Animals, abstraction, arithmetic and language
|July 24, 2016||Posted by vjtorley under Animal minds, Intelligent Design|
During the past two weeks, over at Evolution News and Views, Professor Michael Egnor has been arguing that it is the capacity for abstract thought which distinguishes humans from other animals, and that human language arises from this capacity. While I share Dr. Egnor’s belief in human uniqueness, I have to take issue with his claim that abstraction is what separates man from the beasts.
Why the distinction between humans and other animals is real, but hard to express
I have written over a dozen articles in the past, arguing that there is a real, qualitative difference between the minds of humans and other animals. As I’ve argued here, there appear to be several traits which are unique to human beings: self-awareness (as opposed to mere awareness of one’s own body); autobiographical memory (as opposed to a mere memory of past events); theory of mind (or awareness of one’s own mental states and of other individuals’ mental states); the ability to create symbols; the ability to grasp abstract rules (such as algebraic rules, higher-order spatial relations, hierarchical rules, and causal relationships); the ability to distinguish between perceptions and reality (as we do when we say that although the earth looks flat, it is really round); and finally, language. Some of these distinctions are more controversial than others, and I should acknowledge at the outset that many ethologists would disagree with the list I have given. For instance, many ethologists would now be prepared to ascribe a primitive self-awareness to animals (but see here), and some would also contend that non-human animals possess a rudimentary theory of mind (but see here for a contrary view). Additionally, it has been argued that crows are capable of understanding causal relations (although I’m highly skeptical, myself). Notwithstanding these disagreements, there remains an impressive list of real and qualitative differences, which set human minds apart from those of apes and other animals.
It would be very nice if we could summarize all these differences in one word: rationality. This is Professor Egnor’s view: “Human beings are rational animals,” he writes in a 2015 post. What grounds human rationality is our capacity for abstraction: “Human beings think abstractly, and nonhuman animals do not.” Unfortunately, things aren’t that simple. Take the word “rational.” If the term “rational” refers to the ability of an agent to select and employ suitable means in order to obtain its ends or goal, then it is hard to see how we can deny rationality to cockatoos who can crack locks unassisted, without having to be offered a reward at each step along the way, or to New Caledonian crows, such as the amazing 007, who managed to solve a complex eight-stage puzzle in order to get some food, in the course of just three minutes. (Full disclosure here: my Ph.D. thesis in philosophy was on the subject of animal minds.)
Alternatively, we might define “rationality” by raising the bar, and restricting the term to creatures who are not only capable of directing suitable means to their ends, but also capable of using language to explain why they chose these particular means, and not some other ones, in order to attain their ends. (The crows that can solve an eight-stage puzzle in three minutes can’t do that.) If we adopt this move, then we are essentially making language, rather than reason, the hallmark of humanity. All well and good; but what exactly is language?
Language appears to be a trait unique to human beings, yet the question of precisely what it is that defines a language continues to exercise the minds of the world’s best and brightest linguists. There are several features which set human languages apart from other systems of animal communication: open-endedness (the ability to generate an infinite set of utterances from a finite number of symbols); the existence of grammatical categories; modality independence (which means that we can communicate the same message through several sensory channels); recursivity (as shown in the nursery rhyme, “This is the house that Jack built”); and displacement (or the ability to refer to absent objects). Which of these is the defining feature of language? We don’t know, yet. To further complicate matters, the last two features listed are highly controversial, as we’ll see below: displacement occurs in animal communication as well, and not all human languages exhibit recursivity.
What the example of language illustrates is that while the distinction between human and animal minds is both real and quantitative, it cannot be stated concisely. For that reason, in a recent short post, titled, The immateriality of animal consciousness: why I’m agnostic (May 24, 2015), which I would urge Professor Egnor to read, I felt impelled to critique the Aristotelian-Thomistic view, defended by philosopher Edward Feser, that the essential difference between humans and other animals is that only human beings possess “abstract or universal concepts,” which they can combine into propositions, enabling them to “reason from one proposition to another in accordance with the laws of logic.” As I explained, that view is contradicted by the scientific evidence: it’s far too simplistic. To quote from the conclusion to my post:
…[W]hat Aristotelian-Thomistic philosophers contend is that only human beings are capable of reasoning at all; man is simply defined as a rational animal, full stop. The scientific picture is more nuanced; what it tells us is that certain kinds of reasoning are unique to human beings. It follows that science cannot be enlisted in support of Professor Feser’s contention that while some human mental capacities (reasoning, understanding and choosing) are non-bodily capacities which might in principle be capable of persisting after bodily death, all of the mental capacities of non-human animals are bodily capacities, which means that there can be no possibility of a hereafter for Fido.
Professor Egnor is a thinker in the same tradition as Edward Feser. The mathematician Jeffrey Shallit, whose attempts to minimize those differences have (in my opinion) been soundly refuted in Egnor’s recent posts, is an atheistic materialist. Materialists are prone to conclude from the fact that the nature of the mental divide separating humans from other animals cannot be stated concisely, that the divide is not a qualitative but a quantitative one: man and the other animals differ only in degree, and not in kind. However, this conclusion is a complete non sequitur. What I shall be arguing in this post is that we need to steer a via media between the Scylla of denying human exceptionalism and the Charybdis of exaggerating it in a manner which is scientifically indefensible, by resorting to crude over-simplifications such as: “Animals can feel, but only humans can think,” “Humans are governed by reason; animals, by instinct,” and “Only humans possess abstract concepts.”
One might ask why the essential difference between humans and other animals should be so hard to express in human language. The reason is that understanding this difference requires us to understand what kinds of creatures we really aware, which in turn requires us to view ourselves from an external vantage point: something which by definition, we cannot do. Defining humanity is tantamount to putting the human mind in a box: it can’t be done (at least, not by human beings).
Professor Egnor’s “great divide” between man and beast: humans can form abstract concepts, but other animals can’t
Professor Egnor elaborates his view that the capacity for abstraction is unique to human beings in an article titled, Is Your Cat Logical? (July 9, 2016). He writes:
What distinguishes men from animals is this: men, but not animals, can contemplate universals, independently of particulars. Animals cannot contemplate universals. Animal thought is always tied to particular things.…
Animals are capable of thought only about particular things, although animal thought about particular things can be surprisingly sophisticated. This ability of animals (and people) to think about particulars was called sensus communis and identified by Aristotle in De Anima (Book III, Chapter 2, 425a27)… It means the ability to think about particular things known via perception, especially about particulars known by more than one sense, and to integrate perceptions and make inferences about the particulars.
…Animals don’t measure their food. They don’t abstract universal concepts such as volume or weight from their perception of food. But they can compare bowls of food they perceive, and make judgements based on the comparisons. None of this entails abstract thought. All animal thought is tied to particular things.
There has never been a demonstration of an animal who is capable of abstract thought about universals, unlinked to particulars. In fact, an animal cannot think about universals, for the simple reason that animals have no language.
Animals can think about particulars without language, because particulars provide an object that is necessary to have an intentional thought — a thought about something. But without language, animals are incapable of thinking about anything that is not a particular. Animals think about their owner, but not about ownership. They think about food, but not about nutrition.
In a follow-up article, titled, How Could We Know if Animals Can Think Abstractly? (July 17, 2016), Professor Egnor proposes a simple experiment whereby it could be demonstrated that animals possess a capacity for abstract thought:
But the question can be asked: How could we detect abstract thought in an animal?
Here’s how. The experimental design necessary to detect abstract thought in an animal requires a step in the experiment in which the animal would have to think abstractly, without reference to any particular, and act on that abstract thought.
An abstract concept could be selected — say, a mathematical concept like the concept of square root…
[An] ape could be shown a picture of 25 objects, and then shown screens containing different sets of objects — a set of 1, a set of 2, and set of 3, and so on. The animal would be rewarded if he selected the set of 5, which is the square root of 25. The training could be repeated with different kinds of objects, still using 25 and 5, until the animal reliably chose 5 objects when shown 25 objects.
Once the animal reliably selected the set of 5 after being shown the set of 25, the animal could then be trained to select the set of 4 when shown the set of 16. Once that task was learned, the task could be repeated with 2 and 4.
Once these tasks of selecting the set of objects equal to the square root of the reference set was learned for a series of square-root pairs, the animal would be tested with novel square roots. Would the animal select 3 objects when shown the set of 9? Would the animal select 1 object when shown the set of 1? Would the animal select 6 objects when shown the set of 36?
This is a real test of abstract thought, because it would require that the animal comprehend the concept of square root abstracted from particulars, and then apply it to new particulars.
Two important distinctions overlooked by Professor Egnor
It appears to me that Professor Egnor overlooks two vital distinctions, in the foregoing argument. First, he ignores the crucial distinction between a thought which refers to a concept, and a thought which is about a concept. In order for animal to possess a concept, all it needs to do is entertain a thought that refers to that concept. There is no need for it to entertain a though about a concept itself.
Take the concept of triangularity. Now consider the following two thoughts, which a human being might have, relating to this concept: (a) This shape is a triangle, and not a square. (b) Triangularity is a property of plane figures with three angles and three sides.
The first thought is not about triangularity as such; nevertheless, it clearly makes reference to the concept of a triangle. Only the second thought is about triangularity. Nevertheless, in order to entertain either thought, you need to possess the abstract concept of a triangle.
I maintain that while only humans can entertain the second thought, at least some non-human animals (e.g. African grey parrots such as Alex and Griffin) are probably capable of entertaining the first thought (for reasons I’ll discuss below). That being the case, it follows that these animals possess the abstract concept of triangularity.
But in order to entertain even the first thought, animals need to be capable of abstracting the concept of a triangle from the particulars that they are exposed to. What they do not need to do is entertain thoughts about triangularity, which make no reference to any particular triangle.
The second distinction overlooked by Professor Egnor is that between a concept which is independent of any particular object, and a concept which is independent of every particular object. When I entertain the abstract concept of a triangle, for instance, there is no particular triangle that my concept is tied to: I might be thinking of one which is drawn on a blackboard, or one which I see in the night sky (e.g. the northern constellation Triangulum), or even one which exists only in my mind’s eye. However, I cannot entertain the concept of a triangle without having some triangle in mind. When I picture a triangle, for instance, I usually picture a yellow equilateral triangle, about 10 centimeters wide at the base, set against a blue cloth background.
So when Professor Egnor writes that “There has never been a demonstration of an animal who is capable of abstract thought about universals, unlinked to particulars,” I have to point out that there has never been a human being who is capable of abstract thought unlinked to particulars, either. St. Thomas Aquinas himself taught that human thought requires an image or phantasm; and if an image is not a particular, then what is?
I’d also like to comment briefly on Professor Egnor’s remarks regarding the sensus communis, which allows animals to think about particulars. Philosopher Christian Helmut Wenzel handily summarizes Aristotle’s views on the sensus communis, in an illuminating little essay:
It helps us to become aware of the differences between the proper sensibles [e.g. colors, tastes and sounds – VJT], and it helps us to become aware that we perceive. We somehow “see” that we see and hear and touch and taste. There must be a higher unit, where the different perceptions meet, and where we “perceive” and “assert” the differences. It must be one thing that judges and distinguishes (krinein) between the many. Here thinking and perceiving meet and Aristotle’s “sensus communis” plays a role in this. Today we might also want to speak of consciousness at this point.
This is all well and good, if the task of Aristotle’s “sensus communis” is simply to enable us to distinguish between the objects of the various senses. But it will not help us explain how an animal makes the judgement that this object is the same as that one, or that this object is round (and not triangular). In order to do that, an animal needs to already possess the concepts of “sameness” and “roundness” – neither of which are given to it by its senses. To acquire these concepts, the animal needs to engage in a process of abstraction.
I’d now like to return to Egnor’s example about squaring. As it happens, I have a long and abiding interest in squaring, which is why I’ve written a short appendix to this post, titled, “The Joys of Squaring,” for those who are uninitiated to this pastime. But the concept of a square is an arithmetic concept which most children do not master until the age of eight or nine – and which some adults, who are capable of using language, never master at all. I certainly don’t believe that any non-human animals are capable of squaring, but there is abundant evidence that they can master simpler abstract concepts, such as the concepts of “same” and “different.” What kind of animals are we talking about here? Bees. Yes, bees.
Can animals master abstract concepts?
In order to illustrate how bees can be taught to acquire the concept of “sameness,” I’d like to quote from a blog article titled, Thinking bees and the concept of “sameness” (February 22, 2012) by Dr. Sedeer el-Showk, an evolutionary biologist living in Morocco, and the owner of a blog called Inspiring Science. After noting that “the ability to categorize objects or experiences as ‘the same’ or ‘different’ has generally been considered a relatively advanced cognitive capacity relegated solely to vertebrates,” Dr. El-Showk goes on to describe how a 2001 study in Nature by Martin Giurfa et al. revealed that honeybees actually possess this capacity:
The researchers used a stimulus-matching experiment in order to test the ability of honeybees (Apis mellifera) to distinguish same from different. Individual bees were introduced into a Y-shaped maze with a particular cue at the entrance; in order to reach the reward, the bee had to follow the fork with the same cue. For example, if the bee saw a yellow patch at the entrance, it would have to go down the arm with a yellow patch (instead of a blue one) to get the sugar reward. There were several pairs of cues: blue vs. yellow colour; vertical vs. horizontal grating; radial vs. linear patterns; and lemon vs. mango scents.
The bees were initially trained with one pair of cues; over time, they showed improvement in their performance, which means they were learning that the reward was at the end of the arm which matched the cue at the entrance. Since it’s possible that they had simply learned the correct response for different experimental setups, the researchers required the bees to successfully transfer their learned behaviour to a new set of cues in order to demonstrate a general conception of “sameness”. For example, bees which had learned to match colours were tested with horizontal vs. vertical gratings. If the bees really had learned that they should follow the same cue that was at the entrance, they should have been able to generalize that to a new type of cue. On the other hand, if they had simply memorized the correct response for different configurations, they would have to learn the correct response from scratch again.
The researchers found that the bees were consistently able to successfully transfer their learning to new pairs of cues. Bees trained to match colours in order to find a reward were able to generalize this behaviour to other visual cues, such as patterns; in fact, they even generalized the concept of “sameness” to a different sensory modality — bees which were trained to match odours proved able to also match colours.
The researchers also conducted the same experiment with the reward being at the end of the arm with the different (non-matching) cue; again, bees were able to learn to distinguish between the first pair of cues and were able to transfer that knowledge (“pick the different stimulus”) to a new set of cues, although in this case the researchers only checked visual cues.
These experiments show that bees are able to not only learn to distinguish between cues, but to transfer this knowledge to new cues which weren’t part of the training. Bees are capable of conceptualizing the categories of sameness and difference and classifying their experiences accordingly; they are able to do this between different aspects of a particular sense (colour vs. pattern) and also between different kinds of senses (visual vs. olfactory).
Alert readers will have noticed that the Nature article by Martin Giurfa et al., which Dr. El-Showk reviewed on his blog, was published back in 2001. Have any follow-up studies been conducted? Yes, they have. I’d now like to quote from the abstract of a 2012 article in the Proceedings of the National Academy of Sciences, titled, Simultaneous mastering of two abstract concepts by the miniature brain of bees by Aurore Avarguès-Weber, Adrian G. Dyer Maud Combea and Martin Giurfa (PNAS vol. 109 no. 19, 7481–7486, doi: 10.1073/pnas.1202576109):
Sorting objects and events into categories and concepts is a fundamental cognitive capacity that reduces the cost of learning every particular situation encountered in our daily lives. Relational concepts such as “same,” “different,” “better than,” or “larger than” — among others — are essential in human cognition because they allow highly efficient classifying of events irrespective of physical similarity. Mastering a relational concept involves encoding a relationship by the brain independently of the physical objects linked by the relation and is, therefore, consistent with abstraction capacities. Processing several concepts at a time presupposes an even higher level of cognitive sophistication that is not expected in an invertebrate. We found that the miniature brains of honey bees rapidly learn to master two abstract concepts simultaneously, one based on spatial relationships (above/below and right/left) and another based on the perception of difference. Bees that learned to classify visual targets by using this dual concept transferred their choices to unknown stimuli that offered a best match in terms of dual-concept availability: their components presented the appropriate spatial relationship and differed from one another. This study reveals a surprising facility of brains to extract abstract concepts from a set of complex pictures and to combine them in a rule for subsequent choices. This finding thus provides excellent opportunities for understanding how cognitive processing is achieved by relatively simple neural architectures.
This recent experimental evidence showing that even the humble honeybee can learn to master two abstract concepts at the same time, and that it can apply these concepts to novel stimuli, should suffice to refute Professor Egnor’s claim that abstraction is a capacity unique to human beings. Nor will an appeal to sensus communis help him, as this capacity merely distinguishes different between sensory modalities. It has nothing to do with the acquisition of new concepts, which an animal then has to apply to novel stimuli.
Can animals master mathematical concepts?
Alex (pictured above), the famous African gray parrot trained by Dr. Irene Pepperberg, died at the age of 31 in 2008. By that time, it had mastered a small but significant set of mathematical concepts:
According to Pepperberg who is a faculty member at Brandeis University, Alex was able to identify 50 different objects, seven colors and shapes, and quantities of up to six. Alex also understood the concept of bigger and smaller and same and different. (NPR report, November 12, 2008.)
But that was not all that Alex could do:
He could say how many items were in a group for collections up to six items, even if the objects were scattered around a tray (Pepperberg 1987a). Even more remarkable was Alex’s ability to accurately count specific items in what is called a confounded number set, which are items that vary in more than one characteristic. For example, a set might consist of two types of objects, balls and keys, that appear in two colors, red or blue. When presented with all these objects mixed together on a tray, Alex could say the number of items of a specific type and color, such as the number of blue keys. He responded correctly to these types of questions 83% of the time (Pepperberg 1994). He could even add up the total from two sequentially presented collections (Pepperberg 2006).
(Perspectives on Animal Behavior, by Judith Goodenough, Betty McGuire, Elizabeth Jakob. John Wiley & Sons Inc., Hoboken NJ, 2010.)
A report by Ewen Callaway in Nature News (February 20, 2012), published shortly after Alex’s death, lent further support to the claim that Alex possessed genuine mathematical abilities:
Even in death, the world’s most accomplished parrot continues to amaze. The final experiments involving Alex – a grey parrot (Psittacus eithacus) trained to count objects – have just been published.
They show that Alex could accurately add together Arabic numerals to a sum of eight and three sets of objects, putting his mathematical abilities on par with (and maybe beyond) those of chimpanzees and other non-human primates. The work was just published in the journal Animal Cognition.
To sum up: the experimental evidence that African grey parrots possess simple abstract concepts of number of shape is well-documented and has been carefully reviewed. If Professor Egnor considers such evidence for abstraction in parrots to be unreliable, he owes us an explanation why.
As if this were not enough, it turns out that parrots perceive triangles in much the same way as we do. Studies with another African grey parrot named Griffin have shown that these birds can recognize triangles from their corners alone. Like humans, parrots can perceive the Kanisza triangle (pictured above, courtesy of Wikipedia):
When he looks at a Kanizsa triangle, the famous optical illusion made up of three Pac-Man figures facing each other, Griffin doesn’t just see three figures converging on each other. He sees a triangle.
That might not seem significant, until you realize that Griffin is a parrot.
Despite a visual system vastly different from that of humans, the bird can successfully identify Kanizsa figures and occluded shapes, said Ken Nakayama, the Edgar Pierce Professor of Psychology, and Irene Pepperberg, a research associate in the Psychology Department, co-authors of a study on the subject. The findings, they said, suggest that birds may process visual information in a similar way to humans. (The parrot knows shapes, Harvard Gazette, July 19, 2016.)
Can monkeys master basic mathematics?
There is also tentative evidence that monkeys may be capable of simple addition, although I should point out that the monkeys sometimes make rather odd mistakes when adding, which are presently unexplained. In a recent article titled, Monkeys Can Do Basic Math Using Symbols (April 21, 2014), RealClearScience writer Alex Berezow describes an experiment conducted by a team of Harvard and Yale researchers, suggesting that monkeys can not only add dots on a screen, but can also add Arabic numbers, and even Tetris-like symbols:
The monkeys were given a touch-screen device that was divided in two halves. In the first stage of the experiment, the monkeys had to determine which side had the greater amount. First, they had to examine dots. Second, they had to examine Arabic numerals (1-9) or letters (which represented numbers 10-25). Finally, they had to examine Tetris-like symbols which represented numerical values. To keep the monkeys playing, they were given drops of liquid treats…
In the second stage of the experiment, the monkeys were again prompted to choose the greater of two values. This time, one side displayed two symbols (“addends”). The monkeys had to determine if the sum of the addends was greater than the single value on the other side of the screen. It took them several weeks to get the hang of this, but they eventually caught on…
To determine if the monkeys were actually doing math or were simply recalling memorized patterns, the researchers tested the monkeys with another addition task utilizing the Tetris-like symbols. If they were memorizing symbols, it should take the monkeys just as long as it did previously to determine the correct answers. However, they were much faster at learning this task. This implies that pattern memorization is an unlikely explanation. Instead, the monkeys had transferred the skill of arithmetic to evaluate the Tetris-like symbols.
More evidence in favor of calculation rather than memorization occurred when the monkeys were presented with choices such as 5 + 7 versus 8.… Even on this difficult task, the monkeys more quickly learned how to pick the correct answer using the Tetris-like symbols, lending further support to the conclusion that the monkeys had learned arithmetic with the numerals/letters and transferred the skill to a new set of symbols.
As impressive as these results are, the monkeys’ arithmetic was not terribly accurate if the compared values were similar. For example, choosing between 4 + 6 and 9 was a bit too difficult.
Can monkeys do math? I don’t know, but I certainly wouldn’t rule it out. At any rate, the foregoing experiment meets Professor Egnor’s conditions for a scientific test of the claim that non-human animals are capable of abstraction. The available evidence tentatively suggests that they are. If Professor Egnor disagrees with this conclusion, then he should propose an alternative hypothesis, which accounts for the evidence equally well.
Professor Egnor’s sweeping claim about animal communication: animals communicate only about the here-and-now
In another article over at Evolution News and Views, Professor Egnor addresses the question of whether any non-humans animals possess language. Here, he is on much more solid ground: a decisive majority of ethologists would agree that non-humans animals lack this capacity. The more interesting question is: why do they lack it? In an essay titled, Do Animals Have Language? (July 12, 2016), Professor Egnor sets forth his view, which is that animals are only capable of communicating signals about objects which they perceive, here and now. Language, on the other hand, requires the use of abstract signs (called designators), which are not tied to the here-and-now:
…A sign presents to the mind an object other than itself for attention…
There are, for our purposes, two kinds of signs — signals and designators. Signals are objects or events that draw attention to something else in physical or temporal proximity.…
Designators are words that refer to the objects that they name. Designators are arbitrary, in the sense that there is no intimate linkage between the designator and the object designated… Designators are abstract.
Signals are not language. Human language is an abstract process that relates designators in grammatical relations to objects designated.
…The sounds and gestures that wild and domesticated animals use to express emotions and desires are signals, not designators, and these expressions are not language…
In captivity, under laboratory conditions, animals can be trained to use signals (which they use naturally) in patterns that mimic designators and mimic language. But none of it is actual use of abstract designators to produce genuine language. Animals never use signals to designate any object that is not in the animal’s perceptual realm. Animal signals refer to objects of perception and emotional states and the like. Animals do not signal abstract concepts.
Professor Egnor makes two claims here: first, that language is a trait unique to human beings; and second, that what distinguishes linguistic signs from other signs is the fact that they can be used to refer to absent objects. It is the second claim which I find highly questionable.
Can animals communicate about absent objects?
While Professor Egnor is correct in his assertion that language is unique to human beings, his claim that animals never refer to objects which they cannot perceive, but only to objects present to them here and now, turns out to be empirically false. During the past couple of years, it has been shown that chimpanzees can communicate to one another about objects which are absent. The evidence is admirably summarized in an article by Heidi Lyn et al., titled, Apes Communicate about Absent and Displaced Objects: Methodology Matters (Animal Cognition, January 2014; 17(1): 10.1007/s10071-013-0640-0):
Displaced reference is the ability to refer to an item that has been moved (displaced) in space and/or time, and has been called one of the true hallmarks of referential communication. Several studies suggest that nonhuman primates have this capability, but a recent experiment concluded that in a specific situation (absent entities) human infants display displaced reference but chimpanzees do not. Here we show that chimpanzees and bonobos of diverse rearing histories are capable of displaced reference to absent and displaced objects. It is likely that some of the conflicting findings from animal cognition studies are due to relatively minor methodological differences, but are compounded by interpretation errors. Comparative studies are of great importance in elucidating the evolution of human cognition, however, greater care must be taken with methodology and interpretation for these studies to accurately reflect species differences.
Many previous studies have found that chimpanzees communicate about visibly displaced objects (e.g. Leavens et al. 2004; Woodruff and Premack 1979) and show excellent understanding of displacement in a variety of experimental contexts (reviewed by Call 2001). For instance, Woodruff and Premack (1979) showed that chimpanzees informed a human about the location of hidden food, and also developed deceptive communications if that human had proven untrustworthy. Some nonprimate vertebrates, such as dolphins, sea lions, and parrots (e.g. Herman and Forestell 1985; Pepperberg 1999; Pepperberg and Gordon 2005; Schusterman et al. 1993), also have been found to refer to absent entities. In the most extensive experimental test concerning absent reference, Herman and Forestell (1985) showed that a bottlenose dolphin could respond to one paddle to indicate the presence of an object and to another to indicate its absence. Therefore, it is clear from existing literature that apes and other mammals have the ability for displacement.
It is notable that many of the subjects in the above examples have been given prior experiences relevant to displacement. Chimpanzees and bonobos that have acquired symbol systems routinely exhibit displacement in their daily linguistic communication, much the same as human children when they develop language (e.g. Brakke and Savage-Rumbaugh 1996; Gardner et al. 1989; Lyn 2008; Rumbaugh 1977; Savage-Rumbaugh et al. 1993; Savage-Rumbaugh et al. 1986; Savage-Rumbaugh et al. 1978). In all of these cases, apes refer to items that are out of view. In many cases, comprehension and use of displacement is both referential and flexible (Lyn 2012; Lyn et al. 2011; Lyn et al. 2010; Lyn 2010)…
In a recent comparative experiment, Liszkowski et al. (2009) concluded that chimpanzees do not have displaced reference but that human 12-month-old infants do, and attributed this either to the apes’ lack of language or lack of prerequisite socio-cognitive skills…
The results of the current study demonstrate that, contrary to the findings of Liszkowski et al. (2009), apes do communicate about absent entities: the apes in the current study gestured to the empty container in both conditions, when the desired object was visibly displaced, but still present (as in the Liszkowski et al. (2009) “absent” referent condition) as well as when it was absent altogether (our Truly Absent condition).
In summary: the experimental evidence appears to strongly favor the view that apes are quite capable of referring to absent objects.
Metaphysics: the real reason why animals can’t use language?
In a more recent article, titled, Are Birdsongs Language? (July 18, 2016), Professor Egnor develops his view that the capacity to use a language presupposes a capacity to follow grammatical rules, which in turn presupposes the ability to understand metaphysics, since the grammar rules we use mirror the metaphysical structure of the world around us: a world of things, or substances (nouns), which instantiate universal properties (adjectives or predicates) – e.g. “The ball is spherical.” In other words, the real reason why animals can’t use language is that they can’t do metaphysics:
Grammar is a rule-based framework on which words are arranged. All human grammar (and all grammar is human) is a metaphysical template on which words are hung. By metaphysical, I mean that the elements of grammar — nouns and verbs and adjectives and predicates and the like — categorize words metaphysically. Nouns (grammar) are substances and particulars (metaphysics). Verbs (grammar) are identity or change (metaphysics). Adjectives and predicates (grammar) are universals (metaphysics). Grammar maps words to reality by imposing a metaphysical structure on language.…
Grammar is a deep structure that maps language to reality. It is, as Chomsky has noted, inherent to man. We are born with the capacity for grammar — it is not learned by either imitation or instruction. Grammar, like language, is in our soul, so to speak.
Professor Egnor is a fan of Chomsky’s theory of “universal grammar.” That’s perfectly fine, but readers should be aware that Chomsky’s theory remains a highly controversial one (see here for a summary of some common criticisms). One leading critic, linguist Daniel Everett, argues in a recent article in the Guardian (March 25, 2012) that “universal grammar doesn’t seem to work, there doesn’t seem to be much evidence for [it]. And what can we put in its place? A complex interplay of factors, of which culture, the values human beings share, plays a major role in structuring the way that we talk and the things that we talk about… Humans are a social species more than any other, and in order to build a community, which for some reason humans have to do in order to live, we have to solve the communication problem. Language is the tool that was invented to solve that problem.”
Everett describes a language called Pirahã, spoken by a tribe in the Amazon, which possesses a number of highly unusual features:
Pirahã just seems to have so many unique characteristics. Things that we didn’t expect. I mean the absence of numbers, the absence of counting and colours, the absence of creation myths, and the refusal to talk about the distant past or the distant future. A number of things like this, including, the special characteristic of recursion, the ability to keep a process going in the syntax forever.
No recursion? That flat-out contradicts Chomsky’s theory. And no numbers or colors? Very strange. However, it needs to be borne in mind that only one linguist has mastered the Pirahã language, and that it may turn out to be a degenerate language.
On the other hand, more recent research appears to lend support to Chomsky’s theory of universal grammar, according to a 2015 study described in an online article by Dana Dovey titled, Noam Chomsky’s Theory Of Universal Grammar Is Right; It’s Hardwired Into Our Brains (MedicalDaily, December 7, 2015):
In an email to Medical Daily, [researcher David] Poeppel explained that although it is difficult, if not impossible, to prove theories, the data ascertained in his research supports crucial aspects of Chomsky’s theory, namely that listeners build abstract, hierarchical constituent structures of linguistic information.
“I’d say, on balance (comparative language research, language acquisition research, these kinds of brain data) the empirical research favors the Chomskyan view, as unpopular as it is,” Poeppel wrote.
Poeppel also recognized the controversy in his finding, seeing as the preferred view is that grammar is achieved by using acoustic cues such as intonation, and statistical cues, like word transition.
“However, we demonstrate that linguistic structure building happens in absence of those cues — so grammar based structure building must exist,” Poeppel said. “That is, in brief, the controversy.”
Chomsky did not hold that language is a mirror of the metaphysical structure of the world
What Professor Egnor overlooks, however, is that even if Chomsky’s theories are fully vindicated, Chomsky himself did not hold that language reflects the metaphysical structure of the world. As Robin Allott points out in an interesting essay titled, Language as a Mirror of the World: Reconciling picture theory and language games, “certainly Chomsky’s Universal Grammar proposes the biological innateness of grammatical aspects of language but this is a long way removed from of any idea of syntactic structures as mirroring the world.” Even supposing there to be a one-to-one isomorphism between the grammatical structures of our language and the metaphysical structure of the world around us, we could still ask: is this because our language mirrors the world, or because the metaphysical structure we impose on the world reflects the language that we use? (Allott’s own view, which he defends in his essay, is that “the capacity of language to mirror reality, to reflect the perceived world in which we find ourselves, derives from the structuring of the human brain and specifically from the way in which the human brain plans and executes action, the total motor control system. By way of the motor system, the words mirror the objects or actions which they refer to.”)
It would be difficult to conceive of a more fundamental metaphysical distinction than that between an agent and his/her action. This distinction is said to underlie the distinction between a noun and a verb: agents are substances or entities (and hence nouns), while actions are “doings,” and hence verbs. Not surprisingly, virtually every language on the planet makes a distinction between nouns and verbs.
But there’s an exception to every rule, and it appears that there is at least one language which makes no such distinction, as we learn from an article in The Economist titled, Babel’s children (The Economist, January 8, 2004):
It is hard to conceive of a language without nouns or verbs. But that is just what Riau Indonesian is, according to David Gil, a researcher at the Max Planck Institute for Evolutionary Anthropology, in Leipzig. Dr Gil has been studying Riau for the past 12 years. Initially, he says, he struggled with the language, despite being fluent in standard Indonesian. However, a breakthrough came when he realised that what he had been thinking of as different parts of speech were, in fact, grammatically the same. For example, the phrase “the chicken is eating” translates into colloquial Riau as “ayam makan”. Literally, this is “chicken eat”. But the same pair of words also have meanings as diverse as “the chicken is making somebody eat”, or “somebody is eating where the chicken is”. There are, he says, no modifiers that distinguish the tenses of verbs. Nor are there modifiers for nouns that distinguish the definite from the indefinite (“the”, as opposed to “a”). Indeed, there are no features in Riau Indonesian that distinguish nouns from verbs. These categories, he says, are imposed because the languages that western linguists are familiar with have them.
The moral of the story is twofold: first, we should beware of assuming too readily that we can grasp the underlying metaphysical structure of the world around us, since it may turn out to be less comprehensible to us than we think, at reality’s ultimate level; and second, we should beware of assuming that all cultures view reality in much the same way as ours does. Evidently, some don’t.
There are profound differences between humans and other animals, and the Appendix I have attached to this post offers a perfect illustration of this point. No other animal could write about the joys of squaring. But we must not oversimplify these differences, and we should always resist the temptation to accept easy formulations of the distinction between man and beast (such as “Only humans engage in abstract thought”), even when they are propounded by some eminent philosopher (such as Aristotle). In the end, precisely what it is that makes us human remains a mystery, and probably always will.
APPENDIX: The Joys of Squaring
|m = 12 = 1|
|m = 22 = 4|
|m = 32 = 9|
|m = 42 = 16|
|m = 52 = 25|
(Image courtesy of Wikipedia – VJT.)
There are several reasons why squaring is a good idea.
First, you can use it as a substitute for standard multiplication. The reason is that the product of any two odd numbers or any two even numbers can be represented as the difference between two squares. Mathematically speaking: (a+b)x(a-b)=a2-b2. For instance, suppose you want to multiply 54 by 36. You could say: 54×30=1,620, and 54×6=324, so 54×36=1,620+324=1,944. Or you could split the difference between 54 and 36, which gives 45, and you could calculate the answer like this: 54×36=(45+9)x(45-9)=2,025-81=1,944. Sometimes the second way of computing the answer is faster.
What if you’re multiplying an odd number by an even number, or vice versa? What you need to do is subtract one from the higher of the two numbers, leaving you with two numbers which are both odd or both even. You can then calculate their product as the difference between two squares, and finally, add the lower number back on to the total, to get your final answer. For instance, 48×93=(48×92)+48, which is equal to ((70-22)x(70+22))+48, or 4,900-484+48, or 4,416+48=4,464.
The second reason why I’d recommend squaring is that it’s a good way of checking a multiplication calculation, independently. Let’s say you want to multiply 57 by 35. You might do it the normal way (57×30=1,710, 57×5=285, 1,710+285=1,995), but then you might wonder if you made a mistake. You can verify your computation independently, by recalculating it as the difference between two squares: 57×35=(46+11)x(46-11)=2,116-121=1,995.
Finally, squaring is a great way to keep your mind sharp. On top of that, it’s fun. In fact, it’s so much fun that it can become addictive. For instance, you might end up squaring car number plates as the cars go by, while you’re walking down the street. Be careful: never do this while crossing the road or driving.
How to square numbers in your head
In order to square large numbers in your head, you need to be able to do four things:
(a) you need to know the square of every integer from 1 to 100
(e.g. you need to know that 742 is 5,476);
(b) you need to know the double of every integer from 1 to 100
(e.g. you need to know that 74×2=148);
(c) you need to know the triple of every integer from 1 to 100
(e.g. you need to know that 74×3=222); and
(d) you need to be able to instantly subtract from 100 every integer from 1 to 99
(e.g. you need to know that 100-74=26).
I do not suggest that you try to memorize tables of integers and their squares, doubles, triples and complements to 100. Instead, you can acquire this knowledge through continually practicing with the one- and two-digit numbers which you encounter in everyday life. For example, you might be walking along a street, past a house whose house number is 43. You say to yourself: “43 squared is 1849 (an easy number to remember, as it’s the date of the California gold rush). Double 43 is 86, and triple 43 is 129. 43 from 100 is 57.” You might then try to do the same thing with the number 57: “57 squared is 3249 (not so easy to remember, but it’ll stick in your memory, with time and practice). Double 57 is 114, and triple 57 is 171. 57 from 100 is of course 43.”
Squaring the slow way
There is a simple formula for squaring any multi-digit number:
For instance, 6432=6002+(2x600x43)+432=360,000+(2x43x600)+1,849
Adding a2 and b2 is easy, since the non-zero digits will always be in separate columns. Thus 360,000+1,849 is just 361,849. The tricky part is the middle term, 2ab. This is where you need to be able to cut corners. The ability to double and triple two-digit numbers comes in very handy at this point.
For instance, how do you solve 2x43x600? You could say: 43×6 is double 43×3, or double 129. Since you already know that double 29 is 58, it’s easy to see that double 129 must be 258. And since you’re trying to multiply 43 by 600 (not 6), you need to remember to put two zeroes after your answer (to make it 25,800). Then you need to multiply that number by 2, because the middle term when squaring (a+b) is not ab, but 2ab. Once again, this is easy: double 25,000 is 50,000 and double 800 is 1,600, so you get 51,600.
Now you have three numbers: 6002 [which is 360,000], (2x600x43) [which is 51,600] and 432 [which is 1,849]. It might be tempting to add the first and the last numbers, before adding the middle term, but I would personally recommend going from left to right, because it’s less confusing: 360,000+51,600=411,600, and 411,600+1,849=413,449.
[Note: when adding from left to right, you’ll need to constantly “look ahead” and be prepared to carry over when necessary, but you’ll pick up this ability automatically, with practice. The advantage of going from left to right is that it gives you a feeling for how roughly large the answer will be, from the get-go.]
Squaring with short-cuts
The method of squaring described above will yield the right answer, but it’s rather dull and pedestrian. Now I’d like to reveal some useful short cuts:
Squaring numbers in the 100s (from 101 to 199): EASY
Split the number into two parts: 100 plus the last two digits – e.g. 184=100+84. Double the last two digits – e.g. double 84 is 168. Add that figure to 100 – e.g. 100+168=268. Multiply that total by 100 – e.g. 268×100=26,800. Finally, add this number to the square of the last two digits – e.g. 26,800+842=26,800+7,056=33,856.
With practice, you should be able to perform this kind of computation in about one second.
Putting it mathematically: (100+b)2=(100×100)+(2x100b)+b2=[(100+2b)x100]+b2
Squaring numbers in the 200s (from 201 to 299): MEDIUM
Split the number into two parts: 200 plus the last two digits – e.g. 257=200+57. Discard the 200: you won’t need it again. Add 100 (not 200) to the last two digits, and multiply the sum by 100, then double the result, and double it again – e.g. 100+57=157, 157×100=15,700, 2×15,700=31,400, and 2×31,400=62,800. Add that number to the square of the last two digits – e.g. 62,800+572=62,800+3,249=66,049.
Putting it mathematically: (200+b)2=2002+(2x200xb)+b2=4x(10,000+(100xb))+b2
This formula is not too hard to apply, if you’re good at doubling, and if you know your squares up to 100 squared.
Squaring numbers in the 300s (from 301 to 399): HARD
It can be shown that (300+b)2=150,000+(1,000xb)+(300-b)2. The number (300-b) can be rewritten as (200+c), where c=100-b. Then we get (300+b)2=150,000+(1,000xb)+2x2x(100x(100+c))+c2. But calculating (300+b)2 in this manner is more error-prone than the “slow way” described above, so I’d be inclined to recommend the slow way for numbers in the 300s.
Squaring numbers in the 400s (from 401 to 499): EASY
Split the number into two parts: 400 plus the last two digits. Take the last two digits, add them to 150, and multiply the result by 1,000. Finally, subtract the last two digits from 100, square the result, and add that to the number you obtained previously.
Adding 150 to a number is really easy, and whacking on three zeroes at the end is even easier. Once you know how to instantly subtract a one- or two-digit integer from 100, and square the result, the rest is simple, too.
With practice, you should be able to perform this kind of computation in about one second.
Putting it mathematically:
Squaring numbers in the 500s (from 501 to 599): EASY
Split the number into two parts: 500 plus the last two digits. Take the last two digits, add them to 250, and multiply the result by 1,000. Finally, square the last two digits, and add the result to the number you obtained previously.
With practice, you should be able to perform this kind of computation in about one second.
Putting it mathematically:
Proof: (500+b)2= 5002+(2x500xb)+b2=5002+(1,000xb)+b2=250,000+(1,000b)+b2=[(250+b)x1,000]+b2
Squaring numbers in the 600s (from 601 to 699): MEDIUM
Split the number into two parts: 600 plus the last two digits. Take the last two digits, add them to 350, and multiply the result by 1,000. Finally, add the last two digits to 100, square the result, and add that to the number you obtained previously.
Adding 350 to a two-digit number is really easy, and whacking on three zeroes at the end is even easier. Adding a two-digit number to 100, and squaring the result, is a little harder, but with practice, you’ll get better.
With practice, you should be able to perform this kind of computation in about two seconds.
Putting it mathematically:
Squaring numbers in the 700s (from 701 to 799): HARD
It can be shown that (700+b)2=450,000+(1,000xb)+(200+b)2. But calculating (700+b)2 in this manner is more error-prone than the “slow way” described earlier, so I’d be inclined to recommend the slow way for numbers in the 700s.
Squaring numbers in the 800s (from 801 to 899): MEDIUM
Split the number into two parts: 800 plus the last two digits. Take the last two digits, double them, add the result to 600, and multiply the result by 1,000. Finally, subtract the last two digits from 200, square the result, and add that to the number you obtained previously.
With practice, you should be able to perform this kind of computation in about three seconds.
Putting it mathematically:
Squaring numbers in the 900s (from 901 to 999): EASY
Split the number into two parts: 900 plus the last two digits. Take the last two digits, double them, add the result to 800, and multiply the result by 1,000. Finally, subtract the last two digits from 100, square the result, and add that to the number you obtained previously.
With practice, you should be able to perform this kind of computation in about one second.
Putting it mathematically:
Squaring four-digit numbers in your head
The trick here is to split the number into the first two digits (followed by two zeroes) and the last two digits, and use the (a+b)2 rule. For instance, if you need to square 6,374, then you break it up into 6300 and 74. (Stop thinking about commas: they’re a hindrance in calculations of this sort.) Squaring 63 gives you 3969, so squaring 6,300 gives you 39690000. Squaring 74 gives you 5476.
Now for the hard part: you need to multiply 6300 by 74 and double the result. Multiplying from left to right: 63×70=4410, and 63×4=252, so 63×74=4662. Thus 6300×74=466200, and if you double that, you get 932400.
Now for another tricky part: lining up the zeroes. It’s very easy to make a mistake here, if you’re not careful. To compute the result, you need to add 39690000 to 932400 and 5476. The trick is that the last four digits of the second term (932400) need to be added to the last term (5476), while the preceding digits in the second term (i.e. the first two digits, or 93) need to be added to the 3969 in the first term. So we get: (3969+93)0000=40620000, and (2400+5476)=7876.
Putting it all together, we get 40627876. (Luckily, there was no carry-over, as 7876 is less than 10,000.) Now we need to re-insert the commas: 40627876 is 40,627,876.
Squaring six-digit numbers in your head
The trick here is to split the number into the first three digits (followed by two zeroes) and the last three digits, and use the (a+b)2 rule. The good news is that you can use commas during your calculations: they won’t be a hindrance, as they were when squaring four-digit numbers.
Suppose you want to square 385,791. It’s easy enough to calculate that 385,0002=148,225,000,000, and that 7912=(800-9)2=625,681.
Next, we need to multiply 385,000 by 791, and double the result. With a trained eye, however, we can spot an easier way. 385,000×2=770,000. Multiplying 791 by 77 is easy. 79×77=782-12=6083, so 790×77=60,830. 1×77=77, so we get 60,907. But we wanted to multiply 770,000 by 791, not 77 by 791, so we need to add on four zeroes, to give us 609,070,000.
Now we have to add the three numbers together: a2+2ab+b2, or 148,225,000,000+609,070,000+625,681. This gives us 148,834,695,681.
Needless to say, multiplying two six-digit numbers places a huge strain on your memory, especially if your visualization skills are poor (as mine are, which is why I’m lousy at chess). What I recommend doing is switching on your “internal tape recorder” while doing these calculations: in other words, you need to say the digits to yourself over and over again, in your head. You may end up forgetting them, in which case you’ll need to go back to where you started. But with practice and persistence, you’ll be able to master this skill.
Other tips: it definitely helps if you tilt your head back and look at the ceiling while calculating. I don’t know why. Saying the numbers aloud to yourself while calculating helps, too. At the very least, you’ll need to say them in your head.
Finally, no matter how good you are, the chances are there’s someone out there, who’s a lot better than you. I pride myself on being quick with figures, but there are people who are over 100 times faster than I am. Some people have incredible visual memories. (I don’t.) Other people (savants) have the curious sensation that their brains are dictating the answers to them automatically, when they calculate, as if the answers are coming to them from outside. (I can’t imagine what that must be like.) In any case, calculating makes your life a lot easier, and squaring is a fun kind of calculation.
Squaring nine-digit numbers in your head
I have squared nine-digit numbers on perhaps a dozen occasions in my life, as a mental challenge, but I wouldn’t recommend it. As I recall, it takes over 20 minutes to compute the answer, because you have to say the numbers to yourself over and over again in your head, so as not to forget them – and even so, you still do. It puts quite a bit of strain on the brain, so I wouldn’t suggest trying it unless you have a masochistic streak, or are extremely talented with figures.
A final warning: squaring can be a highly addictive practice. Don’t try it while driving or while crossing the street, because there’s a real risk that you’ll have an accident.