The world tends to be full of uncomfortable surprises. We humans apply a lot of effort, trying to protect ourselves from these surprises. Much of this effort involves rearranging the world, such as growing food, building houses, etc. We also build up ideas about the world, classifying and naming objects, and noting the regularities in their behavior. Once we know how to predict eclipses they are not so surprising or frightening. While these efforts are successful to a large degree, still rude surprises continue to intervene. On such occasions we might decide to investigate things a bit more deeply, to find the hole to be able to patch it. The world turns out not to be exactly the arrangement of objects that we thought it was. How is it then? Our investigation might proceed in a variety of directions.


We work hard to arrange things in some satisfactory way. At some point we might actually succeed. But the next day, things have changed. The paint peels, the fruit rots, the cloth frays. Youth ages, health turns to sickness, then death. What is the nature of change? How can it be, that what is a fact on one day is not a fact the next day?

The Paradox

Thinking about the nature of change has a very deep history both in the scientific tradition going back to Zeno's paradoxes and in the Buddhist tradition. It is worthwhile to study this matter closely. At one point in time we can truly say, "X is true", and then some time later we can truly say, "X is false". This mystery is traditionally illustrated by a variety of ways to fill in the blank X. Zeno tells the story of Achilles, the fastest human, chasing a tortoise, a much slower animal. When Achilles sets off, the tortoise is on the other side of the field, some 100 yards distant. At that time we would truly say, "Achilles has not caught the tortoise." Some time later we would expect to be able to say, "Achilles has caught the tortoise." In Indian philosophy a classical way to fill in for X is to consider the growth of a seed into a sprout. At first we can say, "The seed has not sprouted," then later we say, "The seed has sprouted."

I would like to continue the discussion in the abstract form of "X" and "not X". This is a common scientific move from concrete to abstract and mathematical. Rather than examing science from a Buddhist perspective, do I not by such a move instead introduce a scientific approach into Buddhism? But such a move is native to Buddhism. The Buddha was asked whether his teaching wouldn't be distorted and falsified if translated into other languages. The Buddha replied, "No," that his teaching, the holy Dharma, should be translated to be made more easily understood by whatever audience was at hand. Buddhism for a scientific audience needs to be translated into a scientific language. Furthermore the great Buddhist philosopher Nagarjuna acknowledges the validity of using concepts for discursive purposes without those concepts themselves being granted any more that provisional validity. So, on with X...

I want to look more closely at how it could be that at one time X is true and then at some later time X is false. There must be some last time when X is true - let's call that A. Similarly there must be some earliest time when X is false, which we can call B. Now, what is the temporal relation between A and B? Is A before B? Is A after B? Are A and B actually the same time? But none of these possibilities makes sense. If A and B are the same time, or if A is after B, then X is both true and false for at least an instant of time, which can't happen. But if A is before B, then there is a time right between these when X is neither true nor false, which can't happen either!

One classical resolution of this paradox is to claim that change cannot actually happen. If X is true at one time, then X is always true. If it looks like X becomes false, then this appearance of becoming is illusory. What really exists, exists permanently, changelessly. The rude surprises of life are in fact illusory. The cure is to free oneself of this illusion, to train oneself and transform oneself so one's experience is only that of the eternal realm of unchanging truth.

Mathematical Physics

At the heart of physics, at the heart of science, lies the differential calculus of Newton and Leibniz. Since its origin some 300 years ago this calculus has been elaborated and refined. Careful methods of reasoning now permit more sophisticated resolutions of the paradox of change.

From a mathematical physics perspective, the paradox arises from a flaw in reasoning. It is not valid to claim that, "There must be some last time A when X is true." Physics models time as real numbers. Real numbers are built up starting with integers like 1, 2, 3, then adding rationals like 1/2 and 2/3, and finally filling in the infinitesimal spaces between neighboring rationals with irrationals like pi and the square root of 2. Not every bounded set of real numbers has a maximal element. For example, consider the set of all real numbers greater than 0 and less than 2. Pick any number N in the set. N must be less than 2, because it is in the set. Pick another number M halfway between N and 2. M is still less than 2, so it is also in our set. M is greater than N, so N could not have be the maximal element of our set. Whatever N we might propose to be a maximal element, we can always construct M in this way so that M is bigger than N yet still in the set, still less than 2. So this is an example of a bounded set of real numbers with no maximal element.

Sets of times, just like sets of numbers, need not have maximal or minimal elements, even when bounded. Since the paradox of change depended on the existence of such maximal elements, their nonexistence resolves the paradox.


Real numbers turn out upon further study to be quite strange objects. As mathematics continues to develop, families of alternative systems with subtly different properties have been explored. For example, model theory opens the door to nonstandard analysis which introduces infinitesimal quantities. Which number system provides the correct treatment for time?

A pragmatic approach to this question is to look at the nature of measurement. To understand better how it is possible for X to be true at one time and then to be false at some later time, let us consider the construction of a measurement device that can tell us at what times X is true and at what times is it false. Whether we are to deal with runners and reptiles or with seeds and sprouts, the modern way to build a measurement device is to translate phenomena into voltages. So let us suppose that we can arrange suitable electrodes and amplifiers so we have a signal on a wire with positive voltage when X is true, otherwise X is false. We want to measure the time when X changes from true to false.

The fundamental problem here is one of converting a continuous quantity to a digital one. A measurement is recorded as a number, for example "1.95". After many measurements and an analysis of experimental procedures, an estimate of experimental error will be added to this measurement when it is reported, for example "+/- 0.01". But it will be enough if our measurement process can somehow generate the simple result "1.95". One approach would be to let some clock run as long as X is true, and turn it off when X becomes false. So the time when X became false can be reported by simply reading the clock after it stops. If the clock uses an analog display, then the experimenter must examine the clock hand position relative to the dial and decide which mark the hand lies closest to.

It is somewhat awkward to have a human decision intervene right at the crucial point where our time measurement is to be performed. This obscures things right where we are trying to make them clear; it seems to resolve the paradox with a mystery. To avoid this obscurity, let us use a digital clock instead. We can just record the digital clock reading directly onto a computer disk to be incorporated into the experimental report without any mysterious human decisions involved. Curiously, it turns out to be impossible to build any such device with complete reliability. Just as a human experimenter might have trouble deciding how to record the time when it falls right between two marks, any physical device will have a sticking point, where things will jam up if X happens to switch from true to false just as the digital clock is flipping digits. This is a classic problem, called "metastability", in the design of digital circuits. Actually the problem is far more widespread than just digital circuit design. Anytime a continuous set of possibilities has to be cut into discrete parts, there is always some bit of trouble.

Consider negotiating an intersection controlled by a traffic light. Most of the time things go smoothly. But every once in a while, the light turns yellow when one is at an awkward spot, not obviously so far along as to easily clear the intersection before the red, but not so distant either to allow for a gentle stop. Indecision arises, to go or to stop? For different folks this point of indecision will arise at different distances from the intersection. But between the go zone and the stop zone there is always difficult boundary. Similarly, when people pass each other moving in opposite directions, there is always the choice of passing on the right or the left. Or which person should pass through the door first?

It may seem peculiar to be using a facet of digital systems design as a means to illuminate the nature of change. But in fact digital systems such as computers were actually first envisioned as purely conceptual devices, to clarify issues in mathematics such as the nature of numbers and methods of reasoning. The engineering of computer systems provides a mirror for the general human effort to build a stable, reliable, predictable world. Often this mirror can show more explicitly and clearly the problems and paradoxes that arise but obscurely and mysteriously in the general effort.

Metastability is the pragmatic manifestation of the paradox of change. It is not possible to build a device that can with perfect reliability report the time at which some feature of the world changes. Rude and awkward suprises seem to be unavoidable!