[Goanet] {Dilip's essays} The finite that hints at the infinite

```Mar 19 2024

This column grew out of an idle mind one morning, looking into my coffee
cup. "How many bubbles?" I wondered. (Idle, yes.) Thing is, it isn't the
actual answer that interests me as much as thinking about how I might try
to find one - and what that process means.```
```
To me, it's a window of sorts into the idea of infinity. It's a thought
I've long wanted to flesh out, and so that's what I did recently. My March
1 Mint column: The finite that hints at the infinite,
https://www.livemint.com/opinion/columns/the-finite-that-hints-at-the-infinite-11709201135481.html

Do you also stare at bubbles in your coffee? I wanna know.

cheers,
dilip

---

The finite that hints at the infinite

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 3000. 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
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

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.

--
My book with Joy Ma: "The Deoliwallahs"