When I was fourteen, my parents sent me to Texas to work on my grandfather’s ranch. The first job would be to paint the fence around the field between his house and the road. When I asked what else I’d be doing, he replied, “Let’s see how long it takes to paint the fence.” Two months later, having painted for eight hours a day, I returned home, the job of painting the fence still not finished.
The word “comprehend” means taking something in all at once. Since flat land let me see that fence all at once, it seemed comprehensible. In fact, if I held my two thumbs in front of my face, I could make the fence seem to fit between them. So thinking it might take a few days to paint the fence seemed reasonable. Problem was, my brain does tricks with perspective. In reality, that field was probably close to ten acres, the fence probably ten football fields long. “Comprehension” of very large things requires large scale trickery. It depends on deception.
I should have known better than to trust my brain about the fence. My teacher Paul Czaja had told our class the story of the Emperor’s chessboard and the grains of rice: the Emperor said that if a single grain of rice were placed on the first square, two grains on the second square, four grains on the third, and so on, until the 64th square, the final square would require enough rice to stretch to the moon and back seven times.
The story of the Emperor and his grains of rice “wowed” me with the power of exponential growth. I knew the moon was far away, and a grain of rice very small. Stretching back and forth seven times had to make it a very large number indeed.
Of course I wanted to know what the total number of rice grains was in numbers I could understand. Rather than simply give his class the answer, Paul asked us to compute the number ourselves. Our homework was simply to multiply by two sixty-three times.
If Paul had told us that the total grains on the chessboard came to 18,446,744,073,709,551,615, I’d have realized that the number was larger than any I’d ever seen – but my brain would have attempted to make sense of the number’s “bigness” in the same way it had made sense of the fence in Texas – namely, by making it appear far smaller than it really was. The only way to see a huge field was to make it seem small enough to fit between two thumbs. The only way to make such a large number seem comprehensible is to reduce it to funny little shapes on a page that we call numerals.
Upon seeing that large number on the page, my brain immediately perceives it’s large, because it is physically longer than most numbers I see. But how large? If it takes up two inches on the page, my brain suggests it’s twice as long as a one-inch long number, the same way a two-inch worm is twice as large as a one-inch worm. But my schooling tells me that’s wrong, so I count the digits – all twenty of them. Since twenty apples is twice as many as ten apples, and since my brain has spent years making such comparisons, my intuition suggests that a twenty digit number is twice as large as a ten digit number. But I “know” (in the sense of having been told otherwise) that’s not right.
But here’s my real question: if my intuition is wrong, is it even possible for me to “know” how wrong? Does “calculation”: amount to “comprehension”?
Psychologists tell me my brain is useful in this inquiry only because it tricks me into seeing the distorted, shrunken “mini-icon” versions of things – numerals and digits, rather than the actual quantities themselves. If we’re asked to describe what it feels like to be alive for a million years, we can’t. We’ll never be able to. And for the same reasons, it seems to me, we can’t “comprehend” the reality of 264, only the icons we can comprehend – the mental constructs that only work because they’re fakes.
Consider the Emperor’s assertion that the rice would be enough to go to the moon and back seven times. That mental image impressed me. It made the size of 264 seem far more real than it would have if I’d merely seen the twenty digits on a page. But do I really comprehend the distance between the moon and the earth?
The moon’s craters make it look like a human face. I can recognize faces nearly a hundred yards away. So… is my brain telling me that the moon is a hundred yards away?
I’ve seen the models of lunar orbit in science museums – the moon the size of a baseball two or three feet away from an earth the size of a basketball. My brain is accustomed to dealing with baseballs and basketballs. I can wrap my brain around (“comprehend”) two or three feet. So when I try to imagine rice extending between earth and moon seven times, I relate the grains of rice to such “scientific models.”
But is that , too, an exercise designed to trick me into thinking I “understand”? Is it like making a fenced field seem like it could fit between my two thumbs?
I have little doubt that analogies seem to help. Consider that a million seconds (six zeros) is 12 days, while a billion seconds (nine zeros) is 31 years and a trillion seconds (twelve zeros) is 31,688 years. Wow. That helps me feel like I understand. Or consider that a million hours ago, Alexander Graham Bell was founding AT&T, while a billion hours ago, man hadn’t yet walked on earth. A billion isn’t twice as big as ten thousand, it’s a hundred thousand times bigger.
Such mental exercises add to our feeling that we understand these big numbers. They certainly “wow” me. But is this just more deception?
Distrusting my brain’s suggestions, I decide to do some calculations of my own. A Google search and a little math tell me that seven trips to the moon and back would be about 210 billion inches. Suppose the grains of rice are each a quarter inch long. The seven round trips would therefore require 840 billion grains of rice. The math is simple. If there’s anything my brain can handle, it’s simple math. Digits make it easy to do calculations. But does my ability to do calculations mean I achieve comprehension?
Thinking that it might, I multiply a few more numbers. My calculations show me that the Emperor’s explanation of his big number was not only wrong, but very wrong. The number of grains of rice on the chessboard would actually go to the moon and back not seven times, and not even seven thousand times, but more than 150 million times!
Such a margin of error is immense. I don’t think I could mistake a meal consisting of 7 peas for a meal consisting of 150 million peas. I don’t think I could mistake a musical performance lasting seven minutes for a musical performance lasting 150 million minutes. What conclusion should I draw from the fact that I could, and did, fail to realize the difference between seven trips to the moon and back, and 150 million trips?
The conclusion I draw strikes me as profound. It is that I have no real “comprehension” of the size of such numbers. I’ve retold Paul’s chessboard story for fifty years without ever once supposing that the Emperor’s “seven times to the moon and back” might be wrong. Could there be any better evidence that I have no real sense of the numbers, no real sense of the distances involved? The fact that I can’t really appreciate the difference between two such large numbers tells me that I’ve exceeded the limits of any real understanding. My brain can comprehend baseballs and basketballs. Using simple tools like a decimal number system, I can do calculations. But when I try to comprehend the difference between 18,446,744,073,709,551,615 and 18,446,744,073,709,551,615,000,000, am I really able to understand what it means to say that the second number is a million times larger than the first?
I got my first glimpse of the difference between “calculating” and “comprehending” about five minutes into Paul’s homework assignment. Just five minutes into my calculations, my brain was already playing tricks on me. Pencil in hand, there were already too many numbers going around in my head. I was (literally) dizzy from arithmetic overload. I made errors; I began to slow down, to be more careful. The numbers were already absurdly large. Five minutes after starting with a sharp pencil, my numbers were eight digits long; I was multiplying scores of millions, but no matter how slow I went, the frequency of errors increased. After ten minutes, I had to sharpen the pencil because I couldn’t read my own writing. After twenty minutes, my fingers hurt. Soon, I could feel calluses forming.
Since the first eight squares of the chessboard had taken me about fifteen seconds to calculate, I’d unconsciously supposed that doing all eight rows might take me eight times as long. But the absurdity of such a notion was becoming quickly apparent. Looking up at a clock after half an hour, still not quite half way across the chessboard, my inclination, even then, was to think that the second half of the chessboard would take about as long as the first. If so, I’d be done in another half an hour. So I determined to finish. But a couple of hours past my normal bedtime, I assured my mother I’d be finished soon – notwithstanding finger pain that had been crying for me to stop. By the time I finished –about two a.m., the calculations having taken me over five hours to complete despite the fact that I’d long since given up trying to correct errors – my fingers were so painfully cramped it seemed they’d never recover.
In this way, I started to “feel,” to “experience,” the hugeness of the number 18,446,744,073,709,551,615.
By reading this account, some may feel they have a deeper sense of the enormity of 2 to the sixty-fourth power. But I’m willing to bet that if you’ve never done it before, but now actually try to CALCULATE it, yourself, you’ll appreciate the hugeness in ways that symbols on a page – the stuff our brains use – can never convey.
As I look at the digits on the page, my brain is trying to spare me that visit to the land of reality. It strives to shield me from calloused fingers and mental exhaustion with its easy comparisons, its suggestions to count digits, even its way of hearing words and using them to imagine a story about going to the moon and back seven times. In the same way, perspective had tricked me into thinking I understood the length of a fence as if to spare me a sore back, sore feet, sunburn and thirst for two months. But the experience of painting the fence had taught me more about its length than framing it with my thumbs or even counting the posts between the rails I don’t know how long it would have taken to finish painting that fence. I could do the calculations, but only finishing the job would have really made me “comprehend” the time involved, and who knows –I might have died of heat exhaustion before I ever finished.
I should know that the farther away something is, the bigger it must be, if I can see it. But through the deceit called perspective, my brain tells me precisely the opposite: the farther away something is, the smaller it appears. Stars more massive than the sun are reduced to mere pin pricks in the sky. When I remove a contact lens, my thumb look like the biggest thing in the world. What better evidence can there be that our brains are built to deceive us?
My brain (wisely) keeps me focused on things I need, like apples, and on things than can kill me, like woolly mammoths, men with rifles, fast moving cars, or thumbs in my eye. But to do this, my brain necessarily distorts the things that are far away, and the things that are many, and the things that are very much larger than me, because they are things I can do nothing about. In fact, I suspect that If my brain were asked to identify the biggest, most important thing in the world, it would say it was me.
And that might just be the biggest delusion of all.