these models for four. Now, what you have here is explicit symbols for the boxes. There's one symbol for each box. There's no symbol for integers here, or any other quantifiers, and there are just symbols for the individuals. Why am I talking about this, then, as representations of number? Because these models are shot through with numerical content. For one thing, the symbols stand in one‐to‐one correspondence to the entities in the sets. Furthermore, there have to be computations of numerical identity to tell you whether or not to open a new symbol. You have to know that the second cracker is a different one from the first cracker, to update the representation, so that your model will have symbols that stand the one‐to‐one correspondence with the entities in the set. Numerical identity couldn't be a more numerical concept. But furthermore, there are computations defined over these models. This is not dedicated numbers. There are lots of computations. In fact, in the cracker task what they're computing is the total amount of cracker stuff. But in other cases where they're planning reaches into the things to get the things out, the only thing they care about is number and there, they're computing one‐to‐one correspondence. So there are numerical computations – like, "When have I reached in enough so there's no more left?" – that are defined over these models. So there's numerical content in this models, but the content is implicit. It's carried in the computations that allow you to set up the model so they have the right property and that can operate on the models. There are no actual explicit symbols for number here at all. I'm not going to talk about how natural language quantification works because that would take me the rest of the talk and then some. But let's grant also that there's innate support for representations for number marking and natural language quantifiers, singular, plural, things like that. You need representations of sets if you're going to have any representation of number, and all of these systems have representations of sets. So there's intentional mechanisms that have to pick out sets of individuals to be input to these representations and in the analog magnitude system the only summary symbol that you have is a symbol for the approximate cardinal value of that set. There's no symbol set. For parallel individuation there's no symbol that says set. It's just a representation of a set that you've picked out intentionally. And there's no symbols for quantifiers or cardinal values of sets. This is what I mean by saying what the representational systems are like. Now we're in a position to 11
ask, "Are these natural number? There's numerical content, but is it natural number?" No, it's not remotely natural number. Clearly parallel individuation representation as there's no symbols for integers there at all and there's a set size limit of three or four. You can't represent 100 or 7, let alone 10 to the 36 th . Analog magnitude representations, those aren't natural numbers either. At least there are symbols for cardinal values there. But they're just approximate cardinal values. You can't represent exactly 5 or exactly 15 or 32. Furthermore, it obscures the successor relation. You operate over analog magnitudes by computing ratios. You don't experience the difference between seven and eight as the same as the difference between one and two. So you're very unlikely to get from analog magnitudes alone to integers. Linguistic set‐based quantification has much more of the machinery you actually need, but it's not harnessed to natural number. There's no symbol for cardinal values above two or three, and it's not embedded in a system of arithmetical computation. So if it's true that these three systems are the only systems of mental representation that have numerical content before age 2, and if they've been characterized correctly, then we've got our discontinuity. These systems together or in combination cannot represent, you cannot define "seven" in terms of them. To learn a meaning to the word "seven," or to learn the concept "seven," requires creating a representation that plays a role in arithmetic that cannot be defined in terms of the primitives you have. Now this proposal is falsifiable if I've missed some innate machinery relevant to number for which we can give evidence. Then the story is false. But it's not like people haven't looked. These, there's just tons of evidence for, and I think that these are characterized correctly. There's tons of evidence for them. So that's the first thing you do in establishing a discontinuity. You characterize the conceptual systems at the beginning point. You characterize the conceptual system at the end point, in this case our representation of natural number. And you show that they're qualitatively different. They're qualitatively different in their content, in their format and in their computational role. And you show that you can't express the concepts in CS2 by logical combination of the concepts that are available in CS1. That's the important work in establishing a discontinuity. But there are empirical consequences of this. If this is right, these should not be cross‐culturally universal and they are not. And they should be very difficult to learn, it should be hard to learn what the meaning of "seven" is. And it is. So you check those empirical consequences as well. So counting is the first representational system that actually represents at least a finite subset of the natural numbers, one, two, three, four, five, six, seven, 12
eight, nine, ten. Deployed according to Gelman and Gallistel's counting principles, you have to – one, two, three, four, five, six, six, seven, eight, nine, ten – you have to apply the numbers in one‐to‐one correspondence with the individuals in the set you are trying to enumerate. You have to always do it in the same order and then the numeral you get to, four, represents the cardinal value of that whole set and which cardinal value it is, is given by its ordinal position in the list. If you deploy the counting principles that way, this implements the successor function. This guarantees that four is one more than three and five is one more than four, et cetera. That's implicit in it, but it does actually do it, and none of these other systems did. So if this story is right, it should be hard for children to learn how counting works and indeed it is. So here's just quick developmental facts. Children learn to count: recite the list. If you say how many fingers, they go one, two, three, four, five. They do this in enumerate culture if they're being taught to count by their middle‐class parents. They do this shortly after their second birthday. They can count to ten, and they can count ten things, and they always do it satisfying one‐to‐one correspondence and they might not exactly have economical lists, they might very early on count one, two, four, five, seven. But then they always count one, two, four, five, seven and in that case "seven" means "five." They could still be using it to count. However, the fact that they can do that doesn't mean they know what it means. One reason to think maybe they don't know what it means is that they just counted one, two, three, four, five. You say, "How many was that?" They go, one, two, three, four, five. "How many was that?" One, two, three, four, five. It's like "how many?" is a request for that routine. Now, that's not a very strong evidence that they don't understand it, because maybe they think you're asking them because they made a mistake, or maybe they think you have to say all five numbers. So it doesn't show that they don't know what it means. Karen Wynn, who believed that they did know what it means, wanted to give strong evidence that they do. So she devised a very simple task. You have a set of fish. You say, "Can you give me one fish?" Young two‐year‐olds give you one fish. "Can you give me two fish?" They pick up a huge handful of fish and they hand it to you. And you say, "Is that two? Can you count and make sure?" One, two, three, four, five, six. "But I wanted two." So they give you another fish. They haven't the slightest idea what two means. They know what one means, and if you ask for four fish, same thing. They don't give you any more, they don't estimate a bigger amount for four than two. They know that two, three, four, five and six contrast with one in number. They know what one means and they're in that state for nine months before they learn what two means. That's completely inconsistent with knowing how counting represents number. 13
When you learn Russian, all you have to do is identify the count list and then you know which every number in it means. Well these kids have identified these numerals as relevant to number. They know they contrast with the one they know. But they do not know what "two" means for nine months. Then they become two numbers. So now, "Can you give one?" One. "Can you give me two?" They don't count, they just give you two. "Can you give me three?" And again, it's always more than two, it contrasts with the ones they know, but they don't give a bigger number for ten than for three. They're in that state for three or four months. Then they become "three" knowers, then they become "four" knowers and the whole process takes almost two years. This is a hard case of learning. This is not one‐trial learning. This couldn't be less like learning the word "kangaroo" upon first seeing a kangaroo. This is just as exactly as it should be, a hard case, if the story about discontinuity is right, because they can't formulate the meaning of "seven" in terms of the conceptual resources they have. So they can't do hypothesis testing to figure out what "seven" means. So how do they do it? Okay because I'm running late, I'm just going to tell you how they do it. Basically, in all of the case studies in my book where there are discontinuities, both of this kind and also incommensurabilities as in theory changes, there's a learning process that I call Quinian bootstrapping and it works like this. You can learn relations among external symbols directly. Here you do use the commentatorial structure of your language, and also of your conceptual system in general. Those you learn just in terms of each other. These symbols can be entirely meaningless as one, two, three, four, five, six, seven, eight, nine, ten is for a kid at the beginning of counting. It might as well be eeny, meeny, miney, moe. Or they could be partially interpreted. But they don't yet have the meaning they will have at the end of the bootstrapping process. This placeholder structure, however then, can then be used to model, as a whole, some other aspect of reality by processes of analogy. Or other modeling mechanisms, but analogy is a very common one. And that modeling process is the process that both gives the placeholder structure meaning, and allows a whole system of concepts appropriately interrelated to each other to emerge as a whole without ever having been definable in terms of concepts that you had before. So that's sort of overall how these episodes of Quinian bootstrapping work. How does it work in this case? Well the meanings that are given to one, two and three in the subset knower stage, when you're a one knower, two knower, three knower, not given in terms of the count list, there a system, it's a system I called enriched parallel individuation, what you use is the machine that was there in parallel individuation, but you enrich it by making long‐term memory models. So a long‐term memory model that supports the meaning of "three" would be a 14
representation of a set of three. And then it's deployed in the following way. If I ask you, "How many do I have?" If you're a three knower, you've got long‐term memory models for one, two and three, and you ask, "Does any of them stand in one‐to‐one correspondence to the set?" I'm asking you, and you've associated the word with that. I was mentioning this once in a talk and one person raised their hand and said, "That makes sense of what my daughter did when she was late twos. Every time she saw three things she would say, oh, three chalks! Daddy, Mommy, me." Boy, that's just exactly what I'm talking about, right? So she had a representation Daddy, Mommy, me, doesn't by itself represent three. It represents Daddy, Mommy, me but when it's used in this way of being the model, such that anything that is put in one‐to‐one correspondence with that is called "three." Now it becomes a representation of three. I believe, by the way, that's the kind of representation that underlies singulars and duals, and also singular/plural distinction, and singular/dual markers I think is probably the same kind of representation. So that's what's happening during the "subset in r" period. They're building these kinds of models. But those are going to end at three or four because they can't use them to compute a one‐to‐one correspondence of bigger sets because of the limits on parallel individuation. These are genuine representations of exactly three, but they're using the resources of parallel individuation and they're going to have the same limits that parallel individuation has. So first they learn "one," and this is helped by the fact that one is a quantifier, a singular determiner. There's a lot of content that overlaps with that. And at that point, two, three or four mean something like "sum," that's the one knower stage. Then "two" is analyzed as dual marker supported by these parallel individuation models, and "three" is a trio marker, etcetera. Meanwhile, they've learned this placeholder structure, the count list. This placeholder structure has no numerical content in itself at the beginning. But there's a curious coincidence. When you count one‐two‐three, the last word in the count happens to be the same word that means "three" in this parallel individuation thing. Now that's just a really surprising coincidence. But that's the kind of coincidence to lead you to try to figure out: is there anything systematic? Noticing that coincidence leads you to attempt to try to align these two structures. When you try to align these two structures, there is a principle that makes sense of that. Namely, there's an analogy between two really different follows relations, next on the list one‐two‐three. It's an ordered list. That's the only content for the placeholder structure. It's a strictly ordered list, and next in a series of sets related by one, which parallel individuation allows. If you notice that analogy, then you might make the wild induction that four is related to three as three was to two. And you can make the induction five is 15
related to four as four was to three, even though you can't represent the set "five" in the same way you could represent the set "four." And that's the induction that not only do I think kids make, but there's very good evidence that's what separates the subset knowers from the CP knowers. They've made that induction. So the idea is, if X is followed by Y in the counting sequence, adding an individual to an X collection results in what's called a Y collection. Adding an individual is equivalent to adding one, because one represents sets containing a single individual. That gets you, at least for the first time, the new primitive "seven." You now have a new concept, which wasn't defined in terms of the numerical content in parallel individuation and analog magnitudes alone or natural language quantification. You have a new structure, the count list, that you can just use. This induction is captured implicitly in how the count list is used. But the kids understand it quite productively. The new primitives here are five, six, seven, eight, nine, ten, up to the limit of the child's count list. They are not defined alone in terms of the innately specified representations. There is a new representational structure list built, that's the count list, that's the point of bootstrapping, is that you build a new whole system of representations and then give them their meaning together. You're not defining them one by one out of things that you had before. And the numerical content is still implicit here to support this law. There are many more episodes of bootstrapping till you get to the adult concept of natural number. This story exemplifies all of these properties. Two other really short parts to do. I'm going to skip a Fodorian alternative, too bad. If you've read the Piantadosi model, Piantadosi, too bad. I'm going to skip it. Because I want to show you something that you might not have heard before. In my book, I speculate that only humans should be capable of Quinian bootstrapping since it depends upon these external linguistic placeholder structures. There's nothing that requires that the placeholder structures be in external symbols, not logically. They could be in internal symbols. But why would you ever set up a placeholder structure entirely internal to the mind? These come from culture. Basically this is a kind of learning where a new representational system has been culturally constructed, and then the child learns these placeholder structures from the culture. So they have to be external symbols for that. But is it true that only humans can do this? I thought, "Certainly, yes," and I have a colleague, Irene Pepperberg and Alex, who – this was his last experiment and he actually died in the middle of it. But luckily for us, scientifically we had enough data to answer the first question. There were so many more questions to ask that we didn't get to ask. Sadly for Irene, because if you have a parrot that 16
you've worked with eight hours a day for 30 years, it's like losing a spouse or a child. It's very sad that Alex died, and Alex is the only animal that's gotten an obituary in the New York Times. Alex, over 30 years, had been taught a lot about numbers. And already, what he'd been taught, what he did, suggests that he's got the capacities for this kind of bootstrapping. So here's what he'd been taught. First, he was taught, this is halfway in the middle [shows slide]. This is the picture of Alex. First, he was taught to answer questions like, "How many wood?" Which meant a little wood block. "How many key?" With sets of three and four. And once he learned to say "three" and "four" for "How many wood?" and "How many key," he was asked, "How many cotton ball?" "How many candy?" And he could do three and four for all of those things. So the first words he was taught were English words for "three" and "four." He could say "three" and "four." He could also, if you showed him a set of red things and green things and said, "What color three?" He would say, "Red." So he could both produce and comprehend the words "three" and "four" appropriately for sets of three and four. Presumably, this is a rich parallel individuation too. Why she started with three and four, I do not know. But that's what she did. Then she taught him two and six. He added sets of two and six. It took him two months to figure out what "two" meant, and six months to figure out what "six" meant. So at this point you could give him sets from two, three, four and five and he could appropriately tell you how many for any set, or tell you what color a set of two, three, four, five, which one, which set had that. So here's an example. "How many key," he says, "Three." "How many cork," he says, "Two." So he can pick out the appropriate set by the noun and tell you the answer. That's pretty impressive. I mean there's no claim that he's got the noun phrase syntax here, but he's got a lot of the relevant semantics. She then taught him one and six. That took another several months. So now, over a couple years of teaching, he knows the words one, two, three, four, five and six. He doesn't have them in a count routine, because he's never been taught the ordered list. But he knows their cardinal meanings. She then taught him to label Arabic numeral one. So that's what this picture is about [shows slide]. She's giving him the Arabic labels and she's asking him, she says, "Pick up two," and he's picking up the Arabic numeral that is two. So he's just learning the words apply to the Arabic numerals. The Arabic numerals are never paired with sets. So she doesn't teach him that the Arabic numeral two refers to a set of two things. But there is an association, like the one that the child must use to unite counting and their parallel individuation, namely the same word. The word "two" applies to that and the word "two" applies to sets of two. She then, without any further training, gave him a red Arabic numeral four and a green Arabic numeral three and said, "Which color bigger?" Now he had never used the word "bigger" to refer to number, but he had used it to refer to size. So 17
if you have a big stick and a little stick, this one red and this one blue, you say, "What color bigger?" He'd say, "Red." Without any further training at all, he answered, "Which color bigger?" in terms of the numerical values of one, two, three, four, five and six. So he said the color of the Arabic numeral six. "Which color bigger?" "Red," if six was paired with five, four, three, two or one, for any two pairs within that, with no training whatsoever. So already this shows a capacity to integrate representations, and recover, in this – this is actually a metaphorical way. That's a different notion of "bigger" than the "bigger" that he'd ever used that word for. This was all before I came on board. At this point, he knows the words "one" through "six." He knows the Arabic numerals 1 through 6. He knows the Arabic numerals are ordered, and he has to have derived that order from their order in cardinal value, not from their place in the list, because he never was given the list. We were in a position to teach him two new numerals, 7 and 8. So, we taught him a new word for "six" instead of "six," which he pronounced, "si." We changed it to "sisi" because we wanted a second two‐syllable word, because "seven" he called "sinon." And we wanted that because we didn't want to cue him by not waiting till the end of an answer. So he has two two‐syllable words, "sisi" and "sinon." It was easy for him to learn to say "sisi" for "six." "Sinon" was hard to say, "eight" was easy to say. But anyway, it took a whole year to learn to label Arabic numeral 6 sisi, Arabic numeral 7 sinon and Arabic numeral 8 eight. The reason it took so long is this was very boring. What's interesting about these Arabic numerals? Because they don't have any meaning, they're just shapes. And for some reason, the way she teaches him is, she teaches him to say something, then he gets it. But he didn't care about these. But he did, he learned it, after a year, to a criterion where a listener was blind to what he was looking at could identify what he said 100% of the time. So he does eventually learn this. Now he doesn't really have any evidence that "seven" and "eight" are numbers here, because although he was also being taught by "six", he was always taught these in the context of labeling lots of other things that aren't numbers. So he knows these words, but he doesn't know what kind of things they are. So then she teaches him, she just gives him pairs of Arabic numerals, 6 and 7, 7 and 8, 6and 8, and she asks, "Which color bigger?" or "Which color smaller?" for every pair. Now, this is the first time that he's given any evidence there's any relation between six, seven and eight. It took him about an hour to learn this. This was really easy for him to learn. I mean, obviously he had no basis for doing this, because he doesn't know what cardinal values they are. He just has to arbitrary learn that seven is bigger than six, and eight is bigger than seven, and eight is bigger than six, and six is less than seven, and seven is less than eight. But he can learn that. So now he actually has an ordered list from one through 18
eight, and we tested that by, after he learned the 6, 7 and 8, giving him all pairs of 7 and 8 with all the other numbers and he, of course, did those perfectly too. So he now has an ordered list: one, two, three, four, five, six, seven and eight. Question: does he infer from the place in that ordered list that "seven" refers to cardinal value seven, and "eight" refers to cardinal value eight? This is a wildly unwarranted induction, because what we've taught him is consistent with "seven" being a hundred and "eight" being a thousand. All he knows is six is less than seven and seven is less than eight. However, every other time he's ever been taught numbers, it's been off by one from the other numbers he knew. So although it's not a warranted induction, it's a sensible induction. And it's the same sensible induction the children make, which is not a warranted induction in their case either. So how do we test that? We showed him a set of seven and we asked how many. The very first time we showed him a set of seven and asked how many, he looked at her and he said, "Sinon." The very first time we showed him a set of eight and asked him how many, he looked at it and said, "Sinon. Oops, eight." He didn’t actually say oops. But he self‐corrected. Then, over the next couple of weeks, one trial a day mixed in with the experiments he was doing with visual illusions and all kinds of other things – and they were different experimenters every time. Irene, although she does the training, does not do the testing – different experimenters every time had a protocol of what they were supposed to do with him. One of them was to show him a set and ask how many. He gets seven of the twelve where the set of seven is correct, which is significantly more than chance because he always answers six, seven or eight. He gets nine of the twelve where it's eight correct. And we also asked comprehension. Six, seven, eight, nine and ten, mix these sets together, three sets at a time. This is a mess, all intermixed and you say, "What color seven," or "What color eight," or "What color nine – not nine – or, six?" He was 100% correct when probed with seven, and four out of five when probed with eight. But the mistake really wasn't really a mistake, but we – in the paper, we wrote down every single response he did and this is what it was. Eight yellow sticks, six red sticks, "What color eight?" He says, "Six red." So we counted this wrong, but he actually answered something that was true, that namely there were six red. So we repeated, "What color eight?" and he said, "Yellow." So you could say either he got four out of five or five out of five correct. He made the same induction kids do. It had to be the same bootstrapping story. The only thing he had to go for. Is there hypothesis testing here? No. He never got any feedback. He never got any input. He wasn't once showed the pairing between seven and eight and told, "This is seven." He was not corrected when he made it wrong, and he wasn't rewarded when he got it right. 19
So the conclusion. He induced the cardinal meanings for sinon and eight, from knowledge of the meanings of one through six, plus the information six is smaller than seven and seven is smaller than eight. The only way he could be doing this is aligning the same two structures in the way that I say kids are doing it. I've gone too long. I'm going to stop. But there is another example of rhesus macaques doing that kind of alignment, in this case between a list between one and twenty‐one, and quantities to twenty‐one. And again, it just shows that the kind of alignment process that plays a role in the analogy part of this bootstrapping mechanism is widely available. It's plausible that children have it, and a parrot has it, and rhesus macaques have it. It's natural to construct mappings between these long‐ordered lists and sets of quantities—in the case of the rhesus macaque, it's quantities of liquid as well as of number – and to integrate these. So how does bootstrapping work? It creates new primitives by creating new data structures of blank predicates in terms of available common chorial resources, here the ordered list. It uses these to model phenomena already represented, here sets, cardinal values in terms of analog magnitude, parallel individuation, continuous variables such as liquids. By satisfying the constraints given by both structures, the content that's implicit in one's structure can play a role, because basically what you're trying to do, when you do those alignments, is satisfy all the constraints you've got. The numerical content in parallel individuation is only implicit. But in making this system satisfy those constraints, you end up with an explicit representation set. So the numerical content is not coming from nowhere, but it's not constructing it by logical combination from primitives you already have. And that's the bottom line here. So implications for theory of learning? The importance of conceptual role in determining content? In the easy case the conceptual role is innate. In the hard cases the conceptual role is constructive, is partially implemented in the placeholder structure and, of course, it's already present in the parts that are modeled. Finally, I've tried to illustrate all parts of this story with this case study and also give you a sense of how it answers Fodor's challenge to cognitive science. Thanks. [APPLAUSE] Questions. MALE VOICE #1: [off mic] 20
MALE VOICE #2: Regarding the Indigo Buntings. Is it possible, in our struggle to ascertain an explanation for their navigation, that there would be, which would be more likely for me, a sixth sense, which provides these birds with the intelligence to do this? I mean even if they were navigating by Polaris, how would they land at a certain destination? We can't even explain why birds fly to begin with. PROFESSOR CAREY: Sure. I agree with all of that. I agree that this story doesn't explain how they know when they've gotten to the right place at the other end. It's not trying to do that. The only thing it's trying to do is explain how they know which direction to go. The field of ethology actually has attempted to answer all of those questions. Do they know to stop because they've got some sort of analog representation of distance? So when they've gone 3500 miles it says stop? Or is there something about the topography of where they've gotten to that tells them they're in the right place? It's the latter. So the point is: For each question you might want to ask of how they do this whole feat, you can pose alternative hypotheses about the information they're learning, do these kinds of experiments to get at what the answers to those questions are, and then do the next step for saying, "Okay, well, how do they know that information is relevant? Is it innate by itself or did they learn it in some way?" I often do this case study in undergraduate classes because it's so easy to understand, and it's so satisfying, what Emlen showed. I ask people at the beginning what they think the mechanism could be, and they come up with four. They come up with celestial navigation, navigating by the sun, following somebody else who knows – and that's actually not such a good one because even if you're following somebody that knows, how that helps you to look up in the sky and figure out that it's Polaris that the other guy is following is hard. But in fact, we can rule that out, because isolated nestlings who are raised in the planetarium all by themselves, take off in the right direction the first time they fly. So you're not learning it from somebody else. But you do fly in flocks. So maybe you learn when to stop from somebody else. So you take what you know about the circumstances under which they do it to formulate testable hypotheses and test them. But I don't think there's anybody thinks there's a magical sixth sense. I don’t think we need a magical sixth sense. The idea is you can actually answer these questions. It's hard, but what's so nice about this case is how satisfying the answer is. We really do know how they set their direction and that is really cool. But we also know a lot about the other parts too, and that's what the whole field of ethology does. MALE VOICE #2: I was just suggesting that there is a sixth sense and we can't measure or explain it in physical terms. 21
PROFESSOR CAREY: It's possible. No, it's totally possible. It's certainly possible in science, and it's always been so, that there are whole mechanisms and processes that we don't know anything about. We're not even in the right hypothesis space. That's always possible. But the fact is, it seems like we are, because we really answered it for Polaris. We don't need anything magical or that we didn't already know about. There's no evidence that we're coming up to questions that we can't answer yet. There are plenty of examples in cognitive science where it's likely we're missing the important ideas, like, how do we explain consciousness. There we need something that we don't know about yet. But I don't think that's true in these cases. But it's always possible. MALE VOICE #3: In the cases of these inductions, how much do you think that the induction is driven by – an observation either internally or from feedback – that the concepts that are already there insufficient to allow you to answer the question you're being asked. The kid ‐ there's a whole bunch of fish and the person says, "No, no," ‐‐ more information kind of by falsifying the hypothesis. What I know can explain everything I have to deal with. PROFESSOR CAREY: Yes. So it's certainly not in the case of Alex because he never got any feedback at all. MALE VOICE #3: [off mic] PROFESSOR CAREY: He's being asked the question, "How many?" And he knows it's not one, two, three, four, five and six. But that doesn't tell him it's seven. And that's why, of course, we taught him seven and eight because it's certainly natural that if you ask him how many, and he knows it's not one, two, three, four, five, six, if the only other numeral he knew was seven, then it wouldn't be interesting if he said "Seven." But he knew two other numerals and he knew that eight was two more than six and seven was one more. And he had no feedback with respect to that. What I do think that one of the things that motivates…the kids seem perfectly happy when you ask them for two and they pick up the handful and you give them two. And you say, "But I wanted two," now when you say that, you're giving them some feedback that they didn't do it right and they don't know what it is. However, an interesting fact is that the parents who bring their kids into this study are astonished that their kids don't know what counting means because they count with their kids and the kids are counting right. One, two, three, four, five, it doesn't occur to them what the counting means. So I don’t think they've gotten a lot of feedback from their parents that they're using "two" wrong, because the kids don't spontaneously say, "Oh, there are five people here," when there's only three. I mean they don't use numbers if they don't know which it is. If they're two knowers they use "two" correctly. I want 22
two cookies. But they don't say, "I want seven." They don't pick another one randomly. So the parents don't get a lot of evidence that they don't know the meanings of the words, and they're actually astonished to learn it. I mean they are really surprised. And we warn them in advance because otherwise, as soon the kid gets it wrong the parents start to try to elicit the right answer from the kids. So we tell them in the waiting room, "Look, you're going to learn something about your kids here. Believe me, I'll tell you afterwards, this is really interesting, and it's nothing to worry about. And it is really, really interesting and here's why it's really interesting," and they're fine with it. I do think, however, that some kind of internal comprehension monitoring is probably a motivation, not very strong for two‐year‐olds, and certainly not strong for Alex. I mean the fact that Alex and this rhesus macaque do this, first of all, is very surprising to me, was surprising to me. But second of all, actually supports the kind of bootstrapping story that I'm telling. Why Alex has the capacity to do what he does, I don't know. When does a parrot in the wild use bootstrapping to find over external verbal symbols? I mean I assume if rhesus macaques and parrots have it that lots of animals have these capacities. What role they play in the minds of parrots, I don't know. But I do think that, in the case of conceptual development in children, when they have evidence they don't understand something, that is motivation to work on it. MALE VOICE #4: I have a question about the relevance of the fact that it's verbal in the case that you've talked most about and in the case of number. So things are hanging off the verbal count, but there are lots of other things that are not verbal that could have a similar kind of structure. For example, you might have a stereotype route that you follow for foraging, or there's a certain number of berries that you get bushes from. Or there are other aspects of physical environments that provide their own kind of logic. So walking around in geometry, and so on. Do you think that for domains outside of number that for domains we might care about in cognitive development, that it's often for humans going to be language that has this bootstrapping role? Or do you think that language doesn't really have a disproportionate role in this. We can bootstrap off lots of things and, in fact, people do. PROFESSOR CAREY: I think language has a disproportional role, but that's just because one of the sources of setting up these structures is that people tell you them. The structures that allow you to bootstrap the meanings in Newtonian mechanics are people telling you things like, force equals mass times acceleration. They tell you that in language. But do I think it's a privileged role? Absolutely not. It doesn't have to be language, and the other really common one that's used all the time is – well, there's two. There's diagram‐type representations that play these roles, and also mathematical. 23
In the history cases, and also in some of the case studies in my book, mathematical structures are the understood ones that are playing the role to model in some scientific domain, like concepts of weight and density or concepts of mass in physics. The mathematics is the placeholder structure that you're trying to model the physical things in terms of. Basically just think about this: Why is Descartes such an important figure in the history of mind? Well, one thing is Cartesian coordinates. What he did was discover the mapping between geometric representations of functions and algebraic ones. When you construct that mapping – in that case, you had both already, nonetheless – you gain a tremendous amount of understanding by seeing how the geometric ones map onto the algebraic level. That's because things that are implicit in one are explicit in the other. In many cases of conceptual change in science, the abductive guess was that a certain kind of mathematics would be relevant, and then three years are spent trying to do that modeling. When it pays off, often it's wrong, and so we don't hear about it. When it pays off, it's Maxwell. And there it's not language, it's mathematics that's the structure. So there's nothing necessary about its being language. But I think the reason that it's often external symbols, is that the source of these abductive guesses are often coming from other people. Conceptual change happens in human beings. There's a reason science is collaborative. Maxwell did it in one mind, but he had Faraday, who had discovered the phenomena of electromagnetic fields, but he didn't have the concepts. And he had Newton, who discovered the mathematics of mechanics and dynamics. And he, Maxwell, was an expert on the mathematics of Newtonian forces in a fluid medium, which he could never have learned. All of this happens with the cultural construction of knowledge interacting with individual development, both in child development and in Alex and in the rhesus macaques. Of course it's not the ordinary situation there. But people are doing that for you. I think it's because you learn these placeholder structures in a communicative process that they tend to be external symbols. But logically, as I said, they wouldn’t have to be. It would work in the same way. As you said, it's true, there's lots of things that have the structure of an ordered list, if you always walk the same route. But the question is: What would ever lead you to model sets in terms of that? We have the capacity to represent ordered lists for lots of reasons, including that one. But including we use it for language of course all the time, we use order all the time. So yeah, I don’t think that it has to be language. MALE VOICE #4: [off mic] one last question before we ‐‐ [approaching mic]. The various ways in psychology when we interpret it is extremely important for us. But there's also ancient Greek thought that went into these numbers by 24
Pythagoras, and his contribution to triangles, circles, squares, and we know all that. But what do they mean? Keeping that aside, Socrates before he died had asked Plato, sometime in 399 BC, "How does one person and one person come together and become two persons, whereas separately they were one person each, but by their mere juxtaposition they become two persons? I do not understand." What do you say to that? PROFESSOR CAREY: I say that it is all a matter of our construal. I mean the people haven't changed. One of the basic conceptual tricks we have is to represent sets. But you know, I can look at you and think of you as two individuals, or I can think of you as a set of two. Why you sometimes represent two individuals as a member of a single set, I'm not sure there's a general answer to that. But sets can be the units of working memory, the units of visual working memory, the units of the kind of working memory that we studied in old Ebbinghaus‐type working memory. Representations of sets are at the heart of the quantificational structure of language. So the answer to that question is, "What is a set? And why do we represent it?" I don't know what would count as an answer, but that's where the answer is. And that we do is beyond dispute. [APPLAUSE] 25