“As it develops, mathematics moves both towards the abstractions of the mind, and also towards the connection with the world.” Thus begins the section on “Pure and Applied Mathematics” in the ToK Course Companion. (p. 357). The nature of the relationship between the abstract nature of mathematics and “the world” is one of several issues examined in a new book on mathematics by University of Wisconsin mathematician (and “maths journalist”) Jordan Ellenberg. In How Not to Be Wrong: The Power of Mathematical Thinking he takes a fresh look at many of the issues underlying mathematics as an “area of knowledge.”
For those who aren’t inclined to read the whole book, many of the most important ideas in the book are easily accessible through reviews, interviews, online versions of Ellenberg’s introduction to the book, and, perhaps most usefully for a TOK class, a lively and engaging interview on a podcast: Inquiring Minds, Episode 39, June 19, 2014 (approximately 18:10 to 46:20).
Although Ellenberg scrutinizes several issues, three seem particularly good for a TOK class.
1. Redefining mathematical thought
In TOK–and, indeed, in this blogsite -we talk about areas of knowledge in order, amongst other things, to consider their relationship to ways of knowing. The fact is, though, we talk about mathematics. According to Ellenberg, though, in thinking that we’re doing so we’re not getting our classification quite right! On the contrary, says the author, the very act of thinking about math is the math!
Is Ellenberg just being provocative and/or redefining what we mean by the word “mathematics”? Consider what he says:
“To say that there’s math and then there’s thinking about math, like asking how we consider all the hypotheses, is the argument justified, blah de blah blah….no that stuff is the math. The symbolic manipulation, and multiplying numbers together [and so on]…that is to math as typing is to writing. You need to be able to do it. It’s part of math, but exactly what I want to fight is the idea that actually thinking is something that takes place separate from the math. It is the mathematics.”
At the very least, his claim is an excellent starting point for a discussion about mathematics’ “relevance” even to those who don’t employ its tools (i.e. those couldn’t differentiate their way out of a paper bag). After all, in Ellenberg’s terms, we are all doing “mathematics” by the very act of rational thinking (more of a feat, alas, amongst many of the general public and many of the politicians who lead us, than calculating the area under a curve!).
2. Mathematics and certainty
When looking at Areas of Knowledge, many ToK teachers like to treat mathematics first and the arts last. The idea, of course, is that it is easiest to look at issues of certainty, shared knowledge, language, reason, even intuition and imagination in an area of knowledge where the issues are comparatively clear cut, the degree of “objectivity” and certainty highest.
While not exactly contradicting this perception of mathematics, Ellenberg seems to go out of his way to stress its fuzzier side. Consider this part of the interview:
Chris Mooney (interviewer): What you’re saying is we need to be more nuanced and we need to be more able to process uncertainty in how we think about complex topics involving numbers….
Ellenberg: There’s a stereotype that mathematicians are exactly the opposite of people who care about nuance. I think most people if they were to imagine the psychological makeup of the mathematician. . . would say, “Oh, that’s somebody who’s very black and white, who thinks in yes or no, very precise, down to the tenth decimal place terms…”. The mathematical way of thinking…can be like that, but it’s not the only thing it is. We try to valorize [give value to]uncertainty too, and study it, and bring it in[to] our orbit.
Chris Mooney: Calculation sounds very precise and it sounds very rule based and can only lead to one outcome and it satisfies those who want a very clear answer and so that sounds very direct, but what you’re saying is that it’s more about knowing how to analyze a problem, which leads to all different kinds of analysis and knowing which one to apply. …It’s very difficult to do and it’s very nuanced to do.
Ellenberg:. And that’s what makes mathematics an art and not just an algorithm that we mindlessly apply.
In fact, Ellenberg repeatedly emphasizes that “common sense” can–and in some contexts should–trump calculation. In this a case, we might ask whether he is valuing “gut feeling”, intuition to replace objective analysis. If that is true, then is he endorsing the “woo”, non-science-based medicine, “hand waving” and all the other characteristics of those who feel they can ignore scientific results? Should Jenny McCarthy’s “Mommy Instinct” that vaccinations cause autism trump the numerical data?
No. And this is where it is illuminating to look at a third issue he considers:
3. The role played by mathematics in the current crisis in medical science and social science
And the crisis? According to Ellenberg this crisis lies in interpreting data (numbers!) that result from the countless studies that crowd current research.
“I don’t think it’s overstating to say that there’s a kind of crisis in medical science and social science that people are truly feeling unsure about how well the standard techniques for saying does this drug work or does it not work; does this psychological intervention work or does it not work? We have a standard tool box for making these judgments that we’ve had for 70 years and the cracks are starting to show. “
Computers, it seems, are part of the problem:
“You can test 10,000 different things just at the touch of a key stroke”–and, “just by chance, 1 in 20 of them is going to pass the test; you’re going to have a lot of spurious results that give the appearance of being meaningful.”
And this is where our previous point about common sense comes into play. No–Ellenberg is not, after all, saying that irrationality should trump rationality. Quite the reverse: he is warning that mathematical calculations can be misleading–and that, as a result, we need to be meticulous in how we interpret them.
“If you see two studies which give you the exact same numerical data about the effectiveness of some intervention but one of them is about a cancer drug that is sort of in the same family as other drugs and the other is about waving a swatch of branches over the patient, I think it’s totally okay to end up with different opinions about whether the intervention works, even though in some sense the evidence you get is numerically exactly the same. You’re allowed to take into account what you think is crazy and what you think is reasonable…..”
Mathematics and Engagement
Even more powerfully, the interview ends with a reassertion of a principle that should gladden the heart of any student or teacher who sees TOK as a vehicle for separating the cognitive wheat from the chaff, as honing skills for engaging responsibly in “the world”.
In order to emphasize this notion, Ellenberg describes an imaginary “philosophical battle” he creates in his book, between two historical figures. One is former U.S. president Theodore Roosevelt, a gung ho Man of Action, who espouses the principle: “just do it”. The other is the poet John Ashbery, his view summed up in his poem “Soonest Mended” and his seminal evocation of “this action, this not being sure….” (http://www.poetryfoundation.org/poem/177260).
Who wins the debate? Ellenberg cheerfully admits that he had his “thumb on the scale”, and tips the balance in favour of Ashbery.
“But exactly this difference between the Rooseveltian point of view that all of this mathematical stuff is okay but really it’s just book learning and the real stuff is the person who is just willing to charge ahead without thinking so carefully….to say that not being sure, is saying let’s consider, is saying let’s look at what can we prove and what can’t we prove, what have we thought about and what haven’t we thought about–that is a kind of action. It’s the kind of action that mathematicians carry out…. that is the kind of action that has moved the world just as much as armies and planes.”
What better epigraph for a TOK course than “Let’s look at what we can prove and what can’t we prove, what have we thought about and what haven’t we thought about–this is a kind of action”? What better endorsement of thinking than to claim it is “the kind of action that has moved the world just as much as armies and planes.”?
In his introduction to his book, Ellenberg makes a similar and in some ways more valuable point, though with not quite the same orchestral cadence:
The mathematical ideas we want to address are ones that can be engaged with directly and profitably, whether your mathematical training stops at pre-algebra or extends much further. And they are not “mere facts,” like a simple statement of arithmetic — they are principles, whose application extends far beyond the things you’re used to thinking of as mathematical. They are the go-to tools on the utility belt, and used properly they will help you not be wrong.
Mathematicians, take heart! If we return to our opening assertion from the Course Companion, we find in Jordan Ellenberg a wonderfully resonant spokesman for the “relevance” of mathematics as an Area of Knowledge: “mathematics moves both towards the abstractions of the mind, and also towards the connection with the world.”
Eileen Dombrowski, Lena Rotenberg, Mimi Bick. Theory of Knowledge Course Companion, 2013 edition. Oxford University Press, 2013. https://global.oup.com/education/product/9780199129737?region=international
podcast Inquiring Minds, Episode 39, June 19, 2014 (approximately 18:10 to 46:20).
Other interviews and reviews