I have a problem with division. I lose the ability to visualize the remaining pieces after a sequence of repeated operations. I have a similar problem with multiplication, but division will be the focus for now. To keep it simple I’ll use multiples of ten, a standard practice in science. I’m not trying to do math though. My issue is mind management. I’d like to know what I’m doing when I divide. But, I can’t convince myself that I do! I suppose I could just say that infinity puzzles me, both the infinitely large and the infinitely small. I can “see” what I’m doing for a while, but not for long, and not all the way. Infinity, after all, goes beyond “all the way;” in truth, infinity goes on and on and on ad infinitum.
It’s easy enough to know what you’re doing at the start. I take a straight line measuring an inch and divide it by ten. I can see that I have ten pieces 1/10th of an inch in length. Now I divide by 10 again and I have 100 pieces of 1/100th of an inch each. Then I divide each of those 100 pieces by 10 and I get 1000 pieces of 1/1000th of an inch. Do I sense a problem yet? Not really, but I’m starting to feel uneasy. I wouldn’t want to be asked to cut an inch of anything into 1000 pieces, which would get harder still, but not impossible, when the 1000 pieces are divided by 10 into 10,000 pieces 1/10,000th of an inch in length. That step in the shorthand of mathematics is 10-4. 10-5 produces 100,000 pieces 1/100,000th of an inch, and 10-6 results in one million pieces of 1/1,000, 000,000th of an inch.
I’ve tried this exercise with my wife and a few other people. Right about now they say: “Stop it!” “That’s enough!” “My head hurts!” An aside is in order. Lots of people feel that incessant division of this sort is physically and mentally taxing. It’s actually experienced as pain, as an assault on body and mind as well as individual rights. In experimental psychology the concept of “discrimination threshold” or “discrimination limen” refers to a psychological threshold above or below which a difference in a stimulus is noticed. For example, an experimenter increases the weight of an object incrementally until it reaches a level the subject notices as a step difference by saying, “that’s heavier!” Each increase had been heavier, but this increment in weight is the first one noticed. The physical weight perceived as greater is thereafter used to mark the psychological awareness threshold. Such thresholds are important in setting up airplane cockpits, and for the design of congenial machinery. I suspect that a trillion, 10-12, is the discrimination limen above which, or is it below which, I lose faith in what I’m doing when I divide. At that point I find the thought irresistible that the inch has disappeared for good; no way can that sucker be divided into more than a trillion things. It took guts, I tell myself, to persevere this far. My unwilling co-experimenters find that their cut-off threshold is 10-6, a measly million, at which point they get up and leave the room with their heads hurting. The experimental psychologist might be impressed by such threshold differences, but any good high school math or science student will denounce the handwringing, call us all wusses, and announce with disgust that division, we should know, goes on to infinity, correction, goes on infinitely.
That comment leads to another aside. I think I know what happens to smug students here. When mathematics students get to 10-9, or 10-10, they too give up believing that they could do the manual work of cutting an inch into countable pieces. They don’t really think they could cut that inch into a billion miniscule pieces. At some level, at some threshold, they give up understanding division in visual work terms and just pay attention to the math operation. In other words, ten (10) and its exponents (2, 3, 4, and more) are psychologically comforting operations. Once you understand the logic, it’s no more trouble to count 10, 10-2, 10-3, 10-4, 10-5, 10-6, 10-7, 10-8, 10-9, 10-10 than it is to count 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. No big deal, it is just mathematics. Forget whether you believe it, disbelieve if you want, but really, give up the angst, just do the math.
But I don’t want to be comforted just yet. So, please refuse the comfort blanket. Keep your eye on the inner experience, and continue with the pain. As it happens, 10nth power has a long history and is in such wide use in all sorts of scientific and engineering endeavors that its levels have names that go on—no surprise here—endlessly. For example, 1075 is called quattuorvigintillion in the U.S.A lexicon, but dodecilliard or duodecilliard in the UK system. And 10303 is neatly called centillion in the USA but known awkwardly as guinguagintilliard in the UK. I can’t find the name for 10500th, but I do know that this is the estimated number of universes posited by physicists who give credence to the multiverse theory, the theory that many universes may exist. I find that fact somewhat comforting, that there are other possible universes in such absurd abundance, but why should I feel comforted? Is it that life could go on that way? But how could it, we’re speaking of other universes, and for all we will ever know, we only have this precious one to live in!
Since we’re on the subject, you should know too that 10100th is everywhere called googol, including in the USA and UK, which renders 10googol a number called googolplex and 10googolplex a number known as googolplexplex, and yes, I too am feeling the pain now, 10googolplexplex produces a number known as a googolplexplexplex. Just give me one more and I’ll stop. On the minus or small sizes, a googolplexplexplexminix, I kid you not, is 1/10th googolplexplexplex! Oh, I see how it must work with 10500, of multiverse fame. Wouldn’t that be five googols, or half a googolplex?
So here is the question. What image of the product should I have after dividing our one inch by a googolplexplexplexminex? What do I comprehend if I convince myself that I comprehend that? At some point, I lose faith in the existence of that inch and those pieces. Sand on the beach or specks of milled flour wouldn’t be abundant enough, even though these have been ground rather than cut. Nothing I know of will compare. What, I ask you, would work for you? At some point we decide, do we not, that Zeno must have been right: the hare can’t possibly catch the tortoise in this kind of a divisive numbers game. This is Alice in Wonderland stuff. Lewis Carroll, where are you when we need you!?
But wait; hold on a moment, that argument can’t be right. A distinction must be made between things, actual physical objects, and schemes to measure them. I have been asking about the fate of an inch in an infinite division game using 10nth power. But all I’m really doing is applying one measurement scheme to another measurement scheme, an inch in one to 10n in another. Of course the inch never goes away and pieces remain, no matter the extent of serial division. All that is being done are mathematical transformations. I’m making statements about the durability of a one-inch long object in the physical world on the basis of mathematical possibilities. Such real world objects, say a length of string or wood, will actually disappear after a series of repeated divisions, and disappear at specific points in the series, at points which chemists and physicists would be needed to identify for us. We know as a fact that sawdust results from cutting wood, and that the cutting stops with the sawdust. We may decide to combine, liquefy, and compress the stuff, but we don’t, like idiots, go on cutting the sawdust. Numbers may be infinitely divisible, but objects are not. At some point objects decease into remnants or transform into another kind of object. Substances also transform chemically with heat, from solids to liquids and gasses, and back around again. It requires energy to divide objects. Objects don’t so much disappear as reappear as something else. Given the distinction between dividing things and dividing numbers, a faster runner can pass a slower one, and the rabbit can catch and pass the tortoise, no matter how fervently the distance between them is repeatedly halved.
Perhaps I should pursue my discomfort with the infinitesimal— my psychological pathology with division—in a different way, while continuing in the same general direction. Divided objects don’t usually cease to exist when they disappear. The dividing of objects doesn’t typically end with sawdust. A massive amount of reality disappears with repeated division and doesn’t go away. It is just too small to see. The foundation reality of the world is not only beyond the seen, and beyond the scene, but infinitesimally miniscule. Oh My! So that is why googolplexplexplexminex might be needed! It ‘s needed to apprehend and understand reality.
What’s the smallest real thing in the universe? From what physicists now know, the answer is probably quark or electron; they are about the same size. Keeping things simple, three roundish quarks, bonded in a rough triangle, constitute both protons and neutrons, which, bonded together themselves, make up the nucleus of the atom; this is true for each and every atom, for all atoms. Protons hold a positive electrical charge while neutrons have no charge. When the nucleus is joined through electromagnetism with a circling electron, with the electron’s negative charge attracting it toward the positive charge of the proton, we have the general model of the atom. One electron circling a nucleus of one proton is a hydrogen atom. Two electrons circling a nucleus of two protons and two neutrons is a helium atom. The number of protons and neutrons in a nucleus distinguish the different kinds of atoms in the Table of Elements. The number of protons provides an element’s Atomic Number. The number of protons and neutrons together in the nucleus provides its Atomic Mass Number. Thus, an atom can be visualized as a number of protons joined with an equal or close to equal number of neutrons in its nucleus, and orbited by an equal or close to equal number of electrons. The radii and diameters of atoms are inexact due to the uncertainty principle, which specifies that the location of a particle cannot be determined exactly without losing certainty of its momentum, and vice versa, the momentum of a particle cannot be exactly determined without losing certainty of its location. You might also want to know that the outer area of the atom in which the electrons circle in beautiful waves around the nucleus, in orbits of varying and increasing energy, one electron for every proton in the atom, is huge compared to the tiny nucleus with its protons and neutrons, both constituted of quarks.
I must stop here for another aside, a confession really, because I feel myself transgressing a threshold having to do with authenticity and professional identity. I sense this violation as I write explanations about the composition of atoms. You see, I know little to nothing about math and physics. I have no credential in either field. I was a lousy student of math and physics, chemistry too, when I studied these subjects in school 55 years ago. I took algebra I three times and passed the course only once, and I’ve never received a grade higher than B in a science class. I started serious reading in these subject areas only a year ago, and I’ve had no professional practitioners with whom to discuss my reading. So you must not believe a word I say. I’m a complete novice. You should check out all of this information with reputable sources. I’ve been doing my own checking, of course, and I have some doubts about what I’ve been saying. So you should too. There, I feel better, if only momentarily.
What I’ve been trying to do in my novice status is establish that the atom, any atom, is beyond tiny, that its nucleus of protons and neutrons is vastly tinier still, and that its quarks and electrons are absurdly miniscule. Yet the particles in this miniscule world are real! They exist, and so do more than forty other particles that are getting splattered around at the Large Hadron Collider (LHC) in Switzerland where streams of protons are fired around in a circle against each other at speeds close to the speed of light. The size of these particles, any of them, can only be described in multiples of ten, and only then with certain scales. You are going to need scales, all stated in multiples of ten, to study and visualize the reality of the atomic world, and, also, of course, to visualize the molecular and cellular worlds of chemistry and biology. And you’re going to need them if you’re going to be a surgeon using nanotechnology. If the word nanotechnology has been sneaking into your consciousness, you might look it up and see how its system of measurement works.
Let me see if I can say this fairly clearly: a human body of 70 kilograms is estimated to have roughly 7 x 1027 atoms, that’s 7,000,000,000,000,000,000,000,000,000 atoms, and the smallest particle within the atom, say an electron or quark, is approximately 10-15 attometers (am). Oh, 1 attometer is 10-18 meters (m), one meter is 3.2804 feet, and one inch is .02450 meter (m). A nanometer (nm) is 1 billionth of a meter, that is 1 x 10-9th (m). Dimensions in atomic measurement are often expressed on a nanometer scale. In summary, Atoms are real, infinitesimal, and beyond our eyesight. So is the substance of much of the physical world.
As you’ve learned, I’m having problems overcoming anxiety as I write on this topic, residual, I presume, from my poor record on these subjects in school. Perhaps my dislike and lack of competence in math and science was due to the mental baggage I brought to the subject. Perhaps that is what happens to many American students. Our politicians have made our students’ failures to succeed at science and math shameful, a well publicized national disgrace. Well, shame on them for laying the blame on us.
I think I see my problem clearly now. I was taught by adults to study these subjects for impersonal and external reasons—to pass tests, to get a job, to go to college, to enter a profession, for the good of the country—and I took the cultural message to heart. I thereby burdened myself with duty, and experienced anxiety, terror, and pain. School often made me sick. I didn’t know that the physical world I lived in was my world. I didn’t own the subjects I studied. I tried to master the subject matter, but did not succeed.
It would have been better to learn for personal, internal reasons, as I do now: from interest, curiosity, wonder, surprise, and just plain fun. The spirit of physics and mathematics is most authentically felt when the potent playfulness of the human mind is engaged in the eternal wonders and mysteries of the physical world. We do not often enough invite students to mathematics and science for the puzzles, delights, and paradoxes of the physical world. Rather, we invite them to school as little workers and future producers in an economic world, as prospective employees in the “global economy.” Endless economic callouts to youth can be more than anxiety producing; they can be gut wrenching and soul killing.
I complain for youth, not for myself. I have come to love the peculiar odyssey in which mathematicians and scientists are engaged. I find following their pursuits thrilling, freeing, and nurturing. Even when I can’t fully decode the equations, I feel lucky to have found my way into their fascinating playhouse. I hope that parents, educators, scientists and politicians can concert their efforts to extend a more personal and spiritual invitation to American students. Such an approach is needed to counter-balance and supplement the economic invitation we extend now. Personal engagement is best! Focus, diligence, logic, imaginative thinking, persistence, and hard work are the rest.
Invitation extended, let’s return to the issue of the disquieted mind and the disappearing fragment that nevertheless retains existence and identity as an object dissolves from sight deep into the infinitesimal. What do scientists do when the thing they study disappears? Why don’t they go mad from fright of the unseen?
In answer to the first question, scientists don’t panic, or double up in anxiety, as I once was prone to do, just because they can’t see. Rather they get excited and motivated. The temporarily sightless scientist figures out other ways of seeing, and other ways of measuring what they see. All of the tools of scientists, including the mental and mathematical tools, can be thought of as other ways of seeing. Devices to extend eyesight are built: telescopes, microscopes, electron microscopes, antennae, radio towers, satellites, observatories, colliders, reflection screens, and the like. Scales are designed to measure objects of like size in diminished ranges. Smaller chips are invented to build computers with hugely increased computing power. Clever experiments are devised and conducted. Theories are conceived to unify previously conflicting facts into consistent, testable explanations. Tested theories are used to frame new questions for research. Glimpses of information gathered in experiments, and aided by theory, are snapped into neat and elegant equations. Colorful language, analogies, and metaphors—such as “quark,” “particle zoo,” and the like—are evoked to translate scientific shoptalk to audiences of laypeople. Scientists dream. Scientists scheme. Scientists imagine. Scientists invent clever research tricks.
Understanding physical reality, one might say, is fundamentally a light game, and the most exciting task is to figure out alternative ways of seeing when the eyeglasses currently available no longer suffice. Did you know that visible light makes up only a small fraction of the spectrum of electromagnetic radiation? Radiation—radio waves, microwaves, infrared light, visible light, ultraviolet light, x-rays, gamma rays— is all light! Radiation is another name for light.
Scientists aim to see using all the light there is available. Think of X-Rays. Think of the radio telescope. Scientific knowledge is, in a real sense, light made visible, a previously unseen world of light brought to light.
Eye metaphors, it therefore makes sense, are central to what scientists do. Listen to them talk. Watch and listen to your own talk. “I see what you’re saying;” “Focus on this fact.” “That’s a good way of looking at it.” “From Einstein’s perspective;” “Taking the frame of reference of Newton;” “From his point of view;” “She oversees the project.” “He supervises the data collection process.” “Visualize events this way:” “Here’s my vision of how it works;” “Imagine this;” “Here’s an image;” “One shouldn’t overlook.” “Picture this scene.” Do you see what I mean? “ “That’s a great insight!” “Upon reflection.” In hindsight.” “A light bulb went off.” “Here’s a bright idea.” “I saw it in a flash.”
Why don’t scientists go nuts for lack of immediate vision, certainty, and knowledge of the unknown? The general answer, I think, is they pursue questions that entice them and avoid overwhelming themselves with the terrible burden of knowing everything at once. One intriguing problem at a time is good enough to live on mentally, emotionally, and spiritually until its answer educes another interesting question. They don’t fall into the trap of thinking they should or could read and know everything and thereby answer all questions. They don’t pretend to be God. They thrive in uncertainty and in the quest. The scientist is in spirit a pilgrim. One exciting problem is enough to start. Trial and error, experimentation, theorizing, the imagining of scenarios and systems, are ways to proceed. Proving an idea wrong through experiment is progress, as is proving an idea right. One good idea, question, or finding leads to the next, and that one could be more exciting still. Science walks a road of discovery
Children may be bored to death by dreadful courses. They could be ruled over today by the burden of passing all those courses and tests, and of securing those diplomas and degrees. But then tomorrow, presto, the student looks at the sun, her sun, his sun, and asks how it could be that a furnace like that doesn’t burn up in a year or two? How can that familiar orb have been warming this place for billions of years? If that question captures that kid, a life of science and mathematics is well underway! If only we could help our children see that! They will have seen a light to lead them into and through the unknown. I hope they have confidence to think large. I advise them to give up the fear of thinking small. I’m not going to do that anymore, and neither should you.
Will Callender, Jr.©
February 9, 2012