Math Parables 3: The Fable of the Fearsome √2
Allow me to tell you The Fable of the Fearsome √2, a proud irrational number with an unsettlingly sinister story behind it.
Feel free to share this story with the little children whom you tuck in. Please note that this is, like any respectable fairytale, the stuff of legend. Furthermore, as is a storyteller’s prerogative, I’ve taken a few minor liberties—mostly with respect to vocabulary—in retelling the legend.
Once upon a time there lived a philosopher named Pythagoras. He was a very clever philosopher, so much so that he’s usually remembered as a mathematician.
He’s famous—even today, though he was born long, long ago in the sixth century BC—for giving us a useful and important spell . . . excuse me, tool . . . in mathematics called the Pythagorean Theorem. I’m sure you’ve heard of it: It's a fantastic method for doing amazing things with triangles. Basically, the Pythagorean Theorem tells us that for any self-respecting and well-trained right triangle, the sum of the sides squared equals the hypotenuse squared.
Pythagoras was not only an important and famous man, he was rather popular. As a matter of fact, he had a following of people who believed in what he taught and liked to hang around with him. As a result, he founded his own school of thought, based upon the idea that all “reality” is mathematical and that numbers not only have abstract, mystical significance and unique attributes, but that they are “rational.”1
In other words, for Pythagoras numbers were to be reasonable creatures, and make sense to people (in mathematical terms, this meant that they had to be decent whole numbers or fractions, for goodness’ sake). It was extremely important to Pythagoras, and to those who followed him (called Pythagoreans), that people be clear in their thinking, and that reality—as articulated through abstract number—was understandable via human reason.
All went well for a while, with the Pythagoreans having a wonderful time studying and exploring the well-structured and well-behaved world governed by rational numbers. But one day this happy state of affairs was destroyed by the discovery made by a (no doubt well-intending) Pythagorean named Hippasus.
He made a horrible discovery. This terrible event occurred while a bunch of Pythagoreans were out sailing at sea, probably engaging in mathematical musings. The unfortunate Hippasus was—or so some say—simply working through that famous Pythagorean Theorem, using the idea that if you look at the triangle in the simplest way, treating the length of each of its sides as equal to one unit each, then suddenly you encounter something not just unsettling, but in a way horrifically nonsensical (especially for the Pythagoreans who put so much stock in being reasonable).
When you do this, you discover a measurement for the hypotenuse that doesn’t make sense. The word often used is “incommensurable,”2 meaning it doesn’t participate in the same standard of reasonableness that sensible numbers take part in; it isn’t just an outlier, it’s an illegitimate, ill-behaved impostor!
The poor number in this case would be *gasp* “irrational”! And alas, that fearsome number was √2. It is neither a whole number nor a fraction. In other words, it can’t be written as the relationship, or ratio, between two integers.
Although the ancient Pythagoreans wouldn’t have looked at it this way (they didn’t have a concept of decimal numbers), we now view an irrational number as a decimal that is non-repeating and goes on infinitely, and can only be represented by a messy description that looks like this:
√2 is approximately equal to 99 divided by 70, or about 1.4142857 . . .
If you find this a somewhat difficult notion, and you can also agree that this is an unsightly and upsetting thing to look at even for us (especially if we’re expected to use it to arrive at an answer without a calculator), just imagine how the Pythagoreans felt!
Well, the fearsome √2 threatened to turn everything the Pythagoreans believed in upside down. The whole idea of a neatly pigeon-holed and reasonable universe was being undermined. So, legend has it that this upset the Pythagoreans so much that they became enraged and tossed poor Hippasus overboard.
And he drowned. The End.
Now, either poor Hippasus was too clever for his own good, or he followed the lead of mathematics to its inevitable conclusions, which shockingly did not uphold Pythagoras’s lofty, human-centered ideas of sensibleness, or—as I suspect—both. (Don’t forget Gödel’s Incompleteness Theorem, which points to the ground of mathematics as “extra-logical.”)
But whatever the case, here you have many elements of an excellent parable: a utopian world, populated by sensible men, governed by reasonable mathematical entities, suddenly threatened by some kind of malicious being of clearly alien origin. To protect the world as the Pythagoreans knew it, this terrible darkness had to be squelched! Isn’t it unfortunate that in the process, a murder occurred.
Or was it an execution?
Hippasus certainly wouldn’t be the only person ever killed for expressing an inconvenient truth . . . right? I’m fairly certain we all can think of several other prominent examples of this in literature, philosophy, theology, and history.
Was his death justified or not? Did poor Hippasus pay the price for opening the portal to some kind of major revelation, or did he let loose forces from some “dark side” (a sacrifice then being required of the one for the benefit of the many)? Or was there something inherently tyrannical about Pythagoras, who insisted on censorship of the news? And did this tyranny therefore extend to the human construct and articulation of “rationality”?
One could think about this story a lot, and come to various conclusions, some of which would be contradictory. For example, one might ask such questions as: “What does the existence of incommensurability tell us about reality and/or our ability to grasp it?”3 “Is incommensurability malicious or is it an invitation to keep chasing after the truth (as Kepler did, when for years he sought the answer to a mere eight-minute discrepancy in the astronomical appearances and ended up discovering that orbits were elliptical rather than circular)?” “Are the thoughts of men actually reasonable, or are we always just approximating 'true' reason (whatever that is)?” “Why did Hipassus suffer the ultimate punishment?” and “What is justice and who has the right to carry it out?” “How much truth can people bear?” (Remember Arnauld’s point that teaching math is itself a way to develop the precision of thinking that allows the asking and answering of such questions in a productive way.)
The surfacing of such questions—once we have been given permission to ask them in math and science—testifies to the power of parable and story as a teaching tool. Indeed, that this can occur within these fields of thought demonstrates that these areas are not simply inviolate bastions of “hard facts,” but can be as great a spur to asking and answering the profoundest questions about life and the universe as works of literary, visual, or musical art. In my view, this is one of the greatest reasons mathematics was one of the components of the “Liberal Arts.”
So, next time someone asks you why, as a classical educator, you insist on teaching your students fairytales, poetry, and literature, instead of primarily focusing on “STEM subjects”—with the assumption behind the question being “What good will that do?” and “What can students use that for?”—ask them, in turn, to explain how mathematics, with its plethora of imaginary, irrational, hyperreal, and surreal “numbers” is any “better.”
If they answer along the lines that STEM subjects are “better” because students make more money in those areas, or that it is solely through such subjects that our society can “get ahead,” then you have an excellent opportunity to engage in a conversation about the telos of education itself—a discussion every classical educator should delight in having. (If you’d like to explore an example of such conversation, I highly recommend Peter Kreeft’s book, The Best Things in Life.)
Please don’t mistake my lighthearted tone in these articles, or my criticisms of some perspectives shared by vocal STEM advocates, for a dismissal of STEM subjects. There is great value in studying STEM subjects (that is precisely my point), but it’s not because they are less mysterious, ambiguous, or creative than what we call the “arts”; it’s because they are fully the equal partners of those arts in all those ways.
When you study math with your students, invite them not to be tempted to just “use” it, but encourage them to contemplate its language, the stories it tells, and the strange creatures like π (who masquerades as an old familiar friend but who is, don’t forget, “transcendental”) that are to be found within it. Don’t gloss over the truth and let the standard-fare textbooks sweep those fables under the rug!
Seek and find better textbooks and other resources (such as The Joy of Mathematics by Theoni Pappas) that will help you pull out the strange entities that populate the world of math. Then gaze at them, talk about them, track their strange behaviors and their histories, and consider their implications. For the telos of learning mathematics is, as with any realm of human creativity, to be given the capacity and skills to attend to the transcendent such that we perceive it when it crosses the chasms of information, knowledge, and understanding to reach us. All true classical educators should seek that telos. And classical Christian educators know Who the Transcendent is.4
If you do this, you’ll help students see math through new eyes. You’ll achieve several things, among them showing your students the wonder of it; reducing their fear and perhaps even loathing of math; and assisting them in perceiving how math is creative, interesting, and downright fun—even if they never routinely “get” the “right” answers in this world of hyper-sur-reality and never become engineers or physicists. You can facilitate an understanding about how math is indeed relevant to their twenty-first-century existence. Anything which, parable-like, draws us closer to contemplating the deepest questions, and lifts us to the brink of Transcendence, is—and always will be—relevant in the most significant way.5
- Root: ratio => a rational number can be written as the ratio of two integers. The Greeks, however, only understand this ratio in the context of natural or counting numbers.
- An incommensurable number literally means “having no common standard of measurement.” That is, there is no common divisor between it and the other number with which it is being compared.
- “Scientific inquiry rides on a thruway paved mostly by irrational numbers.” Ernest Zebrowski, A History of the Circle (1999), p. 11.
- Mathematics, as an independent system, does not cross the chasm to the Transcendent. Logic, as an independent system, does not cross the chasm to the Transcendent. The “five ways” of Aquinas, as an independent system of rational proofs, do not cross the chasm to the Transcendent. Natural theology, as an independent system, does not cross the chasm to the Transcendent. The chasm is crossed by the Transcendent coming to us, indeed, coming in us in a very personal way. (Read the Greek in John 1:14; the Greek word ἐν should be translated “in,” not “among.” Cf. the Greek text of Galatians 1:15-16.) The Incarnation, the Logos of God who is the Son of the Father, is the chasm crossed. The Incarnation is the Creator redeeming the brokenness of mankind by reorienting our mind, reframing it, so that we think about Creation rightly, so that we think about mathematics rightly, so that we think about logic rightly, so that we think about the “five ways” rightly, so that we think about natural theology rightly.
- Mathematics does take us to the brink by its account of the macro- and micro-cosmos. At these two points, scientists cry, “Why this singularity?” “What is the ground of the intelligibility of the cosmos and our mathematical report on the cosmos?” At these points, the Gospel of the singularity of the Incarnation breathes redeeming life into these questions. The “Why” and “What” are answered by “Who.” It is in the answer to these root questions that the Gospel not only speaks to individuals, but to culture, healing the rift in it caused by the radical secularization of the past 200-300 years, the heritage of the Enlightenment.
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