Mathematical Understanding as Seen Within a Framework of Beauty (part 1)

In a previous series of posts (part 1 being here:

I began to describe how mathematics could be understood as an endeavor of human discovery and invention by showing how form and the pursuit of beauty underlies successes in such efforts. In particular, I focused on the opening of Book I of Euclid’s Elements as an example of how a formalism of a mathematical subject-matter, in this case geometry, first arises by abstraction of form from things we may experience by our senses in the world around us. In the end, I sought to frame the entire work of the Elements within the three characteristic modes of beauty: proportio, claritas, and integritas. Beginning with this post, I aim to explore more generally what mathematics is as a subject-matter of human understanding through the lens of beauty. To that end, I have three goals (1) elaborate on what serves as the underlying subject-matter of mathematics, (2) show how the three modes of beauty ground human understanding in mathematics, and (3) describe how these perspectives of human understanding in mathematics are pertinent to, and in fact undergird, developments in mathematics up to today and beyond.

Imaginative Theology

[I]t must not be supposed that I am in any sense putting forward the imagination as the organ of truth. We are not talking of truth, but of meaning: meaning which is the antecedent condition both of truth and falsehood, whose antithesis is not error but nonsense. I am a rationalist. For me, reason is the natural organ of truth; but imagination is the organ of meaning. Imagination, producing new metaphors or revivifying old, is not the cause of truth, but its condition.
                                C. S. Lewis, “Bluspels and Flalansferes,” Rehabilitations.

John Calvin distrusted speculative theology. He seems to have thought that it was disrespectful. There are many things in theology that are mysteries; they must simply be pondered. The longer you sit with a mystery the more deeply you may move into it, but you will never be able to walk all the way around it, define it, measure it, or solve it. When confronted with a mystery, Calvin always counseled silence. As an example of this, you will look in vain for any description of heaven in his writings, even though he mentions the reality of heaven fairly often. He was not a great fan of “producing new metaphors,” as Lewis advocates doing in the quotation above. He was distrustful of imagination.

There is something admirable about this approach. It is certainly true that when confronted with a mystery we need to speak with humility, and often we end up saying more about what is not true than about what is true. We call this approach the negative way, and it can be very powerful, especially when speaking about God Himself. There are many times when we need to say “God is not this,” or “God is unlike us,” or “God does not share this quality with us,” and then leave it at that.

But that is not quite the same thing as silence. In our present pluralist context, silence communicates something quite other than what it communicated in Calvin’s sixteenth-century European context. John Calvin lived in world where almost everyone believed in God, and there were very few people who wanted to deny the existence of the supernatural, so claiming that something was too mysterious to be understood made sense to most people, even those who didn’t agree with a particular Christian doctrine. In our time and place, Calvin’s sort of silence has a different meaning, suggesting that what we believe is irrational and that we are believing it blindly. In a disenchanted world, people expect to be able to explain and ultimately to control everything around us. In our context, if your only response to a challenge to your faith is “It’s a mystery,” that looks like intellectual surrender.

Beholding the Beauty of the Lord

One thing I asked of the Lord,
that will I seek after:
to live in the house of the Lord
all the days of my life,
to behold the beauty of the Lord
and to inquire in His temple.

“Come,” my heart says, “seek His face!”
Your face, Lord, do I seek.
           Psalm 27:4, 8

The Biblical book of Exodus tells the story of Moses, a man who was born into slavery, saved from death by his parents’ willingness to hide him in a basket floating in a river, rescued by a princess, and raised as a prince. It’s a stirring story that has captured the imaginations of people throughout history. Eventually, Moses claimed his identity as an Israelite, giving up the privilege of being treated as Pharaoh’s grandson and fleeing Egypt in order to live as a shepherd in the desert. There in the desert Moses had an encounter with God.

This is not the first account in the Bible of someone having a direct encounter with God. God spoke to Adam and Eve, to Cain, to Noah, to Abraham and Isaac, Jacob and Joseph. But with Moses, God begins not only to speak but to show Himself. And with Moses, God shares His name.

Beauty as Divine Yearning

My colleague Harry Plantinga, director of the Christian Classics Ethereal Library, recently posted a blog meditating on the Syrian mystic Dionysius’s ideas of yearning for God. Harry connects the experience of yearning with the attractiveness of God’s beauty. He says: 

Paul’s description of the spiritual life is not the slightest bit dispassionate in the modern sense. He describes it as a striving to win the race, a pressing on, a straining forward. He describes a desire for attaining, for becoming, for uniting. “I want to know Christ, and the power of his resurrection, and the fellowship of sharing in his suffering, becoming like him in his death, and so, somehow, to attain to the resurrection of the dead.”
We yearn for the beauty of creation. Beauty is attractive. We long to experience it, to become a part of it. And as we become more spiritual, we long for spiritual beauty. We yearn to know God, to experience him, to become like him.

Read the whole post here:

Reading for Beauty

Sarah Clarkson has a recent blog post with a title that makes a big claim: “We Read for Beauty.” Naturally, I decided I needed to read this one. Here’s what she understands by that claim:

What do I mean by beauty? I mean a bone-deep knowledge of the goodness of the world; the heaven-crammed splendor of creation, the gift of a mystery that is every other human soul we meet, the possibility of redemption, of friendship and laughter, the reality of grace invading every corner of existence. By beauty, I mean an encounter with God ‘at play’ (in Gerard Manley Hopkin’s terms) in the world he created for joy and is redeeming in love.

 She considers examples of reading that have served to “outfit” her own “inner landscape of imagination.” She is especially concerned with what it means to read for beauty in light of what she hopes her children will read and how they will be formed by their reading. 

Read the full post here.

Beauty Ever Ancient, Ever New

I’ve been reading Thomas Williams’ new translation of Augustine’s Confessions (Hackett 2019), and yesterday I came to my favorite passage. It’s a very famous section, which Williams lays out as a poem.

Late have I loved you, beauty so ancient and so new!
Late have I loved you!
And behold, you were within, but I was outside and looked for you there, and in my ugliness I seized upon these beautiful things that you have made.
You were with me, but I was not with you. 
Those things helped me far away from you — things that would not even exist if they were not in you.
You called, you shouted, and you broke through my deafness;
you flashed, you shone, and you dispersed my blindness;
you breathed perfume, and I’d ruin my breath and pant for you;
I tasted, and I hunger and thirst;
you touched me, and I was set on fire for your peace. (Bk10, 27.38)

The Beauty of Education

One of my colleagues at Calvin University, Prof. David Smith, recently sent me a link to a wonderful article he wrote several years ago on how school can be beautiful and education can be beautiful. The article focuses on the thought of Comenius, for whom beauty was an important value. David begins the article with the closing words of Comenius’s Didactica Magna, on which he then comments:

Do Thou, therefore, O Lord our God, give each one of us a joyful heart to serve Thy glory as best he may […] Let Thy work appear unto Thy servants, and Thy glory upon their children. And let the beauty of the Lord our God be upon us; and establish Thou the works of our hands upon us (Psalm xc.16). In Thee, O Lord, have I trusted, let me never be confounded. Amen. 

The discourse is devotional, the sentiments, biblical; the context is educational. The prayer comes from the final page of an enormously influential treatise on school reform penned in the mid-seventeenth century by the Moravian educator, pastor, philosopher, literary pioneer, and general jack-of-all-trades, John Amos Comenius. After a weighty collection of chapters ranging in topic from the profundities of moral and spiritual education to the practicalities of organizing grade levels and textbooks, Comenius closes with a prayer for beauty. For God’s beauty. That it be upon us. “Us” being in this case primarily those who teach the young.

My Favorite Fairy Tale

 Some 30-plus years ago, I preached a sermon on Colossians 1:13-14: “He [the Father] has rescued us from the power of darkness and transferred us into the Kingdom of his beloved Son, in whom we have redemption, the forgiveness of sins.”

In thinking about what it meant to be rescued and transferred from one Kingdom into another, I decided to use as an illustration my favorite fairy tale, “The 12 Dancing Princesses.” This is not the most well-known of fairy tales. It has never been made into a Disney movie. But it has been written and re-written and updated in many ways. The story is that there was once a King who had twelve daughters of whom he was very protective. Every night he would lock them into the long bedroom in which they all slept; every morning he would come and release them. But what he did not know was that every night another door would open in the floor of the princesses’ bedroom revealing a stairway to a magic kingdom where they would dance all night.

What Makes Music Beautiful?

If you look under “music” in our resource section, what you will find so far will be music about beauty, songs that you might want to add to your repertoire if you are leading worship and would like the people at your church to think about beauty a little more often. Hymns and spiritual songs have power to shape our imaginations, and so such resources are important. But here I’d like to think about a more vexing question: what makes music beautiful? And – even more importantly – is it possible to communicate our sense of beauty in music with those who don’t immediately share it? 

One of the premises that drives this website is that beauty is objective. It’s really out there in the world, built into the nature of reality. In fact, the world is overflowing with beauty. As Hopkins says, the world is “charged” with the current of God’s glorious grandeur, which is to say, with His beauty.  Problem is there’s so much beauty that no one person can perceive all of it. Indeed, no one culture can express or record or recognize all of it. So even though beauty is objective, it’s also perspectival. 

Theology as an Exercise in Beauty

For many years, I’ve been trying to teach my introductory theology classes through the lens of beauty. Over the last year or two, I started to write down what I say in class, and then started to write down things I’ve never said in class but probably should.  Writing is a wonderful way to figure out what we think, and I’m trying to figure out what I think about beauty – after thirty years of pondering it – both by writing on this blog and by working on a textbook. The two projects will no doubt overlap from time to time. Like today. 

My hope for my textbook is not that it will use art to illustrate theological ideas. I think that is what many people expect when I tell them what I’m writing, and then they’re confused if I show them a section. For one thing, although art may sometimes be a place where we encounter beauty, not all art is aiming in that direction. And much of our daily experience of beauty isn’t about art at all. It’s about the moment when you catch a slight breeze through the window and feel it on your face, and for just a second you become aware of the wonderful gift of moving air. Or the moment when your hand brushes against your sweater, and you touch it again just for the pleasure of its softness. Or the moment when you’re reading the Bible and are overwhelmed by some phrase that you may have read many times, but that – this time – lights you up from within.