Author kevinbinz

Polytheistic Roots of Israelite Religion

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Part Of: Demystifying Religion sequence
Followup To: Yahweh and the Levites
Content Summary: 2000 words, 10min read.

Introduction

Is the Hebrew Bible monotheistic?  

We might be tempted to say yes after reading Isaiah 44:6 “I am the first and I am the last; besides me there is no God”.

But the situation is more complicated. The Hebrew Bible is also replete with polytheism. A few examples:

  • “Do you not possess that which Chemosh, your god, has given you? So shall we possess what Yahweh has given us.” Judges 11:24
  • “Who is like Yahweh among the gods?” Exodus 15:11
  • “The people of Judah have as many gods as they have towns.” Jeremiah 11:13

We also see middle ground staked out between these two positions. For example, the original audience of the book of Deuteronomy is often exhorted not to follow after other gods, without it ever being asserted that these gods did not exist or were not real. This is known as monolatrism (“single worship”).

Which belief came first?

Last time, we showed how Yahweh was originally a god of metallurgy in northwest Saudi Arabia. Today, we will work with the framework that Yahweh was introduced to Israel in a five-stage process:

  1. Traditional Polytheism. The earliest Israelites worshipped creator god El, his wife Asherah, and his sons e.g., Baal.
  2. Incorporation. Yahweh was incorporated as a 2nd tier god in El’s pantheon.
  3. Elevation. Yahweh and El are identified as the same deity.
  4. Monolatrism. A new Yahweh-only movement emerges, and the gods of the second tier are denied.
  5. Monotheism. Gods of other nations are denied, Yahweh’s power is deemed universal in scope.

Why did Yahweh worship progress along this trajectory? As we shall explore next time, as with the theocracies of surrounding nations, changes in the religious landscape have strong, robust correlates in the sociopolitical life.

Today I’d like to focus on a different, simpler topic. We shall turn to archaeology and cultural anthropology to explore expressions of polytheism within the Hebrew Bible. Many of my readers already know that the text acknowledges (polemicizes against) polytheistic practices. Less well-known are examples of celebration (bald assertions of polytheistic beliefs) and assimilation (Yahweh “adopts” the roles and characteristics of rival deities). 

Monotheism_ Five Stages (2)

 

Let’s review the deities in El’s pantheon, and their appearance in the Hebrew Bible.

A Disclaimer

For many modern readers, polytheism is a term loaded with negative connotation. Partisans use it as a weapon. Attackers point to continuities between Israelite religion & polytheism, and defenders point to instances where Israelite rhetoric polemicizes against polytheism. But all ideological innovations have both features.

More to the point, those who spend time interacting with polytheism understands how earnestly it grapples with the same aspects of the human condition as other strands of religious expression. Polytheism must be encountered on its own terms. To weaponize is to misunderstand.

The important thing to bear in mind in the following, is that underneath the images and icons of religious expression lie a particular group of people, responding to social and political pressures in thoroughly understandable ways. My experience has been, the more time you spend in someone else’s culture, the easier it becomes to empathize with their plight.

El

Israelite Polytheism_ El

At some point in its history, El was identified with Yahweh as the same god.

This equation is expressed clearly in Exodus 6:2-3. “And God said to Moses, “I am Yahweh. I appeared to the patriarchs as El, but by my name Yahweh I did not make myself known to them.” Other Biblical material asserts this equation. Joshua 22:22 states “the god of gods is Yahweh”. Judges 9:46 refers to “El of the covenant”.

The Yahweh-alone movement vigorously condemn prominent Canaanite gods… except El. There are zero condemnations of El in the Hebrew Bible. This makes sense if Yahweh was ultimately identified with this Canaanite creator-god. What’s more, archaeological evidence suggests that the Yahweh religious centers in Shiloh and Bethel were originally a place of El worship.

El and Yahweh are attributed same characteristics. El is depicted as a wise old man with a beard eg “You are great, O El, and your hoary beard instructs you”. Yahweh is described in the same terms (Daniel 7:9, Job 36:26, Habakkuk 3:6). Like “Kind El, the Compassionate”, Yahweh is a “merciful and gracious god”. The description of Yahweh’s dwelling place as a tent (Psalms 15:1, 27:6, 91:10) recalls the tent of El in the Canaanite narrative of Elkunirsa. Finally, both Yahweh and El are said to dwell amidst cosmic waters (Isaiah 33:20-22, Ezekiel 47:1-12, Zechariah 14:8).

Just as Zeus had a council, or assembly, of other gods, so too does Yahweh. The Hebrew Bible is overflowing with references to Yahweh’s (El’s) assembly. See for example Psalm 89:6-8, Zechariah 14:5, 1 Kings 22:19, Isaiah 6:1-8, and Jeremiah 23:18,22.

Baal

Israelite Polytheism_ Baal

Worship of Baal can be dated back to the foundation of Israelite societies. This can be seen in onamatology, the study of proper names. Names in the Ancient Near East tend to have a theophoric component: usually a suffix that honors a deity. Yahwistic names include Josiah, Jehu (note the “J” sound); Baal-oriented names include e.g., “Zerubabbel”. In addition to hundreds of icons devoted to Baal worship, we also see Ba’al theophoric names as common in the Levant in this time period.

Yahwistic prophets of this period reserve the most vitriol for Baal worship. Why? Because the Omride dynasty (including King Ahab & Jezebel) erected a temple to Ba’al. While the cult of Yahweh continued in the northern kingdom, Baal was perhaps elevated as the patron god of the northern monarchy, thus creating some sort of theopolitical unity between the kingdom of the north and the city of Tyre.

Indeed, there is some evidence that the cult of Baal and Yahweh got conflated in the north. Hosea 2:16-24 suggest that some northern Israelites did not distinguish between Yahweh and Baal. The religious sanctuaries in the Israelite cities of Dan and Bethel centered around golden calves; this iconography strongly parallels that of Baal. Finally, the redundancy in 1 Kings 16:32 was almost certainly a scribe glossing over the original text, “altar for Baal in temple of Yahweh”.

To induce the Israelites to stop worshipping Baal, the imagery of Baal was adopted by the Yahweh cult. The Baal Cycle, ancient mythology on the scale of the Epic of Gilgameth, has four literary themes for the storm god. Here are those themes, along with the Biblical text which mirrors them.

  1. The march of the divine warrior (Psalm 104:3 “He makes the clouds his chariot, and travels along on the wings of the wind”)
  2. The convulsions of nature as the divine warrior manifests his power (Judges 5:5, Hab 3:10)
  3. The return of the divine warrior to his holy mountain to assume divine kingship (Isaiah 31:4)
  4. The utterance of the divine warrior’s voice from his palace provides rains that fertilize the earth (Jeremiah 10:13)

Yahweh is also depicted as defeating Baal’s classic enemies:

  • Baal/Yahweh defeats a seven headed dragon, Leviathan, and River (CAT 5.1, Psalm 74:13-15).
  • Baal/Yahweh defeats Sea (KTU 1.14, Psalm 89:10).
  • Baal/Yahweh defeats Death/Mot (KTU 1.4 VIII-1.6, Isaiah 25:8).

Asherah

Israelite Polytheism_ Asherah

El’s wife was named Asherah. When Yahweh was identified with El, did he also inherit his wife? In the blessings of Joseph, Genesis 49:25 contains language specific to the Asherah cult “blessings from Breast-and-Womb”. The Bible further admits that the Israelites frequently worshipped a “Queen of Heaven” (Jeremiah 7:18, 44:17-25). Indeed, 2 Kings 21:7 tells us that worship of Asherah happened within the Temple itself. Finally, archaeology has uncovered several icons with the inscription “Yahweh and his Asherah”. This evidence cumulatively suggests that, in early forms of Israelite religion Yahweh was believed to have a wife.

Israelite polytheism_ Yahweh and his Asherah

The push towards monolatrism led to the eviction of the Asherah cult, whose memory may be preserved in Zechariah 5:5-11. But this eviction created a deficit of femininity to Israelite religious expression. To compensate, the Biblical writers began attributing feminine attributes to Yahweh (Isaiah 49:15, 46:3, 44:2,24, 42:14). Asherah-like characteristics also appear in the goddess of Wisdom in Proverbs 8.

Astral-ification

There is extensive evidence for worship of an astral deity (sun god) in Jerusalem.  And Jerusalem is presumably the site that Yahweh was identifed with El. Since the Ugaritic texts hint that El’s family was astral in character, it is not unthinkable that Yahweh was viewed similarly.

  • Proper names. A certain number of proper names are constructed from the root ‘-w-r (“shine, gleam, light”). These include Uriyyah (“Yhwh is my light”) the name of one of David’s generals, Neriyahu “Yhwh is my lamp”, Yizrayah “Yhwh gleams”, minister of Hezekiah, and dozens more.
  • Archaeology. Many pieces of material evidence, including many seals found in Jerusalem with image of the sun, or the sun god in the form of a wing bed scarab.
  • Biblical affirmations. Job 38:6-7 may attest to Israelite recognition of astral deities “Who sets its cornerstone when the morning stars sang together, and all the divine beings shouted for joy?” Similarly Judges 5:20 features conflict in the astral plane “the stars fought in the heavens”.
  • Biblical acknowledgements. Ezekiel 8:16 has Israelites worshipping sun gods. So does 2 Kings 23:5,10-11 and Zephaniah 1:4-5.
  • Biblical Incorporation. The story of Sodom and Gomorrah reflects astral themes, where the divine punishment is meted out at the moment when the sun rises. It is even possible that the two messengers and the deity in the story represent the sun god and his two acolytes. Psalm 19:4-6 and Psalm 84:11 also shows Yahweh taking on astral qualities.

Other Deities

The Ugaritic texts mention hundreds of Canaanite gods. The Bible only criticizes two of them: Ba’al and Asherah. What gives?

The Biblical authors conflates Asherah and Astarte, and conflates multiple male god as “the Baals”.  Despite this, there is only evidence of ~10 gods worshipped in early Israel. This is also true amongst Israel’s neighbors. It appears that the religious landscape of Iron Age Canaan was simply less diverse than Bronze Age Ugarit.

Do we see evidence for these gods in the Bible, despite their not being named in that text?

Anat. Known for her savagery, Anat worship involves a celebration of gore. “Knee-deep she gleans in warrior blood, neck-deep in the gore of soldiers, until she [Anat] is sated with fighting.”  While no evidence of Anat-worship exists in ancient Israel, these divine themes have strong parallels in the Biblical text. The Bible describes heaps of copses, drinking blood, devouring flesh, and swords dripping with viscera.

Astarte. In the Bible, the Name of Yahweh is described in personal terms. The divine name acts as a warrior (Isaiah 30:27) and possesses martial qualities such as radiance and strength (Psalm 29:1-2). The warrior goddess Astarte bears the title “name of Baal”. This designation of Astarte and her martial character and special relationship to the god Baal approximate the martial character of the name, and its special relationship to Yahweh as warrior god. Further evidence for this hypothesis has been adduced from the Elephantine papyri. 

Similar lines of argument can be made for entities like Light and Truth of Psalm 43:3.

Angels. The lowest tier of the Israelite pantheon also went through alterations. As the Ugaritic texts show, the lowest tier involved a number of deities who served in menial capacities. A common task for such gods was to act as messenger, the literal meaning of the English word “angel”. Certainly angels are not regarded in later traditions as gods. But they were in early traditions.

Takeaways

This post provides evidence for a simple point. Polytheistic expression (not just condemnation!) occurs in the Hebrew Bible.

These expressions are best explained by the Yahweh cult shifting away from its traditional pagan roots, and towards a monolatrist (worship one god) and later monotheist (acknowledge one god) understandings.

As we will see next time, the reasons why Yahweh worship proceeded in this interesting (but not original) trajectory, are fairly easy to understand.

Yahweh, god of metallurgy

kevinbinz 2 Comments

Part Of: History sequence
Content Summary: 2200 words, 11 min read.

Where, and how, was the god of Judaism first worshiped?

Yahweh was originally a god of metallurgy in northwest Saudi Arabia. 

Rethinking the Israelite origin story

First, a mass exodus of two million people (six hundred thousand fighting-age men) is vanishingly unlikely. If it was historical, we would expect:

  1. physical debris from the pilgrimage, at any of the thirty locations they are said to have stopped.
  2. archaeological evidence of a dramatic demographic shift in the highlands of Israel.
  3. inclusion in the (otherwise quite voluminous) records of the Egyptian border guards
  4. Egyptian texts discussing the new political situation (since the Egyptians had control over, and military outposts throughout Canaan)

How much of the above evidence do we have? Zero! Recall that absence of evidence can (and in this case does) mean evidence of absence. The very first piece of evidence aligns with the Biblical text is from 1000 BCE, where the Tel Dan stele affirms the existence of the “house of David”.

Second, the conquest narrative is non-historical. Most cities listed as razed in the Joshua narrative show evidence of uninterrupted prosperity in the archaeological record. And the three (out of thirty-one!) cities that do show interruption have not been localized to Israelite violence.

Third, until 700 BCE Judah is a much smaller political force than it makes itself to be. One demonstration of the small scale of this society is the request in one of the Armarna letter sent by the king of Jerusalem to the pharaoh that he supply fifty men “to protect the land.” Another letter asks the pharoah for one hundred soldiers to guard Megiddo from an attack by his aggressive neighbor, the king of Shechem. (Finkelstein, pp78). These letters date to the 14th century BCE. But the population in the intervening time period does not change much. Until 700 BCE, Judah’s population totaled no more than twenty settlements with a population of roughly 30,000. Only after the fall of Israel did Judah experience a population boom and full statehood.   

The Israelite people were indigenous Canaanites.

So where did the Israelite people come from? The Israelite people were originally Canaanite pastoralists who, in 1300 BCE. changed their economic strategy in response to worsening conditions. There is substantial evidence for this hypothesis

  • Ecological: we now know that the Late Bronze Age collapse (a dark age from 1200 – 900 BCE) was caused primarily by climate change-driven famine. The pastoralist strategy can only be successful if neighboring agriculturalists have surplus wheat available to trade. When that surplus dried up, former pastoralists are forced to grow their own wheat, and adapt a hybrid lifestyle.
  • Linguistic: Hebrew and Canaanite language are increasingly indistinguishable the further back you go in the Iron Age.
  • Material culture: Israelite and Canaanites shared the same building plans, pottery designs, village layouts, cooking habits …
  • Historic repetition: Canaanite pastoralists had twice before settled the highlands, but the previous two attempts had eventually failed.

We can also see when these highlands settlements began to slowly differentiate themselves from their “parent” lowland cities. First, the highland settlements did not consume pork (pigs were available for food in all regions of Canaan). Second, the highland peoples seemed to go identify themselves by the name “Israelite”, earliest mention of which is in the Merneptah stele (1204 BCE).

Since Israelites were indigenous Canaanites, we know they share the same culture. But did they start out worship the same gods?

The first Israelites worshiped the pantheon of El

In Egyptian mythology, the most powerful god was Ra. In Babylon, it was Marduk. In Greece, it was Chronus.

Monotheism_ Greek Pantheon

In Canaan, the chief god was El. El’s wife was Asherah, and his sons include Ba’al and Anut. The Canaanite pantheon is well-understood from the discovery of the Ugaritic texts.

In most English translations of the Hebrew Bible, you will see frequent use of the words “God” and “Lord”. The Hebrew terms for these phrases are more literally translated “El” and “Yahweh”. They are used so interchangeably in the Hebrew Bible that you would think them synonyms.

  • Names. The very name “Israel” means “house of El”. In contrast, later Israelite names have “Yahweh”-based suffixes e.g., Jehu. Further, most Israelite cities were named after the gods in El’s assembly.  The god Anat was honored in the city of Anathoth, the place of origin of the prophet Jeremiah. The god Dagan in Beth-Dagan. The god El in Beth-El. The god Shamash in Beth-Shamash. The god Shalimu in Jerusalem.
  • Ritual systems. The priestly system laid out in Leviticus is very nearly copy-and-pasted from the Ugaritic sacrificial system.
  • Legal codes. the Covenant, Holiness, and Deuteronomic law codes share strong parallels with surrounding Canaanite legal systems.
  • Iconography. A seal found in Jerusalem in a tomb of the seventh century shows a solar god flanked by two minor gods: “Righteousness” and “Justice”

There are also expressions of polytheism throughout the Hebrew Bible. For example,

  • “Do you not possess that which Chemosh, your god, has given you? So shall we possess what Yahweh has given us.” Judges 11:24
  • “Who is like Yahweh among the gods?” Exodus 15:11
  • “The people of Judah have as many gods as they have towns.” Jeremiah 11:13

In part two of this series, we will see hundreds more data establishing Israel’s traditional religion as polytheism.

The original Yahweh cult was a Shasu religion located in southern Edom

Recognized for their goatees and hair held back in a hairband, the Shasu nomads were well-known to the Egyptian authorities. They conducted copper mining in the wilderness, and also were quite successful camel breeders. The Bible uses the terms Edom, Teman, and Midianite interchangeably. Egyptian descriptions of the Shasu geographically overlap the Biblical land of the Midianites.

Okay. So how do we know that the Yahweh cult originated with the Shashu people?

  • Four of the oldest texts in the Bible tell us so. See Deut 33:2, Judges 5:4-5, Habakkuk 3:3 and Isaiah 63:1.
  • Special treatment of Edom. The Bible repeatedly condemns the gods of the Ammonites, the Moabites, and the Sidionites, but never the god of Edom. Deut 23:7 calls Edomites the “brothers” of the Israelites. Edom’s patriarch Esau is said to be the brother of Israel’s patriarch Jacob. The Bible makes a point of not mentioning Qos, the national god of Edom. We have evidence that Qos was a rather late theological development in Edom. Given this evidence, it is plausible to assume that Yahweh was worshiped in Edom and Qos stepped in only when Yahweh became the national god of Israel/Judah.
  • Archaeology.  Two Egyptian inscriptions, one dated to the period of Amenhotep III (14th century BCE), the other to the age of Ramesses II (13th century BCE), refer to “Yahweh in the land of the Shasu”. We also have one 9th century BCE text at Kuntillet Ajrud which refers to “Yahweh of Teman”.

Yahweh was first worshiped as a god of metallurgy

Gods in the ancient worlds were given a specific set of powers. For reasons we will get into next time, Yahweh in the Bible is attributed the attributes of many kinds of gods: he exhibits power of the storm, of the sun, and even of femininity. But if we limit our search for descriptions of God in Midianite territory, we see the following picture:

For more information, I recommend Amzallag, 2009. Yahweh, the Canaanite God of Metallurgy?

The founder of Judaism, Moses, was said to be a Midianite

Moses is described as having settled down with the Midianite people (the Shasu). His wife Zipporah and two sons were Midianite. What’s more: Moses’ father-in-law Jethro is called a priest. A priest of what god? Well, in Exodus 18:12, Jethro (and not Moses) is portrayed initiating a sacrifice to Yahweh. The Biblical editors seem uncomfortable with this tradition, for they later interjected a confession of faith on Jethro’s lips, which very much mirrors other such confessions. All of this suggests that Moses’ Midianite father-in-law was a priest of Yahweh. In fact, he seems to have spiritual authority over Moses in this passage.

The E source is replete with this kind of claim. We first meet Moses in Midian (no claims of him being born in Egypt, in this document). Moses’ response to Yahweh’s call, “Who am I that I should bring the Israelites out of Egypt?” would be a fair question for a man in Midian. E also claims he cannot go to Egypt because he is “heavy of tongue”. Traditionally interpreted as a speech defect, this phrase only occurs in one other place in the Hebrew Bible, where it means cannot speak the language. Finally, E also claims that the Midianites are direct descendents of Abraham.

While two Levite sources admit Moses’ Midianite connection, P actively tried to hide it. In the P source, has absolutely nothing about his ever being in Midian. Nothing about a Midianite wife, a priest father-in-law, nothing about his sons. Two books later, the P source injects a (blood-curdling) story designed to vilify the Midianites. Moses himself gives the order to kill all of the Midianite women. And this source does not include the little fact that Moses has a wife who happens to be a Midianite woman. The fact that the P source tries to deny the Midianite connection suggests the underlying claim is historical.

One does not need to take a position on the historicity of Moses, or of a mini-Exodus, to consider the above evidence. Even if he was entirely fictional, the fact that Israelite priests portrayed Moses as a Midianite is significant.

Yahweh was introduced to Israel as a second tier deity (a member of El’s family)

This can be seen in Deuteronomy 32:8-9, where El gives each of his sons a nation to rule over:

When El gave the nations their inheritance, when he divided all mankind, he set up boundaries for the peoples according to the number of the sons of El. For Yahweh’s portion is his people, Jacob his allotted inheritance.

In Psalm 82, we see Yahweh not at the head of the pantheon, but later asked to assume the job of all gods. “Yahweh stands in the divine assembly of El. Among the divinities, he pronounces judgment… Arise O Yahweh, judge the world; for You inherit all the nations.” Genesis 49:24-25 and Numbers 23-24 also view YHWH and El existing as distinct deities.

We have seen how Yahweh was first worshiped in Midian, and not Israel. Concurrently, El was worshiped in the land of Israel.

Then, when Yahwism emigrated to Israel (incorporation), Yahweh was not recognized as a god of gods. Rather, Yahweh was elevated to this position (equated with El) as the nation of Judah transitioned towards statehood.

As we will see next, worship of Yahweh emerged gradually, in five stages:

Monotheism_ Five Stages (2)

Takeaways

Here’s what we covered today:

  • The Israelite origin story is largely a patriotic fiction.
  • The Israelite people were indigenous Canaanites.
  • The first Israelites worshiped the pantheon of El.
  • The original Yahweh cult was a Shasu religion located in southern Edom
  • Yahweh was first worshiped as a god of metallurgy
  • The founder of Judaism, Moses, was said to be a Midianite
  • Yahweh was introduced to Israel as a second tier deity (a member of El’s family)

Until next time.

The Documentary Hypothesis

kevinbinz 3 Comments

Part Of: Demystifying Religion sequence
Followup To: A Secret in The Ark
Content Summary: 1900 words, 19min read.

Who Wrote The Hebrew Bible?

A close reading of the Hebrew Bible reveals the existence of doublets: two stories that describe the same event. A few examples:

  • Abraham’s covenant (Genesis 15:1-21 and 17:1-27),
  • Jacob becoming Israel (Genesis 32:25-33 and 35:9-15),
  • Yahweh summons Moses (Exodus 3-4 and 6:2-30)
  • Water in the wilderness (Exodus 15:22b-25a and 17:1-7)

Dozens of these doublets appear throughout the first five books of the Hebrew Bible (also known as the Torah). Traditionally, the Torah is thought to have a single author, and doublets like these were explained as either a) different events, or b) same event but with different emphases.

But what if these doublets exist because the Torah has multiple authors?

Let’s look deeper.

Source Identification as Unsupervised Learning

In principle, how might we discern between a single- and a multi-author book?

The Clustering Method. Let’s conjecture two sources (clusters) and, for each sentence, assign it either Cluster 1 or Cluster 2. We have complete freedom in our assignments. We want to chose clusters that maximize the coherence within each source, and also maximize the difference between the sources.

  • If the clusters are not very different, there is probably only one author.
  • If they are very different, we can safely conclude two authors.

For readers familiar with machine learning: this is unsupervised learning – searching for latent variables that best explain our data.

A Tale of Two Books

Suppose you encounter a book you have never read before, originally written in English by a single author. Call this Book A.

But you don’t know if Book A has one or two authors! To find out, you might use the Clustering Method.

What happens if you look at every sentence in Book A, and try to make each source-cluster as different as possible. Even for books written by a single author, the resultant source-clusters could be contrived to be truly different. For example, you could put all optimistic sentences in one bucket, and all pessimistic sentences in the other. But even though the texts feel a little different, they don’t differ that much (after all, a single person wrote both!)

In contrast, imagine you come across another book, Book B, replete with doublets. You break those doublets into clusters, and discover the following facts:

  1. Dialect. One cluster uses an antiquated dialect of English (e.g., Shakespearean), the other a modern dialect (e.g., African-American Vernacular English).
  2. Terminology. One cluster consistently uses the word “soda”, the other consistently uses the alternative, “pop”. 
  3. Consistent Content. One cluster is very interested in economic issues. The other is more interested in rehashing political debates.
  4. Narrative Flow.  Reading each cluster as a standalone book tends to smooth out non-sequiturs, and generally improve the sense of narrative flow.
  5. Inter-Source Relationships.  Imagine Book B is situated in an anthology with other books (B2 and B3) of unknown authorship. These other books are kinda dissimilar  from B. But B2 has lots in common with Cluster 1, and B3 sounds like it shares an author with Cluster 2. 
  6. Historical Grounding. Given the above information, we can make a pretty good guess as to identify of both authors, and why they got merged into a single anonymous volume.

On this evidence, it seems very unlikely that there is a single author of Book B. Instead, most people would indeed accept that this document has two different authors.

The Hypothesis: Five Sources

The Hebrew Bible is like Book B. Only, instead of two distinct authors, we have identified five. These are the Jahwist source (J), the Elohim source (E), the  Priestly source (P), the Deuteronomist source (D), and the Redactor (R). This is the Documentary Hypothesis.

We will explore the different personalities of these authors in more detail next section; for now, I want to briefly describe their contributions to the Torah from a textual perspective:

Documentary Hypothesis_ Source Distribution

And here is the timeline on which our source documents were authored, where the final redactor R (Ezra) compiled the final JEPD product.

Documentary Hypothesis_ Composition Timeline (2)

Evidence For The Hypothesis

How do we know all of this? On the following grounds:

  1. Dialect. Sources J and E are written in the Hebrew of the 10th BCE. In contrast, P and D are written in 8th century BCE.
  2. Terminology. A couple examples. Source D alone use of the phrase “with all your heart and with all your soul”. Source P uses all 100 instances of the word “congregation”, and 67 out of 69 examples of the work “chieftain”. Here are more examples:

Documentary Hypothesis_ Terminology (1)

  1. Consistent Content.
    • The Revelation of God’s Name.  According to J, the name YHWH was known since the earliest generations of humans. But in E and P it is stated just as explicitly that YHWH does not reveal this name until the generation of Moses.
    • Sacred Objects.
      • Tabernacle: P discusses the Tabernacle 200 times, it receives more attention than any other subject. It is never mentioned in J or D. E mentions it three times.
      • The Ark: J identifies the ark is identified as crucial to Israel’s travels and military successes; it is never mentioned in E.
      • Urim and Thummim: P mentions Urim and Thummim. J, E, and D never do.
      • Cherubs: P and J invoke cherubs. E and D never do.
      • Miracles: E has miracles performed by Moses’ staff. P uses Aaron’s staff.
    • Priestly Leadership. In P, access to the divine is limited to Aaronid priests. There is no talk of dreams, angels, talking animals, judges, and very few mentions to prophets. These themes are developed almost exclusively in J, E, and D.
  2. Narrative Flow.  Reading J, E, D, and P as standalone narratives tends to remove non-sequiturs and contradictions, and generally improve the sense of narrative flow. Want to see this for yourself? Go compare the original composite story of Noah, and contrast it with the original two stories (the original stories were weaved together by a later redactor).
  3. Inter-Source Relationships. Source D shares the same tone, emphases, and worldview as the book of Jeremiah. Source P resonates strongly with the book of Ezekiel. Finally, Sources J and E mirrors the book of Hosea.
  4. Historical Grounding. This is the most exciting piece of evidence, for reasons I will more fully explore next time. Suffice to say that we can localize each source to the historical context in which it was written. We have evidence suggesting that J and E were composed during the divided monarchy, before Israel fell in 722 BCE. J is written from a Southern perspective (in Judah), E is written from a Norther perspective (in Israel). After the fall of the northern kingdom, many Israelites fled to Judah. Because the old tribal disputes had faded in importance, J and E were combined into a JE narrative. The Priestly source P was an alternative telling of JE written in 8th century Judah. Finally, the first iteration of Deuteronomy was composed during the reign of King Josiah (641 BCE), just 20 years before the Babylonian exile (622 BCE).

I’ll let Richard Elliott Friedman wrap up this section.

Above all, the strongest evidence establishing the Documentary Hypothesis is that several different lines of evidence converge. There are more than thirty cases of doublets: stories or laws that are repeated in the Torah. The existence of so many overlapping texts is noteworthy itself. But their mere existence is not the strongest argument. One could respond, after all, that this is just a matter of style of narrative strategy. Similarly, there are hundreds of apparent contradictions in the text, but one could respond that we can taken them one by one and find some explanation for each contradiction. And, similarly, there is a matter of the texts that consistently call the deity God while other texts consistently call God by the name YHWH, to which one could respond that this is simply like calling someone sometimes by his name and sometimes by his title.

The powerful argument is not any one of these matters. It is that all these matters converge. When we separate the doublets, this also results in the resolution of nearly all the contradictions. And when we separate the doublets, the name of God divides consistently in all but three out of more than two thousand occurrences. And when we separate the doublets, the terminology of each source remains consistent within the source. And when we separate the sources, this produces continuous narratives that flow with only a rare break. And when we separate the sources, this fits with the linguistic evidence, where the Hebrew of each source fits consistently with what we know of the Hebrew in each period. And so on for each of the categories that precede this section.

The name of God and doublets were the were the starting-points of the investigation into the formation of the Bible. But they are not major arguments or evidence in themselves. The most compelling argument is that all this evidence of so many kinds comes together so consistently. To this day, no one known to me who challenged the hypothesis has ever addressed this fact.

Open Questions

Most scholars agree with the broad picture of four sources (J, E, P, D) and two redactions (JE and JEPD). There does exist considerable controversy at finer levels of detail. The four most contentious mini-debates I know of are as follows:

  • While there is consensus on the dating of J, E, and D, the dating of P is somewhat controversial (700 vs 500 BCE).
  • The exact relationship of J and E is at times hard to work out, particularly because E has less material than J. Were parts of E ejected during the redaction process of JE? Or was E composed as a supplement to J, and not a standalone work?
  • It is hard to make out how the two redaction processes actually worked. The Hebrew Bible is the very first example of prose writing in the entire world (earlier writing was entirely poetic).
  • There is consensus that J, E, P, and D were authored long after the events that they describe. They were undoubtedly influenced by early oral traditions. However, the extent of continuity and historical memory transferred from these oral traditions is in some doubt.

Takeaways

I was raised in an evangelical household, which means that growing up, I have read the Hebrew Bible (known to Christians as the Old Testament) cover-to-cover several times. I found such reading difficult. Some of this was mere cultural distance: a kid in the 20th century CE is three millenia removed from Canaanite culture in which the Bible was written.

But for me, the Hebrew Bible feels much easier to understand in light of the Documentary Hypothesis.

  • Contradictions are explained.
  • The within-source stories flow much better.
  • It is easier to understand the narrative discontinuities in the composite.
  • The diverse perspectives can be situated within their originating cultural milieu.

I wish more people knew about the Documentary Hypothesis for these reasons. Or better yet, could look at the labelled sources of the Hebrew Bible online. But for now, if you’d like to read the Hebrew Bible yourself, with labelled sources, the best way to do this is simply to purchase a book like, The Bible With Sources Revealed for a copy of the complete Torah, color coded by authorship.

Until next time.

An Introduction to Language Models

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Part Of: Language sequence
Content Summary: 1500 words, 15 min read

Why Language Models?

In the English language, ‘e’ appears more frequently than ‘z’. Similarly,  “the” occurs more frequently than “octopus”. By examining large volumes of text, we can learn the probability distributions of characters and words.

Language Models_ Letter and Word Frequency

Roughly speaking, statistical structure is distance from maximal entropy. The fact that the above distributions are non-uniform means that English is internally recoverable: if noise corrupts part of a message, the surrounding can be used to recover the original signal. Statistical structure is also used to reverse engineer secret codes such as the Roman cipher.

We can illustrate the predictability of English by generating text based on the above probability distributions. As you factor in more of the surrounding context, the utterances begin to sound less alien, and more like natural language.

Language Model_ Structure of English

A language model exploits the statistical structure of a language to express the following:

  • Assign a probability to a sentence P(w_1, w_2, w_3, \ldots w_N)
  • Assign probability of an upcoming word P(w_4 \mid w_1, w_2, w_3)

Language models are particularly useful in language perception, because they can help interpret ambiguous utterances. Three such applications might be,

  • Machine Translation: P(\text{high winds tonight}) > P(\text{large winds tonight})
  • Spelling correction: P(\text{fifteen minutes from}) > P(\text{fifteen minuets from})
  • Speech Recognition: P(\text{I saw a van}) > P(\text{eyes awe of an})

Language models can also aid in language production. One example of this is autocomplete-based typing assistants, commonly displayed within text messaging applications. 

Towards N-Grams

A sentence is a sequence of words \textbf{w} = (w_1, w_2, \ldots, w_3). To model the joint probability over this sequence, we use the chain rule:

p(\text{this is the house})

= p(\text{this})p(\text{is}\mid\text{this})p(\text{the}\mid\text{this is})p(\text{house}\mid\text{this is the})

As the number of words grows, the size of our conditional probability tables (CPTs) quickly becomes intractable. What is to be done? Well, recall the Markov assumption we introduced in Markov chains.

markov_assumption

The Markov assumption constrains the size of our CPTs. However, sometimes we want to condition on more (or less!) than just one previous word. Let v denote how many variables we admit in our context. A variable order Markov model (VOM) allows v elements in its context: p(s_{t+1} | s_{t-v}, \ldots, s_{t}). Then the size of our CPT is n=v+1, because we must take our original variable into account. Thus an N-gram is defined as a v-order Markov model. By far, the most common choices are trigrams, bigrams, and unigrams:

Language Models_ Ngram comparison (1)

We have already discussed Markov Decision Processes, used in reinforcement learning applications.  We haven’t yet discussed MRFs and HMMs. VOMs represent a fourth extension: the formalization of N-grams. Hopefully you are starting to appreciate the  richness of this “formalism family”. 🙂

Language Model_ Markov Formalisms (1)

Estimation and Generation

How can we estimate these probabilities? By counting!

ngram_v2

Let’s consider a simple bigram language model. Imagine training on this corpus:

This is the cheese.

That lay in the house that Alice built.

Suppose our trained LM encounters the new sentence “this is the house”. It estimates its probability as:

p(\text{this is the house})

= p(\text{this})p(\text{is} \mid \text{this})p(\text{the} \mid \text{is})p(\text{house} \mid \text{the}) 

= \dfrac{1}{12} * 1 * 1 * \dfrac{1}{2} = \dfrac{1}{24}

How many problems do you see with this model? Let me discuss two.

First, we have estimated that p(\text{this}) = \dfrac{1}{24}. And it is true that “this” occurs only once in our toy corpus above. But out of two sentences, “this” leads half of them. We can express this fact by adding a special START token into our vocabulary.

Second, recall what happens when language models generate speech. Once they begin a sentence, they are unable to end it! Adding a new END token will allow our model the terminate a sentence, and begin a new one.

With these new tokens in hand, we update our products as follows:

Language Models_ Sentence Estimation (1)

A couple other “bug fixes” I’ll mention in passing:

  • Out-of-vocabulary words are given zero probability. It helps to add an unknown  (UNK) pseudoword and assign it some probability mass.
  • LMs prefer very short sentences (sequential multiplication is monotonic decreasing). We can address this e.g., normalizing by sentence length.

Smoothing

In the last sentence in the image above, we estimate p(END|house) = 0, because we have no instances of this two-word sequence in our toy corpus. But this causes our language model to fail catastrophically: the sentence is deemed impossible (0% probability).

This problem of zero probability increases as we increase the complexity of our N-grams. Trigram models are more accurate than bigrams, but produce more p=0 events. You’ll notice echoes of the bias-variance (accuracy-generalization) tradeoff.

How can we remove zero counts? Why not add one to every word? Of course, we’d then need to increase the size of our denominator, to ensure the probabilities still sum to one. This is Laplace smoothing. 

Language Model_ Laplace Smoothing

In a later post, we will explore how (in a Bayesian framework) such smoothing algorithms can be interpreted as a form of regularization (MAP vs MLE).

Due to its simplicity, Laplace smoothing is well-known  But several algorithms achieve better performance.  How do they approach smoothing?

Recall that a zero count event in an N-gram is not likely to occur in (N-1)-gram model. For example, it is very possible that the phrase “dancing were thought” hasn’t been seen before. 

Language Model_ Backoff Smoothing

While a trigram model may balk at the above sentence, we can fall back on the bigram and/or unigram models. This technique underlies the Stupid Backoff algorithm.

As another variant on this theme, some smoothing algorithms train multiple N-grams, and essentially use interpolation as an ensembling method. Such models include Good-Turing and Kneser-Ney algorithms.

Beam Search

We have so far seen examples of language perception, which assigns probabilities to text. Let us consider language perception, which generates text from the probabilistic model. Consider machine translation. For a French sentence \textbf{x}, we want to produce the English sentence \textbf{y} such that y^* = \text{argmax } p(y\mid x).  

This seemingly innocent expression conceals a truly monstrous search space. Deterministic search has us examine every possible English sentence. For a vocabulary size V, there are V^2 possible two-word sentences. For sentences of length n, our time complexity of our brute force algorithm is O(V^n).

Since deterministic search is so costly, we might consider greedy search instead. Consider an example French sentence \textbf{x} “Jane visite l’Afrique en Septembre”. Three candidate translations might be,

  • y^A: Jane is visiting Africa in September
  • y^B: Jane is going to Africa in September
  • y^C: In September, Jane went to Africa

Of these, p(y^A|x) is the best (most probable) translation. We would like greedy search to recover it.

Greedy search generates the English translation, one word at a time. If “Jane” is the most probable first word \text{argmax } p(w_1 \mid x), then the next word generated is \text{argmax } p(w_2 \mid \text{Jane}, x). However, it is not difficult to contemplate p(\text{going}\mid\text{Jane is}) > p(\text{visiting}\mid\text{Jane is}), since the word “going” is used so much more frequently in everyday conversation. These problems of local optima happen surprisingly often.

The deterministic search space is too large, and greedy search is too confining. Let’s look for a common ground.

Beam search resembles greedy search in that it generates words sequentially. Whereas greedy search only drills one such path in the search tree, beam search drills a finite number of paths. Consider the following example with beamwidth b=3

beam_search

As you can see, beam search elects to explore y^A as a “second rate” translation candidate despite y^B initially receiving the most probability mass. Only later in the sentence does the language model discover the virtues of the y^A translation. 🙂

Strengths and Weaknesses

Language models have three very significant weaknesses.

First, language models are blind to syntax. They don’t even have a concept of nouns vs. verbs!  You have to look elsewhere to find representations of pretty much any latent structure discovered by linguistic and psycholinguistic research.

Second, language models are blind to semantics and pragmatics. This is particularly evident in the case of language production: try having your SMS autocomplete write out an entire sentence for you. In the real world, communication is more constrained: we choose the most likely word given the semantic content we wish to express right now.

Third, the Markov assumption is problematic due to long-distance dependencies. Compare the phrase “dog runs” vs “dogs run”. Clearly, the verb suffix depends on the noun suffix (and vice versa). Trigram models are able to capture this dependency. However, if you center-embed prepositional phrases, e.g., “dog/s that live on my street and bark incessantly at night run/s”, N-grams fail to capture this dependency.

Despite these limitations, language models “just work” in a surprising diversity of applications. These models are particularly relevant today because it turns out that Deep Learning sequence models like LSTMs share much in common with VOMs. But that is a story we shall have to take up next time.

Until then.

 

[Video] An introduction to reinforcement learning

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Part Of: Reinforcement Learning sequence

Sorry it’s been so long since my last post!  I’ve been teaching a Deep Learning class, based on Andrew Ng’s Coursera specialization.  Don’t worry, my other lectures will ultimately be cleaned & shared here too 🙂

This talk covers the mathematical intuitions of RL, which draws from content relating to Markov Chains and Markov Decision Processes. It also contains some novel material, including my thoughts on how RL compares with other machine learning techniques.

 

The Relational Sphere Hypothesis

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Part Of: Demystifying Sociality sequence
Followup To: The Three Spheres of Culture
Content Summary: 1700 words, 17 min read

A Theory of Relationship Dynamics

How can we make sense of social life? Let’s start by considering a simple cup of coffee.  

  1. In my own house, I can just help myself to as much as I want, sharing with others in the framework of “what’s mine is yours.”  
  2. Or my friend can get me a cup of coffee in return for the one I got for him yesterday, so we take turns or match small favors for each other.
  3. At Starbucks, I buy my coffee, using price and value as the framework.
  4. To my children, however, none of these principles apply. To them, coffee is something that only “big people” are allowed to drink: It is a privilege that goes with social rank.

What is true of a humble cup of coffee is true of the moral dilemmas surrounding major policy questions such as organ donation. Decisions have to be made, and there are again four fundamental ways to make them:

  1. Should we hold a lottery, giving each person an equal chance?  
  2. Should we somehow rank the social importance of potential recipients?
  3. Should we sell organs to the highest bidder?  
  4. Or should we expect everyone in a local community to give freely, offering a kidney to anyone group member in need?

(The above excerpt is from [FE] )

Relational Models Theory (RMT) proposes that these four social categories are exhaustive and culturally universal. Human interactions are complex, and typically use more than one of the above processes. But every relationship, in every culture, seems to be some combination of the following:

  • In Communal Sharing (Communality), people are viewed as equals oriented around some particular identity. This can include being in love, sports fans, and co-religionists.
  • In Authority Ranking (Dominance), people are situated in a hierarchy where superiors are deferred to, respected, and in some cases obeyed.
  • In Equality Matching (Reciprocity),  people are interested in restoring balance, turn-taking, and making sure everyone is treated fairly. 
  • In Market Pricing (Exchange), relationships are governed by quantitative, utilitarian concerns such as prices, exchanges, or cost-benefit analyses.

We can use relational models to explain a wide swathe of social phenomena:

  • Some examples of norm violation are in fact category errors. For example, we would interpret a situation such as the price of our meal is two hours on dishwasher duty as a conflation of Market Pricing vs. Equality Matching.
  • Some (but not all) examples of taboo trade-offs are in fact category errors. The Finite Price of Human Life thesis feels counterintuitive because it pits our Market Pricing versus the sacred values held by Communality.
  • Humans often use indirect speech acts to reconcile relationship types with semantic content.Rather than saying e.g., “pick me up after work”, we often say things like, “If you would pick me up after work, that would be awesome”. While more verbose, the latter expression feels more polite because it is couched in a Communality frame, rather than signaling Dominance.

In addition to its explanatory reach, multiple strands of evidence come together in support of  Relational Model theory:

  • Factor analysis. If you ask people to describe their relationships, you can see whether your theory predicts statistical patterns in their responses. When RMT was compared with other taxonomies (and there are a lot of them), RMT starkly outperforms its competitors. 
  • Ethnographies. RMT was invented by anthropologist Alan Fiske to capture regularities he saw across different cultures. For example, he found examples of marriage treated as Dominance, as Market Pricing, etc – but never a fifth type. A number of cross-cultural studies indicate that the four relational models constitute a human universal.
  • Social errors. When people misremember a person’s name, it tends to be a person with whom they share the same relationship type. For example, if you flub the name of your boss, you are more likely to say the name of someone else in a position of authority over you.
  • Brain studies.  In the cortex, the default mode network is universally acknowledged to perform social processing. But within this specialized region, different subregions are activated when processing e.g., Communality vs Reciprocity relationships.

The Relational Sphere Hypothesis

Human societies can be conceived as operating in three spheres: markets, governments, and communities. The Cultural Sphere Hypothesis holds this trichotomy to be fundamental, and exhaustive of social space.

Relational Models_ Cultural Regime Dissociations (4)

There seems to be a relationship between the cultural spheres and relation models. But there are three spheres vs four models. What gives?

Things become more clear when we remember that market- based economies were invented during the Neolithic Revolution, with the dawn of agriculture. Before this inflection point in history, transactions took place with gift economies.

This suggests that the Market Pricing relational model is evolutionarily recent: before the invention of agriculture, it simply did not exist.

Relational Model Theory_ Models vs Spheres (3)

I call this particular mapping from relational models to cultural spheres the Relational Sphere Hypothesis (RSH). It is an intertheoretic reduction: it purports to be a significant join point between micro- and macro-sociality.

RSH predicts that three out of four relational models can be traced back to the birthplace of Homo Sapiens. Thus, we should expect predecessors for these relationship categories in primate societies! And we find precisely that:

  • Dominance models are expressed in the dominance hierarchy (where physical dominance slowly gave way to symbolic dominance).
  • Communality models are expressed in kin selection (where attachment to and care for relatives was slowly extended towards e.g. close friends).
  • Reciprocity models are expressed in reciprocal altruism (where increasingly large delays between favor-transactions became possible).

I have argued elsewhere that the dual-process models so popular in today’s moral psychology can be captured in the interactions between (cortical) propriety frames and (subcortical) social intuitions. These two systems comprise the building blocks of sociality. RSH dovetails nicely with this dual process account, as it perceives categories within these systems, each with its own distinctive logic:

rmt_categorization

With the exception of Sanctity, these subconscious social intuitions arguably exist in primates. For example, here is evidence that rhesus monkeys have strong intuitions about Fairness:

A New Kind of Social Network

The Relational Sphere Hypothesis can be further illustrated by social networks: graphs where nodes are individuals, and edges are relationships. These kinds of models are very common across many disciplines that study aggregate social phenomena; for example evolutionary game theorists. A social network may look something like this:

Relational Models_ Aggregated Social Networks

But relationships inhabit different categories. We can express this fact by coloring edges according to their relational model:

Relational Models_ Complete Social Network (2)

Note that some nodes (e.g. A and B) are connected by more than one color. This signifies that the relationship between A and B features both Communality and Dominance.

From this more complete picture of human relationships, we can derive our cultural spheres by examining the (mono-color) subgraphs:

Relational Models_ Social Network Subgraphs (2)

Sphere Evolution & Competition

Political, social, and economic institutions have dramatically changed across the course of human history. As we saw in Deep History of Humanity, the evolution of our species can be usefully divided into three time periods:

Relational Models_ Sphere Evolution (1)

 

The Sphere Competition Conjecture comprises a set of informal intuitions that relational models “competes for our attention”: gains in one sphere are often accompanied by losses in another.

Let me illustrate this conjecture with examples. 🙂

Social vs Economic spheres

  • The religious instinct is etched deeply into the hominid mind, and evidence for shamanic animism dates back to the advent of behavioral modernity. Modern religion is located squarely within the Social sphere. But what caused its institutionalization, the invention of the full-time religious specialist: the priest? Religious institutions were founded during the transition from gift economy to market economies. For the first time in history, material wealth mattered more in transactions than interpersonal reputation. With the Social sphere threatening to collapse, perhaps it is not a coincidence that it was at this moment in history that religion became more explicitly social.
  • Some existential philosophers argue that the industrial revolution, with its obscenely large increase in Economic productivity, has correlated with a weakening of Social values, as witnessed empirically by the rise of materialism. Perhaps the malaise and cynicism of postmodernity can be explained by the weakening of the ties of community.
  • The custom of tipping can be conceived as an organ of Sociality, that feels misplaced in today’s Market-oriented economy. This institution shows no signs of abating (for example, Uber recently rescinded its no-tipping policy). Perhaps the reason this Social technology persists, while others have disintegrated, is because tipping solves the principal agent problem: customer service is otherwise not factored into the price, because that information is not easily available to management.  
  • Product boycotts are another example of Social outrage affecting Economic markets.

Social vs Political.

  • Another important event in the history of religion is the transition to universal religions: where the concerns of the gods and the consequences of moral violations were imbued with an aura of the eternal. Anthropological evidence clearly suggests that universal religions succeeded because they facilitated larger group sizes.
  • Corruption is often treated as a political problem, but in fact bribery and collusion both require high amounts of social capital.
  • In American history, political partisanship has been most severe in the 1880s, and at present. Both then and now are periods of an intense drought of social capital. Further, participation in voting strongly correlates with vibrant community and civic life. We might conjecture that weaker communities are more vulnerable to partisanship infighting. This conjecture is aligned with the oft-cited observation that partisanship tends to correlate with moderates abandoning the political arena.

Economic vs Political.

  • Capitalist Peace Theory formalizes the observed inverse relationship between free trade and international conflict. On this hypothesis, one of the strongest predictors of war is resource acquisition, and the risk-benefit calculus changes (improves) substantially with the removal of tariffs.

Economic vs Political vs Social.

  • The Size of Nations Hypothesis is the idea that the size of nation (Political) is driven by two competing factors: larger nations are able to produce public goods more efficiently (Economic), but conversely their populations are more heterogenous and thereby less cohesive (Socially).

Some of the phenomena described above have been extensively studied by social scientists. However, to my knowledge, no extant models robustly capture the doctrine of relational model theory. Perhaps the next generation of formal models will do better.

Recommended Resources

[Excerpt] The Three Spheres of Culture

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The Three Sphere Hypothesis

Most people agree that human societies operate in different contexts: markets, governments, and communities. The Three Sphere Hypothesis holds that this trichotomy is fundamental and exhaustive of social space. What’s more, these spheres interact. Neither markets nor governments nor communities can be analyzed thoroughly without understanding their dependence upon, and their effects upon, the others.

Relational Models_ Cultural Regime Dissociations (4)

[Excerpt] Intellectual History of the Hypothesis

Source: Wicks (2009). A Model of Dynamic Balance among the Three Spheres of Society

Social scientists – including economists – as well as journalists and others, often refer to “the economic, political, and social conditions” underlying any particular situation, but usually without any further analysis of what these terms imply, and how they relate to each other.

Apparent references to these three spheres pop up – in both popular and technical literature – almost everywhere. It can be a fun game, like “whack-a-mole”:

  • Where and how will the three spheres “pop up” in this or that text?
  • And, given any set of three social attributes that do “pop up”, can they be seen in some way as representing the three spheres?

Etzioni (1996:122) speaks of “three different conditions: paid, coerced, or convinced”; Etzioni (1988) explores motivations in the community sphere at length.

Personalist economics, based on Catholic theology, also recognizes three organizing principles: competition, intervention, and cooperation (Jonish and Terry, 1999:465-6; O’Boyle, 1999:536-7, 2000:550-51).

Hirschman (1992) referred to three social mechanisms: exit, voice, and loyalty. Though all three can apply in varying ways to each sphere, exit refers primarily to the market sphere where, in a competitive situation, one has unlimited choice of buyers or sellers, so can “exit” from any one. Voice might refer primarily to the political sphere, where one can attempt to influence results by persuasion, and loyalty to the community sphere – though one could argue the other way as well.

Streeck and Smitter (1985:1) refer to these “three basic mechanisms of mediation or control” (Ouchi, 1980) as spontaneous solidarity, hierarchical control, and dispersed competition.

Friedland and Alford (1991:39) refer to three domains with different “logics of action”: In the marketplace, we are more likely to base our actions on individual utility and efficient means; in the polity, on democracy and justice; and in the family, on mutual support.

Van Staveren (2001:24) asserts that “three values appear time and again in economic analysis: liberty, justice, and care. Markets tend to express freedom, states to express justice, and unpaid labor to express care among human beings.” She notes (p. 213) that Ayres (1961:170) asserted a similar set of core human values: “freedom, equality, and security”. Van Staveren (p. 203) also notes:

  • the form that these values take: exchange, redistribution, and giving;
  • the locations where they operate: market, state, and the care-economy; and
  • the corresponding virtues: prudence, propriety, and benevolence.

She further asserts that there are “distinct emotions and forms of deliberation as well”.

Mackey (2002:384) refers to “economic, political, and social problems” in Saddam’s Iraq; elsewhere (p. 181) she uses a different order, referring to “the new political, social, and economic paradigm” (an order which Rothstein and Stolle, 2007:1, also use); and yet elsewhere (p. 49) she notes that something “meant more socially, politically, and economically”. The order of expression doesn’t seem to matter, to Mackey or to most other authors, and one can easily find the other three permutations as well (e.g., Friedman, 2000:131; Giddens and Pierson, 1998:89; Sage, 2003).

But the community sphere is often ignored, and thus is sometimes considered third (Adaman and Madra, 2002). In political theory, the “Third Way” (Giddens, 1998) represents an alternative to either markets or governments, focused more in communities.

Waterman (1986:123) asserts “three freedoms: economic, political, and religious (conscience)”; and Hobson (1938/1976:52) refers to “the democratic triad of liberty, equality, fraternity”.

As some of these examples illustrate, a wide variety of words are used to refer to the three spheres, as in the title of the book (cited by Bennett, 1985) Mexico: Catholicism, Capitalism, and the State, or when

  • Mackey (2002:217) discusses “political, economic, and… cultural control”;
  • Bowles (1998:105) refers to “states, communities, and markets”;
  • Wright (2000:211) refers to “governance, moral codes, and markets”;
  • Mauss (1925/1967:52) refers to the “law, morality, and economy of the Latins” and to “the distinction between ritual, law, and economic interest”;
  • Yuengert (1999:46) discusses “free markets circumscribed within a tight legal framework, and operating within a humane culture”;
  • Polanyi (1997:140), in discussing “economic life”, refers to “freedom under law and custom, as laid down and amended when necessary by the State and public opinion”.

In The Foundations of Welfare Economics (1949:230), Little points out that “if a person argues that a certain change would increase economic welfare, it is open to anyone to argue that it would decrease spiritual or political welfare.”

This tripartite taxonomy has been used by economists since Adam Smith who, of course, had first written The Theory of Moral Sentiments (1759/1982) about communities and social goods, then The Wealth of Nations (1776/1976) about markets, economics. But he was planning a third major work – which was never completed – on the political system (Smith, 1759/1982:342 and “Advertisement” therein).

Minowitz (1993) uses the same tripartite taxonomy twice (in varying order) in the title of his book: Profits, Priests, and Princes: Adam Smith’s Emancipation of Economics from Politics and Religion.

The English economist and theologian Philip Wicksteed referred to “business, politics, and the pulpit” in his book of sermons titled Is Christianity Practical? (1885/1920, referenced in Steedman 1994:83). In discussing Wicksteed’s work, Steedman (p. 99) also refers to “potatoes, politics, and prayer”. Similarly, Hobson (1938/1976:55) referred to “the purse, power, and prestige of the ruling classes in business, politics, and society”. Success itself is often defined as “wealth, fame, and power” (Bogle, 2004:1; Carey, 2006), or sometimes as “money, status, and power”.

A similar tripartite taxonomy – perhaps Marxian – of firms, social classes, and states, can easily be seen as referring to the three spheres.

According to Trotsky (1957:255), communism would demonstrate that the human race had “ceased to crawl on all fours before God, kings, and capital” (quoted by Minowitz, 1993:240).

A variety of sources also provide evidence of an apparently widespread belief that the three spheres are both fundamental and exhaustive of social space. Michael Novak refers to the “three mutually autonomous institutions: the state, economic institutions, and cultural, religious institutions” as “the doctrine of the trinity in democratic capitalism” (Abdul-Rauf, 1986:175; also Neuhaus, 1986:517).

Dasgupta (1993:104) notes “one overarching idea, that of citizenship, with its three constituent spheres: the civil, the political, and the socio-economic.”

Meyer et al. (1992:12) assert that “individuals must acquire the means to participate effectively in the economic, social, and political life of the nation.” In the same work, Wong (1992:141) makes it clear that these three spheres are considered exhaustive by referring to “all social domains… economy… polity… and… cultural system”.

Polanyi (1997:158) describes the Russian Revolution and the Soviets’ “project for a new economic, political, and social system of mankind”.

Shadid (2001:3) points out that “political Islam, or Islamism…suggests an all-embracing approach to economics, politics, and social life.”

Dicken (2007:538) says that “corporate social responsibilities span the entire spectrum of relationships between firms [and] states, civil society, and markets.”

 

The Deep History of Humanity

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Part Of: Anthropogeny sequence

Human Milestones

A graphic I created summarizing key cultural and biological milestones.

Human Deep History_ Master Timeline (3)

Note that time is situated on a logarithmic scale. Full resolution image here.

Hominid Phylogeny

Of course, the hominid line began diverging genetically from that of other primates around 7 million year ago.

Human Deep History_ Homo Sapiens Phylogeny (3)

Image from Berkeley’s Understanding Evolution. Full resolution image here.

Out of Africa

Finally, here is the geography & timeline of the emigration waves out of Africa, courtesy of Huffington Post and National Geographic.

Human Deep History_ Out of Africa (1)

A couple facts that provide context on our journey out of Africa:

Related Content

This post bears on the history of human- and hominid-like species.

The Symmetric Group

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Part Of: Algebra sequence
Followup To: An Introduction to Geometric Group Theory
Content Summary: 1800 words, 18 min read

On Permutations

In the last few posts, we have discussed algebraic structures whose sets contain objects (e.g., numbers). Now, let’s consider structures over a set of functions, whose binary operation is function composition.

Definition 1. Consider two functions f and g. We will denote function composition of g(f(x)) as f \bullet g. We will use this notation instead of the more common g \circ f.  Both represent the idea “apply f, then g“. 

Consider G = (\left\{ f(x) = 2x, g(x) = x+1, h(x) = 2x + 2, i(x) = 2x+1 \right\}, \bullet). 

Is this a group? Let’s check closure:

  • g \bullet f = 2(x+1) = h \in G
  • f \bullet g = (2x)+1 = i \in G
  • f \bullet h = 2(2x)+2 \not\in G

Closure is violated. G isn’t even a magma! Adding j(x) = 4x+2 to the underlying set exacerbates the problem: then both f \bullet j \not\in G and g \bullet j \not\in G.

So it is hard to establish closure under function composition. Can it be done?

Yes. Composition exhibits closure on sets of permutation functions. Recall that a permutation is simply a bijection: it re-arrange a collection of things. For example, here are the six possible bijections over a set of three elements.

Symmetry Group_ Permutation Options

Definition 2. The symmetric group S_n denotes composition over a set of all bijections (permutations) over some set of n objects. The symmetric group is then of order n!.

The underlying set of S_{3} is the set of all permutations over a 3-element set. It is of order 3! = 3 \times 2 \times 1 = 6.

This graphical representation of permutations is rather unwieldy. Let’s switch to a different notation system!

Notation 3: Two Line Notation. We can use two lines to denote each permutation mapping \phi. The top row represents the original elements x, the bottom represents where each element has been relocated \phi(x).

Symmetric Group_ Two Line Notation

Two line notation is sometimes represented as an array, with the top row as matrix row, and bottom denoting matrix column. Then the identity matrix represents the identity permutation.

Definition 4. A cycle is a sequence of morphisms that forms a closed loop. An n-cycle is a cycle of length n. A 1-cycle does nothing. A 2-cycle is given the special name transposition. S_3 has two permutations with 3-cycles: can you find them?

Theorem 5. Cycle Decomposition Theorem. Every permutation can be decomposed into disjoint cycles. Put differently, a node cannot participate in more than one cycle. If it did, its parent cycles would merge.

Notation 6: Cycle Notation. Since permutations always decompose into cycles, we can represent them as (A_1\ A_2\ \ldots\ A_n), pronounced “A_1 goes to A_2 goes to …”.

Symmetric Group_ Cycle Notation (1)

Cycle starting element does not matter: (A\ C\ B) = (C\ B\ A) = (B\ A\ C).

The Cycle Algorithm

It is difficult to tell visually the outcome of permutation composition. Let’s design an algorithm to do it for us!

Algorithm 7: Cycle Algorithm. To compose two permutation functions a(S) and b(S), take each element s \in S and follow its arrows until you find the set of disjoint cycles. More formally, compose these functions x times until you get (a \bullet b)^x(s) = s.

Here’s a simple example from S_{3} = ( \left\{ 0, 1, 2, 3, 4, 5 \right\}, \bullet ).

Symmetric Group_ Cycle Algorithm S3 Ex (2)

Make sense? Good! Let’s try a more complicated example from S_4.

Symmetric Group_ Cycle Algorithm S4 Ex (1)

A couple observations are in order.

  • 1-cycles (e.g., (A)) can be omitted: their inclusion does not affect algorithm results.
  • Disjoint cycles commute: (B\ D),(A\ C) = (A\ C),(B\ D). Contrast this with composition, which does not commute (B\ D) \bullet (A\ C) \neq (A\ C) \bullet (B\ D).

Now, let’s return to S_{3} for a moment, with its set of six permutation functions. Is this group closed? We can just check every possible composition:

Symmetric Group_ Permutation Composition

From the Cayley table on the right, we see immediately that S_{3} is closed (no new colors) and non-Abelian (not diagonal-symmetric).

But there is something much more interesting in this table. You have seen it before. Remember the dihedral group D_{3}? It is isomorphic to S_{3}!

Symmetric Group_ D3 S3 Isomorphism (3)

If you go back to the original permutation pictures, this begins to make sense. Permutations 1 and 2 resemble rotations/cycles; 3, 4, and 5 perform reflections/flips.

Generators & Presentations

In Theorem 7, we learned that permutations decompose into cycles. Let’s dig deeper into this idea. 

Theorem 8. Every n-cycle can be decomposed into some combination of 2-cycles. In other words, cycles are built from transpositions.

The group S_{3} = \left\{ 0, 1, 2, 3, 4, 5 \right\}, \bullet) has three transpositions 3 = (A\ B), 4 = (B\ C), and 5 = (A\ C).

Transpositions are important because they are generators: every permutation can be generated by them. For example, 1 = 3 \bullet 4. In fact, we can lay claim to an even stronger fact:

Theorem 9. Every permutation can be generated by adjacent transpositions. Every permutation S_{n} = \langle (1 2), (2 3), \ldots, (n-1\ n) \rangle.

By the isomorphism S_3 \cong D_3 , we can generate our “dihedral-looking” Cayley graph by selecting generators 1=(A\ B\ C) and 3 = (B\ C).

But we can use Theorem 9 to produce another, equally valid Cayley diagram. There are two adjacent transpositions in S_{3}: 3 and 4. All other permutations can be written in terms of these two generators:

  • 0 = 3 \bullet 3.
  • 1 = 3 \bullet 4.
  • 2 = 4 \bullet 3.
  • 5 = 3 \bullet 4 \bullet 3. 

This allows us to generate a transposition-based Cayley diagram. Here are the dihedral and transposition Cayley diagrams, side by side:

Symmetric Group_ Cayley Diagram (4)

We can confirm the validity of the transposition diagram by returning to our multiplication table: 1 \bullet 3 = 5 means a green arrow 1 \rightarrow 5.

Note that the transposition diagram is not equivalent cyclic group C_6, because arrows in the latter are monochrome and unidirectional.

We’re not quite done! We can also rename our set elements to employ generator-dependent names, by “moving clockwise”:

Symmetric Group_ Abstract Cayley Diagram

We could just as easily have “moved counterclockwise”, with names like 3 \mapsto r, 1 \mapsto rl. And we can confirm by inspection that, in fact, rl = lrlr etc.

Using the original clockwise notation, one presentation of S_3 becomes:

S_3 = \langle l, r \mid r^2 = e, l^2 = e, (lr)^3 = e \rangle

Towards Alternating Groups

Any given permutation can be written as a product of permutation. Consider, for example, the above equalities

  • 1 = rl = lrlr = (lr)^3rl. These have 2, 4, and 8 permutations, respectively. 
  • 5 = lrl = lrl^3 = l^3r^3l^3. These have 3, 5, and 9 permutations, respectively.

Did you notice any patterns in the above lists? All expressions for 1 require an even number of transpositions, and all expressions of 5 require an odd number. In other words, the parity (evenness or oddness) of a given permutation doesn’t seem to be changing. In fact, this observation generalizes:

Theorem 10. For any given permutation, the parity of its transpositions is unique.

Thus, we can classify permutations by their parity. Let’s do this for S_3:

  • 0, 1, 2 are even permutations.
  • 3, 4, 5 of odd permutations.

Theorem 11. Exactly half of S_n are even permutations, and they form a group called the alternating group A_n. Just as \lvert S_n \rvert = n!, A_n has \dfrac{n!}{2} elements. 

Why don’t odd permutations form a group? For one thing, it doesn’t contain the identity permutation, which is always even.

Let’s examine A^3 = (\left\{ 0, 1, 2 \right\}, \bullet ) in more detail. Does it remind you of anything?

It is isomorphic to the cyclic group C_3 ..!

Symmetric Group_ C3 A3 Isomorphism (1)

We have so far identified the following isomorphisms: S_3 \cong D_3 and A_3 \cong C_3. Is it also true that e.g., S_4 \cong D_4 and A_4 \cong C_4?

No! Recall that the \lvert D_n \rvert=2n and \lvert C_n \rvert = n. Only n \neq 3, these sets are not even potentially isomorphic.  For example:

  • \lvert D_4 \rvert=2 \times 4 = 8 \neq 24 = 4! = \lvert S_4 \rvert.
  • \lvert C_4 \rvert= 4 \neq 12 = \frac{24}{2} = \lvert A_4 \rvert.

For these larger values of n, the symmetric group is much larger than dihedral and cyclic groups.

Applications

Why do symmetric & alternating groups matter? Let me give two answers.

Perhaps you have seen the quadratic equation, the generic solution to quadratic polynomials ax^2 + bx + c = 0.

x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Analogous formulae exist for cubic (degree-3) and quartic (degree-4) polynomials. 18th century mathematics was consumed by the theory of equations: mathematicians attempting to solve quintic polynomials (degree-5). Ultimately, this quest proved to be misguided: there is no general solution to quintic polynomials.

Why should degree-5 polynomials admit no solution? As we will see when we get to Galois Theory, it has to do with the properties of symmetric group S_5.

A second reason to pay attention to symmetric groups comes from the classification theorem of finite groups. Mathematicians have spent decades exploring the entire universe of finite groups, finding arcane creatures such as the monster group, which may or may not explain features of quantum gravity.

One way to think about group space is by the following periodic table:

Symmetric Group_ Periodic Table

Image courtesy of Samuel Prime. 

Crucially, in this diverse landscape, the symmetric group plays a unique role:

Theorem 12: Cayley’s Theorem. Every finite group is a subgroup of the subgroup S_n, for some sufficiently large n.

For historical reasons, subgroups of the symmetric group are usually called permutation groups.

Until next time.

Wrapping Up

Takeaways:

  • The symmetric group S_n is set of all bijections (permutations) over some set of n objects, closed under function composition.
  • Permutations can be decomposed into disjoint cycles: cycle notation uses this fact to provide an algorithm to solve for arbitrary compositions.
  • All permutations (and hence, the symmetric group) can be generated by adjacent transpositions. This allows us to construct a presentation of the symmetric group.
  • Permutations have unique parity: thus we can classify permutations as even or odd. The group of even presentations is called the alternating group A_n.
  • It can be shown that S_3 \cong D_3 and A_3 cong C_3. However, for larger n, the symmetric and alternating group are much larger than cyclic and dihedral groups.

The best way to learn math is through practice! If you want to internalize this material, I encourage you to work out for yourself the Cayley table & Cayley diagram for S_4. 🙂

Related Resources

For a more traditional approach to the subject, these Harvard lectures are a good resource.