Why Liberal Education Must Include Cultivate the Capacity for Formal Abstraction

Anthony Mansueto

In a New York Times op-ed dated 29 July 2012, Andrew Hacker, Professor Emeritus of Political Science at Queens College, argues that the insistence of high schools and universities alike that their students master algebra before graduating is fundamentally mistaken –and one of the principal causes of the failure of the US educational system to retain and graduate more students. His argument is fairly straightforward. On the one hand, he says, most adults, even active, engaged citizens and those in highly skilled professions, do not actually need or use algebra. On the other hand, he presents an impressive array of statistics showing that it is, precisely, an inability to pass algebra that is preventing our students from completing their high school and university studies. He advocates substituting in the place of algebra courses in quantitative reasoning, “citizen statistics” teaching how, for example, the Consumer Price Index is computed, and courses in the history and philosophy of mathematics.

The issues which Professor Hacker raises are important and go the heart of current debates around both the nature and aims of liberal education and the reasons why so many of our students fail to complete their high school and university studies. Unfortunately, by presenting the alternative as between continuing to do what we are currently doing (which obviously isn’t working) and abandoning a critical part of what has historically been regarded as integral to life as a free human being and citizen, he obscures the real nature of the issues at hand and excludes possible avenues forward.

To begin, let me explain why algebra is important –and not only for those who will use it, or more advanced mathematics dependent on it, directly in their future work. It is through the study of algebra that we have historically cultivated the capacity for formal abstraction –the ability to leave behind the particular determinations of things and consider only the logical relations between explicitly specified aspects of their definitions. It is not only mathematical physics and its derivative disciplines (the whole of modern science and technology) which depend on such reasoning. So do the humanities and social sciences, all of which make formally rational arguments for their claims. And this is especially true of philosophy and theology which, because they engage directly fundamental questions of meaning and value, lie at the core of a liberal education. I often explain philosophy to my beginning students as “word algebra” and tell them that doing philosophy means, fundamentally, resolving the difficult and mysterious questions, such as the existence of God or the possibility of grounding moral judgment, by moving symbols around on paper using a handful of formal rules which are not too different from those taught in algebra courses –and believing that the results, if not definitive, are authentically liberating and enlightening. In order to be able to make rationally autonomous judgments regarding fundamental questions of meaning and value, and thus have the ability to live as a free human being and engaged citizen in a democracy in which questions are settled by rational discourse, it is necessary to have mastered formal abstraction and indeed for it to have become second nature.

This said, two points are in order. First, I would argue that many of the problems of the modern world are a result of a certain idolatry of formal abstraction. We assume that modern science, which actually provides only a very rigorous formal mathematical description of the universe, actually explains it and displaces the higher, transcendental abstraction employed by philosophy and theology in addressing questions of meaning and value. It does not. The result is a great deal of confusion and unwarranted despair at the spiritual implications of modern science –as well as unwarranted attacks on scientific results which are, within their proper sphere, at least well founded and often definitive.

At a more practical level, we have turned over the management of much of our economy to “quants” and technical analysts who develop sophisticated mathematical algorithms which they then use to guide investment strategies and manage risk, algorithms which have no way of knowing what allocation of resources actually best promotes human development and civilizational progress (or what constitutes such development and progress) and which have been shown to be ignorant of important, concrete, on the ground facts, attention to which historically kept capitalism, for all its problems, from becoming utterly irrational. There is considerable evidence that recent financial crises, including “Black Monday” in 1987 and the financial crisis of 2008 were significantly exacerbated, if not actually caused, by mindless reliance on such algorithms.

In order to understand the limits of formal abstraction, however, we must understand what it is and what it can do and why it is so attractive. And that means mastering it.

Given this, the question is how best to help as many people as possible master this discipline. It should be clear by now that simply requiring students to take courses in algebra and then either fail or be “passed on” isn’t working. And in fact it hasn’t worked for a very long time. It is just that in the past, when liberal education was the preserve of the aristocracy and large Capital, those who failed to master it (along with most of the other disciplines taught at great universities and liberal arts colleges) simply didn’t graduate and went on to take up positions of power and privilege which were effectively hereditary or, somewhat later, took “gentleman’s C’s” and received diplomas, lest they look too bad by comparison with the poor scholars they then hired as advisors and administrators. (It is important to remember that “social promotion” was applied first not to the poor or ethnic minorities but to the rich who endowed and effectively controlled the universities).

So what do we do instead? As an advocate of a question-centered approach to liberal education, I am sympathetic to Professor Hacker’s call for courses which provide students with a basis in experience for understanding what formal abstraction is and with an understanding of its civilizational significance. But I would add two caveats. First, the experience in question must actually be that which has historically enabled people to engage in formal abstraction. Alexandr Luria’s 1928 study of the social conditions for cognitive development in USSR showed rather definitively that only people who are engaged in sophisticated if/then reasoning in a complex market society actually develop the ability to reason formally. So it is not just a question of making formal operations accessible or showing how they are relevant. It is a question of giving people the experience of making important decisions in complex situations in which they must abstract from particulars and identify what criteria make a decision reliable and valid. Second, the courses in question must actually get people to the point of engaging in formal abstraction and formal operations, not just to the point of appreciating their civilizational significance. This means, at some point, thematizing the principles of such abstraction which are, in fact those if not of algebra then of abstract higher mathematics generally. In other words, while we might start with concrete questions we must eventually get students to ask and answer such questions as “What is a Number?” and “What makes a mathematical formalism valid?” This actually takes them beyond algebra (which as taught in most places is simply the application of the Laws of Arithmetic in the general case) to a consideration of for what categories these laws are valid and thus to a rigorous course in the foundations of mathematics, something even most scientists and engineers never get.

It is not possible to spell out in detail in this context just how this might be done, and I am guessing that there are many variants which would work reasonably well. I have had a great deal of success, for example, simply telling students that we are going to talk about “relationships.” If this means relationships between people, then it is obvious right away that laws like those of arithmetic don’t apply. The fact that Tom loves Mary and Mary loves Joe does not mean that Tom loves Joe. Numbers are things which have had enough of their particularity stripped away that such laws do apply. And so students gradually get a sense of what is involved in formal abstraction. Concrete mathematics like what they learned in grade school deals with operations on particular numbers. It is basically counting using some short cuts. Abstract mathematics, such as algebra, deals with numbers or other mathematical categories in general and asks what laws apply and what makes operations on and propositions regarding such relationships valid.

By creating appropriate bases in experience, and then leading students through the exploration first of concrete and then of progressively more abstract questions we can actually teach not only the formal abstraction which is so critical to understanding how the world works, but also the transcendental abstraction they need in order to make rationally autonomous decisions regarding fundamental questions of meaning and value.

I should close by saying that not everyone who begins this journey is going to complete it. In fact no one completes it. Liberal education has never had the 100% success rate which legislators and funders are currently demanding. Even at the best institutions only a handful of people become fully capable of making rationally autonomous judgments regarding questions of meaning and value. And even they are only at the beginning of a very long journey. Those who really master formal abstraction do so only to understand that there is a higher sort of abstraction (what I call transcendental abstraction) which asks about meaning and purpose. That is what Godel’s Incompleteness Theorems are all about. And those who master transcendental abstraction and its practical applications find themselves set on the path of higher spiritual disciplines which stretch us beyond the merely human. We are all pilgrims. Our journey is endless and our destination unknown. That is what makes it so endlessly interesting.

That said, we do make progress and we can find ways to help those joining this path to make more progress than they otherwise might. It is in that spirit that I offer these cautions and these suggestions and invite further deliberation on the role of mathematics in humanity’s long march from slavery to freedom.