Recursive Mapping of Reality

How Place Hierarchies Interact with Place-Object Maps to Enable Abstract Cognition

Recursion. This word is unfamiliar to most people, even educated people. They know that events can recur, that is, history can repeat. But they are unaware that without recursion, the mathematics of the infinite would collapse, computers could not be programmed, the grammar of most languages could not be formally described, and, potentially, human thought would be impossible.

Recursion occurs when something is defined in terms of itself. Recursion is fundamentally about referential loops. A visual example would be what happens when two mirrors are pointed at each other. Elaborating on this idea, in his book I am a Strange Loop (2007), Douglas Hofstadter exhibits fascinating images obtained when a video camera is pointed at a TV screen. The screen displays dynamically changing sequences of images that cannot be easily predicted in advance, illustrating the power of recursion to generate complexity from simple inputs. Hofstadter goes on to argue that consciousness itself is a self-referential loop, a type of recursion.

A formal rule is recursive if its interpretation requires it refer to itself. A list of rules is recursive if there is any circular path through the rules, for example, if Rule 3 depends on Rule 5, which in turn depends again on Rule 3. As it turns out, the mathematical concept of infinity can be formalized recursively through two rules: (1) there is at least one number and (2) every number has a successor. As soon as one number exists, then there is another number after it, and another number after that, ad infinitum. This example shows the power of recursion: it reaches infinity in finitely many steps.

Recursion in mathematics and computer science is uncontroversial. Within linguistics, however, recursion is a fiery topic of debate. Most linguists agree that the interpretation of language can utilize recursive processes. It is easy to see that given any English sentence, a new sentence can be made by prepending the words I think that…, which proves that there is no longest English sentence despite the fact that there are only finitely many words and finitely many grammatical rules. Ergo, English has recursive syntax. But a variety of contradictory statements have been made about the relationship between language and recursion. These statements include the following:

1. All recursion in the human mind is essentially linguistic.

2. Recursion in the brain has a linguistic origin.

3. Recursion is a preexisting cognitive feature co-opted by language.

4. All languages have recursive syntax.

5. There may exist a language without recursive syntax.

Noam Chomsky, the paragon of 20th century formal linguistics, has made statements that can be construed as acknowledging or affirming each of these statements at various times, illustrating the impossibility of disagreeing with Chomsky. Daniel Everett caused a firestorm when he claimed in a 2005 paper that the grammar of an Amazonian language called Pirahã could be described without any recursive structure whatsoever. He has subsequently argued that recursion is independent of syntax in the brain and may be optionally used for syntax or not (Everett, 2017). Those interested in the debate between Everett and supporters of the traditional Chomskyan perspective should read Everett (2017b) and Nevins et al. (2009).

If language is not the seat of recursion in the brain, then what is? Although it would be possible for the brain to have multiple mechanisms for recursion with only one being used for language, recursion is complicated enough that one would not expect it to develop independently on multiple occasions. So without further evidence, Occam's razor leads us to expect that recursion should be instantiated by just one mental mechanism. I will now propose such a mechanism based on processing of place maps.

Concept Recursion from Hierarchical Place Maps

The place-based model of recursion is fairly simple. The last post discussed how each place is associated with a map filled with objects. If each place is also an object, then a place map can be filled with places, which are filled with places, which are filled with places, and so on. That is, place maps could form the basis of recursion. Certainly humans think about places in this way: a map of the world consists of countries, a country consists of states or provinces, a state consists of counties, a county contains cities, a city has buildings, a building has rooms, a room might have a map of the world, which has countries…

How powerful is this place-based recursion? Place-based recursion could provide the very threads from which conscious thought is woven. By abstracting places to serve as concepts, the mind could leverage place maps to serve as conceptual maps, and recursion on places would become recursion on concepts. By allowing concept maps to refer within the map to the same concept being mapped, concept maps could implement recursive knowledge graphs with spatial branching. The two core requirements for this proposal to work are (1) that neurobiological place maps should be used to represent the layout of concepts as well as places, and (2) that places must themselves be identified as objects.

The first core requirement has a rich history in linguistics, psychology, and neuroscience, buttressed by robust experimental support. The idea that cognitive maps could guide behavior was expressed by Edward Tolman as a counter to Skinnerian behavioralists based on experiments with rats learning mazes (Tolman, 1948). More recently, Howard Eichenbaum has shown that place map machinery is used to represent social relationships and not just navigation (Eichenbaum, 2015). Eichenbaum and Cohen (2014) propose explicitly that humans navigate concepts in the same way that they navigate places, providing substantial experimental support. Christian Doeller has also argued that humans manage concepts using cognitive maps (Bottini and Doeller, 2020). These claims are supported by evidence that human brains have place and grid cells not only in the hippocampal formation (as in rats) but in the frontal cortex as well (Staudigl et al., 2018), where conscious reasoning is supposed to occur (Dehaene, 2020). Such concept maps have been proposed in linguistics as well, most notably by Mark Johnson and George Lakoff in their books, including Philosophy in the Flesh (1999), where they call them image schemata. The Mental Spaces of Gilles Fauconnier (1985) are similar, though they are not explicitly spatial. Thus although the idea of conceptual structures being organized as spatial maps is not yet accepted as established fact, there is ample reason to expect that it might be one day.

The possibility for places to serve as objects in a cognitive map is less supported at present. I will briefly discuss some results regarding place recognition. Perschetti and Dilks (2018) showed via fMRI that at least two different neural regions are active when categorizing a visual scene versus when imagining navigating the scene, with place recognition implicating the parahippocampal place area (PPA) and navigation the occipital place area (OPA). This fact accords with the distinctions between the allocentric and egocentric place systems, since the PPA works on spatial layout and is next to the hippocampus, which is the locus of the allocentric system, whereas OPA provides navigational information and is along the egocentric pathway, closer to the vision processing regions. In a separate paper, the same authors argue that in fact PPA is involved in categorizing scenes rather than recognizing landmarks, which they tested by comparing two each of coffee shops, hardware stores, gyms, and dentist offices, demonstrating that activity in PPA was identical within instances of the same category but another area, the retrosplenial cortex (RSC), did distinguish individual instances (Persichetti and Dilks, 2018b). Thus there is neurobiological support for grouping places by type as well as for distinguishing individual place instances.

I am not aware of neurobiological studies supporting my claim that each place can be treated as an object by the brain, nor can I at present suggest a simple experiment for testing this claim. One would need to look for a translation of place coding into object processing regions. One of the difficulties is that we must consider a possibility that objects generally might be coded by a detailed neural firing pattern independent of topological position in the cortex. If this were the case, there would be a kind of neural embedding language such that isomorphic firing patterns in different parts of the brain would mean the same thing. Such embedding languages characterize the behavior of many modern AI systems today and would be very difficult to detect in the brain.

Although neuroscience cannot resolve the claim at present, there is abundant linguistic evidence that the brain can treat objects as places and vice versa. It just as easy for us to imagine walking around in a coffee shop from the inside as it is to imagine watching from the outside as a huge giant picks up the entire coffee shop. The fact that the same coffee shop can serve as a place to be navigated in one instance and an object to be grasped in another indicates the kind of identification between place and object is natural enough for the human mind.

Concept Diagrams

Now let us consider what place-based recursion could accomplish at a conceptual level. Above, I mentioned the possibility of knowledge graphs with spatial branching. Metaphorically, a concept is a place. Places in the brain have a spatial layout, but this layout is only partially based on distance. The brain relies heavily on spatial categories. Categorical spatial relationships indicate general direction (left, right, up, down, in front of, behind) and rough proximity (near, far, around, on, off). These topics are addressed in detail by cognitive scientist Steven Pinker in his book Stuff of Thought (2007) and by linguist Leonard Talmy in The Targeting System of Language (2017).

In speaking of categorical spatial representations, I should emphasize that the brain also maintains non-categorical representations of space, which are necessary for task performance. That is, we should presume at least for the time being that the brain can also construct continuous spatial maps with accurate distances. Without such maps it is difficult to see how artists could paint realistic paintings, but at the same time the rarity of talented artists with an eye for realism may point to limitations on exact spatial representations among the population generally. There is also a hemispheric bias, with categorical representations being more characteristic of the left hemisphere. All evidence suggests the left brain also predominantly handles language and linguistic representations, and hence linguistic concept maps would be expected to prefer categorical rather than continuous layouts within conceptual maps (Peer et al., 2015).

The brain also seems to alternate between one-dimensional and two-dimensional representations of space; there may also be three-dimensional representations in the temporal lobe, but these are uncertain (Bottini and Doeller, 2020). I will refer to these map-like representations as concept diagrams below.

The choice of the word diagram needs some motivation. A two-dimensional representation in the brain is somewhat like a photograph, but more like a stick drawing (as in stick figures) in that the brain mainly preserves spatial relationships such as what touches what, what things are in front, behind, or to the left or right, what is inside versus outside, what is close or far, while mental processing smudges out indications of exact spatial extent (5 cm, 1 mile, etc.). That is, all the pieces of a scene are present in the correct positions with respect to each other, but some parts might be exaggerated or not drawn to scale. A stick figure as a representation of a human is a perfect example of the kind of information that would be retained: a rough circle for a head with lines for the torso, legs, and arms. One should wonder at the fact that we accept stick drawings as a sufficient symbolic representations, given that they look nothing like the visual stimuli that occasion them. This conundrum is resolved if we accept that our conceptual maps actually do resemble stick figures.

One-dimensional representations in the brain are used for time, movement, or even linear comparison. When represented spatially, people typically demonstrate them as running along a left-right or up-down axis isolated from an underlying two-dimensional representation (Bottini and Doeller, 2020). A one-dimensional diagram could be temporal, in which case the elements in the diagram are ordered according to categories of before and after, potentially including information about temporal spacing, such as right before or long after. But one-dimensional diagrams can also be truly spatial, allowing movement both forward and backward.

Elements in a diagram, whether one- or two-dimensional, imply an absolute, allocentric order simply because of their spatial origin. But people move through space, and the order in which elements of the space are encountered is dependent on the path that is traveled. Thus the interaction between different instances of the same abstract place, such as two distinct coffee shops, could potentially be normalized by forcing the identification of homeomorphic paths. I do not have a good analogy for this process; those familiar with analytic topology will recognize some elements of homology in my description. But I do not believe that the diagrams as they occur in the brain are adequately represented either by graphs, images or line drawings. And given the neuroscience is not yet understood, I do not wish to commit to a particular form for these concepts, as a commitment would presume details that are likely to change. I am simply suggesting that the spatial and temporal relationships of elements in diagrams is primarily determined by how they influence navigation in space.

Some spaces, and hence some diagrams, have forced navigational patterns. For example, temporal diagrams must be experienced forwards in time, never backwards. Spatial diagrams in one dimension, however, might allow free navigation among the elements using the suppressed second dimension. Thus I might trace a line with my hands and say, “here is an apple, here is pear, and there a banana”, which creates a space of three fruits with an order, left to right. But then I might use my hands to point first to the apple, then the banana, and finally the pear, establishing a new order through gesture, which is a form of navigation in space. Similarly, I might draw the line on paper with symbols for each fruit, and then draw arrows though the second dimension to indicate a traversal path among the main points.

Figure 1 shows some examples of the relationships contained in diagrams. These diagrams are not intended to be particularly formal, but I have included some visual elements to aid understanding. In Figure 1(a), we see a diagram of a prototypical coffee shop, with a barista, coffee machines, couches, tables, and more, all arranged spatially. Figure 1(b) illustrates the hierarchical nature of diagrams by zooming in on one of the coffee shop tables. Notice that this diagram would likely be a subdiagram of a general diagram for tables, but its presence in a coffee shop would constrain its content to those elements consistent with a coffee shop. Thus in actual encounters with diagrams, the context of the encounter conditions and constrains the realization of the diagram to be consistent with other recently encountered places and objects. I do not assume that all people represent the prototypical coffee shop with the same mental elements. For some, the presence of an electrical outlet might be a necessary element of the diagram. For others, the outlets might be subsumed into the general concept of a wall. How people parse places into elements may vary from person to person, particularly as one moves from concrete places, where our experiences are largely similar, to abstract concepts, where our experiences differ substantially.

Figure 1(c) shows a progressive decomposition of human body, from a stick figure, to a detail of a head, to the layout of a face. Again, an exploration path through this concept hierarchy would introduce constraints. The observation of one blue eye will mean that the other eye should also be blue once observed, heterochromia being a rather rare trait. To foreshadow, for linguists, this kind of contextual constraint should evoke syntactic and phonological notions of attribute spreading, such as those supposedly propagated through governance structures. Perhaps linguistic notions of binding structures might have a broader cognitive significance than commonly realized.

In one-dimensional diagrams, I have drawn a horizontal line to indicate the single dimensionality. I have also included arrows to indicate order. Figure 1(d) shows a temporal episode through the sequence of events that constitute a concrete act of giving. The rightward arrow shows that the sequence must happen from left to right as the gift (Object) passes from giver (Person 1) to recipient (Person 2). The navigational flow in Figure 1(d) is restricted by time, as shown by the arrow.

In Figure 1(e), an integer number line is shown. The left and right arrowheads indicate that one may journey in either direction, but one cannot reach, say, -2 from 2 without passing through 0. Navigation may move backward or forward, but may not pass through the second dimension. By contrast, Figure 1(f) shows an unordered line. The elements are presented in an order, but may be navigated in any order, as one might do with a checklist.

Again, these diagrams are not intended to constitute a formal theory, but rather to illustrate the possibilities of the framework of concept diagrams. I note that none of the diagrams I have shown are particularly abstract, but it is well known how to represent abstract knowledge in graphical forms similar to these concept diagrams, as has been done in ontologies for decades now. I will eventually address the issue of abstraction through embodied analogy in later posts.

Those familiar with the image schemata of Mark Johnson and George Lakoff (as in Philosophy in the Flesh, 2007 and earlier writings) may wonder how concept diagrams differ from image schemata. The true answer is that I don't know yet, and if I find they do not differ substantially then I will shift to the established term of image schema. My current explanation is that whereas image schemata extract diagrammatic representations of the visual field, the concept diagrams I am describing arise from allocentric navigation of place. I am unsure of how image schemata would be applied to the sense of hearing, for example, much less to senses such as touch or proprioception, and I expect to model these in conceptual diagrams. However, the allocentric view of a place emerges from the senses including vision, and there may not be a material difference in the end. Additionally, it may be that at a cognitive level, being a visually oriented species, we understand many of our other senses through visual analogies, as when we say that a musical note is high. The resolution of this question is not yet clear to me.

When speaking of diagrams, I want to emphasize that I am not proposing that all people model all situations with the same concept diagrams. Different people may generate different diagrams that break up physical or conceptual space in different ways. The same person may even generate different diagrams in response to the same stimulus at different times, even at times not far removed from each other. This capability is known by the name of Gestalt, which refers to the ability to perceive a thing as a whole rather than as a deterministic sum of its parts, including switching among distinct interpretations in the face of ambiguity.

Traversal as Instantiation

The mind needs a way to treat similar places as the same while maintaining the distinction between two different places of the same type. Recall how the Perschetti (2018b) reference indicated that the parahippocampal region responded to different coffee shops in the same way, whereas the retrosplenial cortex responded distinctly to each coffee shop. Here we have the distinction between categories of place (a coffee shop) and specific places (your favorite coffee shop). So too we must address how concept diagrams filled in with general concepts (coffee shops in general) are understood when experienced as specific instances (that coffee shop you really like).

Computer scientists familiar with object-oriented programming will recognize in this description the difference between a class and an instance of the class. For the uninitiated, a class is an abstract description that defines a set of things, whereas an instance is one of the things in the set. A class would be the set of all coffee shops, whereas an instance would be your favorite coffee shop. In my proposal, concepts are more like classes. What, then, are the instances of those classes?

A place map encodes the abstract contents of a place. The actual contents of the place are discovered by exploring the place. To explore a hierarchy of places, say, a neighborhood of houses, one could tour the first house in the neighborhood, then exit to the street and enter another house. When in a house, the house map is used and elaborated for each instance. When in the street, the neighborhood map is active. Thus the hierarchical neighborhood map is explored by traversing the map. The experience of the traversal instantiates the map.

Human beings also explore photographs in the same way we explore spaces (Staudigl et al., 2018). Our eyes do not move smoothly across an image but rather jump from point to point. These jumps are called saccades. Figure 2 illustrates how eyes saccade over an image of a face. Each point in the right panel is the locus of a saccade, and the connect lines represent the path followed by the eyes between saccades. We can see that the path generated by the saccades draws a sort of map, tracing out the eyes, nose and mouth as well as the boundary of the face. We might say that the eyes have traversed the face, instantiating a concept diagram similar to that on the right side of Figure 2(c).

Now abstract concepts cannot be physically traversed, but we can imagine traversing them. I suggest that instances of abstract concept diagrams are created by mental traversal. This mental traversal is a controlled simulation that is trained by the embodied acts of spatial exploration and visual saccades. Just as we control which rooms of a house our bodies enter, so too we control which aspects of a concept our minds explore. The activity of thinking is primarily the exploration of a hierarchical conceptual space.

In order to present the idea of concept diagrams, I have drawn a distinction between places and concepts. But in fact, no such distinction need exist in the mind. From an embodied perspective, places are just concepts that can be physically and not only mentally explored. As soon as the simulation exists — even without hierarchy or recursion — the mind can visit imagined places through its planning machinery. But these imaginations can never fully escape their embodied origin.

Recursion in Syntax and Morphology

If recursion in the brain truly has its origins in spatial navigation, how then do recursive structures in language arise? I claim that syntax and morphology can be implemented through traversal of a collection of one-dimensional concept diagrams of the form described above. This traversal is in form highly similar to the structures proposed by linguists over the past half century and yet subtly different in distinct ways. If this view is correct, then the emergence of recursion in language can also be explained as a relatively straightforward replication of existing neural machinery for spatial navigation.

The categorical version of time conceives of time as a sequence of events. Syllables, words, morphemes, and phrases can be all be recognized as temporally bounded events. These words can be joined into phrases that can be joined into other phrases ultimately forming sentences that join into conversations and discourses. Generative syntacticians in the Chomskyan tradition deal with rule systems such as

1. S → NP VP

2. NP → Det VP

3. VP → VP “and” VP

4. VP → V NP

where S stands for sentence, N for noun, V for verb, Det for determiners like the, NP for noun phrase, and VP for verb phrase. Each rule specifies a transformation that expands or rewrites a symbol. Symbols that do not appear on the left are terminal symbols that must be replaced by words. By starting with the sentence symbol S, following any one of the applicable rules to expand each expansible symbol and finally replacing each terminal symbol with a suitable word, a sequence can be generated that represents a valid sentence. The rule [VP → VP “and” VP] is recursive. To illustrate these rules, we can use this toy grammar to generate a sentence:

1. S

2. NP VP

3. Det N VP

4. Det N VP “and” VP

5. Det N V NP “and” V NP

6. Det N V NP “and” V Det NP

7. The man walked the dog and mowed the lawn.

These expansions are usually drawn in tree form, and finding a tree structure to match a given sentence is called parsing the sentence.

Each of the four rules above can be adequately represented by a temporal concept diagram, and the parse can be achieved by navigating these concept diagrams in response to the incoming stream of words, as shown in Figure 3. Sentence parsing can thus be achieved by a traversal of a conceptual hierarchy; that is, I propose that syntax is an analogue of place navigation in the same way that abstract thought might be analogous to place navigation. An example of such a traversal is illustrated in Figure 3 below. As each word comes in, it triggers a syntactic concept. The word the triggers the concept Determiner, which in turn triggers the larger concept Noun Phrase. The word man triggers both the concept Noun and the concept Verb (as in These sailors man the guns when the ship is under attack), but since the concept Noun Phrase is active, the word man is contextually interpreted as a Noun. Next the word walked brings in the concept Verb, which further triggers the concepts Verb Phrase} and Sentence. The interpretation of the verb phrase via the rule [VP → V NP] is provisionally adopted as the more common case. The noun phrase the dog is handled similarly, and all goes well until the word and is encountered, which forces reanalysis to select the verb phrase expansion rule [VP → VP “and” VP]. This reanalysis happens in the same way as the analogous mental process that might occur to correct one's sense of position when one explores a maze and finds he is not where he thought he was.

These temporal diagrams are adequate for English, which has strict rules for word order, but the system is easily adapted for languages with flexible word order as well. Rather than using temporal concept diagrams for syntactic rules, these languages might use unordered one-dimensional concept diagrams, with case marking to resolve the order. An example in Latin is shown in Figure 4. Here, the high-level frame is a simple rule that recognizes a sentence as a subject (Nominative), indirect object (Dative), direct object (Accusative), and verb. Each element can simply be recognized as it arrives. There are a few things to notice here. First, this sequence is not truly unordered. The word order just given is the most common sequence in Latin, and should be thought of as a default. To order the sentence otherwise often indicates that focus is being given to the word put at the front of the sentence. This metaphor of focus is exactly the same as the importance given to the person at the head of a table, or the emphasis put on what an important person does first upon entering a public event. In all cases, this focus is a reflection of the order of traversal within a space.

One of the things that has always bothered me about generative syntax is the “no crossing branches” constraint, which roughly states that a sentence should be parsed into a tree in such a way that the lines of the tree can be arranged so as not cross each other. At least superficially, this constraint is routinely violated in classical Latin poetry. For example, the first line of Ovid's Metamorphoses runs as

    In nova fert animus mutatas dicere formas
    corpora...

    The spirit leads [me] to speak of forms changed into new 
    bodies...

Here, the word nova meaning “new” describes corpora meaning “bodies”, and both are contained in the prepositional phrase into new bodies. Yet this phrase is spread out across opposite ends of the sentence, with the subject, verb, and object all sandwiched in between the preposition and its object. We know nova belongs with corpora from the -a at the end of the words. Traditional generative grammars would assume the existence of transformation rules to move in nova to the right place and so avoid crossing branches, but this interpretation is badly strained, because the prepositional phrase is split in the middle. One could beg off the example by claiming that the true grammar is being ignored due to poetic license, but that explanation doesn't address how the phrase is comprehensible in the first place. Further, poetry rearranges words for beauty or impact, not for the sole purpose of confusing the reader.

In a place-traversal system, the “no crossing branches” constraint is unnecessary. In a building, a person could start in one room looking at a particular object, exit to look at many other rooms, and then return to the first room to observe another object in that room. The case declensions in Latin serve as traversal markers. Treating the sentence as a building, the phrase in nova creates a room for the prepositional phrase, the verb fert (“brings”) indicates that the room formed by the prepositional phrase has been exited, and the -a at the end of corpora, which otherwise does not fit in the sentence, tells the listener to return to the room where nova was observed, filling in the placeholder left by nova and completing the idea in such a way that the two key concepts of formas (“shapes”) and corpora (“bodies”) are placed in the key positions at the end and beginning of a poetic line, respectively. It would not have worked to put corpora at the beginning of the first line, because to do so would have caused confusion, given that corpora could be interpreted as the object of fert. The loose syntactic structure of the poetry is not a feature of generative transformations, but rather indicates the flexibility of structure within a case-based syntax. This flexibility can be accommodated when syntax is regarded as abstract place traversal.

I wish to call attention to another aspect of treating syntax as as an analogy of traversal. Recall how the saccade trajectories in Figure 2 jumped from one eye to the next. If the first eye were brown, a person would experience surprise on observing that the second eye were blue. This surprise illustrates a phenomenon known in syntax as binding. The first brown eye binds the second eye to be brown as well, because the eyes should agree, that is, they should be the same color. Such binding is necessary and natural when interpreting a scene as a hierarchy of objects. Binding phenomena also occur in language. The word these must be followed by a plural noun; the presence of these binds the noun to a plural form. Similarly, if the main verb of a sentence is past tense, then any subordinate verb must use a past tense as well, another instance of binding. What eye example means is that binding is not just a syntactic issue; it is a cognitive issue more generally.

I argue that binding is a general property of feature propagation through concept hierarchies during navigation. Observation of a brown eye instantiates a brown-eyed face, and a brown-eyed face should not have a blue iris in the other eye because faces generally have eyes of the same color. The traversal of the concept hierarchy from brown-eye to brown-eyed face generates a prediction of a brown iris in the other eye. Likewise, observation of the word these instantiates a plural noun phrase, for which the head noun must be plural, as in these clouds. The presence of the plural these binds the word clouds to be plural as well. These sorts of feature binding constraints are a common aspect of many formal approaches to syntax, especially phrase-structure grammars.

Some conclusions follow immediately from the proposal of syntax as place traversal that are worth noting explicitly. First, in terms of linguistic processing, there is no reason to distinguish the mechanisms of morphology from those of syntax, nor even those of phonetics. Once a phoneme is extracted from the stream of sound, then all further operations with respect to linguistic form, though they may happen in different parts of the brain, can be modeled in the same way, consisting of repeating entry, exit, and reentry to and from concept diagrams representing syllables, morphemes, parts of speech, phrases, sentences, and utterances. Each of these are recognizable as temporal concept diagrams.

A second, more subtle observation is that the combination of syntax as place traversal with semantics as simulation produces an interpretation of language in which syntax and semantics are processed simultaneously. With a little reflection, we all know this to be true, because we understand portions of the sentences spoken to us before they are complete. Thus semantic and syntactic operations must happen in parallel.

It is implausible that semantic and syntactic operations are using the same concept diagrams; that would be wildly inefficient, even exponential in complexity, if we had to have a unique noun-phrase structure for every noun in the lexicon. But if semantics and syntax are simultaneously traversing different concept diagrams, then the brain must have a facility for multiple parallel traversals of separate concept banks, at least to separate syntax from semantics, but likely for other purposes as well. Semantics itself is broader than language and is likely not limited to just one such stream but might traverse many aspects of meaning in parallel.

The parallel execution of syntax and semantics could not proceed with each one isolated from the other. When listening, the syntax must determine the structure of what is to be interpreted. But semantics may also guide the syntax to choose between ambiguous parsing choices (cf. Flying planes can be dangerous, Chomsky, 1965). And when the semantics of an online parse indicate that the parse is incorrect, both the semantic and the syntactic traversals must unwind to the last point of agreement for a retraversal in light of the newly obtained information. Both must unwind, because otherwise the semantic simulation could not proceed. The fact of parallel, lockstep traversal reflects the unity of attentional focus; most humans cannot think about two things at once.

In this proposal, then, the difference in character between the method of syntactic processing and the method of semantic processing is minimal. Both are mental traversals of a concept hierarchy, with semantics traversing distilled sensory experiences and syntax traversing distilled temporal forms.

This brief excursion has sketched an approach to syntax and morphology that I believe to be novel at least with respect to the idea of syntax as space traversal both during speaking or listening (imagine that speaking is like giving a tour of your own home, whereas listening is receiving a tour of someone else's house). Syntax was the most salient focus of linguistics as a discipline during the latter half of the 20th century, and many tens of thousands of pages have been written on these topics. I have merely scratched the surface of what needs to be addressed to fill out this proposal. For example, the syntax-as-traversal proposal should share with phrase structure grammars a heavy focus on the lexicon to drive syntactic structures, since places are recognized through the objects they contain and phrases are recognized by the words that constitute them. We will return to this topic of syntax after some time, but first we must complete our exposition of other aspects of language and cognition, after which we will be ready to address the joint operation of semantics and syntax properly.

Evolution of Recursion and Language

Treating recursion as traversal of an abstract place hierarchy, we can make a proposal for how both recursion and recursive language structures may have arisen. Primates and rodents move through space using allocentric navigation maps implemented within the hippocampal formation by assemblies of specialized cells. In humans, we see that copies of such assemblies have migrated into the frontal cortex; I do not know whether primates other than humans exhibit this migration. We have evidence that humans use spatial navigation mechanisms to navigate conceptual spaces as well.

We can presume that at some point in our lineage, navigation mechanisms were replicated outside of the hippocampus and employed for purposes beyond immediate spatial navigation. One possible advantage conveyed by this replication would be the ability to engage in long-range navigation using expanded spatial representations, which would enable rapid adaptation to new environments, promoting survival in the face of environmental challenges such as food scarcity. Many people have proposed that the human ability to adapt rapidly to new environments is a core driver of human evolution. The Sahara pump theory regards the alternation of wet and dry cycles in North Africa as a key stimulus for animal migrations, and some have suggested that these cycles forced early humans to adapt rapidly to changing environmental conditions. The ability to search for food over a wide range based on long-range navigation would certainly provide early hominids with a competitive advantage.

These expanded navigation systems would be more efficient if they were hierarchical rather than flat. Doubling the size of a flat navigational system only doubles the extent of space that can be modeled, while adding a single level of hierarchy would square the extent. Hence we should expect a hierarchical arrangement for the replicated navigation mechanisms, and of course these expectations are coherent with the fact that modern humans do use hierarchical models of space, though the method behind this hierarchy remains unclear.

A sufficiently capacious system for spatial navigation could use its extra capacity to model tools as well as places by treating tools as places. Abstract modeling of tools would lead to developments in technology that would allow bands of early humans to dominate and outlive their neighbors. The conception of tools as places opens the way to ideas that refer to themselves, like a matrushka doll or the layers of an onion. Once the expanded navigation systems were used to model tools, then the mental capacity for recursion was likely present.

The progression from concrete tools to abstract ideas relies on the existence of additional properties beyond recursion, specifically on explicitly analogical cognitive processes that we will explore eventually. These analogical processes probably developed in stages subsequent to the initial recursive facilities, because they rely on recursion. Turner and Fauconnier (2002) discusses how these stages might have built on each other. But the development of more sophisticated analogies and metaphors is somewhat independent of recursion in language. A human without full capabilities in these areas could likely still speak about relatively concrete phenomena, such as the outcome of a hunt or how to make a spear.

Recursive language processing could have been added to the human repertoire at any point through a single replication of the abstracted space navigation systems, or by the gradual cooption of prior such replications initially used for abstract concepts. Either way, the result would be a language module not initially distinct in form or methods from other systems for conceptual navigation. It would eventually be tweaked slightly to prefer processing of the kinds of structures observed in language, perhaps resulting in the claimed language preferences of Wernicke's area and Broca's area. Quite separately, the evolution of language would be further supported along the way by changes in the human vocal tract and increased emphasis on categorical spatial and temporal processing in the left hemisphere of the brain, as discussed in Everett (2017).

My proposal of syntax as abstract place traversal yields a synthesis of two points of view, first that language uses dedicated neural machinery and second that language is a cognitive process not significantly distinct from other cognitive processes. Although I have explicitly invoked a language module, this language module would emerge directly out of human navigational capabilities and only add incremental and adaptive capabilities. The emergence of such a language module follows relatively standard evolutionary processes and avoids requiring a miraculous act. This thesis remains consonant with Pinker's idea of the Language Instinct (1994) in its broadest sense, since it does propose dedicated, left-lateralized neural machinery for processing of syntax and morphology. However, my proposal does not support the idea of Universal Grammar except in a vague sense, given that the syntactic structures my proposal would allow are relatively unbounded.

Conclusion

This post has developed a hypothesis that recursion in the brain grew out of abstract spatial navigation depending on two features, namely, place hierarchies and the equation of places and objects. Although this post is quite long, I have done little more than sketch out the implications and capabilities of this hypothesis in skeletal form. A wide range of questions have been left unaddressed. What are the exact mechanisms of representing place hierarchies in the brain? What is the difference, if any, between concept diagrams and the image schemata of Johnson and Lakoff? Which branches of formal syntax are compatible with the syntax-as-place-traversal approach, and what challenges does traditional syntax raise that the hypothesis must address? And perhaps most importantly, the traversal of a concept diagram hierarchy requires a control policy that makes decisions about which concepts to visit. How is this control policy to be learned and executed?

To address these questions, I must move slowly and build up piece by piece. What I am communicating here is merely a tentative hypothesis, perhaps in need of major revisions, or perhaps incorrect entirely. In the next post, I will turn away from these deep questions and return to the basic parts of speech and sentence categories, first discussing the adjectives, then prepositions and verbs. I hope you are enjoying the process as it unfolds. Please let me know what you think in the comments.

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