Let’s get small, part one

You have before you a piece of paper. Being a philosopher and a scientist (for the present, we will take those words to be synonyms), you use your senses to analyze this piece of paper. You see that it is white. You see that it is flat. Seeing a measuring device, you use it to determine that the paper is eight and one half inches long and eleven inches wide—or, if you prefer, 261 mm (millimeters) x 279 mm. Measuring the thickness of the paper is not so easy, but you are a clever philosopher and scientist. You stack one hundred sheets of paper and find that the stack is one-half inch tall, or 1.2 mm. Therefore, you know that one sheet of paper is one-two-hundredth of an inch thick, or 0.012 mm—about the smallest size your eyes can see or your fingers can feel.

Scent and taste do not reveal useful information about the paper, but your hands tell you that the paper is smooth, but the edges and corners feel sharp. Left alone, the paper makes no noise. With your hands, though, you can cause it to make noises when you flap it in the air, when you crumple it, or when you tear it.

Tearing the piece of paper gives you a new thought. You now have two pieces of paper, both remaining white and flat, with the same thickness and width, but neither as long as the original. How long can this process continue? Can you continue to tear the paper into smaller and smaller pieces? And will each piece remain a piece of paper? Modern science teaches us about molecules and atoms while we are young, but for centuries philosophers and scientists were lacking that information. For centuries they wondered how small a piece of paper could remain paper, and what it might be if it was no longer paper.

With a microscope, you can see that paper consists of fibers. Perhaps with tiny, delicate tools, you could isolate one fiber from the paper and chop it into shorter and shorter lengths. Even this experiment will not answer the age-old question about what tiny parts might make up a piece of paper. Logically, three possibilities exist. Perhaps the process can continue forever—however small a piece of paper you have, you can divide it into two smaller pieces. Perhaps the process reaches a limit—a small particle of paper exists that cannot be divided into smaller pieces. Or perhaps at some point we will find tiny pieces that are no longer paper, but are something else, ingredients, elements of which paper is made. By tearing and shredding the paper, we will never determine which of these results is real.

We set the paper aside for a little while, and instead we consider a drop of water. How big is a drop? For convenience, we will define one drop as one twentieth of a milliliter (0.2 mL) or one one-hundredth of a teaspoon. Modern science tells us that one drop of water contains 1.5 sextillion molecules of water. Sextillion is a real number, unlike jillion or zillion. It is written as a one followed by twenty-one zeroes. When dealing with huge numbers or tiny numbers, scientists prefer to use “scientific notation”—in the case of one sextillion, writing the number as 1 x 10²¹. Another shortcut is to use special measurements, such as nanometers or Angstroms. To try to put this number into perspective, though, let’s take that drop of water and divide it in half. Then divide the half-drop in half again, and do so a third time, a fourth time, a fifth time, and on to a tenth time. Now we have a speck of water that is about as high and wide and thick as the thickness of one piece of paper—and it still contains 1.5 quintillion molecules of water.

With special instruments, we continue dividing that one speck of water—slightly less than one thousandth the original drop—in half, and divide that half in half, until we have done that process another ten times. The invisible bit of water we have now is one millionth the size of the original drop, and it contains 1.5 quadrillion molecules of water. Repeat the process another ten times, and what we have is one billionth the size of the original drop and contains 1.5 trillion molecules of water. Now we are getting to numbers we recognize—at least if we pay attention to the national budget. Billions and trillions are somewhat familiar. Along the way, we may begin to appreciate just how tiny one molecule of water happens to be.

But another thing has happened. By the eleventh or twelfth division of that drop of water, what we had left was not really water. It still contained water molecules—an unimaginarily huge number of molecules—but that water was no longer wet. Drop it on your skin, and you would not feel it. Drop it into a glass of water, and you would not hear it land or see the ripples. It takes an enormous number of molecules of water to be sensed as water, just as it takes an enormous number of molecules of chlorophyl before we can see any green in a leaf.

We will return to the water again and will look at its molecules and consider even smaller parts of the molecule. But, first, we will experiment again with the paper. J.

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