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/) In order to capture, in the most general way, what is to be meant by quantity, we will first aim to give criteria for how quantity operates in the most basic way so that it is clear how to recognize it in action. To that end, we will want to answer:

  • What do we first understand when we understand quantity?
  • How does quantity bear upon being in order for that being to be quantitative?
  • How is quantitative being related to created being when created being is understood in its intelligibility?

In the course of addressing these questions, we will see that quantity is essential in framing mathematical objects in order to understand them. In particular, it will be the case that a being, understood as parts ordered in proportio, becomes truly understandable when it is understood quantitatively.

Initially, we will understand something to be a quantity if it satisfies one or more of the following

  • Being that which may be increased or diminished (L. Euler[i])
  • Being in accord with Euclid’s Common Notions[ii]
    • Things equal to the same thing are also equal to one another.
    • And if equal things are added to equal things then the wholes are equal.
    • And if equal things are subtracted from equal things then the remainders are equal.
    • And things coinciding with one another are equal to one another.
  • Being that by which things may be:
    • Distinguished (as things of different quantity must in some way differ).
    • Limited (for example, in their size, capacity, dimensions, etc. in being finite).
    • Inclined towards (as in a way of comparing with something other considered as fixed).[iii]

Here we see quantity in terms of what it is (quantity per se) or how quantity pertains to something else (by bearing quantity or by being quantitative). We note that with respect to these characteristics, quantity can be understood as forming an order in which

  • To be the same per se, as well as in proportio, is by equality. In general, to be in proportiois to either by being the same per se, or by being the same by increase, or by being the same by diminishment (the latter two providing the notions of inequalities).
  • Therefore, change in quantitative being occurs operationally by being increased or diminished (that is, by addition or subtraction).

The first types of quantity that fit these criteria, and, in fact, motivate them, are

  • Multitude or number (discrete forms of quantity)
  • Magnitude (continuous forms of quantity).

The most fundamental kind of number is that of the natural number system:

0, 1 = 0+1, 2 = 1+1, 3 = 2+1, 4 = 3+1, …

In terms of being an order, an axiomatic system, perhaps akin to that developed by Giuseppe Peano, can be provided from its first principles. In particular, addition satisfies associativity and commutativity, adding 0 fixes the number, while adding 1 uniquely provides for and determines the successor. Moreover, the set of natural numbers is well ordered in the sense that the entire system is completely determined by starting with 0 and then successively adding 1. In this way, we may understand the natural number system as an order in which the intrinsic proportio between natural numbers n and m is if either n = m or m is reached from n by successive addition of 1 (in modern symbolism n ≤ m).

            Moving to things understood as created beings, in viewing a multitude of them we may, by counting them, consider that multitude as participating in the order of the natural number system. Actually, though, there are two levels to the ordering in play with respect to these things viewed as a multitude. In the first level, the things are seen, by abstraction, as individual parts forming a finite collective whole. By counting, there is a preliminary proportio in which the individual things are seen in succession, one after another. This provides a preliminary order that the multitude of things is to be understood as participating in. On a subsequent level, the natural numbers are seen as related to, or by assignment with, or corresponded with, the natural numbers in an accordance between the quantitative being of the natural numbers and the order of the multitude of individual things. This will be understood by saying that the order participates in the quantitative being by measurement and that, therefore, the first order is a measurable order. It is now our goal to describe more generally what constitutes quantitative being and measurable orders and how they are understood as further related to created being by abstraction.

            In order to arrive at a more comprehensive understanding of quantitative being, we will follow a course as developed by Bernard Lonergan in his magnum opus Insight[iv]. In chapter 1 of this work, Lonergan begins his study of mathematical objects by considering the natural number system (see above) as the objects 0, 1, 2, 3, … together with the operations =, + and × satisfying certain rules (read first principles). From this start, we may begin to construct more general systems of quantities by introducing inverse operations to enrich the natural numbers in stages. In particular:

  • Introducing the operation –, as the inverse of +, in turn introduces negative numbers. Combining – with =, +, × while enlarging the operational rules enlarges the natural numbers to the integers.
  • Introducing the operation ÷, as the inverse of ×, in turn introduces reciprocals of non-zero numbers. Combining ÷ with =, –, + and × while enlarging the operational rules enlarges the integers to the rational numbers.
  • Introducing √, as the inverse of ( )2(so that n2 = n×n), introduces irrational numbers for √ operating on positive rational numbers. Combining √ (and inverses of ( )r for general r) with the other operations, as well as enlarging the rules of operations, expands the rational numbers by including the irrational numbers to form the real numbers.
  • Including √ operating further on negative numbers introduces imaginary numbers which, when included with the real numbers, along with expanding the rules of operations, results in the complex numbers.

This hierarchically enriching construction of ever more sophisticated systems of numbers is referred to by Lonergan as higher viewpoints. Now, this hierarchy can be further refined as a hierarchy of orders in which the particular operations, along with their rules of combination, are prioritized first. The objects, then, that the operations are to act on are what is, either specifically or generically, understood only secondarily as being what is subject to those operations in accord with those rules that govern them. For example, the operations that act on the integers (=, +, -, ×), along with rules governing them, will also be the operations and rules that occur for polynomials (with a fixed number system serving as coefficients). In modern parlance, things having this order are called rings. Or, by considering the operations that act on the rational numbers (=, +, -, ×, ÷), together with rules that govern them, also serve the same for the real and complex numbers. Generically then we arrive at what is currently known as being a field. Now, the structure of being a field and the structure of being a ring are hierarchically related by abstraction: by abstracting from the rules of operations that govern and relate =, +, -, ×, ÷ only those rules of operations that govern and relate =, +, -, ×, a field may be understood as a ring. Furthermore, such hierarchical relationships for structures can be extended to that of orders by the introduction of proportios that preserve structures. For example, in the modern parlance of category theory, the concept of morphism can serve as this type of proportio.

            We are now in a position to give a proper notion of both quantitative order and quantitative being. First, a quantitative being is to be understood as a structure in which its parts are referred to as quantities and which consists of:

  1. rules governing and so defining operations on the quantities, and
  2. operations proceeding from some quantities to others and so relating and defining them[v].

It should be noted that while quantitative beings are understood to be structures some may be non-trivially orders. For example, the real numbers can be considered as an order with ≤ as a proportio.

Next, a quantitative order then is an order whose parts are quantitative beings in which

  1. the rules defining and governing the operations are the same across parts, and
  2. proportiosbetween parts preserve structure (or possibly order, if such is present).

It should be noted that, as above, quantitative beings may be related hierarchically through higher viewpoints while quantitative orders can be understood as hierarchically related by abstraction. Moreover, the two types of hierarchies may be seen together in related ways: if quantitative being X’ arises as a higher viewpoint of the quantitative being X which, in turn, is a part of the quantitative order O  and O’ is the resulting quantitative order that Y is a part of then O is hierarchically related to O’ by abstraction and X and X’ are in proportio in O.

            Since quantity is so open to being understood by its systematic nature, quantitative being becomes the ideal mode by which intelligible matter may be understood. Therefore, it becomes important to identify how intelligible matter, understood as order, is made quantitative.  To that end, we will say that a quantitative being measures an order provided:

  • to each part of the order there corresponds a unique quantity that belongs to the quantitative being,
  • parts that are the same in the order correspond to equal quantities in the quantitative being, and
  • parts in proportio in the order correspond to quantities in proportio in the quantitative being (provided the quantitative being itself is understood as an order).

Conversely, we will say that an order is measured if there is a quantitative being that measures it (and, thus, that order participates in that quantitative being). Finally, we define a measurable order to be an order which has the property of being measured. It is to be noted, though, that, in being a measurable order, the quantitative being by which the order is measured is not unique. In fact, there may be quite a number of distinct ways to measure a measurable order, though they will be related in proportio within an appropriate quantitative order by virtue of them all being measures of an individual order (which can be deduced from the Principle of Dependent Existence).

            With this brief sketch on how quantity is to be understood generally and how it bears upon order in terms of being measurable, we may now refine our understanding of what constitutes the subject-matter of mathematics. We first summarize how we arrived at this subject-matter as follows:

  1. Begin by identifying the subject-matter as pertaining to the intelligible matter of an individual created being, arrived at by abstraction.
  2. As pertaining to intelligible matter:
    1. the subject-matter may be studied as being independent (Principle of Independent Intelligibility) and, therefore, as pertaining to created being in its potentiality;
    2. it is understood in act as a being and so conforms to the First Principles of Being;
    3. in its materiality, it is understood as parts unified into a whole.
  3. By being unified into a whole, the parts are understood as ordered in being distinguished as same or different and related in proportio.
  4. As an order, it is to be understood through quantities by being measured.

From this summary, we now arrive at our refined definition of mathematics as being the science (scientia) of measurable orders 

            With our definition of mathematics now in place,  in part 3:

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

we will next move to examine how human understanding comes to arrive at the truths of mathematics. This we will do through the second Thomistic mode of beauty: claritas.

[i] Leonard Euler, Elements of Algebra

[ii] Euclid, Elements Book I

[iii] These three particular criteria come from Saint Bonaventure’s exegesis of the passage “but thou hast ordered all things in measure and number and weight.” Wisdom of Solomon 11:20c in his Itinerarium in Mentis Deum 1, 11

[iv] Bernard Lonergan, Insight

[v] Ibid X.8

 

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