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 moving from *claritas* to *integritas* in mathematical understanding, the act of the will in paying loving attention to the particular subject-matter continues in love to communicate the truths found and how they may be known to be truths. In that spirit, this movement involves articulating the parts that underlie the whole that express a mathematical truth in a medium of sensible matter that consist in some or all of

- Symbolic or iconic representations of the objects and operations.
- Articulation of important terms and definitions
- Articulation of the rules governing operations
- Stated first principles which are either axioms or statements which were previously demonstrated as true in other works.
- Articulation of one or more novel propositions together with demonstrations meeting the standards for proof.
- Examples and/or sample calculations that either support the content or provide speculations of potential new results or directions for future discoveries.

A work expressing mathematical truths in a sensible medium and containing some or all of these conditions will be called a **rendering**. With renderings, the mathematician may, depending upon his or her experience, study them to gain a deeper understanding of the orders and *proportios* that pertain to the mathematician’s particular subject-matter. The symbolic or iconic elements are the sensible subjects that become intelligible species in the mind by abstraction. Definitions, operations and the rules that govern them, and first principles, both explicitly stated and implicitly understood, are abstracted and incorporated into the mind’s engagement in discursive reasoning. From here, the movement can be made to contemplative reasoning in which novel patterns and order arise in the light of being and unity, resulting in *claritas* through insights.

We now identify *integritas* in mathematical understanding as the degree to which a rendering exhibits each of the following criteria

*Verisimilitude*: Statements exhibited in the rendering are supported to be true by way of either proper referencing, valid and verifiable demonstration, or by giving solid evidence supporting novelties to be discovered.*Economy*: A rendering of a subject-matter uses a minimum amount of basic material to express that subject-matter.*Fittingness:*Arrangements within a rendering reveal inter-connected truths of the subject-matter with a minimal number of extraneous elements.*Simplicity*: truths of the subject-matter articulated within the rendering are transparently revealed as they are developed from the organization of the basic material, when understood.*Perspicacity*: The measure of the rendering’s ability to address previously unanswered questions and provide insightful openings to existing or novel directions.

Now, the career of a mathematician is one of growing experience, first, in understanding in ever greater depths the truths of his or her subject-matter through the learning of the underlying order through renderings others have produced. Next, growth occurs in achieving ever deeper insights through experimenting and developing novel truths and theories and, lastly, by producing renderings of these truths and theories as a way of verifying and communicating them to others. In aiming to achieve these over time, a mathematician is developing his or her **craft**. The development of a such craft, through the lens of beauty, is one in which *proportio*, *claritas*, and *integritas* is

“… a marriage of poetic craft and contemplative vision that captures traces of eternity’s radiance in fugitive splendors here below by translating our tacit knowledge of the eternal forms into finite objects of reflection, at once strange and strangely familiar.”[i]

To capture the content of this marriage between contemplation and craft, we may follow Saint Anselm of Canterbury’s criteria for the development of the contemplative life (as portrayed by Hans Urs Von Balthasar[ii]):

- Developing the craft: The search for beauty begins through the loving act of laboring in perfecting one’s craft. In seeking to know a subject-matter in greater depth and breadth, it is essential to begin by gaining ever greater mastery of its features, first in its most basic principles and techniques then moving to understanding deeper structures as greater intuition is developed.
- Perfecting the craft: From this labor with developing an understanding of the basic materials of the subject-matter, insight,
*intellectus*, is developed to begin to see depth and novelty in the orderly arrangements of things within the subject-matter. Thus, with greater insight into the being and unity that underlies the subject-matter comes a greater contuition of its source in Primary Being. - Delighting in the craft: The consummation of the movement from basic to deeper understanding that fulfills a vision of deeper forms within orderly arrangements gives witness to the good of the whole, brings with it a sense of joy and a desire to continue perfecting one’s craft.

In fact, as a way of characterizing the development of a craft, these criteria can pertain to the craft of any profession. As it pertains to the profession of the mathematician, we may provide some further characteristics. In particular, in the process of developing a subject-matter of mathematics through renderings, a consistent language is needed that ties these renderings together as a further criterion for the *integritas* of that subject-matter. From a historical perspective, Ladislav Kvasz[iii] provided the following criteria for measuring the extent a language captures the underlying order that grounds that subject-matter’s understanding:

*Logical power*– how complex formulas can be proven in the language.*Expressive power*– what new things can the language express, which were inexpressible in previous stages.*Explanatory power*– how the language can explain the failures which occurred in the previous stages.*Integrative power*– what sort of unity and order the language enables us to conceive there, where we perceived just unrelated particular cases in the previous stages.*Logical boundaries*– marked by occurrences of unexpected paradoxical expressions.*Expressive boundaries*– marked by failures of the language to describe some complex situations.

And so, in participating in the craft of a mathematician, research in the order of the subject-matter is aimed at creating renderings as a movement from *claritas* to *integritas*. Growth in the practice of this craft is achieved by the production and advance in the language, which the renderings are articulated in and which is common to other renderings produced by fellow mathematicians seeking to develop the same subject-matter. These renderings seek to develop the subject-matter so as to both increase the languages powers, as listed above, through deepening the understanding of the order, and determining more clearly the boundaries. Conversely, there is the aim to refine or even revolutionize the language in order to address and/or move those boundaries to increase the scope and strength the language is able to exercise in its particular powers upon the subject-matter.

Finally, it is in the unified acts of bringing *proportio*, *claritas*, and *integritas* to light that beauty can gauged in the discoveries and inventions of mathematicians as exhibited in the renderings they produce. These are presented as objective characterizations for beauty, but we may also identify a subjective standard which resonates with the variety both within mathematics and by the mathematicians practicing each of their pursuits in the subject-matters, aims, and methods that they choose to pay loving attention toward. This subjective standard may again be taken from works of Thomas Aquinas:

“Beauty and goodness in a thing are identical fundamentally; for they are based upon the same thing, namely, the form, and consequently goodness is praised as beauty. But they differ logically, for goodness properly relates to appetite (goodness being what all things desire); and, therefore, has an aspect of an end (an appetite being a kind of movement towards a thing). On the other hand, *beauty relates to the cognitive faculty, for beautiful things are those which please when seen*. Hence beauty consists in due proportion; for the senses delight in things duly proportioned, as in what is after their own kind — because even sense is a sort of reason, just as is every cognitive faculty. Now since knowledge is by assimilation, and similarity relates to form, beauty properly belongs to the nature of a formal cause.”^{[iv]}

Here, I include a good part of this passage in order to reinforce beauty’s important relationship to form and goodness. Each mathematician, in terms of their chosen subject-matter, possess both a talent for understanding it, as a reflection of being a particular participant in the goodness of creation, and a love for developing knowledge about it, reflecting the pleasure received from the discernment of the order that underlies it. Furthermore, the motivations and methods that different mathematicians engage in in seeking beauty in their subject-matter can be categorized to a certain extent as follows[v]:

- Explorers: Discoverers of objects with prescribed properties that can be either:
- Gems: wholly new and possess novel features, or
- Maps: give novel perspectives to known objects via certain types of relations

- Alchemists: Finding connections between two areas of mathematics that had not been previously connected.
- Wrestlers
- Focusing on relative sizes and strengths of this or that object.
- Focusing on estimates or bounds in terms of quantities.

- Detectives: Pursuing the most difficult, deep questions, seeking clues and following trails, confident a solution can be found. These include:
- Strip miners: Uncovering a hidden layer underneath a visible superficial layer in order to solve a problem. The hidden layer is often more abstract.
- Baptizers: Naming something new, making explicit a key object that has often been implicit earlier but whose significance is clearly seen only when formally defined and given a name.

- Naturalists: Focus on uncovering hidden abstract mathematical layers that underly what is revealed speculatively or by experiment pertaining to physical phenomena as discovered and theorized by natural scientists.

In conclusion, beauty arises in the pursuit of mathematics as a marriage of the mathematician’s desire of and love toward her subject-matter in understanding it in ever greater breadth and depth and the contemplative vision that seeks the being and unity that underly that subject-matter. This beauty is itself a reflection of the *arche* that grounds it in Primary Being as its source, and is revealed in its breadth and depth in the *proportios as* seen in *claritas* and rendered in *integritas*.

[i] From David Bentley Hart’s “A Perfect Game,” First Things

[ii] H. Von Balthasar, *Glory of the Lord II: Studies in Theological Style: Clerical Styles*

[iii] Ladislav Kvasz, *Patterns of Change*, Birkhauser

[iv] Thomas Aquinas, *Summa Theologica* I.5.4 *ad* 1 (*italics* mine)

[v] See David Mumford’s Beauty & Math & Brain Areas at http://www.dam.brown.edu/people/mumford/blog/2015/MathBeautyBrain.html