Blog

Mathematical Understanding as Seen Within a Framework of Beauty (part 4)

Following upon my previous posts, where I began to explore the possibility of framing mathematical understanding within the Thomistic modes of beauty proportio and claritas, I now consider the third mode of beauty, that of integritas. This mode brings beauty in mathematical understanding to its fulfillment and serves to complete and unite the other two modes. For it is in the crafting, rendering, and communicating mathematical truths that integritas is to be sought and developed. In being a measure of beauty, integritas measures the extent to which a work, rendering, or communication expresses a subject-matter of mathematics in such a way as to uncover ever deeper proportios and orders, enable ever more profound claritas to be revealed, and so allow for ever greater works to be developed which, in turn, are themselves measured for their individual integritas. Furthermore, it is in the acts of verification through defining, axiomatizing, and producing proofs that we move from claritas to integritas. For, it is by the movement toward integritas that the proportios form order, as seen with claritas, and become formalized in order to be communicable to others. In the end, the aim is the advancing of mathematical knowledge to those within the community of mathematicians.

(Here is the link to part 3:

https://thinkingbeautifully.org/mathematical-understanding-as-seen-within-a-framework-of-beauty-part-3/)

Mathematical Understanding as Seen Within a Framework of Beauty (part 3)

In the first two parts of this series, I began outlining how mathematics and mathematical understanding can be framed within the Thomistic modes of beauty: proportio, claritas, and integritas. In particular, I defined mathematics as the science whose subject-matter is measurable orders: objects understood as parts united into whole, having a distinction of same or difference, related in proportio, and analyzed by measuring through quantities (broadly understood). In this part, we will begin a focus on how human understanding is an essential component to the development of mathematics by highlighting how the modes of claritas and integritas are integrally involved. The main point to be made is how understanding is arrived at in mathematics as it pertains to the disclosure of order and to the discernment of truths that pertain to it. The key here will be to articulate how being underscores both the unity of parts into a whole and our grasp of it.

(Here is the link to part 2:

https://thinkingbeautifully.org/mathematical-understanding-as-seen-within-a-framework-of-beauty-part-2/)

Mathematical Understanding as Seen Within a Framework of Beauty (part 2)

In part 1 of this series, I began to explore how mathematical reasoning can be understood within a Thomistic framework of beauty as expressed in the modes proportio, claritas, and integritas. In that post, I arrived at an initial description of the subject-matter of mathematics as pertaining to quantitative being understood as parts ordered into a unified whole and related to each other through a proportio. Objects of mathematics themselves are to be understood per se as ordered and potentially related to each other within a further ambient order. As objects of human reasoning, they arrive to our cognitive awareness as beings of our imagination, by means of abstraction from created beings, as received by our senses. Thus, they are understood as formations of intelligible matter. This type of matter is what underlies a created being’s size, shape, etc. and is therefore understood through quantities. Furthermore, changes to an object that preserves its form are understood through such quantities. In this post, my aim is to continue this exploration by defining what is meant by quantity, describe what underlies its nature, and examine how it relates to quantitative being in terms of its ordering.

(Here is the link to part 1:

https://thinkingbeautifully.org/mathematical-understanding-as-seen-within-a-framework-of-beauty-part-1/)

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

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

https://thinkingbeautifully.org/form-beauty-and-euclids-elements-part-1/)

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: https://life.ccel.org/strain-forward/

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.