Mastery of Mathematics for Accomplished Non-Mathematicians
How high functioning in other intellectual disciplines can be an obstacle to learning mathematics
In a famous and controversial public lecture in 1959, the scientist and author C. P. Snow lamented the increasing polarization of western intellectual activity into two distinct “cultures”: science, engineering and mathematics on the one hand, and what he called literary humanities on the other. Snow argued that the phenomenon was particularly pernicious, because the ranks of the political class were increasingly drawn from the latter, leading to an under-representation of scientific insight in the ruling elite — a predicament cast into vivid contemporary relief in the light of diverse national responses to the unfolding of the COVID-19 pandemic.
Snow is dismissive of those who ridicule the supposed illiteracy of scientists, but who themselves are unable to describe, say, the second law of thermodynamics. But in this article I will argue that high-functioning in the literary humanities (as well as the social sciences) can actually present a substantial impediment to the mastery of the mathematical language that mediates scientific knowledge.
It is my professional privilege to teach high-functioning and highly focussed individuals from a wide range of backgrounds the mathematics they need both to model business critical decisions and to use mathematical models to inform strategic and policy choices. In the course of this instruction, I have noticed a number of fundamental differences in perspective and disposition between graduates from different disciplinary backgrounds. The identification and recognition of these paradigmatic distinctions has been instrumental in helping those individuals master those foundational mathematical concepts that were otherwise so elusive to them.
The desolate loneliness of mathematical limbo
Mathematics shares with many intellectual disciplines the feature that its theoretical constructs are established from individual statements, but that those individual statements can only be understood with reference to the theoretical whole. Many disciplines square this so-called hermeneutic circle by a process of sketching and refinement: starting with a crude delineation of the whole and then developing the full, nuanced theoretical conception through a process of iterative refinement.
Mathematical exposition, on the other hand, very often reverses this process. We start with tiny atoms of excruciatingly correct meaning, which we develop through a process of accretion and eventual coalescence, only at which point a conception of the whole body of theory rapidly emerges from the murk. These magical moments of mathematical revelation come at the price of a long sojourn in a shadowy void of provisional, patchy apprehension.
This contrast in the acquisition of concepts clearly has profound consequences for the experience of learning. Even a provisional, rudimentary understanding gives the comfort of context to the learning mind — a frame of reference on the journey to a more complete proficiency. But much mathematical explication is content to leave its students in conceptual limbo, where the faltering steps to a far-off enlightenment feel like unmotivated moves in an arcane Alice-in-Wonderland game and only the subsequent immediate steps are illuminated with clarity.
True, when these game-like moves coalesce into a higher mathematical perception, the sense of transcendence is intoxicating, but traversing that murky lacuna of contingent comprehension demands patience and, critically, confidence.
The vicious circle of mathematical self-doubt
Everyone who has studied mathematics has experienced the chill of mathematical limbo, but my experience is that graduates of other disciplines, accustomed to prompt acquisition of perhaps only provisional apprehension of a theoretical whole, find the cold all the more acutely uncomfortable on account of its unfamiliarity. For them, that long interlude without even a glimpse of the big picture can be intensely disconcerting if not downright terrifying.
All disciplines demand patience, but coupled with the pervasive persuasion that the world is broadly divided into those who can and those who can not, we can not wonder that many falter and lose faith on the way.
Naturally, want of mathematical confidence quickly creates a vicious circle. Lacking that confidence, we try to shortcut the careful steps to mathematical enlightenment, we lose our way and our already wounded mathematical ego takes a further lashing. On the other hand, armed with confidence, we forbear; we’ve been here before. The final revelation is all the sweeter for the longer time lingering in limbo and our confidence (and our joy in the subject) is bolstered.
This is, I believe, the mechanism by which the apparent dichotomy in mathematical accomplishment comes about, not primarily through differences in ability, but through differences in confidence.
Mathematical explanations are not causal
From someone whose primary professional activity is to teach the mathematical modelling of causal relations, this might seem like a controversial, not to say contrary statement. But while mathematics can be used to describe causal relations — that is to say mathematical systems can be put into correspondence with real-world relationships — those systems, in contrast to the theoretical systems of many other disciplines, are not themselves causally constructed.
In these other disciplines, and to a large extent in ordinary everyday discourse, to explain a phenomenon is to give an account of the causes of that phenomenon. These causal accounts can be more or less sophisticated and nuanced, but we understand them as causal because we can imagine alternative accounts where certain causes in the account were otherwise than they were given and the phenomenon either did not occur or unfolded in a different way.
Mathematical explanations are fundamentally different because no part of a mathematical system can be otherwise than it is given without changing the entire system as a whole. At the local level, the “rules” of mathematics can seem rather arbitrary, and often unnecessarily pedantic. They take their authority from their coherence with the mathematical system as a whole (as well as, perhaps, with their correspondence with real-world relations), but even perceived in their entirety, they can feel contrived and contingent, more like an elaborate and terribly sophisticated game than the necessary truths of causal exegesis.
The human mind is wired by evolutionary necessity to seek out causal relations in the world, so that we can intervene and influence outcomes in favour of our survival. Graduates of disciplines that abound in this kind of causal description are accustomed to feed their minds with rich feasts of causal explanatory fodder. After such abundant nourishment, mathematical explanation can feel rather thin and unsatisfactory and not like explanation at all.
I have many times experienced students who have fully internalized a body of mathematical knowledge, can reproduce results and derivations and are fully capable of answering any reasonable problem described by the field. And yet, in a kind of reverse Dunning Kruger effect, they believe themselves to lack understanding. They are unwilling to call their understanding by that name, because accustomed to richer fare, they do not perceive mastery of the mathematical game as understanding at all.
Conclusion
There’s much more, of course. The manifold poverties of mathematical pedagogy are enough to fill several articles in themselves, but a principle failing seen at all levels of teaching and in teachers of all abilities is the perpetuation of the myth of mathematical aptitude — that either you have it or you don’t. This belief intensifies the vicious circle of mathematical self-doubt and is all too often reinforced by an approach to mathematical teaching that emphasizes the discovery of mathematical talent over its cultivation.
The successful scholar of other disciplines may feel that it is only reasonable that their mind, in having a bent to the disciplines in which they have enjoyed success, should also bend away from the mathematical discipline in which they struggle.
We can not change the fundamental nature of mathematical knowledge, nor the way it is mediated and acquired. However, with patience and faith and with a recognition of the peculiar challenges wrought by the thwarted expectations derived from accomplishment in other fields, we can broaden the propagation of a mathematical paradigms and in doing so help to soften the delineation of Snow’s “Two Cultures”, which unfortunately is as sharply drawn today as it was sixty years ago.
