[Excerpt] How Language Evolved

Part Of: Language sequence
See Also: When Language Evolved
Excerpt From: (Johansson 2011) Constraining the Time When Language Evolved
Content Summary: 1600 words, 16 min read

The evolution of language had to involve at least a new ability to map concepts to sounds and gestures and to use these communicatively. But language actually consists of a good deal more than this: First, there is phonological structure—the systematized organization of sounds (or, in sign languages, gestures). Second is morphology—the internal structure of words, such that the word procedural can be seen as built from proceed plus -ure to form procedure, plus -al to form procedural: [[[proceed] [-ure]] [-al]]. Third is syntax, the organization of words into phrases and sentences.

One way to form plausible hypotheses about evolution is through reverse engineering: asking what components could have been useful in the absence of others. A primitive system for communicating thoughts via sound or gestures is useful without phonology, morphology, or syntax. The latter components can improve an existing communication system, but they are useless on their own. So if the components of language evolved in some order, it makes sense that the connection between phonetics and meaning came first, followed by these further refinements.

A system with a linear grammar would have words— that is, stored pairings between a phonological form and a piece of conceptual structure. The linear order of words in an utterance would be specified by phonetics, not by syntax. The individual words would map to meanings, but beyond linear order, there would be no further structure—no syntactic phrases that combine words and no morphological structure inside words (such as in the word procedural).

Language Evolution_ Linear vs Recursive Grammar (1)

Indeed, we can find evidence for linear grammar in many different contexts.

  1. As the early stages of contact languages, pidgins are often described as having no subordination, little or no morphology, no grammatical words like the, and unstable word order governed primarily by semantic principles like agent before action. If the context permits, the characters in the action can be left unexpressed. For instance, if the context had already brought the boy to attention, the speaker might just say girl kiss, which in English would require a pronoun—The girl kissed him. From the perspective of linear grammar, we can ask: Is there any evidence that pidgins have parts of speech like nouns and verbs, independently from the semantic distinction between individuals and actions? Indeed, there is no evidence for syntactic phrases, beyond semantic cohesion. Pidgin grammars are a good candidate for real-world examples of our hypothesized linear grammar.
  2. For a second case, involving late second language acquisition, Wolfgang Klein and Clive Perdue did a multilanguage longitudinal study of immigrants learning various second languages all over Europe. They found that all speakers achieved a stage of semiproficiency that they called the Basic Variety. Many speakers went on to improve on the Basic Variety, but others did not. At this stage, there is no inflectional morphology or sentential subordination, and known characters are freely omitted. Instead, there are simple, semantically based principles of word order including, for instance, agent before action.
  3. A third case is home signs, the languages invented by deaf children who have no exposure to a signed language. Susan Goldin-Meadow has shown that they have at most rudimentary morphology; they also freely omit known characters. In our analysis, home signs only have a semantic distinction of object versus action, not a syntactic distinction of noun versus verb. Word order is probabilistic and is based, if anything, on semantic roles. Homesigners do produce some sentences with multiple verbs, which Goldin-Meadow describes as embedding. We think these are rudimentary serial verb or serial action-word constructions, without embedding, sort of like the compound verb in English expressions such as He came running. So this looks like a linear grammar with possibly a bit of morphology.
  4. Another case is village sign languages, which develop in isolated communities with a significant occurrence of hereditary deafness. A well-known example is Central Taurus Sign Language (CTSL), spoken in two remote villages in the mountains of Turkey. CTSL has some minimal morphology, mostly confined to younger speakers. But there is little or no evidence for syntactic structure. In sentences involving one character, the word order is normally agent + action, and two-character sentences are normally (optional) agent + patient + action: girl ball roll. But if a sentence involves two animate characters, so that semantics alone cannot resolve the potential ambiguity, word order is not very reliable. For instance, girl boy hit is a bit vague about whether the girl hit the boy or vice versa, requiring a huge reliance on pragmatics, common knowledge, and context. In fact, there is a strong tendency to mention only one animate character per predicate, so speakers sometimes clarify by saying things like Girl hit, boy get-hit. So CTSL looks like a linear grammar, augmented by a small amount of morphology. Similar results have been obtained in Al-Sayyid Bedouin Sign Language (ABSL) and the earlier stages of Nicaraguan Sign Language.
  5. These less complex systems are not confined to emerging languages; they also play a role in language processing. Townsend and Bever (2001) discuss what they call semantically based interpretive strategies that influence language comprehension. In particular, hearers tend to rely in part on semantically based principles of word order such as agent precedes action, which is why (in our account) speakers have more difficulty with constructions such as reversible passives and object relatives, in which the agent does not precede the action. Similarly, Ferreira and Patson (2007) discuss good enough parsing, in which listeners apparently rely on linear order and semantic plausibility rather than syntactic structure. It is well known that we see similar though amplified symptoms in language comprehension by agrammatic aphasics. Finally, Van der Lely and Pinker (2014) argue that a particular population of children with specific language impairment behave as though they are processing language through something like a linear grammar. The literature frequently describes these so-called heuristics as something separate from language. But they are still mappings between phonetics and meaning—just simpler ones.
  6. We have also encountered a full-blown language whose grammar appears to be close to a linear grammar: Riau Indonesian, a vernacular with several million speakers, described by Gil (2005, 2009). Gil argues that this language has no syntactic parts of speech and no inflectional morphology such as tense, plural, or agreement. Known characters in the discourse are freely omitted. Messages that English expresses with syntactic subordination are expressed in Riau paratactically, with utterances like girl love, kiss boy. The word order is quite free, but agents tend to precede actions, and actions tend to precede patients. This collection of symptoms again looks very much like a linear grammar. Hence, this is a language virtually all of whose grammar is syntactically simple in our sense. Similar results obtain for the Piraha language, whose non-recursivity is well explained by the linear grammar theory as well.
  7. Another kind of linear grammar—that is, a system that relies on the linear order of the semantic roles being expressed to form conceptual relations—surfaces when people are asked to express actions or situations in a nonlinguistic task, such as in gesture or act-out tasks. Overall, there is a vast preference to gesture, or act out, the agent first (e.g., girl), and then the patient (e.g., boy). The action is usually expressed last (kiss), but when there is a potential ambiguity, people like to avoid it by expressing the action in the middle, between the agent and patient. Crucially, the ordering preferences in these tasks are remarkably stable, independently of the ordering preferences in test subjects’ native language. That seems to indicate that the capacity to map certain semantic notions to certain linear orders is at least partly independent from language itself.
  8. As a final case, traces of something like linear grammar lurk within the grammar of English! Perhaps the most prominent case is compounding, in which two words are stuck together to form a composite word. The constituents may be any part of speech: not just pairs of nouns, as in kitchen table, but also longbow, undercurrent, castoff, overkill, speakeasy, and hearsay. The meaning of the composite usually includes the meanings of the constituents, but the relation between them is determined pragmatically. Consider examples like these:
      • collar size = size of collar
      • dog catcher = person who catches dogs
      • nail file = something with which one files nails
      • beef stew = stew made out of beef
      • bike helmet = helmet that one wears while riding a bike
      • bird brain = person whose brain is similar to that of a bird

    The second noun usually determines what kind of object the compound denotes; for instance, beef stew is a kind of stew, whereas stew beef is a kind of beef. But this can be determined solely from the linear order of the nouns and needs no further syntax.

To sum up, remarkably similar grammatical symptoms turn up in a wide range of different scenarios. This suggests to us that linear grammar is a robust phenomenon, entrenched in modern human brains. It provides a scaffolding on top of which fully syntactic languages can develop, either in an individual, as in the case of the Basic Variety, or in a community, as in the case of pidgins and emerging sign languages. Furthermore, it provides a sort of safety net when syntactic grammar is damaged, as we have seen with aphasia and specific language impairment. We have also seen that it is possible to express a great deal even without syntax, for example in Riau Indonesian—though having syntax gives speakers more sophisticated tools for expressing themselves.

Language Evolution_ Linear Grammar

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[Excerpt] When Language Evolved

Part Of: Language sequence
See Also: How Language Evolved
Excerpt From: (Johansson 2011) Constraining the Time When Language Evolved
Content Summary: 900 words, 9 min read

Speech is not impossible with an ape vocal tract, but merely less expressive, with fewer vowels available. Furthermore, the vocal tract in living mammals is quite flexible, and a resting position different from the human configuration does not preclude a dynamically lowered larynx, giving near-human vocal capabilities, during vocalizations.

Adaptations for speech can be found in our speech organs, hearing organs, the neural connections between these organs, as well as the genes controlling their development.

  • Speech organs. The shape of the human vocal tract, notably the permanently lowered larynx is very likely a speech adaptation, even though some other mammals, such as big cats, also possess a lowered larynx. The vocal tract itself is all soft tissue and does not fossilize, but its shape is connected with the shape of the surrounding bones, the skull base and the hyoid. Already Homo erectus had a near-modern skull base, but the significance of this is unclear, and other factors than vocal tract configuration, notably brain size and face size, strongly affect skull base shape. Hyoid bones are very rare as fossils, as they are not attached to the rest of the skeleton, but one Neanderthal hyoid has been found, as well as two hyoids from Homo heidelbergensis, all very similar to the hyoid of modern Homo sapiens, leading to the conclusion that Neanderthals had a vocal tract adequate for speech. The hyoid of Australopithecus afarensis, on the other hand, is more chimpanzee-like in its morphology, and the vocal tract that reconstruct for Australopithecus is basically apelike.
  • Hearing organs. Some fine-tuning appears to have taken place during human evolution to optimize speech perception, notably our improved perception of sounds in the 2-4 kHz range. The sensitivity of ape ears has a minimum in this range, but human ears do not, mainly due to minor changes in the ear ossicles, the tiny bones that conduct sound from the eardrum to the inner ear. This difference is very likely an adaptation to speech perception, as key features of some speech sounds are in this region. The adaptation interpretation is strengthened by the discovery that a middle-ear structural gene has been the subject of strong natural selection in the human lineage These changes in the ossicles were present already in the 400,000-year-old fossils from Spain, well before the advent of modern Homo sapiens. These fossils are most likely Homo heidelbergensis. In the Middle East, ear ossicles have been found both from Neanderthals and from early Homo Sapiens, likewise with no meaningful differences from modern humans.
  • Lateralization. There is no clearcut increase in general lateralization of the brain in human evolution — ape brains are not symmetric — and fossils are rarely undamaged and undistorted enough to be informative in this respect. But when tools become common, handedness can be inferred from asymmetries in the knapping process, the usewear damage on tools, and also in tooth wear patterns, which may provide circumstantial evidence of lateralization, and possibly language. Among apes there may be marginally significant handedness, but nothing like the strong population-level dominance of right-handers that we find in all human populations. Evidence for a human handedness pattern is clear among Neanderthals and their predecessors in Europe, as far back as 500 kya, and some indications go back as far as 1 mya. To what extent conclusions can be drawn from handedness to lateralization for linguistic purposes is, however, unclear.
  • Neural connections. Where nerves pass through bone, a hole is left that can be seen in well-preserved fossils. Such nerve canals provide a rough estimate of the size of the nerve that passed through them. A thicker nerve means more neurons, and presumably improved sensitivity and control. The hypoglossal canal, leading to the tongue, has been invoked in this context, but broader comparative samples have shown that it is not useful as an indicator of speech. A better case can be made for the nerves to the thorax, presumably for breathing control. Both modern humans and Neanderthals have wide canals here, whereas Homo erectus has the narrow canals typical of other apes, indicating that the canals expanded somewhere between 0.5 and 1.5 million years ago.
  • FOXP2. When mutations in the gene FOXP2 were associated with specific language impairment, and it was shown that the gene had changed along the human lineage, it was heralded as a “language gene”. But intensive research has revealed a more complex story, with FOXP2 controlling synaptic plasticity in the basal ganglia rather than language per se, and playing a role in vocalizations and vocal learning in a wide variety of species, from bats to songbirds. Nevertheless, the changes in FOXP2 in the human lineage quite likely are connected with some aspect of language, even if the connection is not as direct as early reports claimed. Relevant for the timing of the emergence of human language is that the derived human form of FOXP2 was shared with Neanderthals, and that the selective sweep driving that form to fixation may have taken place more than a million years ago, well before the split between Homo Sapiens and Neanderthals.

No single one of these indications is compelling on its own, but their consilience strengthens the case for some form of speech adaptations in Homo Heidelbergensis.

As the speech optimization, with its accompanying costs, would not occur without strong selective pressure for complex vocalizations, presumably verbal communication, this implies that Homo erectus already possessed non-trivial language abilities. While Homo erectus did not possess our species’ ability for ratcheting (cumulative) culture, it did exhibit art and sufficient skills to construct watercraft.

The Walking Ape

Part Of: Anthropogeny sequence.
Content Summary: 1600 words, 16 min read

For all his noble qualities, godlike intellect, and exalted powers, man still bears in his bodily frame the indelible stamp of his lowly origin.

– Charles Darwin, Descent of Man

Setting The Stage

Common descent denotes the discovery that all species are related: that living organisms reside in a single tree of life. Homo Sapiens is no exception. We diverged from other hominoids (great apes) some 7 mya. During that time period, fossils more than 6,000 individuals from dozens of bipedal ape species.

Bipdality_ Hominin Phylogeny

Today, we explore why apes became bipedal. But first, the evolution of apes.  

Primate Evolution

Primates are mammals with flat nails instead of claws, grasping hands and feet, a highly developed visual system. They are highly iteroparous (long juvenile period) and have large brains to support the complex needs of group living. Primates are known for their symbolic dominance hierarchy, friendship mediated by grooming and mindreading (making inferences about the mental state of their peers).

Apes are primates that hang from branches (no tail), and even larger brains that promote behavioral flexibilities. Apes are known for coalitional warfare, group-specific cultural behaviors, flexible group signaling (e.g., mobbing), and tool-making.

The primate lineage emerged in the Paleocene (60 mya); apes in the Miocene (20 mya).

Bipedality_ Geological Periods (1).png

Without a tail (and in a “dead-end” body plan that precludes growing it back), apes increasingly relied upon behavioral flexibility to mitigate their comparative immobility. A monkey is an ecological specialist; the ape lineage was populated by generalists.

Apes flourished in the early and middle Miocene (20-10 mya). But they began to die out, starting in the late Miocene (10 mya). Today, there are hundreds of extant species of monkeys, and only five apes (gibbons, gorillas, orangutans, chimps and bonobos).

Evolution and progress are not synonymous. The ape branch of the tree of life is sparse because we are a failed lineage.

The failure of our ancestors seems to have been driven by a radiation from earlier primates (monkeys) in what can be called revenge of the specialized. It became increasingly difficult for generalized omnivorous species to find niches that were not more effectively exploited by a whole host of small-sized specialist monkeys.

Amidst this harsh inter-primate competition, it is interesting to note that modern apes are substantially larger than their Miocene ancestors. An increase in the body size of living apes and humans may well represent an evolutionary response to competition from monkeys.

We turn now to the question of bipedality. Before we can address why apes stood on two legs, we must first understand the anatomy of bipedality.

The Anatomy of Walking

The main anatomical structure that changed was the pelvis. The pelvis is not a single bone, but rather three bones glued together by cartilage. As we will see shortly, bipedality requires shortening of the ilium. 

Bipedality_ Pelvis Bones (1)

Walking is a pendulum-like motion. Most of the time one foot is off of the ground. This provides a stabilization problem. To solve this, bipedal animals have abductor muscles. You can actually feel these yourself: next time you walk around, feel the muscle on your hips flex (but only the muscle on the side of the weighted foot).

Bipedality_ Abductor Muscles (4)

Abductor muscles aren’t enough, however. In order to further stabilize a two-legged gait, the legs must be brought closer together. Adjusting the femur angle brings the center of gravity closer together:

Bipedality_ Knee

Finally, to improve the energy efficiency of walking, the human foot transitioned from a grasping surface to an energy-transfer platform.

Bipedality_ Foot Differences (3)

We have so far discussed four features of bipedal living. Here is a more complete list:

  1. pelvis shape (smaller ilium)
  2. pelvis musculature (abductor muscles)
  3. femur angle (more “knock-kneed”)
  4. feet (platform instead of grasping tool)
  5. foramen magnum angle (how the skull attaches to the spine),
  6. shape of the spine (bipedal spines are S-shaped), and
  7. reduced arm length (no longer needed to contact the ground)

The definition of hominin is bipedal ape. Little surprise then, that even the earliest hominin (Sahelanthropus Tchadensis) has at  least one feature associated with bipedalism. As we move to more recent species, we can see increasingly “classical” body plans:

Bipedality_ Anatomical Features (1)

Bipedality also explains why human beings suffer from:

  • Lower back pain. For hundreds millions of years, the spine was housed on a horizontal chassis. Switching to a vertical chassis places a lot of pressure on the lower spine. Zebras don’t suffer from lower back pain as much as human beings.
  • Hernias. The strain is not limited to the skeleton. Pressure also dramatically increases in the lower abdomen, causing an unusually high rates of hernias for human beings. In fact, one of the distinguishing characteristics of human beings is our smooth, fatty skin. We preferentially store fat subcutaneously to combat the pressure in our abdomen.

Theories of Bipedality

The fact that African apes became bipedal around 6 mya is not particularly interesting. A more interesting question is why African apes became bipedal. How did bipedality amplify the hominin niche?

There is no shortage of theories. Here are six:

  1. Brachiation (arm-based locomotion via branch-swinging) responsible for the postcranial features we share with apes.
  2. Arboreal apes modified their vertical climbing to walk bipedally along thick branches in the canopy.
  3. Bipedalism emerged from the need to carry babies, food, and other objects back to base.
  4. An aquatic phase of foraging and avoiding predators in water.
  5. Predator avoidance in the savannah with frequent peering over tall grass.
  6. A thermal theory whereby savanna dwellers stand up to keep cool.

These theories leave much to be desired, however.

First, some disregard ecological data entirely. The last two theories rely on the savannah hypothesis: that standing on two legs was made advantageous as forests increasingly disappeared. But the savannah hypothesis is wrong. Bipedalism emerged 6 mya, but the savannah grasslands only appeared 2-4 mya.

Second, they disregard the incrementality of natural selection. Two-legged standing preceded true bipedal walking and should not be lumped with it.  We must conceive of an ape that can stand but not walk (Orrorin tugensis?), and an ape that can walk but not run (Australopithicus afarensis).

More generally, whenever we see a complex adaptive package like walking, it is immediately useful to explore prerequisite abilities. One natural way to conceptualize the increments is as follows:

Bipedalism_ Incremental Improvements (2)

 

The above image identify anatomical increments with each new behavioral capability. 

We are not looking for a single ecological incentive for bipedalism; rather, we need individual motives for each increment in the journey to bipedality.

What kind of niche would reward flexible hips and a straight back?

The Primacy of Ecology

To answer this question, we need to get familiar with African geology and ecology.

As the most common promoter of diversity, allopatric speciation occurs when some population becomes isolated from the broader gene pool. Typically, these episodes are caused by climate change: the species gets “locked in” to a particular area by encroaching deserts, and then expands to surrounding habitats once the desert recedes.

The African continent contains wet-spots (equational rain) and hotspots (deserts). During cold glacial periods, these wet-spots expand along an east-west axis. For warm interglacial periods, the hot-spots expand along a north-south axis.

Bipedality_ African Hot-Dry Cycles.png

There are two primary forests in Africa:

Bipedality_ Two Forests

During the most arid climatic phases, the desert corridor separating these forests would close, leading to genetic isolation and speciation.

Squat Feeding in the Eastern Littorals

What kind of ape would emerge from the Main Forest Block? Such species would remain conservative (change slowly) because their much larger range embraces a much wider range of different types of wooded habitats. In fact, we know that modern-day gorillas derive from this ecosystem.

What kind of ape would be forged by the Eastern Forest Littoral? This smaller, fragmented ecosystem would cause both selection and genetic drift to accelerate. There are several peculiarities to this ecosystem worth pointing out:

Bipedality_ East African Littoral Ecology

In short, apes isolated in East African littoral forests seem likely to have found a niche on the forest floor. The natural distribution of resources favors this interpretation; and the growing competition from monkeys would have made the canopy increasingly infeasible.

These ground apes faced strong selective pressure to improve their foraging efficiency. The chimp pelvis has a very long ilium, which “locks into” the ribcage. There are clear foraging benefits for a reduction in the ilium (flexible waist), and straightening of the back (improved visibility). 

Bipedality_ Squatting Posture.png

In short, the squat-feeding hypothesis explains why flexible hips and straight spines were selected in ground apes of the early Pleistocene.

Other adaptive explanations only become relevant in further increments of the transition to bipedality. In particular, starting around 4 mya, the African continent began to dry. This made fruit increasingly less concentrated, and more seasonal. Locomotion thus became increasingly necessary to get enough calories. 

In modern humans, walking is four times more efficient than chimpanzee knuckle walking. Of course, very ancient hominins like Ardipithecus Ramidus could walk, but were less efficient than the Australopiths (and us, for that matter). But clumsy walking merely needs to improve upon the kinematic efficiency of knuckle walking, which as we have seen is not hard to do.

Bipedalism is not universally advantageous. Hominins like us are half as fast as other apes, and we have lost the ability to gallop. Greatly reduced ability to change direction while running.  The earliest bipeds probably avoided open habitats because of their increased vulnerability to predation, preferring forest and riverine habitats instead.

Bipedalism_ Ecological Pressures (2)

The facilitation of walking and running was not the ecological reason why our ancestors began the journey towards bipedality. But once they started on this particular anatomical pathway, these applications became possible. Thus, it is only with hindsight that we can say that the ultimate worth of standing up, the hidden evolutionary prize, was the ability to find the way out of a sort of ecological cul-de-sac.

Concluding Thoughts

The squat feeding theory of bipedality, as well as several of the images of this post, are credited to Jonathon Kingdon, African zoologist and author of Lowly Origin. I highly recommend this text, for those curious to learn more.

Until next time.

An Introduction to Domestication

Part Of: Anthropogeny sequence
Content Summary: 1300 words, 13 min read.

The Domestication Syndrome

Since our emigration out of Africa 70,000 years ago, Homo Sapiens have domesticated many other species, including

  • dogs (18 ka, first domesticated in Germany)
  • goats, sheep (11 ka)
  • cattle, pigs, cats (10 ka)
  • llamas, horses, donkeys, camels, chickens, turkeys (5 ka)
  • foxes (50 years ago)

Consider the domestication of wolves into dogs. An important part of the environment of a species is other species- not merely its predators or pathogens but its symbionts. In this case, canines began to get food from human campsites. Dogs that were less aggressive were (by unconscious preference and conscious intent) more successful at extracting resources. This process is known as artificial selection.

Most ancient dogs kept by hunter-gatherers share a common body shape. More recently however, humans have conducted pedigree breeding: influencing the morphologies of different dog breeds. We have used this power to sculpt breeds as diverse as the Chihuahua and the Great Dane.

The defining feature of domestication is docility: a reduction in reactive aggression. All domesticated species exhibit this feature, in comparison to their wild counterparts. Not all species are capable of this sort of control. For example, humanity has tried for centuries to domesticate big fauna such as zebras, lions, and hippos. However, some breeds have reproductive and aggressive styles that prohibit domestication.

But domestication doesn’t just bring about a change in behavior. It also brings with it a bewildering number of anatomical changes, to essentially all domesticated species. The domestication syndrome include:

  • Docility (agreeableness, reduction in irritability)
  • Depigmentation (especially white patches, brown regions)
  • Floppy ears
  • Shorter ears
  • Shorter jaws
  • Smaller teeth
  • Smaller brains (10-15% reduction in volume)
  • More neotenous behavior (juvenile behavior that extends into adulthood).
  • Curly tails

Most domesticated species express some aspect of the domestication syndrome, as we can see in the following table:

Self-Domestication_ The Domestication Syndrome (1)

Three Theories of Domestication

The sheer complexity of the domestication syndrome requires an explanation. What is the link between floppy ears and docility?

Three hypotheses suggest themselves:

  1. Multiselection. Are the symptoms of domestication all expressions of human preferences? Do we simply like curly tails and floppy ears?
  2. Environment. Is there something about proximity to humans that incentivizes these changes?
  3. Byproduct. When the genes for aggression are altered, does that somehow incidentally cause these other changes?

Animal husbandry practices are lost to the sands of time. Nevertheless, there is a way to test multiselection directly: by creating a domesticated species in the laboratory.

In 1959, Dmitri Belyaev began trying to domesticate silver foxes. He used exactly one criterion for selection: he only bred pups that exhibited the least aggression. Skeptics thought it would take centuries to complete the domestication process. But changes in temperament were seen after only four generations. At twelve generations, “elite” foxes began to emerge with dog-like characteristics: wagging their tails, allowing themselves to be petted etc. At twenty generations, the entire population was considered fully domesticated.

Despite only selecting for docility, Belyaev’s foxes exhibited the full domestication syndrome. The foxes inexplicably developed floppy ears, curly tails, white patches, etc etc. The multiselection hypothesis is false.

Is there something about proximity to humans that selects for the domestication syndrome? The environment hypothesis seems false for two reasons. First, when they return to the wild, domesticated species take a long time reverting their characteristics. In fact, often domestication gives them a selective advantage over their wild cousins. Second, as we will see in the next section, self-domesticated species such as bonobos exhibit the syndrome despite their evolution not being influence by hominids.

The byproduct hypothesis is our only remaining explanation for the domestication syndrome. But what specific system produces these changes? 

The Biological Basis of Domestication

In order to fully explain aggression reduction, we must understand it at a biological level.

The primary basis of aggression reduction is a shrinking amygdala and periaqueductal gray (PAG). These modules comprise the negative valence system which learn which stimuli are negatively-valenced, and forward them to the mobilization system (e.g., snake → bad → run away). Serotonin inhibits the negative valence system, and domesticated animals have much high concentrations of serotonin receptors in these regions. Finally, it appears that these changes mostly act across development. The negative valence system comes online only slowly: there exists a socialization window in the first month of a wolf’s life, where it can learn “humans are okay”. Domestication primarily acts by increasing the socialization window from one to twelve months. If a dog isn’t exposed to a human in its first year, it’s now-active fear system will kick in: it will be wild for the rest of its life.

So what biological system is able to a) expand the socialization window, and b) cause the rest of domestication syndrome? The leading hypothesis involves a feature of development called the neural crest.

A blastocyst has no brain. To correct this unfortunate situation, every vertebrate genome contains instruction for constructing a neural tube. This structure emerges via folding.

The neural crest resides between the epidermis and the neural tube. These neural crest cells (NCCs) then proceed to migrate to a certain number of other anatomical structures to assist development. When the NCC migration malfunctions, the resultant disease is called a neurocristopathy. Many neurocristopathies result in outcomes similar to the domestication symdrome! For example, here is the effect of piebaldism:

Self-Domestication_ Piebalism

The mild neurocristopathy hypothesis (Wilkins et al, 2014) holds that domestication syndrome is a byproduct of changes to the NCC migration pattern.

Self-Domestication_ Mild Neurocristopathy Hypothesis

The hypothesis, however, is not very detailed (how exactly is NCC migration changed? What are the genomic and epigenomic contributions?). It is more of a promissory note than a mechanistic account. And there are other holistic hypotheses on offer, including genetic regulatory networks (Trut et al 2004) and action of the thyroid gland (Crockford 2000). It seems clear that, in the coming decades, a detailed mechanistic theory of domestication will emerge to vindicate the byproduct hypothesis.

Two Kinds of Domestication

Natural selection explains why the “design requires a designer” trope is obsolete. For the same reason, domestication can occur in the absence of a domesticator. More precisely, change in a species ecological niche can itself select against aggression.  Because aggression is very relevant to survival, we see plenty of species that have increased, and plenty that have decreased their rates of aggression. We call those less aggressive species self-domesticated: they became more peaceful in the absence of humans. What’s more, these species also exhibit the domestication syndrome.  

Another example is embedded in Foster’s Rule. Islands tend to be geologically more recent than continents, so their populations derive from the continent rather than vice versa. Islands tend to have fewer predators, but also fewer resources. Reduced predation increases the size of small animals (e.g., dodos evolved from pigeons), but limited resources decreases the size of big animals (e.g. the 3ft tall dwarf elephant).  

Self-Domestication_ Foster's Rule

Because islands have fewer predators, they also tend to have higher population densities; as such, reactive aggression is a less useful strategy. Selection favors the less aggressive. And we can see the domestication syndrome in island species. For example, the Zanzibar red colobus monkey has diverged from the continental red colobus along the same trajectory as dogs diverged from wolves.

Other examples of self-domestication can be found with group size reduction (ungulates, seals) and low-energy habitats (extremophile fish).

Finally, bonobos provide a particularly relevant example of self-domestication. Because food is more plentiful (don’t have to compete with gorillas for vegetation), females can spend time close to one another. Proximity produces bonding, and female coalitions exert pressure on bonobo behavior.

  • In chimps, bullying women increases reproductive success. Chimps will systematically beat up all females in their group as a coming-of-age ritual.
  • In bonobos, female coalitions retaliate against male aggression, making it unprofitable. Sexual selection then acts against reactive aggression.

So we can see that domestication (i.e., reduction in aggression) can come in two flavors: traditional vs self-domestication.

Self-Domestication_ Categories of Aggression Reduction (1)

As we will see next time, Homo Sapiens is yet another example of a self-domesticated species. See you then!

Related Resources

  • Wilkins et al (2014). The “domestication syndrome” in mammals: a unified explanation based on neural crest cell behavior and genetics

[Excerpt] Replicators and their Vehicles

Original Author: Richard Dawkins, The Selfish Gene
See Also: [Excerpt] The Robot’s Rebellion
Content Summary: 800 words, 4 min read

The First Replicator

Geochemical processes gave rise to the “primeval soup” which biologists and chemists believe constituted the seas some three to four thousand million years ago. The organic substances became locally concentrated, perhaps in drying scum round the shores, or in tiny suspended droplets. Under the further influence of energy such as ultraviolet light from the sun, they combined into larger molecules. Nowadays large organic molecules would not last long enough to be noticed: they would be quickly absorbed and broken down by bacteria or other living creatures. But bacteria and the rest of us are late-comers, and in those days large organic molecules could drift unmolested through the thickening broth.

At some point a particularly remarkable molecule was formed. We will call it the Replicator. It may not necessarily have been the biggest or the most complex molecule around, but it had the extraordinary property of being able to create copies of itself.

A molecule which makes copies of itself is not as difficult to imagine as it seems at first, and it only had to arise once. Think of the replicator as a mold or template. Imagine it as a large molecule consisting of a complex chain of various sorts of building block molecules. The small building blocks were abundantly available in the soup surrounding the replicator. Now suppose that each building block has an affinity for its own kind. Then whenever a building block from out in the soup lands up next to a part of the replicator for which it has an affinity, it will tend to stick there. The building blocks which attach themselves in this way will automatically be arranged in a sequence which mimics that of the replicator itself. It is easy then to think of them joining up to form a stable chain just as in the formation of the original replicator. Should the two chains split apart, we would then have two replicators, each of which can go on to make further copies.

Replicator Competition

The primeval soup was not capable of supporting an infinite number of replicator molecules. For one thing, the earth’s size is finite, but other limiting factors must also have been important.

But now we must mention an important property of the copying process: it is not perfect. mistakes will happen. I hope there will be no misprints in this book, but if you look carefully you may find one or two. We do not know how accurately the first replicator molecules made their copies. Their modern descendants, the DNA molecules, are astonishingly faithful compared with the most high-fidelity human copying process, but even they occasionally make mistakes, and it is ultimately these mistakes which make evolution possible. Mistakes were made, and these mistakes were cumulative.

Replicators with a comparatively worse design must actually have become less numerous because of competition, and ultimately many of their lines must have one extinct. There was a struggle for existence among replicator varieties. They did not know they were struggling, or worry about it; the struggle was conducted without any hard feelings, indeed without feeling of any kind. But they were struggling, in the sense that any mis-copying which resulted in a new improved level of stability, or a new way of reducing the stability of rivals, was automatically preserved and multiplied.

This process of replicator improvement was cumulative. Ways of increasing stability and of decreasing rivals’ stability became more elaborate and more efficient. Some of them may even have ‘discovered’ how to break up molecules of rival varieties chemically, and to use the building blocks so released for making their own copies. These proto-carnivores simultaneously obtained food and removed competing rivals. Other replicators perhaps discovered how to protect themselves, either chemically, or by building a physical wall of protein around themselves. This may have been how the first living cells appeared.

Replicator Self-Improvement

Replicators began not merely to exist, but to construct for themselves containers, vehicles for their continued existence. The replicators that survived were the ones that built survival machines for themselves to live in. The first survival machines probably consisted of nothing more than a protective coat. But making a living got steadily harder as new rivals arose with better and more effective survival machines. Survival machines got bigger and more elaborate, and the process was cumulative and progressive.

Was there to be any end to the gradual improvement in the replicators’]techniques? What weird engines of self-preservation would the millennia bring forth?  Four thousand million years on, what was to be the fate of the ancient replicators?

They did not die out, for they are past masters of the survival arts. But do not look for them floating loose in the sea; they gave up that cavalier freedom long ago. Now they swarm in huge colonies, safe inside gigantic lumbering robots, sealed off from the outside world, communicating with it by tortuous indirect routes, manipulating it by remote control..

They are in you and in me; they created us, body and mind; and their preservation is the ultimate rationale for our existence. They have come a long way, those replicators. Now they go by the name of genes, and we are their survival machines.

An Introduction to Geometric Group Theory

Part Of: Algebra sequence
Followup To: An Introduction to Abstract Algebra
Content Summary: 1500 words, 15 min read

An Example Using Modular Addition

Last time, we saw algebraic structures whose underlying sets were infinitely large (e.g., the real numbers \mathbb{R}). Are finite groups possible?

Consider the structure ( \left\{ 0, 1, 2, 3 \right\}, +). Is it a group? No, it isn’t even a magma: 2 + 3 \not\in \left\{ 0, 1, 2, 3 \right\}! Is there a different operation that would produce closure?

Modular arithmetic is the mathematics of clocks. Clocks “loop around” after 12 hours. We can use modulo-4 arithmetic, or +_{4}, on  \left\{ 0, 1, 2, 3 \right\}. For example, 2 +_{4} 3 = 1.

To check for closure, we need to add all pairs of numbers together, and verify that each sum has not left the original set. This is possible with the help of a Cayley table. You may remember these as elementary school multiplication tables 😛 .

Geometrical Group Theory_ C4 Cayley Table Modular Arithmetic

By inspecting this table, we can classify Z_4 = ( \left\{ 0, 1, 2, 3 \right\} ), +_{4}).

  1. Does it have closure? Yes. Every element in the table is a member of the original set.
  2. Does it have associativity? Yes. (This cannot be determined by the table alone, but is true on inspection).
  3. Does it have identity? Yes. The rows and columns associated with 0 express all elements of the set.
  4. Does it have inverse? Yes. The identity element appears in every row and every column.
  5. Does it have commutativity? Yes. The table is symmetric about the diagonal.

Therefore, Z_4 is an abelian group.

An Example Using Roots of Unity

Definition 1. A group is said to be order n if its underlying set has cardinality n.

So Z_4 is order 4. What other order-4 structures exist?

Consider the equation i^4 = -1. Its solutions, or roots, is the set \left\{ 1, i, -1, -i \right\}. This set is called the fourth roots of unity.

So what is the Cayley table of this set under multiplication R_{4} = ( \left\{ 1, i, -1, -i \right\}, *)? In the following table, recall that i = \sqrt{-1}, thus i^2 = (sqrt{-1})^2 = -1.

Geometric Group Theory_ C4 Cayley Table Roots of Unity (1)

Something funny is going on. This table (and its colors) are patterned identically to Z_4! Recall that a binary operation is just a function f : A \times A \rightarrow A. Let’s compare the function maps of our two groups:

Cyclic Groups_ Binary Operation as Function (2)

These two groups for structurally identical: two sides of the same coin. In other words, they are isomorphic, we write Z_{4} \cong R_{4}. Let us call this single structure C_4.

But why are these examples of modular arithmetic and complex numbers equivalent?

One answer involves an appeal to rotational symmetry. Modular arithmetic is the mathematics of clocks: the hands of the clock rotating around in a circle. Likewise, if the reals are a number line, complex numbers are most naturally viewed as rotation on a number plane.

This rotation interpretation is not an accident. It helps use more easily spot other instances of C_4. Consider, for instance, the following shape.

Geometric Group Theory_ Rotational Symmetry Object

On this shape, the group of rotations that produce symmetry is W_4 = (\left\{ 0, 90, 180, 270 \right\}, \text{rotate}). Inspection reveals that this, too, is isomorphic to C_{4}!

Towards The Presentation Formalism

We describe C_3 as a cyclic group, for reasons that will become clear later. 

Theorem 2. For every cyclic group C_n, there exists some generator g in its underlying set such that every other set element can be constructed by that generator.

Definition 3. When a generator has been identified, we can express a group’s underlying set with generator-dependent names. Two notation are commonly used in practice:

  1. In multiplicative notation, the elements are renamed \left\{ e, r, r^2, r^3 \right\}, where r is any generator.
  2. Similarly, in additive notation, the elements become \left\{ e, r, 2r, 3r \right\}.

Geometric Group Theory_ Multiplicative and Additive Notation (1)

These two notation styles are interchangeable, and a matter of taste. In my experience, most mathematicians prefer multiplicative notation.

What generators exist in C_4? Let’s look at our three instantiations of this group:

  • In modular arithmetic, you can recreate all numbers by 0 + 1 + 1 + \ldots. But you can also recreate them by 0 + 3 + 3 + \ldots.
  • In complex numbers, you can visit all numbers by multiplying by i, or multiplying by -i. Only -1 fails to be a generator.
  • In our rotation symmetry shape, two generators exist: clockwise 90 \circ rotation, and counterclockwise 90 \circ rotation.

For now, let’s rename all elements of C_{4} to be C_4 = (\left\{ 0, 1, 2, 3 \right\}, +)  = \langle 1 \rangle = \langle 3 \rangle.

Okay. But why is 2 not a generator in C_4?

Theorem 4. For finite groups of order n, each generator must be coprime to n. That is, their greatest common divisor \text{gcd}(g, n) = 1.

  • 2 not a generator in C_4 because it is a divisor of | \left\{ 0, 1, 2, 3 \right\} | = 4.
  • What are the generators in C_5? All non-identity elements: C_{5} = \langle 1 \rangle = \langle 2 \rangle =  \langle 3 \rangle =  \langle 4 \rangle.
  • What are the generators in C_6? Only 1 and 5: C_{5} = \langle 1 \rangle = \langle 5 \rangle.

We just spent a lot of words discussing generators. But why do they matter?

Generators are useful because they allow us to discover the “essence” of a group. For example, the Rubik’s cube has 5.19 \times 10^{20} configurations. It would take a long time just writing down such a group. But it has only six generators (one for a 90 \circ rotation along each of its faces) which makes its presentation extremely simple.

Another way to think about it is, finding generators is a little bit like identifying a basis in linear algebra.

Towards Cayley Diagrams

Definition 5. We are used to specifying groups as set-operator pairs. A presentation is an generator-oriented way to specify the structure of a group. A relator is defined as constraints that apply to generators. A presentation is written \langle \text{generators} \mid \text{relators} \rangle

  • In multiplicative notation: C_4 = \langle r \mid r^3 = e \rangle.
  • In additive notation: C_4 = \langle r \mid 3r = e \rangle.

The = e suffix is often left implicit from presentations (e.g., C_4 = \langle r  \mid r^n \rangle) for the sake of concision.

Definition 6. A Cayley diagram is used to visualize the structure specified by the presentation.  Arrow color represents the generator being followed.

Note that Cayley diagrams can be invariant to your particular choice of generator:

Geometric Group Theory_ Cayley Diagram (1)

The shape of the Cayley diagram explains why C_3 is called a cyclic group, by the way!

With these tools in hand, let’s turn to more complex group structures.

Dihedral Groups

Cyclic groups have rotational symmetry. Dihedral groups have both rotational and reflectional symmetry. The dihedral group that describes the symmetries of a regular n-gon is written D_{n}. Let us consider the “triangle group” D_{3}, generated by a clockwise 120\circ rotation r and a horizontal flip f.

With triangles, we know that three rotations returns to the identity r^3 = e. Similarly, two flips returns to the identity f^2 = e. Is there some combination of rotations and flips that are equivalent to one another? Yes. Consider the following equality:

Geometric Group Theory_ Rotation vs Reflection Equivalence (2)

Analogously, it is also true that rf = fr^2.

Definition 7. Some collection of elements is a generating set if combinations amongst only those elements recreates the entire group.

Cyclic groups distinguish themselves by having only one element in their generating set. Dihedral groups require two generators.

We can write each dihedral group element based on how it was constructed by the generators:

D_n = \left\{ e, r, r^2, \ldots, r^n-1, f, rf, r^2f, \ldots, r^{n-1}f \right\}

Alternatively, we can instead just write the presentation of the group:

D_{3} = \langle r, f \mid r^3 = 1, f^2 = 1, r^2f = fr, rf = fr^2  \rangle.

We can visualize this presentation directly, or as a more abstract Cayley graph:

Geometric Group Theory_ Dihedral Groups Intro (3)

The Cayley table for this dihedral group is:

Geometrical Group Theory- Dihedral Cayley Table

This shows that D_3 is not abelian: its multiplication table is not symmetric about the diagonal.

By looking at the color groupings, one might suspect it is possible to summarize this 6 \times 6 table with a 2 \times 2 table. We will explore this intuition further, when we discuss quotients.

Until next time.

Wrapping Up

Takeaways:

  • Finite groups can be analyzed with Cayley tables (aka multiplication tables).
  • The same group can have more than one set-operation expressions (e.g., modular arithmetic vs. roots of unity vs. rotational symmetry).
  • Generators, elements from which the rest of the set can be generated, are a useful way to think about groups.
  • Group presentation is an alternate way to describing group structure. We can represent presentation visually with the help of a Cayley diagram.
  • Cyclic groups (e.g., C_3) have one generator; whereas dihedral groups (e.g., D_3) have two.

Related Resources

  • This post is based on Professor Macaulay’s Visual Group Theory lectures, which in turn is based on Nathan Carter’s eponymous textbook.
  • Related to this style of teaching group theory are Dana Ernst’s lecture notes.
  • If you want to see explore finite groups with software, Group Explorer is excellent.
  • For a more traditional approach to the subject, these Harvard lectures are a good resource.

An Introduction to Abstract Algebra

Part Of: Algebra sequence
Content Summary: 1200 words, 12 min read

A Brief Prelude

Recall that a set is a collection of distinct objects, and a function f: A \rightarrow B is a mapping from the elements of one set to another. Further, in number theory we can express numbers as infinite sets:

  • The natural numbers \mathbb{N} = \left\{ 0, 1, 2, 3, \ldots \right\}.
  • The integers \mathbb{Z} = \left\{ \dots, -2, -1, 0, -1, -2, \ldots \right\}.
  • The rational numbers \mathbb{Q} = \left\{ x | x = p/q, p \in \mathbb{Z}, q \in \mathbb{Z}, q \neq 0 \right\}.
  • The real numbers \mathbb{R}.

The Axioms of Addition and Multiplication

In elementary school you learned that a+b = b+a, for any two integers. In fact there exist five such axioms:

  • Closure. \forall a, b \in \mathbb{Z}: a + b \in \mathbb{Z}.
  • Associativity\forall a, b, c \in \mathbb{Z}: (a + b) + c = a + (b+c).
  • Identity. There exists an element 0 such that, \forall a \in \mathbb{Z}: 0 + a = a + 0 = a.
  • Inverse. \forall a \in \mathbb{Z} there exists an element \boldsymbol{-a} such that a + (-a) = (-a) + a = 0.
  • Commutativity. \forall a, b \in \mathbb{Z}: a + b = b + a.

These axioms encapsulate all of integer addition. We can represent “integer addition” more formally as a set-operator pair: (\mathbb{Z}, +)

Likewise, you have surely learned that a \times b = b \times a. Multiplication too can be described with five axioms:

  • Closure. \forall a, b \in \mathbb{Z}: a \times b \in \mathbb{Z}.
  • Associativity\forall a, b, c \in \mathbb{Z}: (a \times b) \times c = a \times (b \times c).
  • Identity. There exists an element 1 such that, \forall a \in \mathbb{Z}: 1 \times a = a \times 1 = a.
  • Inverse. \forall a \in \mathbb{Z} there exists an element \frac{1}{a} such that a \times \frac{1}{a} = \frac{1}{a} \times a = 1.
  • Commutativity. \forall a, b \in \mathbb{Z}: a \times b = b \times a.

These axioms encapsulate all of integer multiplication. We can represent “integer multiplication” more formally as a set-operator pair: (\mathbb{Z}, \times)

Towards Algebraic Structure

Did the above section feel redundant? A lesson from software engineering: if you notice yourself copy-pasting, you should consolidate the logic into a single block of code.

Let’s build an abstraction that captures the commonalities above.

Definition 1. A binary operation is a function that takes two arguments. Since functions can only map between two sets, we write f : A \times A \rightarrow A.

Examples of binary operations include +, \times, \text{etc}. Note that a \times b is just shorthand for the more formal \times(a, b). Note that the operation symbol \times is just a name: we could just as easily rename the above function to be f(a, b), as long as the underlying mapping doesn’t change.

Definition 2. Let arity denote the number of arguments to an operation. A binary operation has arity-2. A unary operation (e.g., sin(x)) has arity-1. A finitary operation has arity-n.

Definition 3. An algebraic structure is the conjunction of a set with some number of finitary operations, and may be subject to certain axioms. For each operation in an algebraic structure, the following axioms may apply:

  • Closure. \forall a, b \in \mathbb{A}: a \bullet b \in \mathbb{A}.
  • Associativity\forall a, b, c \in \mathbb{A}: (a \bullet b) \bullet c = a \bullet (b \bullet c).
  • Identity. There exists the element \boldsymbol{e} such that, \forall a \in \mathbb{A}: \boldsymbol{e} \bullet a = a \bullet \boldsymbol{e} = a.
  • Inverse. \forall a \in \mathbb{A} there exists an element \boldsymbol{a^{-1}} such that a \bullet \boldsymbol{a^{-1}} = \boldsymbol{a^{-1}} \bullet a = \boldsymbol{e}.
  • Commutativity. \forall a, b \in \mathbb{A}: a \bullet b = b \bullet a.

Algebraic structures are a generalization of  integer addition and integer multiplication. Our  (\mathbb{Z}, +) and (\mathbb{Z}, \times) tuples actually comprise parameters that specify an algebraic structure.

As soon as we define algebraic structures, we begin to recognize these objects strewn across the mathematical landscape. But before we begin, a word about axioms!

The Axiomatic Landscape

Consider algebraic structures that exhibit one binary operation. These structures may honor different combinations of axioms. We can classify these axiom-combinations. Here then, are five kinds algebraic structures (“Abelian” means commutative):

Abstract Algebra- Structure Names

Of course, more esoteric options are available, including:

Abstract Algebra- Other Structure Names (1)

Of all these structures, groups are the most well-studied. In fact, it is easy to find it is not uncommon to of people conflating groups vs algebraic structures.

Definition 4. An algebraic structure is group-like if it contains one 2-ary operation. If it has more than one operation, or operation(s) with a different arity, it is not group-like.

All of our examples today count as group-like algebraic structures. There is also a large body of research studying algebraic structures with two operations, including ring-, lattice-, module-, and algebra-like structures. We will meet these structures another day.

Examples of Group-Like Structures

We saw above that the integers under addition (\mathbb{Z}, +) and multiplication (\mathbb{Z}, \times) are abelian groups. A similar finding occurs when you switch to the reals, or rationals, or natural numbers.

But addition and multiplication are not the only possible binary operations. What about subtraction (\mathbb{Z}, +)? Well, that is only a magma. Closure is satisfied, but all other axioms are violated (e.g., associativity (4 - 2) - 2 \neq 4 - (2-2)) and commutativity (4 - 2 \neq 2 - 4). Likewise, the natural numbers under subtraction are not even a magma: 2 - 4 \not\in \mathbb{N}.

All of our examples so far have groups encapsulating sets of numbers. But groups can contain sets of anything! Let’s switch to linear algebra. What about the set of all n \times n matrices under matrix multiplication?

  • Does it have closure? Yes. Matrix multiplication yields another n \times n matrix.
  • Does it have associativity? Yes. Matrix multiplication is associative.
  • Does it have identity? Yes. The identity element is the matrix I = [ \begin{smallmatrix}1 & 0\\0 & 1\end{smallmatrix}].
  • Does it have inverse? No!  Some n \times n matrices have determinants of 0. Thus, not all members of our set are invertible.

We can now identify this algebraic structure. The set of all n \times n matrices under matrix multiplication is a monoid.

But what if we limit our set to be all n \times n matrices with non-zero determinants? Well, that is a group (the inverse exists for all members). More formally, that set forms the basis of the general linear group GL_{n}(\mathbb{R}). Why isn’t it abelian? Because matrix multiplication is not commutative.

These five examples provide a glimpse into the landscape of algebraic structures. Our recipe is simple:

Take any set and operation that you care about. Classify the resultant algebraic structure by examining which axioms hold.

With these tools, we can begin to build a map of algebraic structures:

group_disc

Takeaways

  • Multiplication and addition share a remarkable number of properties, including closure, associativity, identity, inverse, and commutativity.
  • An algebraic structure (set-operation pair) generalizes the similarities in the above examples.
  • Algebraic structures can have more than one operation. Group-like structures are those with only one (binary) operation.
  • Once you can know about algebraic structures, you can find examples of them strewn across the mathematical landscape.

Until next time.