My wife’s social science PhD is as confusing as my maths dissertation — do I have the high ground?

Mathematics as a language

My wife submitted her PhD this week. I’m thrilled for her, I really am. Her years of toil are neatly packaged in a 300-page dissertation on…I’m not entirely sure, actually. Something about authoritarian regimes and the psychology of revolutions, that much I know. But despite reading and re-reading the abstract, I honestly cannot summarise her thesis in plain words.

The thesis is written in English, allegedly, although it includes terms like “self-understandings” — I didn’t even know that was a thing, much less that it could be pluralised. Her sentence structure reads like it is designed to mystify the layperson.

My wife is keen to remind me of two things. First, she was pursuing an academic project, not a New York Times bestseller. This is the style expected of the social sciences. Second, my own doctoral thesis in mathematics was hardly an accessible read — she could barely scrape past the acknowledgements. Indeed, I struggled to make head or tail of it years after the fact, owing partly to its intense style. At the time, I justified it as convention, but later recognised that my arguments could be elucidated with more deliberate use of language.

As you have no doubt gleaned, dinner talk in the Mubeen household can get intense (and yes, a tad tedious). But I am left wondering if I can reclaim the moral high ground. As a mathematician, am I exempt from the same standards of clarity that I demand of social scientists? Is the language and style of mathematics necessarily complex, or could we work harder on our communication?

My wife’s main defense for writing the way she does is that she is communicating abstract ideas that can only be done justice with a formal style. Her style and structure creates meaning in a way that simple, clear language can not achieve. But she also accepts that the social sciences has gone into overdrive with formal language, to the point where keeping it simple is scorned upon. Formal expression is now a firmly embedded convention and it takes a brave soul to relax their language.

Is the same true of mathematics? In one sense, mathematics is an invented language that has been carefully crafted to make sense of the world. Recall Galileo:

“[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.”

We converse in mathematics using selected representations that compress vast amounts of knowledge into succinct forms. My wife counted every one of her 297 pages. With the deliberate placement of 3 digits, she has a good handle on the size of her thesis (she is also keen to remind me that it is more than double the length of my own effort — as if size matters.) Almost unconsciously, she has adopted the conventions of our decimal system — grouping her pages into hundreds, tens and ones. It would take a mischeivous mind to number their pages any differently. All of us, even social scientists, draw on the lean representations of mathematics to navigate our everyday lives.

But the language of mathematics can also confound. Just ask the reams of mathematicians who have struggled to grasp Shinichi Mochizuki’s 500-page proof of the ABC conjecture, which seeks to establish conditions under which the equation a + b = c holds. Written in 2012, it is not just the length of Mochizuki’s paper that has tormented the mathematical community, but also its language. Mochizuki has invented a language within a language, creating new notation to build up towards his argument. Even by the standards of professional mathematics, it is proving to be a chore — so much so that many mathematicians are reluctant to devote their time to getting to grips with a new language that may or may not resolve the conjecture. Some respite was offered recently when a colleague of Mochizuki presented a 300-page human-friendly summary of the proof. In fact, it may not be humans who grapple with these highly abstract, highly complex areas of maths — that may become the preserve of computers. For now at least, the humans are in the game, but are desperately grasping to understand the rules.

Looks easy, but wait until you see the ‘proof’

Most of us are safe from the sanctuary of these complex branches of maths, and language should never be a reason for struggle at school. The physicist Richard Feynman actively campaigned against maths and science textbooks that made these subjects more difficult than necessary by employing redundant vocabulary. That was half a century ago, yet school maths sounds much the same today.

Students are often expected to absorb mouthfuls of terms and definitions that distract from the underlying concepts. When learning division, they encounter remainders, which is useful and intuitive enough to warrant a place in their memory. But they are also forced to learn the fine details of dividends and quotients, neither of which is vital to their understanding.

Later, students will encounter quadratic equations, which are daunting my name alone. But ‘quadratics’ comes from the Latin word, ‘quadrao’, which means ‘to make square’ —how appropriate. Etymology can be profoundly insightful in explaining where these otherwise archaic sounding terms come from. And when solving quadratics, we might be asked to ‘complete the square’, a straightforward sounding term that utterly perplexes students when the algorithm is presented in purely abstract form. Brett Berry’s piece illuminates both the method and the name. We may be stuck with mathematical conventions, but when we dig beneath the surface, we usually find meaning.

Social science writing seems caught in the trappings of academia; it would be a shame if maths also degraded into a confused mesh of unexplained vocabulary. If mathematics really is the language of the universe, then we may take more care with the words and representations we choose to communicate mathematical ideas. Maths was invented to bring clarity and succinctness to our thinking. While research mathematics may have to succumb to alien notation, school maths should sound as simple as ABC, and not as complex as its purported proof. I don’t know, maybe maths educators just need more “self-understanding” to appreciate their responsibility to students.