A Math Teacher Looks at the Heart of Geometry

With the move from the old HogwartsProfessor site over to Substack, I have enjoyed the recent newsletters that have (re)introduced our foundational principles of interpretation and evaluation, particularly with respect to the Perennialist or Coleridgean view of knowledge. In support of these, my aim here will be to present some of those same ideas in a new light.

This past year was my first full year teaching at a Classical charter high school here in Texas. From my previous posts through Hogwarts Professor, one would suspect that the subject I teach would be English or Classics. In fact, the subject I teach is Mathematics, particularly Geometry and Precalculus. No, this is not an applied-for-the-defense-against-the-dark-arts-post-but-got-potions-instead situation; my studies in college were in Philosophy and Mathematics. Perhaps more surprising is that I find the subject a place where something like the Heart/Intellect of the doctrines of the Perennialists and Samuel Taylor Coleridge can be simply explored and explained. Where in the supposedly dry and emotionless subject of Geometry do I find the understanding that only the Heart brings? What is a romantic poet doing being referenced in a math classroom? Permit me to demonstrate.

Define a Cat

            Within the first week of my Geometry class, as we are getting started on the introductory chapter, I have my students grab dictionaries from off the shelf. I propose the following scenario to them. Suppose that an alien has just shown up. It doesn’t understand English and is unfamiliar with most of the sorts of plants and animals that are on Earth. They have come across our word cat and are eager to know what it means. Our tool to explain this to the aliens: the dictionary. At this point I have them look up the word cat, and they return with (here from Merriam-Webster’s Collegiate Dictionary, 11^th edition): “a carnivorous mammal (Felis Catus) long domesticated as a pet and for catching rats and mice.” The aliens receive our definition and ask us to define carnivorous, mammal, domesticated, and pet (just to get started). At this point I divide up the classroom, having each group look up one of these words. They give me their definitions, and I start drawing out the web of dozens of new words that we have introduced just to clarify our one starting word, cat. After a few rounds of this, including running into a few words that use other words we are trying to define in their definitions, I stop the process and ask my students to identify the problem. One suggested that this is a never-ending process. I respond that this is part of the problem. The problem of definition is this: either we define all words in terms of one another, which produces circular definition (a word defined with words that are themselves ultimately defined by the first word), or we just leave a small handful of starting words undefined, defining all the remaining words from them. But how are we going to show the alien what a cat is, if we take either of these paths to definition? In one, the alien is going to keep asking for definitions and get frustrated when we use circular definition or be left unsatisfied that we have not actually told it anything because we have left the foundational terms without meaning.

The way we teach the aliens what a cat is is this: we go grab a cat and show the fluffy critter to the aliens. We may then proceed, without defining the word cat, in explaining other things, like lions. We choose a small set of undefined, directly experienced terms, and ground all other definition upon them. Direct interaction with whatever things our starting undefined terms refer to is how we can make sense of language at all. We do not know fundamental essences by definition, but by entering into relationship with the thing. In Geometry (at least as presented in the textbook I teach from), we start with three undefined terms: point, line, and plane. No amount of reasoning or defining in Geometry will give you the meaning of these terms, one must understand them directly. And unlike a cat, this direct experience will not come from the senses, but from working long enough with the ideas that one begins to see what they refer to.

            What I don’t tell my students is that this is a reworked version of the argument from St. Augustine’s De Magistro (On the Teacher). In this dialogue Augustine guides his son Adeodatus through a similar argument. If all words are defined in terms of one another, how do we ever enter language at all? (Surprise! We are the aliens.) Augustine’s answer: we have the Logos in the mind as an Inner Teacher that illuminates to us the natures of fundamental terms, brings us into relationship with their meaning. By this, we come to direct knowledge of the Divine Ideas, not through definition, but through building a relationship with the things our undefined terms refer to. This is precisely the role Perennial and Romantic understandings of knowledge give to the Intellect or Heart, that faculty by which one comes into relationship with those things reason and definition cannot reach.

            In the second unit of the year, I introduce them to the rules of logic. We need two premises, major and minor, correctly set up, in order to be able to derive any new conclusions. (For example: 1. If it is a cat, then it is fluffy. 2. Puff is a cat. Therefore: Puff is fluffy.) We have assumed for the logic to work that the two premises are true. How did we know those to be true? Well, we proved them from previous deductions. Where did we get the truth of the premises for those deductions? It is often at this point (depending on the class) that a student suggested that we were effectively trying to define a cat again. We have run into the problem of deduction: either the argument runs on forever, back to further and further earlier premises, or we have accidentally let in a circular argument somewhere, trying to prove a proposition from itself. We therefore need to introduce Postulates or Axioms, statements that we take as true without proof, as needed at the foundation of our argument. A question I then guide my students to consider for most of the first semester: how do we know that our Postulates are true?

            Various answers can be given: that these are social conventions that we have agreed upon, that these are just the rules of an arbitrary game we have decided to play, or that their truth is in some sense directly revealed to the mind. Given the secular nature of the school I teach at, I present all of these options, while mentioning that the last is to me the most convincing. That God illuminates our minds through the Heart, prior to any defining and reasoning, and that this gives us certainty in our Postulates. That done, we may mechanically reason out formal proofs to better understand how these things we have understood behave. (This necessary leap from reasoning back to intellect is central to the meaning of Plato’s analogy of the divided line in Republic Book 6, and I would argue that this is ultimately the kind of leap that was meant by Kierkegaard in his famous expression about a “leap of faith.” Not an irrational leap contrary to reason, but an exercise of the Heart when reason can lead to no conclusion.)

            Geometry here provides, merely to get one of the most precise systems of mere reasoning off the ground, a testing ground for the action of the Intellect. One must begin Geometry with a deep exercise of the imagination, so that one can even begin to understand what is being discussed by our undefined terms and postulates. This is then the basis for the Heart/Intellect to grasp the nature of the undefined terms and the truth of the postulates.

Living Books: A Journey to Flatland

            Charlotte Mason was a philosopher of education who, strongly influenced by Coleridge, suggested that the main task of education was to bring the child into relationship with Living Ideas (archetypes in a Platonic sense). The main way to do this was by introducing them to what she called “Living Books,” which are those books that engage the imagination by being presented narratively, introducing the child to Living Ideas, awakening the curiosity of the child so that they have an interest in gaining the knowledge of facts associated with the central idea, and that shape the child towards virtue. The core principle here is this: humans know narratively and by the Intellect first, and only secondarily by reason. Present them with stories that bring them into relationship with the Ideas, that build up their imagination and Intellect, and you will have educated them better than any pragmatic or technical education would ever have a chance of doing.

So, how does one teach Geometry using Living Books? One way I found was by use of a particular story, Edwin A. Abbott’s Flatland. During our mini-unit on Non-Euclidean Geometry, I showed them a half-hour educational adaptation of the book. It tells the story of A. Square, a resident of a two-dimensional universe. He only understands length and width, having no concept of height.

One day he is visited by a Sphere, who has come to explain the existence of a third dimension. The Square only sees a cross section of the Sphere as he passes through Flatland, and so believes the sphere to be a circle (a ruling-class figure in Flatland) that somehow possesses the power to appear and disappear out of nowhere. The Sphere tries to reason from analogy, from point to line to square to cube, but the Square still does not comprehend.

The Sphere then forces the Square out of the plane of Flatland. From there he has an all-seeing view of what had been his world. After having three dimensions explained and demonstrated to him, the Square challenges the Sphere to show him the fourth dimension, which must exist by that very same analogical reasoning. The Sphere, having no patience for this, puts the Square back in Flatland. The book concludes with the Square preaching the doctrine of three dimensions before getting arrested and imprisoned for life by the tyrannical circles. (The half-hour movie version I show gives the story a slightly happier ending.)

            The students, at least this year, greatly enjoyed the story (enough that my presenting it to them made my school’s yearbook as a notable event for the Freshmen this year). That they were getting a geometrical version of Plato’s Allegory of the Cave was not immediately obvious to them (enough of them caught on later to this connection). It led them to think about what things we might understand that we, for one reason or other, cannot experience. C.S. Lewis referenced Flatland in several places throughout his works, notably in his essay “Bluspels and Flalansferes,” part of his extended argument with Owen Barfield about the place of imagination in the quest for truth, citing it as a brilliant and illuminating metaphor that had been of use in understanding certain geometric concepts.

Flatland presents Geometry in its Ideas, in its quest to find out the natures of Points, Lines, and Planes through the imagination needed to find postulates (the addition of extra dimensions of space can only be established in new postulates, it is not a rationally derivable addition) and to reason boldly from them. When I had presented this to my students, they now had a story to point back to, a handhold for their imagination to think through much more complicated concepts (for example, my discussion of cross-sections was simplified later in the year by having my students think back to what our friend A. Square would see as these three dimensional figures passed through Flatland). This story had given them true symbols by which they could understand the world.

            This transcendence of the merely rational, even in this most rational of subjects, made the subject come alive for a number of these students. The very foundations of the subject, postulates and undefined terms, preach the need for an illuminating Logos that “shows the cat” to us aliens, that establishes a relationship between us and the thing known.

            One of the best ways to judge a book is to see if it brings you into relationship with Living Ideas, if it is one of Charlotte Mason’s Living Books. They will delight as well as inform. They will train the reader in the use of Heart so that they may be purified more and more into the likeness of God, knowing the world as God knows it. Flatland does that, even if just for the dry mathematical ideas of Geometry; Rowling’s work likewise brings us directly to the Idea that the union of minds in Augustine’s Inner Teacher, in the Inner Light, has the power to overcome death and loss. The movement of Heart towards God and his Ideas, be they Geometry or the Communion of Saints, is a thing I deem worthy of learning and of teaching.

Pseudo-Related Postscript on A.I. Chatbots

            We were recently treated to predictions about the upcoming seventh Cormoran Strike novel as worked out by A.I. generation—very fun content. This is a subject I have recently been thinking and experimenting with, as the school newspaper at my school sponsored a Socratic discussion about the challenges A.I. poses. In preparation, I spent some time with ChatGPT, trying to see where its limitations were to be found. And so, I took the basic postulates of Geometry and translated them into a coded language known only to me, exchanging nonsense words for the undefined terms. It was able to do some logical derivation correctly. After trying its ability to do proofs I told it that I had been testing it, that there was a particular field of mathematics that I was dealing with. It was unable to guess that these were geometric axioms.

If I had presented these to a human mathematician, this would have been easy to guess. In order for a human to reason well, we need to jump to the Idea, figure out exactly what is being referred to, however encoded, we have to do the Heart-work of relating to a thing, and then we can judge and use the axioms. The computer was incapable of making that leap, as it is merely rational, merely presenting what it has calculated to be the most likely correct response to a prompt. This Tin Man has no Heart, and the current wizards seem a little hiding-behind-the-curtain-y.