Not long ago, I missed a meeting of friends discussing computers in mathematics. The e-mail that called us together led off with the quote, “We are on the threshold of a revolution in mathematics—how we think about it, how we practice it, and how we learn it.” (Paul Lutus, 1985.)
Everyone is aware of the impact of computers’ on our economy and daily life. And they have indeed caused and are causing a revolution in mathematics, or at least in the practice of mathematics. They have allowed mathematicians (and scientists and engineers) to investigate mathematical problems and physical systems of great variety and hitherto unimaginable complexity. They have also generated strong currents of thought to the effect that reality is at base a digital creation (see, for example, Stephen Wolfram’s A New Kind of Science) and that we should be able to solve all our important mathematical problems by some form of simulation. That is, by brute-force number-crunching. (Of course there are still mathematical and physical problems that are too tough for our computers to crack—such as weather prediction beyond a few days—but the answer to this is obvious: More power! )
However if we cast our mental gaze back over the history of mathematics from, say, the Nineteenth Century till now, I think that there was another mathematical revolution that occurred earlier, that was even more fundamental. Something on the order of the invention of relativity or quantum theory. With one interesting difference—that by and large, aside from the mathematicians themselves, no one realizes this profound revolution took place.
Now the reader may feel this claim a bit odd: How can anyone overlook a revolution as fundamental as relativity or quantum theory? And just what is this revolution?
As for how it could be overlooked, why do people know about relativity and quantum theory? After all, the vast majority of people (and this includes academics aside from scientists in those fields) really have little understanding of them. They do, however, have a strong awareness of relativity and quantum theory and of their importance. This is due to the great stream of publications and presentations that have been made over the years on these topics and the fact that one cannot ignore such striking “fruits” as nuclear weapons and laser discs.
Mathematics on the other hand, lacks such a stream of publicity. And although it has an abundance of “fruits”—things like functional analysis and topology—they tend to be edible only by scientists and engineers (and a few other specialist types such as economists and statisticians). So the general public remains pretty much oblivious of the triumphs of mathematics (with perhaps the exception of fractals).
As for what sort of revolution we are talking about, a good article on this is A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today by Frank Quinn. But let me try to present this idea from a personal perspective.
My own mathematical initiation occurred at the University of Florida and Florida State University in the 1960’s during what was still perhaps the Golden Age of pure, abstract mathematics. What I learned in high school—algebra and trigonometry—had little bearing on the experience I am trying to get at and did not do much to prepare me for it. Nor would it have helped if I had learned calculus in high school. My only foreshadowing occurred in tenth grade geometry; but this was at a time when high school geometry was still a demanding theorem-proof sort of course.
Even at the beginning in university, through calculus and differential equations, I really had no inkling of the “revolution” we are discussing. After all, calculus and differential equations are courses taught to a lot of people besides mathematicians—engineers, statisticians, physicists, etc. They tend to be dominated by a desire to compute things, not by an interest in deep understanding.
Things began to change when I took my first real math course, which, as I recall, was abstract algebra. And as I went on to other math courses, things got weirder. I had little awareness at first that I was the beneficiary of an intellectually cataclysmic event that had occurred in the second half of the Nineteenth and beginning of the Twentieth Centuries. But one thing did impress me mightily, and that was that I was having to learn to think in a really strange and different way.
Now when I say this, I am not talking about learning exotic new concepts such as transfinite cardinals or Banach spaces or topologies. I mean that I had to learn to control my thoughts in a different and much more careful—excruciatingly so! —way than I was used to.
For one thing, words had to be used exactly the way they were defined during whatever the current discussion was. If, for example, a duck was defined to be something that had feathers, two legs, and quacked, then the fact that it might have been constructed in a factory using a marine landing craft for a body and weighed three tons was totally irrelevant. So long as feathers had been attached, it had two legs (hydraulic or not), and quacked, then by heavens, it was a duck!
To put this another way, words ceased to be things such that, “Everyone knows what that means! ” Words were now precisely what we said they were, nothing more and nothing less. And any would-be-mathematician who forgot that would find him- or herself sliced into epsilon-small bits.
For another thing, every assumption had to be laid out clearly and explicitly. Or else, everyone worked from a body of assumptions that was felt to be crystal clear to the participants. One might, for example, start with the assumptions of Euclidean plane geometry. Or with the properties of the real numbers which, it was felt, all the participants knew so well that no real disagreement could arise as to what they were.
After this, every step in an argument had to follow by sheer logic and nothing else.
Therefore one could not reason by analogy. (“We know that all objects have locations. The number 3 is an object, therefore 3 must be located somewhere.”)
Nor by pictures. (“Look, if we draw the line from A to B, we can see that it has to intersect the line that runs from C to D.”)
Nor by intuitively obvious “facts” brought in from outside the system. (“We know that any event A must have a cause B. That in turn must have a cause C. And C is caused by D and so forth. Now if we follow this chain of causation backward, we must eventually come to some event E which has no cause. Let us call this event God.”)
From my point of view, the heart of the revolution I am talking about was (and is) this way I had to learn to think about things. Let us call this the non-Euclidean revolution. This is because the birth of non-Euclidean geometries was one of its most visible products.
Let me point out two of the most striking “fruits” of this revolution.
The first and one of the most useful is this: Mathematics as we understand it today, is not about any thing.
To understand what I mean by that, consider, for example, the theory of what mathematicians call groups.
A group is a collection G of “objects” and a “binary operation” * on G that satisfies certain axioms. When we say that we have a binary operation * on G, what we mean is this: Given any pair of elements a and b from the set G, there is a way to “combine” them to form a new element a*b of G. The axioms for this binary operation (the properties it is assumed to have) are these:
1. For all elements a,b,c of G, we have
a*(b*c) = (a*b)*c.
2. There exists an element e of G having the property that for all a in G,
a*e = e*a = a.
This e is called an identity element of the group.
3. For all elements a of G, there exists an element b such that
a*b = b*a = e.
One instance of a group is Z, the integers, the set of whole numbers, positive, negative, and zero,
…, –3, -2, -1, 0, 1, 2, 3, 4, …
with the binary operation being addition. The element e in this case is 0, the additive identity.
Another instance of a group is found by taking a polygon P in three dimensions and letting G be the set of rotations and reflections that take P onto itself. Suppose that a is the rotation by 90 degrees about the x-axis and b is the rotation by 90 degrees about the z-axis. Then a*b can be taken to be one rotation of P followed by the other (say the a rotation first). In general, the * operation in this setting is meant to be the composition of the transformations of P, that is, one transformation followed by the other one. It is not hard to convince oneself all the axioms of a group are satisfied. The transformation e is the one in which P is neither reflected nor rotated, the identity transformation (that is, you do nothing at all to P).
We might think from our first example, with the integers, that a*b=b*a for groups in general since we know that a+b=b+a when we add numbers. However in the second example (transformations of a polygon), one can show instances in which a*b does not equalb*a. That is, the property of commutativity does not follow from the axioms of a group. This is an instance when we see that we have to be careful not to let unspoken assumptions (what “everyone knows is true”) creep into our thoughts.
We have given two examples of groups but there are tons more. For example, if we look at maps of countries or the world, we know there are many ways to set up coordinate systems on a map and specify the locations of cities, geological sites, and so forth. The transformations between different coordinate systems can be treated as a group with the operation a*b amounting to the application of the transformation a followed by the transformation b.
The examples we have given of groups are instances of the idea of a group. (I think another nice way to describe them is as incarnations of the group idea.) But the thing to notice here is that there is a vast and complex literature on the theory of groups. That literature is not meant to be about any particular example or incarnation of group theory. Any result which is proved about groups can automatically be applied to all of them. In this sense, group theory is not about any particular group but about the idea that stands behind all of them.
This generality applies to many areas of mathematics. If, for instance, we study the theory of Hilbert spaces, we find that possible solutions of differential equations can be treated as a Hilbert space, and so can probability amplitudes in quantum mechanics. Thus anything we can establish about Hilbert spaces in general tells us something about differential equations, quantum mechanics, and other areas as well.
So precisely because mathematics is not about any thing, it tells something useful about many different things. Sometimes we do not know what an area of mathematics is telling us about things till after the mathematics is invented. This was the case for the theory of tensor analysis (invented to do calculus and geometry on curved spaces of any dimension) when it was discovered to be just the tool to develop general relativity.
The second “fruit” of the non-Euclidean revolution that I want to point out is that its very explicit statement of its assumptions leads one to question those assumptions. The outstanding example of this is non-Euclidean geometries.
In Euclidean geometry, if L is a line and p is a point not on L, then there can be one and only one line L’ through p that is parallel to L, and L’ will never meet L.
This is the parallel postulate of Euclidean geometry. It is an explicit assumption about that geometry. But logically there are other possibilities. One of these is that L and L’ do meet; this leads to a new and non-Euclidean geometry. Another possibility is that there may be many lines passing through p that are parallel to L and never meet it; this leads to a different non-Euclidean geometry.
This invention of non-Euclidean geometries has turned out to be very fruitful area of mathematics and has also been quite useful in physics.
So a most marvelous quality of this fundamental, non-Euclidean revolution in mathematics is not simply that it enables one to think outside the box but that it forces one to actually see the box and measure its dimensions. This is tremendously liberating for the creative mind. For the greatest confining power of a box occurs when we do not know it is there.