The proofs of the formula are in fact many and varied; the first one found by Google is at:
http://www.artofproblemsolving.com/LaTeX/Examples/AreaOfACircle.pdf

Don't worry, that's not the elegant one.

There are many proofs that don't (directly) involve the use of calculus, and Wikipedia gives a good sample of them:

http://en.wikipedia.org/wiki/Area_of_a_disk

of which the rearrangement proof is perhaps the most elegant.  Another presentation of this proof is given here (along with Archimedes' equally elegant derivation of the volume of a sphere):

http://www.mathreference.com/geo,circle.html

Yesterday I came across an approach that to me seems even simpler, based on a post at:

http://foxmath.wordpress.com/2008/06/24/perimeter-area/

This shows that for any regular polygon with an area equal to its circumference, the length of the apothem (the red line in the diagram to the left) is 2.  This is immediately obvious from the fact that the area of each individual triangle is equal to the base length, when the height equals 2.

In the limit as the number of sides of a regular polygon tends to infinity the polygon approaches a circle, and the length of the apothem approaches the radius of the enclosing circle.  It therefore follows that the area of a circle of radius 2 is equal to its circumference; i.e. 4.pi.

A circle of radius R may be scaled to radius 2 by multiplying the radius by 2/R.  The radius of this circle is then 4pi x (R/2)^2 = pi.R^2.

Finally a "wordless" proof provided by the people at SSSF:

http://www.maa.org/pubs/Calc_articles/ma018.pdf

The story of mathematics is the story of interesting people. What a shame it is that our children see only the dry remains of these people’s passion. By learning math history, our students will see how men and women wrestled with concepts, made mistakes, argued with each other, and gradually developed the knowledge we today take for granted.

In a previous article, I recommended books that you may find at your local library or be able to order through inter-library loan. Now, let me introduce you to the wealth of math history resources on the Internet.

[These will also be added to my math resources page.]

Math history on the Internet

Quicklinks for easy browsing:

• Most valuable sites
• General resources
• Math topics & significant individuals
• Specific cultures or time periods
• Math history for elementary/middle-school students

Math history > Most valuable sites

The MacTutor History of Mathematics Archive
My favorite place to begin any foray into math history. Highlights include:

• An Overview of the History of Mathematics
• Biographies Index
• History Topics Index
• Famous curves index
• Mathematicians of the day
• A Time Line of Mathematicians

La Habra High School's Math History Timeline
Math discoveries, publications, and other tidbits --- from paleolithic number bones to the present.

• Pre-historic and Ancient Times 1,000,000 B.C. - 500 A.D.
• Middle Ages 500 - 1400 A.D.
• Renaissance 1400 - 1550 A.D.
• Reformation 1517-1598 A.D.
• Baroque Era 1600-1700 A.D.
• Enlightenment 1700-1789 A.D.
• Age of Revolutions 1789-1848 A.D.
• Age of Liberalism 1848-1914 A.D.
• 20th Century ... 1914-present A.D.

[Back to Math History quicklinks.]

Math history > General resources

Biographies at Wolfram MathWorld
Long, long list, and each biography is linked to explanations of the mathematician's major discoveries.

Biographies of Women Mathematicians
Indexed alphabetically, chronologically, and by country of birth. Includes modern news tidbits, too.

A Completely Inadequate Bibliography of the History of Mathematics
"Most of the following books are aimed at the professional non-mathematician (i.e., someone to whom the land of mathematics is an interesting place to visit, but you wouldn't want to live there)."

Convergence
An online magazine from the MAA: "Where mathematics, history, and teaching interact."

Fred Rickey's History of Mathematics Page
Includes Teaching a Course in the History of Mathematics and An Annotated Bibliography.

Galileo and Einstein: Overview and Lecture Index
Lecture notes on the history of math and physics.

Mathematicians of the 17th and 18th Centuries
Adapted from A Short Account of the History of Mathematics, by W. W. Rouse Ball.

Mathematicians of the African Diaspora
Black men and women of mathematics, in history and in the present.

Mathematical Quotation Server
I love quotations! No matter what I want to say, somebody else has probably already said it better.

Math Forum History Listings
"651 items found." No, I have not checked them all. Go browse for yourself!

Math History and Mathematicians Pages
Julie Brennan at Living Math is building an index of links to biographical information, famous quotes, activities and book suggestions to accompany a homeschool math history course. [Sample lessons.] The last time I visited, she had almost finished the first year's listings.

Philosophy of Science
Many assorted links to readings for a college class, including several chapters from String, Straightedge & Shadow.

[Back to Math History quicklinks.]

Math history > By topic

Abacus: The Art of Calculating with Beads
The abacus through history, how to make and use an abacus, and classroom ideas.

Archimedes
"This site is a collection of Archimedean miscellanea under continual development." See also: Archimedes' Approximation of Pi, and The Archimedes Palimpsest.
Edited to add: Prehistoric Calculus: Discovering Pi.

Carl Friedrich Gauss
Detailed biography, quotations, and more.

Earliest Known Uses Of Common Mathematical Symbols and Words
Research is ongoing (I found a page that had been modified last week), so don't assume that a citation is the earliest use unless indicated as such.

Euclid’s Elements
David E. Joyce brings the text of Euclid's 13 Books to life with Java applets. See also: An Introduction to the Works of Euclid.

Famous Problems in the History of Mathematics
This site includes problems, paradoxes, and proofs that have inspired mathematicians through the ages, plus links for further exploration.

A Golden Sales Pitch
"There is little evidence to suggest that the golden ratio has any special aesthetic appeal... When a myth is repeated over and over, it begins to sound like truth."

The History of Measurement
"There were unbelievably many different measurement systems developed in early times, most of them only being used in a small locality."

Hypatia of Alexandria
Lots of links, including The Primary Sources for the Life and Work of Hypatia of Alexandria. See also: Hypatia, the First Known Woman Mathematician.

MacTutor Topical Indexes

• Algebra
• Analysis
• Geometry and Topology
• Numbers and Number Theory
• Mathematical Astronomy
• Mathematical Physics

Mathematical games and recreations
"The whole history of mathematics is interwoven with mathematical games which have led to the study of many areas of mathematics."

The Mathematical Problems of David Hilbert
With a link to Hilbert's 1900 address to the International Congress of Mathematicians in Paris, surely the most influential speech ever given about mathematics. Wolfram MathWorld has an annotated list of all 23 problems.

MathPages History Topics
A wide assortment of tidbits for advanced students.

Slide Rule History
"The slide rule has a long and distinguished ancestry … from William Oughtred in 1622 to the Apollo missions to the moon."

Who was Fibonacci?
"A brief biographical sketch of Fibonacci, his life, times and mathematical achievements."

[Back to Math History quicklinks.]

Math history > Cultures or time periods

Ancient Africa
Part of the Mathematicians of the African Diaspora website.

History of Egyptian and Mesopotamian Mathematics Page
An excellent resource for my Alexandria Jones stories.

History of Mathematical Education
What topics of mathematics have been taught in different cultures and time periods? Why have these changed?

MacTutor Mathematics in Various Cultures

• Ancient Babylonian mathematics
• Ancient Egyptian mathematics
• Ancient Greek mathematics
• Arabic mathematics
• Chinese mathematics
• Indian mathematics
• Mayan mathematics
• American mathematics
• Mathematics in Scotland

Mathematics in Specific Cultures, Periods or Places
A short collection of links. This site also contains: Websites relevant to the History of Mathematics.

Mesopotamian Mathematics
"From the earliest tokens, through the development of Sumerian mathematics to the grand flowering in the Old Babylonian period, and on..."

[Back to Math History quicklinks.]

Math history > For elementary/middle-school students

Adding with the Abacus
"What did people do to save time working out more difficult problems before the calculator existed?"

Ancient Greek Mathematics
Selections from String, Straightedge & Shadow:

• Chapters 8, 9: Thales
• Chapters 11, 12: Pythagoras and his Theorem
• Chapter 13: Platonic Solids
• Chapter 14: The Irrationals
• Chapter 15: The Golden Mean
• Chapter 16: Archimedes
• Chapter 17: Eratosthenes

Archimedes & Large Numbers
A brief look at Archimedes, Avogadro, and Cantor. See Approximating Pi for an interactive demonstration.

Calendars
"Calendars were one of the earliest calculating devices developed by civilizations."

Egyptian Math
Could you survive in the world of Egyptian numerals and mathematics? [Note to teachers: The Egyptian Math Worksheet Creator looks like fun!]

Eratosthenes' sieve
Click on any number, and all its multiples (except the number itself) will disappear from the chart. See also: Murderous Maths Prime Numbers Page.

Eureka! The Achievements of Archimedes
Click "next" to read the pages one by one, or browse through the Index.

Famous Problems in the History of Mathematics
This site includes problems, paradoxes, and proofs that have inspired mathematicians through the ages, plus links for further exploration.

Fibonacci Activities
For explanations and more fun, see: Fibonacci Numbers and Nature.

Solid Gold Gnarly Math: The Gnarly Gnews
Free bi-monthly newsletter of math history with a twist of humor.

History of Fractions
"Did you know that fractions as we use them today didn't exist in Europe until the 17th century?"

History of Measurement
To work effectively and share goods fairly, people had to find ways to measure their stuff. See also: Measure for Measure.

Leonardo da Vinci Activity
"Is the ratio of our arm span to our height really equal to 1?" See also: Teacher Lesson Plan and Leonardo's Vitruvian Man.

Negative Numbers
"Among the earliest people to use negative numbers in calculations were the ancient Chinese." See also: The History of Negative Numbers.

Pascal's Triangle
Lessons and links for all grade levels. See also: All You Ever Wanted to Know About Pascal's Triangle.

Pi, a Very Special Number
Over the centuries, mathematicians kept looking for better values for pi.

Platonic Solids
With printable nets, so you can make your own models. Part of the wonderful Maths is Fun site --- take some time to explore!

Pythagoras
"Pythagoras believed that everything in the world could be explained by numbers." See also: All Is Number.

ThinkQuest History of Mathematics
Brief overview of math history, with biographies of influential mathematicians and short online quizzes.

Women in Maths
"Ever wondered why stories about mathematicians always seem to be about men? ...There were a few women who dared to go against the flow."

[Back to Math History quicklinks.]

:: :: :: :: :: :: :: :: :: :: ::

• A Very Short History of Mathematics
• Hooray for (Math) History
• An Ancient Mathematical Crisis
• The Puzzling Pythagorean Pebbles
• The secret of Egyptian fractions

Sábado no Archimedes tem Cecília Massa e Banda cantando músicas de

compositores e intérpretes como Djavan, João Bosco, Milton Nascimento,

Tom Jobim, Gonzaguinha, Elis Regina, Maria Rita, entre outros.

A banda é formada por:

Cecília Massa - Voz
Daniel 'Pezim' - baixo
Josué - Piano
Giovani - Bateria

Início: 21:30
Couvert: 3,50

Chopperia Archimedes:
Rua Tiradentes, 121 - Fabrício tel: 3312 2050
(na esquina da Tiradentes com a Padre Zeferino)

Comunidade da Choperia Archimedes no Orkut
http://www.orkut.com/Community.aspx?cmm=1007853 ]]> http://alamanach.wordpress.com/?p=83 Fri, 20 Jun 2008 11:57:01 +0000 alamanach http://alamanach.wordpress.com/?p=83 In an obscure corner of the Humanities, they study a place called Serindia. Serindia appears on no map, as it had no political identity. It was a region defined by a common and very unusual artistic culture. It was a fusion of two great cultures, actually, and it produced some stunningly beautiful artworks. Serindia had no definite geographical boundaries, it was just a curious borderland.

Right now I am south of Serindia, in a place that has served as a crossroads between culures since the time of Alexander the Great. Travellers to Serindia use to pass through here. There have been great treasures here, though little fusion. You could say I am on the highway, along which goods were transported, not in the foundries in which they were made. Art was traded here, but that doesn't mean that artistry was mixed.

That road long ago turned to dust, and you won't find much high culture moving through here now. But if we move ourself back in time, to happier days, we can look to the west, and see an ancient event unfold that has meaning for this place today.

The Siege of Syracuse, 212 BC, was a minor footnote in the Second Punic War. While Hannibal was making a holy terror of himself, marauding through Italy, King Hieron of Syracuse broke with Rome and sided with the Carthaginians. Being an island seaport town, Syracuse soon had 60 Roman warships bearing down on it, under the command of General Marcus Claudius Marcellus.

The Romans, of course, possessed one of the most powerful armies in history-- even Hannibal was eventually defeated by them. They were the acme of physical might. But little Syracuse had something it could pit against them; Archimedes, one of the greatest minds in human history. Archimedes discovered how levers worked, he wrote the law of bouyancy that we still use today, and he came breathtakingly close to inventing calculus-- 1900 years before Isaac Newton. He found the value of Pi, the square root of three, and the volume of a sphere relative to a cylinder that contains it. He was the acme of intellectual ability. Hieron assigned Archimedes to the defense of Syracuse against Marcellus, in a perfect test case of brains versus brawn.

Marcellus' ships approached Syracuse. From some distance out, ballistic weapons of various sorts struck at them with unnerving accuracy. But the fleet pushed on, and got inside the range of these weapons. There are reports-- bordering on the implausible, but reports all the same-- that Archimedes burst ships into flames with some kind of death ray. Using large copper mirrors in a parabolic array to focus the heat of the sun, this just might have been possible. How many ships he supposedly took out by this means is not known, but in any case the advance continued.

Archimedes' final weapon forced Marcellus into a tactical retreat. As ships reached the city wall, a giant mechanical arm reached out from inside the city. The clawed arm plucked a ship from the water, hoisted it up, and shook it. Romans fell to their deaths, and the arm cast the entire ship away like a piece of debris. It then reached for another ship...

Marcellus fled, and saw that a naval attack was impossible. He retreated, grumbling about the mathematician "who plays pitch-and-toss with our ships." The Romans would have to find another way. What resulted was a long, slow, and comparatively dull siege that lasted two years. The Romans circled Syracuse and camped far outside its walls, hoping to starve their enemies out. They could not get close enough to stop all resupply, though one imagines that they did make life uncomfortable.

In the end Syracuse fell. Brawn won. The victory was not dramatic or inspiring; the Romans simply had to be patient more than anything. And as they swept into the city, Marcellus' instructions were clear: he wanted the mathemtician taken alive. Unfortunately, Archimedes was struck down anyway by some anonymous soldier who didn't know who he was. Thus ended a brilliant career.

The battle of Syracuse was not an abberation; might is stronger than thought. Archimedes taught us about the lever, and showed us how it can generate incredibly large amounts of force from a tiny force input. But even the cleverest lever needs that initial tiny input. Thinking alone cannot provide it. Somewhere, there must be muscle. Neuroscientist Charles Scott Sherrington recognized this, with his famous quote, "To move things is all mankind can do, and for such, the sole executant is muscle, whether in whispering a syllable or in felling a forest."

Physical force rules the day, and whoever rules physical force controls the state. Max Weber identified the state as having a monopoly on the legitimate use of violence, the ends to which this violence is put being a lesser concern. As an American, I notice many Americans (and many other first-worlders, for that matter) tend to forget this. For us, physical violence has been under firm control for so long-- the question of who holds the sword has been resolved for so long-- that we forget it was ever a point of discussion. Thus we talk about pens being mightier than swords. We no longer remember what swords do.

In 1990, I was in Kathmandu, Nepal, having arrived just before pro-democracy riots broke out. My hotel was two blocks from the royal palace, and we could her shouting and gunfire all day long. Some 150 protestors were shot. That night, the Nepalese Army rolled in, and established martial law and a 24-hour curfew. For five days, nothing moved. No one was allowed on the streets, the employees at our hotel could not go home, and the Army didn't really care if we were running low on food. Nobody was moving and nobody was getting hurt.

(As an aside, I should mention that Nepal was something of a democracy already, the rioters were calling for a democracy which would suit them better. That, and the Queen of Nepal at the time was extremely unpopular. As far as I am aware, there were no extrajudicial punishments of key dissidents or other human rights abuses during the martial law period-- just no freedom of movement. There were, as I recall, two one-hour windows opened on day four; this saw no violence, and the curfew ended the next day.)

The multi-party democracy that the protestors were calling for was eventually implemented, and today even the Nepalese monarchy has been discarded. The pen brought about those changes. But those changes happened within the context of a state with a firm grip on the sword. Violence did not spread for five days after the riots, violence was smothered for five days after the riots. That peace-- extreme as it was-- had as its guarantee the might of physical force.

Nepal is to the east of my location. I am in Kandahar, Afghanistan, and last week, the Taliban demonstrated that there is no monopoly on violence here. In a dramatic night attack, they struck the main prison with a massive truck bomb, killing 15 policemen and freeing some 1600 prisoners-- 400 of whom were fellow Taliban. The next morning, there was no martial law, no massive military presence, no material restriction on movement. I was reminded instatly of Kathmandu, and I was depressed by the contrast. The Afghan government could not secure Kandahar simply because it does not have the manpower. It is too weak, in the most physical of terms. We have been on edge for the last few days, waiting to see if the Taliban will attempt to retake the entire city-- the Afghan army would be unable to stop them.

And we find that a pen is useless here. The Taliban refuse to negotiate. This may be for ideological reasons on their part, though it may be for practical reason, too; why negotiate with an enemy when he is too weak to oppose you? The Taliban have nothing to gain by talking; they can simply take by force anything they want.

Physical might is not limited to acts of violence. Afghanistan needs peace, but it also needs roads, and buildings, and irrigation, and so on. That is why I am here; 30 years of war have reduced the Silk Road to a legend, the loya jirga to an anachronism, and the Durrani Empire to a page in history. Afghanistan, once prosperous, is nearly the poorest country on Earth. Even its own artistic culture is forgotten, with the present generations having never learned about Afghanistan's great poets or its traditional folklore-- or Serindia. If this is to change, if civilization is to succeed here, then physical might is needed. Might to repel the Taliban, but might also to shape the land, and to build a country. 

They want me to help with things like irrigation canals. One cannot rebuild canals through words; somewhere, there must be muscle. Afghanistan is in such a primordial state that we can see clearly here the primacy of the sword over the pen. We would do well to remember that the rest of the world, though positioned to enjoy the luxury of discourse, rests on the same rough foundation of physical might.

Image copyright 2000-2008, Metropolitan Museum of Art. Reproduced in accordance with their guidelines, as described here. See also www.metmuseum.org.

John Napier foiled a thief with the aid of logic and a black rooster. For this and other acts of creative problem solving, his servants and neighbors suspected him of witchcraft.

What does this have to do with mathematics?

Math was Napier’s favorite hobby. He invented logarithms to help people handle large numbers easily, and he even created a calculator out of a chessboard. [See how it works: addition, subtraction, multiplication.]

More stories from math history

Isaac Newton caused a UFO scare by flying a kite that carried a lantern. As a boy, Newton was a poor student who became a scholar only to show up the class bully.

Maria Agnesi solved math problems while sleepwalking. When she got stumped, she left the problem on her desk and went to bed. The next morning, she found the correct solution neatly written on her paper.

After teaching calculus to her younger brothers, Agnesi wrote what became Europe’s most popular calculus textbook for the next 50 years. But there was something she loved more than math --- she longed to become a nun, and she devoted much of her life to helping the poor and homeless.

Sofia (Sonya) Kovalevskaya became intrigued with math from reading her bedroom wall, papered with old calculus lecture notes. To escape Russia, where women were not allowed to study mathematics, she arranged a marriage of convenience. When her husband died, she struggled on as a single mother. Kovalevskaya won a prestigious prize for original mathematical research --- and her paper was so brilliant that the judges increased the prize to nearly double what they usually awarded.

The key to understanding

The story of mathematics is the story of interesting people. They faced the normal challenges of daily life as well as the creative challenges of mathematical imagination. For some, calculation and problem solving seemed as natural as breathing. Others worked for years in fits and starts before reaching a solution. Some had long and happy lives. Others died tragically young.

What a shame it is that our children see only the dry remains of these people’s passion. Worksheet exercises are the bare, abstract skeletons of what were once living puzzles.

As Victorian-era math professor James Glaisher said, “I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history.”

Math and history --- what can they possibly have in common?

After all, history is all about kings and wars, while math is numbers and rules. Isn’t it?

“Biographical history, as taught in our public schools, is still largely a history of boneheads: ridiculous kings and queens, paranoid political leaders, compulsive voyagers, ignorant generals --- the flotsam and jetsam of historical currents,” according to popular math writer Martin Gardner. “The men who radically altered history, the great scientists and mathematicians, are seldom mentioned, if at all.”

It does not have to be that way for our children. By teaching math history, we can help our students build a mental picture of the ebb and flow of ideas through the centuries. They will see how men and women wrestled with concepts, made mistakes, argued with each other, and gradually developed the knowledge that today we take for granted.

“I will not go so far as to say that to construct a history of thought without profound study of the mathematical ideas of successive epochs is like omitting Hamlet from the play which is named after him. That would be claiming too much,” wrote Alfred North Whitehead, a pioneer of mathematical philosophy. “But it is certainly analogous to cutting out the part of Ophelia. This simile is singularly exact. For Ophelia is quite essential to the play, she is very charming… and a little mad.”

A “living books” approach

Most homeschool teachers, whatever our curriculum or schooling approach, understand the importance of teaching with “real” books. We read aloud biographies, historical fiction, or the classics of literature. We scour library shelves for the most creative presentations of scientific topics that interest our children. We encourage our high school students to go back to the original documents whenever possible.

And we teach math with a textbook.

One reason for this imbalance is that most of us never learned math history ourselves. We may not even be aware that math has a history. Our teachers made it seem like something handed down from on high, to be accepted and memorized --- and never to be challenged.

Fortunately, when we decide to embark on a tour of math history, we won’t have to go it alone. Several talented and knowledgeable guides are available. Some of them may be sitting on the shelf at your local library right now, just waiting to lead you along the way…

Math history in pictures

You can begin exploring the excitement of mathematics with your children through picture books. What's Your Angle, Pythagoras? offers a fanciful look at the childhood of that famous mathematician. The Librarian Who Measured the Earth tells how Eratosthenes calculated the circumference of the earth using sunlight and shadows. Dear Benjamin Banneker skips forward in history to examine the life of a self-taught African-American astronomer and mathematician.

For beginning readers: A Fly on the Ceiling will make children laugh while they learn about René Descartes, the father of analytic geometry. Ben Franklin and the Magic Squares tells a lively story about one of old Ben's favorite pastimes.

Treat your older children (and yourself) to a couple of our family favorites. Archimedes and the Door of Science describes the life and discoveries of one of the greatest mathematicians who ever lived. Carry On, Mr. Bowditch follows the inspiring life of an 18th-century American hero whose mathematical studies saved the lives of countless sailors.

Edited to add: The Wonderful World of Mathematics will give your children a great overview of math in many cultures from Ancient Egypt to the Industrial Revolution. It is out of print, but used copies are available, or you may be able to get it through your library. [Thanks to Brian Foley for reminding me of this book in the comments below.]

Meet the men (and women) who made math history

Mathematicians Are People, Too [and Volume 2] features short, fictionalized vignettes for teachers to read aloud to elementary or middle school students. The authors have also written a three-volume reference series called Historical Connections in Mathematics: Resources for Using History of Mathematics in the Classroom [Volumes II, III], which offers basic facts and anecdotes about each mathematician, without fictional elaboration, and includes related worksheets.

Older students (and adults) will enjoy Famous Problems and Their Mathematicians. Another combination of anecdotes and activities, the book touches on many ideas that have challenged mathematicians for centuries.

If you want more complete biographies, start with Of Men and Numbers: The Story of the Great Mathematicians. For junior high or older students, try Men of Mathematics, a classic of math history that has been called over-romanticized, but I have not heard of anyone who didn’t enjoy it.

For balance, you may want to sample Women in Mathematics or Math Equals.

And those are just a few of the books available. Indeed, there is a lot more to the history of mathematics than most people ever suspect.

Edited to add: The World of Mathematics looks like a wonderful resource, and I can't wait to get it from my library. Editor James Newman has collected four volumes of articles by eminent mathematicians and other thinkers "from A'hmose the Scribe to Albert Einstein, presented with commentaries and notes." [Thanks to G Johnson for recommending this book in the comments below.]

And here are some of my all-time favorites

William Dunham writes for the general adult audience, but high school students will find his books pleasant reading. The Mathematical Universe offers an A-to-Z smorgasbord of math topics and people. Journey Through Genius follows the development of several discoveries by the great masters of mathematics.

Also for older readers, Keith Devlin provides insight into historical and modern math for a general audience in Mathematics: The Science of Patterns. This book covers a wide range of topics and gives readers an idea of what modern mathematicians do for a living.

Coming soon to "Let's Play Math!" blog

The Internet contains such a wealth of math history resources that they require a post of their own. I plan to work on that for next week [Now posted here.] --- and will, I think, be adding a math history section to my resources page as well.

Finally, my math newsletter always used to include historical tidbits and quotations. I have been republishing the Alexandria Jones stories, but I've fallen behind in the history department. Over the summer, I hope to catch up on my backlog.

:: :: :: :: :: :: :: :: :: :: ::

• Historical Tidbits: Alexandria Jones
• Egyptian Geometry and Other Challenges
• An Ancient Mathematical Crisis
• A Very Short History of Mathematics
• Rewriting the History of Math

The first was a really excellent piece by Eileen Myles, about a notebook she lost on a trip to Canada. It's a fascinating essay in a number of ways, but especially for its discussion of how a writer's view of her own writing is changed by the deposit of her papers in a special collections library. As she writes:

The problem with writing on the plane is not your neighbor. It's your own growing sense that these mango-toned reflections at dawn over Buffalo will be read by someone you never met. They will meet this.... A notebook is the definition of private writing — private living. It's precareer and postcareer in that it's the only writing only you know as long as there is a you. And that excites me anew. There being a space of knowing apart from any selling, sharing, even making. Just sketching out — OK, I have to use my favorite new theory word: episteme... The word felt like god. It means the possibility of discourse.... It's all that my notebook gets told.

Apart from being written in this really incredibly skillful stream-of-consciousness that alleviates whatever annoyance I usually have about autobiographical writer-writing-about-writing pieces, the essay touches on a lot of issues I'm really interested in but haven't read much about: air travel and its weirdness and beauty; lost books, lost words, and the places they go, the spaces they occupy, the ways that they return to "nature" (Myles is fantastic on this); especially the relationship between working writer and archive. How does a writer maintain a sense of privacy, knowing all of her creative work is supposed to end up being read? How does that sense of one's own importance — all you produce is valuable and worthy of preservation — affect one's future work, one's sense of privacy, one's record keeping or lack thereof? Most uncomfortably for a librarian: is preservation necessarily a good thing? Has the mania for the literary archive gone too far? Are we, the archivists and special collections librarians of the world (and especially the U.S.), intruding too much into the ongoing creative lives of our creative thinkers? Do we need to back off? (There's a conference touching on these issues later this year at the Ransom Center in Austin -- the institution spurring much of the current mania.)

Then there's an essay on Aby Warburg, the brilliant, occasionally insane art historian. He founded the Warburg Institute in London. He was the oldest son of an extremely wealthy banking family, and made a deal with his younger brother that the younger brother could take control of the family business so long as he agreed to buy Aby whatever books he wanted for the rest of his life. He set about doing just that, and organized his library on "the law of the good neighbor." As Leland de la Durantaye explains, "the various sections and the books within them were arranged as a function of their ability to engage with the books on either side of them." Here, then, is a personal library the likes of which Anne Garreta wrote about so well in "On Bookselves" (see my earlier entry "The Dream of Total Recall"). Warburg also worked on a massive project, called Mnemosyne, throughout his life: in it (as I understand), disparate images were juxtaposed to follow the path of themes, motifs, and ideas throughout the history of art. I want to read some of Warburg's stuff now.

Then there's Avi Davis's "The Brain and the Tomb," about the Archimedes Palimpsest, the manuscript of Archimedes's work which was (partially) scratched out and written over by a Greek monk in the thirteenth century. Of course I love palimpsests: there's no better physical metaphor for the dense, confusing, complicated paths that history takes, the ways that ideas are undervalued, written over, reevaluated, belatedly treasured. As Davis points out, very little has been written about the visible text of the palimpsest, the Greek prayers, which are now being ignored as squadrons of scholars pore over the Archimedes text beneath. We're always looking one way, missing what's under our noses as we sniff after some other "more important" idea or sensation; Warburg was on to this, and so is Myles, searching for authentic experience and immediate, personal contact with her own thoughts, ideas, life (harder than it sounds). Of course, this is why librarians preserve, this is why we fear the discarded: one day it will be wanted, you see, but it will be lost — and the episteme it may have made possible will be impossible for the lack of its existence.

http://de.wikipedia.org/wiki/Quadratzahl

Eine Quadratzahl ist eine Zahl, die durch die Multiplikation einer ganzen Zahl mit sich selbst entsteht. Beispielsweise ist 12 x 12 = 144 eine Quadratzahl. Die ersten Quadratzahlen sind

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …

Die Bezeichnung Quadratzahl leitet sich von der geometrischen Figur des Quadrats her. Die Anzahl der Steine, die man zum Legen eines Quadrats benötigt, entspricht immer einer Quadratzahl. So lässt sich beispielsweise ein Quadrat mit der Seitenlänge 4 mit Hilfe von 16 Steinen legen.

Aufgrund dieser Verwandtschaft mit einer geometrischen Figur zählen die Quadratzahlen zu den figurierten Zahlen, zu denen auch die Dreieckszahlen und Kubikzahlen gehören. Diese Begriffe waren schon den griechischen Mathematikern der Antike bekannt.

Eigenschaften

• Gerade Quadratzahlen sind das Quadrat gerader Zahlen.
• Ungerade Quadratzahlen sind das Quadrat ungerader Zahlen.
• Jede Quadratzahl hat eine ungerade Anzahl von Teilern.

Formeln zum Generieren von Quadratzahlen

Jede Quadratzahl n² ist die Summe der ersten n ungeraden Zahlen.

Dieser Artikel basiert auf dem Artikel http://de.wikipedia.org/wiki/Quadratzahlaus der freien Enzyklopädie Wikipedia und steht unter der GNU-Lizenz für freie Dokumentation. In der Wikipedia ist eine Liste der Autoren verfügbar.

http://de.wikipedia.org/wiki/Primzahl

Eine Primzahl ist eine natürliche Zahl mit genau zwei natürlichen Zahlen als Teiler, nämlich der Zahl 1 und sich selbst. Das Wort Primzahl kommt aus dem Französischen (nombre premier) und bedeutet: ’die ersten Zahlen’. Primzahlen sind also 2, 3, 5, 7, 11, … Die fundamentale Bedeutung der Primzahlen für viele Bereiche der Mathematik beruht auf den folgenden drei Konsequenzen aus dieser Definition:

• Primzahlen lassen sich nicht als Produkt zweier natürlicher Zahlen, die beide größer als eins sind, darstellen. Diese Eigenschaft kann auch als Definition des Begriffes Primzahl verwendet werden.
• Lemma von Euklid: Ist ein Produkt zweier natürlicher Zahlen durch eine Primzahl teilbar, so ist bereits einer der Faktoren durch sie teilbar.
• Eindeutigkeit der Primfaktorzerlegung: Jede natürliche Zahl lässt sich als Produkt von Primzahlen schreiben. Diese Produktdarstellung ist bis auf die Reihenfolge der Faktoren eindeutig.

Eine natürliche Zahl größer als 1 heißt prim, wenn sie eine Primzahl ist, andernfalls heißt sie zusammengesetzt. Die Zahlen 0 und 1 sind weder prim noch zusammengesetzt.

Bereits die antiken Griechen interessierten sich für die Primzahlen und entdeckten einige ihrer Eigenschaften. Obwohl sie über die Jahrhunderte stets einen großen Reiz auf die Menschen ausübten, sind bis heute viele die Primzahlen betreffenden Fragen ungeklärt.

Über zweitausend Jahre lang wusste man keinen praktischen Nutzen aus dem Wissen über die Primzahlen zu ziehen. Dies änderte sich erst mit dem Aufkommen elektronischer Rechenmaschinen, wo die Primzahlen beispielsweise in der Kryptographie eine zentrale Rolle spielen.

Liste der Rekordprimzahlen nach Jahren

Zahl	Ziffernanzahl	Jahr	Entdecker (genutzter Computer)
217 - 1	6	1588	Cataldi
219 - 1	6	1588	Cataldi
231 - 1	10	1772	Euler
(259 - 1)/179951	13	1867	Landry
2127 - 1	39	1876	Lucas
(2148+1)/17	44	1951	Ferrier
180·(2127-1)2+1	79	1951	Miller & Wheeler (EDSAC1)
2521-1	157	1952	Robinson (SWAC)
2607-1	183	1952	Robinson (SWAC)
21279-1	386	1952	Robinson (SWAC)
22203-1	664	1952	Robinson (SWAC)
22281-1	687	1952	Robinson (SWAC)
23217-1	969	1957	Riesel (BESK)
24423-1	1332	1961	Hurwitz (IBM7090)
29689-1	2917	1963	Gillies (ILLIAC 2)
29941-1	2993	1963	Gillies (ILLIAC 2)
211213-1	3376	1963	Gillies (ILLIAC 2)
219937-1	6002	1971	Tuckerman (IBM360/91)
221701-1	6533	1978	Noll & Nickel (CDC Cyber 174)
223209-1	6987	1979	Noll (CDC Cyber 174)
244497-1	13395	1979	Nelson & Slowinski (Cray 1)
286243-1	25962	1982	Slowinski (Cray 1)
2132049-1	39751	1983	Slowinski (Cray X-MP)
2216091-1	65050	1985	Slowinski (Cray X-MP/24)
2216193-1	65087	1989	„Amdahler Sechs“ (Amdahl 1200)
2756839-1	227832	1992	Slowinski & Gage (Cray 2)
2859433-1	258716	1994	Slowinski & Gage (Cray C90)
21257787-1	378632	1996	Slowinski & Gage (Cray T94)
21398269-1	420921	1996	Armengaud, Woltman (GIMPS, Pentium 90 MHz)
22976221-1	895932	1997	Spence, Woltman (GIMPS, Pentium 100 MHz)
23021377-1	909526	1998	Clarkson, Woltman, Kurowski (GIMPS, Pentium 200 MHz)
26972593-1	2098960	1999	Hajratwala, Woltman, Kurowski (GIMPS, Pentium 350 MHz)
213466917-1	4053946	2001	Cameron, Woltman, Kurowski (GIMPS, Athlon 800 MHz)
220996011-1	6320430	2003	Shafer (GIMPS, Pentium 4 2 GHz)
224036583-1	7235733	2004	Findley (GIMPS, Pentium 4 2,4 GHz)
225964951-1	7816230	2005	Nowak (GIMPS, Pentium 4 2,4 GHz)
230402457-1	9152052	2005	Cooper, Boone (GIMPS, Pentium 4 3 GHz)
232582657-1	9808358	2006	Cooper, Boone (GIMPS)

Dieser Artikel basiert auf dem Artikel http://de.wikipedia.org/wiki/Primzahlaus der freien Enzyklopädie Wikipedia und steht unter der GNU-Lizenz für freie Dokumentation. In der Wikipedia ist eine Liste der Autoren verfügbar.

http://de.wikipedia.org/wiki/Kreiszahl

Die Kreiszahl π (pi) ist eine mathematische Konstante; ihr numerischer Wert beträgt

π = 3,14159...

Sie beschreibt in der Geometrie das Verhältnis des Umfangs eines Kreises zu seinem Durchmesser. Dieses Verhältnis ist unabhängig von der Größe des Kreises. Die Kreiszahl wird mit dem kleinen griechischen Buchstaben pi (π) bezeichnet, dem Anfangsbuchstaben des griechischen Wortes περιφέρεια periphereia (Randbereich) bzw. περίμετρος perimetros (Umfang). Die Bezeichnung pi (π) erschien erstmals 1706 in dem Buch Synopsis palmariorum mathesos (zu Deutsch etwa: Eine neue Einführung in die Mathematik) des aus Wales stammenden Gelehrten William Jones (1675–1749). Die Kreiszahl π wird auch Archimedes-Konstante oder ludolphsche Zahl (nach Ludolph van Ceulen) genannt.

Die ersten 100 Nachkommastellen

Da π eine irrationale Zahl ist, lässt sich ihre Darstellung in keinem Stellenwertsystem vollständig angeben: Die Darstellung ist stets unendlich lang und nicht periodisch. Die ersten 100 dezimalen Nachkommastellen lauten

π = 3,14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 …

Dieser Artikel basiert auf dem Artikel aus der freien Enzyklopädie Wikipedia und steht unter der GNU-Lizenz für freie Dokumentation. In der Wikipedia ist eine Liste der Autoren verfügbar.

]]> http://newleafnews.wordpress.com/?p=45 Tue, 10 Jun 2008 16:10:17 +0000 New Leaf News http://newleafnews.wordpress.com/?p=45 by Margaret Lukens, New Leaf + Company

When the water spray hits your face, does your brain go into overdrive? Some people get their very best ideas in the shower. Here are two ideas for catching those “shower thoughts”.

At just 4 ½ by 3 ½ inches, the miniature diver’s slate doesn’t have a lot of real estate, but it’s enough to keep some priceless thoughts from going down the drain. The cost at scuba.com is about $30.

My thanks to Jeri Dansky and her blog Jeri’s Organizing & Decluttering News for this smart idea.

For the real shower-stall Archimedes, here’s another method. Though cheaper, simpler, and more effective than the dive slate, it is not for the fastidious. In fact, I hesitate to share it. Okay, you’ve pried it out of me: keep a supply of Crayola washable markers in the shower and just write on the shower door.

Got any ideas for capturing shower thoughts? Record them here before they evaporate.

It has become de rigueur in the last century, the last philosophical century, to approach thought via a limit-concept, and our philosophies are marked by the tropes of this finitude. In short, thought is tragic; or put slightly differently, thought is Heideggerian. Tragic since the ultimate horizon of thought and being are found in the human. The question that arises in this paper is whether the next philosophical century is possible. Is a non-tragic thought possible? Or put slightly differently, is it possible for a thought to exist that does not simply reject the previous conditions of thought but instead pushes these conditions to their most radical conclusion?

The difficulty that arises is from where a non-tragic thought could find its bearings. Tragic thought does not simply affect the current position of philosophy, but instead transforms the entire history of philosophy, through Heidegger’s appropriation of ancient thought through his studies from Parmenides to Aristotle up through Augustine. What is needed, then, is a position within the stream of philosophy that is not already appropriated by the tragic moment in thought. It is the wager of this paper that that position is the one occupied by Titus Lucretius Carus, and his magnificent De Rerum Natura (On the Nature of Things ). Lucretius is, quite simply, unthinkable from the perspective of tragic thought, from Heidegger’s thought, and thus remains an obscure figure in the history of philosophy, at best. More importantly, Lucretius continually runs ahead of Heidegger and thus he appears to be beyond the tragic era. What Lucretius’ philosophy provides is the possibility to re-think the tropes of thought that have been brought forth by the tragic century and thus, allows for the beginning of a non-tragic thought.

Ultimately, tragic thought operates via a particular relationship between philosophy and science; it is an assertion that science must take philosophy as its condition. This point is made explicitly in Heidegger’s earliest work, whether philosophy is seen as a fundamental ontology versus science’s regional ontology or philosophy as critical science with science as positive science. In his lecture course on Kant’s Critique of Pure Reason, Heidegger states,

…whereas the physicist defines what he understands by motion and circumscribes what place and time mean—whereby he relies in part on ordinary concepts—still, however, he does not make motion’s way of being a theme of his investigation. Rather he examines only certain movements. The physicist does not inquire into the ownmost inner possibility of time, but rather uses time as that with respect to which he measures motion…The scientific methods have been developed precisely in order to explore beings. But they are not suited for examining the being of these beings. If this is to happen, then what we need is not to objectify a being, e.g., the existing nature as a whole, but the ontological constitution of nature or the being of that which exists as historical.

In the course the previous year, Heidegger states, “…the other sciences, mathematics, physics, history, philology, linguistics, do not begin by asking what is mathematics (etc.); instead they just set about their work, they plunge into their subject matter…In the very essence of all these sciences, in the fact that they are positive sciences, versus philosophy, which we call the critical science.” For Heidegger, science is only capable of dealing with beings but is incapable of understandings its own foundations in being. Philosophy, on the other hand, is capable of providing this foundation via its role as fundamental ontology. In short, science remains lacking, existing only as a possible philosophy at best.

But what is the effect on science from this approach? At the surface level, this requirement is little more than a version of the Kantian transcendental, a search for the conditions of possibility for any science. But what must be recognized, and this is the contention of this paper, is that this seemingly innocuous requirement is a poison pill and that science’s acceptance of philosophy as its condition is nothing short of the destruction of science itself, and the casting of philosophy into the sea of idealism. Why is this the case?

Heidegger begins by asserting the necessity of science’s acceptance of philosophy as its condition, pointing to antiquity for proof of this relationship. He then adds that what philosophy, as the “freest possibility of human existence” provides is the “most original and necessary relationship” to being. But since Heidegger’s understanding of being is its openness, its givenness, or its manifestation, then he has affirmed himself within the lineage of Kant, and thus, science, with its condition set by philosophy, loses its ability to touch the real, the reality indifferent to dasein or the human. But such a loss cannot be overcome by science and so results in science’s utter destruction. Being, for Heidegger, cannot be separated from its relationship to the human, imagined as dasein, and thus remains a humanistic idealism masquerading as a radical philosophy.

But what if the relationship is reversed? What if science becomes the condition of a philosophical thought? It is already in Plato where this understanding of philosophy is put into practice. Mathematics and specifically its non-empirical status serves as the student’s, and our own, path from becoming to being. It is mathematics that sets itself against doxa, against the endless assertion of one’s opinion and perspective. Let us look at Plato’s Republic Book 7:
Think a little and you will see that what has preceded will supply the answer; for if simple unity could be adequately perceived by the sight or by any other sense, then, as we were saying in the case of the finger, there would be nothing to attract towards being; but when there is some contradiction always present, and one is the reverse of one and involves the conception of plurality, then thought begins to be aroused within us, and the soul perplexed and wanting to arrive at a decision asks 'What is absolute unity?' This is the way in which the study of the one has a power of drawing and converting the mind to the contemplation of true being.

What then of Lucretius, this obscure Latin poet philosopher? The insight is Michel Serres’, found in his extended study on Lucretius, The Birth of the Physics . Serres rejects the notion that ancient atomism, specifically Lucretius’, is naïve or non-mathematical. Instead, he asserts that Lucretius’ most fundamental insight, the swerve of the clinamen, operates via a specific dialogue with the mathematics of Archimedes, specifically the proto-calculus of infinitesimals. “Not only did the atom have to be born by way of the treatment of curved elements, in the irrational and differential, or by way of the indefinitely divisible…This is because the angle of contingency may not be subdivided: it is demonstrably minimal. It is null, but without the lines which form it overlaying one another.” Let us look to Lucretius’ actual text to see this for ourselves.

It begins with the laminar flow, the gravitational descent of atoms in the perfect parallel order, then “at absolutely unpredictable times” the atom swerves off its path, slightly, by “only an infinitesimal degree,” a deviation of the smallest possible angle. And thus the lines of the laminar flow become spirals, at utterly incomprehensible moments since they are moments without witness or givenness, as the clinamen begins to wobble, order is broken, the void is joined by the reality of the atom. Atom meets atom, eventually, as the spiral forms a cone and force of gravity is transferred via an encounter. As Althusser will say in his rediscovery of the secret history of materialism: “it is clear that the encounter creates nothing of the reality of the world, which is nothing but agglomerated atoms, but that it confers their reality upon atoms themselves, which without swerve and encounter, would be nothing but abstract elements…” The swerve of the clinamen, long dismissed as “puerile” or a “most monstrous absurdity” or worse yet, used to introduce free will, should be seen as the most basic mathematized thought of a temporality without givenness.

We should pause here just to note the radical difference between this mathematical beginning and the other ancient model that has received so much attention in our tragic era, namely the one discovered in Plato’s Timeaus. The point is not primarily a question on the value of chaos versus order, although this certainly plays a role, but instead the relationship between the world and the human. In Plato, the world is placed into order by the demiurge because of its moral purity, which is to say, because of the coming human soul that the demiurge is preparing. Whereas in Lucretius, the human, the product of the random combination of atoms, is ill-suited for the world, since it exists without reason, and is incapable of forming itself as dominion. Lucretius’ approach is thoroughly scientific; Plato is unable to break from the doxa of Greek religion. To paraphrase Lautréamont, “he who knows and appreciates (mathematics) no longer wants the goods of the earth and is satisfied with (its) magical delights…(the demiurge) only offers (Plato) illusions and moral phantasmagoria.”

What, then, is the effect of such a reversal of the basic relation between science and philosophy? What becomes of the tropes of finitude that we philosophers have been unable to operate without? Let us look at two of these tropes, death of the mortal man and death of the immortal god, and how these are transformed in Lucretius’ philosophy. Heidegger, in the Beitrage, writes, “The uniqueness of death in human Da-sein belongs to the most originary determination of Da-sein, namely to be en-owned by be-ing itself in order to ground its truth (openness of self-sheltering). What is most non-ordinary in all of beings is opened up within death’s non-ordinariness and uniqueness, namely be-ing itself, which holds sway as estranging.” Or in Being and Time, we are presented with “With death, Dasein stands before itself in its ownmost potentiality-of-being. In this possibility, Da-sein is concerned about its being-in-the-world absolutely…As the end of Da-sein, death is the ownmost nonrelation, certain, and as such, indefinite and not to be bypassed possibility of Da-sein. ” In short, death is an event since it pronounces the highest possibility of Dasein and with this, death displays the ultimately groundedness of Da-sein as the ab-ground, the abyss. Death reveals the truth of being to Dasein; this truth being the original givenness of the world, or what Heidegger comes to call the “other beginning.” This other beginning is not the material conditions for a world but instead the ultimate human centeredness, or Dasein centeredness if you prefer, of being itself.

This being said, there is a remarkably different approach to death in the work of Lucretius. He states, “Death, then, is nothing to us and does not affect us in the least, now that the nature of the mind is understood to be mortal.” The most important difference between the tragic approach and Lucretius’ is that Lucretius robs death of its status of the singularity. Death, according to Lucretius, is a property of the body, as it is a property of all bodies. A property according to Lucretius “is what cannot under any circumstances be severed and separated from a body without the divorce involving destruction.” Death then is a property of all bodies, save one, the atom. Humans die, but so do plants and planets. Heidegger magnifies the importance of death since the revealedness of being is directly correlated to Dasein. No such relationship is presupposed by Lucretius, since his mathematics has presented him with the possibility of a thought without human witnesses. Mathematics is, after all, “an instance of stellar and warlike inhumanity.” We should be wary of making the mistake of supposing that Lucretius’ view on death is an historical curiosity, and mere replication of the zeitgeist of Roman philosophy, a masculine warrior’s response to the inevitable. While it is true the Stoics dismissed death as a concern, they also offered themselves an escape clause via their insistence that the gods would honor their triumphs. Lucretius, as will discuss in the next section has no such clause.

What then of the death of the immortal god? The death of God or, to use Heidegger’s later formulation, the fleeing of the Last God is the condition of possibility for world. “The last god has its essential swaying within the hint, the onset and staying-away of the arrival as well as the flight of the gods who have been…In such essential swaying of the hint, be-ing itself comes to fullness. Fullness is preparedness for becoming a fruit and a gifting…Here the innermost finitude of be-ing reveals itself: in the hint of the last god.” Once again we return to the centrality of the human, Da-sein. The fleeing of the Gods reveals be-ing as gift, whether delivered or delayed, it makes no difference. Heidegger has asserted the impossibility of any thought beyond the manifestation of world to Dasein, and tragic thought receives its trope. This position is not unique to Heidegger and is found both before his explicit creation of tragic thought and in those tragic thinkers who have carried in his wake. We can, for instance, see this beginnings of this approach to God’s absence in the sorrow of Nietzsche’s madman, who wonders what humanity will do now that it as lost its greatest treasure. Also, we have a more recent contribution to this thought in the work of Simon Critchley, specifically his newest book Infinitely Demanding. In this work, Critchley locates the motivating force of philosophy itself in the disappointment that the philosopher feels in losing God. What we get in all three of these thinkers is the arrival of Aristotle’s tragic hero, the great man who has suffered a reversal of fortune.

Lucretius, on the other hand, commits the unforgivable sin of rejecting God at its premises, and therefore God’s death as logically impossible. There is simply no role preserved for the gods in Lucretius’ philosophy, since the beginning of the reality occurs via mathematics or physics rather than divine commandment. “For it is inherent in the very nature of the gods that they should enjoy immortal life in perfect peace, far removed and separated from our world; free from all distress, free from peril, fully self-sufficient, independent of us, they are not influenced by worthy conduct nor touched by anger.” It is simply logically impossible that the gods are here in our world or were ever here. To say otherwise to assert the importance of the human and its perspective, but humanity has never been a hero, is not touched by destiny, and therefore, lacks the fortune that vanity perceives.

What is important is to see that these tropes of tragic thought were not selected at random; instead they are the central coordinates of tragic thought itself. The traditional relationship in western philosophy between humanity and God has focused on the “great chain of being,” with the immortal god seating comfortable at the top, well above mortal humans. But the death of god has brought about a collapse of this chain. What has been missed by tragic thought is effect that this event has on the rest of the chain. We have been told repeatedly the loss of God solidifies our finite position, and has engendered disappointment, etc. But the dyad of immortal/mortal can only be maintained so long as the point of comparison, here infinity, survives. The ultimately effect of the loss of God is not the absolutizing of the finitude of be-ing, but instead the infinitizing of reality itself. Much has been made by Alain Badiou of Georg Cantor’s discovery of infinity as number, but we can already see the idea of numbered infinity at play in Lucretius. From a distance the sheep bleed into one another and appear as a mass of white upon the hill, moving as one, and as it is with the sheep, it is with atoms. Once again, it is Archimedes and his Sand Reckoner, which provide the first hints at the infinite, not as divine, but as number.

One possibility that confronts us is to assume that non-tragic thought is simply the opposite of tragic thought. This would be a mistake. Non-tragic thought is tragic thought taken to its most extreme possibility, which is the full embrace of finitude. Contrary to the approach of contemporary thought, human finitude does not operate as a limit to thought but instead allows thought to get beyond the human perspective; the radical acceptance of finitude makes it possible that, to quote Ray Brassier, “Philosophy should be more than a sop to the pathetic twinge of human self-esteem.”

As we have already discussed, Heidegger sees death as the ultimately potentiality of Dasein. The failure that Heidegger succumbs to is the inability to fully embrace this potentiality for all possible Dasein. Heidegger touches twice on this possibility but abandons the path both times to return to the singular death of a particular Dasein. The analysis of the being-toward-death of Dasein begins as the death of others. This first path is found to be limited since “in dying, it becomes evident that death is ontologically constituted by mineness and existence.” And so, although “every Dasein must itself actually take dying upon itself…”, Heidegger fails to truly grasp the “every” of finitude, and continues to think death only as a singular occurrence of a singular Dasein. When Heidegger does approach the concept of death in the collective, it is only in relation to the “the they.” But the “the they” can only think the death of the generic Dasein, “one dies”; and thus, “the they” fails to grasp the real occurrence of death for themselves.

The problem for Heideggerian thought is that commits a similar error. If “the they” cannot think its own death, Heidegger cannot get beyond his own death, which is the death of a specific Dasein. The proper procedure would be the combination of both approaches to death, that is, the recognition that one does indeed die, but also that this one is always a specific one, a Dasein. Thus we fully accept the possibility that “every Dasein” will die, not someday, but at the specific point of extinction, at the arrival of the stellar explosion. This extinction, which is first properly thought by Lucretius, is the absolute exhaustion of the human given.

Now the aged plowsman shakes his head and time after time sighs that his hard labor has all come to nothing…His gloomy sentiments are echoed by the planter of the old and shriveled vine who deplores the tendency of the times…Only he fails to grasp that all things gradually decay and head for the reef of destruction, exhausted by long lapse of time.

From the thought of the absolute finitude of all possible Dasein we arrive at the possible thought of a thinking beyond the existence of Dasein, a point when “the earth be confounded with the sea, and the sea with the sky.” And as we can now think the beyond of Dasein, we can also think the before of Dasein. Through a radical embracing of finitude, a radicality absent from Heidegger and his followers, thought is severed from the limits of humanity and becomes thoroughly inhuman.

Ultimately, tragic thought, in general, and Heidegger’s thought, in particular, remains little more than elaborate and hyper-stylized astrology. In Quentin Meillassoux’s After Finitude, he calls into question Kant’s relationship to his so-called “Copernican revolution” and instead, asserts that what critical and, most importantly for us, post-critical philosophy represents is a “Ptolemaic counter-revolution.” If Galileo’s great insight was to expel humans from the center of reality, then it was Kant’s greatness that returned us to our place. We can even see in Ptolemy the desire to supplement the science of astronomy with a proper metaphysical outlook, so as to understand the effects of the “ambients.” And thus we today are the inheritors of a thought that places the cosmos back in orbit around us, a thought that can only be tragic since it assumes that the movements of the planets are forever tied to the whims and desires of the human ego. Therefore, non-tragic thought is that thought that is willing to go all the way to the end and accept the utter meaninglessness of our very existence and in so doing re-affirm philosophy’s materialist binding to science.

Here's what I do. I go to sleep. I kid you not. But before going to bed, I make sure I've spent a reasonably long period of time fully immersed in the problem, reading up everything I can about it, and stuffing myself with information until I'm at bursting point. I try very hard to solve the problem when compos mentis. If I fail, I sleep on it. Literally zzzzz.

In the middle of the night I might be rudely awakened by the answer trying to break free from my subconscious. Sometimes when wafting in that twilight zone of half wake half sleep, the solution bursts unceremoniously into my hebetudinous mind. Occasionally, I might be taking a stroll, having a shower, chatting with friends, or sitting on my throne performing royal duties, when quite out of the blue, a bolt of lightning strikes.

The common denominator is that when the solution comes, it comes. It almost always arrives unexpectedly, without warning, like a terrorist. And often when I am most relaxed, not thinking any thoughts, non compos mentis. Fortunately I don't take long baths, so I don't have Eureka moments where I leap out of the tub and run down the streets naked.

Archimedes (287-212 BC) had just gotten into his bath which had too much water because it overflowed, when a small region in the right hemisphere at the anterior superior temporal gyrus lit up with electrical activity. He suddenly realised that water displacement could be used to work out the volume and density of the king's crown. Archimedes shouted "Eureka" (I have found it), and forgetting all sense of decorum, ran home naked, whereupon his wife gave him an earful.

Psychology professor Richard Wiseman found that 89% of people obtained their top ideas outside the office. Most when in a relaxed state of mind, such as waking up from sleep. Wiseman, said: "These new results illustrate how our minds are often most creative when we relax and take time away from everyday pressures... In our dreams we produce unusual combinations of ideas that can seem surreal, but every once in a while result in an amazingly creative solution to an important problem". He pointed out that his ressearch showed the relative ease with which ideas are produced, but suggested that bosses should alter the employees' working habits to aid creativity.

I would second that! "Boss, let me sleep on the problem a bit longer!"