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QuoteFranco Oshiro
Mr. Kavanagh
Philosophy .1
27 April 2008
     Everything around us must be looked at with amazement, for it all has roots in mathematics.  It is the same mathematics that started evolving at the birth of mankind.  It has changed us.  The unifying principles of mathematics have influenced our thoughts and the way we look at the world.  Our modern marvels can be traced back to our ancestors and the way they looked at mathematics.  In particular, the mathematics of Ancient Greece, has influenced the western world, due to the fact that Greece was the birthplace of western thought..  Ancient Greek mathematics has technologically changed the western world’s perception of the world through its evolution.
     Prior to delving into its evolution and impact, an essential question is:  Are mathematics and numbers special to humans?  It is a known fact that animals can differentiate between one and two.  A starving dog will eagerly approach a bowl with fifty treats rather than a bowl with ten.  In Gilbert White’s book, The Natural History of Selborne, White comments that a plover, a type of shorebird, can count how many eggs it has laid.  “[Gilbert White] secretly removed one egg each day from a plover’s nest - and how the mother persisted in laying an extra egg every day to make up the original total” (McLeish 7).  The main difference, however, is that humans have learned to think of numbers abstractly.  (McLeish 7).
     McLeish states that it is through communication with other people that we learn to think abstractly.  By having one unifying symbol, word or mental image, we can let go of the solid objects and think in the abstract (7-8).  McLeish gives us an example of the abstract methods of counting through Lewis Carroll poem, “The Walrus and the Carpenter”:
   "The time has come," the Walrus said,
   "To talk of many things:
   Of shoes--and ships--and sealing-wax--
   Of cabbages--and kings-- (Carroll)
First, a set is introduced, “many things,” this set is then broken up into elements:  the shoes, ships, sealing wax, cabbages, and kings.  We connect the solid number of objects to the set as a whole.  As one new element is introduced, the amount of objects in the set increases.  The elements, of course, are miscellaneous objects.  By being in a group of “things”, they enter into an obscure and abstract form.  Since these “things” are in the world of the abstract, so must the number of “things.”  Therefore, we only see these numbers as increasing, or, in other words, a count.  Finally, “we can use the set of number words, devoid of any concrete content or reference, to describe the miscellaneous set, and we can point to each object in the set, identifying it in a unique way.”  We can say that the set contains five objects with the third object being the sealing wax, all without seeing the concrete set.  (McLeish 8-9).
     The evolution of counting and mathematic dates back to prehistoric times.  David M. Burton comments, “anthropologists tell us that there has hardly been a culture, however primitive, that has not had some awareness of number…” (1).  This is due to the fact that people “felt the need to enumerate their livestock, or objects for barter, or the passage of days” (1).  The most prominent way of keeping count was through simple tallying, “to match the collection to be counted with some easily employed set of objects” (2).  This was done through objects, such as pebbles, sticks, and notches on objects.  It has been discovered that people from the Old Stone Age used the tally system to keep count.  For instance, a wolf’s shinbone with tally marks was found in 1937 in Czechoslovakia that dated back to 30,000 B.C.  (2-4).  Similar tools were also found near the Nile, dating 8500 B.C.  The tally technique was not restricted to the Europe-Asia-Africa landmass; the Incans from Peru used tools called quipus, ropes that kept the count in knots.  These tools were so important that each city a person was assigned that would interpret and tie these knots.  (5-6).
     The prevalence of numbers in human history is clear.  However, where did we start thinking about numbers and what they mean?  We have found evidence that we indeed first learned to count with solid objects and converted the experience to something abstract.  Numbers became an abstract thought that helped early man survive by allowing him to keep count of days, livestock, etc.  (Burton 1).  For western civilization, it was not until around 500 B.C. that Thales of Miletus, the “father of deductive reasoning,” introduced a much deeper understanding of mathematics. (Skinner 42).  
     Thales of Miletus was born around 625 B.C. and died around 547 B.C.  During his youth, Thales studied Geometry in Egypt and Astronomy in Babylon.  In Commentary on the First Book of Euclid’s Elements, Proclus said, “Thales was the first to go into Egypt and bring back this learning [geometry] into Greece.  He discovered many propositions himself and he disclosed to his successors the underlying principles of many others…” (Burton 93).  Some of these propositions that Thales made were:
   1. Every angle inscribed in a semicircle is a right angle.
   2. A circle is bisected by its diameter.
   3. The base angles of an isosceles triangle are equal.
   4. If two straight lines intersect, the opposite angles are equal.
   5. The sides of similar triangles are proportional.
   6. Two triangles are congruent if they have one side and two adjacent angles respectively    equal.  (Burton 94).
Although these propositions may seem simple, one must remember that mathematics was just in its infancy.  These were the baby steps of deductive reasoning.  Thales new ideas of Mathematics changed the thought process of many Ancient Greek thinkers.  “They valued the subject because it was the ideal preparation for philosophy, it granted understanding of the structure of the universe, and it was delight food for minds that enjoyed intellectual challenge.”  (Kline 139).  
     Through this concept of deductive reasoning, Thales attempted to explain matter.  He stated that the first form of matter was water, since it was the only known substance that could be in the form of a solid, a liquid, and a gas.  Other thinkers joined Thales, but argued that there were other prime elements:  earth, air, and fire.  These were the first endeavors that tried to “…explain all the variety of nature by something within nature and not outside it.”  (McLeish 76).
     One of Thales most prominent students was Pythagoras.  Pythagoras was born in Samos, a Greek Island, circa 569 B.C. and died circa 500 B.C.  His father was a foreign merchant who had become a Greek citizen.  Since Pythagoras’s father was a very successful merchant, Pythagoras was allowed to have an excellent education, which included studies in philosophy, physical education, and music.  (Anderson and Stephenson 9-11).  Thales heavily inspired Pythagoras by advising him to go to Egypt and study mathematics and astronomy (O’Connor).
     Pythagoras journeyed to Egypt around 535 B.C. (O’Connor).  In Egypt, he learned about “astronomy and geometry, and being initiated in no casual or superficial manner in all the mysteries of the Gods” (Iamblichus).  In 515 B.C., Egypt fell under the rule of Darius I, a Persian leader (Gascoigne).  Pythagoras was sent to Babylon along with other Egyptian priests (Livio 24).  Iamblichus described him as “overjoyed to associate with the Magi, who instructed him in their venerable knowledge, and in the most perfect worship of the Gods.”  While in Babylon, he also studied “arithmetic, music, and all the other sciences” (Iamblichus).
     Anderson and Stephenson state that, at the age of fifty-six, Pythagoras went back to Samos and tried to establish a school (12).  Later, Pythagoras moved to Crotona, in Southern Italy, and established another school there (O‘Connor).  Anderson and Stephenson  suggest that Pythagoras migrated to Croton in order to escape tyranny (13).  However, Iamblichus said that it was due to the Simians themselves:
           …the method of teaching he wished to introduce was the symbolical    
                  one, in a manner similar to that in which he had been instructed in
                  Egypt. This mode of teaching, however did not please the Simians,
                  whose attention lacked perseverance.  (Iamblichus)
Furthermore, according to Iamblichus, Pythagoras wanted to follow the footsteps of other philosophers and teach in foreign lands.  Through the years at Samos and Crotona, Pythagoras gained many followers, who were known as the Pythagoreans. (Livio 25).  
     “Mathematics,” which means “that which is learned,” was coined by Pythagoras (Livio 26).  During his life, Pythagoras discovered many mathematical principles which the Pythagoreans later shared with the world.  He separated numbers into primes and perfect numbers, and also into odd and even numbers.  (Skinner 16).  Furthermore, he was the one that introduced the famous “Pythagorean Theorem” to the western world, even though “he probably learned this theorem … in Egypt or Babylon” (17).  The theorem states “The length of the hypotenuse2 = side a2 + side b2” (17).  By following this rule, the Pythagoreans discovered irrational numbers, such as the square root of 2.  Since they were not whole and rational, these numbers were called “unmentionables,” and some said that if one mentioned such numbers, one would be punished by the gods.  (Anderson and Stephenson 16).  Lastly, Pythagoras discovered that a triangle’s angles added up to 180º, and that “a polygon with n sides has a sum of interior angles 2n - 4” (O’Connor).
     Pythagoras and his followers had five basic rules.  The first stated that the universe was created through a divine plan.  The true reality of that universe, however, was not physical, but spiritual.  “…It consist of the ideas of number and form.  The ideas are divine concepts, superior to matter and independent of it” (McLeish 97).  The second stated that the soul was made up of “self-moving numbers.”  The souls are eternal and can move through and exist in different bodies.  The third stated that there is an inner harmony in the world, which was made up by opposites.  Pythagoras acknowledged ten fundamental opposites.  They were:  “odd/even; male/female; good/evil; wet/dry; right/left; rest/motion; hot/cold; light/dark; straight/curved; limited/unlimited” (McLeish 97)  The fourth stated that relationships were the most important thing in life.  “Men and women should live an ascetic life in a sharing group devoted to the rearing of children in harmony with the divine plan.”  The fifth, and most important rule, stated:  “The divine ideas, which created and maintain the universe, are those of numbers” (McLeish 97-98).  Pythagoras declared that in order to reach goodness and perfection people must study mathematics.  (McLeish 97-98).
     In addition to coining the word “mathematics,” Pythagoras also coined the word “philosophy,” or “love of wisdom” (Livio 26).  Pythagoras was a lover of wisdom, and he believed that the universe and mathematics were connected.  “He was convinced that the divine principles of the universe, though imperceptible to the senses, can be expressed in terms of relationships of numbers” (Knierim).  In other words, Pythagoras suggested that everything had a numerical relationship with everything else.  Everything is “… measurable and commensurable or proportional in terms of number” (Pythagoreanism).  Mathematical ideas are thought to be “the most concrete and yet the most abstract form of reasoning” (Skinner 6).  Pythagoras even went as far as to say that mathematics was “a purifier of the soul” (Knierim).
    One of the Pythagoreans, Theodorus of Cyrene, later became the teacher of one of the most influential philosopher in western thinking, Plato.  The inscription on the entrance of Plato’s school, the Academy, said, “Let no one destitute of geometry enter my doors” (Livio 63).  In Plato’s opinion, mathematics was necessary for a leaders and philosophers.  (Livio 63-64).  Plato’s perception of the world echoed back to the teachings of Thales.  He used the idea of the four basic elements, earth, water, air, and fire.  This idea was developed by Empedocles, who in turn used Thales’s ideas about the first form of matter.  Plato united this idea with the idea of the “atom,” individual particles that make up matter, which was developed by Democritus.  The new theory that Plato came up with stated that:
   Earth is associated with the stable cube, the ‘penetrating’ quality of fire with the pointy
   and relatively simple tetrahedron, air with the ‘mobile’ appearance of the octahedron, and
   water with the multifaceted icosahedrons.  The fifth solid, the dodecahedron, was
   assigned by Plato to the universe as a whole.  (Livio 68).  
This was the beginning of the idea that the universe is made up of unique atoms.  (68-69).
     Today, humanity has developed a more sophisticated idea of how the universe exists.  Matter is made up of molecules, which are made up of atoms.  Although Plato’s view seems to be a primitive version of atoms, atoms only hold into account the first four Platonic shapes, squares, tetrahedron, octahedron, and icosahedrons, since the essence of both is to be a simple building blocks of the physical world.  However, the fifth element is still up for questioning.  This is the same question of “quintessence” that people pondered during Thales’s time.  Recently, in 1998, astronomers discovered that the universe was expanding at an accelerating rate.  This means that there is a force that is stronger than gravity that is causing the acceleration.  Physicists have named this force “dark energy.”  This idea of “dark energy” is an evolved idea that started back with Plato’s fifth element, the dodecahedron, which represented the universe as a whole.  (Livio 68-69).  
     Following Plato’s contribution to mathematics and its relation to our world, the next prominent mathematician that appeared was Euclid of Alexandria around 300 B.C.  There is not a lot of information on his actual life, however, his literary work are most famous in geometry classes today.  For twenty centuries, introduction to geometry classes have been taught using the ideas and principles that appear on the first six books of Euclid’s Elements.  (Burton 153-154).
     Euclid’s Elements included:  the fundamentals of geometry: theories of triangles, parallels, and area, geometric algebra, theory of circles, constructions for inscribed and circumscribed figures, theory of abstract proportions, similar figures and proportions in geometry, fundamentals of number theory, continued proportions in number theory, number theory, classification of incommensurables, solid geometry, measurement of figures, and regular solids (Euclid).  Book I opens with the Postulates and the Common Notions.  These are the universal truths and cannot be disproved.  The Postulates were:
   1. A straight line can be drawn from any point to any point.
   2. A finite straight line can be produced continuously in a line.
   3. A circle may be described with any center and distance
   4. All right angles are equal to one another.
   5. If a straight line falling on two straight lines makes the interior angles on the same side
   less than two right angles, then the two straight lines, if produced indefinitely, meet on
   that side on which are the angles less than two right angles. (Burton 156-157).
The Common Notions were:
   1. Things which are equal to the same thing are also equal to one another.
   2. If equals are added to equals, the wholes are equal
   3. If equals are subtracted from equals, the remainders are equal.
   4. Things which coincide with one another are equal to one another.
   5. The whole is greater than the part.  (Burton 157).
With these 10 statements, Euclid was able to make 465 propositions. (Burton 156-157).  However, most of the 465 propositions seemed very common sensical.  For instance, “the Epicureans, a school of Greek philosophers, picked on Euclid's theorem that the sum of two sides of a triangle is greater than the third side.  Said the Epicureans, ‘Any ass knows this theorem.  If fodder is placed at one point and the ass is at another, the ass does not traverse two sides of a triangle to reach the fodder but goes directly to it’” (Kline 76).  Although the propositions were straightforward, the mere fact that they were simple strengthened the truth behind them.  This allowed the more complex proposition to have a stabler base.  (Kline 76-77).
     The question on many of Greek’s minds was, “Why?”  What was the use of proving such propositions?  First of all, there were the practical uses of such proofs.  Humans could use these proof to “shed light on natural phenomena” (Kline 77).  People could measure how to build bridges, how to cross rivers, etc.  These were, and still are, the practical uses of Euclid’s Elements.  However, there are much deeper functions for it.  One can obtain satisfaction by merely proving a complex geometrical problem.  For some people, reaching this satisfaction brings them happiness.  (Kline 77).  Furthermore, these propositions allowed us to understand the physical world in which we live in.  
   Geometry and numbers are sacred because they codify the hidden order behind creation.  
   They are the instruments used to create the physical universe.  Simplicity in number,
   fraction and ratio provide the harmony and intellectual rigor of both the universe and the
   geometry of Euclid and his fellow Greek geometers (Skinner 15).
     Shortly after Euclid‘s death, one of the last influential Greek mathematicians was born.  Archimedes was born about two generations after Euclid, around 287 B.C. and died around 212 B.C.  Although he spent most of his life in his hometown, Syracuse, Archimedes studied mathematics and science in Alexandria.  In 212 B.C. Archimedes defended Syracuse against the invading Roman, and he died by the hand of one of the Roman soldiers.  (Burton 214).
     Archimedes invented the Archimedean screw, and perfected the ideas of levers and pulleys.  
The Archimedean screw was invented to haul water up from the canals in Egypt.  It consists of a tube and an inner screw that pushes the water up when it is turned.  This device is still used today to pump water out of places such as mines and boats.  Although he did not invent them, Archimedes also showed much proficiency with levers and pulleys.  Such engineering feats included launching a large ship and many contraptions used to defend Syracuse from the Romans. Archimedes proclaimed, “Give me a place to stand and I will move the Earth.” (Burton 215).  
     One of the more famous contribution that Archimedes offered was “Archimedes’ Principle.”  The story behind this discovery starts with King Hiero of Syracuse.  The king had obtained a crown that was thought to be made of gold.  However, he was not sure if the crown was made of pure gold or an alloy.  Archimedes came to his aid, and formulated an idea that concluded that if the crown was made of pure gold, an equal weight of gold would dissipate the same amount of water since they would share the density.  In the end, the crown dissipated less water than an equal weight of silver, but more water than an equal weight of gold;  the crown was an alloy.  However, Archimedes did not stop there, he wanted to know exactly how much of the crown was silver and gold.  To do this, Archimedes used a series of algebraic equations with unknowns.  This process is called elementary algebra.  (Kline 63-68).
     The aim of the elementary algebra is to obtain a usable result using unknown variables.  After a quick look, one can see that elementary algebra behaves like a machine.  “The mechanization of processes that have to be used repeatedly is of course a great gain, since one does not have to think about them.  They become habitual like washing and dressing” (Kline 68).  However, this mechanic aspect of algebra does not show the glamour that mathematicians today admire.  
   The individual techniques of algebra are like single notes selected at random from large
   and magnificent musical compositions.  When struck in succession these notes are
   discordant.  However, these same techniques, employed in the investigation of more
   significant undertaking, help to form mooi patterns of reasoning.  (Kline 68).
In other words, algebra must have reasoning and purpose in order for it to be meaningful.  Algebra is not about the means, the mechanical and tedious work, but the ends, reason and purpose.  (Kline 68-69).
     The elementary algebra presented a new way to look at thought:  calculative thinking and meditative thinking.  Although at the time, these terms were not used, the essence behind algebra greatly resembles some of Heidegger’s ideas, a 20th century philosopher.  By doing the algebra alone without purpose, one would be practicing calculative thinking.  On the other hand, if one has a purpose and passion for the ends of the mechanical means, one would be practicing meditative thinking.  
   Calculative thinking computes.  It computes ever new, ever more promising and at the    same time more economical possibilities.  Calculative thinking races from one prospect to    the next.  Calculative thinking never stops, never collects itself.  (Heidegger).
Unfortunately, Heidegger suggests that human kind has become strictly calculative.  In other words, most people now calculate like a machine without meditating on their actions.  (Heidegger).  Even though Aristotle allowed algebra to become a tool for calculative thinking, it is very doubtful that he himself was a calculative thinker.  Aristotle used the tools of elementary algebra to enjoy and learn about himself and discover more about the physical world.  (Kline 68).
     With Archimedes came the end of Greek mathematics. In 211 B.C. Syracuse had fallen to the Romans.  Soon after in 146 B.C. Greece was fully conquered.  By 30 B.C. the Roman Empire was fully established.  With this new culture mathematics ceased to a halt.  “…although the Roman and Greek civilizations existed over roughly the same centuries… in all that time there appeared no Roman mathematician of note” (Burton 234).  The Romans simply used the formulas to create structures, but never strived to understand the theories behind them.  (Burton 233-234).
     Thus, the evolution of mathematics laid dormant until it was reawakened later on in history.  This short evolution of mathematics, however, was profound.  First, Thales struggled to discover the first principles of math.  Pythagoras followed, making a theory that everything was made of numbers.  Plato added to Pythagoras theory by combining it with Thales theory of matter and the theory of the atom.  Euclid extended this notion of smaller objects that make up and support a bigger object to construct Elements.  Finally, Archimedes used Euclid’s theories and knowledge to invent new things and make mathematics have a purpose.  Through this evolution, human kind has found a new perception to our world.

Work Cited
Anderson, Margaret J., and Karen F. Stephenson.  Scientists of the Ancient
          World.  Berkeley Heights:  Enslow Publishers, Inc., 1999.

Burton, David M. The History of Mathematics. Boston: Allyn and Bacon, Inc., 1985.

Carroll, Lewis. "The Walrus and the Carpenter." 1872. 16 May 2008    <http://www.jabberwocky.com/carroll/walrus.html>.

Euclid. "Euclid's Elements." June 1997. 16 May 2008    <http://aleph0.clarku.edu/~djoyce/java/elements/toc.html>.

Gascoigne, Bamber. "History of Egypt." HistoryWorld. 15 Dec. 2007
   <http://www.historyworld.net/wrldhis/PlainTextHistories.asp?groupid=260
   &HistoryID=aa28>.

Iamblichus. "The Complete Pythagoras." The Complete Pythagoras. 2
          Dec. 2007 <http://www.completepythagoras.net/>.

Kline, Morris. Mathematics and the Physical World. New York: Thomas Y. Crowell
   Company, 1959.

Knierim, Thomas. "Pythagoras." The Big View. 2007. 2 Dec. 2007
          <http://www.thebigview.com/greeks/pythagoras.html>.

Livio, Mario.  The Golden Ratio.  New York:  Broadway Books, 2002.

McLeish, John. The Story of Numbers. New York: Fawcett Columbine, 1991.

O'connor, J. J., and E. F. Robertson. "Pythagoras of Samos." Jan. 1999.
          JOC/EFR. 2 Dec. 2007 <http://www-groups.dcs.st-and.ac.uk/~history/
          Biographies/Pythagoras.html>.

"Pythagoreanism." 14 Sept. 2001. Mizii Inc. 2 Dec. 2007
          <http://cyberspacei.com/jesusi/inlight/philosophy/western/
          Pythagoreanism.htm>.

Skinner, Stephen. Sacred Geometry Deciphering the Code. New York: Sterling
   Publishing Co., Inc., 2006.

you will read this... kay?

cause its fucking 2:10 am, and i just finished this fucking research paper
congratulate me or something idk
gah gonna go to sleep now... night

http://ikariam.com/

my friend recommended this to me :
seems really nice for something to do for 10 minutes of your day and let it sit there

basically you have like a village or something and you gather resources, but it takes hours and stuff to build/research/etc., so there isnt a lot of interaction.

kind of lame, but good for shits and giggles i guess.

http://www.youtube.com/watch?v=UTby_e4-Rhg
_poogle_

http://www.youtube.com/watch?v=ayZGZ2NSxs4

...you mexicans

now, educate me and tell me what you're celebrating

http://dontstayvirgin.movielol.org/main2.php

help this girl achieve the world record for most stds

so im doing my hw, and right behind me my brothers are watching tv...
when this commercial came up...
http://www.youtube.com/watch?v=GDdNsomWEno

Discuss

Heidegger's point of view:
thats shit, we mustn't rely on technology to make us happy, although we can still use it.  We must look into ourselves.  Meditative thinking > Calculative thinking. (super over-simplification)

So i got accepted to Boston University, Northeastern, WPI, and Boston College
(got rejected from Cornell, Harvard, and Brown i suck at life , but "wait-listed" at Tufts :/)
however...
im thinking of going to BC since i've heard from my friends that its the best in my list right now.

so whats up clock crew, did some of you gets get your college letters this year?