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Sunday, October 27, 2024

Christianity and paganism

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Christianity_and_paganism
The Triumph of Christianity over Paganism, a painting by Gustave Doré (1899)

Paganism is commonly used to refer to various religions that existed during Antiquity and the Middle Ages, such as the Greco-Roman religions of the Roman Empire, including the Roman imperial cult, the various mystery religions, religious philosophies such as Neoplatonism and Gnosticism, and more localized ethnic religions practiced both inside and outside the empire. During the Middle Ages, the term was also adapted to refer to religions practiced outside the former Roman Empire, such as Germanic paganism, Egyptian paganism and Baltic paganism.

From the point of view of the early Christians, these religions all qualified as ethnic (or gentile, ethnikos, gentilis, the term translating goyim, later rendered as paganus) in contrast with Second Temple Judaism. By the Early Middle Ages (800–1000), faiths referred to as pagan had mostly disappeared in the West through a mixture of peaceful conversion, natural religious change, persecution, and the military conquest of pagan peoples; the Christianization of Lithuania in the 15th century is typically considered to mark the end of this process.

Early history

Early Christianity arose as a movement within Second Temple Judaism and Hellenistic Judaism, following the teachings of Jesus of Nazareth. With a missionary commitment to both Jews and Gentiles (non-Jews), Christianity rapidly spread into the greater Roman empire and beyond. Here, Christianity came into contact with the dominant Pagan religions. Acts 19 recounts a riot that occurred in Ephesus, instigated by silversmiths who crafted images of Artemis, and were concerned that Paul's success was cutting into their trade. These conflicts are recorded in the works of the early Christian writers such as Justin Martyr as well as hostile reports by writers including Tacitus and Suetonius.

The pattern for the Roman state's response to what was seen as a religious threat was established in 186 BC. Roman officials became suspicious of the worshippers of Dionysus and their practice of Bacchanalia because it "took place at night" (also a later Christian practice). Magic and secret plots against the emperor were seen as products of the night. Bacchic associations were dissolved, leaders were arrested and executed, women were forbidden to hold important positions in the cult, no Roman citizen could be a priest, and strict control of the cult was thereafter established. In the first century of the common era, there were "periodic expulsions of astrologers, philosophers and even teachers of rhetoric... as well as Jews and...the cult of Isis". Druids also received this treatment, as did Christians.

Persecution of early Christians

Christianity was persecuted by Roman imperial authorities early on in its history within the greater empire. By the early part of the 2nd century AD Christians were no longer viewed as forming a breakaway sect of Judaism, but were considered as belonging to just another of many foreign cults which had infiltrated the Empire. They gradually became conspicuous by their absence from festival activities where ritual sacrifices for the health of the emperor and well-being of the empire took place, behavior that carried a "whiff of both sacrilege and treason".

Persecution under Nero, 64–68 AD

The first documented case of imperially supervised persecution of the Christians in the Roman Empire begins with Nero (37–68). In AD 64, a great fire broke out in Rome, destroying portions of the city and economically devastating the Roman population. Nero himself was suspected as the arsonist by Suetonius. In his Annals, Tacitus (who claimed Nero was in Antium at the time of the fire's outbreak), stated that "to get rid of the report, Nero fastened the guilt and inflicted the most exquisite tortures on a class hated for their abominations, called Christians, or Chrestians, by the populace" (Tacit. Annals XV, see Tacitus on Jesus). Suetonius, later to the period, does not mention any persecution after the fire, but in a previous paragraph unrelated to the fire, mentions punishments inflicted on Christians, defined as men following a new and malefic superstition. But Suetonius did not specify the reasons for the punishment; he just listed the fact together with other abuses put down by Nero.

Persecution from the 2nd century to Constantine

Drawing of the Alexamenos graffito, c. 200

The Persecution in Lyon was preceded by mob violence, including assaults, robberies and stonings (Eusebius, Ecclesiastical History 5.1.7). Further state persecutions were desultory until the 3rd century, though Tertullian's Apologeticus of 197 was ostensibly written in defense of persecuted Christians and addressed to Roman governors.

Although there was sporadic local persecution, there was no empire-wide persecution of Christians until the reign of Decius in the mid-3rd century AD. A decree was issued requiring public sacrifice, a formality equivalent to a testimonial of allegiance to the Emperor and the established order. Christians who refused were charged with impiety and punished by arrest, imprisonment, torture, and/or executions. Some Christians complied and purchased their certificates, called libelli, which certified their compliance; others fled to safe havens in the countryside. Several councils held at Carthage debated the extent to which the community should accept lapsed Christians.

The Diocletianic Persecution

The Christian Martyrs' Last Prayer by Jean-Léon Gérôme (1883)

The persecutions culminated with Diocletian and Galerius at the end of the third and beginning of the 4th century. Beginning with a series of four edicts banning Christian practices and ordering the imprisonment of Christian clergy, the persecution intensified until all Christians in the empire were commanded to sacrifice to the gods or face immediate execution. This persecution lasted until Constantine I, along with Licinius, legalized Christianity in 313. The New Catholic Encyclopedia states that "Ancient, medieval and early modern hagiographers were inclined to exaggerate the number of martyrs. Since the title of martyr is the highest title to which a Christian can aspire, this tendency is natural". Attempts at estimating the numbers involved are inevitably based on inadequate sources.

The failure of the Great Persecution of Diocletian was regarded as a confirmation of a long process of religious self-assertion against the conformism of a pagan empire. Freedom to assert a belief not recognized by the State was won and held. 'However much Christian churches and states may have sinned in later times by their religious coercion, the martyrdoms of the Roman Persecutions belong to the history of freedom.

Prohibition and persecution of paganism in the Roman Empire

Triumph of Christian religion (over paganism) by Tommaso Laureti (1582), Vatican Palace

According to Rodney Stark, since Christians most likely formed only sixteen to seventeen percent of the empire's population at the time of Constantine's conversion, they did not have the numerical advantage to form a sufficient power–base to begin a systematic persecution of pagans. However, Brown reminds us "We should not underestimate the fierce mood of the Christians of the fourth century", nor should it be forgotten that repression, persecution and martyrdom do not generally breed tolerance of those same persecutors. Brown says Roman authorities had shown no hesitation in "taking out" the Christian church which they saw as a threat to the peace of the empire, and that Constantine and his successors did the same for the same reasons. Rome had been removing anything it saw as a challenge to Roman identity since Bacchic associations were dissolved in 186 BCE. That military action against a mystical religion became the pattern for the Roman state's response to anything it saw as a religious threat. That position of the state toward internal threats did not change once the emperors were Christian.

Within this environment, Christians of the fourth century also believed the conversion of Constantine (traditionally 312) showed that Christianity had triumphed over paganism (in Heaven) and little further action against pagans was necessary; everything was done but the sweeping up in the Christian view. As a result, the fourth century included a focus on heresy as a higher priority than paganism. According to Brown, "In most areas, polytheists were not molested, and apart from a few ugly incidents of local violence, Jewish communities also enjoyed a century of stable, even privileged, existence".

After Constantine, except for the brief period of Julian's rule, paganism never regained its previous status as a state religion. Yet despite its inferior status in the Christian Empire, paganism still existed and was practiced. Up to the time of Justin I and Justinian, there was some toleration for all religions; there were anti-pagan and anti-heretical laws, but they were not generally enforced. Thus, up through the sixth century, there still existed centers of paganism in Athens, Gaza, Alexandria, and elsewhere.

Constantine

The Edict of Milan of 313 finally legalized Christianity, and it gained governmental privileges, such as tax exemptions to Christian clergy, and a degree of official approval under Constantine. Constantine destroyed a few temples and plundered more, converted others to churches, and neglected the rest; he "confiscated temple funds to help finance his own building projects", and in an effort to establish a stable currency, so he was primarily interested in hoards of gold and silver, but he also confiscated temple land; he refused to support pagan beliefs and practices while also speaking out against them; he periodically forbade pagan sacrifices and closed temples, made laws that threatened and menaced pagans while other laws markedly favored Christianity, and he personally endowed Christians with gifts of money, land and government positions.

As the emperor, he openly supported Christianity after 324, but there are indications he also remained tolerant of pagans. He never engaged in a purge. Opponents' supporters were not slaughtered when Constantine took the capital; their families and court were not killed. There were no pagan martyrs. Laws menaced death, but during Constantine's reign, no one suffered the death penalty for violating anti-pagan laws against sacrifice. There is no evidence of judicial killings for illegal sacrifices before Tiberius Constantine (574–582), and many temples remained open into the reign of Justinian I (527-565). Peter Leithart says of Constantine that, "He did not punish pagans for being pagans, or Jews for being Jews, and did not adopt a policy of forced conversion." Pagans remained in important positions at his court. Constantine ruled for 31 years and never outlawed paganism.

All records of anti-pagan legislation by Constantine are found in the Life of Constantine, written by Eusebius, as a kind of eulogy after Constantine's death. It is not a history so much as a panegyric praising Constantine. The laws as they are stated in the Life of Constantine often do not correspond, "closely, or at all", to the text of the Codes themselves. Eusebius gives these laws a "strongly Christian interpretation by selective quotation or other means". This has led many to question the veracity of the record, and whether, in his zeal to praise Constantine, Eusebius generously attributed actions to Constantine that were not actually his.

Gratian

In 382, Gratian was the first Roman emperor to formally, in law, divert into the crown's coffers those public financial subsidies that had previously supported Rome's cults; he appropriated the income of pagan priests and the Vestal Virgins, forbid their right to inherit land, confiscated the possessions of the priestly colleges, and was the first to refuse the title of Pontifex Maximus. The colleges of pagan priests lost privileges and immunities. He also ordered the Altar of Victory to be removed again.

Gratian and Ambrose, the Bishop of Milan, exchanged multiple letters and books on Christianity, and the sheer volume of these writings has often been seen as evidence that Gratian was dominated by Ambrose. Ambrose, therefore, was the 'true source' of Gratian's anti-pagan actions. McLynn finds this unlikely and unnecessary as an explanation: Gratian was, himself, devout, and "The many differences between Gratian's religious policies and his father's, and the shifts that occurred during his own reign, are to be explained by changed political circumstances [after the Battle of Adrianople], rather than capitulation to Ambrose".

Modern scholars have noted that Sozomen is the only ancient source that shows Ambrose and Gratian having any personal interaction. In the last year of Gratian's reign, Ambrose crashed Gratian's private hunting party in order to appeal on behalf of a pagan senator sentenced to die. After years of acquaintance, this indicates Ambrose did not personally feel he had enough influence to take for granted that Gratian would grant a request to see him. Instead, Ambrose had to resort to such maneuverings to make his appeal. Gratian's brother, Valentinian II, and his mother disliked Ambrose, but Valentinian II also refused to grant requests from pagans to restore the Altar of Victory and the income of the temple priests and Vestal Virgins.

Between 382 and 384, there was yet another dispute over the Altar of Victory. According to the Oxford Handbook of Late Antiquity, Symmachus requested the restoration of the altar that Gratian had removed and the restoration of state support for the Vestals. Ambrose campaigned against any financial support for paganism throughout his entire career, and anything like the Altar that required participation in blood sacrifices was anathema. Ambrose responded to Symmachus' and his arguments prevailed; the requests were denied. Pagans became outspoken in their demands for respect, concessions and support from the state, voicing their resentment in historical works, such as the writings of Eunapius and Olympiodorus.

After Gratian, the emperors Arcadius, Honorius and Theodosius continued to appropriate for the crown the tax revenue collected by the temple custodians, though this may have been more about the empire's ongoing financial difficulties than religion. Urban ritual procession and ceremony was gradually stripped of support and funding. Rather than being removed outright though, many festivals were secularized, and later, these were incorporated into a developing Christian calendar (often with little alteration). Some had already severely declined in popularity by the end of the 3rd century.

Theodosius

In the Eastern Empire, up until the time of Justinian, the Byzantine emperors practiced a policy of tolerance toward all religions. This pertained to both devotions to the Greco-Roman gods and the religion of barbarians living within the empire. Although there were anti-pagan laws, there is no record of their harsh punishments ever being enforced. As the eastern emperor, Theodosius seems to have practiced this same type of cautious policy from the beginning of his reign. Theodosius declared Nicene Christianity the official religion of the empire, though this was aimed more at the local Arians in Constantinople than the pagans. For pagans, he reiterated his Christian predecessors' bans on animal sacrifice, divination, and apostasy, but allowed other pagan practices to be performed publicly and temples to remain open. He also turned pagan holidays into workdays, but the festivals associated with them continued.

There is evidence that Theodosius took care to prevent the empire's still substantial pagan population from feeling ill-disposed toward his rule. Following the death in 388 of his praetorian prefect, Cynegius, who had, contrary to Theodosius' spoken policies, vandalized a number of pagan shrines and temples in the eastern provinces, Theodosius replaced him with a moderate pagan who subsequently moved to protect the temples. During his first official tour of Italy (389–391), the emperor won over the influential pagan lobby in the Roman Senate by appointing its foremost members to important administrative posts. Theodosius also nominated the last pair of pagan consuls in Roman history (Tatianus and Symmachus) in 391.

Classicist Ingomar Hamlet says that, contrary to popular myth, Theodosius did not ban the Olympic games. Sofie Remijsen [nl] indicates there are several reasons to conclude the Olympic games continued after Theodosius and came to an end under Theodosius II instead. Two scholia on Lucian connect the end of the games with a fire that burned down the temple of the Olympian Zeus during his reign.

Legislation

A number of laws against pagan sacrifice, and against heresy, were issued towards the end of Theodosius' reign in 391 and 392.

Anti-pagan legislation reflects what Brown calls "the most potent social and religious drama" of the fourth-century Roman empire. From Constantine forward, the Christian intelligentsia wrote of Christianity as fully triumphant over paganism. It did not matter that they were still a minority in the empire, this triumph had occurred in Heaven; it was evidenced by Constantine; but even after Constantine, they wrote that Christianity would defeat, and be seen to defeat, all of its enemies – not convert them. As Peter Brown says, "Conversion was not the principal aim of a social order that declared the God-given dominance of Christianity".

The laws were not intended to convert; "the laws were intended to terrorize... Their language was uniformly vehement, and... frequently horrifying". Their intent was to reorder society along religious lines with the 'triumphant' Christian church in charge, and pagans and Jews at the outskirts of influence, so that laws could be made that were sufficiently intimidating to enable Christianity to put a stop to animal sacrifice. Blood sacrifice was the element of pagan culture most abhorrent to Christians. If they could not stop the private practice of sacrifice, they could "hope to determine what would be normative and socially acceptable in public spaces". Altars used for sacrifice were routinely smashed by Christians who were deeply offended by the blood of slaughtered victims as they were reminded of their own past sufferings associated with such altars.

Blood sacrifice was a central rite of virtually all religious groups in the pre-Christian Mediterranean, and its gradual disappearance is one of the most significant religious developments of late antiquity. ... Public sacrifices and communal feasting had declined as the result of a decline in the prestige of pagan priesthoods and a shift in patterns of [private donations] in civic life. That shift would have occurred on a lesser scale even without the conversion of Constantine... It is easy, nonetheless, to imagine a situation in which sacrifice could decline without disappearing. Why not retain, for example, a single animal victim in order to preserve the integrity of the ancient rite? The fact that public sacrifices appear to have disappeared completely in many towns and cities must be attributed to the atmosphere created by imperial and episcopal hostility.

By the time the pro-pagan emperor Julian (r. 360–363) made his trip through Asia Minor to Antioch to assemble an army and resume war against Persia, opposing sacrifice had become the norm among the people. Julian reached Antioch on July 18 which coincided with a pagan festival that had already become secular: it did not include sacrifice. Julian's preference for blood sacrifice found little support, and the citizens of Antioch accused Julian of "turning the world upside down" by reinstituting it, calling him "slaughterer". "When Julian restored altars in Antioch, the Christian populace promptly threw them down again". Julian succeeded in marching to the Sassanid capital of Ctesiphon, but during the Battle of Samarra, he was mortally wounded. The facts of his death have become obscured by the "war of words between Christians and pagans" which followed. It was "principally over the source of the fatal spear... The thought that Julian might have died by the hand of one of his own side... was a godsend to a Christian tradition eager to have the apostate emperor accorded his just deserts. Yet such a rumor was not solely the product of religious polemic. It had its roots in the broader trail of disaffection Julian left in his wake".

One of the first things that is important about this, in Malcolm Errington's view, is how much anti-pagan legislation was applied and used, which would show how dependable the laws are as a reflection of what actually happened to pagans in history. Brown says that, given the large numbers of non-Christians in every region at this time, local authorities were "notoriously lax" in imposing them. Christian bishops also frequently obstructed their application. The harsh imperial edicts had to face the vast following of paganism among the population, and the passive resistance of governors and magistrates, thereby limiting their impact. Limiting, but not eliminating impact altogether, as Anna Leone says, "Temple closures and the prohibition of sacrifices had an impact... After AD 375 the majority of [pagan] religious offices disappear completely from the epigraphic record".

Secondly, the laws reveal the emergence of a language of intolerance. The legal language runs parallel to the writings of the apologists, such as Augustine of Hippo and Theodoret of Cyrrhus, and heresiologists such as Epiphanius of Salamis. Christian writers and imperial legislators both drew on a rhetoric of conquest. These writings were commonly hostile and often contemptuous toward a paganism Christianity saw as already defeated.

Lastly, on the one hand the laws, and these Christian sources with their violent rhetoric, have had great influence on modern perceptions of this period by creating an impression of continuous violent conflict that has been assumed on an empire-wide scale. Archaeological evidence, on the other hand, indicates that, outside of violent rhetoric, there were only isolated incidents of actual violence between Christians and pagans. Non-Christian, (non-heretical), groups such as pagans and Jews enjoyed a tolerance based on contempt through most of Late Antiquity.

Temple destruction

It has been common for much of scholarship to attribute rampant temple destruction to Theodosius through his prefect, Maternus Cynegius. Cynegius did commission the destruction of temples, using the army under his control and nearby monks, especially in the territory around Constantinople in the diocese of Oriens (the East). Peter Brown says that in 392, inspired by the mood created by Cynegius, Theophilus of Alexandria staged a procession ridiculing statues of pagan gods. Political complications contributed to turning it into a riot, and the Serapium in Alexandria, Egypt was destroyed. Some scholars think this is when the philosopher Hypatia was killed, (though there is evidence this happened in 415 instead). These examples were seen as the 'tip of the iceberg' by earlier scholars who saw these events as part of a tide of violent Christian iconoclasm that continued throughout the 390s and into the 400s.

Problems with this view have arisen in the twenty-first century. Archaeological evidence for the violent destruction of temples in the fourth century, from around the entire Mediterranean, is limited to a handful of sites. Temple destruction is attested to in 43 cases in the written sources, but only 4 of them have been confirmed by archaeological evidence.

In Gaul, only 2.4% of over 500 known temples and religious sites were destroyed by violence, some of it barbarian. In Africa, the city of Cyrene has good evidence of the burning of several temples; Asia Minor has produced one weak possibility; in Greece the only strong candidate may relate to a barbarian raid instead of Christians. Egypt has produced no archaeologically confirmed temple destructions from this period except the Serapeum. In Italy there is one; Britain has the highest percentage with 2 out of 40 temples.

Trombley and MacMullen say part of why discrepancies between literary sources and archaeological evidence exist is because it is common for details in the literary sources to be ambiguous and unclear.  For example, Malalas claimed Constantine destroyed all the temples, then he said Theodisius destroyed them all, then he said Constantine converted them all to churches. "According to Procopius, in the 530s Justinian destroyed the temples of Philae widely identified as the last bastion of paganism in Egypt. But no priests are attested to after the 450s, Christianity was thriving there from the early fourth century, and the temples themselves are among the best preserved in the ancient world".

Archaeology suggests that religious buildings were subject to three different directions of change during the imperial period: early abandonment, destruction and re-use. The financial struggles begun in the third century continued on into the fourth century to negatively impact available funding to maintain the large temple complexes and their festivals. Lower budgets, with less spent on statuary, monuments, and simple maintenance, meant the physical decline of urban structures of all types. Many Temples were left to fall into disrepair and in many instances, such as in Tripolitana, this happened before any Christian anti-pagan legislation could have been a factor.

Progressive early decay was accompanied by an increased trade in statuary and salvaged building materials, as the practice of recycling became common in Late Antiquity, resulting in their complete destruction and removal. "Even churches were reused in similar ways". Some temple restorations took place throughout the imperial period, but there is no evidence of state participation or support. Restorations were funded and accomplished privately.

Overall data indicates that a number of elements coincided to end the Temples, but none of them were strictly religious. The economy, necessity, and political expressions of power were the primary driving forces for the destruction and conversion of pagan religious monuments. Lavan says: "We must rule out most of the images of destruction created by the [written sources]. Archaeology shows the vast majority of temples were not treated this way".

Temple conversion

Some scholars have long asserted that not all temples were destroyed but were instead converted to churches throughout the empire. According to modern archaeology, 120 pagan temples were converted to churches in the whole empire, out of the thousands of temples that existed, and only about 40 of them are dated before the end of the fifth century. R. P. C. Hanson says the direct conversion of temples into churches did not begin until the mid fifth century in any but a few isolated incidents. In Rome the first recorded temple conversion was the Pantheon in 609. None of the churches attributed to Martin of Tours can be proven to have existed in the fourth century.

Rival literature

Some pagans blamed the Christian hegemony for the 410 Sack of Rome, while Christians in turn blamed the pagans, provoking Saint Augustine, a Christian bishop, to respond by writing The City of God, a seminal Christian text. It is alleged that Christians destroyed almost all pagan political literature and threatened to cut off the hands of any copyist who dared to make new copies of the offending writings. Yet there is no evidence any Christian in authority ever "actually punished the expression of pagan sentiments" and there is no known prosecution of any pagan work. Many pagan poets and writers were popular among the still classically educated Christian elite, for example, Seneca was referenced 13 times in Augustine's City of God.

Mob violence

Mob violence was an occasional problem in all the independent cities of the empire. There were no police forces as such. Taxes, food and politics were common reasons for rioting. Religion was also a factor though it is difficult to separate from politics since they were intertwined in all aspects of life. In 361, the murder of the Arian bishop George of Cappadocia was committed by a mob of pagans, although there is evidence he had cruelly provoked them; the conflict over the Serapeum involved both a Christian and a pagan mob; the Jews and the Christians each gathered to fight in 415, although the sources indicate it was the upper levels of the Jewish community who decided to massacre the Christians after Cyril made serious threats to their leadership. A Christian mob threw objects at Orestes and, finally, Hypatia was killed by a Christian mob though politics and personal jealousy were probably the primary causes. Mobs were composed of lower-class urban dwellers, upper class educated pagans, Jews and Christians, and in Alexandria, monks from the monastery of Nitria.

Pagan influences on early Christianity

Art

Hermes Kriophoros

The early Christians adapted many elements of paganism. Ancient pagan funeral rituals often remained within Christian culture as aspects of custom and community with very little alteration. A type of song sung at death, the ritual lament, is one of the oldest of all art forms. As soon as death was imminent, the ritual began, then came the "struggle of the soul" and prayer for the dying. John Chrysostum gives a vivid account of the dying soul seeing angels and demons – "account books in hand" – struggling against each other in a contest for possession of the dying person's soul. Macarius of Egypt (fourth century) writes of such a contest which is only resolved by the intervention of the person's guardian angel – which is roughly parallel to Plato's daimon.

Spontaneous lamentation would break out among those present once the struggle of the soul was over. All evidence suggests this was a violent display of grief – the laceration of the cheeks, tearing one's hair, and the rending of garments along with the wailing of the lament song. The church saw this immoderate behavior as improper for people who believed death was not the end, so they attempted to moderate it by singing Psalms, with two groups of singers on opposite sides chanting an antiphonal lament, with rhythm, harmony and order instead. However, this too is similar to the pagan lament sung for Achilles and one suggested by Plato for his Examiners in the Laws.

Pagans and Jews decorated their burial chambers, so Christians did as well, thereby creating the first Christian art in the catacombs beneath Rome. This art is symbolic, rising out of a reinterpretation of Jewish and pagan symbolism. Christian piety infused the symbols with its own fresh interpretation. Christian art had something fundamentally new to say as it gave visual expression to the conviction that the human soul can be delivered from death to an everlasting life. Neither Judaism nor any pagan religion had previously made such a claim. "The Jewish faith puts little emphasis on immortality, and pagan beliefs about the afterlife were vague, uncertain, and sometimes dismal".

Noah catacomb (orans)

While many new subjects appear for the first time in the Christian catacombs – i.e. the Good Shepherd, Baptism, and the Eucharistic meal – the Orant figures (women praying with upraised hands) probably came directly from pagan art. Pagan symbolism in the form of Victories, cupids, and shepherd scenes are scattered throughout the catacombs.  Jewish and pagan use of sheep and goats, birds in a tree or vine, or eating fruit, especially grapes, seven steps leading up to a tomb, a pair of peacocks, the Robe of sanctity, the reading of scrolls, are all found in pagan art and adapted in the Christian art to express the hope of immortality in Christian terms. Pagan sarcophagi had long carried shells, and portraits of the dead often had shells over the head of the dead, while some put a shell over a grave. Christians and Jews adapted the convention, identifying it with another symbol – the halo. For the Christians who made the catacombs, these symbols were necessary to convey their message.

Many previously pagan holy places were converted to Christian use. In 609 Pope Boniface IV obtained leave from the Byzantine Emperor Phocas to convert the Pantheon in Rome into a Christian church, a practice similar to that recommended eight years earlier by Pope Gregory I to Mellitus regarding Anglo-Saxon holy places, in order to ease the transition to Christianity. According to Willibald's Life of Saint Boniface, about 723, the missioner cut down the sacred Donar's Oak and used the lumber to build a church dedicated to St. Peter. Around 744, Saint Sturm established the monastery of Fulda on the ruins of a 6th-century Merovingian royal camp, destroyed 50 years earlier by the Saxons, at a ford on the Fulda River.

Calendar

Many names for months and days of the week – even the concept of a seven-day week – were borrowed from Roman paganism.

In its first three centuries, Christianity did not celebrate the birth of Christ. Birthdays were seen as pagan, no one knew Jesus's true birthdate, and many early church fathers were against the idea. The earliest source giving December 25 as Jesus's birthdate is the Chronograph of 354, which liturgical historians generally agree was written in Rome in AD 336. A supposedly earlier reference by Hippolytus of Rome is considered a later interpolation. In the ancient Roman calendar, December 25 was the date of the winter solstice.

A widely-held theory is that the Church chose December 25 as Christ's birthday (Dies Natalis Christi) to appropriate the Roman winter solstice festival Dies Natalis Solis Invicti (birthday of Sol Invictus, the 'Invincible Sun'), held on the same date. This festival had been instituted by the emperor Aurelian in AD 274. Gary Forsythe, Professor of Ancient History, says "This celebration would have formed a welcome addition to the seven-day period of the Saturnalia (December 17–23), Rome's most joyous holiday season since Republican times, characterized by parties, banquets, and exchanges of gifts". Historian Stephen Nissenbaum says this choice was a compromise with paganism, arguing there is no avoiding "Roman midwinter parties and Christianity's conscious decision to place a Christmas celebration right in the middle of them" as part of that compromise. Many observers schooled in the classical tradition have noted similarities between the Saturnalia and historical revelry during the Twelve Days of Christmas and the Feast of Fools. William Warde Fowler notes: "[Saturnalia] has left its traces and found its parallels in great numbers of medieval and modern customs, occurring about the time of the winter solstice."

Some way or another, Christmas was started to compete with rival Roman religions, or to co-opt the winter celebrations as a way to spread Christianity, or to baptize the winter festivals with Christian meaning in an effort to limit their [drunken] excesses. Most likely all three.

John the Baptist was said to be six months older than Jesus, thus the Church began holding the feast of the Nativity of John the Baptist on 24 June, the Roman date of the summer solstice. John the Baptist "was understood to be preparing the way for Jesus", with John 3:30 stating "He must increase, but I must decrease". The sun's height in the sky and length of the day begins to decrease after the summer solstice and to increase after the winter solstice. Thus 'Johnmas' was held at midsummer and 'Christmas' at midwinter. With the spread of Christianity, some of the local Germanic Midsummer traditions were incorporated into St John's Eve festivities.

Another theory, first proposed by French writer Louis Duchesne in 1889, is that Christmas was calculated as nine months after a date chosen as Christ's conception: March 25, the Roman date of the spring equinox. This is based on a belief that the spring equinox was the day of God's act of Creation.

While attending Crozer Theological Seminary c.1950, Martin Luther King Jr. wrote an essay titled "The Influence of the Mystery Religions on Christianity". King noted that the place at Bethlehem selected by early Christians as Jesus's birthplace was an early shrine of a pagan god, Adonis. After the Bar Kokhba revolt (c. 132–136 CE) was crushed, the Roman emperor Hadrian converted the Christian site above the Grotto into a shrine dedicated to the Greek god Adonis, to honour his favourite, the Greek youth Antinous.

Influence on early Christian theology

Justin Martyr was a pagan who became a Christian around 132. In his First Apology, Justin used the concept of the "Logos" as a way of arguing for Christianity to non-Jews. These references demonstrate that Justin's knowledge of Stoicism was the knowledge of an ordinary man of his time in ordinary conversation, and that it is unlikely he ever studied Stoicism. However, he calls himself a Platonist, his references to Plato are much more detailed, and parallels to Plato's writings can be found in Justin's, though they do not suggest direct influence. Since a Greek audience would accept references to Greek philosophy, his argument could concentrate on identifying the Logos with Jesus. Scholars generally recognize that Clement went much farther, perhaps the farthest "any Orthodox Christian ever did in his appropriation and use of Hellenistic philosophical and ethical concepts for the expression of his Christian faith".

Augustine of Hippo

Augustine of Hippo (354–430), who ultimately systematized Christian philosophy after converting to Christianity from Manichaeism, wrote in the late 4th and early 5th century: "But when I read those books of the Platonists I was taught by them to seek incorporeal truth, so I saw your 'invisible things, understood by the things that are made'." Until the 20th century, most of the Western world's concept of Manichaeism came through Augustine's negative polemics against it. According to his Confessions, after eight or nine years of adhering to the Manichaean faith (as an "auditor", the lowest level in the sect's hierarchy), he became a Christian and a potent adversary of Manichaeism. When he turned from Manichaeism, he took up skepticism. In AD 386, he published Contra Academicos (Against the Skeptics). J. Brachtendorf says Augustine used the Ciceronian Stoic concept of passions to interpret Paul's doctrine of universal sin and redemption.

The influence of paganism can also be found in the development of many non-Orthodox theologies such as the Cathars of the Middle Ages. Cathars were dualists and felt that the world was the work of a demiurge of Satanic origin. Whether this was due to influence from Manichaeism or another strand of Gnosticism, and whether it was originally Zoroastrian has been impossible to determine. The Bogomils and Cathars, in particular, left few records of their rituals or doctrines, and the link between them and Manichaeans is unclear. Regardless of its historical veracity the charge of Manichaeism was leveled at them by contemporary orthodox opponents, who often tried to fit contemporary heresies with those combated by the church fathers. Not all Cathars held that The Evil God (or principle) was as powerful as The Good God (also called a principle) as Mani did, a belief also known as absolute dualism. In the case of the Cathars, it seems they adopted the Manichaean principles of church organization, but none of its religious cosmology. Similarly, Priscillian and his followers apparently tried to absorb what they thought was the valuable part of Manichaeism into Christianity.

Christianization during the European Middle Ages

Anatolia

A safe area for the pagans was the city of Harran which, Despite the persecution of its pagan inhabitants by Byzantine Emperor Maurice, remained a largely pagan city well into the early Islamic period. When the city was besieged by the armies of the Rashidun Caliphate in 639–640, it was the pagan community that negotiated its peaceful surrender. Under the subsequent rule of the caliphates, Harran became a major settlement within the Diyar Mudar region and retained a significant degree of autonomy. During the First Fitna, the people of Harran sided with Mu'awiya I over Ali at the Battle of Siffin in 657, which allegedly resulted in a brutal retaliation by Ali, who massacred much of the population.

Under the Umayyad Caliphate (661–750), Harran prospered and was selected as the capital by the last Umayyad caliph, Marwan II, from 744 to 750. This move may have been influenced by the city's pagan sympathies and its strategic position near the empire's eastern provinces. The city's prominence under Umayyad rule saw it grow as a cultural and scholarly center, with the establishment of the first Muslim university in 717 under Umar II, attracting scholars from across the Islamic world.

Although Harran lost its capital status under the Abbasid Caliphate, it continued to flourish, particularly during the reign of Harun al-Rashid (786–809), when its university became a key center for translation and intellectual activity. The local religion, blending elements of Mesopotamian paganism and Neoplatonism, persisted into the 10th century, though periodic decrees enforced conversions to Islam, especially under Al-Ma'mun in 830. Nonetheless, Harran retained its heterogeneity, with a population that included Muslims, Christians, Jews, and a variety of other religious groups.

Anglo-Saxons

Part of seventh-century casket, depicting the pan-Germanic legend of Weyland Smith, which was apparently also a part of Anglo-Saxon pagan mythology

The most likely date for Christianity getting its first foothold in Britain is sometime around 200. Recent archaeology indicates that it had become an established minority faith by the fourth century. It was largely mainstream, and in certain areas, had been continuous.

In early Anglo-Saxon England, non-stop religious development meant paganism and Christianity were never completely separate. Arthur Weston wrote that, "When Gregory the Great was taking steps for the conversion of the heathen Saxons, he is said to have warned his missionaries not to interfere with any traditional belief or religious observance which could be harmonized with Christianity".

Tell Augustine that he should be no means destroy the temples of the gods but rather the idols within those temples. Let him, after he has purified them with holy water, place altars and relics of the saints in them. For, if those temples are well built, they should be converted from the worship of demons to the service of the true God. Thus, seeing that their places of worship are not destroyed, the people will banish error from their hearts and come to places familiar and dear to them in acknowledgement and worship of the true God.

Further, since it has been their custom to slaughter oxen in sacrifice, they should receive some solemnity in exchange. Let them therefore, on the day of the dedication of their churches, or on the feast of the martyrs whose relics are preserved in them, build themselves huts around their one-time temples and celebrate the occasion with religious feasting. They will sacrifice and eat the animals not any more as an offering to the devil, but for the glory of God to whom, as the giver of all things, they will give thanks for having been satiated. Thus, if they are not deprived of all exterior joys, they will more easily taste the interior ones. For surely it is impossible to efface all at once everything from their strong minds, just as, when one wishes to reach the top of a mountain, he must climb by stages and step by step, not by leaps and bounds.

Richard A. Fletcher suggests that holy wells developed out of a like adaptation.

The word Easter is linked, by a single documentary source, to an Anglo-Saxon goddess, though the roots of the Easter celebration predate Christian contact with the Anglo-Saxons. In his eighth-century The Reckoning of Time the venerable Bede wrote that "Ēosturmōnaþ was Anglo-Saxon for 'Month of Ēostre', the month that corresponded to April, so-named "after a goddess of theirs named Ēostre, in whose honour feasts were celebrated in that month". The German cognate of the goddess Eostre is called Ostara, and likewise the word for Easter in German is Ostern. Richard Fletcher, however, speculated that the name Easter might come from the Anglo-Saxon eastan, meaning east.

The conversion of the Anglo-Saxons was begun at about the same time in both the north and south of the Anglo-Saxon kingdoms in two unconnected initiatives. Irish missionaries led by Saint Columba, based in Iona (from 563), converted many Picts. The court of Anglo-Saxon Northumbria, and the Gregorian mission, who landed in 596, did the same to the Kingdom of Kent. They had been sent by Pope Gregory I and were led by Augustine of Canterbury with a mission team from Italy. In both cases, as in other kingdoms of this period, conversion generally began with the royal family and the nobility adopting the new religion first.

The conversion of Æthelberht, king of Kent is the first account of any Christian bretwalda conversion and is told by the Venerable Bede in his histories of the conversion of England. In 582 Pope Gregory sent Augustine and 40 companions from Rome to convert the Anglo-Saxons. "They had, by order of the blessed Pope Gregory, brought interpreters of the nation of the Franks, and sending to Æthelberht, signified that they were come from Rome, and brought a joyful message, which most undoubtedly assured to all that took advantage of it everlasting joys in heaven, and a kingdom that would never end with the living and true God." Æthelberht was not unfamiliar with Christianity because he had a Christian wife, and Bede says that there was even a church dedicated to St. Martin nearby. Æthelberht was converted eventually and Augustine remained in Canterbury.

Reliquary of St. Oswald, Hildesheim

After his death, King Oswald of Northumbria came to be regarded as a saint, and the spot where he died was associated with miracles. Reginald of Durham mentions one, saying that Oswald's right arm was taken by a raven to an ash tree, which gave the tree ageless vigor; when the bird dropped the arm onto the ground, a spring emerged from the ground. Both the tree and the spring were, according to Reginald, subsequently associated with healing miracles. Aspects of the legend have been considered to have pagan overtones or influences and may represent a fusion of his status as a traditional Germanic warrior-king with Christianity. The cult surrounding him gained prominence in parts of continental Europe.

Some time prior to 655, Œthelwald of Deira gave Chad of Mercia land upon which to build a monastery. According to Bede, Chad felt it necessary to fast for forty days in order to cleanse the place. This ritual purification indicates that the new monastery was likely built on the site of a pre-Christian cult.

Greece

Pagan Continuity in Mani and Mistra (800-1100) Christianity was introduced late in Mani, with the first Greek temples converted into churches during the 11th century. Byzantine monk Nikon "the Metanoite" (Νίκων ὁ Μετανοείτε) was sent in the 10th century to convert the predominantly pagan Maniots. Although his preaching began the conversion process, it took over 200 years for the majority to accept Christianity fully by the 11th and 12th centuries. Patrick Leigh Fermor noted that the Maniots, isolated by mountains, were among the last Greeks to abandon the old religion, doing so towards the end of the 9th century:

Sealed off from outside influences by their mountains, the semi-troglodytic Maniots themselves were the last of the Greeks to be converted. They only abandoned the old religion of Greece towards the end of the ninth century. It is surprising to remember that this peninsula of rock, so near the heart of the Levant from which Christianity springs, should have been baptised three whole centuries after the arrival of St. Augustine in far-away Kent.

According to Constantine VII in De Administrando Imperio, the Maniots were referred to as 'Hellenes' and only fully Christianized in the 9th century, despite some church ruins from the 4th century indicating early Christian presence. The region's mountainous terrain allowed the Maniots to evade the Eastern Roman Empire's Christianization efforts, thus preserving pagan traditions, which coincided with significant years in the life of Gemistos Plethon.

Saxons

The conversion of the northern Saxons began with their forced incorporation into the Frankish kingdom in 776 by Charlemagne (r. 768–814). Thereafter, the Saxon's Christian conversion slowly progressed into the eleventh century. The Saxon conversion was difficult for a number of reasons including that their pagan beliefs were so strongly tied to their culture that conversion necessarily meant massive cultural change that was hard to accept. Their sophisticated theology was also a bulwark against an immediate and complete conversion to Christianity.

Saxons had gone back and forth between rebellion and submission to the Franks for decades. Charlemagne placed missionaries and courts across Saxony in hopes of pacifying the region, but Saxons rebelled again in 782 with disastrous losses for the Franks. In response, the Frankish King "enacted a variety of draconian measures" beginning with the massacre at Verden in 782 when he ordered the decapitation of 4500 Saxon prisoners offering them baptism as an alternative to death. These events were followed by the severe legislation of the Capitulatio de partibus Saxoniae in 785 which prescribes death to those that are disloyal to the king, harm Christian churches or its ministers, or practice pagan burial rites. His harsh methods of Christianization raised objections from his friends Alcuin and Paulinus of Aquileia. Charlemagne abolished the death penalty for paganism in 797.

Scandinavia

The first recorded attempts at spreading Christianity in Norway were made by King Haakon the Good in the tenth century, who was raised in England. His efforts were unpopular and were met with little success. In 995 Olaf Tryggvason became King Olaf I of Norway. Olaf I then made it his priority to convert the country to Christianity. By destroying temples and torturing and killing pagan resisters he succeeded in making every part of Norway at least nominally Christian. Expanding his efforts to the Norse settlements in the west the kings' sagas credit him with Christianizing the Faroes, Orkney, Shetland, Iceland, and Greenland. After Olaf's defeat at the Battle of Svolder in 1000 there was a partial relapse to paganism in Norway under the rule of the Jarls of Lade. In the following reign of Saint Olaf, pagan remnants were stamped out and Christianity entrenched.

Northern Crusades

Armed conflict between the Baltic Finns, Balts and Slavs who dwelt by the Baltic shores and their Saxon and Danish neighbors to the north and south had been common for several centuries. The Christianization of the pagan Balts, Slavs and Finns was undertaken primarily during the 12th and 13th centuries, in a series of uncoordinated military campaigns by various German and Scandinavian kingdoms, and later by the Teutonic Knights and other orders of warrior-monks. It was during these Northern Crusades that armed conversion of paganism first became a part of Christianity.

Dragnea and Christiansen indicate the primary motive for these wars was the noble's desire for territorial expansion and material wealth in the form of land, furs, amber, slaves, and tribute. Medieval historian and political scientist Iben Fonnesberg-Schmidt says, the princes were motivated by their desire to extend their power and prestige, and conversion was not always an element of their plans. However, conversion was part of the language for all these invaders, and conversion was almost always by the direct use of force or the indirect force of a leader who had converted and required conversion of his followers as well. There were often severe consequences for populations that chose to resist. For example, the conquest and conversion of Old Prussia resulted in the death of much of the native population, whose language subsequently became extinct.

"While the theologians maintained that conversion should be voluntary, there was a widespread pragmatic acceptance of conversion obtained through political pressure or military coercion." The church's acceptance of this led some commentators of the time to endorse and approve it, something Christian thought had never previously done. Dominican friars helped with this ideological justification. By portraying the pagans as possessed by evil spirits, they could assert the pagans were in need of conquest, persecution and force in order to free them; then they would become peacefully converted. Ideals of peaceful conversion were rarely realized in these crusades; monks and priests had to work with the secular rulers on their terms, and the military leaders seldom cared about allowing the time necessary for peaceful conversion.

Absorption and erasure of pagan traditions

The practice of replacing pagan beliefs and motifs with Christian, and purposefully not recording the pagan history (such as the names of pagan gods, or details of pagan religious practices), has been compared to the practice of damnatio memoriae.

Saturday, October 26, 2024

History of group theory

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/History_of_group_theory

The history of group theory, a mathematical domain studying groups in their various forms, has evolved in various parallel threads. There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry. Joseph Louis Lagrange, Niels Henrik Abel and Évariste Galois were early researchers in the field of group theory.

Early 19th century

The earliest study of groups as such probably goes back to the work of Lagrange in the late 18th century. However, this work was somewhat isolated, and 1846 publications of Augustin Louis Cauchy and Galois are more commonly referred to as the beginning of group theory. The theory did not develop in a vacuum, and so three important threads in its pre-history are developed here.

Development of permutation groups

One foundational root of group theory was the quest of solutions of polynomial equations of degree higher than 4.

An early source occurs in the problem of forming an equation of degree m having as its roots m of the roots of a given equation of degree . For simple cases, the problem goes back to Johann van Waveren Hudde (1659). Nicholas Saunderson (1740) noted that the determination of the quadratic factors of a biquadratic expression necessarily leads to a sextic equation, and Thomas Le Seur (1703–1770) (1748) and Edward Waring (1762 to 1782) still further elaborated the idea. Waring proved the fundamental theorem of symmetric polynomials, and specially considered the relation between the roots of a quartic equation and its resolvent cubic.

Lagrange's goal (1770, 1771) was to understand why equations of third and fourth degree admit formulas for solutions, and a key object was the group of permutations of the roots. On this was built the theory of substitutions. He discovered that the roots of all Lagrange resolvents (résolvantes, réduites) which he examined are rational functions of the roots of the respective equations. To study the properties of these functions, he invented a Calcul des Combinaisons. The contemporary work of Alexandre-Théophile Vandermonde (1770) developed the theory of symmetric functions and solution of cyclotomic polynomials. Leopold Kronecker has been quoted as saying that a new boom in algebra began with Vandermonde's first paper. Similarly Cauchy gave credit to both Lagrange and Vandermonde for studying symmetric functions and permutations of variables.

Paolo Ruffini (1799) attempted a proof of the impossibility of solving the quintic and higher equations. Ruffini was the first person to explore ideas in the theory of permutation groups such as the order of an element of a group, conjugacy, and the cycle decomposition of elements of permutation groups. Ruffini distinguished what are now called intransitive and transitive, and imprimitive and primitive groups, and (1801) uses the group of an equation under the name l'assieme delle permutazioni. He also published a letter from Pietro Abbati to himself, in which the group idea is prominent. However, he never formalized the concept of a group, or even of a permutation group.

Galois age fifteen, drawn by a classmate.

Évariste Galois is honored as the first mathematician linking group theory and field theory, with the theory that is now called Galois theory. Galois also contributed to the theory of modular equations and to that of elliptic functions. His first publication on group theory was made at the age of eighteen (1829), but his contributions attracted little attention until the posthumous publication of his collected papers in 1846 (Liouville, Vol. XI). He considered for the first time what is now called the closure property of a group of permutations, which he expressed as

if in such a group one has the substitutions S and T then one has the substitution ST.

Galois found that if are the n roots of an equation, there is always a group of permutations of the r's such that

  • every function of the roots invariable by the substitutions of the group is rationally known, and
  • conversely, every rationally determinable function of the roots is invariant under the substitutions of the group.

In modern terms, the solvability of the Galois group attached to the equation determines the solvability of the equation with radicals.

Galois was the first to use the words group (groupe in French) and primitive in their modern meanings. He did not use primitive group but called equation primitive an equation whose Galois group is primitive. He discovered the notion of normal subgroups and found that a solvable primitive group may be identified to a subgroup of the affine group of an affine space over a finite field of prime order.

Groups similar to Galois groups are (today) called permutation groups. The theory of permutation groups received further far-reaching development in the hands of Augustin Cauchy and Camille Jordan, both through introduction of new concepts and, primarily, a great wealth of results about special classes of permutation groups and even some general theorems. Among other things, Jordan defined a notion of isomorphism, although limited to the context of permutation groups. It was also Jordan who put the term group in wide use.

An abstract notion of a (finite) group appeared for the first time in Arthur Cayley's 1854 paper On the theory of groups, as depending on the symbolic equation . Cayley proposed that any finite group is isomorphic to a subgroup of a permutation group, a result known today as Cayley's theorem. In succeeding years, Cayley systematically investigated infinite groups and the algebraic properties of matrices, such as the associativity of multiplication, existence of inverses, and characteristic polynomials.

Felix Klein
Sophus Lie

Secondly, the systematic use of groups in geometry, mainly in the guise of symmetry groups, was initiated by Felix Klein's 1872 Erlangen program. The study of what are now called Lie groups started systematically in 1884 with Sophus Lie, followed by work of Wilhelm Killing, Eduard Study, Issai Schur, Ludwig Maurer, and Élie Cartan. The discontinuous (discrete group) theory was built up by Klein, Lie, Henri Poincaré, and Charles Émile Picard, in connection in particular with modular forms and monodromy.

Appearance of groups in number theory

Ernst Kummer

The third root of group theory was number theory. Leonhard Euler considered algebraic operations on numbers modulo an integer—modular arithmetic—in his generalization of Fermat's little theorem. These investigations were taken much further by Carl Friedrich Gauss, who considered the structure of multiplicative groups of residues mod n and established many properties of cyclic and more general abelian groups that arise in this way. In his investigations of composition of binary quadratic forms, Gauss explicitly stated the associative law for the composition of forms. In 1870, Leopold Kronecker gave a definition of an abelian group in the context of ideal class groups of a number field, generalizing Gauss's work. Ernst Kummer's attempts to prove Fermat's Last Theorem resulted in work introducing groups describing factorization into prime numbers. In 1882, Heinrich M. Weber realized the connection between permutation groups and abelian groups and gave a definition that included a two-sided cancellation property but omitted the existence of the inverse element, which was sufficient in his context (finite groups).

Convergence

Camille Jordan

Group theory as an increasingly independent subject was popularized by Serret, who devoted section IV of his algebra to the theory; by Camille Jordan, whose Traité des substitutions et des équations algébriques (1870) is a classic; and to Eugen Netto (1882), whose Theory of Substitutions and its Applications to Algebra was translated into English by Cole (1892). Other group theorists of the 19th century were Joseph Louis François Bertrand, Charles Hermite, Ferdinand Georg Frobenius, Leopold Kronecker, and Émile Mathieu; as well as William Burnside, Leonard Eugene Dickson, Otto Hölder, E. H. Moore, Ludwig Sylow, and Heinrich Martin Weber.

The convergence of the above three sources into a uniform theory started with Jordan's Traité and Walther von Dyck (1882) who first defined a group in the full modern sense. The textbooks of Weber and Burnside helped establish group theory as a discipline. The abstract group formulation did not apply to a large portion of 19th century group theory, and an alternative formalism was given in terms of Lie algebras.

Late 19th century

Groups in the 1870-1900 period were described as the continuous groups of Lie, the discontinuous groups, finite groups of substitutions of roots (gradually being called permutations), and finite groups of linear substitutions (usually of finite fields). During the 1880-1920 period, groups described by presentations came into a life of their own through the work of Cayley, Walther von Dyck, Max Dehn, Jakob Nielsen, Otto Schreier, and continued in the 1920-1940 period with the work of H. S. M. Coxeter, Wilhelm Magnus, and others to form the field of combinatorial group theory.

Finite groups in the 1870-1900 period saw such highlights as the Sylow theorems, Hölder's classification of groups of square-free order, and the early beginnings of the character theory of Frobenius. Already by 1860, the groups of automorphisms of the finite projective planes had been studied (by Mathieu), and in the 1870s Klein's group-theoretic vision of geometry was being realized in his Erlangen program. The automorphism groups of higher dimensional projective spaces were studied by Jordan in his Traité and included composition series for most of the so-called classical groups, though he avoided non-prime fields and omitted the unitary groups. The study was continued by Moore and Burnside, and brought into comprehensive textbook form by Leonard Dickson in 1901. The role of simple groups was emphasized by Jordan, and criteria for non-simplicity were developed by Hölder until he was able to classify the simple groups of order less than 200. The study was continued by Frank Nelson Cole (up to 660) and Burnside (up to 1092), and finally in an early "millennium project", up to 2001 by Miller and Ling in 1900.

Continuous groups in the 1870-1900 period developed rapidly. Killing and Lie's foundational papers were published, Hilbert's theorem in invariant theory 1882, etc.

Early 20th century

In the period 1900–1940, infinite "discontinuous" (now called discrete groups) groups gained life of their own. Burnside's famous problem ushered in the study of arbitrary subgroups of finite-dimensional linear groups over arbitrary fields, and indeed arbitrary groups. Fundamental groups and reflection groups encouraged the developments of J. A. Todd and Coxeter, such as the Todd–Coxeter algorithm in combinatorial group theory. Algebraic groups, defined as solutions of polynomial equations (rather than acting on them, as in the earlier century), benefited heavily from the continuous theory of Lie. Bernard Neumann and Hanna Neumann produced their study of varieties of groups, groups defined by group theoretic equations rather than polynomial ones.

Continuous groups also had explosive growth in the 1900-1940 period. Topological groups began to be studied as such. There were many great achievements in continuous groups: Cartan's classification of semisimple Lie algebras, Hermann Weyl's theory of representations of compact groups, Alfréd Haar's work in the locally compact case.

Finite groups in the 1900-1940 grew immensely. This period witnessed the birth of character theory by Frobenius, Burnside, and Schur which helped answer many of the 19th century questions in permutation groups, and opened the way to entirely new techniques in abstract finite groups. This period saw the work of Philip Hall: on a generalization of Sylow's theorem to arbitrary sets of primes which revolutionized the study of finite soluble groups, and on the power-commutator structure of p-groups, including the ideas of regular p-groups and isoclinism of groups, which revolutionized the study of p-groups and was the first major result in this area since Sylow. This period saw Hans Zassenhaus's famous Schur-Zassenhaus theorem on the existence of complements to Hall's generalization of Sylow subgroups, as well as his progress on Frobenius groups, and a near classification of Zassenhaus groups.

Mid-20th century

Both depth, breadth and also the impact of group theory subsequently grew. The domain started branching out into areas such as algebraic groups, group extensions, and representation theory. Starting in the 1950s, in a huge collaborative effort, group theorists succeeded to classify all finite simple groups in 1982. Completing and simplifying the proof of the classification are areas of active research.

Anatoly Maltsev also made important contributions to group theory during this time; his early work was in logic in the 1930s, but in the 1940s he proved important embedding properties of semigroups into groups, studied the isomorphism problem of group rings, established the Malçev correspondence for polycyclic groups, and in the 1960s return to logic proving various theories within the study of groups to be undecidable. Earlier, Alfred Tarski proved elementary group theory undecidable.

The period of 1960-1980 was one of excitement in many areas of group theory.

In finite groups, there were many independent milestones. One had the discovery of 22 new sporadic groups, and the completion of the first generation of the classification of finite simple groups. One had the influential idea of the Carter subgroup, and the subsequent creation of formation theory and the theory of classes of groups. One had the remarkable extensions of Clifford theory by Green to the indecomposable modules of group algebras. During this era, the field of computational group theory became a recognized field of study, due in part to its tremendous success during the first generation classification.

In discrete groups, the geometric methods of Jacques Tits and the availability the surjectivity of Serge Lang's map allowed a revolution in algebraic groups. The Burnside problem had tremendous progress, with better counterexamples constructed in the 1960s and early 1980s, but the finishing touches "for all but finitely many" were not completed until the 1990s. The work on the Burnside problem increased interest in Lie algebras in exponent p, and the methods of Michel Lazard began to see a wider impact, especially in the study of p-groups.

Continuous groups broadened considerably, with p-adic analytic questions becoming important. Many conjectures were made during this time, including the coclass conjectures.

Late 20th century

The last twenty years of the 20th century enjoyed the successes of over one hundred years of study in group theory.

In finite groups, post classification results included the O'Nan–Scott theorem, the Aschbacher classification, the classification of multiply transitive finite groups, the determination of the maximal subgroups of the simple groups and the corresponding classifications of primitive groups. In finite geometry and combinatorics, many problems could now be settled. The modular representation theory entered a new era as the techniques of the classification were axiomatized, including fusion systems, Luis Puig's theory of pairs and nilpotent blocks. The theory of finite soluble groups was likewise transformed by the influential book of Klaus Doerk and Trevor Hawkes which brought the theory of projectors and injectors to a wider audience.

In discrete groups, several areas of geometry came together to produce exciting new fields. Work on knot theory, orbifolds, hyperbolic manifolds, and groups acting on trees (the Bass–Serre theory), much enlivened the study of hyperbolic groups, automatic groups. Questions such as William Thurston's 1982 geometrization conjecture, inspired entirely new techniques in geometric group theory and low-dimensional topology, and was involved in the solution of one of the Millennium Prize Problems, the Poincaré conjecture.

Continuous groups saw the solution of the problem of hearing the shape of a drum in 1992 using symmetry groups of the laplacian operator. Continuous techniques were applied to many aspects of group theory using function spaces and quantum groups. Many 18th and 19th century problems are now revisited in this more general setting, and many questions in the theory of the representations of groups have answers.

Today

Group theory continues to be an intensely studied matter. Its importance to contemporary mathematics as a whole may be seen from the 2008 Abel Prize, awarded to John Griggs Thompson and Jacques Tits for their contributions to group theory.

Giambattista della Porta

From Wikipedia, the free encyclopedia
 
Giambattista della Porta
Giambattista della Porta in 1589
Bornlate 1535
Died4 February 1615 (aged 79)
Naples, Kingdom of Naples
NationalityItalian
Scientific career
FieldsOccultism, astrology, alchemy, mathematics, meteorology, and natural philosophy

Giambattista della Porta (Italian pronunciation: [dʒambatˈtista della ˈpɔrta]; 1535 – 4 February 1615), also known as Giovanni Battista Della Porta, was an Italian scholar, polymath and playwright who lived in Naples at the time of the Renaissance, Scientific Revolution and Counter-Reformation.

Giambattista della Porta spent the majority of his life on scientific endeavours. He benefited from an informal education of tutors and visits from renowned scholars. His most famous work, first published in 1558, is entitled Magia Naturalis (Natural Magic). In this book he covered a variety of the subjects he had investigated, including occult philosophy, astrology, alchemy, mathematics, meteorology, and natural philosophy. He was also referred to as "professor of secrets".

Childhood

Giambattista della Porta was born at Vico Equense, near Naples, to the nobleman Nardo Antonio della Porta. He was the third of four sons and the second to survive childhood, having an older brother Gian Vincenzo and a younger brother Gian Ferrante. Della Porta had a privileged childhood including his education. His father had a thirst for learning, a trait he would pass on to all of his children. He surrounded himself with distinguished people and entertained the likes of philosophers, mathematicians, poets, and musicians. The atmosphere of the house resembled an academy for his sons. The members of the learned circle of friends stimulated the boys, tutoring and mentoring them, under the strict guidance of their father.

In addition to having talents for the sciences and mathematics, all the brothers were also extremely interested in the arts, music in particular. Despite their interest, none of them possessed any sort of talent for it, but they did not allow that to stifle their progress in learning theory. They were all accepted into the Scuola di Pitagora, a highly exclusive academy of musicians.

More aware of their social position than the idea that his sons could have professions in science, Nardo Antonio raised the boys more as gentlemen than as scholars. Therefore, the boys struggled to learn to sing, as that was considered a courtly accomplishment of gentlemen. They were taught to dance, ride, perform well in tournaments and games, and dress well. The training gave della Porta, at least earlier in his life, a taste for the finer aspects of privileged living.

Scientific disciplines

From De humana physiognomonia, 1586

In 1563, della Porta published De Furtivis Literarum Notis, a work about cryptography. In it, he described the first known digraphic substitution cipher. Charles J. Mendelsohn commented:

He was, in my opinion, the outstanding cryptographer of the Renaissance. Some unknown who worked in a hidden room behind closed doors may possibly have surpassed him in general grasp of the subject, but among those whose work can be studied he towers like a giant.

Della Porta invented a method which allowed him to write secret messages on the inside of eggs. Some of his friends were imprisoned by the Inquisition. At the gate of the prison, everything was checked except for eggs. Della Porta wrote messages on the eggshell using a mixture made of plant pigments and alum. The ink penetrated the eggshell which is semi-porous. When the eggshell was dry, he boiled the egg in hot water and the ink on the outside of the egg was washed away. When the recipient in prison peeled off the shell, the message was revealed once again on the egg white. De Furtivis Literarum Notis also contains one of the earliest known examples of music substitution ciphers.

In 1586 della Porta published a work on physiognomy, De humana physiognomonia libri IIII (1586). This influenced the Swiss eighteenth-century pastor Johann Kaspar Lavater as well as the 19th-century criminologist Cesare Lombroso. Della Porta wrote extensively on a wide spectrum of subjects throughout his life – for instance, an agricultural encyclopedia entitled "Villa" as well as works on meteorology, optics, and astronomy.

Phytognomonica, 1588

In 1589, on the eve of the early modern Scientific Revolution, della Porta became the first person to attack in print, on experimental grounds, the ancient assertion that garlic could disempower magnets. This was an early example of the authority of early authors being replaced by experiment as the backing for a scientific assertion. Della Porta's conclusion was confirmed experimentally by Thomas Browne, among others.

In later life, della Porta collected rare specimens and grew exotic plants. His work Phytognomonica lists plants according to their geographical location. In Phytognomonica the first observation of fungal spores is recorded, making him a pioneer of mycology.

His private museum was visited by travellers and was one of the earliest examples of natural history museums. It inspired the Jesuit Athanasius Kircher to begin a similar, even more renowned, collection in Rome.

Pioneering scientific society

Della Porta was the founder of a scientific society called the Academia Secretorum Naturae (Accademia dei Segreti). This group was more commonly known as the Otiosi, (Men of Leisure). Founded sometime before 1580, the Otiosi were one of the first scientific societies in Europe and their aim was to study the "secrets of nature." Any person applying for membership had to demonstrate they had made a new discovery in the natural sciences.

The Academia Secretorum Naturae was compelled to disband when its members were suspected of dealing with the occult. Della Porta was summoned to Rome by Pope Gregory XIII. Though he personally emerged from the meeting unscathed, the Academia Secretorum Naturae disbanded. Despite this incident, della Porta remained religiously devout and became a lay Jesuit brother.

Della Porta joined The Academy of the Lynxes in 1610.

Technological contributions

Chemical apparatus for a still from De distillatione, 1608

His interest in a variety of disciplines resulted in the technological advances of the following: agriculture, hydraulics, Military Engineering, instruments, and pharmacology. He published a book in 1606 on raising water by the force of the air. In 1608 he published a book on military engineering, and another on distillation.

Additionally, della Porta perfected the camera obscura. In a later edition of his Natural Magic, della Porta described this device as having a convex lens. Though he was not the inventor, the popularity of this work helped spread knowledge of it. He compared the shape of the human eye to the lens in his camera obscura, and provided an easily understandable example of how light could bring images into the eye.

Della Porta also claimed to have invented the first telescope, but died while preparing the treatise (De telescopiis) in support of his claim. His efforts were also overshadowed by Galileo Galilei's improvement of the telescope in 1609, following its introduction by Lippershey in the Netherlands in 1608.

In the book, della Porta also mentioned an imaginary device known as a sympathetic telegraph. The device consisted of two circular boxes, similar to compasses, each with a magnetic needle, supposed to be magnetized by the same lodestone. Each box was to be labelled with the 26 letters, instead of the usual directions. Della Porta assumed that this would coordinate the needles such that when a letter was dialled in one box, the needle in the other box would swing to point to the same letter, thereby helping in communicating.

Religious complications

A Catholic, della Porta was examined by the Inquisition in the years prior to 1578. He was forced to disband his Academia Secretorum Naturae, and in 1592 his philosophical works were prohibited from further publication by the Church; the ban was lifted in 1598. Porta's involvement with the Inquisition puzzles historians due to his active participation in charitable Jesuit works by 1585. A possible explanation for this lies in Porta's personal relations with Fra Paolo Sarpi after 1579.

Playwright

The 17 theatrical works that have survived from a total of perhaps 21 or 23 works comprise 14 comedies, one tragicomedy, one tragedy and one liturgical drama.

Comedies

  • Lo Astrologo;
  • La Carbonaria;
  • La Chiappinaria;
  • La Cintia;
  • Gli Duoi fratelli rivali;
  • La Fantesca;
  • La Furiosa;
  • Il Moro;
  • L'Olimpia;
  • I Simili;
  • La Sorella;
  • La Tabernaria;
  • La Trappolaria;
  • La Turca

Others

Although they belong to the lesser-known tradition of the commedia erudita rather than the commedia dell'arte - which means they were written out as entire scripts instead of being improvised from a scenario - della Porta's comedies are eminently performable. While there are obvious similarities between some of the characters in della Porta's comedies and the masks of the commedia dell'arte, it should be borne in mind that the characters of the commedia erudita are uniquely created by the text in which they appear, unlike the masks, which remain constant from one scenario to another. Indeed, the masks of the improvised theatre evolved as stylised versions of recurring character types in written comedies. One of Della Porta's most notable stock characters was the parasito or parassita, a gluttonous trickster whose lack of moral scruples enabled him to pull off stunts that initially might risk bringing the plot crashing down, but ended up winning the day in unexpected ways. The term parasito was translated by John Florio in his Italian to English Dictionary first published in 1598 as a smell-feast, a flatterer, a parasite, a trencherd or bellie friend, one that saieth and doeth all things to please the humor of another, and agreeth unto him in all things to have his repast scotfree. Perhaps the best example of this type is Morfeo in the comedy La Fantesca.

Censorship in the United States

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