We continue to sketch an account of creational thinking as a way of understanding mathematics and its relationship to reality. In our narrative so far, we had begun to develop a perspective of objects of reality as creative by adopting a hylomorphic view of them as the sensible and intelligible united in matter. As objects of thought and understanding, they are unified wholes of data, what Lonergan refers to as things. In reference to the object itself, as received from by the senses, we will continue to refer as formations[i]. In reasoning about a formation through its thing in the mind, the key characteristic we identified as pertaining to its intelligible structure is that it is ordinable. To understand what that means is that it is first and foremost intelligible above and beyond the sensible. Furthermore, the ordinable is arrived at by being separated from the sensible through abstraction. What is ordinable in a formation is still discerned in a material way through quantities which constrain and are constrained by the formation’s essence. These are the measured quantities and correlations which are termed explanatory conjugates (a la Lonergan). We now seek to give an account of how such conjugates become framed within scientific theories and, ultimately, become further framed within mathematical theories. In Przywara-ian terms, this is to move from the morphological to the eidetic (from the morphe to the eidos). To that end, we will be adopting Lonergan’s account of enriching abstraction.
We begin our journey in developing a notion of creational thinking by giving an account of physical objects in hylomorphic terms and describe how by abstraction human understanding distinguishes such an object in terms of the sensible and the intelligible. In the course of laying out such an account, we will be expanding upon our previous descriptions, which was largely dependent upon the thought of Thomas Aquinas, by channeling the thought of Bernard Lonergan, through his magnum opus Insight[i], and Erich Przywara, through his magnum opus Analogia Entis[ii].
In a previous series of posts (beginning here https://thinkingbeautifully.org/mathematical-understanding-as-seen-within-a-framework-of-beauty-part-1/), I described a particular perspective on thinking beautifully in mathematics. In that description, I aimed to channel the thinking of Medieval thinkers such as Thomas Aquinas and Bonaventure, but also by incorporating the thinking of such twentieth century philosophers as Bernard Lonergan and Michael Polanyi. Unfortunately, my grasp on the latter two thinkers was not sufficient to capture the depths of their ideas in developing the theory of mathematical knowledge that I am aiming for. I have since dug deeper in both of these authors’ works on human understanding, particularly Lonergan’s Insight, and I want to now expand upon my previous forays by articulating how mathematical knowledge fits into a larger mode of human understanding which seeks to see the human knower and knowable reality as united and understood in that unity as a divine creation. To that end, I will also be bringing in as an additional conversation partner, along with Lonergan, Erich Przywara through his magnum opus Analogia Entis.
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:
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:
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:
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.
[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.
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.
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/