It's the Thought That Counts

If you glance quickly at a pile of identical items—let’s use bananas—you might be able to tell how many of them there are in an instant. It turns out that our brains are wired to identify very small numbers of things without taking the time to count them.

This is called subitizing, and most of us can do this with numbers up to four (although sometimes 3 is a more realistic cutoff). Our brains recognize patterns, so we don’t have to spend any mental energy to know whether there’s one, two, or three of something (and sometimes four). We simply know it.

There’s another sort of quick visual reckoning we do well, and it’s a game we might call “more or less.” We humans are able to identify that this pile with ten bananas is bigger than the pile with five bananas, so there must be more bananas in the bigger pile. When we start adding bananas to both sides of the scale, though, it starts to get a lot tougher for us.

Imagine two piles of bananas. One has 10 bananas and the other has 9. Could you tell the difference at a glance? How about if the piles were 99 and 100 bananas, respectively?

No, in order to do this, most of us will need to count. This means looking at individual bananas, one at a time, and then adding them all up in our minds.

Or does it? Is that what’s really going on in there when counting takes place? That’s what we’re here to think about today.

In order to count, you need to have three key components. First, you need to have one-to-one correspondence, where “one” means one individual item, “two” means “two things”, and so on. You have to define these words, and then make sure they always go in that particular order.

You also need to always go in a particular order with these numbers. If you can’t keep this order straight, it just won’t work. The order 7, 8, 9, 10… is a great little sequence that gives us a joke about how ten is now scared since seven ate nine, but never the other way around.

Lastly, you need to understand that the last number you identify actually represents the number of things there are. In other words, if you say, “one, two, three!” and then can’t get to four, there are three things there, not two or one. This one’s called cardinality.

It is this combination of a fixed order with cardinality that makes correspondence counting work so well.

So, there you have it! That’s how everyone counts, and there’s nothing else to say.

Well, okay. There’s more to say. For one thing, not everyone actually learns to count this way. Do you remember how you memorized what “three” or “four” really meant? I’m pretty confident that you looked at your fingers as you said each word.

You might even use part of this skill set today—by counting things with your fingers. If this seems like something no reasonably intelligent person might do, keep in mind that cognitive offloading is a very real thing, and your mind isn’t contained within your skull. That’s your brain, but your mind is way, way bigger than just your brain.

Also: don’t judge me. Counting on my fingers is useful sometimes!

Additionally, it’s not necessarily true that everyone counts this way. One common technique for smaller numbers is to visualize little groups, like a set of three of four items. If you can see that three items tend to form the rough shape of a triangle, and if you can envision that four items form a four-sided shape (quadrilateral), you might have another way to visualize smaller numbers.

Some people are very good at grouping these little clusters into shapes, so you might have a new triangle shape made up of 3 little triangle shapes, for 9 items total. The number of people who can do this decreases with the number of items involved, as you might imagine. There really was a guy who could count toothpicks, like the scene in Rain Man where Dustin Hoffman instantly counts 246 toothpicks on the ground, but we’re talking about a handful of humans alive.

Language plays a big role in the way we think about numbers, too. Remember how we were talking about how you have to encode a word for each numerical concept, like “one” means “one thing”? There’s a nifty trick some languages use to make this easier, by using the base to name those numbers.

Imagine if you had to memorize every digit from one to one thousand. Now, I’m not suggesting memorizing a string of digits, but instead a thousand different words for each of these first thousand digits. Instead of “five hundred forty one”, you have flergle, and instead of “five hundred forty eight”, you have bazglooble.

Since most of us has ten fingers and ten toes, it makes sense that we’ve settled on math based on tens, although that certainly doesn’t need to be the case—there’s nothing inherent about the number ten that makes mathematics work any more than, say, the numbers two or twelve.

The ancient Babylonians used sixty. I’m just saying, sometimes we should be grateful for the little things in life.

That base-ten system means we don’t really have to memorize a thousand names for numbers in order to count to a thousand. Instead, we just throw the new base up there at the front of it, so you know the word for 541 even if you’ve never heard of a flergle before.

Some languages don’t bother with a framework like this. Indigenous cultures with little connection to the modern world will have words for many and few, but nothing above a certain level, like the Pirahã in the Amazon, who are said to have no way to express specific number quantities of any sort.

This doesn’t seem to bother them too much. You might want to use specific, precise numbers for stuff like record-keeping or recording transactions, but with a smaller number of transactions and less overall complexity, the Pirahã navigate daily life without the need for 541, or flergle, or any other big numbers.

You might have already noticed that English is far from perfect in this regard. Take “seventeen” for instance. That’s not bad! You just need to know that “teen” means to add ten to the first part of the word. Not too shabby there, English!

Oh, but eleven? Twelve? Those aren’t words as much as they are shenanigans.

I’ve already gone on rants about the French method of describing numbers as you approach 100, although this method, too, makes sense if you look at the context in which it arose.

Language and culture can modify the way we think about counting, but at its core, we are limited by our capacity to identify bigger quantities of things without counting them out, one at a time. Think about it: what’s the highest number of object you can reliably identify without counting them out?

Also: do you ever use your fingers when you’re counting?