The Elements of Euclid: A Prolegomena

If you spend enough time in Masonic discussion groups, you will find Masons who are interested in Kabbalah, in astrology, in alchemy, in Tarot and many other arcane sciences. Some are very knowledgeable in these fields, and are keenly aware of how these areas of knowledge shape Masonic ritual; but many more, while they might be eager to learn, do not yet know much about these fields. Do these fields have an influence on Masonic ritual? Quite clearly they do, although more as allusions than as central sources. These ideas are very attractive to the esoteric Mason, and the tantalizing hints in our ritual dropped by Preston, Webb, Cross, Gleason and others make the Masonic study of these fields something like a treasure hunt.

And yet, there are things explicitly in our ritual that are of profound spiritual importance that we seem not to notice. We don't have to speculate as to whether the authors of our ritual intended for us to contemplate these ideas; they tell us directly to study them. I would like to see more works of Masonic spirituality take advantage of what is explicitly in our ritual for topics of study that will both enlighten us and make us better Masons.

Tom Worrell's essay, A Spiritual Vision of the Liberal Arts and Sciences, published in Ahiman: A Review of Masonic Culture and Tradition, Vol. 1 is a good example of this. We all know that Fellows of the Craft are told during their Passing that they are to study the Seven Liberal Arts and Sciences for the rest of their lives, and to thereby improve themselves in Masonry. But how many do? The essay gives us an inroad into starting a study of these arts and sciences, and an excellent explanation of the spiritual imperatives for doing so.

In that spirit, I would like to engage in an extended study of The Elements of Euclid. Although our ritual talks about Pythagoras, it mentions the 47th Problem of Euclid (actually, Proposition 47 of Book I of The Elements). Until about 1900, an educated man studied The Elements in every society that had access to the text. We know that the Greeks, Romans, Persians, Arabs, Turks and Europeans studied The Elements, and there are translations and commentaries on Euclid in every major language in the Indo-European language family. Therefore, pretty much all the framers of Freemasonry studied Euclid. Preston tells us that of the Seven Liberal Arts and Sciences, Geometry is most essential to Masonry. He even goes so far as to say that Geometry is Masonry. No Mason can claim ignorance of Geometry, and yet few Masons actually study Geometry. I would like to change that. I would like to use my years of mathematics education, both as a student and as a teacher, to encourage Masons (and people in general) to gain a grasp of this subject so central to both Operative and Speculative Masonry, and give them a hand getting started.

To this end, I'm going to be blogging about The Elements a lot for a while. I'm also going to be preparing a Euclid class. While I encourage Masons to attend, there will be nothing in my Euclid class that would violate my Obligation for a non-Mason to hear, and so I'm going to open it to whomever wants to attend, and do the work. Over the next few months, I am going to blog about Euclid, and especially The Elements, a lot. If you want to follow along, I cannot recommend more highly the three-volume paperback translation by Sir Thomas L. Heath, published by Dover. I am not sure a more exhaustive annotated version exists in English. I will also be using an online version. My goal is to make reading Euclid easier for someone without a background in geometry, so that they can tackle this work of immense profundity.

Deep Consciousness and Waking Consciousness

 My background is in mathematics, where there are axioms (which are as few as possible) and there are proofs based on rigorous definitions. And nothing else. You don't really get to even make a conjecture without overwhelming evidence, and that conjecture is worthless compared to a proof. I try very hard to liberally interject conditionals into factual statements: "it appears that", "you might want to consider that", "conceivably", etc.

But I also realize that I can use different epistemologies for different purposes. It has become useful to me to distinguish between truth and fact, making a distinction between the epistemological aims of my spiritual journey from the epistemological aims of my scientific journey, which are not always congruent. When I was working on my math Ph.D., I would work on very difficult math proofs, the kind of problems that took weeks to solve. I would sit down for four-hours stretches, and sometimes make a breakthrough, but often not. I would have to live with the problem for long stretches of time.

In solving a difficult problem, one's mental hygiene needs to be immaculate. I would have every definition and theorem that might be useful to its solution memorized, not merely by rote, but so that I could take those theorems and definitions apart and put them back together again in my mind. Even still, often I would run through scenario after scenario, speculation after speculation, and test them over hours and even days. After much exertion, rest, exertion and rest, I would sometimes suddenly have an insight, and the whole solution would be revealed to me in its entirety, in a flash. It was orgasmic. Sometimes it would take me hours to unpack what I had received and put it in a form that was intelligible to other mathematicians (i.e. writing up the solution), but when I got one of those flashes, the problem was solved in my mind.

I talked to my advisor about this (who, incidentally was a staunch atheist), and he told me that, in his observation, humans in their waking consciousness are just not very smart. We can be disciplined and rigorous, and we can work hard at something. When we work hard on a problem, some deeper consciousness within notices that we are struggling with an idea, and decides that maybe that idea is interesting, and begins to pay attention to that idea. That deeper intelligence is very smart, and it can solve these kind of problems almost instantly. It transmits a solution to the waking consciousness in a very concentrated burst, and then is gone, leaving the waking consciousness to shape that burst into something that works for it.

That is an astonishing description of a mental process, and it should be rightly met with deep skepticism. Had I not experienced it directly over and over again, I would probably dismiss it out of hand. When I was a grad student, I discovered a book by Jacques Hadamard called "The Psychology of Invention in the Mathematical Field." Hadamard was a student of Einstein and Poincaré. He interviewed them and other mathematicians about how they create proofs, and their answers are quite similar to the above. Almost an oracular relationship takes place, where the mathematician courts a muse, who provides answers after sufficient supplication. That's very strange, but the testimony of some of the most brilliant minds of the twentieth century attest to this peculiar mental phenomenon.

When I was studying for my comprehensive exams, I would study by reading old exams going back twenty years, and try to solve those problems in preparation for my exam. These problems were so hard that if I spotted them for the first time while sitting for the exam, I might not be able to solve them. So I studied by solving as many of these problems as I could, and hoping that by memorizing their solutions, I could solve something similar if it appeared on the exam. I was studying for an algebra exam, and I worked on one problem for four hours straight, and got nowhere with finding a solution. It was 1:30 AM, and I was exhausted, so I dragged myself away from my desk and put myself to bed. Asleep, I dreamed the solution to the problem. I felt a blissful glow in the dream, and in the midst of that feeling, a voice said, "That's the real answer to the problem. Wake up, schmuck, and right it down."

I did, and went back to sleep. When I woke up the next morning, I looked at what I had written, and it was a concise and elegant solution to the problem, a six-line proof. Astonishingly, that problem, nearly verbatim, showed up on my exam, and I gave the dream solution. My professor later told me that I gave the most elegant solution to the problem he'd seen in twenty-five years of grading.

That's very strange, and it has bothered me for a long time. Poincaré describes something very similar in his anecdote about creating the Theory of Automorphic Forms. Recall that the word "genius" originally referred to a spirit creature that a person could communicate with who would give him knowledge. Socrates, in Plato's "Trial of Socrates", refers to a "daimonion" who always gave him advice and had never failed him. I think Socrates was playfully exteriorizing a mental process that he relied on, but a mental process that transcended the ego-driven consciousness that is the waking state for most people. The direct experience of such mental phenomena has convinced me that there is more to consciousness than the waking state, and made me eager to explore these different states of consciousness.

But it is unsatisfactory to describe these mental workings as either "belief" or "knowledge". Something else is going on. The flashes I received could have been mental garbage, and indeed, they were worthless before I was able to shape them back into mathematical proofs, which shaping was the result of my training as a mathematician. The speculative burst of insight that caused Picasso to paint Guernica, or caused Miles Davis to record Bitches Brew would have been worthless if bestowed upon someone with no particular talents.

Going back to my split between "fact" and "truth", I would say that the proof I discovered is a fact. The description of these mental processes is a truth for me. Truth is somewhat subjective, and fact is objective. That does not make truth inferior to fact; in this story, the fact comes out of the truth, but the truth was only useful for me, whereas the fact anyone could use. Everyone understands (or should understand) that it takes rigor to come up with facts. I would assert that it takes similar rigor to handle truths, and that most people fall upon their truths almost haphazardly, without scrutiny, mental discipline, or circumspection.

The Reason Rally

About the Reason Rally this weekend: it is disingenuous for atheists to think that by rejecting one axiom, they have reason on their side. Reason is a tool that everybody has access to. It is a powerful servant but a lousy master, since it wants nothing, cares for nothing, and wills nothing into being. There is reason behind rejecting theism, but there is reason behind rejecting oysters, and ultimately both reasons are aesthetic (unless one has a shellfish allergy).

To use an analogy from mathematics, believing in God is like using the Axiom of Choice, and rejecting God is like rejecting the Axiom of Choice. The majority of mathematicians embrace the Axiom of Choice, even though it is fraught with logical paradoxes, the same way that the majority of people are theistic, even though theism is fraught with logical paradoxes. Rejecting the Axiom of Choice provides its own sets of challenges, and makes it hard to engage with the majority of mathematicians who use it, when discussing mathematics governed by the Axiom of Choice. Rejecting theism makes it challenging to engage in ontological issues with the majority who are theistic. Ultimately, the existence of God is axiomatic, as it is unprovable, and for a theist, a first ontological principle.

In mathematics, there is a conceit that all of mathematics can be based on a set of axioms called ZFC, or the Zermelo-Frankel axioms, with the Axiom of Choice. Some try to build all of mathematics based on ZF¬C, or the Zermelo-Frankel axioms without the Axiom of Choice, because they regard those paradoxes that come with the Axiom of Choice as a refutation of the Axiom of Choice. Right now, it is very hard to be a ZF¬C mathematician. It might not always be so difficult, but currently the Axiom of Choice permeates a lot of mathematics, including algebra, analysis, measure theory, geometry, topology, set theory, logic, and without it, it is harder to do math. I see atheism as parallel to ZF¬C. It challenges an assumption that a lot of people live by. Those who evangelize atheism are demanding that people abandon a basic assumption about the way existence works.

There are brilliant atheists and stupid atheists. There are compassionate atheists and cruel atheists. There are moral victories and atrocities committed by atheists just as there are by theists. I don't believe that atheists should be discriminated against at their workplaces, in acquiring housing, or in their daily lives, and I am sorry if atheists feel oppressed, and will do what I can to help alleviate their oppression wherever appropriate. One big assumption of the Reason Rally is that there are many atheists who are living closeted lives as atheists. The assumption is that there are many people who would be vocal atheists, but are afraid of losing their jobs and families if they "come out" as atheists.  As people who put their trust in reason and science, I'm sure they have sociological data to back up their claims, which I encourage them to publish. I feel sympathy for these people, assuming they exist in the numbers claimed by their vocal advocates. Nobody should deny their natures because of outward pressure to conform. They are justified in convening and encouraging like-minded others to join them.

But if they think they have a monopoly on reason merely by rejecting the existence of God, they forfeit the very reason they claim to monopolize.

Masonic Quadrivia

While the word trivia has come to mean "useless knowledge", its original meaning was nearly the opposite. A similar word is elementary, which has come to mean "easy" rather than fundamental. Euclid's Elements is not an easy book to understand, and Jerrold Marsden's Elementary Classical Analysis is a book that someone who just finished a year of university calculus with an A grade would have difficulty understanding fully (which does not mean that it is not a great book).

The opposite of trivial is non-trivial, a word which can have savagely ironic meaning. I heard a lecture by the great number theorist Ken Ribet, in which he sketched Andrew Wiles' proof of Fermat's Last Theorem. Fermat first stated the conjecture, without proof, in 1637. Fermat claimed "I have discovered a truly marvellous proof of this, which this margin is too narrow to contain." For more than 350 years, hundreds of mathematicians scrambled to find any proof, to no avail. Finally in 1993, Andrew Wiles submitted a proof about 150 pages long, in which several new branches of mathematics had been invented just to prove this result. The proof was flawed, and it took Wiles and a graduate student two years to fix the flaws in his theory of Galois Representations, which trimmed the proof down to a manageable 100 pages or so. In 1995, a team of experts declared that this new proof was correct. At his lecture, Ken Ribet wrote the proposition of Fermat's Last Theorem on a blackboard, and then said, "The proof of this result is non-trivial."

But nobody uses the adjective quadrivial. The Quadrivium consists of Arithmetic, Geometry, Music and Astronomy. Arithmetic is the study of number. Geometry is the study of number in space. Music is the study of number in time. Astronomy is the study of number in space and time. These subjects were considered to be more difficult than the subjects in the Trivium.

By Arithmetic, the modern subject of Number Theory would be more appropriate. This includes the theory of prime numbers, factoring, divisibility, continued fractions (a fun subject hardly anyone studies anymore), Diophantine equations (or algebra restricted to whole numbers). The modern subject includes a lot of cryptography. If you use a credit card on the internet, chances are that it is being encrypted with a public key encryption as well as symmetric encryption algorithms, which involve factoring massively large prime numbers. Not many masons still use the Pig-Pen Cipher, but any simple substitution cipher can be cracked in milliseconds by a decent computer. Because of the easy availability of computers millions of times more powerful than the computers that fly the Space Shuttle, almost anyone has access to cryptography so powerful that no government or military agency can crack it within the time frame of human lives. Depending on your temperament, that's either thrilling or terrifying.

Geometry is what is today called Synthetic Geometry, which starts with Euclid's Elements, but continues into a study of Conic Sections. It could be argued that, after Newton, this would include Analytic Geometry and the Calculus, especially after Descartes. Nobody today studies Conic Sections the way the Greeks taught the subject, and the way that it was studied up until Kepler. A little bit of algebra makes a very difficult subject much simpler, and most high school students study Conic Sections using algebra, rather than straight-edge and compass.

Music would today be called Music Theory, especially the theory of harmony and harmonics. Pythagoras' treatment of music was mathematical in nature, showing the ratios of the lengths of strings of various musical intervals. A taut string one half the length of a given taut string will sound at one octave higher in pitch. Considering the original string to be the tonic, a string 3/4 the length will sound at a major fourth, and one 2/3 as long will sound at a major fifth. One 8/9 the length will be a whole step higher. The classical theory of music uses these ratios to derive scales, modes, harmonies, counterpoint, and other ideas in music theory.

Astronomy would today probably include Physics. We know so much more than the ancients did about this subject that the modern subject is totally different from the classical subject. The ancients believed that the sun orbited the earth, which seems pretty obvious except that it is not true. Explaining why this is not true is not easy (informal exercise for the reader: come up with a convincing argument why the earth actually orbits the sun, rather than the reverse. It is much harder than you think!). Because there's no immediate reason or evidence for the actual state of affairs, the ancients kept the geocentric model. To improve upon it without giving up the basic premise, they had to come up with little mini-corrections, called epicycles, to get the theory to match the data. Mercury, for example, has an orbit that requires Einstein's General Theory of Relativity to plot accurately. From the point of view of all the planets and the sun orbiting the earth, it appears to stop in its orbit, and go backwards from time to time, then stop again, and go forwards again. The culmination of ancient knowledge about Astronomy is the Almagest of Claudius Ptolemy, which is Arabic for the The Great Book (Al- usually means "the" in Arabic). It fails in the accuracy department, but is very elegant. The Catholic Church fused the Almagest into their cosmology in medieval times, and both Saint Thomas Aquinas and Dante treat this theory as a general fact. Copernicus discovered that the math gets much easier if you assume that the earth and other planets orbit the sun. He was careful to warn his readers that his model did not describe reality, but was a convenient fiction to make the calculations easier. When Galileo dared to assert that the solar system actually was heliocentric, the Holy Inquisition invited him to tour their torture chambers and inspect the instruments there.

Masons are told that of the Quadrivium, we should have an especial love for Geometry ("or Masonry" the Preston-Webb Monitor cheekily asserts). As one of the few Freemasons I know who can take a compass and straight-edge and use them to construct and prove the 47th Problem of Euclid, I'm aware that not all masons take this admonition to heart. It is worth pointing out that the future President Garfield, while Chaplain of Garrettsville Lodge #246 in Ohio, invented a new proof of the Pythagorean Theorem.

The Trivium and the Quadrivium together comprise the Seven Liberal Arts. Today we use the term Liberal Arts to mean the Humanities. Ask a Liberal Arts major to reduce a rational number to a continued fraction, to inscribe an ellipse in a parallelogram, to compose a melody in G Mixolydian, or to predict the next perihelion of Venus, and you might get smacked for your trouble.

I'm not as firmly convinced that the study of the Quadrivium is as central to Freemasonry as the study of the Trivium. While masons had a large influence on the creation of the Royal Society, and much of the Scientific Revolution, there's not much of the Quadrivium that is essential for every modern mason to learn. Certain numbers show up again and again in our rituals, and an understanding of these numbers and their relationships is very illuminating. I personally believe a mason should know the proof of the 47th Problem of Euclid, and know why the dimensions of the lodge room are what they are, but one can be a fine mason without that knowledge. At least one person in a lodge should know how to play the organ or a similar instrument, because every lodge needs music. While our ritual mentions a few heavenly bodies, I'm not sure a mason needs to know more about them than a basic public school education teaches. If anything, the symbolic meaning of these things is more important than a scientific understanding of them.

Every mason must be a friend of science, an ally of true knowledge. A mason sees no conflict between science and religion. We understand that the Grand Architect of the Universe leaves information for us to perceive for our own edification, and if a passage in a particular Volume of Sacred Law contradicts what the GAOTU is showing us, we know how to sort out these little discrepancies without insulting either source of knowledge. The value in studying the Quadrivium comes from a improved epistemology: by learning a myriad of things we can know, we learn how it is we actually know what we know.

Introduction (this is long)

I'm an IT professional, specializing in health care technology. In my past, I have been a roadie for a rock band, a published poet, an uchi deshi (live-in martial arts student) for an aikido sensei, an editor, a teacher, a Doctoral student in pure mathematics, a sarcastic video store clerk, a house painter, a sandwich maker, a camp counselor , a yacht club land activities manager, a project manager for a doomed project, and a t-shirt vendor in a Grateful Dead parking lot. I've been all over the world, and all over the USA.

I was born into a secular Jewish family. I have been a Quaker, a Unitarian, a Wiccan, a Taoist, and a Thelemite before learning about the religion of my birth and taking Judaism seriously. My mother's mother's side of the family have been secular Jews for five generations. My grandmother has a letter from her grandfather where he met a Hasidic Jew on a train, and asked him a myriad of questions about Judaism, because he knew almost nothing about the religion. My mother's father, on the other hand, was religious, and was a Freemason. He was a surgeon, and was a pillar of his community. He founded a synagogue, and was one of very few Jews in 1940s Massachusetts to be invited by Christians to speak at their churches. He studied Torah and Talmud, and died five years before I was born.

For most of my father's family, Judaism was cultural, a matter of protection rather than of faith. My late great-uncle Morris became very religious later in his life, but he seemed almost an anomaly. My late great-uncle Paul was a Freemason, but I did not know this until a few months ago, when his daughter showed me his Mason's ring.

Both of my parents are B'nei Mitzvah, but both quit their religious education, and along with it, their religious devotion, quite soon afterwards. When I was ten, I was given the option of either Hebrew School or pee-wee hockey, and I chose pee-wee hockey (I was a goalie). I never considered Judaism, even though I learned a fair amount of Kabbalah while studying Aleister Crowley. The ten s'firot were the first ten words of Hebrew I learned, except for the Yiddish Hebrew I picked up from my extended family: mishpachah, mazal, mitzvah, etc.

In my twenties, I read Tom Robbins' Still Life With Woodpecker, in which he advises that, rather than run away and try some flavor of Eastern spirituality, disaffected Westerners would do better to find the spirituality of their ancestors. I knew that some day I would probably become a religious Jew, decades before it happened.

As it was, I found aikido and mathematics, and lost them both, leaving a serious vacuum in my spiritual life. I earned my black belt in January 2000. I earned an MS in pure mathematics in May 2002. While aikido is grounded in the body, it is very spiritual, and while pure mathematics is rooted in the mind, it too can be very spiritual. I worked on Riemann Surfaces, which are locally two-dimensional, but sit in four-dimensional space. Spending hours every day mentally in four dimensions does something to one's perception of this mundane three-dimensional world. Working in complex numbers makes real numbers seem limited and small. Aikido teaches about the true nature of conflict, and how nature handles conflict in ways that are sometimes far gentler than the way that humans handle conflict (and sometimes far crueler). The shore does not resent the sea, even though the sea acts upon the shoreline relentlessly, changing it, shaping it, sometimes obliterating it completely. Meditating upon how one can be gentler makes one perceive levels of gentleness inconceivable to most.

I stopped my aikido training when my responsibilities at graduate school became too intense. By sacrificing my aikido practice, I got a 4.0 GPA at school. I went on to a PhD, which I worked on for two years. My Master's advisor was a fierce polymath, an astonishing mind with an intense curiousity for all aspects of math and physics. He would often give an hour-long proof in a lecture without ever looking at his notes. He had a ruthless sense of discipline, giving 45 minute quizzes with only 15 minutes to finish them. I adored him. I found nobody remotely like him in my Doctoral program, and I despaired. I struggled through two years, and became horribly depressed. In my program, I could find nobody who saw math as spiritual. There were a few students who were adept, but none were inspired. There was too much pressure to specialize, too much hostility between the pure mathematicians and the applied mathematicians, too much childish nonsense that was totally undignified. By the end of my time in my program, I was absolutely miserable. I gained weight, became a recluse, and fell apart.

After dropping out, I stumbled through a bunch of teaching jobs. I taught at a pre-Civil War military academy in the South, a school for emotionally disturbed rich kids, my mom's old boarding school. I edited math textbooks, and later math websites, but all the while I was thirsty for a meaningful life. During a span of unemployment, I switched my computer to Linux, and taught myself some basic computer skills. Enough to get an entry-level job as a healthcare technology analyst. I paid off my debts, and finally had breathing room to take spiritual inventory.

I had gained 100 pounds above my leanest adult weight. I had pre-diabetic symptoms, was having fainting spells, high blood pressure, sleep apnea, and was deeply unhappy.

Around this time, I read a book about Reconstructionist Judaism. I found it challenging, but it didn't really hit the nail on the head for me. I then found a book by Rabbi Zalman Schachter-Shalomi, Jewish with Feeling, and it deeply resonated within me. Reb Zalman described a religious life that was deeply spiritual, and yet totally livable. Flexible, funny, affirming, inclusive, contemporary, without the rigid dogmas of more Orthodox Judaism. I viisited six or seven synagogues in the area, and found the one I currently belong to, run by a student of Reb Zalman's.

Around the same time, the Grand Lodge of Massachusetts was airing the controversial "Ben Franklin" television commercial. Just previously, my grandmother had commented that my late grandfather would have been proud of my recent religious choices. Watching that commercial, I knew I wanted to be a Freemason like my grandfather. I did a web search. There were two lodges in my town, but only one had a website. I called the lodge secretary, and met with him that weekend in the lodge. He showed me around, and gave me a history of the lodge. When he showed me the diagram of the 47th Problem of Euclid on the Eastern wall, I was hooked. I petitioned the lodge, and was entered, passed and raised the very next time the lodge offered the degrees.

Since then, I earned my 32nd degree in a Scottish Rite one-day class. I imagine that the Royal Arch and the Shrine are in my future, but not too soon. I want to work in my Blue lodge, and earn the Scottish Rite degrees I missed. I would love to participate in the degree rituals, should I be found worthy to do so.

I have lost nearly 50 pounds in the last year, and then regained 15 around the holidays. Since I lost my job two weeks ago, I have been in the gym every day I can. I intend to join an aikido dojo when I reach 200 pounds, if I can afford it. I take longer with my shacharit (morning prayers), and have even started minchah (afternoon prayers) on the days where I'm not too busy. I go to every Lodge of Instruction, and will be giving a talk at the May 2009 Lodge of Instruction, at my lodge, on the 47th Proposition of Euclid, and its Masonic meaning. I don't expect to be unemployed for very long, but I might have to settle for a staffing position instead of a permanent position, at least in the short run. The economy is in a shambles, but healthcare technology may not be as bad as other sectors. I'm deeply grateful for how fortunate I am in this life, and thank God every day for my blessings.