Word-nerd fun: Galaxy

Photo of the Milky Way
taken by my son John
10 November 2023

Yesterday I was reading Chaucer’s delightful Parliament of Fouls. Early on, the narrator says he fell asleep reading Cicero’s “Dream of Scipio,” and he retells the story for the reader. In the dream, Scipio meets his grandfather who “showed him the Galaxy.”

Then today I was reading The Marriage of Philology and Mercury, a 5th century Latin work by Martianus Capella. In Book II, Philology is ascending through the heavenly spheres on her way to the wedding. When she reaches the upper limits of the cosmos, the narrator says, “The incandescence of a milk-white river gradually flowed down from the burning stars. Full of joy and thanksgiving she turned toward the Galaxy where she knew that Jove had assembled the divine senate.”

I wondered whether the word “galaxy” was used in the original Latin so I looked it up and sure enough it says, “iter in Galaxeum flectit.” Galaxias is itself a borrowing from Greek.

The story behind the word galaxy is a fun one, so I thought I’d share it.

Prior to modern astronomy, Galaxy simply meant the Milky Way, γάλα [gala] being the Greek word for milk. The story is that after Zeus’s son Heracles was born to the mortal woman Alcmene, Zeus waited till his wife, Hera, was asleep and put the newborn to nurse from her so he could partake of the divine quality of her milk and become divine himself. When Hera woke up and found she was nursing a strange infant, she thrust him away causing milk to spurt from her out into the heavens, and that’s the origin of the Milky Way.

If you study the word “galactic,” you’ll see “lac” in the middle of it. This is because the Latin root lac, milk, seems to have come from an earlier word which has been reconstructed as either *g(a)lag- or *g(a)lakt-. In English, we get from this root lactate, latte, and even lettuce.

Our word milk comes from the Indo-European root melg- which is a verb and means “to rub off,” and also “to milk.” It’s related to the word emulsion which comes from the Latin emulgeo, “to milk out.” In English, the verb milk seems to predate the noun milk. It’s been used from the beginning to refer specifically to human or animal milk. Incidentally, mammals and the mammary glands are named after mamma, which means mother in English, but it means breast in Latin. They both come from the same root, ma-, which means mother, and gives us the Greek Maia (good mother) and also maieutic, which means to act as a midwife, and is the word 17th century philosophers used to describe the Socratic method.

The Old English language also used the word milk to refer to the milky juice you can get from a plant. The 2nd century text Herbarium by Pseudo-Apuleius describes getting milk from wyrte/wort, meaning from a plant (think of the “wort” in St. John’s Wort). “Almond milk” has been used since at least the 1300s, so keep this in mind next time one of your friends scoffs at people talking about almond milk, and insists that milk can only properly refer to that which is produced by the mammary glands. :-D

 

* In linguistics, the asterisk before a word means that the word can’t be found written anywhere, so scholars have made an educated guess about what it probably was.

Winter rest

 

I was going to write a beautiful post about the gloriously busy holiday season and the need for rest before I go back to teaching again in February, but I’m too tired. The days are cold, and dark, and dreary. Nature is saying, “Rest,” so I will rest.

Work, glory, leisure, worship, and love

Buy at Alibris.com

At the beginning of his Leisure: The Basis of Culture, Joseph Pieper quotes this passage from Plato ...

But the Gods, taking pity on mankind, born to work, laid down the succession of recurring Feasts to restore them from their fatigue, and gave them the Muses, and Apollo their leader, and Dionysus, as companions in their Feasts, so that nourishing themselves in festive companionship with the Gods, they should again stand upright and erect.
[Laws, 2.653c-d]

... which reminds me of this poem:

“Unstooping”
~Walter de la Mare (1873-1956)

Low on his fours the Lion
Treads with the surly Bear;
But Men straight upward from the dust
Walk with their heads in air;
The free sweet winds of heaven,
The sunlight from on high
Beat on their clear bright cheeks and brows
As they go striding by;
The doors of all their houses
They arch so they may go,
Uplifted o’er the four-foot beasts,
Unstooping, to and fro.

And of course the very word “leisure” reminds me of this poem:

“Leisure”
~William Henry Davies (1871-1940)

What is this life if, full of care,
We have no time to stand and stare.

No time to stand beneath the boughs
And stare as long as sheep or cows.

No time to see, when woods we pass,
Where squirrels hide their nuts in grass.

No time to see, in broad daylight,
Streams full of stars, like skies at night.

No time to turn at Beauty’s glance,
And watch her feet, how they can dance.

No time to wait till her mouth can
Enrich that smile her eyes began.

A poor life this is if, full of care,
We have no time to stand and stare.

We know from the creation account in Genesis and from the Ten Commandments that we’re supposed to rest and worship, and we learn in the Psalms that this leisure properly grows from a delight in God and his love for us.

Domine, Dominus noster
~Psalm 8

O LORD, our Lord, how excellent is thy name in all the earth!
who hast set thy glory above the heavens.

Out of the mouth of babes and sucklings
hast thou ordained strength because of thine enemies,
that thou mightest still the enemy and the avenger.

When I consider thy heavens, the work of thy fingers,
the moon and the stars, which thou hast ordained;

What is man, that thou art mindful of him?
and the son of man, that thou visitest him?

For thou hast made him a little lower than the angels,
and hast crowned him with glory and honour.

Thou madest him to have dominion over the works of thy hands;
thou hast put all things under his feet:

All sheep and oxen,
yea, and the beasts of the field;

The fowl of the air, and the fish of the sea,
and whatsoever passeth through the paths of the seas.

O LORD our Lord, how excellent is thy name in all the earth!

Worship, leisure, standing upright crowned with glory, having dominion... It all comes together in this Psalm.

This is such an essential part of what it means to be a human that even the pagan philosophers and poets wrote about it to the best of their understanding. They knew that all of creation, beauty, glory, and rest were the gifts of Divine Love.

“Address to Venus”
~Lucretius (ca. 99 B.C. - ca. 55 B.C.)

Delight of Human kind, and Gods above;
Parent of Rome; Propitious Queen of Love;
Whose vital pow’r, Air, Earth, and Sea supplies;
And breeds what e’r is born beneath the rowling Skies:
For every kind, by thy prolifique might,
Springs, and beholds the Regions of the light:
Thee, Goddess thee, the clouds and tempests fear,
And at thy pleasing presence disappear:
For thee the Land in fragrant Flow’rs is drest,
For thee the Ocean smiles, and smooths her wavy breast;
And Heav’n it self with more serene, and purer light is blest.
For when the rising Spring adorns the Mead,
And a new Scene of Nature stands display’d,
When teeming Budds, and chearful greens appear,
And Western gales unlock the lazy year,
The joyous Birds thy welcome first express,
Whose native Songs thy genial fire confess:
Then savage Beasts bound o’re their slighted food,
Strook with thy darts, and tempt the raging floud:
All Nature is thy Gift; Earth, Air, and Sea:
Of all that breathes, the various progeny,
Stung with delight, is goaded on by thee.
O’er barren Mountains, o’er the flow’ry Plain,
The leavy Forest, and the liquid Main
Extends thy uncontroul’d and boundless reign.
Through all the living Regions dost thou move,
And scattr’st, where thou goest, the kindly seeds of Love:
Since then the race of every living thing,
Obeys thy pow’r; since nothing new can spring
Without thy warmth, without thy influence bear,
Or beautiful, or lovesome can appear,
Be thou my ayd: My tuneful Song inspire,
And kindle with thy own productive fire;
While all thy Province Nature, I survey,
And sing to Memmius an immortal lay
Of Heav’n, and Earth, and every where thy wond’rous pow’r display.
To Memmius, under thy sweet influence born,
Whom thou with all thy gifts and graces dost adorn.
The rather, then assist my Muse and me,
Infusing Verses worthy him and thee.
Mean time on Land and Sea let barb’rous discord cease,
And lull the listening world in universal peace.
To thee, Mankind their soft repose must owe,
For thou alone that blessing canst bestow;
Because the brutal business of the War
Is manag’d by thy dreadful Servant’s care:
Who oft retires from fighting fields, to prove
The pleasing pains of thy eternal Love:
And panting on thy breast, supinely lies,
While with thy heavenly form he feeds his famish’d eyes:
Sucks in with open lips, thy balmy breath,
By turns restor’d to life, and plung’d in pleasing death.
There while thy curling limbs about him move,
Involv’d and fetter’d in the links of Love,
When wishing all, he nothing can deny,
Thy charms in that auspicious moment try;
With winning eloquence our peace implore,
And quiet to the weary World restore.

[This content was originally part of my Poetry Month 2011 series.]

Current reading and a plea for help

Active:

C.S. Lewis
Prince Caspian*
Preface to Paradise Lost
The Discarded Image*

Plato, Republic

Ovid, Metamorphoses

Ramon Lull, Book of Knighthood and Chivalry

Eugene Vodolazkin, Laurus

Mary Jo Tate, Flourish



Sidelined (it’s been a few weeks, but they’re still sitting out):

Dorothy Sayers, all the Lord Peter novels and short stories in chronological order*

Ariosto, Orlando Furioso



Reference (cluttering up my desk area because I keep dipping into them):

Lewis, Studies in Medieval and Renaissance Literature

Northrup Frye, The Secular Scripture

E.M.W. Tillyard, The Elizabethan World Picture

Stratford Caldecott, Beauty for Truth’s Sake


To Be Read (too numerous to list)

[*denotes re-read]

Clearly I need to come up with some sort of a reading plan. I have a loose plan, in that I do my Bible reading when I first wake up, and read Ovid during breakfast. After that though, it’s as my whimsey takes me.

Also I need a decent system for keeping track of teaching/writing notes and ideas. Normally, everything is just in my head, or I’ll write down sketchy ideas on random pieces of paper, but then I don’t have a system and things get lost. If you have any advice, please share!

Classical education and philosophy turn up in such interesting places

Some day I hope to have time to read a bit of Confucius.  Having read a bit of Plato and a lot of Lewis, I'm discovering that Confucius ought to be part of the Classical tradition.  I wonder if Asian Christians read him like we read Plato and Aristotle.

Last night the two oldest girls and I watched the Korean film "The Fatal Encounter," starring Hyun Bin as King Jeongjo.  If you have any familiarity with K-drama at all you'll know who that is, but for those of you who don't I'll give a bit of background info.

King Jeongjo of Joseon

Jeongjo became king in 1776 on the death of his grandfather, King Yeongjo, who had been forced by one political faction to put to death his own son, Crown Prince Sado (Jeongjo's father), for siding with the other political faction.  So it was a mess he inherited, and it is said that on the day he took the throne, he sat there and said to everyone in the court, "I am the son of the late Crown Prince Sado," which was practically a declaration of war against the faction that had had his father put to death.

The movie covers the 20-hour period leading up to an assassination attempt the year after Jeongjo became king.  I'm not sure whether the event is actually historical, but it's a wonderful piece of storytelling, beautifully filmed, well-acted, and I'm going to show this one on our next family movie night.

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Well, I said all that to say this -- a few minutes into the film, Jeongjo quotes a passage from Confucius' book The Doctrine of the Mean:

What Heaven confers is called Nature.

Accordance with this Nature is called the Way.

Cultivating the Way is called Education.

Isn't that perfect?


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The literal title is "The King's Wrath,"
which I like much better than the one under which
it was marketed to English speaking audiences.

This isn't a movie review, but I thought I'd mention to anyone who's interested in watching it that it would probably be a PG-13 on account of lots of blood during the assassination attempt, plus some blood and implied cruelty of a horrific nature when one person is accused of treason and is "questioned," and a few scenes involving cruel treatment of slaves.

Math update: Struggling through a dry spell

I was working on  the third installment of my Squaring the Circle series, in which I’d intended to relate what I’d learned about the nature of pi – I said I was pretty excited about it, and I really was!  I only wish I’d written that section this spring when I was still glowing from that Eureka! moment.  It’s a lot harder to write about all these weeks later, so instead* I’m reading the whole chapter on the Ancient Greeks in A History of Mathematics – before I just read the sections dealing specifically with measuring a circle.

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But here’s something fascinating about this chapter – I met nearly all of these people when I read John Mark Reynolds’ When Athens Met Jerusalem this spring and summer.  Did you know that Thales, the first philosopher, was also the first mathematician?  I didn’t, which is why I was so surprised when I wrote this post on Math and Philosophy.

All the early mathematicians were philosophers.  In fact, it is said that Pythagoras is the one who coined the words philosophy, “love of wisdom,” and mathematics, “that which is learned.” 

“Perhaps the most striking characteristic of the Pythagorean order was the confidence it maintained in the pursuit of philosophical and mathematical studies as a moral basis for the conduct of life.” (Merzbach and Boyer, p. 44)

More interesting passages from this chapter:

“ . . . mathematics was more closely related to a love of wisdom than to the exigencies of practical life.”

“Greek mathematics, in its earlier stages, frequently came closer to the ‘modern’ mathematics of today than to the ordinary arithmetic of a generation ago.”

“[T]he Pythagoreans not only established arithmetic as a branch of philosophy; they seem to have made it the basis of a unification of all aspects of the world around them.”

“The point of view of the Pythagoreans seems to have been so overwhelmingly philosophical and abstract that technical details in computation were related to a separate discipline, called logistic. This dealt with the number of things, rather than with the essence and properties of number as such, matter of concern in arithmetic. That is, the ancient Greeks made a clear distinction between mere calculation, on the one hand, and what today is known as the theory of numbers, on the other. . . .  [T]he early Ionian and Pythagorean mathematicians [have] the primary role in establishing mathematics as a rational and liberal discipline.”
That’s from the first half of the chapter.  There are sections coming up headed “Mathematics and the Liberal Arts,” and “The Academy,” so I’m looking forward to what else the authors have to say about that.

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I had a brief moment of elation today when one of my friends shared a math quiz on Facebook. 

There was one moment of panic involving multiplying a four-digit number by a two-digit one – the rest I was able to do in my head, but I had to pull out pen and paper for this one.  Oh, and there was one other that threw me, but my violin daughter was watching over my shoulder at that moment and said, “Think of it this way . . .” and then I was able to do it in my head.

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Here’s an interesting article you should read – Research on the Teaching of Math, which I think is a reprint of the appendix of the same name in Harvey and Laurie Bluedorn’s book Teaching the Trivium. They cover what they’ve learned of the history of teaching math, research on brain development, and suggestions for what to do with your own children.

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* Here’s a bonus article:  Why Writers Are the Worst Procrastinators.  I printed it out two days ago, but I’ve been avoiding reading it because I’m afraid it’ll be depressing.
:-p

Beauty for Truth’s Sake, Chapter 3: The Lost Wisdom of the World

“this world of patterns and relationships”

The Quadrivium

The assumption of this system of education was that by learning to understand the harmonies of the cosmos, our minds would be raised toward God, in whom we could find the unity from which all these harmonies derive: Dante’s “love that moves the sun and the other stars.” Thus the quadrivium would prepare the ground for the study of the highest contemplative sciences: philosophy and theology.

The idea that the cosmos is built on mathematical harmonies, and that numbers themselves can be a path to God, flowed from Pythagoras and Plato down to the Middle Ages, where it influenced the cathedral builders and later the artists of the Italian Renaissance. It was also one of the essential factors in the birth of science . . . .

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“Numbers are a map of the beautiful order of the universe, the plan by which the divine Architect transformed undifferentiated Chaos into orderly Cosmos. Cultures didn’t necessarily learn this from each other but only had to look at numbers and their relationships to see how they revealed harmonious models which are the same everywhere and at all times [quoting Michael S. Schneider in Constructing the Cosmological Circle].”

Yet our present education trends to eliminate the contemplative or qualitative dimension of mathematics altogether, reducing everything to sheer quantity. Mathematics is regarded as a form of logical notation, a mental tool with no relation to truth except the fact that it assists us in manipulating the world. This elimination of the symbolic dimension of mathematics is largely responsible for the divorce of science from religion, and art from science.

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Rose Window, Chartres
image via Wikipedia (cropped)

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The Pythagorean-Platonic tradition at the core of the Liberal Arts developed the following symbolic associations with the natural numbers:

One

The Unity of being, transcending all that exists. It is often represented by a circle, or else by a point. One is the number that when “squared,” i.e., multiplied by itself, produces itself. Symbolically, One is not the first in a series of numbers, but the number-beyond-number that includes all others, equivalent in that sense to the modern conception of infinity . . . .

Two

If one is the source and archetype of Unity, two is the beginning of Diversity. It represents polarity and division, and also feminine receptivity and fruitfulness. In a Christian context it often signifies the separation of matter and spirit. Duality can also symbolize the beginning of the process of creation, which in the book of Genesis is described as taking place through a series of separations or polarizations (heavens and earth, light and dark, etc.). The division of Adam’s unity into duality gave us male and female . . . .

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In nature there is no zero. Some writers on number symbolism therefore regard it as an interloper, whose introduction as a placeholder led to the loss of awareness of the symbolic properties of number, and especially of Unity (displacing it from its position at the beginning of the number series), creating a framework for the development of atheism . . . .

Personally, I am not so sure. Zero could also be taken as the ground of being, and a symbol for the return to one. Perhaps the mistake lay not in introducing zero, but failing to read it symbolically.

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I’m two days late, but I’m linking to Dawn’s Wednesdays with Words anyway.  Be sure to read her quotes, and check out all the links.

Wednesays with Words: Beauty for Truth's Sake

[I wrote this up and scheduled it for the next Wednesday before Cindy announced that she's leaving the blogging world.  I'm keeping the title because, well . .  y'all know why.]

From Chapter 1:  The Tradition of the Four Ways

"The process of education requires us to become open, receptive, curious, and humble in the face of what we do not know."

"[P]hilosophy is a preparation for dying; or rather, for dying well."

"An integrated curriculum must teach subjects, and it must teach the right subjects, but it should do so by incorporating each subject, even mathematics and the hard sciences, within the history of ideas, which is the history of our culture."

"Beauty is the radiance of the true and the good, and it is what attracts us to both."

"[Beauty] is, we should add, difference or otherness held in a unity that does not destroy uniqueness. As Hart explains, if the Trinity were instead a Duality, God would not be love but narcissism, and beauty would lose its radiance. It is the Holy Spirit, the fact that true love is always turned away from itself, pouring itself out for others, that makes it open and radiant, and creates room in the Trinity for the creation itself, as well as for all the suffering and all the sacrifice that creation involves. The Trinity is the home of the Logos and the shape of love. These are the high secrets of our Western tradition, and together they offer the key to its renewal."


Update:

Dawn at Ladydusk is continuing Cindy's WWW link-up.  Yay, Dawn!  Check her blog for more quotes.

CiRCE Conference 2014 -- some tidbits

Pre-conference session with John Mark Reynolds

Worry less about the condition of your child's soul and more about yours -- he's going to imitate you, you know.

Your calling is towards your own relationship with Christ.

“The important thing is to prepare your kids to die well.  That’s the only thing they are all going to do.”

Find a good teacher and you'll find a place where they "waste time."

The only way to teach math is the Platonic way -- you poke them.

The 1940s and 50s changed the way math was taught -- didn't need kids to understand it, just be able DO it efficiently so that they could become scientists and engineers and help us beat the Russians.

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Teaching is leading the student from the known to the unknown according to the student's nature. (Andrew Kern)

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I could listen to John Hodges quote Shakespeare and Scripture and talk about music all day long.  Y'all have to listen to the Brahms' requiem Ein Deutsche Requiem (op 45), especially if you're familiar with traditional requiems.

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Piety, Imitation, and Art:  A Circle of Paideia -- Ravi Jain

Gymnastic (training the body) and Music (tuning the heart) are the foundation of education.  By "Music" he means the nine Muses, so practically everything I've been accustomed to think of as an education (literature, history, and the rest) are actually the foundation, not the education itself!

In teaching math, you should recapitulate with your students the narrative and discoveries of mathematics, so that they're as close as possible to the wonder, awe, and mystery of Math.

Math is concerned with proofs -- this is the Logos, the unifying principle of math.

Let your students come up with their own proofs.

Textbooks help mediate the conversation [i.e. don't abandon them, but don't follow them slavishly].

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The Three Paths to Wisdom -- Paula Flint

"By three methods we may learn wisdom; first by reflection which is the noblest, second by imitation which is the easiest, and third by experience which is the most bitter." (Confucius)

Nature is the first classroom.

Reflection comes after experience and imitation -- it's a harder level and takes time.

In imitation we benefit from past accomplishments.

As teachers and parents we need to think in terms of building good habits.  Good habits make life easier, freer.  It's not about punishment -- it's about helping children live a good life.

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St. Augustine and the Dialectic of Desire -- Wes Callihan

Cicero had a daughter named Hortensia who was a famous orator -- she spoke before the Senate!  Cicero wrote a book named after her, Hortensio.  It was an encouragement to philosophy, and sadly, is lost.

Practically everything you know and believe, you believe by faith. If you had to prove everything you'd never get anywhere. We are finite beings, so it's our nature to live by faith.

"Beatrice" is not the thing the soul is made for -- she is the reminder of the thing the soul is made for.  [This reminds me of CS Lewis's Surprised by Joy.]

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With Hobbits on the Road to Wisdom -- Andrew Seely

Homer wrote great works that are for his time -- we have to educate ourselves for them to have a real affect on us.  But Tolkien is for our time -- we don't have to do any groundwork before coming to his work.

All the stories told during The Lord of the Rings were Frodo's education, to that he could make the right decision at Parth Galen.  Frodo has a new heart which is trustworthy.  When he pities Gollum, he is able to follow his heart instead of his old ideas of justice.

The temptation for anyone who has made their home life beautiful is to cut themselves off from the rest of the world.

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[More later.  I'm tired now.]

Have you heard of Coursera?

Another new term you might have seen is MOOC:  Massively Open Online Course, which is what Coursera offers.

From the website:  “Coursera is an education platform that partners with top universities and organizations worldwide, to offer courses online for anyone to take, for free.”

I’ve taken three classes over the last couple of years, Introduction to Mathematical Thinking, The Ancient Greeks, and Reason and Persuasion: Thinking Through Three Dialogues by Plato.

The format for each is similar.  There are lecture videos to watch and homework assignments and tests.  There are discussions forums available.  Most classes have some standard you have to meet in order to receive certification. You get out of it what you put into it, and that will vary according to your goals, how much time and effort is demanded by the class, and how much time and effort you can afford to devote to it.

Once you’ve taken a course you’ll always have access to the archives so you don’t have to finish by the deadline, and you can listen to the lectures over again if you like.

The only class for which I met the standard for certification was The Ancient Greeks.  My goal for this class was to get a solid overview of the period – the flow of events and ideas.  Of course, we’ve touched on Ancient Greece over the years in our home school, reading Plutarch’s Lives together and Rosemary Sutcliff’s adaptations of The Illiad and the Odyssey.  I cut my teeth on Edith Hamilton’s Mythology, and I’ve read some of her other books, like The Echo of Greece, but I lacked a cohesive understanding of this period, and that’s what this course offered.  I did not have time to read everything on the syllabus, but I feel like I’ve gotten a decent start on the topic, a foundation to build on in the future, which is what I wanted.

My goals for the Reason and Persuasion course were to read The Meno at a deeper level than my previous effort, to learn more about Socrates and the Socratic Method, and to be introduced to philosophy.  Since the class covered three of Plato’s dialogues and spent two weeks covering recent developments in philosphy, a lot of what was taught was outside the scope of my interest.  I watched all the introductory material, read The Meno and The Euthyphro and listened to their lectures, and listened to the lectures on The Republic. 

I did not write the paper, or listen to the later lectures, and I only took two tests, but I feel the course was a success because I met my own goals.  Near the end of the course, I even decided to buy the textbook.  I really like the translation and I want to reread the dialogues regularly in the future, especially The Meno, since it’s the one that sets the standard for interacting with a student.  Also, I really appreciated the instructor’s style and insight and wanted to finish reading his commentary.  (The textbook was available online for free in PDF format, but I find that a difficult format to get much use out of.  I want a book I can curl up in bed with or carry with me when running errands.  I want to be able to underline and write notes, and flip quickly through pages to find things.  My brain simply has not adapted to e-readers – not for a purpose like this, anyway.)

The one class where I did not meet my own goal was Introduction to Mathematical Thinking.  That class was brutal.  The pace was too rapid for me.  It took me ages to do my homework and the only way I could understand what was going on was by asking the members of my study group to walk me through things.  They were very kind, and one in particular, Denise Gaskins of Let’s Play Math, was very patient and helpful with her clear explanations, but I was falling farther and farther behind everyone else, and I finally gave up. 

I’m not completely discouraged though – I believe that the time I spent in that class won’t go to waste, even it just sits in my mind and turns to compost.  When I took it in the fall of 2012, it was seven weeks long but since then they’ve reformatted it into a ten-week class and have offered it twice more.  I’m interested in trying it again in the future, but that will depend on... well, life.

Math and Philosophy

Don't you love it when seemingly unrelated things come together?

A few years ago, I heard (or read, I don't remember which) Andrew Kern say that teachers ought to read The Meno, one of Plato's dialogues, regularly in order to become a better teacher.  I printed it out and tried reading it, but seriously.  It was heavy going and I didn't get very far.

When I noticed that Coursera was offering a class called Reason and Persuasion: Thinking Through Three Dialogues by Plato I immediately checked to see which three dialogues, and sure enough, The Meno was one of them, so I signed up.  The class started last month and it's been a huge help.  We spent two weeks on Euthyphro, and now I'm near the end of our second week of Meno.

Before starting this class, I had no idea that math would come up, but it plays an important role in The Meno, which you'll already know if you've ever read it.  The dialogue discusses the questions What is virtue? and Is virtue the sort of thing that can be taught?  But right smack in the middle of this dialogue is a geometry lesson.

While discussing this, the Coursera instructor mentions an essay by Bertrand Russell, "The Study of Mathematics," and quotes from it:

One of the chief ends served by mathematics, when rightly taught, is to awaken the learner's belief in reason, his confidence in the truth of what has been demonstrated, and in the value of demonstration. This purpose is not served by existing instruction; but it is easy to see ways in which it might be served. At present, in what concerns arithmetic, the boy or girl is given a set of rules, which present themselves as neither true nor false, but as merely the will of the teacher, the way in which, for some unfathomable reason, the teacher prefers to have the game played.

I don't know about you, but once I got to fourth grade math where they were actually teaching me new stuff (everything before that was pretty much just me getting to show off what I'd already figured out on my own in the normal course of childhood) this is exactly what math felt like to me -- arbitrary rules.

Russell goes to to critique my tenth-grade geometry class.  Do you remember geometry?  In that class they drive you insane by making your prove things that don't need to be proven -- you can tell just by looking at them.  Russell says this is a huge mistake:

[T]he learner should be allowed at first to assume the truth of everything obvious, and should be instructed in the demonstrations of theorems which are at once startling and easily verifiable by actual drawing . . . . In this way belief is generated; it is seen that reasoning may lead to startling conclusions, which nevertheless the facts will verify; and thus the instinctive distrust of whatever is abstract or rational is gradually overcome. *

This is what happens during the geometry lesson in The Meno.  Socrates calls over a slave boy, who has had no instruction in geometry, and draws a square that's two feet long on each side, and asks the boy what the square's area is.  The boy says that it is four square feet.

Next Socrates asks the boy what he would do in order to make a square whose area is twice this one.  The boy says that he would double the length of each side of the square.  They draw that out, but of course that turns out to make a square that's sixteen square feet, so the boy tries again -- three feet on a side, which still makes too big a square.

Of course, you probably know that the square root of 8 is not a rational number -- it's not something that can be expressed as a fraction.  It's less than 3, but it's more than 2 3/4.  In Socrates' time they didn't have a way of expressing that number since they weren't using the decimal system, so how in the world can this poor slave boy come up with the right answer?

Turns out it can be done and Socrates helps the boy figure out how to do it.  It's pretty clever, and I'll leave it to your imagination.**

The next passage quoted in my class was this one:

What is best in mathematics deserves not merely to be learnt as a task, but to be assimilated as a part of daily thought, and brought again and again before the mind with ever-renewed encouragement. Real life is, to most men, a long second-best, a perpetual compromise between the ideal and the possible; but the world of pure reason knows no compromise, no practical limitations, no barrier to the creative activity embodying in splendid edifices the passionate aspiration after the perfect from which all great work springs. Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos, where pure thought can dwell as in its natural home, and where one, at least, of our nobler impulses can escape from the dreary exile of the actual world.

A lot of my reading the last couple of years about what math is has been hinting at this kind of thing.  Math is ennobling.  This was so exciting that I decided to find and read the entire essay.  Even if I hadn't had this introduction Russell would have won me over in his first paragraph:

In regard to every form of human activity it is necessary that the question should be asked from time to time, What is its purpose and ideal? In what way does it contribute to the beauty of human existence? As respects those pursuits which contribute only remotely, by providing the mechanism of life, it is well to be reminded that not the mere fact of living is to be desired, but the art of living in the contemplation of great things. [Emphasis added]

If you've read the John Gould Fletcher quote in my sidebar you'll know why that phrase resonates with me.

Our imaginations are meant to live in a kind of temple, and originally it was this idea that gave shape to education, or what we would call a Classical Education.  But in recent generations why we're teaching what we're teaching has been forgotten, or, as Russell puts it:

Dry pedants possess themselves of the privilege of instilling this knowledge: they forget that it is to serve but as a key to open the doors of the temple; though they spend their lives on the steps leading up to those sacred doors, they turn their backs upon the temple so resolutely that its very existence is forgotten, and the eager youth, who would press forward to be initiated to its domes and arches, is bidden to turn back and count the steps.

I haven't finished The Meno yet, so I've only got an inkling of what geometry has to do with virtue, but dear Lord, please save me from being a dry pedant!

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*Here, Russell is discussing teaching geometry with Euclid.  Euclid starts off with a long list of definitions and postulates, which is a fine way to organize the information, but a lousy way to teach it -- not that I know what the best way is.  I'm just reporting what I've learned so far.  :-p

**Hint:  It has to do with triangles.