Are we about to see advances in mathematics come to an end? Until last year, I would have said no. Now I am not so sure.
Given the degree to which the advances in science, engineering, technology, and medicine that created our modern world have all depended on advances in mathematics, if advances in mathematics were to come to an end, then it's hard to see anything ahead for society other than stagnation, if not decline.
How likely is this to occur? I don't know. The escape clause in this year's Edge Question is that word "should." Had the question been, "What are we worried about?" I suspect any honest answer I could have given would have focused on personal issues of health, aging and mortality; mathematics—dead or alive—would not have gotten a look in.
Sure, there are many things going on in society that disturb me. But it's an inevitable consequence of growing older to view changes in society as being for the worse. We are conditioned by the circumstances that prevailed during our own formative years, and when those circumstances change—as they must and should, since societies are, like people, living entities—we find those changes disconcerting. Things inevitably seem worse than they were during our childhood years, but that reflects the fact that, as children, we simply accepted things the way they were. To let a disturbed sensation as an adult turn into worry would be to adopt an unproductive, ego-centric view of the world.
That said, we are reflective, rationale beings that have some degree of control of our actions, both individual and collective, and it's prudent both for ourselves and for others, including our descendents, to read the signs and try to judge if a change in direction is called for.
One sign I came across unexpectedly last year was a first hint that mathematics as we know it may die within a generation. The sign was, as you may suspect, technological, but not the kind you probably imagine.
Though the 1960s and 70s saw an endless stream of commentators lamenting that the rise of calculators and computers were leading to a generation of mathematical illiterates, all that was happening was a shift in focus in the mathematics that was important. In particular, a high level of arithmetical ability, which had been crucial to successful living for many centuries, suddenly went away, just as basic crop-growing and animal rearing skills went away with the onset of the industrial revolution. In the place of arithmetic skills, a greater need for algebraic thinking arose. (The fact that, in many cases, that need was ill met by being taught as if it were arithmetic, does not eliminate the new importance of algebraic skills in today's world.)
On the other hand, there is no denying that advances in technology change the kind of mathematics that gets done. When calculating devices are available to everyone on the planet (North Korea excepted), which is almost certainly going to occur within the next ten years or so, the world is unlikely to ever again see the kinds of discoveries made in earlier eras by mathematical giants such as Fermat, Gauss, Riemann, or Ramanujan. The notebooks they left behind showed that they spent many, many hours carrying out long hand calculations as they investigated primality and other properties of numbers. This led them to develop such a deep understanding for numbers that they were able to formulate profound conjectures, some of which they or others were subsequently able to prove, a few that remain unproved to this day, the Riemann Hypothesis being the most notable.
There are likely to be properties of numbers that will never be suspected, once everyone has access to powerful computing technology. On the other hand, there is also a gain, since the computer has given rise to what is called Experimental Mathematics, where massive numerical simulations give rise to a different kind of conjecture—conjectures that likely would not have been discovered without powerful computers. (I described this recent phenomenon in a book with Jonathan Borwein a few years ago.)
So, technology can certainly change the direction of mathematical discovery, but can it really cause its death? In September last year, I caught the first glimpse of how this could occur. I was giving one of those (instantly famous) Stanford MOOCs, teaching the basic principles of mathematical thinking to a class of 64,000 students around the world.
Since the course was on university-level mathematical thinking, not computation, there was no possibility of machine-graded assignments. The focus was on solving novel problems for which you don't have a standard procedure available, in some cases constructing rigorous proofs of results. This kind of work is highly creative and intrinsically symbolic, and can really be done only by covering sheets of paper (sometimes several sheets of paper) with symbols and little diagrams—sometimes using notations you devise specially in order to solve the problem at hand.
In a regular university class, I or my TAs would grade the students' work, but in a MOOC, that is not feasible, so I made use of a method called Calibrated Peer Evaluation, whereby the students graded one another's work. To facilitate the anonymous sharing of work, I asked students to take good quality smartphone photos of their work, or scan their pages into PDF, and upload them to the course website, where the MOOC software would organize distribution and track the grades.
Very early one, a few students posted questions on the MOOC discussion forum asking if they could type up their work in LaTeX, a mathematical typesetting program widely used in advanced work in mathematics, physics, computer science, and engineering. I said they could. In fact, I had used LaTeX to create all of the MOOC resources on the website, to create the weekly assignment sheets, and to self-publish the course textbook.
Now, LaTeX is a large, complex system with an extremely long and steep learning curve, so I was not prepared for what happened next. From that first forum post on, hardly anyone submitted their work as an image of a handwritten page! Almost the entire class (more precisely, the MOOC-typical ten percent who were highly active throughout the entire course) either went to tortuous lengths to write mathematics using regular keyboard text, or else mastered sufficient LaTeX skills to do their work that way. The forum thread on how to use LaTeX became one of the largest and most frequently used in the course. I shudder to think of the amount of time my students spent on typesetting; time that would have been far better spent on the mathematics itself.
We have, it seems, become so accustomed to working on a keyboard, and generating nicely laid out pages, we are rapidly losing, if indeed we have not already lost, the habit—and love—of scribbling with paper and pencil. Our presentation technologies encourage form over substance. But if (free-form) scribbling goes away, then I think mathematics goes with it. You simply cannot do original mathematics at a keyboard. The cognitive load is too great.
The increasing availability of pens that record what is being written may, I suppose, save the day. But that's not substantially different from taking a photo or scan of a page, so I am not so sure that is the answer. The issue seems to be the expectation that our work should meet a certain presentation standard. In a world dominated by cheap, sophisticated, presentation technologies, paper-and-pencil work may go the way of the DoDo. And if that happens, mathematics will no longer advance. As a living, growing subject, it will die. RIP mathematics? Maybe. We will likely know for certain within twenty years.