by Jeroen Bouterse
It’s my favorite topic of the year, I tell the kids, before scanning my conscience for signs that I have just lied to them. No, this feels about right, and in any case, they didn’t need my reassurance: after a dry unit about ratios, at my mention of the word “probability” I can practically hear the sound of neurons firing. Whether they know of any contexts in which chance turns up? Boy, do they. My twelve-year old students inform me about dice, cards, spinners, lotteries, casinos, about your chances of survival in Squid Game, about unearthing rare minerals in Minecraft, and about whether the stuff you want to buy is actually in stock because sometimes they run out and there’s no telling in advance when. Also the weather.
I love introducing them to the math of probability precisely because it is a topic that they have already thought about so much, but that I know many of them have incoherent intuitions about. Not that I don’t, of course, but I am a little bit ahead of them and I know from experience what I am working with. Somebody will opine that the chance that it’s going to rain next month is “fifty-fifty” because it either will or it won’t. Somebody will say that there is a 1/7 chance that someone’s favorite day of the week is Thursday. Somebody will reason that since the highest sum you can roll with two D6-dice is a 12, the chance you roll a 10 is 1 in 12, which…
Oh wait, is it? Anyway, you get the point: the process feels properly Socratic, taking slightly muddled concepts that students already feel strongly about, and providing the right nudges to make them reconsider some of those concepts and make others click together. It is a significant source of satisfaction every year to see just how fast ‘the group’ moves from guessing that a sequence of three coin flips has six possible outcomes (unless it can land on its side, which they also unerringly point out), to calculating the number of four-digit pin codes where all the digits have to be different.
It’s also becoming a little predictable, now that I teach this level for the third time. Though I was surprised to learn that this cohort of young teenagers had already binged Squid Game, I don’t generally expect them to say anything unexpected about mathematics. Until, last week, we were classifying certain things according to likelihood. I always include an apodictic claim like “1 + 1 = 2” in the mix, and students usually say that its probability is, indeed, 1. This time, I wondered out loud why we are so sure of this.
I expected them to say something like: it just can’t be different, what else would 1 + 1 be? Or, perhaps, to read this as a prompt to be skeptical, and doubt out loud if we can actually ever be a 100% certain about anything. Instead, a student raised his hand and said: “Because we created it.” Asked to elaborate, he explained that people made math, so of course we know how it works.
It’s the first time I heard this explanation for our confidence in mathematical truths inside the walls of a secondary school. I’m curious now whether it is more widespread than I might have thought; whether I’m wrong to assume my students to be natural Cartesians, perceiving the objects of mathematical knowledge to have been put in place by some higher authority, being either doubtful or confident about their own capacity to attain accurate knowledge about them, but experiencing themselves as having no say in what the objects are. Perhaps this is not always how they feel after all.
The view that we understand what we create was famously put forward by the early modern thinker Giambattista Vico (with whom this website has an indirect but special relationship through Finnegans Wake – see Bill Benzon’s essay earlier this year). In his book On the most ancient wisdom of the Italians, Vico claimed that for ancient Latins, what was true (verum) was interchangeable with what was made (factum). To him, this suggested a metaphysical principle: creating something and understanding it go together, and in some deep sense amount to the same thing. This implies that we can never have perfect knowledge of nature, whose creator is outside of us: only God truly defines, makes, and knows things. In Vico’s eyes, Descartes’ attempt to model knowledge of nature on mathematics was therefore a bit of a category mistake: Descartes rightly thought that mathematical knowledge was certain, but he mistakenly decided that its certainty rested on its method, and was therefore transferable to other domains of life by approaching these with the same tools. Vico mocked what mathematics had done to some philosophers, who thought that they could settle anything by applying definitions, postulates, axioms and conclusions to it.
According to Vico, the causes for the certainty of mathematics were altogether different. The reason we have perfect knowledge of measure and number is that they are our own inventions. We create “from no substrate the point, the line, and the plane, as if from nothing and as if they were things.” It stands to reason, then, that our understanding of it has nothing to do with our understanding of nature. To Vico, math is rather our most coherent form of fiction.
To the extent that he believes math manipulates objects that exist only as arbitrary definitions, having no meaning or relation to the world, Vico might today be classified as a formalist, his perspective being equally vulnerable to Russell’s objection that it comes with “the disadvantage of failing to explain the application of numbers in counting”. However, it seems more accurate to see his philosophy of mathematics as sui generis, founded on a metaphysical association of creation and understanding that is probably less viable now, in a post-theistic world where there is no God in whom the two come together. In any case, both professional mathematics and philosophy of mathematics have moved on happily without Vico. High-school mathematics, however, has its own phenomenology, and I wonder what contemporary classroom practices and experiences might ground a Vician perspective on it.
Vico believed that math, after being safely isolated from other knowledge, should be taught in the spirit of creativity: not by analysis but by composition, “so that we do not just discover the truth, but make it.” Parallels between him and 20th-century constructivist pedagogy (especially Piaget) have often been drawn, and some constructivists explicitly enlist Vico to argue that knowing and learning are creative activities. On the other hand, a classroom where students are engaging in hands-on activities, learning to use mathematical tools to navigate the real world, in many ways runs counter to Vico’s insistence to let math be fiction and life be real. He fought against math creeping into practical matters, which were the proper domain of philosophy:
“Philosophers have had no function in the world except to make the nations among whom they flourished affluent, skillful, able, acute, and reflective, so that men became open-minded, quick, magnanimous, imaginative, and prudent in their active life. Mathematics, on the other hand, has promoted in men a sense of order and developed a sense of beauty, fitness, and consistency. When the community of letters was first established, philosophers contented themselves with probabilities and left it to the mathematicians to treat truth.”
He might have been horrified to discover that the uncertainties, ambiguities and negotiations of everyday life had become subject to quantification, and that we even encourage students to think of them in this way.
Perhaps it is a more traditional aspect to learning math that does the trick of establishing a sense of authorship; something closer to the mathematical experience itself (though one of them was an outstanding mathematician and the other was not) that Descartes and Vico were trying to establish the philosophical meaning of. It is the sense that we can actually be there to witness a mathematical fact being established. We don’t have to take the textbook’s word for anything. We need only ourselves – our imagination, and some clear thinking. Everything outside of us – the teacher, the compass, the textbook and the whiteboard – exists to assist these things; the moment that something clicks is something private and intimate, something that is happening inside our minds without feeling like it is just happening to us or being done to us. It feels like we are in control.
In his Mathematician’s Lament, Paul Lockhart wants to establish math as a creative art, where “things are what you want them to be”, but where your designs, once defined into existence, might talk back, and do things you don’t like. It’s an expansion of the divine creation analogy that Vico could have appreciated. Lockhart then describes how we may come to see that an acute triangle takes up half of the rectangular box that it just fits in, by dividing it into two right-angled triangles. He reflects:
“Now where did this idea of mine come from? How did I know to draw that line? How does a painter know where to put his brush? Inspiration, experience, trial and error, dumb luck. That’s the art of it, creating these beautiful little poems of thought, these sonnets of pure reason. There is something so wonderfully transformational about this art form. The relationship between the triangle and the rectangle was a mystery, and then that one little line made it obvious. I couldn’t see, and then all of a sudden I could. Somehow, I was able to create a profound simple beauty out of nothing, and change myself in the process. Isn’t that what art is all about?”
Lockhart’s general argument is that school, in treating the subject as a handmaiden to science, strangles the art and poetry out of mathematics, and that without those it stops being math altogether. As a secondary school teacher, I am both acutely aware of the danger and a little offended by the suggestion. I believe that I am empowering my students, rather than suffocating them. I know that guided insights can feel like your own. And I suspect that the relatively orderly, responsible, careful step-by-step approach of a school curriculum may actually encourage a sense of ownership, if it doesn’t degenerate into assembly-line style practices. If things go more or less right, we are building an edifice in such a way that, though the teacher may have a fuller grasp of the basic blueprints, there are no bricks being laid behind students’ backs, without them present. They are included in the process, in such a way that they can speak about it in the first person plural. If math is an art, as Lockhart argues, it is not just a matter of private genius, of inexplicable flashes of inspiration; it is also a tradition, the slow building of a structure with which they can come to identify.
I mentioned that in practice, the ways in which students navigate this structure, or the places where they seek to build annexes and extensions to it, often feel predictable after a few years. Not that they don’t ask insightful and imaginative questions or make apt and piercing observations, but their individual forays always proceed from what is still a relatively small encampment, into parts of the landscape that are already familiar to me. It’s the opposite of teaching history, which I used to do and where any follow-up question that a child could think of could stump me, and where in general any grasp I imagine to have on its structure and meaning feels tenuous and vulnerable – even though Vico would insist that human history ought to be understandable too, being after all made by people.
I don’t know how special mathematical knowledge is, metaphysically speaking, but I know that school math is special phenomenologically speaking, and I believe this is in part because it manifests itself as relatively manageable and self-contained. I don’t doubt that we could arrange the subject in such a way that it becomes porous, open to the real world, with the same vanishingly small volume-to-surface ratio as any other subject (such as history), still teachable but dazzling. I’m glad we didn’t.
Or mostly didn’t; as a counterpoint and a warning, treatment of probability distributions and statistical tests in high school is, as far as I have seen, an alienating exercise in button-pressing in which everybody’s worst fears are realized – where every sense of creativity, understanding and authorship is abandoned, and math remains only as a method, a tool for extending certainty into the realm of micro-economics; where thinking about probability has suddenly turned into deciding whether the manufacturing procedure for nuts and bolts should be adapted, based on this distribution of measurements of 100 nuts and 100 bolts.
With all due respect to everybody who has aligned their own interest in puzzle-solving with technological and industrial progress, it would be heartbreaking if school mathematics became merely a matter of (in Lockhart’s words) “preparing tomorrow’s workforce today”. This is why probability and statistics are my least favorite topic in the curriculum; it’s where math stops being centered around co-creating an abstract world of our own, and starts being a tool in production processes put in place by others.
Vico would have had a thing or two to say about that, too; I have barely scratched the surface of the ways in which his criticism of methodical and unimaginative thinking, his insistence on the natural relation between creators and the things they make, resonate with later Marxist critiques of capitalism. I have not even mentioned the pronounced motif of class struggle in his philosophy of history. Luckily we don’t need to rely on Vico for all of that; for some of it, we can also watch Squid Game.
 Giambattista Vico, On the most ancient wisdom of the Italians unearthed from the origins of the Latin language (1710). Transl. L.M. Palmer (Cornell University Press 1988) p. 45.
 Ibid., 51.
 Bertrand Russell, My philosophical development, ch 10.
 Vico, On the most ancient wisdom of the Italians, 104.
 Ibid., 182 (second response to his critics).
 I was almost going to pull a Heideggerian etymological trick on you, and point out that in Dutch and German, the word for being there to witness something (‘meemaken/mitmachen’) literally translates to ‘co-creating’.
 https://www.maa.org/external_archive/devlin/LockhartsLament.pdf, page 4.
 Ibid., p. 5.
 Ibid., p. 12.