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> Pythagoras, Metaphysics of Numbers
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post Mar 11, 2005, 08:20 PM
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Aristotle tells us in Metaphysics (I,5) that the Pytagoreans were the first to advance the study of mathematics and are known as the first mathematics research society. So many discoveries were made by them: the existence of prime #'s, the distinction between odd and even numbers. the graphing of a regular polyhedron, and most of the theorems of elementary geometry.

In the history of the field, the earlier number systems of the ancient Egyptians and Babylonians were more arithmatic than mathematical. The preceding systems were basically methods of practical land measurement. The Pythagoreans were the first to elevate this arithmetical geodesy to the level of mathematics and geometry.
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post Mar 16, 2005, 10:44 AM
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The Pythagoreans also proved that there are an infinite number of primes.
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post Apr 23, 2005, 05:38 PM
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http://www.crcsite.org/numbers.htm

Perceptions of Numbers

by H. Peter Aleff


Modern questions about number

Numbers were and are astonishing entities, entirely intangible but more enduring and reliable than the fleeting reality that hides them to unaware minds. To appreciate what the ancients saw in them, it may be best to begin with the questions some modern thinkers are asking about their nature. For instance, the astronomer and cosmologist John D. Barrow muses:

“Why does the world dance to a mathematical tune? Why do things keep following the path mapped out by a sequence of numbers that issue from an equation on a piece of paper? Is there some secret connection between them; is it just a coincidence; or is there just no other way that things could be? (...) Down the centuries there have been those who saw in mathematics the closest approach we have to absolute truth (...). Its very structure forms a model for all other searches after absolute truth.”[1]

Comparably, the searcher-for-absolute-truth-mathematician yet occasionally inventive chronicler of mathematical history Eric Temple Bell discussed in his book on Pythagoras what numbers might be. He concluded that

“[Whether numbers were discovered or invented] is the oldest and the simplest of all questions regarding the nature of mathematical truths. History gives no universally accepted answer to it.” [2]

Bell was a gifted story-teller and went here for dramatic effect. The question may be simple, but old it is not.

Until the current century, people rarely, if ever, doubted the once universally accepted bedrock principle that numbers and mathematics were “out there” and not invented by humans. Here are some comments on this subject which the mathematician Bonnie Gold wrote recently in a review of two books on the philosophy of mathematics:

“In the early years of this century, Platonism (by which I mean the belief that mathematics is the science of certain mind-independent, non-physical objects with determinate properties) was dethroned as the dominant philosophy of mathematics.

Since then, there’s been a struggle to replace it with an alternative that avoids the philosophical problems of Platonism while accurately reflecting the working mathematician’s daily experiences of doing mathematical research. None of Platonism’s immediate successors -- logicism, formalism, intuitionism -- has proved satisfactory. (...) In the last 25 years, new candidates for philosophies of mathematics have become popular, including fictionalism, conventionalism, structuralism, and social constructivism.”[3]

Proponents of the last-named among these philosophies argue, for instance, that the equation 2 + 2 = 4 is only a convention our grade school teachers bullied us into accepting as a law of nature, and that mathematical objects are social entities in the same way as monetary systems or political institutions.[4]

They also propose that mathematical objects get constructed by the community of mathematicians and then only take on a sort of life on their own in the minds of its members.

This may of course all well be so, but it makes one wonder where these objects go to survive when the knowledge about say, the golden ratio and the pentagram, gets lost and then rediscovered. How long can these objects stay outside of minds, and do they get enough food, water, and air there to continue their life while waiting to snatch another mind as their new host?

Why would a different community living at a different time come up again with the very same mathematical objects that some other people had already created?

And why do we laugh about the lawgivers in Indiana who tried in 1896 to legislate a value of 3.2 for the circle constant pi since it would be easier to use than the longer and more complicated traditional number which was produced strictly by and for impractical mathematicians?[5]

Such modern thinkers may reject the Platonist belief in some abstract, non-physical and non-psychological realm of numbers and mathematics where constants and pentagrams and other such constructs exist by themselves and outside of minds.


Ancient status of NUMBER

On the other hand, people in the highly religious environment of the ancient Levant had their then unquestioned certitude: NUMBERs belonged to the divine domain. Obviously, they had to be the first things made to then serve as helpers or tools for the rest of creation. No animals with four legs or people with two of those and five toes on each foot could be shaped before those numbers existed to build them with.

We owe much of our pre-Platonic written documentation of that once common “Platonist” belief to the ancient Mesopotamians because their baked- clay tablets survived better than the parchments and papyri of their neighbors. Another source for ancient attitudes towards numbers are the traditions about the Greek Pythagoras (around 580 to 500 BCE) and his teachings because this founder of a number-venerating religion had acquired many if not all elements of his doctrine in Phoenicia, then above all in Egypt, and finally in Mesopotamia.

The apparent absence of direct evidence from Egypt or Canaan does not mean said neighbors did not share very similar basic beliefs even though they left us little or no written testimony to that effect.

All civilizations of the ancient Near East flourished on a common cultural ground from which they drew many shared ideas and core convictions. For instance, winged disks represented the sun as royal and/or divine emblem on Hittite palace portals, Egyptian stelae, and Canaanite seals, as well as in Assyrian and Babylonian art. The same image appears also in Bible verses such as Malachi 4:2:[6]

“the sun of righteousness shall rise with healing in his wings”.

Conceptions of number are rooted deeply in that same shared substratum of basic ideas beneath the superficial cultural differences. It seems therefore more likely than not that the Phoenician and Egyptian beliefs about the nature of the number world had grown on similar spiritual soil as those of the ancient Mesopotamians.


Mesopotamian number gods

In those Mesopotamian beliefs, numbers were the most basic concepts of existence. Each deity had and/or was his or her number, and numbers were gods. The names of the gods could even be written as numbers. For instance, a personal name from the Third Dynasty of Ur reads “My-god-is-50.”[7]

Mathematics was above all a priestly science, embedded in and driven by a culture of number mysticism. The historian of science André Pichot describes this once common attitude towards numbers:

“Just as Mesopotamian astronomy was inseparable from astrology, Mesopotamian mathematics was inseparable from number mysticism.

(...) Numbers do not present themselves to our senses as unequivocally as geometric figures do. They are by far more abstract and are not part of our perceivable environment. Yet these abstractions exist and, moreover, they have certain regular properties.

Numbers cannot be connected with the concrete reality the way geometry is, and so they easily slide into the supernatural. They are not in nature but determine its manifestations through certain ratios and relationships; from this follows that they rank above nature. (...)

On the other hand, these “super-natural” and “nature-ruling” numbers are also a means to understand that nature. (...) [The Mesopotamians sought not mathematical rules or proofs] but technical recipes as well as certain number ratios without any direct use; this quest and this knowledge belong more to the domain of magic than to that of science. We find therefore besides the mathematical role of the numbers also purely mythical aspects which are equally interwoven with their use. (...)

This number mysticism is not to be confused with another kind of mysticism: in Greek and Hebrew writing, the numbers are represented by letters and not by symbols of their own, so that each word, particularly names, has its number, obtained, for instance, by adding the numerical values of the individual letters. (...) This [gematria] is fundamentally different from the number mysticism in which the numbers represent supernatural and nature-dominating entities; the latter can be considered a precursor of mathematical physics and even of mathematical rationalism in its broadest sense.”[8]

Many modern scholars believe that this number mysticism was the major motivation behind the Mesopotamians’ deep and ancient interest in the manipulation of numbers. The ancient sages’ search for the relationships between these supernatural but predictable entities contributed greatly to their development of mathematical skills.

Their looking for patterns in the number world anticipates the modern “alternative philosophy of mathematics”[9] which defines its subject as “the science of pattern”[10] . The search is the same even though the ancients asked different questions and looked for different patterns, and even though they interpreted some of their results in mytho-logical instead of modern-logical terms.

Here is how Pichot summarized his comments on the Mesopotamian number mysticism from which I quoted above:

"Just as the introduction of metalworking did not make chemistry any more rational but rather enriched magic by a few concepts and mythology by sundry smith gods, so the manipulation of numbers did not produce a unified corpus of rationally founded laws -- mathematics -- but remained a computing technique with a mystical background.

We should not expect rational explanations from this approach to numbers; its explanatory contribution is on the same level as that of alchemistic magic or of smith gods. It is moreover characteristic for this non-rational use of numbers and their relationships that each number is linked to a god and derives its value and position in the number system from the position of that god in the mythology. The structure of the number world corresponds therefore much more to a hierarchy of gods in heaven than to a system based on rational mathematical laws.”[11]


Pythagorean number veneration

The same ideas resonate also in the doctrine of Pythagoras where NUMBER had created and continued to rule the cosmic order. NUMBER was divine and functioned as “the principle, the source, and the root of all things”[12]. It permeated the world, intangible and unseen but as real and live to Pythagoreans as radio waves are to us.

NUMBER was even more essential to them than electromagnetic radiation is to us: electromagnetism is only one of currently four forces that hold our world together, whereas Pythagoreans held that NUMBER was the One and Only universal principle, the fabric of all there is and can be.

In Plato’s Pythagorean-influenced Timaeus, NUMBER was the World Soul from which the Demiurge had fashioned all of Creation. NUMBER wove together Form and Matter through its mathematical harmony and ruled everything, from the string lengths of musical instruments to the human emotions their sounds created and guided.

Even abstract concepts such as light and dark or good and bad were NUMBERS, and of course, so were the planetary motions that produced by their numerical ratios the famous though inaudible “Music of the Spheres”. NUMBER was the unchanging reality behind the illusion of the ever-changing phenomena we perceive.


Indo-European number rituals

The mystic status of numbers which led to this intense concern with their properties and relationships seems to have existed also, even before Sumerian times, in the beliefs of the neighboring Indo-Europeans.

Modern scholars interpret similarities between word roots as signs of deep and original connections, just as ancient sages had long done with similarities between the sounds of words, or between the ways to write them. Based on this principle, some of the moderns have shown that the religious view of numbers among speakers of Indo-European languages goes back to the prehistoric period when the words for their relationships formed.

Here is what David R. Fideler says about these early word roots in his introduction to Guthrie's “The Pythagorean Sourcebook and Library”:

“Cameron, in his important study of Pythagorean thought, observes that harmonia in Pythagorean thought inevitably possesses a religious dimension. He goes on to note that both harmonia -- there is no “h” in the Greek spelling -- and arithmos appear to be descended from the single root “ar”.

This seems to ‘indicate that somewhere in the unrecorded past, the Number religion, which dealt in concepts of harmony or attunement, made itself felt in Greek lands. And it is probable that the religious element belonged to the arithmos - harmonia combination in prehistoric times, for we find that ritus in Latin comes from the same Indo-European root’.”[13]

Such traces of early reverence for invisible but knowable Numbers suggest that if some ancient mathematicians were aware of the major constants, they might have ranked these mysterious “super-numbers” even higher than the natural numbers. They would have assigned them important religious and symbolic roles, and they would have explored their properties and permutations as a means to understand the relationships among the gods they represented or were.




Copyright © 2003 bv H. Peter Aleff

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Footnotes :

[1] John D. Barrow: “Pi in the Sky - Counting, Thinking, and Being”, Oxford University Press, Oxford, 1992, page 4.

[2] Erik Temple Bell: “The Magic of Numbers”, 1946 and 1974, edition consulted Dover, New York, 1991, pages 20 and 21.

[3]) Bonnie Gold: Review of “Social Constructivism as a Philosophy of Mathematics” by Paul Ernest and of “What is Mathematics, Really” by Reuben Hersh, The American Mathematical Monthly, April 1999, pages 373 to 380.

[4] For instance, Paul Ernest in: “Social Constructivism as a Philosophy of Mathematics”, State University of New York Press, 1998, per Bonnie Gold in the above review, page 375.

[5] Jan Gullberg: “Mathematics, From the Birth of Numbers”, W.W. Norton & Co., New York, 1997, pages 95 and 96.

[6] See, for instance, the biblical king Hezekiah’s stamp on jar handles mentioned in Frank Moore Cross: “King Hezekiah’s Seal Bears Phoenician Imagery”, Biblical Archaeology Review, March/April 1999, pages 42 to 45 and 60, see page 45 left.

[7] Simo Parpola: “The Assyrian Tree of Life: Tracing the Origins of Jewish Monotheism and Greek Philosophy”, Journal of Near Eastern Studies, Volume 52, July 1993, Number 3, pages 161-208. see page 182 left and note 87.

[8] André Pichot: “La naissance de la science”, Gallimard, Paris, 1991, edition consulted “The Birth of Science -- from the Babylonians to the early Greeks”, Wissenschaftliche Buchgesellschaft, Darmstadt, 1995, pages 92 to 94. (In German, my translation)

[9] so dubbed by Bonnie Gold, in the above review, American Mathematical Monthly, April 1999, page 379.

[10] as promoted in another book title by “Making the Invisible Visible” author Keith Devlin: “Mathematics: The Science of Pattern: The Search for Order in Life, Mind, and the Universe”, Scientific American Library, 1997.

[11] André Pichot: “The Birth of Science ...”, page 92. (in German, my translation)

[12]) David R. Fideler, in his “Introduction” to Kenneth Sylvan Guthrie, compiler and translator: “The Pythagorean Sourcebook and Library”, Phanes Press, Grand Rapids, Michigan, 1988, page 21 bottom, quoting Theon of Smyrna.

[13] David R. Fideler: “Introduction” to Kenneth Sylvan Guthrie: “The Pythagorean Sourcebook and Library”, Phanes Press, Grand Rapids, Michigan, 1987, citing Alister Cameron: “The Pythagorean Background of Recollection”, see note 30 on page 51 middle.
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post Apr 23, 2005, 05:41 PM
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One of the most important things humans ever did was invent (or was it discover?) numbers. It may seem odd to describe numbers as an important discovery or invention, but at some point someone must have realised that one sheep and another sheep made something more precise than 'some' sheep.

The invention of the counting numbers predates written history, but several numbers have since been added to the simple set of counting numbers: 1, 2, 3, 4...

For starters, it wasn't until 200 BCE that someone got an inkling that there ought to be a number to represent nothingness.

It was the Babylonians that first got this inkling, but it took another thousand years before the Hindus finally nailed down the idea, symbol and name for this new invention.

The word zero is ultimately derived from the Hindi for 'void' (sunya). Negative numbers had a similarly drawn-out birth. The slowness of the acceptance of negatives and zero is one of the reasons that the Western calendar lacks a year nought: 1 BCE changes to 1 CE, much to the suprise of today's more numerate historians.

Pythagoras, father of mathematics, ran a secret society of mathematicians. Women were welcomed into the 'brother'hood.

۰۱۲۳۴۵۶۷۸۹
०१२३४५६७८९
0123456789
The numerals we now use are ultimately derived from the Arabic numeral system (top row), via the Indian (Devanagari) numerals (middle row).

So, Indians invented zero, with a little help from the Babylonians, so creating the integers, ... -3, -2, -1, 0, 1, 2, 3, and from them the fractions, 1/2, 2/3, 345/2345.

However, being the endlessly inventive apes we are, there were plenty of other numbers yet to be discovered.

Pythagoras (born in 569 BCE in Samos, now Greece, died 475 BCE) was the leader of a highly secretive mathematical community, who was so horrified when it was proved that irrational numbers (like π, 3.142... infinite decimals that cannot be represented as simple fractions) existed that it is said that he had its discoverer Hippasus (c. 500 BCE) murdered.

It would appear that the Indian Aryabhatiya (475-550), who independently derived a value of π at 3.1416, seems to have been unworried that it might be irrational, which was very rational of him.

Similar problems arose when Georg Cantor (1845-1918) discovered that infinity ∞, hitherto thought of as a grubby concept to be swept under the table, came in several different sizes. Jealous rivals eventually sent Cantor mad, and he died in a lunatic asylum in 1918.

Much of the European Renaissance was spent relearning what the Greeks had already discovered in and around 400 BCE. The works of Socrates (469-399 BCE), Plato (427-347 BCE), Euclid (325-265 BCE) and Aristotle (384-322 BCE) covered fields of human endeavour as diverse as geometry, ethics, classification and logic. Their rediscovered works didn't just excite the fledgling scientist of southern Europe. Over four hundred years before, the Arab philosopher Abu Yusuf Al-Kindi (801-873), working in what is now Iraq, spent his life expounding Greek philosophy to an Islamic audience, and was the world's first cryptanalyst, inventing a way of cracking the letter-substitution cipher that until then had been assumed secure. Some hundred years later, the Iranian Al-Razi (864-930) worked on both medicine and alchemy: the word for one third of science, chemistry, is derived from the Arabic words al kimiya (alchemy).

Mikolaj Kopernik (Copernicus, 1473-1543), whom many don't realise was Polish, and the Italian Galileo Galilei (1564-1642) noticed that the Sun, rather than the Earth, was the centre of what we now call the solar system. Galileo was so forthright in his opinion that he was eventually declared a heretic by the Roman Inquisition and died under house arrest, an insult for which the Roman Catholic Church didn't quite apologise in 1992. Sometimes science isn't just about grant awarding councils and towing the line of dogma.

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post Apr 23, 2005, 05:44 PM
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Before the Secret Brotherhood was disbanded, its members really thought they had grasped the key to the cosmos.

Then everything collapsed. Their whole scheme was destroyed by a fatal discovery, and the Order itself was destroyed by traitors and mob violence. Yet as we retell the somber tale, we will find that it was not a complete tragedy after all, for the Pythagoreans did enjoy their cosmic key briefly. This key was not found in abstract shapes alone nor in music, nor in the stars, but in one factor that-they believed-linked all of these: number.

"Himself" had said it: "Everything is number!"

So they followed Pythagoras teaching that the universe was ruled by whole numbers That did not mean numbers for ordinary counting or calculating. What interested them was the nature of a number itself odd even, divisible, indivisible and the relations between numbers. This was their arithmetike. And they applied it to their other three fields, and found startling number patterns in each.

In music, for instance, a sensational discovery about the relations of whole numbers and musical intervals was attributed to Pythagoras himself.

One legend said that on his long voyages he listened to the music of flapping sails, and the wind whistling and whining through the ship's rigging and playing a melody on the ropes. And that he decided then and there to investigate the connection between the tempest of sounds and the vibrating strings.

Another version said that he was strolling through the village of Croton, deep in thought, listening to the musical sounds of hammers striking anvils in a blacksmith's shop; when suddenly, tripping on a taut string that some children had stretched across the street, he got the inspiration for an experiment.

But the most popular story told that the idea came to him straight from the stringed lyre of his "father" Apollo, who was also the god of music.

Anyway, Pythagoras experimented with stretched strings of different lengths placed under the same tension. Soon he found the relation between the length of the vibrating string and the pitch of the note. He discovered that the octave, fifth, and fourth of a note could be produced by one string under tension, simply by "stopping" the string at different places: at one-half its length for the octave, two-thirds its length for the fifth and three-fourths its length for the fourth!

Other musical innovations were credited to him, such as a one-string apparatus for the study of harmonics. But his great discovery was the tetrachord, where the most important harmonic intervals were obtained by ratios of the whole numbers:

1, 2, 3, 4. The Secret Brotherhood gave this fourfold chord mystical significance and used to say: "What is the oracle at Delphi? The tetrachord! For it is the scale of the sirens."

And the Pythagoreans even used it for their astronomy. In the relation of number and music, they believed they had found the pattern that guided the "wandering" planets through the heavens. They pictured the sun and the planets as geometrically perfect spheres, moving through the visibly circular sky on perfect circular orbits, separated by harmonic ratios-musical intervals! Theirs was a vision of time and space revealed in lines, tones, and mathematical ratios. And they even imagined the brilliant planets emitting harmonious tones, the so-called "music of the spheres."

But it was in the connection of number and geometry, their two completely mathematical subjects, that the Pythagoreans were on surest ground.

Numbers, they had discovered, whole numbers, actually had geometric shapes. There were triangular numbers, square numbers, pentagonal numbers, rectangular numbers, and so on.

This was no wild fantasy, like the singing planets. It was a real mathematical discovery, and came from the circumstance that they did not do their number work by writing the numbers at all. Instead, they placed pebbles on the sand, like the reckoners. But the Pythagoreans placed their pebbles in patterns, adding extra rows for each number. Their two most important series were the square numbers and the triangular numbers.


The most important number of all, to the Pythagoreans, was the fourth triangular number, 10. For it was made up of 1 + 2 + 3 + 4. They called it the "Sacred Tetractys, " swore by it in their oaths, and attached marvelous properties to it, as "the source and root of eternal nature."

Everything fitted perfectly: the Tetractys, the tetrachord, the four regular solids representing the four "elements," inscribed in a dodecahedron representing the celestial sphere. But it was all too pat a jumble of luck, imagination, serious mathematical experiments, and old number magic from the East. just as the Pythagoreans thought they were getting more and more evidence that number was everywhere, the whole system broke down. The entire connection between geometry and number-the foundation of their thinking-was shattered by one disastrous experiment.

Presiding was Hippasus of Metapontum, whose name was to loom dark in their future of the Brotherhood. The idea was simply to find the numbers that matched the sides of the two right triangles with which Pythagoras had first demonstrated his theorem-the Egyptian triangle and the one from the tiled floor.

Of course, the Egyptian rope-triangle worked perfectly: its 3-4-5 sides made a beautiful Pythagorean series. They indicated the intervals with pebbles. Now what about the right triangle from the Greek tile design, where the two sides were equal?

Suppose each side had a length of 1 unit-that would require 1 pebble. Then for the hypotenuse-how many pebbles should they put there? Well, the sum of the squares on the sides would equal the square on the hypotenuse. Therefore,

12 = 1 (square on one side)
and 12 = 1 (square on other side)

and 1 + 1 = 2,


so 2 is the square on the hypotenuse. And the hypotenuse is the square root of 2.

But what was the square root of 2?

It couldn't be a whole number, since there is no whole number between 1 and 2.

Then was it a ratio of whole numbers between 1 and 2? Hopefully, they tried every possible ratio, multiplying it by itself, to see if the answer would be 2. There was no such ratio.

After long and fruitless work, the Pythagoreans had to give up. They simply could not find any number for the square root of 2. We write the answer as 1.4141..., a continuing decimal fraction, but they couldn't do that since they had no concept of zero and of decimals. They could draw the hypotenuse easily, but they could not express its length as a number. It was "unutterable"-"unspeakable"!

Horrified, the Pythagoreans called the square root of 2 an irrational number. After that, they found other irrationals and swore to keep them secret, for the discovery of these "irrationals" wrecked their entire beautifully constructed system of a universe guided by whole numbers. The breakdown in their mystical morale was followed by the breakup of the Secret Brotherhood itself.

In this final demolition, Hippasus played a decisive role, though his own fate is shrouded in mystery. The Order was already in trouble. Bitter resentment had grown up against its secrecy and exclusiveness, and riots of villagers had driven it out of Croton. Pythagoras himself had died on a neighboring island. And now mobs of "democrats" began to attack the aristocratic Pythagorean societies everywhere.

Against this background, Hippasus took a step that was regarded by the conservative members as sheer betrayal. He broke the oath of secrecy and revealed their most closely guarded discoveries-the dodecahedron and the irrationals. When they promptly expelled him, he set himself up as a public teacher of geometry.

The traitor's punishment was swift and terrible. He was very shortly drowned in a mysterious "accident" at sea, and strange rumors circulated. Some said that a storm had struck his ship as a direct vengeance from the gods; others, that he had been pushed overboard by agents of the Secret Brotherhood. But Hippasus' death was to no avail. The harm was already done to the Order of Pythagoreans, though the discovery of irrational numbers eventually worked for the good of mankind.

The remaining secret groups soon collapsed, torn by outer violence and inner dissensions. And more and more "mathematicians" followed Hippasus' example and came out to earn a living as teachers. Pythagoras' idea had been demolished: no longer was there a closed Brotherhood of followers, bound together by a mystical belief in a cosmos ruled by number. Yet his ideals lived on in this broader field. He had pursued knowledge for its own sake, loving wisdom for itself. He knew learning could be shared without diminishing, that it lasts through life and immortalizes the learned after death. And the destruction of the Order gave his legacy to the world.

Geometry was now out in the open-and it was the new Pythagorean geometry. True, mathematics was still mixed with some magic: number mysticism, cosmic ideas about the regular solids. Burt there was, in addition, the famous theorem and its applications, the careful study of shapes, the theory of numbers, and the discovery of irrationals.
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post Apr 23, 2005, 11:27 PM
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[a Joke]
Interesting, now there is a new "Key of Cosmos": 911.
Using this number any good "Magi" can Ping the Mankind, as one can find that Number Matrix in actually anybody's brain.
A new way how to hack the world.
Funny is not it?
[end of the Joke]
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Yours,
Enki
24 April, 2005 AD

PS: BTW have you watched Constantine movie (by Fransis Lourence), I am going to watch it today. laugh.gif
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post Apr 24, 2005, 07:52 AM
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"What is the oracle at Delphi? The tetrachord! For it is the scale of the sirens."
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post May 11, 2005, 08:34 AM
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QUOTE (Trip like I do @ Apr 24, 07:52 AM)
"What is the oracle at Delphi? The tetrachord! For it is the scale of the sirens."

What do you mean by that man?
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post May 12, 2005, 03:35 PM
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First of all, the basis for Greek scale construction was the tetrachord....

So the tetrachord designates 4 notes, of which two are fixed and two are moveable....

The fixed notes are those bounding the tetrachord, which are always assumed to be the interval of the Pythagorean 'perfect 4th', with the ratio 3:4. It's the position of the two moveable notes that was argued about so much, and which makes this stuff so interesting to tuning theorists.
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post Feb 28, 2008, 05:12 AM
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[quote][quote name='Trip like I do' date='Mar 11, 2005, 08:20 PM' post='48335']
Aristotle tells us in Metaphysics (I,5) that the Pytagoreans were the first to advance the study of mathematics and are known as the first mathematics research society. So many discoveries were made by them: the existence of prime #'s, the distinction between odd and even numbers. the graphing of a regular polyhedron, and most of the theorems of elementary geometry.

In the history of the field, the earlier number systems of the ancient Egyptians and Babylonians were more arithmatic than mathematical. The preceding systems were basically methods of practical land measurement. The Pythagoreans were the first to elevate this arithmetical geodesy to the level of mathematics and geometry.
[/quote]

maths at tarporley is crap we dont have good teachers just 1s dat piss u off
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post Feb 28, 2008, 05:45 AM
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You could use a few more years in school, preferably boarding school with corporal punishment. tongue.gif
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