The History of Western Philosophy
Bertrand Russell
Ch. XXIV Early Greek Mathematics and Astronomy
The Greeks took a little from the Egyptians and a little from the Babylonians and then developed some sound mathematics.
Euclid was still regarded as the sole textbook on geometry when Bertrand Russell was a schoolboy.
"Euclid Elements is certainly one of the greatest books ever written, and one of the most perfect
monuments of the Greek intellect. It has, of course, the typical Greek limitations: the method is
purely deductive, and there is no way, within it, of testing the initial assumptions. These
assumptions were supposed to be unquestionable, but in the nineteenth century non-Euclidean
geometry showed that they might be in part mistaken, and that only observation could decide
whether they were so" (Russell, 211)
Like Plato, Euclid had a contempt for the practical utility of his work.
The Buridan's ass argument: two bundles of hay were placed at equal distances to the right and left of the donkey. Therefore, he died of hunger. That could be mined for an interesting short story idea.
In astronomy the Greeks made leaps and strides, with some even moving away from an anthropocentric universe. Of course, it wasn't quite heliocentric yet, but the notion of earth being just one of the planets moving in space paved the way from a heliocentric theory.
Heraclides, a pupil of Plato's school postulated that the earth rotates once on its own axis every twenty-four hours and discovered that Venus and Mercury revolve around the son.
Aristarchus of Samos (310 - 230 BC) "is the most interesting of all ancient astronomers, because he advanced the complete Copernican hypothesis, that all the planets, including the earth, revolve in circles around the sun, and that the earth rotates on its axis once in twenty-four hours" (Russell, 214).
Seleucus, Aristarchus' successor fully adhered to this theory as well, but then was not adopted by any other ancient astronomer.
Eratosthenes estimated the earth's diameter within 50 miles of the actual diameter.
Showing posts with label Euclid. Show all posts
Showing posts with label Euclid. Show all posts
Thursday, September 19, 2019
Tuesday, August 4, 2015
Axiomatic Geometry - The Classical Mind
The Classical Mind
W.T. Jones
1980
Axiomatic Geometry
During the 6th century BC the Greeks began to take what knowledge of Egyptian practical geometry they had and systematize it into a system for the desire of pure knowledge, that is, they didn't study it for any particular utility.
The way axiomatic geometry worked for the Greeks was to start at self-evident principles and then to use logical deductive reasoning to come up with axioms. These axioms were then combined logically to create further axioms and conclusions.
Euclid in 300 BC published Elements. "In Euclid's book, the first theorems are proved by showing that they follow directly from the axioms; then, by judicious combinations of already proved theorems with the axioms, more complex theorems are derived. In this systematic way Euclid was able to 'demonstrate' a large body of geometrical knowledge" (Jones, 20).
The effect that axiomatic geometry had on the Greek thought was far more profound than shaping mathematics. It focused Greek thought so deeply on reasoning that reason became preeminent and far more important than sensory perception.
W.T. Jones
1980
Axiomatic Geometry
During the 6th century BC the Greeks began to take what knowledge of Egyptian practical geometry they had and systematize it into a system for the desire of pure knowledge, that is, they didn't study it for any particular utility.
The way axiomatic geometry worked for the Greeks was to start at self-evident principles and then to use logical deductive reasoning to come up with axioms. These axioms were then combined logically to create further axioms and conclusions.
Euclid in 300 BC published Elements. "In Euclid's book, the first theorems are proved by showing that they follow directly from the axioms; then, by judicious combinations of already proved theorems with the axioms, more complex theorems are derived. In this systematic way Euclid was able to 'demonstrate' a large body of geometrical knowledge" (Jones, 20).
The effect that axiomatic geometry had on the Greek thought was far more profound than shaping mathematics. It focused Greek thought so deeply on reasoning that reason became preeminent and far more important than sensory perception.
Wednesday, August 30, 2006
Homer, Hesiod, The Milesians, Heraclitus & Xenophanes
(Originally written August 30, 2006 in Book 10)
Book 10: History of Philosophy I
This note book, or at least the beginning of this notebook is dedicated to the readings for History of Philosophy I.
The Classical Mind
W.T. Jones
Chapter One: "Pre-Socratic Philosophy"
When did Western Philosophy begin?
-In the 6th century B.C. with Thales. Thales is the father of Greek and thus, Western philosophy.
-"But, the history of philosophy itself does not have a definitive starting point" (Jones, 1).
It's impossible to pinpoint a time when there was no philosophy; thus no pinpoint of a beginning. Philosophy began as a gradual continuum and exists in that state today.
God and Nature in Homer
Zeus is not the omnipotent God in Christianity. He is "dominated by the members of his household" (Jones, 4).
The Homeric State was a monarchy, but no king was to be absolute. The nobility and warrior class played a part in limiting the king's power. But, a king often overrode the nobility and nobody complained.
Nature was not viewed as cyclical by Homer. Yes, there were regular occurrences, but when irregularities occurred they were attributed to the gods.
The gods were impulsive and that nature was the cause of life's irregularities.
While the gods were impulsive they operated under some form of rationale. (i.e. they punished Achilles for his excessive anger).
Lack of moderation and hubris (insubordination) were great iniquities for Homer. Arrogance came out of men of excess or hubris. Arrogant men were rarely tolerated by the gods.
In the Illiad Achilles is punished for his lack of moderation: Agamemnon is not punished for his infliction of misery and suffering, but for his violation of the god's regulations.
Homer does not make the gods to be moral role models. He does not believe the divine rules were established by the gods for the benefit of man.
The gods were egotistical, lustful, selfish, vain, unscrupulous, dishonest and childish.
Homer's men worship the gods because they were powerful, not because they are good.
Worship is a business transaction between a god and a man.
Homer's gods are causal agents. They are the cause of the regular state of affairs and the disruptive irregularities of life as well.
Above the dos is a blind will known as fate. Fate was above both gods and men. It was impersonal and unresponsive to anybody's volition.
God and Nature in Hesiod
Hesiod was probably from the 8th century B.C.
To Hesiod, Zeus was justice and justice would eventually punish the wicked and make restitution for the abused.
Homer saw the greatest sin of man being hubris (insubordination) or "not knowing one's place". Hesiod contrastingly saw man's worst iniquity as the oppression of the weak by the strong.
Zeus was moral and powerful enough to enforce the divine law in Hesiod, not the impotent, immoral god of Homer.
"Fate became the concept of a pervasive moral law" in Hesiod. (Jones, 6).
Hesiod sets man apart from the rest of nature. The Greeks took pride in their humanity and believed that humans had to live up to a higher standard than animals, a human responsibility.
Moderation is one of the great differentiators between humans and animals.
Despite Hesiod's innovations, his cosmogony is very anthropomorphic.
Thales
Thales was from Miletus, a Greek colony in Ionia (on the coast of Asia Minor)
He predicted an eclipse in 585 B.C.
He believed water was the cause of all things and that all things were filled with god or filled with soul.
Thales marked a chance for science to be founded; prior to him everything was attributed to the gods. Thales' belief of water as the cause of all things shifts causation from the supernatural to the natural and thus, under the scope of science.
Thales is remembered because he was the first man to answer the question: Why do things happen as they do? without giving a total answer of because of the gods...
Thales assumed that:
1) There is some one thing that was the cause of everything else (monism is born)
2) That the cause was in fact a thing or the "ultimate stuff"
3) The ultimate stuff is active and contains an internal principle of change
The third assumption is what Thales meant by stating that things are full of gods. Gods were causal agents in Ancient Greece. Thus, Thales was stating things happen because "things" change. Since things were causal agents and the common religion held gods to be the sole causal agents, by believing things to be causal agents he called them gods.
Thales marks a chance for an expansion of knowledge for Ancient Greece. It is actualized when Thales' successors acutely critiqued each others' hypotheses and moved from the stage of myth to philosophy.
Anaximander
Anaximander was a Milesian and a younger contemporary of Thales.
He wrote a book and a single sentence survived: "From what source things arise, to that they return of necessity when they are destroyed; for they suffer punishment and make reparation to one another for their injustice according to the order of time" (Jones, 11).
Anaximander agreed with Thales in that:
1) There is a one "stuff"
2) There is a process by which this one stuff becomes the many individual things that exist.
He also added a third:
3) This process is a necessary one
HE believed this first stuff was not water or any other element, but something preceding the elements and this substance was "boundless"
Anaximander contended that water couldn't be the first substance because things that are dry would have to be made of water. This is a logical contradiction and Anaximander exploited it.
Anaximander noticed that the world existed in opposites: hot-cold, wet-dry, etc. Each of these opposites would become its own opposite.
A substance returns to the boundless and reemerges as the opposite or some in betwixt substance.
He believed the world to be overall disorderly and felt the orderliness of the world was an abnormality and needed an explanation.
Anaximander put forth a precursor to modern evolutionary theories to explain how the boundless stuff became the many things in existence.
Anaximenes
Anaximenes was the third Milesian philosopher and critiqued Anaximander's theory the way Anaximander critiqued Thales' theory.
Like the other two Milesians, Anaximenes believed in a one stuff.
Anaximenes doubted the boundless stuff because it had no characteristics. He rejected it because Anaximander's view held conflicting positions: a material evolutionary process and a boundless infinite stuff are incompatible.
He explained that qualitative changes were changes in the destiny of air:
Air dilated becomes fire
Air condensed becomes Wind
Wind condensed becomes water
Water condensed becomes earth
Earth condensed becomes rock
Neither Thales nor Anaximander nor Anaximenes knew anything of scientific experimentation and thus never tested their theories.
The Milesian philosophers were ignorant of scientific method (as were all Greeks) but the Milesians laid the foundation for the scientific method.
Heraclitus
Heraclitus was another Ionian.
HE held that the world was composed of an everlasting fire.
Heraclitus believed the problem of one substance becoming many substances was an insoluble one, as long as the "one substance" is taken to be a material thing.
He held that his one substance, the everlasting fire, was not a material thing, but a process of perpetual change.
Heraclitus proposed everything was in a constant state of flux. He then stated things appear to be still, but that is because the flux is constant. Heraclitus developed the notion of appearances and reality.
Moral and Social Theory
The Milesians were focused on the nature of the physical world. Heraclitus was primarily concerned with man and his destiny.
He believed the social world existed in flux, just like the physical world.
He held war to be the "king of all" and believed strife to be justice. He held this because without strife there can be no peace.
Strife was the hidden tensions between opposites.
His contempt oft he public masses can be summed up in his aphorism: "Asses would rather have straw than gold" (Jones, 17).
Religion
The men who accepted the Milesian science rejected the old religion of Homer and Hesiod. Heraclitus was Milesian in thought and looked down on the public for their adherence to the old religion.
Heraclitus was not an atheist. He believed god was connected to the everlasting fire. He believed this god to be indifferent to men.
His rejection of a personal god, but acceptance of the universal nature of god laid the foundation for the concept of natural law.
Despite his rejection of common Greek religion and a personal deity, his theological concepts of "logos" and man's chief good as being "listening to the logos" were absorbed into Stoicism and Christianity.
Logos was the process for Heraclitus.
He was both a scientific and religious philosopher.
Xenophanes
Xenophanes discovered every man makes god in his own image.
Xenophanes was an Ionian and followed Heraclitus in his religious views.
Xenophanes believed in one God who is not in form or thought like men. This God sees all. But, he was probably still a pantheist like Heraclitus.
The religion of Xenophanes and Heraclitus cut away at the ancient myth religion of Greece and caused the undermining of the Greek society. The collapse of the ancient religion led to the decline of morality in Ancient Greece.
Axiomatic Geometry
The Greeks took a basic geometric understanding from he Egyptians and developed it.
Axiomatic Geometry slowly formed over the 6th and 5th B.C. The Earliest complete text is Euclid's Elements written about 300 BC.
Axiomatic Geometry starts with self-evident statements (axioms) then uses logical reasoning to produce theorems. Theorems are then combined with one another to form more complex theorems and so on. Euclid had a large demonstrable book by the end of his reasoning.
Axiomatic geometry pushed Greeks to rely on reasoning rather than sense perception.
Euclid's theorems were not challenged until the 19th century. "Most of Western Philosophers have regarded it as the paradigm of human reasoning" (Jones, 20).
Book 10: History of Philosophy I
This note book, or at least the beginning of this notebook is dedicated to the readings for History of Philosophy I.
The Classical Mind
W.T. Jones
Chapter One: "Pre-Socratic Philosophy"
When did Western Philosophy begin?
-In the 6th century B.C. with Thales. Thales is the father of Greek and thus, Western philosophy.
-"But, the history of philosophy itself does not have a definitive starting point" (Jones, 1).
It's impossible to pinpoint a time when there was no philosophy; thus no pinpoint of a beginning. Philosophy began as a gradual continuum and exists in that state today.
God and Nature in Homer
Zeus is not the omnipotent God in Christianity. He is "dominated by the members of his household" (Jones, 4).
The Homeric State was a monarchy, but no king was to be absolute. The nobility and warrior class played a part in limiting the king's power. But, a king often overrode the nobility and nobody complained.
Nature was not viewed as cyclical by Homer. Yes, there were regular occurrences, but when irregularities occurred they were attributed to the gods.
The gods were impulsive and that nature was the cause of life's irregularities.
While the gods were impulsive they operated under some form of rationale. (i.e. they punished Achilles for his excessive anger).
Lack of moderation and hubris (insubordination) were great iniquities for Homer. Arrogance came out of men of excess or hubris. Arrogant men were rarely tolerated by the gods.
In the Illiad Achilles is punished for his lack of moderation: Agamemnon is not punished for his infliction of misery and suffering, but for his violation of the god's regulations.
Homer does not make the gods to be moral role models. He does not believe the divine rules were established by the gods for the benefit of man.
The gods were egotistical, lustful, selfish, vain, unscrupulous, dishonest and childish.
Homer's men worship the gods because they were powerful, not because they are good.
Worship is a business transaction between a god and a man.
Homer's gods are causal agents. They are the cause of the regular state of affairs and the disruptive irregularities of life as well.
Above the dos is a blind will known as fate. Fate was above both gods and men. It was impersonal and unresponsive to anybody's volition.
God and Nature in Hesiod
Hesiod was probably from the 8th century B.C.
To Hesiod, Zeus was justice and justice would eventually punish the wicked and make restitution for the abused.
Homer saw the greatest sin of man being hubris (insubordination) or "not knowing one's place". Hesiod contrastingly saw man's worst iniquity as the oppression of the weak by the strong.
Zeus was moral and powerful enough to enforce the divine law in Hesiod, not the impotent, immoral god of Homer.
"Fate became the concept of a pervasive moral law" in Hesiod. (Jones, 6).
Hesiod sets man apart from the rest of nature. The Greeks took pride in their humanity and believed that humans had to live up to a higher standard than animals, a human responsibility.
Moderation is one of the great differentiators between humans and animals.
Despite Hesiod's innovations, his cosmogony is very anthropomorphic.
Thales
Thales was from Miletus, a Greek colony in Ionia (on the coast of Asia Minor)
He predicted an eclipse in 585 B.C.
He believed water was the cause of all things and that all things were filled with god or filled with soul.
Thales marked a chance for science to be founded; prior to him everything was attributed to the gods. Thales' belief of water as the cause of all things shifts causation from the supernatural to the natural and thus, under the scope of science.
Thales is remembered because he was the first man to answer the question: Why do things happen as they do? without giving a total answer of because of the gods...
Thales assumed that:
1) There is some one thing that was the cause of everything else (monism is born)
2) That the cause was in fact a thing or the "ultimate stuff"
3) The ultimate stuff is active and contains an internal principle of change
The third assumption is what Thales meant by stating that things are full of gods. Gods were causal agents in Ancient Greece. Thus, Thales was stating things happen because "things" change. Since things were causal agents and the common religion held gods to be the sole causal agents, by believing things to be causal agents he called them gods.
Thales marks a chance for an expansion of knowledge for Ancient Greece. It is actualized when Thales' successors acutely critiqued each others' hypotheses and moved from the stage of myth to philosophy.
Anaximander
Anaximander was a Milesian and a younger contemporary of Thales.
He wrote a book and a single sentence survived: "From what source things arise, to that they return of necessity when they are destroyed; for they suffer punishment and make reparation to one another for their injustice according to the order of time" (Jones, 11).
Anaximander agreed with Thales in that:
1) There is a one "stuff"
2) There is a process by which this one stuff becomes the many individual things that exist.
He also added a third:
3) This process is a necessary one
HE believed this first stuff was not water or any other element, but something preceding the elements and this substance was "boundless"
Anaximander contended that water couldn't be the first substance because things that are dry would have to be made of water. This is a logical contradiction and Anaximander exploited it.
Anaximander noticed that the world existed in opposites: hot-cold, wet-dry, etc. Each of these opposites would become its own opposite.
A substance returns to the boundless and reemerges as the opposite or some in betwixt substance.
He believed the world to be overall disorderly and felt the orderliness of the world was an abnormality and needed an explanation.
Anaximander put forth a precursor to modern evolutionary theories to explain how the boundless stuff became the many things in existence.
Anaximenes
Anaximenes was the third Milesian philosopher and critiqued Anaximander's theory the way Anaximander critiqued Thales' theory.
Like the other two Milesians, Anaximenes believed in a one stuff.
Anaximenes doubted the boundless stuff because it had no characteristics. He rejected it because Anaximander's view held conflicting positions: a material evolutionary process and a boundless infinite stuff are incompatible.
He explained that qualitative changes were changes in the destiny of air:
Air dilated becomes fire
Air condensed becomes Wind
Wind condensed becomes water
Water condensed becomes earth
Earth condensed becomes rock
Neither Thales nor Anaximander nor Anaximenes knew anything of scientific experimentation and thus never tested their theories.
The Milesian philosophers were ignorant of scientific method (as were all Greeks) but the Milesians laid the foundation for the scientific method.
Heraclitus
Heraclitus was another Ionian.
HE held that the world was composed of an everlasting fire.
Heraclitus believed the problem of one substance becoming many substances was an insoluble one, as long as the "one substance" is taken to be a material thing.
He held that his one substance, the everlasting fire, was not a material thing, but a process of perpetual change.
Heraclitus proposed everything was in a constant state of flux. He then stated things appear to be still, but that is because the flux is constant. Heraclitus developed the notion of appearances and reality.
Moral and Social Theory
The Milesians were focused on the nature of the physical world. Heraclitus was primarily concerned with man and his destiny.
He believed the social world existed in flux, just like the physical world.
He held war to be the "king of all" and believed strife to be justice. He held this because without strife there can be no peace.
Strife was the hidden tensions between opposites.
His contempt oft he public masses can be summed up in his aphorism: "Asses would rather have straw than gold" (Jones, 17).
Religion
The men who accepted the Milesian science rejected the old religion of Homer and Hesiod. Heraclitus was Milesian in thought and looked down on the public for their adherence to the old religion.
Heraclitus was not an atheist. He believed god was connected to the everlasting fire. He believed this god to be indifferent to men.
His rejection of a personal god, but acceptance of the universal nature of god laid the foundation for the concept of natural law.
Despite his rejection of common Greek religion and a personal deity, his theological concepts of "logos" and man's chief good as being "listening to the logos" were absorbed into Stoicism and Christianity.
Logos was the process for Heraclitus.
He was both a scientific and religious philosopher.
Xenophanes
Xenophanes discovered every man makes god in his own image.
Xenophanes was an Ionian and followed Heraclitus in his religious views.
Xenophanes believed in one God who is not in form or thought like men. This God sees all. But, he was probably still a pantheist like Heraclitus.
The religion of Xenophanes and Heraclitus cut away at the ancient myth religion of Greece and caused the undermining of the Greek society. The collapse of the ancient religion led to the decline of morality in Ancient Greece.
Axiomatic Geometry
The Greeks took a basic geometric understanding from he Egyptians and developed it.
Axiomatic Geometry slowly formed over the 6th and 5th B.C. The Earliest complete text is Euclid's Elements written about 300 BC.
Axiomatic Geometry starts with self-evident statements (axioms) then uses logical reasoning to produce theorems. Theorems are then combined with one another to form more complex theorems and so on. Euclid had a large demonstrable book by the end of his reasoning.
Axiomatic geometry pushed Greeks to rely on reasoning rather than sense perception.
Euclid's theorems were not challenged until the 19th century. "Most of Western Philosophers have regarded it as the paradigm of human reasoning" (Jones, 20).
Saturday, July 29, 2006
The 12th Century (C)
(Originally written July 29, 2006 in Book 5)
Roscelin recanted the heresy at Rheims then fled to England. While in England he attacked Anselm, then fled to Rome.
Roscelin's writings are all lost except for a letter to Abelard on the Trinity. Roscelin mocks Abelard in the letter.
Ueberweg describes Roscelin as not a very nice man.
Roscelin's teachings are known through the writings of Abelard and Anselm.
Anselm states that Roscelin believed universals are "breath of voice" or that they physically occur when we speak them. Russell contends that Roscelin would not have believed something so foolish. Anselm states that Roscelin also believed man is not a unity, but a collection under a common name. Anselm stated that Roscelin believed reality is what is sensible. Roscelin believed (according to Anselm) that a whole is merely a word, but reality rests in the parts of a whole. He was led by this view to extreme atomism and troubles in dealing with the Trinity.
He held that the Godhead was three distinct substances and thus there are three Gods. He stated this because if they are one substance then all three had been incarnate in Jesus Christ. He couldn't accept that. But this is what he recanted in Rheims.
Abelard was Roselyn's pupil. He was born near Nantes in 1079. He first studied under the realist William of Champeaux in Paris. He taught at the Paris cathedral school where he argued against William's realism and compelled his former teacher to change his views. He left Paris to study theology under Anselm of Leon then returned to Paris in 1113. Upon returning he became widely popular as a teacher and became the lover of Heloïse, niece of Canon Fulbert. Canon Fulbert castrated him and he joined the monastery at St. Denis.
Abelard was vain, disputatious and tempts. After being castrated he was angry and humiliated. (Linehan - what a shock!)
He was condemned for an unorthodox book on the Trinity in 1121.
He then became the abbot of Saint Gildas in Brittany for four years. He hated it because he saw the monks as uncivilized.
In 1141 he was again condemned at the insistence of St. Bernard. He retired to Cluny and died in 1142.
Abelard's most famous book is "Yes and No". It is a dialectical argument for and against many theses. It did not however produce any conclusions.
Abelard held that, apart from Scriptures, dialectic is the sole road to truth.
He considered logic to be the chief science.
He is best known for logic and theory of knowledge.
Universals - "what can be predicated of many different things" (Russell, 430).
Abelard held that we do not predicate a thing, but only a word. He was a nomalist in this sense.
He fought realism.
He regards Platonic Ideas as the pattern of creation in the mind of God.
St. Bernard charged him without fully understanding Abelard's teachings. The heresies Bernard charged him with are:
1) Treating the Trinity like an Arian
2) Treating grace like a Pelagian
3) Treating the Person of Christ like a Nestorian
4) Being a heathen by making Plato to be a Christian
5) Destroying Christianity by maintaining that God can be understood through reason alone.
In all reality it was his hostile disposition towards clergy that caused his heretical label and not his teachings.
Abelard was more dialectical than any others at his time.
1) The school of Chartres was a humanistic movement
2) Abelard of Bath translated Euclid due to a renaissance of mathematical learning
3) There was a strong mystical movement headed by St. Bernard.
St. Bernard was not an intellectualist, but was extremely influential.
1) His father was a knight in the first Crusade
2) He was a Cistercian monk
3) He was the abbot at Clairvaux
4) He was influential in ecclesiastical politics
5) He fought antipopes
6) he combatted heresy in Northern Italy and Southern France
7) He brought philosophers down with the weight of orthodoxy
8) He set into motion the second Crusade
"Although a politician and a bigot, he was a man of genuinely religious temperament" (Russel, 439).
Bernard's reverence of the Papacy increased the power of the Pope.
Roscelin recanted the heresy at Rheims then fled to England. While in England he attacked Anselm, then fled to Rome.
Roscelin's writings are all lost except for a letter to Abelard on the Trinity. Roscelin mocks Abelard in the letter.
Ueberweg describes Roscelin as not a very nice man.
Roscelin's teachings are known through the writings of Abelard and Anselm.
Anselm states that Roscelin believed universals are "breath of voice" or that they physically occur when we speak them. Russell contends that Roscelin would not have believed something so foolish. Anselm states that Roscelin also believed man is not a unity, but a collection under a common name. Anselm stated that Roscelin believed reality is what is sensible. Roscelin believed (according to Anselm) that a whole is merely a word, but reality rests in the parts of a whole. He was led by this view to extreme atomism and troubles in dealing with the Trinity.
He held that the Godhead was three distinct substances and thus there are three Gods. He stated this because if they are one substance then all three had been incarnate in Jesus Christ. He couldn't accept that. But this is what he recanted in Rheims.
Abelard was Roselyn's pupil. He was born near Nantes in 1079. He first studied under the realist William of Champeaux in Paris. He taught at the Paris cathedral school where he argued against William's realism and compelled his former teacher to change his views. He left Paris to study theology under Anselm of Leon then returned to Paris in 1113. Upon returning he became widely popular as a teacher and became the lover of Heloïse, niece of Canon Fulbert. Canon Fulbert castrated him and he joined the monastery at St. Denis.
Abelard was vain, disputatious and tempts. After being castrated he was angry and humiliated. (Linehan - what a shock!)
He was condemned for an unorthodox book on the Trinity in 1121.
He then became the abbot of Saint Gildas in Brittany for four years. He hated it because he saw the monks as uncivilized.
In 1141 he was again condemned at the insistence of St. Bernard. He retired to Cluny and died in 1142.
Abelard's most famous book is "Yes and No". It is a dialectical argument for and against many theses. It did not however produce any conclusions.
Abelard held that, apart from Scriptures, dialectic is the sole road to truth.
He considered logic to be the chief science.
He is best known for logic and theory of knowledge.
Universals - "what can be predicated of many different things" (Russell, 430).
Abelard held that we do not predicate a thing, but only a word. He was a nomalist in this sense.
He fought realism.
He regards Platonic Ideas as the pattern of creation in the mind of God.
St. Bernard charged him without fully understanding Abelard's teachings. The heresies Bernard charged him with are:
1) Treating the Trinity like an Arian
2) Treating grace like a Pelagian
3) Treating the Person of Christ like a Nestorian
4) Being a heathen by making Plato to be a Christian
5) Destroying Christianity by maintaining that God can be understood through reason alone.
In all reality it was his hostile disposition towards clergy that caused his heretical label and not his teachings.
Abelard was more dialectical than any others at his time.
1) The school of Chartres was a humanistic movement
2) Abelard of Bath translated Euclid due to a renaissance of mathematical learning
3) There was a strong mystical movement headed by St. Bernard.
St. Bernard was not an intellectualist, but was extremely influential.
1) His father was a knight in the first Crusade
2) He was a Cistercian monk
3) He was the abbot at Clairvaux
4) He was influential in ecclesiastical politics
5) He fought antipopes
6) he combatted heresy in Northern Italy and Southern France
7) He brought philosophers down with the weight of orthodoxy
8) He set into motion the second Crusade
"Although a politician and a bigot, he was a man of genuinely religious temperament" (Russel, 439).
Bernard's reverence of the Papacy increased the power of the Pope.
Friday, July 14, 2006
Greek Math and Astronomy
(Originally written July 14, 2006 in Book 4)
The History of Western Philosophy
Bertrand Russell
Chapter 24 - Early Greek Mathematics and Astronomy
While the Greek advancements in philosophy, literature, and art can be judged on personal preferences, their advancements in geometry is unquestionable. Mathematical demonstration originated nearly entirely in Greece.
The Greeks discovered irrationals.
Then Eudoxus (408-355 BC) discovered the geometrical theory of proportion, (a:b=c:d, if ad=bc)
Archimedes was able to define π as being less than 3 1/7 and greater than 3 10/71.
Euclid, the preeminent geometer in Greece was from Alexandria and lived shortly after the death of Aristotle. Euclid studied geometry and mathematics for reasons most Greeks did, a pure love of theory and not for any practical usage.
Centuries of observations by the Egyptians and Babylonians laid the foundation for the marvelous advancements made in Astronomy by the Greeks.
Anaximander was the first to suppose that nothing held the earth up.
Pythagoras was the first to suppose the earth to be spherical.
Anaxagoras discovered the moon shines via reflection. (Greeks also believed that the sun shined via reflection as well).
Heraclides discovered the the earth rotated once a day on its axis.
Seleucus estimated the lunar month within one second of error.
Despite being off in figures, Ancient Greek astronomy was done correctly according to scientific standards. Had they possessed better tools of precision they may have nailed the numbers down exactly.
The History of Western Philosophy
Bertrand Russell
Chapter 24 - Early Greek Mathematics and Astronomy
While the Greek advancements in philosophy, literature, and art can be judged on personal preferences, their advancements in geometry is unquestionable. Mathematical demonstration originated nearly entirely in Greece.
The Greeks discovered irrationals.
Then Eudoxus (408-355 BC) discovered the geometrical theory of proportion, (a:b=c:d, if ad=bc)
Archimedes was able to define π as being less than 3 1/7 and greater than 3 10/71.
Euclid, the preeminent geometer in Greece was from Alexandria and lived shortly after the death of Aristotle. Euclid studied geometry and mathematics for reasons most Greeks did, a pure love of theory and not for any practical usage.
Centuries of observations by the Egyptians and Babylonians laid the foundation for the marvelous advancements made in Astronomy by the Greeks.
Anaximander was the first to suppose that nothing held the earth up.
Pythagoras was the first to suppose the earth to be spherical.
Anaxagoras discovered the moon shines via reflection. (Greeks also believed that the sun shined via reflection as well).
Heraclides discovered the the earth rotated once a day on its axis.
Seleucus estimated the lunar month within one second of error.
Despite being off in figures, Ancient Greek astronomy was done correctly according to scientific standards. Had they possessed better tools of precision they may have nailed the numbers down exactly.
Saturday, July 1, 2006
Chapter 3 - Pythagoras
(Originally written July 1, 2006 in Book 2)
The History of Western Philosophy
Bertrand Russell
1974
Chapter 3 - Pythagoras
Pythagoras was one of the most important men intellectually of all time. Mathematics as a demonstrative deductive argument began with Pythagoras.
Native of Samos circa 532 BC, Pythagoras was either a son of an important citizen or the child of Apollo.
Pythagoras disliked Polycrates, the tyrant of Samos and his artistic, immoral, pirate government and left. Pythagoras settled in Croton in Southern Italy, one of the most populated and greatest Greek cities in Italy. The Greek cities in Italy fought each other fiercely. Croton was known for its medicine.
Pythagoras founded a society of disciples that became influential in Croton. Eventually the citizens turned agains them and Pythagoras and his followers left for Metapontion Calso in Southern Italy where he died.
While he quickly became a legend after his death, he in reality founded a school of mathematics.
Pythagoras founded a religion based on the transmigration of souls and the sinfulness of eating beans.
Here are some of Pythagoras' rules:
"1. To abstain from beans
2. Not to pick up what has fallen.
3. Not to touch a white cock.
4. Not to break bread.
5. Not to step over a cross bar.
6. Not to stir the fire with iron.
7. Not to eat from a whole loaf.
8. Not to pluck a garland.
9. Not to sit on a quart measure.
10. Not to eat the heart.
11. Not to walk on highways
12. Not to let swallows share one's roof.
13. When the pot is taken off the fire, not to leave the mark of it in the ashes, but to stir them together.
14. Do not look in a mirror beside a light.
15. When you rise from the bed clothes, roll them together and smooth out the impress of the body" (Russell, 31-32).
Pythagoras believed:
1. The soul is immortal
2. The soul transformed into other kinds of things
3. Reincarnation of the soul
Like St. Francis, Pythagoras is reported to have preached to animals.
Pythagoras' society:
1. held men and women as equals
2. held common property
Three types of men according to Pythagoreans:
1. Low - the buyers and sellers
2. Middle - Participants
3. High - Onlookers/contemplators
Pythagoras was a disinterested scientist. Pythagoras devoted himself to the joy of enlightenment of theory and mathematics.
Pythagoras based his life and ethical theory on contemplation, pure thought.
Pythagoras' influence was huge on a number of people and movements: Plato, Descartes, Kant, Euclid, Christian Theology and Personal Religion.
Religion has been influenced by mathematics ever since Pythagoras. Kant, Plato, Aquinas, Augustine, Spinoza and others blended religion and reasoning.
Christianity derived the theory of Christ as the Word and led theology to produce rational proofs for God and morality because of Pythagoras.
The History of Western Philosophy
Bertrand Russell
1974
Chapter 3 - Pythagoras
Pythagoras was one of the most important men intellectually of all time. Mathematics as a demonstrative deductive argument began with Pythagoras.
Native of Samos circa 532 BC, Pythagoras was either a son of an important citizen or the child of Apollo.
Pythagoras disliked Polycrates, the tyrant of Samos and his artistic, immoral, pirate government and left. Pythagoras settled in Croton in Southern Italy, one of the most populated and greatest Greek cities in Italy. The Greek cities in Italy fought each other fiercely. Croton was known for its medicine.
Pythagoras founded a society of disciples that became influential in Croton. Eventually the citizens turned agains them and Pythagoras and his followers left for Metapontion Calso in Southern Italy where he died.
While he quickly became a legend after his death, he in reality founded a school of mathematics.
Pythagoras founded a religion based on the transmigration of souls and the sinfulness of eating beans.
Here are some of Pythagoras' rules:
"1. To abstain from beans
2. Not to pick up what has fallen.
3. Not to touch a white cock.
4. Not to break bread.
5. Not to step over a cross bar.
6. Not to stir the fire with iron.
7. Not to eat from a whole loaf.
8. Not to pluck a garland.
9. Not to sit on a quart measure.
10. Not to eat the heart.
11. Not to walk on highways
12. Not to let swallows share one's roof.
13. When the pot is taken off the fire, not to leave the mark of it in the ashes, but to stir them together.
14. Do not look in a mirror beside a light.
15. When you rise from the bed clothes, roll them together and smooth out the impress of the body" (Russell, 31-32).
Pythagoras believed:
1. The soul is immortal
2. The soul transformed into other kinds of things
3. Reincarnation of the soul
Like St. Francis, Pythagoras is reported to have preached to animals.
Pythagoras' society:
1. held men and women as equals
2. held common property
Three types of men according to Pythagoreans:
1. Low - the buyers and sellers
2. Middle - Participants
3. High - Onlookers/contemplators
Pythagoras was a disinterested scientist. Pythagoras devoted himself to the joy of enlightenment of theory and mathematics.
Pythagoras based his life and ethical theory on contemplation, pure thought.
Pythagoras' influence was huge on a number of people and movements: Plato, Descartes, Kant, Euclid, Christian Theology and Personal Religion.
Religion has been influenced by mathematics ever since Pythagoras. Kant, Plato, Aquinas, Augustine, Spinoza and others blended religion and reasoning.
Christianity derived the theory of Christ as the Word and led theology to produce rational proofs for God and morality because of Pythagoras.
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