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:

We will begin by remarking that by claritas we will initially understand to be the light by which things are seen together and in relation to each other. Such a relation can integrate the parts together holistically to a varying range of degrees in terms of how harmoniously such parts hang together. One analogy is as a prism which reveals white light that holds color together in a spectrum. It is by claritas that we will understand to be the act in which objects are seen as parts unified into a whole in accord with a proportio. In unfolding how and in what ways claritas occurs, we will be engaging with both the Augustinian-Bonaventurian tradition and the Aristotelean-Thomistic tradition, as well as Bernard Lonergan’s theory of insight and Michael Polanyi’s theory of tacit knowledge.

First, recall that we presume to arrive at the cognitive content of the objects of mathematics by abstraction from the sensible material of a created being. This content we will refer to as a species or imageof the original being. This species, being strictly part of our imagination, will still be material having intelligible matter as its substrate. In viewing it as an object of mathematics, we seek to understand it as (1) being ordered by (2) measuring it. This way of distinguishing the modes by which we come to understand the object through the species will themselves be reflected by two modes of reasoning in our soul: the ratio inferior (lower reasoning) and the ratio superior (higher reasoning). This distinction in reasoning we draw from the Augustinian-Bonaventurian tradition[i] though we will need to address how this will merge with the views from the Thomistic tradition that we have already espoused.

In reasoning about an object of mathematics, arriving at the understanding of it as ordered will be achieved first by measuring it and then reasoning about it discursively through either (a) what is known as true of the quantitative being that measures it, or (b) by performing calculations in accord with the operations and the rules that govern them as possessed by this quantitative being. This discursive process is a directed, conscious effort by the knower in which the awareness involves an interaction of the senses, by way of reading, writing, consultation, etc., and the ratio inferior by abstraction, memory, and experience in relation to performing calculations, sorting, and making comparisons of the pertinent data. Such data not only becomes the raw material from which order is sought, but, in order to arrive at an understanding of the potential order present, this data must move to the ratio superior in the form of tacit knowledge[ii] where “the presence of lasting shapes differs from ordinary perception only by the fact that it can integrate shapes that ordinary perception cannot readily handle. Scientific knowing consists in discerning gestalten that indicate a true coherence in nature.[iii] We will refer to this mode of reasoning in which the ratio superior is taking priority as contemplation or contemplative reasoning.

This integration of shapes in tacit reasoning is what we shall assert to be the point at which order is discerned and becomes presented to the conscious mind in the ratio inferior.  This presentation is what Bernard Lonergan calls insight[iv] which is “not any act of attention or advertence or memory but the supervening act of understanding”[v] which is the “source … of clear and distinct ideas”, “the act of organizing intelligence” and so is “an apprehension of relations” being both “a priori and synthetic”[vi]. Finally, “insight unifies and organizes” and so is central to the discernment of order within the intelligible matter underlying a species. Now, by what we call claritas will mean the measure of the degree to which an order properly belongs to a species, understood as an abstraction of a potentially created being, as revealed from the unified activities of the ratio inferior and the ratio superior in forming insights obtained from reasoning conducted as a marriage of both discursive and contemplative reasoning.

Now, let’s examine a little closer how claritas arises as a result of this unified activity of the ratio inferior and the ratio superior. First, we identify what it is to be fundamentally sought and understood in determining order as a simultaneity of being and unity. In being most basic, it undergirds not just the mind’s cognitive activities and the unity of our sensual experiences, but the coherence and unity of both taken together by the powers of the soul. Furthermore, in receiving and perceiving of things around us, in their particular natures and the nature of the environments that they belong to, such things as beings and unities are understood as created beings and as participants of the being and unity that constitutes the whole of creation. It is from this vantage point that we will understand our awareness, discernment, and contemplation of the discernibility, intelligibility, and unity of creation itself as grounded in the one, being, unity that grounds both creation and our understanding of it. Moreover, it is this intrinsic being and unity underlying creation which determines its order and goodness. Furthermore, it grounds our perception and understanding of the created order, and of order in general, in being participants of creation. In understanding our cosmos as created, its intelligibility, ordering, and goodness in its being is thus to be understood as having its source in Primary or First Being. This, in turn, grounds our understanding of being, unity, and order in considering it as created, as being a participant in the act of being.

We now identify the movement in arriving at the human understanding of order from the ratio superiorto the ratio inferior in which an insight results from tacit knowledge. Here we will identify this movement as an act of illumination in accord with that developed in the Augustinian-Bonaventurian tradition[vii]. This act occurs in three modes. First, what is primarily tacit in the ratio superior is a knowledge of Primary Being by which all other beings are understood and compared. This knowledge, in being fundamental, is disclosed by way of the memoria in the mind. This is what grounds our sense of and search for resemblance and unity among things in our capacity to reason. By memoria we are able to tacitly access a standard by which being and unity may be discerned. A posteriori, this is seen to be so through, and measured by, the First Principles of Being.

The next mode in the act of illumination is the understanding that occurs in insight. To further elucidate the way this occurs in illumination, we refine our understanding of what undergirds the movement to insight from memoria. In particular, in understanding a species as an abstraction of a potentially created being, the intellect understands that species as a symbolic representation of what that being is in relation to its archetype or exemplar as a participant of Primary Being, which is the ground for its existence. The intellect doesn’t see Primary Being directly, nor its exemplary causes, but, instead, it contuits those causes as it contemplates the species in being an abstraction from created being[viii]. Thus, the movement from contemplation, as regulated through memoria, to insight is fulfilled by bridging a species’ potential for being and unity with how it may be understood in act as a created being by contuition.

The third mode in illumination, is the act itself of seeking to disclose the possible unity and order in or among things by endeavoring to see them in their goodness and as participants of the good order of creation. This directedness of the will toward seeking the good order of things is what is understood to be an act of love and is central in any unified act of discursive and contemplative reason in seeking to discern what it is to be a fitting part of creation. For in seeking the beauty of a thing, in its goodness and order, one must pay loving attention to it.

By relating our search for the nature of order as participating in the search to understand the good order of creation, we recognize that the source of this order is one in whom resides the first principle of existence. This the “I AM” or “He Who Is” (אֶֽהְיֶ֖ה) of Exodus 3:14 (ἐγώ εἰμι in the Septuagint)  which is at once Primary Being and the Pure Act of Being (in Thomas Aquinas’s phrasing, the actus essendi subsistens[ix]—the subsistent act of being— “one of his most beautiful names for God”[x]). Moreover, in that the cosmos is understood as a created order, its being, unity, and order originates in the Pure Act of Being, for

“… in every effect the ultimate end is the proper intention of the principle agent, as the order of an army (is the proper intention) of the general. Now the highest good existing in things is the good of the order of the universe … Therefore, the order of the universe is properly intended by God …”[xi]

It is in our participation of creation that both creation and our reasoning of it are regulated by our participation in the Pure Act of Being. In turn, our reasoning of the order of the cosmos and of order in general is grounded in our memoria and understood in contuition as an image and act of Primary Being. Further, in seeking order by reasoning we seek what is to be a reflection of the logoi of things in creation as grounded in Primary Being as Logos which “before it was merely reduced to a ‘word’ conveying facts, or to ‘reason’ in the philosophical sense, or to ‘principle’ or the ground of ‘logic’”[xii]  the ancients understood as “that power of gathering that brings all things forth into the light of being, holding them together in the unity of the world while also allowing them to shine forth in their separateness”[xiii] and which Prologue to the Gospel of John declares:

“In the origin there was the Logos, and the Logos was present with God, and the Logos was god; This one was present with God in the origin. All things came to be through him, and without him came to be not a single thing that has come to be. In him was life, and this life was the light of men. And the light shines in the darkness, and the darkness did not conquer it … It was the true light, which illuminates everyone, that was coming into the cosmos.” – John 1:1-5, 9[xiv]

In the final part of this series,

I will explore how the mode of integritas in understanding mathematics through beauty is the mode in which the craft of mathematics is exercised in the practice of articulating, rendering, and communicating of mathematical truths.

[i] Our primary source for this perspective will be Saint Bonaventure’s Disputed Questions on the Knowledge of Christ Works of Saint Bonaventure, Vol. IV, Z. Hayes ed. (1995).

[ii] Michael Polanyi, “Logic of Tacit Inference,” Journal of the Royal Institute of Philosophy Vol. XLI, No. 155 (1966)

[iii] Ibid p. 1 (Polanyi’s emphasis)

[iv] Bernard Lonergan, Insight

[v] Ibid p. ix.

[vi] Ibid p. x

[vii] Laura Smit, “In Your Light We See Light: The Continuing Viability of a Christocentric Epistemology” Realism and Antirealism, William Alston ed., Cornell University Press (2002) pp. 167-180

[viii] See Hayes Ch. 4, p.58

[ix] Thomas Aquinas, Summa Theologica I, Q.3, A.4, ad 2

[x] David Bentley Hart, “Beyond disbelief,” New Criterion June 2005

[xi] Ibid I, Q.15, A.2

[xii] David Bentley Hart, “A Philosopher in the Twilight,” reprinted in A Splendid Wickedness and Other Essays, Eerdmans Publishing (2016) p.96

[xiii] ibid

[xiv] David Bentley Hart, The New Testament: A Translation, Yale University Press, 2017

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