Two years ago, Adam, my student and research colleague, rushed into my office. As he began to explain a new idea, a smile spread across my face. He had discovered the missing link in our mathematical proof, a proof we had worked on for a year. Now we were ready to publish a new theorem. Our ideas had become truth.
The truth that Adam and I sought was the same, but our ways of finding, combining, and evaluating ideas differed, and fascinatingly so. Adam was able to draw ideas from lots of sources, whether books, people, or the Internet. He was keen to engage ideas quickly. And he managed it all, it seemed, effortlessly. For Adam, juggling ideas was simply part of contemporary intellectual life.
I am as savvy an Internet searcher as the next guy, and I am capable of handling lots of ideas, too. I chalked up the differences between us to Adam’s abundance of energy and his available time. His way, I thought, was simply the way of a younger me.
Later on, however, I sensed more differences in our assumptions about research. Yes, I explore the Internet daily, but, unlike Adam, I do not expect it to play a major role in an academic project; I tend to consult books and other “received authorities” first. I feel I should read entire chapters, in sequence. I trust authors to choose well in laying out their material.
What I had initially sensed in Adam went beyond a level of energy; it was, in fact, an intellectual disposition. Adam was less shaped by intellectual conventions like mine. He did not assume that only the most reliable sources were useful. He did not find the precise ordering of a presentation essential.
I showed Adam a graduate textbook that I thought might be helpful to our project. What did he do? Just as he might click from Web page to Web page, he flipped through the pages, looking for theorems. If a potentially relevant one used terms or concepts he did not know, he learned about them. Adam’s intellectual delivery system was need-to-know, just-in-time. When it came to manipulating and validating the ideas as they arrived in his mental warehouse, he was proficient.
For smart twenty-somethings and their intellectual fellow travelers, ideas are in the ascendant. Progressively freed from their moorings in books or college courses, ideas are increasingly encountered on their own, without the mediating force of a received authority like a professor, a presenter, or an author. When students like Adam access ideas, they evaluate them quickly for acceptance or rejection.
That is not to say they no longer see synthesis, careful argument, and good judgment as relevant to reaching conclusions. Of course they do. But the pendulum of young people’s attention is swinging toward ideas à la carte.
The consequences for higher education are significant. The pendulum swing enables great creativity—not to mention individualism. At the same time, for students to manage ideas successfully, they need guidance in developing their capacities for argument and judgment. Such guidance is neither easily given nor easily received, but it is essential.
The increasing accessibility of ideas is more of a progression than a rupture. Just as Socrates worried about the written word’s effects on oral culture, and the church feared that Gutenberg’s invention of the printing press would undermine its authority, today we worry about the quality of online information. Technology, certainly, has always been a driving force. Thanks to Google, we engage ideas in remarkable variety: facts, images, and melodies; recipes and background tracks; Wikipedia entries and tweets. Thanks to social media like Facebook and Twitter, our methods of accessing and disseminating ideas continue to multiply.
Technology, however, is not the only force at work, because, once accessed, ideas become shaped through human thought and experience. The rise of restaurants offering small plates of food reflects, in a culinary context, the same far-reaching desire to sample and combine, suggesting that social and other forces are at work as well. In the marketplace, in mass media, even in the religious domain, we have seen our menu of choices increase for some time, and consumerism is only a partial explanation. We are also seeking our own way to construct a considered life. What is different for us today is that the considerations take place more quickly, with less regard for the combinations of the past.
For instance, the practice of revering the perfect reproduction of an artistic or literary work has waned. Where earlier we hailed CD’s and fine audio speakers as the next steps toward the perfect experience, today many of us accept AAC encoding (the form in which your iPod or iPhone usually stores music) as good enough, and children put up with poor-quality resolution for YouTube videos or on-demand Netflix movies. Hi-fi yields to Wi-Fi and convenience. Similarly, the reverence for delicately argued and analyzed theological traditions is on the decline. American Protestant churchgoers, for example, sample churches worrying less about denominational factors than about a suitable mix of location, preaching style, event duration, and child care.
As I saw in Adam, the change in convention from reverence to pragmatism has an intellectual parallel: good-enough researching. As a way of knowing, good-enough researching is now as established as reading the heavens for portents was in antiquity, or listening to experts on TV only a few years ago. As we search the Web, we make hundreds of quick judgments about whether the ideas we receive are good enough to use. But what is good enough?
Adam’s case is instructive, suggesting that ideas à la carte are less dangerous in certain disciplines, like mathematics. Consider the principal fears that trouble observers about the ascendancy of ideas. Nicholas G. Carr, who writes about technology and culture, offers two in his recent book, The Shallows: What the Internet Is Doing to Our Brains (Norton, 2010). With access to so much, he argues, students will, ironically, know less. And they will lose their ability to think deeply. I would add a third: In navigating a galaxy of ideas, students will be less able to determine “truth,” however we define it.
For mathematicians over the past two decades, the first two fears are cause for some, but not great, concern. The first, for example, calls to mind the appearance of calculators in calculus courses. Calculators certainly disrupted established pedagogies; professors were suddenly asked to ensure that students gain facility both with the tool and with fundamental procedures in arithmetic or calculus. Accommodating the new tools in class took time—but we have done it. Easy access to tremendous amounts of information in many fields will very likely be analogous. Students will probably commit less to memory, but they will find it efficient to learn those things well that they have frequent reason to use: the most-important passages from oft-cited works of literature, for example. We will all need to strike that balance.
Similarly, while computers and e-mail present distractions, at least as far as mathematicians go, there has been no substitution for deep thinking in solving problems and proving theorems. No doubt we should add the Internet to the list of potential distractions that college students encounter. But as numerous authors have recorded, we are hardly lacking in work-driven students. Some of them will learn to manage their time better, and some worse; some will think deeply, and others won’t. Remember, when we decry today’s shallow thought: It was just a few years ago that the mathematician Grigori Perelman proved the Poincaré conjecture, posed in 1904, and was awarded the prestigious Fields Medal and the first Millennium Prize (the six other Millennium Prize problems remain outstanding).
About the third concern, the case of mathematics does not provide us with answers. It does, however, shed light on the kind of guidance many of our students need. That is, barring a fundamental inconsistency in mathematics that we have yet to discover, mathematicians find truth by building logical arguments upon those that have gone before. As ideas gain ascendancy, more mathematicians will work like Adam, accepting the results of the past without learning them linearly or only from established textbooks. And probably fewer will learn as many past results as they once did. We might worry that mathematical truth will become more fragile as it is more unbundled. But I believe that mathematicians will have reason to remain confident that new proofs are sound, because in creating new proofs, mathematicians often repeat earlier thinking—and perform an implicit check against error. In many proofs, mathematical objects and theorems come into contact in a variety of ways, and thinking about each point of contact provides the opportunity to perceive a contradiction lying in wait.
Outside of mathematics, arriving at truth requires more than a recipe of logic, analysis, access to what has gone before, and a mix of inspiration and creativity. Analysis and synthesis are still requirements, but they must be rendered compelling by persuasion and tempered by judgment. In an age when ideas are ascendant, but in disciplines with less reverence for established traditions, the intellectual capacity to judge and persuade is what we need to teach students. The basic human questions remain. The challenge is to figure out how to manage and validate ideas—without the logical proofs that mathematicians use.
The challenge is by no means limited to the humanities. Controversies among psychologists about Null Hypothesis Significance Testing—regarding the validity of a long-used technique of statistical inference—are essentially epistemological. So are debates among scientists about the “decline effect” and whether experimental results may be more difficult to replicate over time.
Those concerns are even more pressing outside of disciplinary frameworks. Disciplinary courses offer basic contexts in which to ask what ideas are relevant to the discussion of a topic and in what ways we can justify a conclusion. But too often we respond with answers from convention, basically because we define the discipline that way. The elephant in the room, of course, is that today we increasingly sample ideas across traditional boundaries.
At Davidson College, therefore, over the past several years we have questioned disciplines, their boundaries, and their methods and evidence, both at the level of the overall curriculum and at the level of individual courses. My own proposal to help students filter ideas is that we teach them a simple principle: A necessary part of making an argument is the investigation of potential opposition. At a minimum, every writer or speaker should take one step beyond listing sources, to determine whether the assertions cited are subject to reasonable challenge, and to consider whether those challenges merit further review.
Take the phenomenon of using Wikipedia as a research source. Although we have heard much debate about whether Wikipedia is a suitable resource, the question is still open—and still a thorn in the side of many faculty members. What about approaching the question differently? Wikipedia recently came up as a discussion topic in my section of an interdisciplinary first-year course in the humanities. One student observed that Wikipedia should be considered a fair source of information because it represents a consensus view of a large population. I was suddenly on the edge of my chair. Here was a teachable moment, and I felt that I should have been better prepared.
What that student deserved from me, and what students deserve from me in the future, is not only a response, but also a model of how to approach an issue. What are possible opinions about the utility of the resource, beyond conventional wisdom? What sort of evidence for those sometimes opposing ideas would be compelling? Right or wrong answers may often be elusive, but we cannot avoid confronting the oppositions. Mathematicians have implicit checks against error. Students in other disciplines need such methods as well.
My proposal’s deceptive simplicity hides the difficulty of putting it into practice. It does not admit to uniform application. From high school to graduate school, every level will have its own standards for how to investigate the ideas and arguments of others. And the proposal is no substitute for nuanced, intellectual judgment. It’s just a starting point in meeting the challenge of deciding what is good enough when ideas are newly ascendant.
