Shades Of Reality

Chapter One

The Smoothness Principle

As the sun slowly sinks below the clouds on the western horizon, the sky gradually takes on a new set of colors -- delicate shades of red, orange, and purple -- which gently nudge away the blue in preparation for the coming of the night. Even now you can just about begin to make out one or two stars, and then a few more, and soon even more still. Meanwhile, as you walk along the edge of the water, the sky to the west still continues to glow with the soft golden-red luster of twilight.

The sea breeze has started to become quite cool now, and you wish that you had had the foresight to bring your jacket with you. And since it's getting harder and harder to see where you're walking, you decide that it's time for you too to start heading back home.

The walk back to your house is pleasant but uneventful, and you soon find yourself standing next to the steps of your front porch. As you glance upward, you see thousands of tiny diamonds twinkling in the night sky, and you contemplate the idea that some of those tiny lights are actually even bigger and brighter than our own sun.

I hope that you enjoyed your little trip to the beach and that you didn't get too sunburned. But now I would like to ask you something: When you left home this morning to go to the beach, it was light outside. But by the time you got back to your house, it was already dark. Now, we all know the difference between light and dark. They each represent totally opposite concepts. (In fact the two concepts are about as different from each other as night and day!) So you shouldn't have any trouble answering the following little question: when did the sky become dark?

"It became dark after the sun went down," is probably the answer that you would give. "Everybody knows that!" you say.

True, but could you please be a little bit more specifc? How much after? One second? One minute? One hour? I want to know the exact instant at which the daylight ended -- the moment when "not dark" turned into "dark."

If that question was too tough for you, then let's try a different one (but also about the same beach): where is the boundary line between the beach and the ocean?

"Well, that's an easy one," you say. "The ocean is the part that has the water, and the beach is the part that has the sand. And where they meet, well, that defines the boundary line."

OK, but once again, I'm looking for a more exact answer. For example, how about the sand that is wet? Should it be classified as being part of the ocean because it has water on it, or should it be classified as being part of the beach? And how wet or dry does the sand have to become before it changes classfication? Furthermore, consider water that has washed up onto the beach and formed into little pools. Should such pools be classified as being part of the ocean, or part of the beach?

By now you are probably thinking to yourself: "Who cares? These kinds of questions are completely trite and meaningless. I've got better things to do with my time than to sit around thinking about silly things like finding the exact time of day when the sky gets dark!" And to some extent you're right! The actual answers to the questions (if answers even exist) are totally irrelevant. For example, suppose I were to say to you: "The correct answer to the question (about the sky getting dark) is exactly 27.3 seconds past 6:48 PM." (That's not the answer, but let's pretend that it is.) Of what use would that particular piece of information be?

My sole purpose in asking the questions is to point out an extremely important (but seldom recognized) aspect of reality, which I call the smoothness principle:

Everything in physical reality occurs smoothly.

What the smoothness principle says is that physical changes do not occur instantaneously at some point in time (Figure 1.1). Instead, all changes take place over a period of time, and they take place in a continuous way (Figure 1.2). In other words, the mathematical concept of a step function does not exist in physical reality.

step function	physical reality
(Figure 1.1)	(Figure 1.2)

As another example of the smoothness principle, imagine that you are holding one end of a horizontal ten-foot steel pole and the other end is inside a furnace. Your end of the pole may still be quite cool even though the other end of the pole is red hot. Yet there is no precise location along the pole at which the temperature changes from hot to cool.

Or consider the metamorphosis of a tadpole into a frog. There is no point in time at which anyone can meaningfully say, "One second ago it was still only a tadpole, but now it is definitely a frog."

The smoothness principle may come as a surprise to many readers, and they are undoubtedly already constructing a mental list of "Oh-yeah?--what-abouts." Let me see if I can anticipate some of them.

When electricity flows through the filiment of a lightbulb, the filiment requires a finite amount of time to heat up. The bulb does not just light up "all of a sudden."

Likewise, turning off a lightbulb doesn't instantaneously cause the bulb to become dark. It is easy to verify this for yourself, if you use a clear (not frosted) bulb in which you can see the filiment. Close your eyes while the bulb is on. And then, just as you turn the bulb off, quickly open your eyes. You can actually see the filiment fading out. In fact it still continues to emit light even long after it appears to have reached its "off" state. But this light is in the infrared portion of the spectrum, and the human eye cannot see it.

An explosion is nothing more than an example of combustion, except that it takes place very quickly. Molecules of the explosive combine with molecules of the oxidant and release thermal energy, which in turn, causes other molecules to combine. But it requires a finite amount of time for the individual molecules to interact with each other, so the combustion does not occur all at once. (And even if it did, the expanding ball of hot gas has limits on how fast it can expand.)

By using time-lapse photography, even the flight of a bullet (which is nothing more than the result of a tiny explosion of gunpowder) can be continuously observed.

It might seem intuitive to define a "collision" as being the exactinstant in time when any atom of one of the balls just touches any atom of the other ball. And so the two balls should abruptly change from a state of not-touching to a state of touching at some precise moment.

The flaws in this intuitive definition are that it incorrectly assumes that atoms are objects that have well-defined boundaries, and that the individual atoms of each ball are totally motionless with respect to its neighboring atoms. We know that all atoms are in constant motion because of thermal energy. And quantum theory tells us that their precise locations are slightly "smeared out" (the Heisenberg uncertainty principle). As the two atoms approach each other, they begin to "feel" each other's presence gradually. Therefore, even billiard balls have an extremely slight degree of smoothness to their collisions.

The smoothness principle only applies to physical events. Terms like "Monday" and "Tuesday" are man-made concepts and, as such, can take on whatever attributes we wish to assign them. Therefore the day of the week changes "instantly" simply because we say it does!

It might seem as if I'm being a little nit-picky in some of my justifications, but then I have to be. Because reality doesn't merely "go away" just because things may have gotten too small or too fast for our eyes to see. Still, we like to pretend that it does, so that we can make our world view simpler. "For all practical purposes," is the phrase we like to use. But, in so pretending, it is important not to loose sight of the distinction between the practicality and the reality.

That the smoothness principle may have initially struck you as somewhat surprising, is symptomatic of a phenomenon sometimes referred to as the mismatch problem -- reality is gray, but in our minds the world is black and white. A lightbulb is either on, or it's off. Each person in our society is either an adult, or else they're a minor A window is either open, or it's closed. A body of water is either a lake, or it isn't a lake. If something isn't all, then it's nothing.

In reality, vague boundaries exist between things and not things -- between day and not day -- between lakes and not lakes. At what point does "dry" become "damp," and when does "damp" become "wet"? How does one differentiate between very wet ground and a small puddle of water? What's the difference between a very large puddle and a small pond? What distinguishes a large pond from a small lake? Minnesota claims to be the land of 10,000 lakes. Clearly this is an exaggerated estimate. But if somebody were to try to count them, would the actual number of lakes in Minnesota turn out to be an even number or an odd number?

When does the battery in a flashlight have to be replaced? At what precise instant does the light become dim enough to justify referring to the battery as no longer being "good." A battery starts to become "bad" the first moment you use it. And even a so-called "dead" battery usually has at least a tiny amount of electricity still in it.

Since transitions in the real world don't happen at points, questions like: "At what point did the sky get dark?" are semantically meaningless and self-contradictory (like asking about the distinct smell of an odorless gas, or asking about what it feels like to be dead). Furthermore, since there is no distinct boundary line for the transition, it is difficult to determine if certain specific points in time should be classified as being in the category called "day" or the category called "night." However, at any point in time, the question: "Is it dark yet?" does have a meaningful answer! I will explain what I mean later in the book.

In his book, Fuzzy Thinking (pages 18-19), Bart Kosko defines what he calls the "Fuzzy Principle: Everything is a matter of degree." However, he acknowledges that things from the world of mathematics are not fuzzy. An expression like, "Two plus two equals four," is 100 percent true. But when we move out of the artificial world of math, fuzziness reigns. Kosko writes:

What Kosko is saying is that fuzziness increases as the valence increases. The more options there are (i.e., the more multi the multivalence is), the greater the degree of fuzziness. (Yes, even fuzziness comes in degrees!) But it is incorrect to say that bivalence is the opposite of fuzziness, and that fuzziness "means three or more options." (Non-fuzzy does not suddenly become fuzzy at the three-option boundary line!)

As the number of options increase, the closer they can come to resembling a continuum (the highest degree of fuzziness). And specifying points on a continuum requires the use of precise numerical values. But we all want things to be easy. Few want to deal with exact precision. So we look at reality and try to distill the essence of what we see into simple bivalent all-or-nothing categories, because "yes" and "no" are easier concepts to deal with than numbers, especially fractions. (Remember what part of arithmetic you hated most when you were in school?) And so we trade off reality for simplicity. We round off the fractional values of reality to the nearest "1" or "0" and pretend that that is reality. We pretend that fuzziness doesn't exist.

"The fuzzy principle has emerged from almost three thousand years of Western culture," writes Kosko, "from three thousand years of attempts to deny it, ignore it, disprove it, relabel it, and axiomize it out of existence. But fuzziness remains despite our best efforts to get rid of it."

Find somebody whose birthday is coming up in a few days and ask them how old they are. You'll almost always get an answer like, "Right now I'm still only fifteen. But next week I'm going to be sixteen." They seem to feel that their age is some kind of "title" that becomes conferred upon them once a year on the aniversary of their birth, and that numerical "age" remains with them until their next birthday. But a person's age is not a title. Their age is a measurement of how many years have elapsed since the day of their birth. And a person grows older instant by instant, not year by year. You do not suddenly jump to becoming a year older on your next birthday, any more than history suddenly jumps 100 years at the turn of a century.

Rounding off is something we do almost unconsciously. For example, if we see a glass of water that is 95 percent full we refer to it as being full, even though the water doesn't actually quite reach the brim. Similarly, when we pour the water out of the glass we refer to the glass as then being empty, even though the inside walls of the glass are still wet. Except in special cases, this kind of imprecision just doesn't matter. The rounded off interpretation is sufficient for everyday use.

But if the water level in the glass is not close to one of the two extremes (full or empty), then rounding off becomes an uncomfortable process. If a glass is only halfway filled with water, should we round it up (and claim that the glass is full), or should we round it down (and claim that the glass is empty)? In this case, rounding off just doesn't work. Neither claim comes close enough to describing the reality of the matter. So we are forced into acknowledging the fuzziness -- the fact that the glass is 1/2 full and 1/2 empty.

When we round off, we simplify to the nearest whole number (zero or one). But simplicity comes at a cost. When we simplify, we trade accuracy for convenience. Most of the time, this decrease in accuracy is negligible. However, just remember one simple rule: Everything should be made as simple as it can be, but no simpler!

Americans today even find themselves having to round off their trash! Instead of simply having one generic trash container for disposing refuse, it is now fashionable to have separate containers for different categories of trash (such as "paper only," "glass only," "plastic only," etc.). But what do you do when you want to throw away a glass bottle that has a paper label and a plastic cap? It doesn't completely fit into any of the categories, and at the same time it fits into all of them. And so you round it off; you say that it is mostly glass, so it therefore goes into the "glass only" trash container. (We will talk more about the concept of "categories" in the next chapter.)

In passing, let me just make one amusing observation about the contradictory expression, "rounding off." When we start with a smooth reality (Figure 1.2) and "round it off," we end up creating a step function (Figure 1.1) having square corners!

Two different events almost never happen at exactly the same time. Try to slap the top of a table with your left hand and right hand at exactly the same instant. You might be able to come close. But if you could measure the timing very precisely, you would probably see that you are usually off by quite a few microseconds. And yet we speak of closely occurring events as "happening at the same time." We round off the slight inequality to produce an equality.

Most of the time this kind of rounding off is inconsequential. But now consider the famous "stop sign dilemma:" Whenever two cars approach an intersection controlled by stop signs, the first car to arrive at the intersection has the right-of-way. However, if the two cars arrive at the "same time," then the car to the driver's right has the right-of-way.

The problem, of course, now becomes one of trying to define the boundary line that separates "equality" from "inequality" The driver on the left may have arrived at the intersection a fraction of a second ahead of the other driver. And because he got there first, he claims the right-of-way. But the driver on the right may have called their arrival times a draw, and therefore he claims the right-of-way because his car is the one on the right.

The mathematical concept of "equality" is therefore somewhat fuzzy when applied to everyday language. We will discuss the concept of fuzzy equality further in Chapter Twelve.