I have been told that fear of math is irrational. Perhaps this is so, but it seems very rational to acknowledge one’s algebraic limitations or to express a tested dislike of geometry.
I have also been told that math is good for me. To this I reply that Brussels Sprouts are good for me as well, but no matter how many ways I have tried to eat them (my friends all swear I haven’t yet had them the right way) I always find them completely unpalatable. No amount of lemon juice, olive oil, or salt can make the supposed interior goodness of Brussels Sprouts emerge in their smell or flavor.
No amount of tricks or techniques have ever helped me engage with or enjoy math. I needed something more fundamental – I needed math to change, and this is why Euclid has so surprised me. Euclid changed math for me. He changed my understanding of math. Once easily my worst subject (I managed a D in junior year Trigonometry), I have, in Elements, discovered a poetry and ontology in forms and numbers I did not or could not see before. “Times tables” still give me nightmares, but when I think of Euclid, I think of a great story I want to finish reading. Euclid’s Elements has, beyond the numbers, a literary logic. I think this logic can perhaps best be described by the word “plot.”
What is a plot? In his Poetics, Aristotle discusses plot at length and names it a preeminent aspect of literature – more important than character, exposition, or dialogue. Aristotle tells us that the various components of a plot “must causally relate to one another as being either necessary or probable,” and plot must, by Aristotle’s standard, have a defined beginning, middle, and end.
This is far from the mathematics I was taught in elementary school and failed at in high school. The definitions and propositions of Euclid “causally relate to one another” as the events in a novel. There is exposition, progress and “character development” in Euclid (This is especially evident in the grand finale of Book 1 when the Pythagorean Theorem suddenly emerges out of everything that came before it.) Rather than simply being another piece of information, each concept in Elements takes from and leads to the next, like steps in a dance.
In any story – fiction or non-fiction – the reader is given an exposition: a setting for the narrative which includes time, place, and surrounding events. In a fantasy story this exposition also includes elements that are imaginary, but they nevertheless must be internally consistent. The logic of the fairytale is impossible outside fairyland, but operates rigidly within it.
Elements has this kind of internal integrity. The definitions and axioms define the logic we will use, and the common notions are the givens, the “setting” of Euclid’s story. We never leave this setting or break these laws.
However, there is even more to see here if we carefully consider the original word translated into English as “plot”. Plot translates the muti-functional Greek word mythos – obviously a more rich and nuanced word than “plot” and referring to more than a mere series of events. Mythos is a system of beliefs, values and attitudes of a culture, expressed and taught through myths and the arts. Using this full definition, we could now ask even more probing questions. From “does Elements have a plot?” we could now make the following twofold inquiry:
- Do the parts of Elements causally relate to one another leading to a definite end?
- And does Elements artistically express a set of values or beliefs?
I began to discover an answer to the first question when I saw that the steps of Elements depend on and lead to each other. Plot in that sense is not only clear in Euclid, he might even provide the best example.
Understanding plot as mythos opens a new window onto the whole landscape of math. If Elements has a mythos, shouldn’t we then consider it to be a work of art and not merely (in the understanably repugnant sense) a mathematics textbook? And, how would we determine if it is a work of art? What does a work of art do?
Among many possible effects of art, we can generally affirm that a genuine work of art educates, initiates dialogue and contemplation, and inspires other art.
Has Euclid, in fact, fulfilled these three qualifications? My unexpected experience is that he has, and this broadens Euclid and makes him almost universal – with applications for all the classic liberal arts, including (and I would think especially) Logic, Rhetoric and Grammar. I think we find Euclid at the heart of the visual arts as well. Greek sculpture and painting, along with their descendants in later Roman and Renaissance art, are famously geometric.
But I also want to know what kind of art is Elements? Is an analogy of Euclid with literature (e.g., a novel) too much of an imaginative jump? I’ve already hinted that Elements is a story by seeing in it the suggestion of a plot. Perhaps, comparing Euclid to literature has more to do with how we think rather than what we are asked to think about. Are the structure of Elements and the structure of a typical story necessarily similar because they are created by and being assimilated by similar minds? Does the mind often, if not always, move from the simple to the complex by a series of steps, and all the better if the steps are harmonious and interrelated?
No math textbook has ever invited me to ask these kinds of questions before.
After a little research, I was not surprised to find that Euclid actually has inspired literary as well as geometric art. Two examples are well-known, a third less so. First, Edward Abbot’s Flatland – a political commentary based in a Euclidian universe populated by sentient planes and lines; second, the simple magic of Norton Juster’s The Dot and the Line: A Romance in Lower Mathematics – which has delighted me for years with its serendipitous-yet-subtle portrayal of human behavior.
Finally, Euclid and His Modern Rivals, by none other than Lewis Carroll of Alice in Wonderland fame.
In Carroll’s play, Euclid himself returns to defend Elements against inferior texts. Carroll, known for twisting and making light of the customs of Victorian England, knew that he could only do this twisting from within a stable and literate culture.
The wonder of Wonderland is that it holds a curved mirror up to the real world. But without a good, solid and systematic world to begin with, Alice would have had no way to see the strangeness of the rabbit hole. Euclid, Carroll seems to be saying, is the solid foundation we must begin with, even if our search leads finally to other perspectives.
My appreciation for Euclid leads me to consider that there seem to be two current responses to Euclid. On the one hand, respectfully, some disagree with him. The Non-Euclidian Geometries that emerged in the eighteenth century began with careful study of Euclid and progressed to possible alternatives. This is a very creative approach, another kind of inspiration that seems to give Euclid proper deference and makes him as much a pioneer and philosopher as a mathematician.
On the other hand, though, we can forget him. This seems to be the choice of current educational models. When I was in high school, “Euclidian” was a meaningless adjective attached to the dreaded word “geometry”. But after enjoying him so much as an adult, I’m not surprised that when Euclid is removed from school curricula, students lose both the wonder of his mythos and the amazing flights of intellect (or fantasy!) he might inspire.
Without Euclid, it’s no wonder we’re left with aimless art – paintings and sculptures with no transcendent geometry. Without Euclid, students lose a necessary connection between abstract logic and the solid, physical space in which they live. Without Euclid, we only have artless textbooks and a dis-integrated mathematics with no genuine mythos – no “literary” connection to the world.
The math of Euclid seems very, well, human. If this is so, then fear of math (as math is presented in current schools) might not be so irrational after all – it might be a genuine aversion to inhumanity.