Presented at conference on: "Access to the Abstract." University of Southern Denmark Odense, 30-31 May 2003. http://www.ecs.soton.ac.uk/~harnad/Temp/abstract.htm
(or: Living Within One's Means)
Université du Québec à Montreal
Pensar es olvidar diferencias, es generalizar, abstraer. En el
abarrotado mundo de Funes no había sino detalles, casi inmediatos.
We are accustomed to thinking that a primrose is "concrete" and a prime number is "abstract," that "roundness" is more abstract than "round," and that "property" is more abstract than "roundness." In reality, the relation between "abstract" and "concrete" is more like the (non)relation between "abstract" and "concave," "concrete" being a sensory term [about what something feels like] and "abstract" being a functional term (about what the sensorimotor system is doing with its input in order to produce its output): Feelings and things are correlated, but otherwise incommensurable.
Everything that any sensorimotor system such as ourselves manages to categorize successfully is based on abstracting sensorimotor "affordances" (invariant features). The rest is merely a question of what inputs we can and do categorize, and what we must abstract from the particulars of each sensorimotor interaction in order to be able to categorize them correctly. To categorize, in other words, is to abstract. And not to categorize is merely to experience.
Borges's Funes the Memorious, with his infinite, infallible rote memory, is a fictional hint at what it would be like not to be able to categorize, not to be able to selectively forget and ignore most of our input by abstracting only its reliably recurrent invariants. But a sensorimotor system like Funes would not really be viable, for if something along those lines did exist, it could not categorize recurrent objects, events or states, hence it could have no language, private or public, and could at most only feel, not function adaptively (hence survive).
Luria's "S" in "The Mind of a Mnemonist" is a real-life approximation whose difficulties in conceptualizing were directly proportional to his difficulties in selectively forgetting and ignoring.
Watanabe's "Ugly Duckling Theorem" shows how, if we did not selectively weight some properties more heavily than others, everything would be equally (and infinitely and indifferently) similar to everything else.
Miller's "Magical Number Seven Plus or Minus Two" shows that there are (and must be) limitations on our capacity to process and remember information, both in our capacity to discriminate relatively (detect sameness/difference, degree-of-similarity) and in our capacity to discriminate absolutely (identify, categorize, name),
The phenomenon of categorical perception shows how selective feature-detection puts a Whorfian "warp" on our feelings of similarity in the service of categorization, compressing within-category similarities and expanding between-category differences by abstracting and selectively filtering inputs through their invariant features, thereby allowing us to sort and name things reliably.
Language does allow us to acquire categories indirectly through symbolic description ("hearsay," definition) instead of just through direct sensorimotor trial-and-error experience, but to do so, all the categories named and used in the description must be recursively grounded in direct sensorimotor invariants. Language is largely a way to ground new categories by recombining already grounded ones, often by making their implicit invariant features into explicit categories too.
If prime numbers differ from primroses, it is hence only in the degree to which they happen to be indirect, explicit, language-mediated categories. Like everything else, they are recursively grounded in sensorimotor invariants. The democracy of things is that, for sensorimotor systems like ourselves, all things are just absolute discriminables: they number among those categories that our sensorimotor interactions can potentially afford, no more, no less. A primrose affords dicotyledonousness as reliably (if not as surely) as a numerosity of 6 (e.g., 6 primroses) affords factoring (whereas 7 does not).
Consider the five kinds of (sensorimotor) things that we can do with things: We can see them, recognize them, manipulate them, name them or describe them. "Manipulate" in a sense already covers all five, because manipulating is something we do with things; but let us reserve the word "manipulate" for our more direct physical interactions with objects, such as touching, lifting, pushing, building, destroying, eating, mating with, and fleeing from them. Naming them and describing them is also a thing we do with them, but let us not subsume those two acts under manipulation. Seeing and recognizing are likewise things we do with things, but these too are better treated separately, rather than as forms of manipulation. And "seeing" is meant to stand in for all modes of sensory contact with things (hearing, smelling, tasting, touching), not just vision.
Recognizing is special, because it is not just a passive sensory event. When we recognize something, we see it as a kind of thing (or an individual) that we have seen before. And it is a small step from recognizing a thing as a kind or an individual to giving it a name. Seeing requires sensorimotor equipment, but recognizing requires more. It requires the capacity to abstract. To abstract is to single out some subset of the sensory input, and ignore the rest. For example, we may see many flowers in a scene, but we must abstract to recognize some of them as being primroses. Of course, seeing them as flowers is itself abstraction. Even distinguishing figure from ground is abstraction. Is any sensorimotor event not abstraction?
To answer, we have to turn to fiction. Borges, in his 1944 short story, "Funes the Memorious," describes a person who cannot abstract. One day Funes fell off a horse, and from then onward he could no longer forget anything. He had an infinite rote memory. Every successive instant of his experience was stored forever; he could mentally replay the "tapes" of his daily experience afterwards, and it would take even longer to keep re-experiencing them than it had to experience them in the first place. His memory was so good that he gave proper names or descriptors to all the numbers -- "Luis Melián Lafinur, Olimar, azufre, los bastos, la ballena, el gas, la caldera, Napoléon, Agustín de Vedía" -- from 1 all the way up to enormous numbers. Each was a unique individual for him. But, as a consequence, he could not do arithmetic; could not even grasp the concepts. The same puzzlement accompanied his everyday perception. He could not understand why we people with ordinary, frail memories insisted on calling a particular dog, at a particular moment, in a particular place, in a particular position, by the same name that we call it at another moment, a different time, place, position. For Funes, every instant was infinitely unique, and different instants were incomparable, incommensurable.
Funes's infinite rote memory was hence a handicap, not an advantage. He was unable to forget, yet forgetting, or at least ignoring, is what is required in order to recognize and name things. Strictly speaking, a true Funes could not even exist, or if he did, he could only be a passive sensorimotor system, buffeted about by its surroundings. Borges portrayed Funes as having difficulties in grasping abstractions, yet if he had really had the infinite memory and incapacity for selective forgetting that Borges ascribed to him, Funes should have been unable to speak at all, for our words all pick out abstractions. He should not have been able to grasp the concept of a dog, let alone any particular dog, or anything else, whether an individual or a kind. He should have been unable to name numbers, even with proper names, for a numerosity (or a numeral shape) is itself an abstraction. There should be the same problem of recognizing either a numerosity or numeral as being the same numerosity (numeral) on another occasion as there was in recognizing a dog as the same dog, or as a dog at all.
Funes was a fiction, but Luria described a real person who had handicaps that went in the same direction, though not all the way to an infinite rote memory. In "The Mind of a Mnemonist" (1968) Luria describes a stage memory-artist, "S," whom he had noticed when S was a journalist because he never took notes. S did not have an infinite rote memory like Funes's, but a far more powerful and persistent rote memory than a normal person. When he performed as a memory artist he would memorize long strings of numbers heard only once, or all of the objects in the purse of an audience member. He could remember the exact details of scenes, or long sequences. He also had synaesthesia, which means that sensory events for him were richer, polysensory experiences: sounds and numbers had colors and smells; these would help him remember. But his powerful rote memory was a handicap too. He had trouble reading novels, because when a scene was described, he would visualize a corresponding scene he had once actually seen, and soon he was lost in reliving his vivid eidetic memory, unable to follow the content of the novel. And he had trouble with abstract concepts, such as numbers, or even ordinary generalizations that we all make with no difficulty.
What the stories of Funes and S show is that living in the world requires the capacity to detect recurrences, and that that in turn requires the capacity to forget or at least ignore what makes every instant infinitely unique, and hence incapable of exactly recurring. Although I am not a Gibsonian, I think JJ Gibson's (1979) concept of an "affordance" capture the requisite capacity nicely: Objects afford certain sensorimotor interactions with them: A chair affords sitting-upon; flowers afford sorting by color, or by species. These affordances are all invariant features of the sensory input, or of the sensorimotor interaction with the input, and the organism has to be capable of detecting these invariants selectively -- of abstracting them. If all sensorimotor features are somehow on a par, and every variation is infinitely unique, then there can be no abstraction of the invariants that allow us to recognize sameness, or similarity, or identity, whether of kinds or of individuals.
Watanabe's (1985) "Ugly Duckling Theorem" captures the same insight. He describes how, considered only logically, there is no basis for saying that the "ugly duckling" -- the odd swanlet among the several ducklings in Hans Christian Anderson's fable -- can be said to be any less similar to any of the ducklings than the ducklings are to one another. The only reason it looks as if the ducklings are more similar to one another than to the swanlet is that our visual system "weights" certain features more heavily than others -- in other words, it is selective, it abstracts certain features as privileged. For if all features are given equal weight and there are, say, two ducklings and a swanlet, in the spatial position D1, S, D2, then although D1 and D2 do share the feature that they are both yellow, and S is not, it is equally true that D1 and S share the feature that they are both to the left of D2 spatially, a feature they do not share with D2. Watanabe pointed out that if we made a list of all the (physical and logical) features of D1, D2, and S, and we did not preferentially weight any of the features relative to the others, then S would share exactly as many features with D1 as D1 shared with D2 (and as D2 shared with S). This is an exact analogue of Borges's and Luria's memory effect, for the feature list is in fact infinite (it includes either/or features too, as well as negative ones, such as "not bigger than a breadbox," not double, not triple, etc.), so unless some features are arbitrarily selected and given extra weight, everything is equally (and infinitely) similar to everything else.
But of course our sensorimotor systems do not give equal weight to all features; they do not even detect all features. And among the features they do detect, some (such as shape and color) are more salient than others (such as spatial position and number of feathers). And not only are detected features finite and differentially weighted, but our memory for them is even more finite: We can see, while they are present, far more features than we can remember afterward.
The best illustration of this is the difference between relative and absolute discrimination that was pointed out by George Miller in his famous 1956 paper on our brains' information-processing limits: "The Magical Number 7+/-2". If you show someone an unfamiliar, random shape, and immediately afterward show either the same shape again or a slightly different shape, they will be able to tell you whether the two successive shapes were the same or different. That is a relative discrimination, based on a simultaneous or rapid successive pairwise comparison. But if instead one shows only one of the two shapes, in isolation, and asks which of the two it is, and if the difference between them is small enough, then the viewer will be unable to say which one it is. How small does the difference have to be? The "just-noticeable-difference" or JND is the smallest difference that we can detect in pairwise relative comparisons. But to identify a shape in isolation is to make an absolute discrimination, and Miller showed that the limits on absolute discrimination were far narrower than those on relative discrimination.
Let us call relative discrimination "discrimination" and absolute discrimination "identification." Differences have to be far greater for identifying what something is than for discriminating it from something else that is simultaneously present or viewed in rapid succession. Miller pointed out that if the differences are along only one sensory dimension, such as size, then the number of JNDs we can discriminate is very large, and the size of the JND is very small, and depends on the dimension in question. In contrast, the number of values along the dimension for which we can identify the object in isolation is approximately seven. If we try to subdivide any dimension more finely than that, identification errors grow.
This limit on identification capacity has its counterpart in memory too: If we are given a string of digits to remember we -- unlike Luria's S, who can remember a very large number of them -- can recall only about 7. If the string is longer, errors and interference grow.
Is there any way to increase our capacity to make absolute identifications? One way is to add more dimensions of variation; presumably this is one of the ways in which S's synaesthesia helped him. But even higher dimensionality has its limits, and never approaches the resolution power of the JND. Another way of increasing memory is by recoding. Miller showed that if we have to remember a string of 0's and 1's, then a string of 7 items is about our limit. But if we first learn to recode the digits into, say, triplets in binary code, using their decimal names -- so that 001 is called "one", 010 is called "two," 011 is called "three" etc., and we overlearn that code, so that we can read the strings automatically in the new code, then we can remember three times as many of the digits. The 7-limit is still there, but it is now operating on the binary triplets into which we have recoded the digits: 101 is no longer three items: it is recoded into one "chunk," "five." We have learned to see the strings in terms of bigger chunks -- and it is these new chunks that are now subject to the 7-limit, not the single binary digits.
Recoding by overlearning bigger chunks is a way to enhance rote memory for sequences, but something similar operates at the level of features of objects: Although the number of features our sensory systems can detect in an object is not infinite, it is large enough so that if we see two different objects, sharing one or a few features, we will not necessarily be able to detect that they share features, hence that they are the same kind of object. (This is called "underdetermination" and is related to the so-called "credit assignment problem": how to find the winning feature or rule among many possibilities.) To be able to abstract the shared features, we need identification training, with trial and error and corrective feedback based on a large enough sample to allow our brains to abstract the invariants underlying the variation. The result, if the learning is successful, is that the inputs are recoded, just as they are in the digit string memorization: The objects that are of the same kind, because they share invariant features, are consequently seen as more similar to one another; and objects of different kinds, not sharing the invariants, are seen as more different.
This within-category enhancement of perceived similarity and between-category enhancement of perceived differences is called "categorical perception" (CP). When it is an effect of learning, it is a kind of a Whorfian effect. Whorf (1956) suggested that the way objects look to us depends on how we sort and name them. He cited colors as an example of CP, but the evidence suggests that the qualitative color-boundaries along the visible spectrum are a result of inborn feature detectors rather than of learning to sort and name colors in particular ways. Learned CP effects are subtler, and can only be demonstrated in the psychophysical laboratory, but they work much the way inborn CP does: Some features are selectively enhanced, others are suppressed, thereby bringing out the commonalities underlying categories or kinds. This works like a kind of input filter, filtering out the categories according to their invariant features, and ignoring non-invariant features. Certain neural networks have been proposed as the possible mechanisms for this abstracting capacity, with sensorimotor interactions also helping us to converge on the affordances, resolving the underdetermination and solving the credit-assignment problem.
Where does this leave the concrete/abstract distinction, then? In what sense is a primrose concrete and a prime number abstract? And how is "roundness" more abstract than "round," and "property" more abstract still? Identifying any kind is always based on abstraction, as the example of Funes shows us. To recognize a wall as a wall rather than, say, a floor, requires us to abstract some of its features, of which verticality, as opposed to horizontality, is a critical one here (and sensorimotor interactions and affordances obviously help narrow the options). But in the harder, more underdetermined cases like chicken-sexing, what determines which features are critical? The old joke from Maine is apropos here: "How's your wife?" "Compared to what?"
Although identification is an absolute judgment, in that it is based on categorizing an object in isolation, it is relative in another sense: What invariant features need to be abstracted depends on what the alternatives are. "Compared to what?" The invariance is relative to the variance. Information, as we learn from formal information theory, is something that reduces the uncertainty among alternatives. So when we learn to categorize things, we are learning to sort the alternatives that might be confused with one another. Sorting walls from floors is rather trivial, because the affordance difference is so obvious already, but sorting the sex of newborn chickens is harder, and it is even rumoured that the invariant features are ineffable in that case: They cannot be described. The only way to learn them is through months or years of trial and error experience training at the feet of the grand-master, 7th-degree black-belt chicken-sexers (most of whom, appropriately, live in Japan).
But let us not mistake the fact that it is difficult to make them explicit verbally for the fact that there is anything mysterious about the features underlying chicken-sexing -- or any other subtle categorization. Biederman & Shiffrar (1987) did a computer-analysis of newborn chick-abdomens and identified the invariant features (in terms of elementary features called "geons."). Biederman was then able to teach them explicitly to a sample of novices so that within 10 minutes they were able to sex chicks at the brown-belt level, if not the black belt level. This progress should have taken them months, according to the masters.
So if we accept that all categorization, great and small, depends on abstracting some features and ignoring others, then all categories are abstract. Only Funes lives in the world of the concrete, and that is the world of mere passive experiential flow from one infinitely unique instant to the next. For to do anything systematic or adaptive with the input would require abstraction, whether innate or learned: the detection of recurrence of a thing of the same kind.
What about degrees of abstractness? (Having, with G.B. Shaw, identified the profession, we are now merely haggling about the price.) When I am sorting things as instances of a round-thing and a non-round-thing, I am sorting things. This thing is round, that thing is non-round. When I am sorting things as instances of roundness and non-roundness, I am sorting features of things. Or rather, the things I am sorting are features (also known as properties, when we are not just speaking about them in a sensorimotor sense). And features themselves are things too: roundness is a feature, an apple is not (although any thing, even an apple, can also be a part, hence a fetaures, of another thing).
In principle, all this sorting and naming could be applied directly to sensorimotor inputs; but much of the sorting and naming of what we consider more abstract things, such as numbers, is applied to symbols rather than to sensorimotor interactions with objects. I name or describe an object, and then I categorize it: "A number is an invariant numerosity" (ignoring the variation in the kinds or individuals involved). This simple proposition already illustrates the adaptive value of language: Language allows as to acquire new categories without having to go through the time-consuming and risky process of direct trial-and-error learning. Someone who already knows can just tell me the features of an X that will allow me to recognize it as an X. (This is rather like what Biederman did for his experimental subjects, in telling them what features to use to sex chickens, except that his method was hybrid: It was show-and-telling, not just telling, because he didn't just describe the critical features verbally, but also pointed them out and illustrated them visually. He did not pretrain his subjects on geon-naming, as Miller's subjects were pretrained on naming binary triplets.)
If Biederman had done it all with words, through pure hearsay, he would have demonstrated the full and unique category-conveying power of language: In sensorimotor learning, the abstraction usually occurs implicitly. The neural net in the learner's brain does all the hard work, and the learner is merely the beneficiary of the outcome. The evidence for this is that people who are perfectly capable of sorting and naming things correctly usually cannot tell you how they do it. They may try to tell you what features and rules they are using, but as often as not their explanation is incomplete, or even just plain wrong. This is what makes cognitive science a science; for if we could all make it explicit, merely by introspecting, how it is that we are able to do all that we can do, then our introspection would have done all of cognitive science's work for it. But we usually cannot make our implicit knowledge explicit, just as the master chicken-sexers could not. Yet what explicit knowledge we do have, we can convey to one another much more efficiently by hearsay than if we had to learn it all the hard way, through trial-and-error experience. This is what gave language the powerful adaptive advantage that it had for our species; Cangelosi & Harnad 2001).
Where does this leave prime numbers then, relative to primroses? Pretty much on a par, really. I, for one, do not happen to know what primroses are. I am not even sure they are roses. But I am sure I could find out, either through direct trial and error experience, my guesses corrected by feedback from the grandmasters, and my neural nets busily and implicitly solving the credit-assignment problem for me, converging eventually on the winning invariants; or, if the grandmasters are willing and able to make the invariants explicit for me in words, I could find out what primroses are through hearsay. It can't be hearsay all the way down, though. I will have had to learn some things the hard, sensorimotor way, if the words used by the grandmasters are to have any sense for me. The words have to name categoroes I already have.
Is it any different with prime numbers? I know they are a kind of number. I will have to be told about factoring, and will probably have to try it out on some numbers to see what it affords, before recognizing that some kinds of numbers do afford factoring and others do not. The same is true for finding out what deductive proofs afford, when they tell me more about further features of prime numbers. Numbers themselvesI will have had to learn at first hand, guided by feedback in absolutely discriminating numerosities as provided by yellow-belt arithemeticians -- for here too it cannot be hearsay all the way down. (I will also need to experience counting at first hand, and especially what "adding one" to something, over and over again, affords.)
But is there any sense in which primroses are "realer" than prime numbers? Any more basis for doubting whether one is really "out there" than the other? The sense in which either of them is out there is that they are both absolute discriminables: Both have sensorimotor affordances that I can detect, either implicitly, through concrete trial-and-error experience, guided by corrective feedback (not necessarily from a live teacher, by the way: if, for example, primroses were edible, and all other flowers toxic, or prime numerosities were fungible, and all others worthless, feedback from the consequences of the sensorimotor interactions would be feedback enough); or explicitly, through symbolic descriptions (as long as the symbols are grounded, directly or recursively, in concrete trial-and-error experience; Harnad 1990). The affordances are not imposed by me; they are "external" constraints, properties of the outside world, if you like, governing its sensorimotor interactions with me. And what I do know of the outside world is only through what it affords (to my senses, and to any sensory prostheses I can use to augment them). That 2+2 is 4 rather than 5 is hence as much of a sensorimotor constraint as that projections of nearer objects move faster along my retina than those of farther ones.
Mere cognitive scientists (sensorimotor roboticists, really) should not presume to do ontology at all, or should at least restrict their ontic claims to their own variables and terms of art -- in this case, sensorimotor systems and their inputs and outputs. By this token, whatever it is that "subtends" absolute discriminations -- whatever distal objects, events or states are the sources of the proximal projections on our sensory surfaces that afford us the capacity to see, recognize, manipulate, name and describe them -- are all on an ontological par; subtler discriminations are unaffordable.
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