Human Minds Vs. Large NumbersPosted: November 6, 2013 A few months ago I was playing with my cousin’s kids at a beach- sandcastles and kicking sand and burying feet and the lot. One of the two was three and a half, the other one and a half. Their mother came down with a new set of sand toys and we all got excited. “Do you know which one of these sand toys looks like an animal that doesn’t exist any more?” I asked.
Ellie, the younger one, pointed to the sand toy in the shape of a penguin. “Are you being mischievous?” I asked. “A world without penguins would be a sad one!” I got a seriously blank look in return. Then it hit me- they had no idea what mischievous meant. Time to talk little-kid. “It’s the dinosaur,” I said, “all of the dinosaurs died out a long long time ago.”
“When I was a baby?” Abby asked, the three and a half year old. Uh-oh. No concept of time or numbers, either. “No,” I said, “a long, long, long, long, looooonnnng time ago. Before you were born.”
“When my mommy was a baby?”
Oh gosh. “No, it was before your mom was born, and before your Nonnie was born, and before her grandmother was born. It was before your great, great, great-grandmother was born. It was even before there were people. Do you know what millions are? It was millions and millions of years ago.” Both of them looked thoughtful and perplexed and went back to playing in the sand.
I don’t normally play with little kids, but it was a surprise how much they still had to learn about numbers and time, not to mention vocabulary and dinosaurs. Their little brains were rapidly connecting and growing and pruning new neural networks, but their concepts of numbers and time clearly had some developing yet to do. For Abby, eleven was a bit of a tricky number because she ran out of fingers. I could relate- I run out of space in my brain trying to comprehend the trillions of cells in my body. But my brain isn’t still rapidly developing like theirs are.
An average four year old has developed enough to understand about counting, but to still make some of what are, to adults, obvious mistakes. If you show two glasses holding the same amount of water to a four year old, pour one of them into a taller glass in front of the child, and then ask which if the two glasses are holding more water, the four year old will answer confidently that the taller glass has more water in it.
Psychologists have a name for this stage of brain development: pre-operational, a term coined by developmental psychologist Jean Piaget in the 1950s. A kid at this stage has just begun to grasp the idea of symbols, or mental representations of things that are not immediately present. But they haven’t learned to mentally manipulate these representations, be they numbers or volumes. In other words, they know what symbols are, but they don’t know how to use them to logically solve problems. The video below demonstrates how kids from two to five generally haven’t learned what psychologists like to call “conservation of number, length, liquid, mass, or area.”
The kid in this video isn’t stupid; he’s just pre-operational. He’ll get there. But as a member of the human species, which did most of its evolving in the hunter-gatherer age, he’ll always have trouble concretely conceptualizing the number one million, or one billion, or one sextillion, without contextualizing and contrasting tricks, like the fact that 1 million seconds is equal to 11 days. We can easily recognize up to four objects without counting, knowing what number is in the group in under 250 milliseconds. But as the number goes up, humans’ response time and confidence in their answer decreases, and soon we have to search for patterns, count, or estimate. Infants 4 to 7 months display an innate number sense that must have served our ancestors well; but while they can discriminate between 2 items and 3 items, they can’t tell the difference between 4 and 6 items. That was ok when your tribe and hunting grounds were small.
But humans are beyond that now, not only multiplying at dizzying rates but plumbing the reaches of the universe. We now need to use bigger and bigger numbers on a day to day basis (or much, much smaller, if you’re studying certain types of physics). A million seconds might be 11 days and a fair amount of money, but how much is a billion? Astronomers think there are roughly 70 sextillion stars in the universe. That’s 70,000,000,000,000,000,000,000 stars. Where do I even start if I want to wrap my head around that number?
Several science writers have written about the incomprehensibility of large numbers before me- one of my favorites is about Justin Bieber. Another hypothesizes that as humans deal more and more with bigger numbers, we’ll intrinsically be able to conceptualize them better.
Which brings me to the conclusion of this long and winding piece: are adult humans pre-operational when it comes to very large numbers? Have we yet to grasp them well enough to be able to manipulate and relate them to each other? If we deal in very large numbers every day, does it get easier? My cousins Abbie and Ellie have yet to fully comprehend how long one hundred years is, let alone the millions since the dinosaurs died, but they will. How long, if ever, do you think it will be until adult humans are as fluent in manipulating sextillions as we are twos, tens, or hundreds? Will we ever evolve or adapt to wrap our brains around these huge numbers? I would not mind developing out of a pre-operational stage when it comes to billions, trillions, and sextillions. There is a lot of universe still to explore.