The finite that hints at the infinite

How many grains of sand on our planet Earth? Right at this instant, there is some gargantuan number that answers that question, though I have no idea what it is, nor how to get its value, nor even how to estimate it.
How many grains of sand on our planet Earth? Right at this instant, there is some gargantuan number that answers that question, though I have no idea what it is, nor how to get its value, nor even how to estimate it.

Summary

  • Counting gigantic, but apparently finite, number is a process that will never end, and that endlessness is why it makes sense to think there’s an infinity of them

On my college campus again after several years, I’m consumed—as you can imagine—with nostalgia. Memories at this corner, that hostel looks so different, where did the warden’s home go, and ice-cream for lunch, really?

Some things are the same, others have changed. When I was a student here back in the Neolithic era, there were 12 hostels. Now there are at least two or three more.

That, and some consolidation and addition of new wings, means not just more than 12 hostels now, but room for plenty more students than in my time.

How many students, I wondered idly one morning. Probably 3,000. That’s an estimate, but I can conceive of ways to divine the actual number and, I’m sure, so can you. There is a definite count of students, and we can do that count.

Nostalgia apart, that also got me thinking—I have time on my hands, ok—of other numbers, perhaps a little more difficult to divine.

For example, and staying with the college: how many trees on this campus? I could wander the whole 300+ acres counting, but what’s a shrub, what’s a bush, what’s a plant, what’s a tree?

Deciding those questions is what makes this a tougher enumeration.

Or try this: how many leaves? Ah, now that’s still tougher, in an entirely different way. There’s far too many to count on even one tree, let alone a whole campus full of them.

And yet, think of this: as I write this, there is a number that answers that question. It may be 5,214,786, it may be something else, and I will never know it for sure. But there absolutely is such a number.

You may be wondering, in turn, whether this campus visit has touched me in the head a little. Bear with me, I’m going somewhere with this.

Most mornings, I have a hot cup of coffee first thing.

It wakes me up. I usually whisk the coffee before adding milk, to give it some additional foam. And once in a while, on a slow morning, I look down at the foam and wonder: how many bubbles?

Yes, on very slow mornings.

Still, think of it. There are bubbles in that foam. There are a lot of bubbles, both large and small. Like with the leaves, in fact, there are far too many to actually count.

And yet, there is an actual number that counts the bubbles in there. The foam is contained in the cup, and each bubble is of a finite size.

Thus it stands to reason that there is a finite number of them. Meaning, I could count them if I were so inclined.

Instead, I drink the coffee. The thousands of bubbles included.

What am I getting at here? These may seem to you like pointless musings, and they probably are.

But they intrigue me nevertheless. If not by actually counting, is there a way to estimate the number of bubbles in that cup? Certainly! I could make a quick guess at the size of an individual bubble, and then calculate how many would fit in the cup, in a layer that’s one bubble thick.

It wouldn’t be correct, of course. The bubbles are not all the same size.

They are probably more layers of them than just one. Still, it’s a reasonable estimate. I’ll take it.

There are other, and even more unknowable numbers around us. How many times have you blinked in your life? We could estimate that, given that on average we blink about once every four seconds. That’s 15 times a minute, 900 times an hour, 21,600 times a day, and nearly 8 million times a year.

So if you’re 35 years old, say, you’ve blinked nearly 300 million times.

But wait, surely that’s an overestimate. For we humans sleep too, with no blinking happening then. Typically we sleep for about eight hours, or about a third of the day. So we need to reduce that 300 million count by a third, to 200 million. There you are, a reasonable first-pass attempt at divining this fairly useless piece of numerical information. We can make an estimate because we know it is a finite number and we know the average human’s blink frequency.

But here’s a different number that seems finite to me, in that its magnitude is bounded in the same way as trees on this campus. How many grains of sand on our planet Earth? Right at this instant, there is some gargantuan number that answers that question, though I have no idea what it is, nor how to get its value, nor even how to estimate it.

I could just start counting, of course, next time I’m on a beach. It would take a serious length of time to count the grains in just a simple handful of sand. So you can imagine how long it would take to count the grains on an entire beach, and then on all the beaches in the world.

Yet if this was possible to do, you know there’s a number at the end. The Earth holds exactly that many grains of sand right now. So it’s a finite number. Unimaginably difficult to arrive at, but finite.Yet in that finitude, I want to suggest, is a hint of infinity. Because consider that the act of counting grains of sand will take a very long time, and in that time, even more grains will be formed and we’ll have to count those and while counting them, still more will appear ... on and on without end.

In fact, counting this gigantic but apparently finite number is a process that will never end. And that endlessness is why it makes sense to think that when it comes to grains of sand, there’s an infinity of them.

You could apply much the same reasoning to other apparently finite numbers: the stars in the sky, or the cells your body has ever produced, or the length of the coast between Mumbai and Goa. For one reason or another, it can make a lot of sense to think of those as infinite.

The thing is, it was on this college campus that I first started wondering how to grasp, to understand, the idea of infinity. Ah, the nostalgia! But no, I’m not about to count 5,214,786 leaves here.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun.

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