Unleashing the monsters of mathematics
The mathematical landscape is undeniably beautiful, but monsters lurk within. I did not appreciate the prevalence of these terrifying creatures until I encountered the Takagi function as an undergraduate. To understand why this function qualifies as a monster, we need two notions: continuity and differentiability.
Informally, a function is continuous if it has no sudden breaks. You can sketch it without removing your pen(cil) from the page. Go ahead and draw a continuous function. I bet it looks smooth. Does every point have a tangent? If so, your function is differentiable.
The Takagi function is monstrous because it is continuous everywhere (no breaks) and differentiable nowhere (no points where there is a tangent). Such a combination scarcely seems imaginable, but the Takagi function is very real.
The formal definition of the Takagi function involves infinite series; we’ll skip straight to the visual representation. Let’s give the function a domain of [0,1], which means it only acts on numbers with values between 0 and 1. We can construct the function iteratively, starting with the saw-tooth:
Nothing monstrous there, although the pointy top is not as easy on the eye as one might hope. Next we take a double saw-tooth of half the height (red) and add it to the previous function (dotted), resulting in our second iteration (black):
You might sense where this is heading. The third iteration comes from adding a four-pointed tooth with half the previous height to the previous function:
And again for the fourth iteration:
One observation screams out: our function is getting pointier with each iteration. Takagi dared to continue the process on to infinity, yielding the function that bears his name. The result:
We witnessed the function gradually lose its smoothness in the first few iterations. By the end of the process, no point remains where the Takagi function has a tangent — nowhere differentiable. And yet there are no sudden breaks to the function — everywhere continuous.
That the Takagi function exists unsettles our worldview of mathematics. The functions we meet in school — polynomials, trigonometric functions, exponentials — are chosen largely because they model real-world phenomena. They all share two properties: they are continuous and they are differentiable.
We catch a fleeting glimpse of where things might go awry when asymptotes arise (blow-ups, such as the one on the y-axis for the function y=1/1-x). The saw-tooth itself pops up when we study the modulus function, although it only manifests at a single point. The thought that a function can have this jaggedness at every point is too wretched for school maths to entertain.
Perhaps that’s the way it should be — the Takagi function may just be an anomaly that deserves no more attention in the curriculum than the abominable snowman. Except…the Takagi function is no outlier.
Imagine taking every continuous function on [0,1] — all infinitely many of them — and picking one at random. How differentiable would you expect it to be? Since most of the functions we have studied are differentiable, and it takes a leap of abstraction to concoct an example like the Takagi function, a good bet is ‘very differentiable.’ So I was stunned to learn that the vast majority of continuous functions are nowhere differentiable. Since we are comparing infinite quantities, the notion of ‘majority’ needs careful definition (formally, the set of everywhere differentiable functions forms a meagre subset of the set of all continuous functions.)
By analogy, all the functions you met in school are a mere drop in the ocean of all continuous functions. The ocean is filled with the Takagi function and its ilk; functions that by rights should not even be allowed to exist. How’s that mathematical landscape looking to you now?
A more familiar example involves rational numbers. We have real numbers (all the ones on a standard number line) and, as we all know, there is a particular type called fractions, which consist of one integer divided by another (the second integer is not allowed to be 0). The rational numbers are just the fractions, and they are called rational because they make sense; easy to describe and work with. What of irrational numbers? Are there numbers that cannot be represented as a fraction? One notable consequence of irrational numbers is that their decimal expansions go on forever and never repeat. Pythagoras and others could not tolerate the thought that such beasts existed, but Euclid put the debate to rest by demonstrating the irrationality of √2.
Things get even stranger with the world’s most revered constant, π, which is not only irrational, but transcendental; the name given to a smaller class of irrationals that never arise as solutions to polynomials with whole number coefficients. That takes a few readings to fully grasp, but the gist is that numbers are either rational (fractions) or irrational (like √2), and some of those irrationals, like π, are even weirder.
All this talk of irrationals and transcendentals may seem superfluous; fractions are the only numbers that matter, aren’t they? In the real world, that may be true to an extent. But in the mathematical world, fractions suffer a similar fate to differentiable functions — they are hardly anywhere to be seen. More specifically, the fractions can be enumerated in a list. So while there are infinitely many of them, they are countable (the smallest type of infinity). Cantor ingeniously demonstrated that the real numbers cannot be enumerated; there are too many of them to systematically list. Put together, this means there are far more irrationals than rationals, i.e. it is the irrationals that make the set of real numbers too large to count. A slightly more involved argument shows that the same is true of transcendental numbers. So, if you plucked a number out at random, it is far more likely to take on the unwieldy shape of π than a neatly packaged fraction.
Let’s take stock: at school, we spend almost all of our time studying objects that barely even feature in the mathematical world. Memorising the digits of π is about the extent to which we grasp at irrational numbers, just as the modulus function only hints at the existence of non-differentiable functions. Irrationals and non-differentiable functions are the Dark Matter of mathematics; they populate up most of its landscape, but are hidden from view and largely ignored in mainstream education.
Mathematics has a shock and awe factor that captivates its devoted subjects. Mathematicians are glued to the terrifying intrigue of strange objects. My doctorate centred on a class of functions called unbounded linear operators — the bounded ones are continuous, and not terrifying enough to make for interesting research. The unbounded operators, on the other hand, reigned untold terror on me over four years. I don’t know what I would have done without them.
The school curriculum need not plunge students into the depths of research to underscore the prevalence of mathematical monsters. It is perfectly understandable that fractions (as a basis for arithmetic and other advanced topics) and differentiable functions (as a basis for modelling real-world phenomena) dominate the curriculum agenda. But we do students a disservice by shielding them from the parts of mathematics that are not smooth or orderly. We are slicing off an essential idea in mathematics — that it does not exist simply to make sense of the real world. Among the most profound characteristics of mathematics is that it is bounded only by logic and reasoning, resulting in some of the most astonishing truths that all students should have the opportunity to play with (and be terrorised by).
If we want students to immerse themselves in the world of mathematics, we must teach them to embrace its monsters.
I am a research mathematician turned educator working at the nexus of mathematics, education and innovation.
If you liked this article you might want to check out my following pieces:
You weren’t bad at maths — you just weren’t looking at it the right way
Mental representations in mathematics