Mathematics

Question for Bill Nye

Big Think, an American think tank, invites readers to send in videotaped questions to Bill Nye.  I sent in my question yesterday about how the Greeks viewed the number zero- how they didn't want to adopt it, even though it could have helped them in their math and science, because of what it represented- the void, nothingness,

The Ball that the Greeks Dropped

 The law of three is an example of one of the lost principles of the Western tradition that the Greeks didn't pass on from the Egyptians. 

The law of three is an example of one of the lost principles of the Western tradition that the Greeks didn't pass on from the Egyptians. 

Because it's almost due at the library, I'm reviewing Zero: The Biography of a Dangerous Idea by Charles Seife.  I feel like the book is underrated in its importance.  It got a few remarkably ho-hum reviews from Scientific American and The New York Times when it came out in 2000, but in my view it deserves way more credit.  I don't know if Seife has a mystical bent, but to me, it lends itself to breaking down a big wall between Westerners and their souls.

As you might recall, I learnt about the Enneagram at a monastery in Winnipeg 4.5 years ago from a scholar who is little-known in the Enneagram community, David Walsh.  The context that he gave the Enneagram fascinated me, and that weekend has been emblazoned in my memory for the uniqueness of his approach.  He came at the Enneagram (which is a personality-typing system with a mystical side) from the point of view of the Classics- Pythagoras, Plato, the Enneads and the Divine Forms all led up to the modern-day Enneagram we know today.  (That he and his wife are retiring after teaching the Enneagram for 30+ years without passing on their knowledge to a successor is incredible to me and blows my mind.  I called him earlier this year in January to see if he'd be willing to talk about any of the Classical references further, and he declined politely.) 

Russ Hudson, one of the world's most renowned Enneagram teachers today also comes at it from a Classics point of view, but doesn't get into the Greek contribution as much.  His love affair is with Egypt, who gave the Greeks their ontology.  But we can explain a lot by how today's Western society got its flavor by looking at the Greeks, and that's exactly what Seife does in his book about the number zero, and the Greek philosophers' aversion to the idea it represented.  They limited their numeral system to the numbers 1 to 9 because of scariness of the idea of the void.  If it's possible to have a vaccuum, then the earth may not be the centre of the universe, and what does that mean for the "specialness" of mankind?  Early Greek philosophers worked their way around it, enabling Christianity to subsequently fudge its way around it until the Church was finally forced to deal with it after the dark ages, and it was actually Judaism that showed Christianity a way to work it into its theology. 

Islam and Hinduism were comfortable with zero.  Muslims were using it in their number system via algebra, and Hindus had been grappling with nothingness already way before Algebra came on the scene.

My argument is that the Enneagram's purpose was-- and is-- to show Westerners how to deal with their inner void, if you will.  That's why Enneagram teacher and historian Russ Hudson follows up Gurdjieff's quote, "The earth can only be saved when the energy of the West meets the wisdom of the East", with "But the West isn't ready to meet the East because we don't even know our own mystical traditions, so when we meet, how can we have a coherent conversation?"

When Westerners have been given the tools for approaching -- and integrating-- their inner void, they can start to solve some of the most intractable issues the world faces like terrorism, global warming, and growing divide between the rich and poor.  But until we have the emotional intelligence to approach that void, we'll just keep ramming the same truck into the same brick wall hoping for the same results. 

Seife's book outlines how poorly our post-Egyptian Western heritage handled the idea of the infinity and the void.  I propose that heritage is still having its effects on Westerners' mindset -- and therefore policy -- today. 

A Slow Motion Trainwreck that Ends in the Dark Ages

  Aristotle tutoring Alexander of Macedon.   Image from Wikipedia

Aristotle tutoring Alexander of Macedon.  Image from Wikipedia

Yesterday, I wrote that Aristotle set a trainwreck in motion by ignoring zero; instead of just humbly adopting the baggage-laden number and letting his philosophy-chips fall where they may, he laid out a series of arguments "proving" that Greek mathematicians didn't need zero or the idea of infinitity.  The train got pirated by the earliest Christians who were looking for a systematic way of organizing their "theology" and explaining it to their pagan neighbors who had never heard of a religion with only one god before.  Being in the awkward position of not quite being a philosophy, nor a typical religion with rituals and sacrifices, Christians needed a systematic way of organizing their "beliefs" in order to win converts(Karen Armstrong, 1993, pg. 93), and so they chose Aristotilieanism to help them prove the existence of God.

Alexander the Great's massive empire, in purple.  Click to expand.  Image from Wikipedia.

But before we get into the scene of the accident in the Dark Ages- ground zero if you like- where science and math were relegated to the offices of monks in charge of figuring out when the church was to celebrate Easter, we must first take into account the man who made the proliferation of the Aristotilean way of thinking all over Asia and north east Africa possible.  The Western world would not be, and would certainly not be what it is today without a young man named Alexander of Macedon, better known as Alexander the Great.  From his tiny Greek homebase, after being tutored by Aristotle himself, and once he succeeded his Father Philip, extended his kingdom far and wide by ravaging much of his continent with back-to-back military campaigns, submitting civilization after civilization under Hellenistic rule between 323-30 BC.  According to Wikipedia, this size of his empire measured "5.2 million square kilometers, the largest state of its time".  (By comparison, the USA is just under 10 million square kilometers- STILL.  This guy was under 30.).  What he accomplished in his short career as one of history's most impressive military commanders is either truly gruesome or truly great, depending on whose side you're taking. 

But first we take a look at India, whose Hindus were quite comfortable with the zero, and whose north-west corner was briefly a point of contact between the East and West- The Greeks and the Hindus.

To be continued...

 

You Think You Know Somebody

I was disapointed to read this morning that someone I've always taken for granted, whose wisdom I thought was undisputable, cobbled together a philosophy that suited him because he was afraid of the alternative.  Aristotle.  You think just because somebody's treatises are the very foundation for Western philosophy, and eventually the three major monotheistic religions that emerged from their work, that they're actually going to be good for your civilization.  It turns out, Aristotle was good for us in a way (logic!), but in a stomach-turning twist of irony, he's also answerable for some of the most annoying, intractable issues we face today (religious people who oppose science!). 

The Latest Adopters Ever

On Wednesday, I wrote about an early adopting community in the tech sector that I kind of follow.  They like to be ahead of the curve - way ahead for these guys- so they can know the trends before they happen.  They want to know what to invest in while the valuations are still low. 

The Greeks were the latest adopters ever.  It took them thousands of years to adopt the number zero, even though their neighbors were using it for astronomy, because to them it represented the void and nothingness, which, because their minds couldn't fathom it, they just decided to avoid it.  They much preferred the perfection of perfect ratios anyway.  Instead of implementing the zero, their measurements just became smaller and smaller fractions in tinier and tinier increments. 

Procrastination is a little bit like being terrified of the zero.  Sometimes you have to step into the void and start a new equation.

Numbers and The Thug Life

 This guy.  Charles Seife.  What a writer, but look at him.  Doesn't he look pretty surprised and confused that his picture's being taken?  I think he was walking down some halls of academia and a gang of thugs is following after him, saying, "Hey!  Hey you!  Seife!  Turn around!" and Seife is like "Wh- wh- what's going on?  What do you want from me?"  and the thugs are like, "Don't say anything- just stand against that wall and smile." and Seife is like, holy shit- what are you going to do with my picture? and the thugs are like, "We're gonna put it on the back inside flap of your book jacket.  This guy 'Penguin' said you wanted it against a white background.  Say cheese, mofo.

This guy.  Charles Seife.  What a writer, but look at him.  Doesn't he look pretty surprised and confused that his picture's being taken?  I think he was walking down some halls of academia and a gang of thugs is following after him, saying, "Hey!  Hey you!  Seife!  Turn around!" and Seife is like "Wh- wh- what's going on?  What do you want from me?"  and the thugs are like, "Don't say anything- just stand against that wall and smile." and Seife is like, holy shit- what are you going to do with my picture? and the thugs are like, "We're gonna put it on the back inside flap of your book jacket.  This guy 'Penguin' said you wanted it against a white background.  Say cheese, mofo.

I'm reading this book by Charles Seife called Zero: The Biography of a Dangerous Idea, and it should be mandatory reading for every adult who's in a position to 1) lock someone away, 2) impose a death sentence, or 3) inflict any kind of punishment, because it demonstrates just what kind of clowns we can be when we find an idea threatening to our settled way of thinking.   Humans are capable of some pretty terrifying acts of violence when something threatens their worldview.  Last night I went to go see "The Imitation Game" with my parents, a movie about how Allan Turing invented the computer during World War II under massive pressure.  His job was to decode the German's encrypted messages that got radioed to their warships every day.  After successfully decoding their messages, the war ended, and he went home, only to be convicted of being a homosexual.  He had to take hormonal therapy to reverse his preferences, which led to his committing suicide.  We sure have perfect vision in hindsight, but at the time, we feel like we're being completely rational.

Speaking of being rational, in studying the Enneagram, I've come across Pythagoras a few times, and I just assumed he was this wise old man with a long beard who did math and geometry all day and taught his students about the spiritual quality of numbers.  It turns out he did a lot more than sit around and stoke his beard and think about numbers- there was an ugly side to him, and I don't mean his hypotenuse side.  He sentenced one of his own students, Hippassus of Metapontum to death by drowning (so the legend goes) for revealing to "the outside world" that irrational numbers were a mathematical possibility. 

It was a discovery completely at odds with the spirituality of the brotherhood- their entire worldview-- not to mention their understanding of their own role in the world-- was based on being able to make sense mathematically of their universe. 

For everything in the universe to be govered by ratios, as the Pythagoreans hoped, everything that made sense in the universe had to be related to a nice, neat proportion.  It literally had to be rational. (pg. 35).

As Greek scholars tended to be interested in mathematics-- having learned their math from the Egyptians who were also leaders in science, but moreso in geometry-- many of them studied ratios and "the interchangeability of math, music, and nature".  To them, perfect ratios were a way to connect to divinity.  Even the planets, in their minds, made music in perfect fifths- the most harmonious ratio in music- as they rotated around the earth (that's where the music of the spheres comes from).  So when Hippassus got out a really tiny ruler and realized that the diagonal line connecting two inner corners of a perfect square can't exactly be plugged into an a/b ratio where a and b are commensurable (able to be meaured by a comon yardstick), it didn't make sense.

Irrationality was dangerous to Pythagoras as it threatened the basis of his ratio-universe.  To add insult to injury, the Pythagoreans soon discovered that the Golden Ratio, the ultimate Pythagorean symbol of beauty and rationality, was an irrational number.  To keep these horrible numbers from ruining the Pythagorean doctrine, the irrationals were kept secret.  Everyone in the Pythagorean brotherhood was already tight-lipped - nobody was allowed even to take notes - and the incommensurability of the square root of two became the deepest, darkest secret of the Pythagorean order.." (pg. 37).

Let this be a lesson to everyone in power, or who has any measure of power over anyone- to sure, be bold about our opinions, but to hold them loosely.  We might end up being the next judge who sentences the next Allen Turing to hormonal therapy, or Pythagoras who makes a massive mistake in discounting irrational numbers.

Sacred mathematics is probably like fine wine

  The Enneagram symbol.  It's made up of three parts- the circle, the triangle, and the hexad.  You can see on the hexad, the numbers (if you start at 1) go from 1, 4, 2, 8, 5, 7.   Image from theenneagraminbusiness.com

The Enneagram symbol.  It's made up of three parts- the circle, the triangle, and the hexad.  You can see on the hexad, the numbers (if you start at 1) go from 1, 4, 2, 8, 5, 7.  Image from theenneagraminbusiness.com

As I'm neither a wine afficionado or a mathematician, I can't verify the above claim for absolute certain, but I'm assuming sacred mathematics ranks quite highly on the list of most fascinating phenomena ever.  Plato called number symbolism “the highest level of knowledge”.

At my first workshop on the Enneagram four years ago, our teacher had someone get out their calculator and do this math around the number 9 (and of course, the Enneagram is built around the number 9).  I haven't heard anyone talk about it since, so I was happy to find it in P.D. Ouspensky's book, In Search of the Miraculous.  (He was one of the teachers who recorded Gurdjieff's teachings.) 

So the premise here in the following sequence is that the number 7 was a sacred number as far back as with the Sumerians and the Egyptians.

1/7=.142857...

2/7=.285714...

3/7=.428571...

4/7=.571428...

5/7=.714285...

6/7=..857142...

7/7=.999999...

You can see the 1-4-2-8-5-7 repeating over and over again in the same sequence in these divisions, just starting with a different number, and when you do a little thing the ancient Greeks used to call "theosophical addition" (adding up the sum of the resulting number), you get 9 each time.  I have to read about this tomorrow and will blog about it, but it was just so good to be reconnected with something I'd heard before, and didn't know how to find again.

The Number Zero

This was me working in a claims call centre- the job that made me realize I should probably just stick with right-brained work.

In order for me to understand math, it has to be in the context of a story, preferrably with lots of emotion.  I'm right-brained, so when someone tells me a story where calculations and numbers are needed, but there's elation or heartbreak involved, I have a decent hope of understanding the math. 

That's why, when I first heard about the Enneagram- a personality typing system that ascribes a personality type to each of the numbers between 1 and 9, a spark of interest was lit.  Fours years after my workshop with him, I still remember a line our teacher said that has nipped at my heels this whole time, "When they discovered irrational numbers, there was an inherent horror in this idea that you could be stuck forever in a repeating decimal that never ended", the repeating decimal a metaphor for being stuck in one of the nine personality ruts.  It was an incredible fusion of left- and right-brained principles for me on a topic that meant so much.  So begins today my search for answers around this connection between math and the transformation of our souls.  And in order to understand irrational numbers, first we have to understand the importance of the discovery of zero.

Plimpton 322, the name of the Babylonian clay tablet discovered by George Plimpton in 1922.  One of the oldest and most recognized mathematical discoveries, it dates back to 1800 BC.  Image courtesy of isaw.nyu.edu. 

It's wierd to think that the number 0 didn't always exist- that it had to be thought up by mathematicians in response to a problem they didn't really know how to define until hundreds of years after Christ.  It was the Indians who first solved the riddle in the context of the place-value system- before that, the Babylonians just left a space where the zero should be (Pickover, 80) and Egyptians used the symbol Nfr- the symbol for beauty and completeness- for zero (source).  But that was cuneiform and hieroglypic math respectively- the kind where you draw a line or symbol for one, and two lines or symbols for two, and so on. 

The number system most cultures use today is a place-holder system called the decimal system with a base of ten and a 0 to distinguish the tenth place from the first, a Hindu invention that changed history forever. 

The introduction of the zero into the decimal system ... was the most significant achievement in the development of a number system, in which calculation with large numbers became feasible.  Without the notion of zero, the ... modeling process in commerce, astronomy, physics, chemistry and industry would have been unthinkable.  The lack of such a symbol is one of the serious drawbacks in the Roman numeral system" (mathematician Hossein Arsham quoted in The Math Book by Clifford Pickover).

"Right in the center of this image the characters '270' stand out, surprisingly modern numerals etched in an ancient temple in Gwalior." Image and caption from smithsonianmag.com

The oldest undisputed documentation for the Indians' use of the 0 is in from a tablet in Gwalior, India, dated to 876 AD.  Their mathematicians used a small, raised circle to hold its place.  The Bakhshali manuscript, found around Pakistan in 1881 might be older than that, but nobody agrees on how old it is.  Mathematician Amir Aczel just came out with a book that describes his search for an even older zero, and he says he found it in Cambodia inscribed on a stone listed as K-127, which would date to 683 AD, about 200 years older than the Gwalior one. 

 Fibonacci.  Illustration courtesy of www.storyofmathematics.com

Fibonacci.  Illustration courtesy of www.storyofmathematics.com

From India, the idea of the zero spread to the Persians, then to the Arabs who traded with India, then finally to Europe in the 12th century when the Moors conquered Spain.  It wasn't until Fibonacci, an Italian merchant and mathematician, translated the 9th century works of Mohammed Ibn Musa Al-Khowarizmi into Latin for other businessmen that Europe started to "see the light" and abandon its awkward Roman numeral system (source).  Slowly, over the next three hundred years, the use of the zero spread from its tight bankers' and mercantile circle outward to the common European, enabling Europe to shed the shackles of the Middle ages and enter the thriving Renaissance period.