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.
To begin, we need to first keep in view the perspective that the essence of mathematics, as a subject of human understanding, has been demonstrated to be deeply relevant to the sciences and yet can be studied as independent of any relevance to the physical world around us. In that light, we will take as the things that the objects of mathematics are drawn from to be understood to be created beings. In doing so, we are seeking to understand such things as potentially as part of the physical world by being both intelligible in nature and yet having the features enabling it to be grasped by our senses. As such, we may initially adopt an Aristotelean-Thomistic understanding of things as composed of form and matter in what makes them both sensible and intelligible. From this perspective, we shall adopt the language that the form inheres in the matter of a thing and that the matter serves as the substrate for that thing. The form then determines and limits what a thing is and can be as a created being while the matter provides the basis for variety, change, combining, and dividing.
In order to arrive at the subject-matter of mathematics, we will follow Aquinas[i] in identifying a way to distinguish between the sensible and intelligible features of a thing by characterizing how matter is ordered as a substrate of a thing. In basic terms, in coming to discern a thing, we first understand it as existing. Then we discern it to have shape, size, and other similar features, that is, as bearing quantity. Only next do we discern it in terms of particular qualities, passions, and, motions, such as color, texture, odor, place, time, movement, etc. Now, quantity and qualities in a thing are subject to change in that thing as determined by its material substrate. Such changes, though, are limited by the inhering form and are understood either through our senses, or can be understood by way of intelligence through the imagination. In terms of the latter, we will adopt the distinction that a thing being first understood as existing and then subject to a specified quantity are the criteria for that thing to have individual intelligible matter as its substrate. In terms of the further qualities, etc. a thing possesses, it is understood strictly through the senses and, hence, pertains to individual sensible matter as its substrate. Now, these two types of matter are not distinct as individual matters underlying the thing, rather they are distinct in the order of being and, hence, in understanding. In order to indicate how this distinction is arrived at, we will describe the cognitive act that provides it, namely that of abstraction.
We will understand, following Thomas Aquinas[ii], abstraction to mean “[the consideration] of some attributes of a thing apart from its other attributes, even though all these attributes exist together.”[iii] We will further understand this cognitive act as subject to the following principles:
- Principle of Independent Intelligibility: “The aspect which is considered separate must not depend for its intelligibility on the other aspects from which it is mentally separated.”
- Principle of Dependent Existence: “The aspect which is considered separate cannot be asserted to exist apart from the thing and its other aspects with which it really does exist.”[iv]
We note up front that the mind, by abstracting as subject to both principles, is one that begins with the thing under consideration understood as a created being. Hence, it first understands the thing as existing. Then the Principle of Independent Intelligibility asserts that, under abstraction, the thing may be understood in terms of aspects that belong to it which can be thought independent of other aspects. On the other hand, the Principle of Dependent Existence asserts that aspects properly abstracted from the thing does not form a new thing that exists independent of the original. (A cylindrical mug may have its volume determined by either careful measuring utilizing a measuring cup or by calculating it from its height and base diameter. The two different ways must produce the same quantity.) Nevertheless, in arriving at an object of mathematics, the Principle of Independent Intelligibility gives such objects a type independent existence, but only as an object of the imagination. In this capacity, abstraction will serve to provide the medium for understanding types of mathematical objects by characterizing them through the principles that are understood to govern them. These principles ensure the object in question is understood as an abstraction from created being even if only so potentially. Such mediums provide the suitable contexts for understanding objects of mathematics from a formalist perspective.
We now define an object of mathematics (or mathematical object) to be what arises by abstracting from a thing what is understood solely in terms of individual intelligible matter. This can be characterized further as being substance as a subject of quantity. With respect to what such an object is, as having a mode of existence independent of other qualities, etc., we will understand it to be a quantitative being, that is, a created being understood in terms of intelligible matter by abstraction. Viewed this way, change, variation, combining, and dividing in which form is preserved is understood purely in terms of quantity. In relation to a created being, a quantitative being that can be understood as arising by abstraction from an individual created being is said to be a similitude of that being. In turn, we will say that a thing participates in a mathematical object if that thing is a concrete instance of that object when understood as being that object by abstraction.
Now that we have a definition of mathematical object, we need to unpack how the defining features of such objects are further understood. In particular, we need to delineate an understanding of both intelligible matter and quantity and then describe how the two notions interrelate and interact. Beginning with the notion of intelligible matter, being a type of matter, it is understood to be subject to change, variation, combining, and dividing. As a compound of matter and form, the form limits the ways the object may be subject to such alterations or constructions. In turn, such alterations and construction are solely understood through the quantities the object is subject to. In what way, though, do we understand all this as related to objects of mathematics?
First, an object understood as composed of intelligible matter is subject to the following First Principles of Being:
- Principle of Non-Contradiction: An object cannot at once both be and not-be. Propositionally: what is understood of the object cannot simultaneously be both true and false.
- Principle of Excluded Middle: An object must at once either be or not-be. Propositionally: what is understood of the object must either be true or false.
- Principle of Sufficient Reason: What is understood as being true of an object must have a cause or reason for that truth. Propositionally, such a reason is due to at least one of the following: being by definition, being by first principle, or being by reason of another understood as true.
We note that, while we have distinguished between cause and reason in the condition of sufficiency, we will understand that reasoning is itself a type of cause, namely a formal cause. This type of cause underlies each of these principles, the reasoning from one truth to another, and the disclosure of first principles.
Next, with these first principles understood, we next understand an object composed of intelligible matter in mereological terms. Specifically, we will understand an object with an intelligible material substrate to be a structure[v] provided it is subject to the First Principles of Being and satisfies:
- the object is understood as consisting of parts united into a whole and
- there are relations of same and difference among the parts in such a way that each individual part is the same as itself.
In addition, we will define a structure to be an order if it further satisfies:
- parts are related by a proportio for which
- parts can be the same in proportio in such a way that parts understood as the same are the same in proportio;
- proportio among parts satisfies the transitive property.
In first understanding an object of mathematics as being an order, the Principle of Sufficient Reason ensures that there are principles that govern any understanding of how parts are arranged in proportio within an order. Certain such principles can serve as an axiomatic basis for understanding orders in a formalized way. In particular, we may observe that objects of mathematics of a certain type may be understood as coexisting together and serve as parts determining a whole that itself bears the structure of an order. Here, the intelligible matter that serves as the substrate for this super-order can be understood as a relative type of common intelligible matter which, in turn, serves as the substrate for each individual intelligible matter for the particular objects that occur as the parts. Further, a proportio for this super-order can be viewed as a certain relation between such objects, like being parallel for lines in plane geometry. Certain principles would then serve as axioms that provide, for example, existence conditions for such objects, or for relations between them.
In sum, we have finally arrived at how the first mode of beauty, proportio, can be taken as a primary component in defining and understanding objects of mathematics. Before seeing how the other two conditions, claritas and integritas, will be seen as central to characterizing human understanding of mathematics, we will need to complete our portrait of the objects of mathematics by delineating how and in what ways quantity is integrated into the notion of order. This will further clarify how a mathematical object is to be properly understood as a quantitative being. This we will do in part 2:
[i] Thomas Aquinas, Summa Theologica I q. 85, 1 ad 2; In Boethius’s De Trinitae V, 3
[iii] Thomas C. Anderson “Intelligible matter and objects of mathematics in Aquinas”
[v] See An Aristotelean Realist Philosophy of Mathematics by James Franklin