Pantheons of gods and relations, similarities or differences between religions, deities, and cultures

I will start by noting that several essential gods of different ancient pantheons can be identified as being the same or equivalent deities, with different names (and small variations in their stories).
Some authors identified the supreme god Ahura Mazda with Zeus.
The Egyptian supreme god Amun-Ra was the same as Zeus or Jupiter, also called Zeus Ammon or Jupiter Ammon.
Sometimes there was more than just a “triad” of important gods, such as the Ennead or Great Ennead, a group of nine deities in Egyptian mythology and ancient Egyptian religion, worshipped at the city of Heliopolis.

The Ennead included the sun god Atum; his children Shu and Tefnut; their children Geb and Nut; and their children Osiris, Isis, Set, and Nephthys. The Ennead sometimes included Horus, the son of Osiris and Isis.
Atum was equated with the sun god Ra. In the New Kingdom, when the god Amun rose to prominence he was merged with Ra as Amun-Ra.

In the ancient Canaanite and Phoenician religions, Baal is the equivalent of Zeus, and the god of the sea Yam is the equivalent of Poseidon or Neptune. The god Mot is identified with Hades or Pluto.

The Babylonian king of the gods Marduk was associated with Zeus by the ancient Greeks, and associated with Jupiter by the Romans. There is an ancient Hittite and Hurrian group of texts known as The Song of Kumarbi, “Song of Emergence” or Kingship in Heaven. Scholars have pointed out the similarities between the Hurrian myth and the story from Greek mythology of Uranus, Cronus, and Zeus.

In Hinduism the chief god Indra has many characteristics in common with Zeus.

It is possible to notice that several ancient religions or cultures were connected and had a common origin. One could observe or suggest that these ancient religions and the related ancient stories and mythologies had their origin in the ancient Near East, West Asia and the (eastern) Mediterranean, plausibly based on some events that were rooted in history. A second related source of origin pertains to ancient India and the ancient Hindu religion, since the ancient Indians had many historical interactions with the ancient Persians in terms of cultural exchanges and political influence (particularly during the Achaemenid empire), language, religion, spirituality, etc. The ancient practices, beliefs, ideas and religions spread to the entire Mediterranean region and to the south of Europe, to Greece and Italy. It is plausible that these ancient religions, stories and practices also spread (with some modifications) to the north and to other parts of Europe through a process of gradual cultural diffusion, influencing the formation of ancient Germanic and Norse religion(s) and mythology.

In light of my readings and analysis, it can be said that whether polytheistic or not, religions and cultures are interconnected in one way or another. The ancient allegorical stories of gods and deities that were constructed, embellished and changed with time could be explained reasonably by philosophical doctrines similar to Euhemerism, where these diverse embroidered stories are considered to be rooted in history. A very plausible and a realistic explanation of the origins of ancient religions is that most of them (in the ancient world and in the Mediterranean region and the Near East, and beyond), started mainly with a man who lived in ancient times, most likely about two millennia before the beginning of the Common or Christian era. This man did great and important things, accomplished great deeds, and brought about new innovative teachings and ideas for his time. This man left a lasting impression on those who saw or knew him. When he died he was revered, his life and actions being interpreted in different ways and directions. There were people who revered him a lot, divinized and deified him or started worshipping him.

This man was later called Amun Ra by the Egyptians. He was called Baal by the Canaanites and the Phoenicians, Zeus by the ancient Greeks, and Jupiter by the Romans. He was very likely the same deity as the supreme god Indra of the Hindus (since Zeus and Indra had many similar characteristics), Ahura Mazda of the ancient Persians, as well as other supreme gods in the ancient world.

Regarded as the head of a pantheon (or several related and connected pantheons, with small or minor differences between them), and as the father and ruler of the gods, he, his family and the other related deities had their stories transmitted, retransmitted, interpreted, reinterpreted, and modified over the years and centuries. Supernatural, symbolic and metaphorical elements were added to the stories of the gods with the passing of time. Each generation or group of people interpreted these ancient stories according to their understanding and their cultural and physical environment.

Did this man or person want to be deified or worshipped? He likely wanted to live his life and “do what was needed”, he knew that he was doing important things and that he was going to be remembered, but the deification or worship and the subsequent rituals and religious practices were mostly the result of the interpretations and choices of those who knew him, liked him, and/or followed him.

A thing that has been somewhat forgotten in the last two centuries is that many authors of the past, from the ancient Babylonian author Berosus to Isaac Newton and others, stated or were of the opinion that Zeus or Jupiter was the same person as one of the earliest and most important patriarchs in the Bible, his story having been modified to comply with biblical monotheism.

Another man came and lived about two millennia ago. He did important things, brought about new innovative teachings and concepts of morality for his time, and left a big impression on those who knew him and on his followers. He was subsequently revered, deified and worshipped by his followers, his story being interpreted and explained in various and different ways and directions, with allegorical and supernatural elements added to it. This man is mostly known by the name of Jesus.

About three centuries after his passing, the religion with Jesus at its head gradually replaced the religion having Zeus or Jupiter at its head.

In the same way that the one known as Zeus or Jupiter was mentioned as a very important patriarch or prophet in the monotheistic Bible, Jesus was later mentioned as a very important prophet in the strictly monotheistic Quran.

During the last century and the last few decades, a progressive change in mentalities and a regain of interest in ancient philosophies, ancient cultures, and ancient religions took place. Taking into account the evolution of ideas and the progress and advances in science during the last centuries and decades, there is a possibility that a new transmutation of values is gradually taking place, related to some sort of periodical or cyclical return of historical events, being the inverse of what happened at the start of Christianity two millennia ago. In any case, one will have to wait, see and look at the unfolding of future events.

Fields, rings, groups, their history and why they are called that way in math

In short, mathematicians usually like to use words or concepts (starting from the native language they use in their mathematical work) to describe or define theories or mathematical ”things”. Then these words get translated into other languages, get defined better and more accurately, and become widely known and used.

For more details, let us review how or when mathematicians began using these words, starting with the origin or history of the word field in mathematics:

“The term Zahlenkörper (body of numbers) is due to Richard Dedekind (1831-1916) . Dedekind used the term in his lectures of 1858 but the term did not come into general use until the early 1890s. Until then, the expression used was “rationally known quantities,” which means either the field of rational numbers or some finite extension of it, depending on the context.[…]

Dedekind used Zahlenkörper [literally “number body”] in Supplement X of his 4th edition of Dirichlet’s Vorlesungenueber Zahlentheorie, section 159. In a footnote, he explained his choice of terminology, writing that, in earlier lectures (1857-8) he used the term ‘rationalen Gebietes’ and he says that Kronecker (1882) used the term ‘Rationalitaetsbereich’.

Dedekind did not allow for finite fields; for him, the smallest field was the field of rational numbers. According to a post in sci.math by Steve Wildstrom, ‘Dedekind’s ‘Koerper’ is actually what we would call a division ring rather than a field as it does not require that multiplication be commutative.’

Eliakim Hastings Moore (1862-1932) was apparently the first person to use the English word field in its modern sense and the first to allow for a finite field. He coined the expressions ‘field of order s’ and ‘Galois-field of order ‘ These expressions appeared in print in December 1893 in the Bulletin of the New York Mathematical Society III. 75.”

(Source: https://jeff560.tripod.com/f.html)

In 1871 Richard Dedekind introduced, for a set of real or complex numbers that is closed under the four arithmetic operations, the German word Körper, which means “body” or “corpus” (to suggest an organically closed entity).

In 1881 Leopold Kronecker defined what he called a domain of rationality, which is a field of rational fractions in modern terms. Kronecker’s notion did not cover the field of all algebraic numbers (which is a field in Dedekind’s sense), but on the other hand was more abstract than Dedekind’s in that it made no specific assumption on the nature of the elements of a field. Kronecker interpreted a field such as Q(π) abstractly as the rational function field Q(X).

The first clear definition of an abstract field is due to Heinrich Weber (1893).

Now about the origin of the word ring in mathematics:

In 1871, Richard Dedekind defined the concept of the ring of integers of a number field. In this context, he introduced the terms “ideal” (inspired by Ernst Kummer’s notion of ideal number) and “module” and studied their properties. But Dedekind did not use the term “ring” and did not define the concept of a ring in a general setting.

The term “Zahlring” (number ring) was coined by David Hilbert in 1892 and published in 1897 (in relation to algebraic number theory).

In 19th century German, the word “Ring” could mean “association”, which is still used today in English in a limited sense (e.g., spy ring), so if that were the etymology then it would be similar to the way “group” entered mathematics by being a non-technical word for “collection of related things”. According to Harvey Cohn, Hilbert used the term for a ring that had the property of “circling directly back” to an element of itself.

The image below is a photograph of David Hilbert, from 1912:

(Image source: https://en.m.wikipedia.org/wiki/File:Hilbert.jpg)

The first axiomatic definition of a ring was given by Adolf Fraenkel in 1914, but his axioms were stricter than those in the modern definition. For instance, he required every non-zero-divisor to have a multiplicative inverse. In 1921, Emmy Noether gave the modern axiomatic definition of (commutative) ring and developed the foundations of commutative ring theory in her paper Idealtheorie in Ringbereichen.

The mathematical disciplines or areas of study that influenced the formulation of group theory at the end of the 18th century and in the 19th century were geometry, number theory, and the theory of algebraic equations (related to to the study of permutations).

Moreover,

“Evariste Galois introduced the term group in the form of the expression groupe de l’équation in his ‘Mémoire sur les conditions de résolubilité des équations par radicaux’ (written in 1830 but first published in 1846) Oeuvres mathématiques. p. 417. Cajori (vol. 2, page 83) points out that the modern definition of a group is somewhat different from that of Galois, for whom the term denoted a subgroup of the group of permutations of the roots of a given polynomial.

Group appears in English in Arthur Cayley, ‘On the theory of groups, as depending on the symbolic equation ‘Philosophical Magazine, 1854, vol. 7, pp. 40-47. […] The paper also introduced the term theory of groups. At the time this more abstract notion of a group made little impact.[…]

Klein and Lie use the term ‘closed system’ in their ‘Ueber diejenigen ebenen Curven, welche durch ein geschlossenes System von einfach unendlich vielen vertauschbaren linearen Transformationen in sich übergehen,’Mathematische Annalen, 4, (1871), 50-84. Klein adopted the term gruppe in his ‘Vergleichende Betrachtungen über neuere geometrische Forschungen’ written in 1872 [in relation to the Erlangen program].

Group-theory is found in English in 1888 in George Gavin Morrice’s translation of Felix Klein, Lectures on the Ikosahedron and the solution of Equations of the Fifth Degree.”

(Source: http://jeff560.tripod.com/g.html)

The image below shows a portrait of Évariste Galois (1811-1832):

(Image source: https://en.m.wikipedia.org/wiki/File:Evariste_galois.jpg)

After novel geometries such as hyperbolic and projective geometry (dealing with the behavior of geometric figures under various transformations) had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884.

The convergence of various sources into a uniform theory of groups started with Camille Jordan’s Traité des substitutions et des équations algébriques (Treatise on Substitutions and Algebraic Equations) in 1870.

Walther von Dyck (1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definition of an “abstract group”, in the terminology of the time.

As of the 20th century, groups gained wide recognition by the work of Ferdinand Georg Frobenius and William Burnside
, who worked on representation theory
of finite groups, Richard Brauer’s modular representation theory
and Issai Schur’s papers. The theory of Lie groups, and more generally locally compact groups was studied by Hermann Weyl, Élie Cartan and others.

Additional information about fields, rings, groups and related topics can be found in Wikipedia and other or similar online resources.