On a visit to Kyoto once, I encountered a gate
with a posted sign written in Kanji. I asked the guide what
it meant, and he replied that it could be translated as
"Post no bills." One need only be moderately
astute to realize that the sign itself is a bill and so
violates its own message. To prevent such bills from being
posted, we might consider posting another sign that says,
"Post no 'Post no bills' bills." But this in turn
is a bill, so perhaps we need another sign that says,
"Post no 'Post no "Post no bills" bills'
bills." The problem, as I'm sure you now see, is that
this process leads to an infinite sequence of signs that
would not only cover the gate, but eventually the entire
universe, since each bill is longer than the last. Better,
perhaps, just to allow people to stick a few bills on the
gate.
The self-referential bills are examples of recursion.
Another instance comes from an anonymous parody of the first
line of Edward Bulwer-Lytton's infamous novel, Paul
Clifford:
It was a dark and stormy night, and
we said to the captain, "Tell us a story!" And
this is the story the captain told: "It was a dark and
stormy night, and we said to the captain, 'Tell us a story!'
And this is the story the captain told: 'It was a
dark.…'"
In computational terms,
recursion is a process that calls itself, or that
calls a similar process. In the example of "Post no
bills," the sign is, albeit unwittingly, referring to
itself, whereas in the parody of Bulwer-Lytton's novel, the
story in the story is the same story.
Alternatively, one
can put the definition in dictionary style:
Perhaps it is best not to dwell on this.
I propose
that recursion is a ubiquitous property of the human mind
and possibly the principal characteristic that distinguishes our
species from all other creatures on the planet. Recursion is a
well-known property of language, but I shall argue that the
phenomenon applies to a number of other putatively human
domains, including "theory of mind," mental time
travel, manufacture, the concept of self and possibly even
religion.
Language
Grammatical
rules in language exploit recursion to create the infinite
variety of potential sentences that we can utter and
understand. Perhaps the simplest example is a sentence that is
composed of sentences, according to a rule that might be
expressed as
S → S + S
where
the arrow is a symbol for can be rewritten as, and
S stands for sentence. Rewrite rules of this
form are a conventional way to show how language is
constructed. This particular rule, invoked twice, creates
the following sentence, found in A. A. Milne's Winnie
the Pooh: "It rained and it rained and it
rained." This example, though, is rather trivial, since
it simply amounts to repetition, presumably to create a
sense that it rained for rather a long time, causing Piglet some
ennui.
Recursion can be much more complex than simple
repetition, and it is used in human language to provide
qualification or to add complexity to an utterance. For
example, we can break sentences down into phrases, and then
apply recursive rules to tack phrases onto phrases, or embed
phrases within phrases. There are noun phrases (NP), verb
phrases (VP) and prepositional phrases (PP).
On a visit
to a publishing house in Hove, England, I was greeted by the
publisher with the unlikely sentence "Ribena is trickling
down the chandeliers." (Ribena is the brand name of a
flavored drink in the United Kingdom, and it was indeed
dripping off the light fixtures.) Here, the sentence is
first broken down into an NP ("Ribena") and a VP
("is trickling down the chandeliers"). But the VP
is itself composed of a verb ("is trickling") plus
a PP ("down the chandeliers"), which in turn is
composed of a preposition ("down") plus an NP
("the chandeliers"). The sentence can therefore be
parsed in terms of the following rewrite rules:
1. S → NP + VP
2. NP → noun
3. VP
→ verb + PP
4. PP → preposition + NP
Applied to more complex sentences, such rules involve
recursion. For example, an NP may include aPP,
which may in turn include an NP. In principle, one
could cycle repeatedly through rules 2 and 4. Had the
publisher not been in something of a panic, he might have
elaborated: "Ribena is trickling down the chandeliers
onto the carpet beside my desk." (There was a nursery
upstairs).
Children quickly learn to appreciate the power of
recursion (if not of Ribena), as illustrated by these
sentences from the well-known children's story, "The
House That Jack Built":
1. This is the
house that Jack built. 2. This is the malt that
lay in the house that Jack built.
3. This
is the rat that ate the malt that lay in the house that
Jack built.
4. This is the cat that killed the
rat that ate the malt that lay in the house that
Jack built.
5. This is the dog that worried
the cat that killed the rat that ate the malt that
lay in the house that Jack built.
And
so it goes on. It is important to understand that this is not
simply a matter of adding disconnected elements. New sentences
are added progressively at the beginning, and the rest of
the sentence increasingly qualifies the noun. In the fourth
sentence, for example, the cat in question is precisely the
one that killed the rat that ate the malt, and so on. A cat
that killed a rat that did not eat the malt that lay in the
house that Jack built simply will not do.
The sentences in "The House that Jack Built" are
examples of what is called end recursion, in which
the recursive rule is invoked at the end. The fourth
sentence, for example, begins with the sentence "This
is the cat," but then the relative clause "that
killed the rat" is added to qualify the cat. This
relative clause mentions a rat, and a further relative clause,
"that ate the malt," is added to qualify the rat, and
so forth. In principle, recursive elements can be added
ad infinitumor until your short-term memory can
hold no more.
Another kind of recursion is
center-embedded recursion, in which
constituents are embedded within constituents. For example, in
the third sentence of "The House That Jack Built," one
might wish to make the malt rather than the rat the subject of
the sentence, and so embed a relative clause describing the
malt into a sentence about the malt: The malt that the
rat ate lay in the house that Jack built.
Such embedded phrases, as in this very sentence, are
common. But if phrases are embedded in embedded phrases,
things can get complicated. For example, let's convert the
fourth sentence into one featuring the malt: The malt
that the rat that the cat killed ate lay in the
house that Jack built. Now try the fifth sentence:
The malt that the rat that the cat that the dog
worried killed ate lay in the house that Jack built.
Are you still following me?
That last example is
perfectly grammatical, but more than one level of
center-embedded recursion is hard to follow, for psychological
rather than linguistic reasons. Center-embedding requires a
memory device, such as a stack of pointers, indicating where
to pick up the procedure once an embedded constituent has
been completed. This is not so bad if there is just one
embedded structure, since a single pointer can be held in
memory to show where to pick up the original procedure. With
multiple embedding, you need to keep track of several
pointers, which can overstretch working memory. Examples of
sentences with more than one level of center-embedding are rare
in natural discourse.
In a recent Science
article, psychologist Marc D. Hauser of Harvard University
and his colleagues suggested that recursion is a basic
characteristic that distinguishes human language from all
other forms of animal communication. Chimpanzees and bonobos
have been taught a form of language, sometimes known as
protolanguage, with some of the characteristics of
human language, including the use of symbols to represent
objects or actions, and some ability to combine symbols to
create new meanings. There is no evidence, though, that
these apes use these symbols (or combinations of symbols) in
recursive fashion, to create anything like the unlimited set
of meanings that we humans can create.
What About Birds?
Superficially, at least, the songs emitted by some birds seem
to have something of the complexity of human language.
American ornithologists Jack P. Hailman and Millicent S.
Ficken once argued that chickadee calls have a computable
syntax and therefore qualify as "language." These
calls are made up of four qualitatively distinct sounds,
which we may label A, B, C and D. These
elements are always in the same order, although any element
may be repeated any number of times or may be omitted
altogether. The sequences ABCD, B, BD, AAABBCCCD
are all legitimate. Although there is considerable variety in
such sequences, they do not involve recursion beyond the
simple repetition of elements. Unlike human language, the
sequences can be specified by what is known as a
finite-state grammar, in which the choice of
element at any point in the sequence can be specified by the
preceding element. Thus B may follow A, or
B may be the first in the sequence, but it can
never follow C or D. Of course, the birds may
well have used more complex rules, but we do not need to suppose
more than that they merely stepped from one element to the next,
without any appreciation of what went before or what comes
after. In contrast, human language involves the combining of
constituents into phrases, and the generation of sentences
by recursive rules whereby phrases may be defined in terms
of phrases, and every element in the sequence contributes to
the grammatical construction.
Neuroethologist Timothy Q. Gentner at the University of
California, San Diego, and his colleagues have recently
argued that European starlings can parse sequences of sounds
consisting of as many as four levels of center-embedded
recursion. The starlings were taught to identify sequences
of sounds, drawn from eight sounds classified as rattles,
A, and eight classified as warbles, B. The
sequences were generated either by a finite-state grammar in
which AB sequences were simply repeated up to four
times (as in AB, ABAB, ABABAB or
ABABABAB), or in which AB pairs are
embedded in AB pairs with up to four levels of
recursion (as in AB, AABB, AAABBB, and
AAAABBBB). The actual choices of As and
Bs were generated randomly, so that the birds could not
simply learn particular sequences. Some, but not all, of the
birds learned to discriminate these kinds of sequences from
each other, and also from sequences that did not obey the
rules, suggesting a capacity to understand recursion.
The problem here is that the starlings need not have parsed
the recursive sequences according to the recursive rule. A
simpler solution is simply to count the number of successive
As and the number of successive Bs, and
accept the sequence as belonging to the recursive category
if the two numbers are equal. Such a strategy is probably
not beyond the computational capacity of a starling. There
is abundant evidence for a number sense in birds. For
example, the famous African gray parrot, Alex, raised by Irene
Pepperberg of Harvard University, can count up to six and
understands the concepts of same and different. Starlings are
also known for their intricate songs, suggesting
sophisticated production and perception of sequences. The
final movement of Mozart's Piano Concerto in G Major, K.
453, is said to be based on a song by his pet starling.
There is no suggestion, though, that the songs of starlings,
although Mozartian, are recursive. It should also be
remembered that even humans have considerable difficulty parsing
sentences in which the number of embedded phrases exceeds
two.
Again, we do not know what really goes on in the mind
of a starling, but parsimony dictates that we accept the
simpler explanation for their behavior. The challenge to
show that any nonhuman species can either produce or parse
recursive combinations of elements remains.
Theory of Mind
Recursion
is not restricted to language but applies to other aspects
of human thought. One of these is known as "theory of
mind," which refers to the ability to imagine what might be
going on in the mind of another individual. The mental processes
of thinking, knowing, perceiving or feeling might be
regarded as zero-order theory of mind, and are probably
common to many species. They are not recursive. First-order
theory of mind refers to thinking, knowing, perceiving or
feeling what others are thinking, knowing,
perceiving or feeling, and therefore is recursive. It is
implied by statements such as "I think you must be
thinking that I'm an idiot," or "Ted thinks Alice
wants Fred to stop bugging her."
Are nonhuman species capable of this? How could we know?
The problem is that language itself is well designed to
express recursive ideas, and it is difficult to test theory
of mind without language. So far, no nonhuman animal has
shown us that it has a communication system powerful enough
to reveal theory of mind, so we must rely on nonlinguistic
tests.
One such test involves tactical deception, where one
animal performs some act based on an appreciation of what
another animal might be thinking, or what it might be able
to see. A young chimpanzee might wait until a dominant male
is not looking before stealing food. In a more complex
example, a young male baboon may see another baboon eating
corn. He screams in mock fear, and his mother comes and
chases the other baboon away. The young baboon then seizes the
corn. The question here is whether the young baboon actually
knew what would be in his mother's mind when he screamed or
whether the behavior had simply been learned through trial
and error.
Psychologists Richard Byrne and Andrew Whiten
at the University of St. Andrews in Fife, Scotland,
collected examples of possible tactical deception from
observations made by primatologists in the field and
carefully filtered out those that might depend on simple
trial and error. From a total of 253 cases only 26 observations
passed the test. There were 12 examples from common chimpanzees
and three each from bonobos, gorillas and orangutans. Five
more instances from mangabeys, which are closely related to
baboons, also passed. Nevertheless, Byrne and Whiten
suggested that true tactical deception, requiring theory of
mind, might be restricted to human beings and great apes,
and even among the latter the evidence is not very
compelling. In marked contrast, a search for "tactical
deception" on the Google search engine results in roughly
967,000 responses, most of them relating to human warfare.
Other tests have been proposed for nonhuman species, but
the results are scarcely more compelling. For example,
psychologist Daniel Povinelli and his colleagues at the
University of Louisiana, Lafayette, have shown that
chimpanzees are as likely to beg for food from a person who
is blindfolded, or who has a bucket over her head, as they
are from one who can see, suggesting that the chimps are
incapable of the recursive understanding of whether another
individual can see. Michael Tomasello at the Max Planck
Institute for Evolutionary Anthropology and his colleagues
argue that chimpanzees are actually smarter than that and
under some circumstances do understand what others can see,
although the authors do acknowledge that "chimpanzees
do not have a full-blown, human-like theory of
mind."
If there is doubt over whether great apes are
capable of first-order theory of mind, there is certainly
nothing to suggest that they are capable of higher orders.
Human affairs run easily into many orders of theory of mind,
as is well conveyed in literature and the theater. In Jane
Austen's Pride and Prejudice, Elizabeth Bennet
thinks that Darcy thinks that she
thinks he thinks too harshly of her family. Or
in Shakespeare's Twelfth Night, Maria
foresees that Sir Toby will eagerly
anticipate that Olivia will judge Malvolio
absurdly impertinent to suppose that she
wishes him to regard himself as her
preferred suitor. Each italicized word after the first
indicates another level of recursion.
Theory of mind may
even be a prerequisite for religious belief, according to
evolutionary psychologist Robin Dunbar at the University of
Liverpool. The notion of a God who is kind, who watches over
us, who punishes and who admits us to Heaven if we are
suitably virtuous depends on the understanding that other
beings—in this case a supernatural one—can have
human-like thoughts and emotions. Indeed, Dunbar supposes that
several orders of recursion may be required, since religion is a
social activity, which depends on shared beliefs.
The
recursive loop that is necessary runs something like this: I
suppose that you think that I believe
there are gods who intend to influence our futures
because they understand what we desire.
This is fifth-order recursion. Dunbar himself must have
achieved sixth-order recursion if he supposes all this, and
if you suppose that he does then you have achieved
seventh-order recursion. Call it "Seventh Heaven,"
if you like.
The Self and Mental Time
Travel
The belief that one has thoughts of one's own is theory of
mind about one's self. Philosopher René Descartes is
famous for the phrase "cogito, ergo sum,"
although what he actually wrote was "Je pense, donc
je suis"—"I think, therefore I
am." Descartes took this as the basic proof of his own
existence, because even if he doubted it, doubt was a form
of thinking, so his actual existence was not in doubt. The
phrase is fundamentally recursive, since it implies not just
thinking, but thinking about thinking. The fact that we can
be aware of our own thinking (and not just of our own
thoughts) implies a concept of self.
One way to
investigate whether animals have a concept of self is the
mirror test, first devised in 1970 by psychologist Gordon G.
Gallup Jr., now at The State University of New, York at
Albany. A mark is placed on the animal in such a way that it
can only be viewed in a mirror. The question then is whether
the animal will attempt to remove the mark, or otherwise
indicate that it realizes the mark is on its own body. The
evidence suggests that only dolphins, great apes and
elephants pass the test, which has been taken to mean that
they have a concept of self. The results are controversial,
though, because it might mean no more than that these
animals understand that the object in the mirror corresponds
to their own physical self, but need not mean that
they understand that the physical self is capable of
thoughts or desires.
To be properly recursive, the
concept of self should be involved—that is, not merely
knowing that one is a physical object, but knowing that one
knows, or knowing that one has mental states. There is
little evidence that shows this to be the case for any
nonhuman species.
Another way to test the concept of
self is through the awareness that one can exist at
different points in time. For example, we might remember
what we were thinking or experiencing yesterday, which is
again a recursive process. It indicates that we can not only
understand that we have thought processes in the present, but
that we also had them in the past and will have them in the
future. To extend Descartes' dictum, we might say "I
thought, therefore I was," and "I will think,
therefore I will be." The concept of self can be
extended through time.
The notion of a past self depends
on memory. There are two kinds of conscious memory,
according to the Canadian neuroscientist Endel Tulving.
Semantic memory refers to your storehouse of
knowledge about the world, such as knowing that Wellington is
the capital of New Zealand, or that the boiling point of
water is 100 degrees Celsius. Episodic memory
refers to the particular episodes in your life that you can
relive in your mind. You probably remember what you did
yesterday, not just as a succession of facts, but as events
that you can bring to consciousness and replay in your mind.
Such memories, unlike semantic memories, are recursive because
they involve making mental reference to your earlier mental
self. Recovery of semantic memories implies what Tulving
calls noetic awareness, which is simply knowing,
whereas the recovery of episodic memories implies
autonoetic awareness, which is self-knowing.
Tulving has also claimed that episodic memory is unique to
humans. This is not to deny that other species have
memories, often prodigious ones. Among birds that cache
food, for example, the Clark's nutcracker is among the most
prolific, storing seeds in thousands of locations and
recovering them with high, but not perfect, accuracy. This
does not mean, though, that the bird remembers the act of
caching the food itself; rather, it may simply remember
where the food is located. I believe I know the meanings of
tens of thousands of words, but with very few exceptions I
cannot remember the episodes in which I first encountered
these words.
Clever experiments by Nicola Clayton and
her colleagues at the University of Cambridge have suggested
that at least one species of bird, the Western scrub-jay,
may have more detailed memory than one might have imagined.
The birds can remember where they have stored particular
items, such as worms or nuts, and they will recover one or
the other depending on how long they have been cached. They
generally prefer worms, but they will avoid the nightcrawlers in
favor of nuts if the worms have been cached for too long, and
are therefore likely to have decayed and become unpalatable.
This has been taken to mean that the jays know what
has been cached, where it has been cached and
when it was cached. These three conditions, known
as the WWW criteria, have been claimed by
some to be sufficient evidence for episodic memory in the
scrub-jay—a humbling thought. Even so, this may not be
sufficient proof that the birds relive the act of caching.
The memory for where a food item has been cached might carry a
time tag, equivalent to a "use-by date," which
indicates how long the item has been cached, but this need
not involve a specific memory for the caching episode
itself.
One might stand a better chance of demonstrating
mental time travel in primates than in birds, especially in
our closest nonhuman relatives, the chimpanzee and the
bonobo. The German-American psychologist Wolfgang
Köhler, who is famous for his experiments on
chimpanzees while stationed in the Canary Islands during World
War I, noted that, for all their improvisational skills,
chimpanzees had little concept of the past or the future.
Work over the past 50 years on teaching language-like
communication to chimpanzees and bonobos provides little
reason to question this conclusion. So far, there is no
evidence for the acquisition of tense, nor any sense that
these animals communicate about specific past events or
possible future ones.
Psychologist Thomas Suddendorf at
the University of Queensland has argued that episodic memory
is but one part of a more general capacity for mental
time travel, which includes travel into an imagined
future as well as into the recalled past. Amnesic patients
who lose episodic memory also lose a sense of possible
future events, and children seem to understand the concepts of
past and future at about the same time, around age four.
Indeed, episodic memory may function as not so much a record
of the past as a repository of information about events that
can supply a kind of vocabulary for the generation of future
events. Perhaps this explains why episodic memory is both
unreliable and incomplete, and often a bane to courts of
law. In cases of amnesia, it is typically episodic rather
than semantic memories that are lost. It does not matter
that episodic memory is incomplete and fragile, so long as it
supplies sufficient information to generate plausible and
effective future scenarios. After all, it is the future that
matters to us, not the past.
It is perhaps not an
exaggeration to say that human beings are obsessed with
time, as we ponder the past and plan our futures. We measure
time in seconds, minutes, hours, days, weeks, months, years,
decades, centuries, millennia, eras and eons. We measure it both
back to the past and forward to the future. We extrapolate it
beyond our own life spans, even back to the Big Bang, which
is said to have created the universe. Through time we
understand death, and perhaps because of this we have
appealed to religions for the promise of an afterlife. Time
creates stress, as deadlines draw near, but we can also
appeal to time to heal our woes. When Viola in Twelfth
Night, masquerading as a man, finds herself in an
impossible situation, she is moved to say, "O Time,
thou must untangle this, not I; It is too hard a knot for me
t'untie!"
Language itself is infused with time. We use
many prepositions, such as at, about,
around, between, across,
against, from, to and
through, to apply to time as well as space, and some,
such as since or until, are restricted to
the time dimension. The use of tense allows us to
incorporate time into language, even in recursive fashion.
For example, the past perfect, as in "I had already
eaten," refers to an event that is further in the past
than some reference time in the past, while the future
perfect, as in "He will have arrived," refers to
something that will be in the past at some future date.
Whatever the capacity for nonhuman animals to travel mentally
in time, it seems safe to say that once again the
generative, recursive way in which we imagine events in time
seems to surpass anything that has been demonstrated, or
even hinted at, in our closest primate relatives.
Counting and Tools
Another example of recursion, probably derived from
language, is counting. By using recursive rules, humans can
count indefinitely. All you need is a finite set of digits
and a few simple rules that you can use to progress from one
number to the next. We now know that many animal species can
count, but they can only do so accurately up to some small
value. Even that is not strictly counting but is closer to
the human capacity for subitization, which is the
ability to enumerate at a glance quantities up to about
four. Beyond that, our ability to enumerate without actually
counting is increasingly inaccurate as the actual number
increases. You might estimate the number attending a lecture
as around 75, or the crowd at a football game as 15,000, but in
neither case are you likely to be accurate. Counting, though,
can lend perfect precision up to any number—although
it can of course be time-consuming. Counting is a further
illustration of how recursive principles can vastly increase
the power and capacity of the human mind. More generally,
human computation is recursive. Programmers use subroutines
that call subroutines, and my laptop contains folders that
contain folders that contain folders.
The use and manufacture of tools may also include recursive
components. Tool use itself is not uniquely human. Chimpanzees
use stones to crack nuts and twigs to extract termites from
their holes; they even fashion "spears" to stab
prey. Capuchinmonkeys are exceptional tool users, using
objects in a variety of ways to achieve their own ends. They
use sticks to rake in food, stack boxes to reach food and
even throw objects at people. New Caledonian crows strip and
shape pandanus leaves, or fashion twigs with hooks on the
end, to extract grubs from holes. But human beings are surely
the most prodigious makers and users of tools. The American
comparative psychologist Benjamin B. Beck, an expert on
tool-making behavior, once remarked that "man is the
only animal that to date has been observed to use a tool to
make a tool." This again implies recursion. Modern
technology is at least repetitive, if not always truly
recursive; consider the assembly lines that began with the
Model T Ford. Nevertheless there are wheels within wheels,
engines within engines, computers within computers.
Ultimately, perhaps, we may swamp the globe with the
products of our recursion.
Evolving the Recursive Mind
Human beings may well be unique in our attempts to find
criteria that define our own uniqueness—we think
uniquely, therefore we are unique. Among the characteristics
that are often claimed as uniquely human are language,
theory of mind, the concept of a knowing self, episodic
memory, mental time travel, making tools that make tools and
counting. All, I submit, are unique because of the human
capacity for recursive thought.
Evolutionary
psychologists maintain that the essential characteristics of
the human mind evolved during the Pleistocene, the period
from about 1.8 million years ago until around 10,000 years
ago. During this era, our hominin forebears were
hunter-gatherers, and social bonding and communication became
essential to survival. According to evolutionary psychologists,
such as Leda Cosmides and John Tooby at the University of
California, Santa Barbara, the mind evolved in modular
fashion, with specific modules dedicated to specific
functions, such as language, theory of mind, cheater
detection and romantic love. Because recursion applies to
several domains, it is unlikely that the phenomenon comprises a
module in the sense that evolutionary psychologists use the
term. I suggest rather that it is a mode of computation that
can be applied to several different mental domains.
From around two million years ago, and through the
Pleistocene, the brains of our hominin forebears increased
dramatically in size, to become about three times as large
as expected in a primate of the same bodily size. American
zoologist Richard D. Alexander has suggested not only that
social bonding was necessary to ensure survival in a hostile
environment (shared with killer cats and other dangers), but
also that our forebears faced increasing competition from
their own kind. This led to runaway cycles of Machiavellianism
countered by social bonding and mechanisms for the detection and
expulsion of freeloaders, leading to such complex but
fundamentally social phenomena as language, theory of mind,
religion and wars. It was this complex calculus of social
affairs that may have led to the selection for larger brains
with the capacity for recursive neural systems.
The
expansion of the frontal lobes, in particular, may have been
especially critical. The frontal lobes are known to be involved
in language, theory of mind, episodic memory, and mental
time travel, and these recursive faculties may also depend
on the fact that humans, relative to other primates, undergo
a prolonged period of growth. To conform to the general
primate pattern, human babies should be born at around 18
months, not 9 months. But, as any mother will know, this
would be impossible given the size of the birth canal. The
brain of a newborn chimpanzee is about 60 percent of its
adult weight, whereas the brain of a newborn human is only about
24 percent. Our prolonged childhood means that the human
brain undergoes most of its growth while exposed to external
influences, and so is finely tuned to its environment.
Patricia M. Greenfield of the University of California, Los
Angeles, has documented how children develop hierarchical
representations for both language and the manipulation of
objects at around the same time. Just as they begin to
combine words into phrases and then use those phrases as
units to combine into sentences, so they begin to combine
objects, such as nuts and bolts, and then use the
combinations as objects for further manipulation. Greenfield
argues that both activities depend on a region corresponding
to Broca's area, that part of the cerebral cortex primarily
responsible for the production of language, on the left side
of the brain. She goes on to suggest that this relation
between language and hierarchical manipulation persists into
adulthood, citing evidence that people with Broca's aphasia
are also poor at reproducing drawings of hierarchical tree
structures composed of lines.
In the further course of
development, though, the frontal-lobe structures involved in
recursion may differentiate. Greenfield also refers to
evidence that, in a sample of mentally retarded children,
some were skilled in hierarchical construction but deficient in
grammar, whereas others showed the reverse pattern. She relates
these findings to neuro-physiological evidence that the same
brain area may be involved equally in both functions up to
the age of two. Beyond that age there is increasing
differentiation in the vicinity of Broca's area, such that
an upper region is involved with the physical manipulation
of objects and an adjacent lower region organizes
grammar.
Greenfield's analysis may be widely applicable,
pertaining to the growth and differentiation of a number of
recursive skills, including language, theory of mind,
episodic memory, the understanding of time and object
manipulation. All of these skills seem to emerge early in
childhood, at a time when the brain is growing. The critical
period of postnatal growth is both an evolutionary and a
developmental phenomenon. It probably began to appear as a
characteristic of the genus Homo some two million
years ago and governs the way children acquire skills. This
extended growth pattern takes us beyond simple associative
networks toward more dynamic processors that can parse
hierarchical structures and use rules recursively.
Although recursive skills appear to be somewhat dissociable,
their co-development, and perhaps co-evolution, may be
linked. For example, the emergence of recursive syntax may
have been evolutionarily selected precisely because it maps
onto the recursive structure of theory of mind and allowed
our ancestors to communicate their Machiavellian
thoughts—no doubt to their accomplices, not their
rivals. Theory of mind may be involved in a different way in
language, allowing us to modulate our speech in conformity with
the mental state of the listener. The recursive understanding of
time may have been critical to the evolution of language itself,
which is exquisitely equipped to recount events at different
points in time, and at locations other than the present one.
Recursion, then, is a property that infuses the early
development of basic skills, providing us with the
flexibility and creativity that characterizes the human
mind.