Climbing the ladder
I want to talk briefly about the way that mathematics has continually, across the course of its history, introduced new concepts that seem successively more and more remote from immediate experience. And I want to make a suggestion about why this ascent into abstraction has occurred, and why the act of drawing distinctions between “practical mathematics” and “abstract mathematics” is more problematic than it might at first seem.
We begin with the counting numbers: 0, 1, 2, 3 . . . These are straightforward. It should be clear what I mean when I say “I have three apples,” and why such notions might be useful in the course of everyday life. It should also be relatively clear why the operations called “addition” and “multiplication” might naturally arise in the course of putting these numbers to their natural purpose. (If I have 3 apples and you have 2, then together we have 5: addition. If there are 6 of us, and we each have 4 apples, then all together we have 24: multiplication.)
A problem arises when we want to talk about things that are divisible into pieces. If I have a pie, I could say that what I have is “one” — one pie. But if I cut the pie into eight slices, what do I have now? Each slice is now “one” of something — one slice. But we want to count the slices on the same “scale” on which we counted the pie, so that we can faithfully represent the fact that when we have eight slices, we have only one pie. This leads us to the rational numbers, each of which can be identified by a numerator and a denominator. On a “scale” in which I associate a pie with 1, I could associate the rational number with each slice, which would mean that if I have eight of them (the denominator), I would have one pie (the numerator). The same applies to more complicated rational numbers like : if I have 19 such objects, then I have in total 72 pies. And so forth.
The rational numbers have a certain “continuity” to them that the counting numbers lack. There are no counting numbers between 0 and 1, or between 1 and 2; but given two distinct rational numbers, I can always produce a rational number that lies between them by taking their average (i.e., by adding the two and diving the result by 2). This suggests that the rational numbers are appropriate for quantifying extents — things like lengths, areas and durations, which come in continuous ranges and not in jumps like the counting numbers. Indeed, this appears to work: we can quantify lengths by stating them as fractions of some arbitrarily chosen “unit” length, just as we quantified slices by stating them as fractions of our “unit,” the pie. Likewise for areas, durations, etc.
At this point, we might be led to suspect that the rational numbers are all the numbers we need for the task of describing the world around us. By choosing numerators and denominators as subtly as we please, we can approximate any continuous extent arbitrarily well. But it turns out that this anti-abstract paradise can’t be maintained. It turns out that we are led, inevitably, to the consideration of strange entities called irrational numbers, which will intrude upon our consideration of rational numbers even though they are not rational numbers themselves.
The most famous illustration of this intrusion is the following. It can be shown — through purely geometrical methods that have no inherent dependence on notions of “number” per se — that the lengths and of the two legs of a right triangle are related to the length of the hypotenuse by
This is, of course, the Pythagorean Theorem. Now if the legs are both 1 (by our arbitrarily-chosen length-standard), the hypotenuse must then satisfy
Now it can be shown that this equation has no solution: there is no rational number that satisfies it. (Or, as the more common phrasing goes, “the square root of two is irrational.”)
Well, that seems funny! If we simply made a triangle with sides of length 1, would it not have a hypotenuse of some measurable length? One possible way to resolve this conundrum is simply to declare that we can’t do this: it must be impossible, as a matter of physical law (or something yet deeper), to make a triangle whose sides have length 1. Strange, but conceivable. However, another example will make it clear that this kind of “solution” will never work.
If I drop a ball, it will fall under the influence of gravity, and before it hits the ground its height above the ground will be described by a quadratic function of time, i.e., something like
(For simplicity, I have written this equation so that will be zero at time and will be negative thereafter.) The ball would appear to you and I to fall in a continuous path, picking up speed in gradual fashion as it does so. It seems intuitive, then, that in its downward course, the ball passes through every point in space between its initial position and its final resting place on the floor. But now suppose we ask, “when was the ball at ?” By the hypothesis just stated, the ball should have been there at some time (if that height is above the floor). And the time at which this height is achieved must satisfy
After cancelling the minus sign, we see that this is the same equation we encountered before, the one that has no solution. The ball, apparently, does not cross this point at all: it somehow “skips” it! Or that is what we would say if we were committed to the use of rational numbers to describe its behavior.
Now here, as in the previous example, we could choose to simply conclude that the ball really does behave this way. But now suppose that we change our arbitrary standard of length, so that the unit is half as big — say, we could measure in half-meters instead of meters. Then in the new system, the height under consideration would be “-4” rather than “-2,” and the equation would become
which is clearly solved by . So whether or not the ball “skips” a given height is not an objective, physical fact, for our judgment of where the “skips” occurred will vary with the arbitrary choice of our measurement system!
So much for the use of rational numbers to describe continuous extents. We have seen that, if we restrict our calculations to the rational numbers, those calculations will “prohibit” the existence of certain things — triangles, positions, etc. — against which no physical case can be made. They are not prohibited by reality, only by the mathematics of the rational numbers; the structure of acceptances and rejections does not mirror anything in the real world. To fix this defect, we must introduce new numbers, numbers that correspond to (say) the length of the hypotenuse in our first example above. We know these can’t be rational numbers — they must, then, be something else — they must be “irrational numbers.” Numbers that can’t be stated as the ratio of two counting numbers.
A strange category indeed. Counting numbers were clear to us, rational numbers clear enough through simple examples about pies. But what is an irrational number? There was nothing in our immediate experience to suggest, at first, that such things might be necessary. Indeed, even now their nature remains obscure and their justification frustratingly abstract. (What exactly is the square root of two, if not a ratio? A modern mathematician would tell you that it is a sort of partition of the rational numbers called a “Dedekind cut” — as if that helps!) We have had our first taste of the fruit called abstraction, and from here we can never go back to the paradise of purely intuitive concepts.
In A Mathematician’s Apology, his famous and passionate defense of pure mathematics, the eminent number theorist G. H. Hardy held up the proof of the irrationality of the square root of two as an example of the sort of thing a practical user of mathematics — such as an engineer — could have no use for. He pointed out that any irrational number can be approximated arbitrarily well by a rational number (we can always make our rationals get closer and closer to any given irrational, by something like the process of averaging referred to above). And since all of the quantities that occur in engineering and science have some inevitable degree of measurement error, an arbitrarily good (if still erroneous) approximation is the best one could ever need in such applied subjects. For Hardy, the beginning of abstraction is the beginning of a departure from the physical world, from practicality. It is the first step on a journey into “pure mathematics,” a subject in which concepts are introduced and considered solely for their aesthetic virtues and not for any physical importance they might happen to have.
But Hardy is wrong here. It is not hard to see why. Just ask any engineer or physicist whether they would be willing to do all their calculation in the realm of the rational numbers, and watch them shudder in response. Why will they shudder? Because, counter-intuitive as it might seem, the act of accepting the irrationals into our purview allows us to ignore their existence — and ignoring their existence is, as Hardy was right to assert, what a practical sort of person should want to do. If we try to do math in the rational number system, we will have to constantly check, at every step, to make sure that none of our quantities are irrational. Does our triangle have the wrong sorts of sides? Whoops — it’s prohibited! Better try another one! Will we have to take a square root of two to solve a equation? Whoops — that equation has no solution! And none of this extra trouble reflects anything physical, anything real. Those irrational hypotenuses and solutions are, as Hardy said, arbitrarily close to rational numbers, and if “arbitrarily close” is as close as we need, then we needn’t care that they’re irrational. The irrationals naturally pop up in the course of rational calculations; our only choice is whether we will accept their presence (allowing us to ignore the practically irrelevant rational/irrational distinction entirely), or whether we will shut them out (which requires us to attend to that practically irrelevant distinction). It is, to use a technical term, a pain in the ass to try to do physical calculations while scrupulously avoiding the irrationals.
Thus the introduction of the irrationals is a departure from immediate physical intuition, but not a departure from practicality. Practical considerations led us to it, and although the irrationals themselves are in some sense “practically irrelevant,” it is nonetheless easier to do practical calculations with them than without them. This theme repeats itself again and again in the development of mathematics. At each step up the ladder of increasing abstraction, practicality itself drives the ascent. We are forced to introduce new quantities, remoter from immediate experience than any yet introduced, simply because the contortions required to exclude them from consideration become prohibitively cumbersome. Pace Hardy, the ladder of abstraction is not a trail that runs away from practicality — it is the road that practicality pushes us along, reluctant as we may be.
Another example of this phenomenon is the step from “real numbers” (the name given to the rationals plus the irrationals) to “complex numbers.” This is the point at which a lot of people begin to find the ladder of abstraction a bit silly — they’re willing to swallow irrational numbers (the Pythagorean Theorem is clearly useful, after all), but why on earth would someone make up a square root of negative one? Why can’t mathematicians just accept that there’s nothing you can square to make negative one? (The physical picture that would first come to mind for such a thing would be a square with negative area . . . but nothing can have negative area, so the concept seems physically unmotivated.)
But now look. When considering falling balls, we were led to consider the equation
And we wanted this to have a solution. It was inconvenient, and physically meaningless, to claim that there was no solution to this equation. As it turns out, there are analogous cases — less elementary, but not essentially different — in which we might say the same thing of the equation
In linear algebra — a theory that is crucial to any serious consideration of physical motion, including the sort done in many branches of engineering — we find ourselves considering entities called “matrices” and associating them with “characteristic equations.” Even if a matrix can be described solely in terms of real numbers, its characteristic equation may in general be just about any polynomial equation, and thus may have complex solutions. And the solutions to a matrix’s characteristic equation have a deep connection to the matrix itself, so that the theory even of real matrices cannot be formulated without complex numbers. We could try to state and prove the basic results of the theory in a setting in which there is no square root of negative one, but it would be cumbersome and frustrating; by contrast, the theory in complex numbers is clean and elegant and general (and thus — don’t tell Hardy — useful).
What is the meaning of the ladder of abstraction? It seems clear that somehow the feeling of being led from one rung to the next is not an arbitrary thing. It is somehow “objective” — one feels that an alien species, confronted with the same basic physical reality, would be forced to introduce (for the sake of calculational ease) the same concepts I have gone over in this post. But what sort of “objectivity” is this? One answer is mathematical Platonism, which says that these concepts have some sort of real existence, and that their introduction into our thinking constitutes the discovery of previously existing things, just like the discovery of new substances in physical science. But it is not clear to me whether introducing these sorts of entities is necessary to explain what we want to explain. For we came upon these concepts by wrestling with physical reality. Might it just be the case that these are the structures most apt for describing the reality we happen to live in? (If we were not interested in triangles and falling balls, would we have needed irrationals?) And then might we need no explanation beyond the structure of our own reality?
I do not know the answer. For now, I just want to emphasize that we are pushed along the ladder largely, if not entirely, for practical reasons. The study of reality forces us to invent fantastical objects that seem to have no direct relation to our immediate experience. This is a wonder, and one that doesn’t, I think, get the appreciation it deserves. Pure mathematicians in Hardy’s vein tell us that math is a beautiful road that leads away from practical matters. But if it were simply to be judged on aesthetic merit, it would have to be stacked up against all of our other creations — and in that company, I think it would be found wanting. No, the magic of math is that it seems whimsical, aesthetic, and removed from reality, and yet in its creation we have not relied on our own whims — we have taken dictation from the unquestionable demands of reality itself.