Vampire Numbers

UPDATE: I am quite honored to state that this post has been endorsed by the man who defined Vampire Numbers, Clifford A. Pickover:


Tremble ghouls and boos: It’s a Spherical Cow Halloween! Those of you with timid hearts and fragile minds may not wish to continue. Hark dear reader; I beseech you, if you are adverse to maths macabre, chilling calculations, diabolic divisions, and fearful factoring, read no further. But if you are made of sterner stuff, and you reserve does not pall, tread onwards and I promise you delectably devilish mathematical diversions. For it is well-known by the old folk and those that care to listen that in these nights ‘fore All Hallows’ Eve, ghosts and goblins and bogeymen bewitch this site, whispering forbidden formulae to all fool enough to hear it.

Ah, so you’ve chosen to continue. Your fearlessness shows your true red blood, and here normally that would be a liability – You see we are in the presence of the nosferatu! Aye, the very children of the night! From your pallor, I see that you feel their presence in your very ichor, but your trepidations are unfounded. For, you see these parasites aren’t sustained by the instrument of your pulse; rather they feed off each other. Cannibal bloodsuckers, how horrid! But that is indeed what they are- The dreadful Vampire Numbers!

“What manner of demon is this,” you ask, and so I will tell you: Vampire Numbers were first described by the brave mathematical author Clifford A. Pickover in 1994. Risking life, limb, and sanity he valiantly scrawled this enigmatic message:

If we are to believe best-selling novelist Anne Rice, vampires resemble humans in many respects, but live secret lives hidden among the rest of us mortals.  Consider a numerical metaphor for vampires.  I call numbers like 2187 vampires numbers because they’re formed when two progenitor numbers 27 and 81 are multiplied together (27*81 = 2187).
Note that the vampire, 2187, contains the same digits as both parents, except that these digits are subtly hidden, scrambled in some fashion. Similarly, 1435 is a vampire number because it contains the digits of the progenitors 35 and 4: (35*41=1435).  

These vampire numbers secretly inhabit our number system, but most have been undetected so far.  I believe there are only six four-digit vampires in existence, but have no idea if there are any larger vampire numbers.

The Count, Roger Cruz

Mathematicians have a penchant for games, and a greater penchant for puns, so Mr. Pickover’s modest message spawned some monstrous results. A legion of vampire hunters arose. Their numbers and tenacity would shame even the likes of the Van Helsings and Belmonts. Mathematicians and computer scientists, amateur and professional, jumped on the challenge, and what they found lurking in the integers was more nightmarish than any could have predicted.

Poor Pickover could never have known the multitude of fiends that would be released and the magnitude of their monstrousness when he made his modest post, but they are unleashed now. For example, he was wrong when he predicted that there are only six four-digit vampire numbers, in fact, there are seven. I will show them here, so that you may be better able to protect yourself:

21 x 60 = 1,260
15 x 93 = 1,395
35 x 41 = 1,435
30 x 51 = 1,530
21 x 87 = 1,827
27 x 81 = 2,187
80 x 86 = 6,880

Worse yet, they do come bigger, much bigger. Here is a wicked, six digit vampire:

204 x 615 = 125,460

As they were hunted, our taxonomy improved, and what Pickover had termed progenitors we now call fangs. So, for instance, in the eight digit behemoth:

1,301 x 9,170 = 11,930,170

1,301 and 9,170 are the vampire’s fangs. To summarize, so that you may know how to identify the devils: A Vampire Number is a number that contains every digit of the two numbers multiplied together to create it. More technically, A Vampire Number, v, is a number with an even number n digits that is the product of a pair of numbers, x,y, of n/2 digits each such that appearing in v are all the digits of x and y in any order. To avoid trivialities, a Vampire Number is not allowed to have trailing zeros. Thus, 350 x 410 = 143,500 is not considered a Vampire Number.

The search for Vampire Numbers has led to the discovery of some truly awesome nightmares such as this 100 digit vampire number found on March 9, 1999 by Myles Hilliard:

98765432109876543210987654321098765432108990776898 x 98765432109876543210987654321098765432109967196255 =


However, a word of caution to those who would hunt the unclean themselves, finding ever larger monsters has become a quest, nay an obsession for some, and the world record is currently hard to top. Jens K. Anderson, a braver man than most, currently has the distinction of staring farthest into the abyss. He has seen the beast and lived to tell about it. At 10,060 digits, he has earned his fame. Go here to see the goliath in its lair, but I cannot be held responsible for the consequences to your sanity or your soul.

Anderson’s number is literally indescribably huge. There is not currently a word to classify a 10,060 digit number. It is larger than anything the physical world can offer- the number of atoms in the universe is estimated to be 10^80, an 81 digit number. Conventional naming systems offer a name that is more cumbersome than 10,060 digits, and therefor I propose a new name for numbers that are 10,060 digits. My suggestions are either Vampillion or Vampirillion. Which do you prefer?

Addendum: A heroic reader, Mr. Rossi, observed that I had neglected to properly arm you against these numbers. I have ruminated over this quandary and have devised the following guide for dealing with these abominations. For the benefit of all mankind I preserve here this historic exchange:

Mr. Rossi: First you charm us with a Moobius Mooroboros and then you do this?! I was not prepared for this kind of sucker punch. Not only that but you only mentioned how to identify these horrific integers. What must we do to protect ourselves and our loved ones?

My Response:Traveler, your admonition is well deserved and it is an error I fear may have cost the life and soul of too many a mortal. But you see the wards against the foul things are well known and widely written. Recall Van Helsing’s words (with which I know you are familiar): “I shall cut off her head and fill her mouth with garlic, and I shall drive a stake through her body.”

This shall be our stratagem. Consider the wretched 13,078,260 = 1620 × 8073. First, cut off its head. Now, the Vitruvian proportions tell us that a man’s head is one eighth its height, so from the beast we slice off its first eighth: 13,078,260 becomes 3,078,260. Then we are to fill its mouth with garlic. Garlic is related to the astrological sign ares, which is numerological related to the number nine. So we jam 9 into its gullet: 3,078,260 becomes 39,078,260. Finally we drive a stake through its heart. Take firm hold of one (1) and nail it into its chest: 39,078,260 becomes 390,718,260. Now the beast is good and dead.

This is the first in a series of Halloween posts. If you enjoyed it, you may like the others: Infernal Integers, Zombi(nacci)The Sum of All Fears

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Moobius Mooroboros

Inspired by John Barth’s Frame-Tale, and not being one to shy away from a good pun, I was seized by an idea for a nerdy craft project. Too much exposition will ruin it, so with apologies to Barth and the majestic Bos primigenius, whom I have twice now grotesquely distorted, I present to you Spherical Cow‘s first craft project, Moobius Mooroboros:


Part 1

Part 2

Print these two parts back-back on the same piece of paper, and cut out. Adhere the two ends together giving the strip a half twist. If done correctly you should achieve something like the following:


The Graceful Moobius Mooroboros

However, for those scared of scissors (or cows) you can simulate the experience of owning your own Moobius by clicking HERE (and no, the link is not broken).

For the Creator conceived that a being which was self-sufficient would be far more excellent than one which lacked anything; and, as he had no need to take anything or defend himself against any one, the Creator did not think it necessary to bestow upon him hands: nor had he any need of feet, nor of the whole apparatus of walking; but the movement suited to his spherical form was assigned to him, being of all the seven that which is most appropriate to mind and intelligence; and he was made to move in the same manner and on the same spot, within his own limits revolving in a circle. All the other six motions were taken away from him, and he was made not to partake of their deviations. And as this circular movement required no feet, the universe was created without legs and without feet. – Plato, Timaeus

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The Longest Shortest Story Ever Told

” ‘ ” ‘ ” ‘ ”       ” ‘ ” ‘ ” ‘ “

– John Barth, Menelaiad

I began reading John Barth’s collection of short stories Lost in the Funhouse last night and upon finishing the author’s prefatory note, I encountered a story of staggering simplicity, let unimaginable depth. I have yet to finish it, and doubt I ever will. It so beautiful encapsulates a theme of this site, that I have decided to present the story in full here. Be warned though, once you start reading, you may never finish. I present to you John Barth’s Frame-Tale:

For Clarification: These are two consecutive pages printed on opposite sides of the same sheet.

Frame-Tale Page 1, John Barth

Frame-Tale page 2, John Barth

If you are having trouble visualizing what this is, here is an assembled version: SPOILER.

You will never read this, and thus I can fearlessly admit a grave error: Now that this is on my site,  my previous writings are all in vain, for no one will ever be able to reach them. If you do not hear from me again, you know where I will be. To lose yourself even further read:

Barth, John. “Frame-Tale.” Lost in the Funhouse. New York: Anchor, 1988. 1-2. Print.

To see my modest tribute to this story, see Moobius Mooroboros.


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Hypatia Miscellanea

Author’s Note: This is the third and last of a series of posts. Though they can be read independently, they were written with this order in mind: The Rape of Cassandra, Mice on a Galley: A Review of Agora, Hypatia Miscellanea

The Great Library of Alexandria, O. Von Corven, 2001

Everything is gestation and then birthing.

– Rainer Maria Rilkes

Incredulous you may be, but in my previous two posts, The Rape of Cassandra and Mice on a Galley, I was unable to call upon the whole of the collectanea I assembled for their writing. Before I bring to a close my investigation of Hypatia, her world, and her legacy, I would like to offer several additional facts, observations, and quotations that, though interesting, I was unable to work into the earlier posts. Here then is the Hypatia Miscellanea:

Hypatia’s Name

Agora was original titled “Hypatia”, however Amenabar felt that audiences would have difficulty pronouncing her name, and that the name, “Hypatia”, wasn’t particularly beautiful. The pronunciation is something I should have introduced earlier; several people who have read the post have asked me about it. Her name is pronounced “High-Pay-Shuh”, not “High-Pat-Ia” or “High-Pay-Tia.” As for the film’s title, Agora is probably a more fitting, for, as I discussed, its focus is ever expanding. As for beauty, I think Hypatia is a wonderful name. It is Greek and means “highest.”

Aristarchus of Samos

The first known heliocentric model of the solar system was presented by Aristarchus of Samos (310-230 BCE) a Greek astronomer and mathematician. His original discovery of and presentation of this model is lost to history. What we do know of it comes to us by way of Archimedes (287-212 BCE) who references it in his lyrically titled book The Sand Reckoner in which he wondrously estimates the number of grains of sand it would take to fill the universe ( He arrives at a value of 10^63 grains or  1 viginitillion. For comparison the current estimated number of atoms in the universe is 10^80 or 100 quinvigintillion. I am not making these names up.) In The Sand Rekoner Archimedes writes of Aristarchus:

You (King Gelon) are aware the ‘universe’ is the name given by most astronomers to the sphere the center of which is the center of the Earth, while its radius is equal to the straight line between the center of the Sun and the center of the Earth. This is the common account as you have heard from astronomers. But Aristarchus has brought out a book consisting of certain hypotheses, wherein it appears, as a consequence of the assumptions made, that the universe is many times greater than the ‘universe’ just mentioned. His hypotheses are that the fixed stars and the Sun remain unmoved, that the Earth revolves about the Sun on the circumference of a circle, the Sun lying in the middle of the Floor, and that the sphere of the fixed stars, situated about the same center as the Sun, is so great that the circle in which he supposes the Earth to revolve bears such a proportion to the distance of the fixed stars as the center of the sphere bears to its surface

It is unknown exactly how Aristarchus arrived at a heliocentric theory, but it is theorized that he developed it out of a logical dilemma he had discovered in another one of his works – the only surviving book of Aristarchus, On the Sizes and Distances of the Sun and Moon. In this treatise, he accurately calculates the Moon’s size and distance from the Earth, but incorrectly estimates the Sun’s size and distance from the Earth. He writes that the Sun is about 18 to 20 times further away from the Earth than the Moon, and that it is about 300 times the volume of the Earth. In reality, the Sun’s distance from the Earth is 400 times that of the Moon and its volume is about 1,300,000 times that of the Earth. Aristarchus’s errors were due to poor observation and measurements. His logic and mathematics were sound.

It is thought that when he realized the Sun was so much larger than the Earth, he had trouble reconciling its bulk with its supposed orbit around the Earth, and then contemplated alternative models. As with many astronomers after him, he realized that a heliocentric model provided a simpler mathematical explanation of astronomical observations, and was therefor more likely true. He was right.

On the Sizes and Distances of the Sun and Moon, Aristarchus

Hypatia, Marriage, and Women in the Hellenic World

There is a scene early on in Agora in which Hypatia rejects a suitor by giving him a gift of a blood stained napkin and explaining: “It is the blood of my cycle. You say that you have found harmony in me. Well, I am suggesting that you look elsewhere because I think there is little harmony or beauty in that.”  This is historically accurate. It depicts a well-documented exchange that typifies Hypatia’s consistent rejection of suitors, and she is rumored to have had many suitors. Hypatia never married, and, as far as we know, was never romantically or sexually involved with anyone. This act is very telling of Hypatia and her time.

I do not think that we should regard Hypatia as homosexual or asexual. I do not think that is what her celibacy or maidenhood indicate. Hypatia understood, as many great women have understood throughout history, that marriage represented something of a legal sentence to her. She often claimed that she was married to her studies. She loved her academic life and knew that a marriage in ancient Alexandria would indicate the end of it. In many traditions of marriage, the woman’s identity is subsumed by the institution (consider the implications of the western traditional name change). Many outstanding women have noticed this and refused marriage likewise: Emily Bronte, Florence Nightingale, Clara Barton, and Susan B. Anthony are some examples that spring to my mind. The loss of identity, loss of individuality, loss of family history, and sublimation of independence are unfortunately aspects of the institution of marriage across many cultures and with long histories that we have only recently, as a species, begun to address. Henrik Ibsen perhaps best frames the dilemma for the twentieth and twenty-first centuries in the climax of his landmark 1879 play, A Doll’s House, in which Nora informs her husband that she must leave their marriage to discover herself- this was a shocking scene at the time:

Helmer: You blind, foolish woman!

Nora: I must try and get some sense, Torvald.

Helmer: To desert your home, your husband and your children! And you don’t consider what people will say!

Nora: I cannot consider that at all. I only know what is necessary for me.

Helmer: It’s shocking. This is how you would neglect your most sacred duties.

Nora: What do you consider my most sacred duties?

Helmer: Do I need to tell you that? Are they not your duties to your husband and your children?

Nora: I have other duties just as sacred.

Helmer: That you have not. What duties could those be?

Nora: Duties to myself.

Helmer: Before all else, you are a wife and a mother.

Nora: I don’t believe that any longer. I believe that before all else I am a reasonable human being, just as you are- or, at all events, that I must try and become one. I know quite well, Torvald, that most people would think you right, and that views of that kind are to be found in books; but I can no longer content myself with what most people say, or with what is found in books. I must think over things for myself and get to understand them.

For Hypatia, marriage and reproductive sexuality represented an even grimmer fate. In ancient Rome and Greece, infant mortality rates were exceptionally high. To compensate for this, couples strove for many pregnancies. With each successive pregnancy and labor, the risk to the mother increased. As such married woman tended to die young, leaving there husbands who would often go on to remarry several times. This cycle was aggravated by the fact that only male children were able to inherit. For this reason husbands would insist on children until a male was conceived and then survived the childhood gauntlet of disease.

I have wondered about Hypatia’s judgment of menstruation and the implications of her remarks that it is unharmonious and not beautiful. I think I have come to an understanding of her attitude beyond that of the childish inhibitions regarding menstrual blood that linger in our culture sixteen hundred years after Hypatia’s time and manifest as anomalous blue fluid in commercials for “feminine hygiene products” (the puritanism of the euphemism makes me nauseous). A repugnance for menstruation is a repugnance for a defining characteristic of femininity. If Hypatia’s comment, “I think there is little harmony or beauty in that,” is indicative of a devaluation of herself as a woman, it is important that we understand where she acquired this notion. Hypatia was schooled in classic Greek thought, and her teachers and textbooks were definitively misogynistic. I will let two giants of Greek thought provide support for this assertion:

It is only males who are created directly by the gods and are given souls. Those who live rightly return to the stars, but those who are cowards or lead unrighteous lives may with reason be supposed to have changed into the nature of women in the second generation. This downward progress may continue through successive reincarnations unless reversed. In this situation, obviously it is only men who are complete human beings and can hope for ultimate fulfillment; the best a woman can hope for is to become a man

– Plato (regarding his theory of reincarnation), Timaeus

It is the best for all tame animals to be ruled by human beings. For this is how they are kept alive. In the same way, the relationship between the male and the female is by nature such that the male is higher, the female lower, that the male rules and the female is ruled.

– Aristotle, Politics

The sexism displayed here was not confined to academia. Aristotle and Plato truly believed they were logically describing natural laws when they wrote these passages, and they had much empirical evidence to support their claims for the entire ancient Greek world’s social structure was built on extremely misogynistic principles. Sarah B. Pomeroy fully describes Hellenist gender bias in her 1975 book, Goddesses, Whores, Wives, and Slaves. Women in Classical Antiquity, from which I’ll summarize:

  • Confined within the parental home until a husband was chosen for her- at which time she would be in her mid-teens, he at least fifteen years older- the Athenian woman of the citizen class would then be transferred to the home of her husband where she was to fulfill her principal function, of bearing and rearing children.
  • Of those children (on the average, four or five in number, one or two of whom might die at birth), the sons would be raised within the family – particularly in post-war years when there was a shortage of men – but ordinarily only one daughter, at most, would be reared (infanticide and abandonment of female children was common).
  • Other girl children would probably be exposed; if they did not die, they might be picked up by slave dealers or prostitutes and prepared for a life of slavery, prostitution, or both.
  • Athenian men had a variety of opportunities to satisfy their sexual drive: boys and other men, courtesans or hetairai, prostitutes or their own slave women, and wives. The wife’s function was, however, primarily that of carrying on the family line and tending the family hearth.
  • The wife did not socialize with her husband and his friends; men’s social gatherings, even if held in her own home, were off-limits to her. As for going to the marketplace or communal well, that was an activity reserved for men or for women slaves.

Our fundamental obligation when confronted with such bigotry is to acknowledge and remember it, because the only way to move forward is to know what we are moving from. We are nowhere near journey’s end yet, and we must never content ourselves that we have arrived.

Against the philosophical sexism quoted above, I will offer another philosopher, because philosophers are at their best when they are at their throats. In our corner then is the great philosopher, mathematician, logician, historian, pacifist, and a personal hero of mine: Bertrand Russell:

When we come to compare Aristotle’s ethical tastes with our own … we find … an acceptance of inequality which is repugnant to much modern sentiment. Not only is there no objection to slavery, or to the superiority of husbands and fathers over wives and children, but it is held that what is best is essentially only for the few-proud men and philosophers

Almost every serious intellectual advance has had to begin with an attack on some Aristotelian doctrine; in logic, this is still true at the present day

Aristotle could have avoided the mistake of thinking that women have fewer teeth than men, by the simple device of asking Mrs. Aristotle to keep her mouth open while he counted.


Hypatia (Rachel Weisz) Frantically Tries to Save Books in Agora

Views of Hypatia in the Middle Ages

I have tried to use academically authoritative sources whenever I can, and have only used Wikipedia as a source of personal guidance and inspiration when planning these posts. However I must praise the writers of Wikipedia’s Hypatia entry for cleverly juxtaposing two early historic accounts of her death. I think the selections are very telling of the opposing ways her death was viewed:

Yet even she fell a victim to the political jealousy which at that time prevailed. For as she had frequent interviews with Orestes, it was calumniously reported among the Christian populace, that it was she who prevented Orestes from being reconciled to the bishop. Some of them therefore, hurried away by a fierce and bigoted zeal, whose ringleader was a reader named Peter, waylaid her returning home, and dragging her from her carriage, they took her to the church called Caesareum, where they completely stripped her, and then murdered her by scraping her skin off with tiles and bits of shell. After tearing her body in pieces, they took her mangled limbs to a place called Cinaron, and there burnt them. –Socrates Scholasticus (5th Century)

And in those days there appeared in Alexandria a female philosopher, a pagan named Hypatia, and she was devoted at all times to magic, astrolabes and instruments of music, and she beguiled many people through Satanic wiles…A multitude of believers in God arose under the guidance of Peter the magistrate…and they proceeded to seek for the pagan woman who had beguiled the people of the city and the prefect through her enchantments. And when they learnt the place where she was, they proceeded to her and found her…they dragged her along till they brought her to the great church, named Caesareum. Now this was in the days of the fast. And they tore off her clothing and dragged her…through the streets of the city till she died. And they carried her to a place named Cinaron, and they burned her body with fire. – John of Nikiû (7th Century)


The plan of the thing is always simpler than the thing itself. The last two posts were conceived as a simple rebuttal to the left-brain right brain myth and as review of a math related movie respectively. Only when I began writing the posts did I realize the amount of concerns I wished to address and how I had connected them internally. I do not think experience is compartmentalized (perhaps the impetus for attacking the brain myth), and I truly believe that passing observations (like the layout of a room, the questionable casting of a film, the change of a woman’s last name) do not fall through the vast web of a person’s consciousness without plucking a chord, tangling the connections, or snapping a line. What I’ve presented is clearly not a cohesive theory, but a snapshot of how I conceive the situation, why I do, and the implications for related concepts.

Since the creation of this site, I have wanted to write about Hypatia. I found her on my own in a footnote in a math book several years ago and have wanted to address the injustice I felt at her marginalization. These three posts have been my tribute to her for now, and I hope that with them I have raised awareness of this remarkable woman, even if just by a little. Thank you for reading. – Webster Batista-Lin October 22, 2010

Concern should drive us into action and not into a depression.


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Mice on a Galley: A review of Agora

Author’s Note: This is the second in a series of posts. Though they can be read independently, they were written with this order in mind: The Rape of CassandraMice on a Galley: A Review of AgoraHypatia Miscellanea

“Master,” said he, “we come to beg you to tell why so strange an animal as man was made.”

–Voltaire, Candide

Conspicuously absent from most math classes are the mathematicians. Amidst the problems, exercises, theorems, laws, and rules of thumb, there is little sign of those that pioneered them. This is part of the nature of mathematics. Its study has been a collective work throughout human history, with each generation adding to, enhancing and mending the efforts of its predecessors. By this process, math evolves. Its vocabulary becomes more precise, its symbols more elegant, its methods more efficient, its philosophy more sophisticated, and it becomes purer, its foundations sturdier and its frontiers better defined. It is an edifying, ongoing enterprise. The present works with the past with reverence instead of disdain and neighbors share rather than pillage. As a discipline it moves towards crystalline perfection, but the constant polishing tends to rub smooth idiosyncrasies left by its practitioners. The individual mathematician irons the wrinkles from this area or charts the boundaries of that one and then steps back into shadow, the work taking center stage. In this way mathematics is gloriously indifferent to the mathematicians and the mathematicians respect this. More than respect, many mathematicians find this transcendental, for they are participating in something greater than the individual, greater than the present, and for some, greater than man. As Bertrand Russell remarked, “I like mathematics because it is not human and has nothing particular to do with this planet or the whole accidental universe- because like Spinoza’s God, it won’t love us in return.”

An unfortunate, inevitable effect of this aspect of its nature is that the mathematicians are left out of the classroom, or if they are included, they are relegated to the margins or the footnotes. Their absence, the intentional divorce of mathematics from the physical and temporal world, and the strict formality of its structure mislead many students to the conclusion that math is antagonistic towards them and everyday life. For instance, without proper context Russell’s statement can seem misanthropic. To understand it, it is important to know that Spinoza’s God is benign if not beneficent, and that its presence is something to inspire rather than to fear.

Unfortunately, context is what the novice lacks. To the insider, the absence of the mathematicians may seem something like selfless nobility, to the outsider, it may seem barren. The separation of the math from the mathematician erodes some of the footholds other subjects provide their students. When studying the arts, it is customary to review the biographies of the authors and artists, and understand their times. In the sciences, the students are placed in the laboratory, and constantly reminded of the human beings who conducted the experiments before them. In history, there are only the personalities. But in math, the pioneers have mostly been excised from the curriculum.

The School of Athens, Raphael (1509-1510)

Math’s masterpieces also in some sense marginalize their makers. Anyone can view Raphael’s work and appreciate its beauty his genius and uniqueness. It is tangible and recognizable as an accomplishment. By its very physicality it imprints itself on us; viewing it is a sensation- it is literally impressive. Furthermore, we are always aware that it has an author. It is signed of course, and it shows all the characteristics of Raphael’s style, but more fundamentally, it is simply necessary that it was made by a person. Gazing at it affirms our convictions in human potential. We feel that we can hold it up as an example of achievement, and we are right. Similarly, with literature we can point to the tome or recite the lines and say, “that is a masterwork.” These things are inarguably awesome. To see how math differs, compare The School of Athens to one of math’s greatest masterpieces:

Euler's Identity, Leonhard Euler (1710)

It appeals equally to the mystic, the scientist, the mathematician.

-James Newman & Edward Kasner

I can assure you that this is as beautiful as anything else humanity has or will ever achieve. Look at the disparity between the two works. Raphael’s school can be appreciated by the non-painter, it is inherently arresting. However Euler’s (pronounced “Oiler”) arrangement of seven symbols is not. It is absolutely opaque to the untrained eye. It remains mostly opaque to the mathematician’s eye as well: As the mathematician Benjamin Price said of this formula, “…it is absolutely paradoxical; we cannot understand it, and we don’t know what it means, but we have proved it and therefore we know it must be true.” However, even an incomplete understanding is enough to allow us to recognize that it is immensely wondrous. I would conjecture that for many mathematicians the fact that this formula is true serves as the best argument for the existence of a deity. Indeed, it has been called “God’s Formula.” Yet even with this praise and recognition, Euler’s work denies him for it refuses to be biographical. Most human achievements tell us something about their creators, but math stands alone.

Their absence doesn’t indicate their non-existence though, and math has been host to many fiery passions, however unheralded they may be. The allure of mathematics can easily be missed if it is treated only as a tool of the sciences rather than a human pursuit in and of itself. Unfortunately this distinction is not often made explicit to the student, and the inconsistent and sometimes conflicted curriculum of lower level mathematics conflates application and theory delivering a bland stew palatable to many but satiating to few. This can lead to a profoundly poor conception and impression of mathematics for some students. Occasionally when I encounter a student who has a distaste for mathematics, I will try to place the math in a more humanistic frame: I will remind the student that there were people who spent their lives developing what he is studying, sometimes at the expense of all else, sometimes at the expense of their lives, and this fact informs us that there is something important and captivating in the formula, that there is something worth dying for between the symbols, and perhaps we should stay with them a little longer, perhaps we should examine them a little closer.

I have personally found this mode of thinking very stimulating, and from the conception of this site, it has been one of my goals to express this. It has been one of my goals to put a human face on mathematics and to inspire an interest in mathematics not just through its product, but through its producers. Hypatia’s story and my interpretation of its meaning was my first attempt at this, and I intend it to be the first of a series of miniature biographies. Towards the same goal, I want to investigate the intersection of art and math, and to this end I have tried to include works of visual and literary art in my discussions. With this mission in mind, I would like to spend the rest of this post discussing Alejandro Amenabar’s (Chilean director of Abre los Ojos, The Others, and 2005 foreign language Oscar winner The Sea Within) 2009 film Agora. This is very much connected to my previous post, The Rape of Cassandra, and if you haven’t read that, I would advise you to before continuing.

Agora is Amenabar’s vision of Alexandria in the period beginning with the Serapeum’s destruction in 391 and ending with Hypatia’s assassination in 416. It is noteworthy and deserves are attention for it is surprisingly the only film to ever tell Hypatia’s tale.  But while the tragedy is at its heart, Amenabar and his cowriter and frequent collaborator, Mateo Gil, spend as much time on the social and political history of Alexandria as they do on her biography. The parallels between the turmoil, bigotry, and zealotry of the 4th century and those of our time are the clear focus of the film. This is an ambitious goal, and it is an ambitious film that attempts to recreate the physical reality, thought, and culture of Hypatia’s time. In its reconstruction of Alexandria, it is strikingly successful.

The most impressive aspect of the film is its meticulous attention to detail. In interviews, the cast and crew have repeatedly stressed the amount of research they undertook, and it is apparent. The movie exudes authenticity. Alexandria was a city of landmarks, and several are particularly well rendered: The Serapeum, Hypatia’s classroom, and the small library at the Serapeum are beautifully constructed and filmed. They feel like an amalgamation of ancient Greek and Egyptian sensibilities, and that is perfect for that is exactly what they would have been. The film’s library looks as if it had been lifted from one of the many paintings that have depicted it.

Alexandria is in its twilight during the events of the film, but it had been a jewel during its apex and home to a bustling economy and culture. We see this legacy in the film through fully realized theaters, marketplaces and religious sights. Particularly striking is the Canopic Way, a long avenue of street vendors in the shade of multitudinous cloth canopies stretching between the buildings. Finally, one of the marvels of the ancient world is depicted, The Lighthouse of Alexandria. It is as grand and impressive as it should be, but shot in such a casual style that it never seems forced into the film- We see it when the characters do, and it orients the city. It is treated as part of the city’s skyline, and in doing so it furthers the illusion that this was really filmed in 4th century Alexandria.

Amenabar was insistent that the film be shot on actual sets instead of using green screens, and production designer Guy Dyas (Inception) and his team are to be commended for accomplishing this. Agora is an impressive production, the largest undertaken by Amenabar, and apparently the largest film ever shot on the Island of Malta where most of the photography took place. Impressively it feels like it was shot at location rather than on set. The locations of the film have a sense of weight, age, and physicality that are impressive. Adding to this impression is their dynamism- they feel solid and interactive as if they were really made out of stone and wood rather than simulated. Noteworthy is the artistry of the statues and engravings that adorn the Serapeum and liter Alexandria. It must be an intimidating assignment to emulate the style and craftsmanship of another culture’s art, especially classical art, but the production designers of Agora accomplish this as well as or better than any other film I’ve seen. Accurate costume design completes the recreation of the material culture.

This world is fully populated as well. Legions of extras crowd the scenes. Cultural events are represented well and it is interesting to note that the film features festivals of all three major religions present in Alexandria at this time (Greek pantheism, Judaism, and Christianity). I was especially impressed by the classic Greek theater portrayed in the film for it serves a variety of functions: It is historically accurate as an important part of the ancient Greek religion, it is something of a play within a play for the movie, its gaiety provides a contrast to the piety of the more modern Christianity, and it illustrates class inequality in Alexandria (slaves and the poor are only allowed to watch the theater from behind metal gates rather than join the elite in the auditorium). However, this was not a peaceful time, and the movie never fails to remind us of this. Riots (important to the film’s plot and as a historic detail) are chaotic, violent, and dense. It is a testament to Amenabar that the crowds are so well choreographed and organic.

They are expertly photographed as well. The camera’s movement throughout the film is novel, but it is especially noticeable when capturing the street level chaos. An exceptionally interesting shot occurs during the destruction of the library when the camera is inverted and the world literally turns upside down. The director of photography, Xavi Gimenez (The Machinist), elegantly involves us in Alexandria’s turmoil.

Freedom, Rachel Weisz as Hypatia

The acting is as immaculately accomplished as the setting. Rachel Weisz (Best Supporting Actress The Constant Gardner, The Fountain, The Mummy) plays Hypatia with an intensity and energy that both convinces us of and legitimizes her devotion to her studies and her students. She is eloquent, intelligent, confident, and thoroughly convincing. Her most famous students are present and both are complexly written and well performed. Orestes portrayed by Oscar Isaac (Ridley Scott’s Robin Hood) and Synesius portrayed by Rupert Evans (Hellboy) are both dynamic, compelling characters.

Power, Oscar Isaac as Orestes

Orestes, the prefect of Alexandria at the time of Hypatia’s assassination, is perhaps apocryphally depicted as an enamored student of Hypatia during the film’s first act, but this ahistorical characterization serves the plot and cleverly allows the film to depict more historical events without the cast of characters becoming bloated. We see his development from student to leader, from pagan to Christian, and from naïve to world-weary. When he finally assumes power he is a sympathetic if ineffective and outmaneuvered politician. Isaacs’ performance is seamless throughout this metamorphosis and you never doubt the emotional depth behind his actions. Syenesius plays a smaller but no less important role. Historically he is our best link to Hypatia, and I would have liked to see his reverence for her more fully developed, but what is shown of the character is intelligently and sensitively conveyed by Evans. One of the films best scenes occurs early on when a religious dispute between Orestes (a pagan) and Synesius (a Christian) is mediated by Hypatia through an extraordinarily clever use of Euclid’s First Common Notion: “Things which are equal to the same thing are also equal to one another.” In the scene, the two things are ‘a pagan’ and ‘a Christian’ and the third thing is ‘an atheist.’ It is a surprisingly effective scene that supports the film’s theological theme and allows for mention of classical mathematics

Ambition, Sami Samir as Cyril

Finally Hypatia’s enemies are sympathetic and well developed. Sami Samir (Munich) plays Cyril sincerely and though we must condemn his actions, Samir’s intelligent portrayal of his devotion and confidence forces us to admit that we can understand him. Ammonius, the monk who injured Orestes, is played by Ashraf Barhom, an actor I hadn’t previously seen. Ammonius is given a surprisingly large role in the film and Barhom fills him with such zealotry and conviction that the comparison between Ammonius and present day fundamentalism is inescapable; this is precisely what the filmmakers intend.

Whenever the film adheres to history, it is enormously successful. When it embellishes or changes history however, it falters. There are two characters who I feel detract from the film due to being contrivances of the filmmakers rather than true depictions of historic figures. This is no fault of the actors. Michael Lonsdale, known only to me as Hugo Drax in Moonraker, plays Hypatia’s father, Theon. He is clearly a capable actor, and with the writers he creates a realistic character. However this character is incompatible with my conception of the man. On the film’s official site, Amenabar explains his depiction of Theon: “In the film we portray Theon to be somewhat distracted, an old man reaching the end of his life who sees things happening much too quickly around him.” This is an understandable person, but it hardly seems in character for the man who radically defied society in his mission to liberate his daughter to react to the world in such a docile, futile, and confused manner.

Passion, Max Minghella as Davus

The film’s greatest weakness is an invented character, Davus. Max Minghella (Art School Confidential) who plays Davus shares top billing with Rachel Weisz deservedly. His character is the most compelling character in the film. This is unfortunate. In the film Davus is a slave of Theon’s family and an assistant to Hypatia in her classroom. He is an intelligent, emotional young man who has come to appreciate Hypatia’s teachings and fallen in love with her. However, Christianity appeals to him in its inclusion of slaves and the lower class, and he cannot reconcile the teachings of Hypatia with the orthodoxy of Christianity. As a conflicted character he is naturally compelling, and as an invented character he allows Amenabar and Gil the greatest latitude to invent interesting scenarios and investigate aspects of Alexandria during a time of social and theological revolution. Unfortunately he proves to be the most compelling character of the film and he steals the focus from Hypatia. His story is so inherently interesting and he is such a well-developed character that The Internet Movie Database’s entry for Agora mistakenly summarizes the plot as a “historical drama set in Roman Egypt, concerning a slave who turns to the rising tide of Christianity in the hopes of pursuing freedom while also falling in love with his master, the famous female philosophy professor and atheist, Hypatia of Alexandria.”

Whenever the film invents or changes history it detracts from the actual events. A minor, illogical modification is with the nature of the Christian Church’s encroachment into Alexandria. In reality Christianity was implemented by imperial decree. In the film Christianity is shown developing through proselytization of the lower classes. This change is obviously intended to allow the contrast of different social strata, and depict a more contemporary and familiar form of Chrisitianity, but I think the theocratic nature of the Roman Empire and Christianity’s status as its state religion is an important aspect of the western world’s history.

Ammonius’s story is mostly invented. History records him only as the monk who injured Orestes. In the film he is the leader of Alexandria’s Parabolani. Historically, the Parabolani were a lower class sect of monks that worked almost exclusively with the sick and at times served as bodyguards for bishops. The film does show this aspect, but it also portrays them as violent extremists used as a terrorist group by Cyril. They carry stones and clubs and are shown brutalizing the Jews. In the film it is this group that kills Hypatia. While this is internally logical, and it calls to mind contemporary religious strife, I can find no evidence that the Parabolani were really used in this way.

It is when Hypatia’s biography is modified that the film is most diluted. Amenabar first became aware of Hypatia while studying the history of cosmology, and as such she is principally an astronomer to him. Her discussions of ancient cosmological models are a highlight of the film. I was ecstatic when she discussed Ptolemy and Aristarchus. These geometric models of the universe are bolstered by mention of Appolonius, Euclid, and conic sections – this is great. Hypatia’s journey in the film is an intellectual one as she analyzes these systems and tries to fit them into her Platonic world view. This is a clever choice for it involves her math, science, and philosophy. However, the evolution of cosmological models is Amenabar’s first love and he forces it into the film. Agora’s Hypatia rejects geocentricism and Ptolemy’s model in favor of Aristarchus’s heliocentric model. In pivotal scenes she makes Galileo’s discovery of inertia and Kepler’s discovery of elliptical orbits allowing her to realize the Copernican heliocentric model. This is ahistorical obviously, but worse it confuses the tragedy of her assassination. Heliocentrism was of course a heresy during this time, and her connection to it diminishes the significance of her far more fundamental heresy of being an educated, bold woman.

Hypatia’s death is also significantly altered in a way that I feel diminishes its impact. She is not dragged through the streets to the cathedral, but walked by a Parabolani escort to the abandoned Serapeum. There they intend to skin her, but Davus (who has been freed from slavery, converted to Christianity, and become a Parabolani) convinces his brothers that a more religiously clean execution would be to stone her. Davus cannot bear to see his former teacher and owner, who he loves, die in such a painful way, and to prevent her suffering he suffocates her while the other Parabolani gather stones. He explains to the others that she has fainted from fright, and as they rain stones on her corpse, we follow Davus’s exit. Thus ends the film. This death is obviously more filmable and palpable than the grisly death actually suffered by Hypatia, but its key involvement of Davus again confuses the film’s focus.

Each addition or change is well written and clever, but ultimately serves to undermine the impact of the film. It is already a complicated story that touches on class struggle, religious extremism, intolerance, fanaticism, math, science, philosophy, history, misogyny and sexism, and the filmmakers insistence to involve more concepts and make more parallels between the 4th century and the present ends up creating a less moving, less convincing movie. It is certainly not a bad film; I simply do not think it is as successful as it could have been without the contrivances. Despite these faults, it is a very accomplished production that was rightfully allowed to premiere at the prestigious Cannes Film Festival, and was later recognized in Spain with 13 Goya (Spanish Academy Awards) nominations and 7 wins including well deserved Best Cinematography, Best Screenplay and Best Production Design awards.

Though the film’s potency is diluted by its complexity, its most striking error is one of simplification. Agora is a linguistic anomaly. It is an English language film from a Spanish production company, shot in Malta, with a Greek title, about Egyptians. With such multicultural and multilingual origins and with themes criticizing bigotry and stereotypes one would hope that the film would be free from stereotypes and bigotry itself. Sadly, this is not the case. The film follows an unfortunate trend that seems endemic to historical motion pictures: The protagonists are all played by Caucasians with accents from the British Isles, while the antagonists are portrayed by middle-easterners. This feels glaringly hypocritical of a movie that criticizes such typecasting. Worse, it is unmistakably not accidental. Rachel Weisz (Hypatia), Max Minghella (Davus), and Rupert Evans (Synesius) all hail from Great Britain. Michael Lonsdale (Theon) is French. Oscar Isaac (Orestes) is Guatemalan; however he is affecting a British accent in this film. The clearest antagonists, Cyril and Ammonius are played by an Egyptian (Sami Samir), and an Israeli Muslim (Ashraf Barhom) respectively. Both Samir and Barhom speak with their native ascents.  The obvious reason for this discrepancy are the differences in education level between those educated in the Greek style (Hypatia, Davus, Syensius, Orestes, Theon), and those that are either illiterate (Ammonius) or less educated (Cyril). However, even if this is the rationalization for these casting choices, the educated should be affecting Greek rather than British accents. If the choice was made because the filmmakers and production company did not think the audience would be able to accept or enjoy eloquent, reasonable, and sensitive middle-eastern protagonists, then it is an indictment of us all. Furthermore, if that was the case, it seems that it should have been the responsibility of Ammenabar to challenge that notion as he does other ignorant beliefs in Agora. That the film reinforces such damaging stereotypes diminishes all of its messages because it feels empty in its bigotry. When watching this film remember that all these characters should be middle-eastern and that history owes them a great debt, a debt this film and most films ignore.

Still it is a unique, challenging film with themes that most films do not dare approach. It is a testament to Telecino Cinema and Spain that the film was allowed to be produced and became a Spanish commercial and critical success. However America appears to be too immature and squeamish for this film. Focus Features and Newmarket Films, Agora’s US distributors, achieved only a very small release and did not succeed in adequately publicizing or advertising the film. When Borat’s Sacha Baron Cohen read the screenplay after being approached to appear in the film, he turned down the opportunity  explaining that the story was “too prickly, and would lift sores.” This comment both nullifies any respect I had had for Cohen’s artistic integrity and summarizes Agora’s failure in the US – we may be too reserved and bigoted to accept such a challenging film. Lionsgate, the film’s US DVD distributor, perhaps best illustrates America’s inability to handle the film with the unimaginably juvenile tagline chosen for the DVD: “A Holy War Becomes Hell on Earth.” Bravo, Lionsgate, bravo.

In the end, while I respect the film for its complexity, boldness, and uniqueness, I do not feel that it is successful. It fails to have the emotional resonance that great films do. This is not a failure on behalf of its director, writers, or actors. Neither is this due to the historical inventions or questionable stereotyping. It is ironically due to an artistic decision that proves too successful. Throughout the film, Amenabar uses aerial shots and views of the planet Earth from space for transitions. We are forced to see humans as small, as ant-like. Occasionally the comparison is made explicit through shots of insects. This is a definite artistic decision as explained by Amenabar:

The Agora is the story of a woman, of a city, of a civilization and of a planet. The Agora is the planet upon which we must all live together. We tried to show the human reality within the context of all the species of the Earth, and the Earth within the context of the universe- seeing human beings as ants, and the Earth as just another little ball, spinning beside many other stars.

This is exactly what the film achieves, however with each expansion of focus- going from woman, to city, to civilization, to planet- we become more and more distant from the individuals, we become forcibly divorced from their emotions, and the film loses the intimacy and passion that it needs to truly be effective. And again the mathematician is lost.

Agora (noun): A public meeting place for open discussion. – Root of agoraphobia.

Apollo 17 Anniversary, NASA (2002)

“Master,” said he, “we come to beg you to tell why so strange an animal as man was made.”

“With what meddlest thou?” said the Dervish; “is it thy business?”

“But reverend father,” said Candide, “there is horrible evil in this world.”

“What signifies it,” said the Dervish, “whether there be evil or good? When his highness sends a ship to Egypt does he trouble his hand whether the mice on board are at ease or not?”

-Voltaire, Candide

Hope, George Frederick Watts (1885)


“Agora (2009).” The Internet Movie Database (IMDb). Accessed October 12, 2010.

Agora. Directed by Alejandro Amenábar. By Alejandro Amenábar and Mateo Gil. Performed by Rachel Weisz, Max Minghella. Madrid, Spain: Telecinco Cinema, 2009. DVD.

“Alejandro Amenabar(Agora).” Interview by Tribute Movies. Tribute Movies. Accessed October 12, 2010.

Billington, Alex. “Cannes Interview: Agora Director Alejandro Amenábar.” May 26, 2009. Accessed October 12, 2010.

Fine, Marshall. “Rachel Weisz and Agora.” The Huffington Post. June 1, 2010. Accessed October 12, 2010.

Ágora, La Película. Accessed October 12, 2010.

Healy, Patrick J. “Parabolani.” Original Catholic Encyclopedia. Accessed October 12, 2010.

Holleran, Scott. “Alejandro Amenabar on Agora.” Scott Holleran: Freelance Writer. 2010. Accessed October 12, 2010.

Messer, Ron. “Rachel Weisz Interview AGORA.” Collider. May 28, 2010. Accessed October 12, 2010.

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The Rape of Cassandra

Author’s Note: This is first in a series of posts. Though they can be read independently, they were written with this order in mind: The Rape of CassandraMice on a Galley: A Review of AgoraHypatia Miscellanea

To Athena

(Homeric Hymn to Athena)

I begin to sing of Pallas Athena, the glorious Goddess, bright eyed,

inventive, unbending of heart

pure virgin, savior of cities

courageous, Tritogeneia. Wise Zeus himself bare her

from his awful head, arrayed in warlike arms

of flashing gold, and awe seized all the gods as they gazed.

But Athena sprang quickly from the immortal head

and stood before Zeus who holds the aegis,

shaking a sharp spear: great Olympus began to reel horribly

at the might of the bright-eyed Goddess,

and earth round about cried fearfully,

and the sea was moved and tossed with dark waves,

while foam burst forth suddenly:

the bright son of Hyperion stopped his swift-footed horses a long while, until the maiden Pallas Athena had stripped the heavenly armour from her immortal shoulders.

And wise Zeus was glad.

And so hail to you, daughter of Zeus who holds the aegis!

Now I will remember you and another song as well.



Athena Mourning 450 BCE

ATHENA: Do you know the insult done to me and the shrine I love?

-Aeschylus, Agamemnon

It is as if she is radiating fear, or that anxiety is a virile contagion. Some ancient animal sense, some herd instinct flashes through me. The white tailed doe.

“I’m sorry. I’m a little disorganized,” she needlessly says. “It’s this math. I hate it.”

“Take your time, we have as long as you need,” I respond. I look over the hundred digit poster of pi that’s on the wall. I have looked it over hundreds of times. With some students I tutor, I use the poster as a prompt to talk about infinite, or irrational numbers. Not with her. Despite my curiosity, I do not ask why she hates math. The question has no meaning.

“Give me a paper… a thesis, any day, and I can write it like that,” she snaps her fingers. “Ask me an anatomy question or psychology, and I’ll answer you right away, but math… I don’t know how you do it- ugh.” She is shuffling through her bags looking for her math notebook. Her hands shake, her eyes are watery. She is bloodless. My breath tries to emulate her hurried gasps, and I consciously slow it. Fear feeds on itself.

“Don’t be scared,” I say. “You’ll get through this.” I smile at her.

“Help me,” and she looks up at me. She looks into my eyes, and her eyes are lensed with tears. What I say next will determine whether they flow or recede.

I know of no subject with the possible exception of Speech that can cause such a physiological response in a person. It is worse than fear, it is dread, it is the fear of fear, and the fear of feeling fear. This student is experiencing the exact response she would have to a predator, to a snake. It is no surprise that she has avoided the subject, that it is odious to her. Mathematics has bitten her, has wounded her, and she remembers. She is not alone.

Her kind is legion. Sheila Tobias writes in Overcoming Math Anxiety that this intense fear of mathematics has limited the opportunities of millions of adults and that their negative experiences in mathematics can haunt them throughout their lives. It is part of our American culture that we do not give into fear and that we respond to it with defiance. Mathematics seems to be the exception. We not only accept as fact that math must cause dread, we tell ourselves a myriad of harmful myths to rationalize it.

Chief among these is the lateralization of brain function. This is the Right Brain/ Left Brain myth. There are many brain myths, such as the myth that we only use ten percent of our brains, but I think this one is the most harmful. Stemming from a misunderstanding of the split brain experiments performed by Sperry and Gazzaniga in the 1960s, the Right Brain/ Left Brain myth stipulates that people fall into one of two categories either Right Brain dominant or Left Brain dominant. Right Brain dominated people are said to be more creative and intuitive, whereas Left Brain dominated people are logical and analytic. The dangerous and fallacious conclusion is this: If you are Left Brained, you can do mathematics, if not then you can’t. When asked about astrology, Arthur C. Clark responded “I don’t believe in astrology; I’m a Sagittarius and we’re skeptical.”, and we should receive any such dictatorial theory with the same spirit, but we haven’t, and we don’t.

Roger W. Sperry and Michael S. Gazzaniga discovered some important things. They found out that the left and right sides of the brain process information independently. They discovered that they processed information differently. What they didn’t discover was that people are dominated by one hemisphere or the other.  This “fact” was “discovered” and popularized by the army of self-help writing vultures circling society’s anxieties. What was discovered is something that shouldn’t be surprising- we use both hemispheres of our brains all the time. Our experience of consciousness comes from a complex interaction of both types of data processing. As John McCrone writes in “Right Brain” or “Left Brain” – Myth Or Reality?,“whatever the story about lateralization, simple dichotomies are out. It is how the two sides of the brain complement and combine that counts.” He is perhaps too optimistic with his appraisal the myth when he states: “at least there seems no prospect of a return to the old left-right caricatures that inspired so many self-help books exhorting people to liberate their right-brains and avoid too much sterile left-brain thinking,”(2000).

If it is only a myth, why does it survive and resonate with so many? For the same reason that many myths survive- it provides a simple etiology for the state of the world: “I can’t paint because I’m a left brain person.”, “She always gets lost because she is right brained.”, “Math is for left brain, boring people.” Now, we all have our strengths and weaknesses, and there is certainly such a thing as talent, but we are not, for the most part, dictated by our biology. The myth of brain dichotomies is comforting to many because it provides a simple anodyne for the unsanitary and undemocratic fact that all people are not equal.  It provides the implicit assurance that any weakness one may possess is necessarily offset by a strength. Most harmful is the way it rationalizes and maintains the status quo and prevents challenges to it.

This dichotomy is taken as such a truism, that it is apparent in our very architecture. Here is a map of the tutoring department at the college I work at:

Room 116 is devoted to languages, social science, literature and the subjects falling under Humanities. Room 115 is devoted to mathematics, statistics, and the sciences. There is a wall between them. I realize that, in this photo, the arrangement of the subject is the reverse of that purported in the myth, but remember that the orientation of this map is arbitrary. The thing to note is this- we have so fully incorporated this myth into our culture that we no longer think to question it. We are so often consciously and unconsciously reminded of it that most people take it as fact.

If a lie is repeated enough times it would become widely accepted as truth.

-attributed to Joseph Goebbels.

This myth is most destructive when combined with another pernicious misbelief. Here is a typical list of right brain characteristics (taken from the Herald Sun, October 9, 2009 Right Brain v Left Brain), Right: uses feeling, believes, appreciates, fantasy based, impetuous. Compare this list of attributes to this poem written by Mrs. E. Little in response to the initial stirrings of the Women’s Rights movement and published in Godey’s Ladies Book in 1848:

The Rights of a True Woman

The right to love whom others scorn,

The right to comfort and to mourn,

The right to shed new joy on earth,

The right to feel the soul’s high worth …

Such women’s rights, and God will bless

And crown their champions with success.

When reading this poem, it is important to remember that the rights mentioned in the poem are the only rights assumed by a “true woman.” A “true woman” only has these characteristics. Certainly these are good characteristics, but the author is binding women by these rights and denying her any others. So what is the conclusion of the comparison between poem and the characteristics? The True Woman is Right-Brained.

The synthesis of the Right Brain/ Left Brain myth and the True Woman myth is a truly dangerous and largely unquestioned lie that runs rampant throughout our society. Furthermore, it is a self fulfilling stereotype.

University of California Santa Barbara mathematics professor, John Ernest states the results of a 1976 study of high school students: “[both genders] have a fair amount of trouble doing math, and most of them do not like the subject very much.” If both genders are equally poor at mathematics, then what is the explanation for this:

From the same study: “The difference between them was that boys stuck with math, because they felt their careers depended on it and because they had more confidence than girls in their ability to learn it.”  Why is there this difference in confidence, and where did it come from? I would like to emphasize the point that it isn’t biological: “No one has yet identified a ‘math lobe’ in the brain.” If the answer isn’t internal, it necessarily must be external, and it is: “Researchers can only measure performance on tests, and we know much more now than we used to about how much performance is influenced by beliefs, perceptions, prior experience, and self-esteem.” (Tobias, 1993).  It is societal.

There is a common bias to dismiss the effect society has on educated people. We believe that our knowledge and cleverness buffers us from the stereotypes. We think that because we know words and phrases like peer pressure, discrimination, mores, folkways, and words ending in ‘ism’, we are protected from them, like knowing the names of demons. However this is dismissive and untrue. Temple University mathematics professor, John Allen Paulos, writes in Innumeracy, “whether or not a department has a mathematics or statistics requirement is the most important single determinant of where a woman will attend graduate school to study political science,”(2001).

Math anxiety is a serious thing. It truly affects people’s lives, and I think it is most harmful to women.  In my personal experience, it manifests itself differently in women than in men. The anxiety is most often displayed as frustration and anger in males, and fear and despair in females. From a tutoring perspective, this presents very different challenges. In a tutoring session, anger can be used creatively: The tutoring session can be recast as a confrontation between the student and the problem, and the belligerent side of anger used as a catalyst to get the student to confront and overcome the difficulty.  Fear is far harder to deal with. When a student feels real physical dread of mathematics it is very hard to get the student to refuse the natural instinct to run and to stay with the problem.

I’m going to quote a long passage from Malcolm Gladwell’s book, Outliers: The Story of Success. I encourage you to read the full passage.  Though Outliers is not specifically about mathematics, this passage captures the essence of mathematical learning better than any other I have come across, and I believe its example is invaluable. (For clarification, I have used Wolfram|Alpha to make reproductions of the graphs featured in the book):

A few years ago, Alan Schoenfeld, a math professor at Berkeley, made a videotape of a woman named Renee as she was trying to solve a math problem. Renee was in her mid-twenties, with long black hair and round silver glasses. In the tape, she’s playing with a software program designed to teach algebra. On the screen are a y and an x axis. The program asks the user to punch in a set of coordinates and then draws the line from those coordinates on the screen. For example, when she typed in 5 on the y axis and 5 on the x axis, the computer did this:

At this point, I’m sure, some vague memory of your middle-school algebra is coming back to you. But rest assured, you don’t need to remember any of it to understand the significance of Renee’s example. In fact, as you listen to Renee talking in the next few paragraphs, focus not on what she’s saying but rather on how she’s talking and why she’s talking the way she is.

The point of the computer program, which Schoenfeld created, was to teach students about how to calculate the slope of a line. Slope, as I’m sure you remember (or, more accurately, as I’ll bet you don’t remember; I certainly didn’t), is rise over run. The slope of the line in our example is 1, since the rise is 5 and the run is 5.

So there is Renee. She’s sitting at the keyboard, and she’s trying to figure out what numbers to enter in order to get the computer to draw a line that is absolutely vertical, that is directly superimposed over the y axis. Now, those of you who remember your high school math will know that this is, in fact, impossible. A vertical line has an undefined slope. Its rise is infinite: any number on the y axis starting at zero and going on forever. It’s run on the x axis, meanwhile, is zero. Infinity divided by zero is not a number.

But Renee doesn’t realize that what she’s trying to do can’t be done. She is, rather, in the grip of what Schoenfeld calls a “glorious misconception,” and the reason Schoenfeld likes to show this particular tape is that it is a perfect demonstration of how this misconception came to be resolved.

Renee was a nurse. She wasn’t someone who had been particularly interested in mathematics in the past. But she had somehow gotten hold of the software and was hooked.

“Now, what I want to do is make a straight line with this formula, parallel to the y axis,” she begins. Schoenfeld is sitting next to her. She looks over at him anxiously. “It’s been five years since I did any of this.”

She starts to fiddle with the program, typing in different numbers. “Now if I change the slope that way…minus 1 . .. now what I mean to do is make the line go straight.”

As she types in numbers, the line on the screen changes.

“Oops. That’s not going to do it.”

She looks puzzled.

“What are you trying to do?” Schoenfeld asks.

“What I’m trying to do is make a straight line parallel to the y axis. What do I need to do here? I think what I need to do is change this a little bit.” She points at the place where the number for the y axis is. “That was something I discovered. That when you go from 1 to 2, there was a rather big change. But now if you get way up there you have to keep changing.”

This is Renee’s glorious misconception. She’s noticed the higher she makes the y axis coordinate, the steeper the line gets. So she thinks the key to making a vertical line is just making the y axis coordinate large enough.

“I guess 12 or even 13 could do it. Maybe even as much as 15.”

She frowns. She and Schoenfeld go back and forth. She asks him questions. He prods her gently in the right direction. She keeps trying and trying, one approach after another.

At one point, she types in 20. The line gets a little bit steeper.

She types in 40. The line gets steeper still.

“I see that there is a relationship there. But as to why, it doesn’t seem to make sense to me… What if I do 80? If 40 gets me halfway, then 80 should get me all the way to the y axis. So let’s just see what happens.” She types in 80. The line is steeper. But it’s still not totally vertical.

“Ohhh. It’s infinity, isn’t it? It’s never going to get there.” Renee is close. But then she reverts to her original misconception.

“So what do I need? 100? Every time you double the number, you get halfway to the y axis. But it never gets there…”

She types in 100.

“It’s closer. But not quite there yet.”

She starts to think out loud. It’s obvious she’s on the verge of figuring something out. “Well, I knew this, though… but… I knew that. For each one up, it goes that many over. I’m still somewhat confused as to why…”

She pauses, squinting at the screen.

“I’m getting confused. It’s a tenth of the way to the one. But I don’t want it to be…”

And then she sees it.

“Oh! It’s any number up, and zero over. It’s any number divided by zero!” Her face lights up. “A vertical line is anything divided by zero — and that’s an undefined number. Ohhh. Okay. Now I see. The slope of a vertical line is undefined. Ahhhh. That means something now. I won’t forget that!”

Over the course of his career, Schoenfeld has videotaped countless students as they worked on math problems. But the Renee tape is one of his favorites because of how beautifully it illustrates what he considers to be the secret to learning mathematics. Twenty-two minutes pass from the moment Renee begins playing with the computer program to the moment she says, “Ahhhh. That means something now.” That’s a long time. (2008)

This is the story of a math student. Studying mathematics is a long, difficult process. It is tedium and torture interspersed with sporadic epiphanies of such brilliance that they justify the effort that came before them and renew one’s desire to venture onwards. I often think of studying mathematics as climbing a succession of mountains. The climb is hard and exhausting, but upon reaching a summit, the vista’s beauty inspires you to continue, to attempt the next peak, and with each crest, you see farther and clearer. To understand and enjoy mathematics, this trek is necessary. Imagine the student with math anxiety attempting this journey, sticking with one problem for twenty-two agonizing minutes knowing that she is a right-brained woman in a left-brained man’s field, and throughout it all, her body reminding her that she is in imminent mortal danger.

[Mathematicians] resemble those that gaze out from the tops of high mountains whose summits are lost in the clouds. Objects on the plain below have disappeared from view; they are left with only the spectacle of their own thoughts and the consciousness of the height to which they have risen… -Denis Diderot

There is a nasty and ridiculous response that rears its head whenever someone is audacious enough to suggest that sexism or gender inequality is in fact a thing. It is the inevitable argument by example. It goes like this: “A bias against women does not exist in mathematics; my high school chemistry teacher, Mrs. Representativeforallwomen, was a scientist and a woman.” We should all be able to see the absurdity of the speaker’s conclusion, but to draw attention to that fact I am going to list the first names of the entire 2010 faculty of Stanford University mathematics department, one of the most prestigious mathematics departments in the world. You will see that some of them are indeed women:

Ricardo, Simon, Gregory, Daniel, Isabelle, Emmanuel, Gunnar, Ralph, Brian, Craig, Amir, Persi, Yakov, Solomon, Robert, Søren, Pierre, Eleny, Renata, Vladislav, Yitzhak, Joseph, Steren, Jun, Joan, Tai-Ping, Rafe, Peter, James, Grigori, Maryam, Donald, Robert, George, Lenya, Richard, Leon, Kannon, Ravi, András, Akshay, Brian, Melanie

Out of the 43 faculty members, 6 are women. That’s about 14%. Have I proved anything? No, but it most provoke our curiosity why there is such a disparity between the percentage of the faculty that is female, 14%, and the percentage of people who are female, 51%. If the reason for this schism is not biological, which it isn’t then what is it?

A 2010 report by the American Association of University Women titled Why so Few: Women in Science, Technology, Engineering, and Mathematics provides some answers: “In schools and in homes the environment that is created serves to subtly and perhaps in some cases not so subtly discourages girls or encourages them to focus on other areas, even if they might have a brimming interest and ability in science.” Sheila Tobias provides us with examples of the practical manifestation of this phenomenon:

Fathers… are more likely to help with math homework than mothers. Even teachers, expecting more in mathematics of their boy students than of their girls, ask higher-order questions of boys and encourage males to discover alternative solutions while inhibiting girls’ mathematical creativity by insisting that they follow the rules.(1993)

We do not encourage our female students to continue their studies in mathematics. We don’t provide clear female role models in these areas. We believe myths about women being incapable of math. Consciously or unconsciously we reinforce these attitudes and create an atmosphere that implies math is not for girls.

Most tragic is the fact that these prejudices shape the world in their image. The constant repetition and reinforcement of the notion that math and science are only for the biologically selected discourages and prevents encouragement for many. Fewer women then go into these fields, and the myth appears true. Less women scientists and mathematicians mean less female role models for students considering the fields. It means simple intimidation by gender inequality. There is no shadowy plan against women here; there is simply the self-fulfilling prophecy of some myths run rampant.

My own sister has experienced these prejudices. When she asked her adviser about pursuing science, her adviser advised her against it. The reason? She was a “language person” was the response. Implicit in “language person” is this pernicious logic: language person= right brained= a woman. A person in authority attempts to shape someone’s future based on falsehoods. I wonder how many woman have been turned away from sciences because their advisers, teachers, friends, or parents told them they were “language persons.”

How it Works, XKCD by Randall Munroe

Having failed to embed it( wordpress currently does not support vimeo embedding), I’m placing a link to a video by physicist, educator, and skateboarder, Yung Tae Kim addresses several of the failings of our education system, and provides some suggestions for reform. While these concerns apply to every aspect of education, I feel that they are especially pertinent to science and mathematics education. I think the entire video is brilliant, however if you are pressed for time I would suggest skipping to:

1:40 – a discussion of the problem of depersonalization and class size. Keep in mind how this atmosphere may affect individuals with math anxiety.

5:00 – a shocking statistic regarding the credentials of many science teachers.

10:28 – the negative effects of some common teaching practices. I feel these practices may be especially harmful to women.

Here is Dr. Tae’s Building a New Culture of Teaching and Learning

Why are such harmful attitudes so prevalent in our society? How have we let them flourish? Why aren’t there more women in science and mathematics? Many of these beliefs have become entrenched in our culture through their long history. It is a legacy that begins fourteen hundred years ago with the first recorded female mathematician.

In order to understand something, you must go to its origins.

– Aristotle

St. Paul at Epesus from the Doré Bible, Gustave Doré 1865

In 391, the Christian emperor Theodosius I made paganism illegal. Shortly thereafter, by decree of Patriarch Theophilus, the Serapeum of Alexandria was looted and destroyed.   A temple to the god Serapis, it was renowned in the ancient world. The Roman historian Ammianus Marcellinus, a contemporary to the Serapeum’s destruction, described it as “splendid to a point that words would only diminish its beauty.”  However, its importance in 391 went far beyond its aesthetics. After the inadvertent destruction of the Great Library of Alexandria by Julius Caesar in 31BCE, the Serapeum became the shelter for many of the surviving books. The Great Library of Alexandria was the greatest repository of knowledge, and for a time the most academically vibrant place in the ancient world. It would remain the single greatest accumulation of knowledge achieved until the Renaissance, thirteen hundred years later. With its destruction, human knowledge was set back hundreds of years. It would take millennia for science and mathematics to recover. The destruction of the Serapeum and the surviving books set the recovery back even further.

Watching this tragic event was a 21 year old woman whose brilliance was equal to any displayed in those books, and whose end was just as tragic. Hypatia of Alexandria, history’s first recorded female scientist, was an anomaly of her time.  The ancient view of women was terrible. In Athens, women could not leave their homes, accept inheritance, or buy things and they were bound to marriage for survival. The story of Pandora suggests that not only were women created as a punishment for men, but that women are the cause of all evils in the world. The story of Genesis goes so far as to blame women for the existence of death.  In the Timaeus, when Plato outlines his theory of reincarnation, he tells us that wretched men are punished by being reborn as women: “Of the men who came into the world, those who were cowards or led unrighteous lives may with reason be supposed to have changed into the nature of women in the second generation.” But here in 391, watching the holocaust of the achievements of the ancient world is 21 year old Hypatia. A woman dressed in the strictly male robes of scholars. A woman who had the audacity and keenness to lecture to halls of men about science, astronomy, logic, philosophy and mathematics – subjects considered beyond all reach of the female mind.

Hypatia in The School of Athens, Raphael 1510

There was a woman at Alexandria named Hypatia, daughter of the philosopher Theon, who made such attainments in literature and science, as to far surpass all the philosophers of her own time. Having succeeded to the school of Plato and Plotinus, she explained the principles of philosophy to her auditors, many of whom came from a distance to receive her instructions. On account of the self-possession and ease of manner, which she had acquired in consequence of the cultivation of her mind, she not unfrequently appeared in public in presence of the magistrates. Neither did she feel abashed in going to an assembly of men. For all men on account of her extraordinary dignity and virtue admired her the more.

-Socrates Scholasticus

The vast majority of Hypatia’s work has been destroyed or lost to the tides of history. However, what remains is of such remarkable quality and importance that it is truly surprising that she was indeed more prolific. She was a mathematics professor at the Museum of Alexandria, a position first held by the single greatest mathematics writer of all time, Euclid. This is comparable to the “Newton’s Chair” of physics at the University of Cambridge, formerly held by Stephen Hawking.  Her lectures were extremely well regarded and men would travel vast distances to study under her.

Woman Teaching Geometry in Adelard of Bath's translation of Euclid's Elements 1309-1316

Most of what we know about her personality, reputation and research comes from one of her students, Synesius of Cyrene who became bishop of Ptolemias in Libya. Through his writings we know that Hypatia was involved in the invention of the astrolabe (used for astronomy and navigation), the planesphere (an interactive star chart), the hydroscope (similar to a periscope, used for seeing under the water), and the hydrometer (an instrument for measuring the density/ specific gravity of liquids).

An Astrolabe

Most of her mathematical writings are believed to have been in the form of class notes and textbooks to her students, which have been lost. However two of her treatises have survived. In On the Conics of Appolonius, she popularizes and extends Appolonius work on conic sections. After Hypatia’s death, this area of mathematics would be forgotten and ignored until the seventeenth century. Her most important work is a commentary on Diophantus’ Arithmetica. Diophantus who lived in the third century, is considered “the father of algebra.” He invented the concept of using symbolic notation in mathematics, an invention without which mathematics would be hopelessly impaired. The academic tradition of commentaries required the commentator to copy the entire work by hand adding annotations and enhancement. Of the surviving copies of Diophantus’ Arithmetica, it is theorized that all are derived from Hypatia’s commentaries and therefore include her additions. She is as much an author of the work we now call Arithmetica as Diophantus. Without her work we would no longer have his and science and mathematics would be centuries behind.

Furthermore Hypatia was a philosopher, and she passed along her neoPlatonist beliefs to her students. At the height of her career, letters addressed simply to “The Muse” or “The Philosopher” would be delivered unquestioningly to her. This influence is clear in Synesius, who was a forerunner in combining Platonism with Christianity – a practice which would become dominant in the middle ages, and is still seen today. Central to her philosophy was scientific rationalism, and this was a liability for her because in the fourth century, an ecclesiastical campaign against science would lead to one of the greatest tragedies of the ancient world.

Hypatia by Masolino da Panicale 1428

It is important to remember that Hypatia was unique in her world. There were no other female scientists, lecturers, inventors, mathematicians, or philosophers at that time. As far as we know, there had never been. It was assumed that there couldn’t be. Hypatia’s opportunity to become a scholar and public figure was provided to her by her father, Theon of Alexandria; a man who is remarkable for his belief in and love of his daughter.

Reserve your right to think, for even to think wrongly is better than not to think at all.

-Theon of Alexandria

Theon was himself a distinguished mathematician and astronomer at the Museum of Alexandria. He may even have been its director. His greatest works are his commentaries on Ptolemy’s astronomical work, Almagest (assisted by Hypatia), and Euclid’s Elements (perhaps the most significant work of mathematics ever published). Like Hypatia, it seems that most of his writing was for his students, and for them he made significant clarifications and enhancements to the Elements. He was concerned with the quality of his students, noting that many could not follow geometric demonstrations. He felt that academics was losing its rigor. Perhaps in response to these fears, he took a special interest in his daughter.  He was determined to help Hypatia develop into the “perfect human being.”

Theon made sure that Hypatia received a formal education in the arts, literature, science, and philosophy. He immersed her in the intellectual world of Alexandria, and personally tutored her in mathematics. Theon knew that Hypatia would face harsh criticism due to her gender, and to equip her against these, he had her trained in speech and rhetoric. When her competencies exceeded his, she travelled to Athens to study at the school founded by the philosopher Plutarch.

Theon invented as series of exercises and calisthenics for Hypatia to practice. He cherished her, and tried to cultivate her in all dimensions. Theon’s belief in his daughter should still be inspiring. She may have been naturally intelligent and talented, but it was her father’s support and encouragement that allowed her to blossom. Success does not spring from a vacuum.

All formal dogmatic religions are fallacious and must never be accepted by self-respecting persons as final. –Theon of Alexandria

By the end of the fourth century and into the fifth century, by necessity of Roman law, Roman citizens were converting to Christianity. Throughout the empire, heresies were being purged, and the heretics assassinated. It had become popular propaganda that non-Christian beliefs were weakening Roman character and had to be annihilated. In Alexandria, this was the task of the bishop Cyril.

Saint Cyril of Alexandria by Ignaz Franz Platzer, 1717-1787

Cyril of Alexandria was ideal for this position. He seems to have had a gift for inciting the passions of his followers. In 412, he was consecrated as a successor to his uncle, Theophilus, but only after his supporters rioted violently against his opponent. His first act was the persecution of the Novations (a sect that denied the authority of the pope), and the plunder of their churches. Shortly thereafter, he instructed his followers to drive the Jews from Alexandria.

The expulsion of the Jews incensed the Prefect of Alexandria, Orestes, who moved against Cyril. In defense of the Bishop, five hundred monks travelled to Alexandria from Nitria several miles south of Alexandria. With this influx, Cyril’s power in the city rivaled Orestes’ and things became violent. At one riot, a monk named Ammonius injured Orestes with a thrown rock. Orestes had Ammonius arrested and tortured to death.

Cyril brought Ammonius’ remains to the cathedral, and honored the monk as a martyr. The monk’s death, and Cyril’s fiery passion galvanized his supporters, and rioting in Alexandria became continuous. The most infamous of these riots was led by Peter the Lector in 415. It had a specific target.

Hypatia was the perfect victim. She was a friend and supporter of Orestes. She represented the paganism that the Christians were attempting to purge. She taught the knowledge of the decadent, sinful Greeks. Her Platonism and scientific rationalism were in opposition to the Chrisitian dogma. Most importantly, she was an example of the corrupting influence of that knowledge- She was a woman who didn’t know her place, and she was corrupting the young men of Alexandria.

Hypatia was torn from her chariot by a bloodthirsty mob as she was returning to her home from the Museum. She was stripped of her clothes as they dragged her through the streets and into the Cathedral of Alexandria. On that hallowed ground, she was held by the rioters as they tore her skin from her bones with oyster shells. The strips were burnt piece by piece as she died.

Hypatia by Charles William Mitchell, 1885

Let your women keep silent…-1 Corinthians 14

Cyril would later become Saint Cyril, and go on to author the seventeen volume On Adoration in Spirit and in Truth. Both titles seem tragically ironic to me.

Hypatia’s murder represents more than just the loss of a great human. With her assassination, the academic spirit in Alexandria was broken. Historians mark the end of Greek mathematics with her death. A millennium would pass before science and mathematics regained its vigor. The Dark Ages would soon begin.

Hypatia is often mentioned as an inspiration for woman, and she is. However, I think the truth of her legacy is much darker. There would not be another woman mathematician for over twelve hundred years until Elena Lucrezia Cornaro Piscopia received her doctorate in 1678. Saint Cyril and Peter the Lector’s message seems to have rung loud, clear, and long. They laid the precedent: “It is dangerous for a woman to study science and mathematics, to be a scholar and an author. Society doesn’t want you, and will expel you. It is not natural and not Christian.” Their misogyny and anti-intellectualism have sounded throughout history and still echo today. How many fathers like Theon decided against schooling their daughters out of a fear that they would follow Hypatia’s fate? If only Hypatia were not murdered, perhaps her story would have served as inspiration for thousands of girls instead of as a warning.

Rape of Cassandra, Red figure pottery, C, 370-360 BCE

The most important event of classical mythology occurs during the sack of Troy after the Greeks have succeeded with their gambit of the Trojan Horse. Amidst the chaos of Troy’s collapse and the massacre being carried out by the Greek troops, Cassandra, daughter of Troy’s King Priam, Priestess of Athena, Seer of Troy, and the one Trojan who knew they were doomed, sought sanctuary in the Temple of Athena. There she grasped the statue of Athena with both arms. In classical tradition this should have meant that she was untouchable, that Athena had granted her asylum. But something went horribly wrong. Ajax the Lessor, a hero of Greece, burst into the temple, raped Cassandra at the feet of Athena, and abducted her.

This trespass of one of the fundamental codes of the Hellenistic religions enraged the gods who cursed the Greeks. This curse was the reason the Heroes of Troy met such difficulties in returning home. Hesiod in Works and Days marks this as the event that turned the gods away from men, and ended the age of heroes and mythology. The gods were so disgusted by Ajax actions, that they washed their hands of us. In Hesiod’s words, humanity entered a period of unstoppable degeneration in which “there will be no help against evil.”

Hypatia’s murder has cast a similar shadow. We have all been cursed for what was done in that cathedral. We had the opportunity to mature as a species and make strides for equality and rationality, but it was lost to dogmatic hatred. Our heritage is a sixteen hundred year old handicap, and it is time to make amends. Do not passively allow dogmatic thinking to limit the world we live in – this is our responsibility.

Fable should be taught as fable, myth as myth, and miracles as poetic fancies. To teach superstitions as truths is horrifying. The mind of a child accepts them and only through great pain, perhaps tragedy, can the child be relieved of them. Men will fight for superstition as quickly as for the living truth – even more so, since a superstition is intangible, you can’t refute it, but truth is a point of view, so it is changeable.

-Hypatia of Alexandria


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L’arte Della Cosa

Melencolia I by Albrecht Dürer (1514)

Mathematics is the sister, as well as the servant, of the arts and is touched with the same madness and genius. –  Harold Marston Morse

I had not intended this to be the first essay posted here. Indeed, I had not intended to write this at all. However, while preparing what was to be the first essay, and in doing so delving into many mathematical writings, I noticed a common deficiency in them. This deficiency in mathematical exposition is, I believe, the cause of many difficulties in mathematics education, and the central motive to my creation of this site. It is simply this: Mathematics writing, even popular mathematics writing is horribly insular. There is a fiery vitality to mathematics that is caged behind a web of arcane symbols, exclusive terminology, and elusive concepts. To the adept who has tamed these, the passion is evident; it seeps through the symbols, glows behind the formulas, and thrums in the theorems.  However, to the outsider, or initiate it is often intractable.  I do not believe the nature of this great joy is indescribable. I do not believe that language is so poor a tool that it cannot capture it.

What makes mathematics so vital? What is the nature of its joy? While I have seen the answer to these questions circumscribed and hinted at, I have never seen it clearly articulated. Yet, it is there in mathematical work. Calmly coiled in the rigor, and secretly shared by its enthusiasts, the heart of math’s fierce cadence is constant and insistent, but seldom heralded. Here I will attempt to define it. I do not pretend that the specifics of mathematical fury are constant from person to person, or that they are not intensely personal. My aim here is to identify and illustrate the common thread, to get to the root of mathematical wonder.

We are the bees of the invisible. –Rainer Maria Rilkes

Mathematics isn’t the exercises and word problems in textbooks. It isn’t the +, -,×,÷ of arithmetic, the lines and curves of geometry, the angles of trigonometry, or the sums of analysis. These things, though they are wondrous, are merely humanity’s best, and woefully inadequate hymns to mathematics. To get at the nature of mathematics, let us consider a poem (poetry being, to me, math’s closest analogue):

Sonnet 75

Edmund Spenser

One day I wrote her name upon the strand,

But came the waves and washed it away:

Again I wrote it with a second hand,

But came the tide, and made my pains his prey.

Vain man, said she, that doest in vain assay

A mortal thing so to immortalize,

For I myself shall like to this decay,

And eek my name be wiped out likewise.

Not so (quoth I), let baser things devise

To die in dust, but you shall live by fame:

My verse your virtues rare shall eternize,

And in the heavens write your glorious name.

Where whenas Death shall all the world subdue,

Out love shall live, and later life renew.

Here Spenser confronts the central paradox of humanity, perhaps of all life: The present touches eternity, but it is nothing to eternity. Worse still, perhaps there is no present at all. If all efforts are washed away by the tides of time, were they ever really there? Does the present exist, or is it just an illusion, the fleeting impression of the future becoming the past? How much more truthful were the hour glasses and water clocks, where the unceasing flow of time was permanent and irreversible, than our circular clocks, where the hours will always come again?

It is a major theme of the human experience that when people are confronted with the infinite, the bold respond with dread, horror, and awe, and that the meek turn away and try to forget. There is madness in eternity. It is far too large, we are far too small, and the implication of this is too easily grasped. To frame it mathematically, in calculus, whenever the individual (1) encounters the eternal (∞), we write this:1/∞=0 ; you are nothing when compared to everything. The present is so fleeting, and so small that it may as well have never existed.

Spenser recognizes this. He sees the great threat, and responds heroically as an artist. His weapon against eternity’s indifference is his poem, and he knows it is an effective one, because he recognizes that a poem is something quite special: It is the crystallization of an idea.  Idea is a surprisingly slippery concept, and importantly so. Ideas have no fetters, they are atemporal, and they represent our transcendence over time’s tyranny.

Everything changes, but ideas. They are our one constant, and they let us know who we are. You have never seen the same face in your mirror. What you have seen is a succession of similar masks. Each time you’ve combed you hair, brushed your teeth, and washed your face someone different was watching you. Sisters perhaps, maybe even twins, but surely not the same person. Her hair was a little longer, his skin a little paler, the eyes slightly duller, the wrinkles deeper, the age older, the lighting greener. However, this constant assault by strangers has not driven you to shatter all your mirrors. Somehow you have known that from day to day, hour to hour, minute to minute, the reflections you see are not portraits of different people, but are rather frames of the same person’s film. The totem that has saved you from insanity and your mirrors from destruction is the continuum of your thoughts.

Your thoughts can also resurrect the dead. This was Spenser’s mission. By reading his poem, and mirroring his thoughts at the time of its writing, you have invited Spenser into you. Spenser, his lover, and his love now live in you, and all those that have found something in his sonnet. In a way, he is more alive today, and his love burns brighter than ever, because it is no longer one heart that beats with the vigor of Spenser’s love, but the hearts of the millions that have read the poem, and the millions that will read it.

Spenser’s poem tells the story of much more than a poet on the beach. It is the David and Goliath story of man versus the infinite. It is a great act of defiance that has immortalized two humans, and amplified their love exponentially; a tremendous feat for fourteen lines.

Mathematics is pure poetry. –Immanuel Kant

Perhaps the most well-known mathematical statement is the Pythagorean Theorem: In a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse, or a2+b2=c2. This modest collection of letters and numbers encapsulates something that is truly awesome. Those that understand what this is love mathematics. However, it is the great tragedy of mathematics that the symbols, chosen for their utility, we use to describe and investigate the subject obfuscate by their quotidian nature the enormity of what they represent. To understand the magnificence of a2+b2=c2 , we must analyze the essential feature of mathematics- proof.

Nasir al-Din al-Tusi's Proof of the Pythagorean Theorem (1274)

The name given to this remarkable relationship, the Pythagorean Theorem, indicates that it is something very special. A theorem is a mathematical statement that has been proven. The concept of proof in mathematics is the defining attribute of the subject. It is not used casually, and it has nothing to do with evidence. In our legal system, we follow the Presumption of Innocence, the often stated legal right that the accused is innocent until proven guilty beyond a reasonable doubt. In the sciences we search for empirical evidence that supports a suspected correlation. In neither of these fields is a conclusion considered final and unfalsifiable no matter the extent of the evidence. The “proofs” in human endeavors only assert a high probability, never a certainty.  Though they may loath to admit it, both the lawyer and the scientist are essentially Humean in their outlook. If they are being truthful, they will never contend that they know something, only that they strongly suspect it.

The mathematical proof is not evidence of the validity of a mathematical statement, it is the validity of the statement. When something has been proved its dominion becomes infinite and eternal. a2+b2=c2 is not a characteristic of any one right triangle, or any group of right triangles, or a very likely “law” of right triangles. It is a feature of all right triangles so fundamental that it defines right triangle. There is no need to test each case. It is applicable no matter the triangle. It is as applicable to the triangle formed by your closet floor, broom, and closet wall as it is to the triangle formed by the Earth, Sun and Moon during the third lunar phase.

The theorem’s universal truth is implicitly relied upon by every architect. It is the reason why one carpenter’s square can be used in every project, and each blueprint is not required to “re-invent the wheel” to provide a stable house. By proving the statement, humankind gained infinite knowledge. Mathematicians have been able to tame the numinous.

A thought is an idea in transit. –Pythagoras

It cannot be emphasized enough: We know something about everything. To make something of the infinite comprehensible, concise, and approachable is math’s most remarkable success. However, this joy is merely intellectual, and there is a much more human side to mathematics that is no less unique to the subject.

The Pythagorean Theorem was discovered (or invented) more than two and a half thousand years ago. Throughout those years, it has never lost its vigor. Each generation of mathematicians has learned it, meditated upon it, incorporated it into their mathematical corpus, and added to it. Regardless of time, place, culture, gender, religion, etc. the mathematicians of each age safeguard the work that came before them, and expand upon it. It is a unifier across history, culture and language that has not gone unnoticed by historians. As Edward Gibbon observes in The Decline and Fall of the Roman Empire: “The mathematics are distinguished by a particular privilege, that is, in the course of ages, they may always advance and can never recede.”

In most sciences one generation tears down what another had built and what one has established another undoes. In mathematics alone each generation adds a new story to the old structure. -Hermann Hankel

From a modest seed, counting, an ever growing, enormously complex structure has grown. Branches sprout, expand, branch themselves, but they never break. All the complexity, each new bud, draws its sustenance from the same trunk, and common roots. This map of the infinite, this island in the eternal is humankind’s most amazing creation.

Pythagoras Tree by Murdoc Snook (2009)

This is grand from a cultural perspective, but it is amazingly humbling from a personal one. As a practitioner of math you are constantly aware of the long lineage of great minds that have come before you, each providing a wrung for you to grasp. For the student, the ascent of this mathematical tree is a literal walk through human history. It is a living history that to understand you must participate in. With each new lesson you are confronting and overcoming the challenges of a generation. When I’m studying Euclid, I am quite literally thinking the same thoughts of the ancient Greek geometers. This synchronicity through history, through millennia, binds us together.  It makes personally clear the continuum of human experience, and reminds us of our common position as peers in existence.

The great glory of mathematics is its durative nature; that it is one of humankind’s longest conversations; that it never finishes by answering some questions and taking a bow. – Barry Mazur

A commonly expressed belief about math (traceable to Leibniz, I believe) is that it is a universal language. This is true, but I believe in a way more subtle than is often intended. The great message of all mathematical statements is the latent one that declares, “I’m conscience.” It is no accident that the man who declared, “I think, therefore I am” also revolutionized geometry, and it is no coincidence that the messages we would send into space via Voyager’s golden records would be coded mathematically. Mathematics is our touchstone and litmus test for consciousness, our vanguard against solipsism when seen in others, our Rosetta stone for the dead, and the great thread that ties us together. That is the burning heart, the vibrant core, of math.

Voyager Record Cover, NASA, (1977)

Cogito ergo sum. -René Descartes

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