Friday, November 4, 2011



When you're involved in a dispute, a good tactic is to use the concept of infinity. For example, if you're engaged in the classic "Am not," "Are too" argument, one possibility is to shoot for "am not am not amnotamnotamnotamnotamnot...." A more advanced move, though, is to counter with, "Are too - infinity."

The novice at this style of argumentation might think that's a winning move - that whoever gets infinity out there first has automatically got more ammunition.

However, you might be interested to know that there are different sizes of infinity.
A lot of kids get this concept intuitively. They might say "Am not - infinity times two," for example. Their parents might think they're being silly, because infinity is infinity - ultimately the biggest possible, the most there could be, right? How could you take the most there could be and times it by two? But parents are notoriously silly themselves, because they think they know things when they don't really.
Well, think about this: there are twice as many whole numbers as even numbers, although there are infinitely many of both.
Here's the beginning of the infinite list of whole numbers: 1, 2, 3, 4, 5, 6, 7,...
Here's the even numbers list: 2, 4, 6, 8, 10....
Notice that the even numbers can never overtake the whole numbers - in fact, they're missing one for every two whole numbers, and they always will. The poor even numbers can't hope that at some point the whole numbers will stop and take a break, like the hare racing the tortoise. They never will.

This leaves open all kinds of expert moves in the argument game under discussion: infinity plus infinity, infinity times infinity, infinity to the infinitieth power, and so on.

Some questions to ponder, regarding our mental deficiencies regarding the infinitely large or small.

If you're told that the Universe is infinitely large, you probably find that you can't really grasp that. If you're anything like lots of people, you probably imagine the Universe as kind of like a big balloon - but can't help wondering "what's on the outside of the balloon?" Nothing - and by nothing we don't mean empty space, as in what the Universe is largely made of. There is no outside of the balloon. The balloon is all there is - period.

The way that we can't really get our heads around certain things like this has been called "cognitive closure" - meaning that our minds are simply closed to certain things (or they're closed to our minds). Chimpanzees are cognitively closed with regard to higher math - their brains just can't go there. And we seem to be cognitively closed to very very large and small things.

There's a great explanation for this evolutionarily, involving what the biologist Richard Dawkins has called "middle world" (not Middle Earth). We evolved as medium-sized beings who dealt exclusively with middle-sized objects like rabbits and boulders, not molecules and tectonic plates, so middle-sized things are what our brains can best deal with. We're much better at thinking about things in amounts we would have been called upon to confront, and those amounts fall ridiculously far short of infinity (obviously). In fact, wouldn't any being's world make it so that they were cognitively closed to infinity? It's one of those things we can usefully talk about and utilize, but not really get. Atoms are like this: the electrons, for example, are so different from middle-sized objects that we can only vaguely understand them, or pretend they're like things we can deal with, like planets in orbit around a sun (which we pretend are like other things we can deal with, like styrofoam balls on revolving sticks).

This gets dangerous because of how different we've made our world from middle world. Our brains have not developed to be able to really conceive of billions, or trillions - but incredibly important issues arise because of billions of people and trillions of dollars. We find it easy to just let these things slide past our minds, whereas the $1 value menu at the fast food restaurant is all too much in our grasp.


Thinkling is a resource for those interested in helping their kids think philosophically. Philosophy is a way of looking at the world that involves depth, meaning, critical thinking, and purposeful direction in thinking. Thinkling is committed to the idea that kids, their parents, and the world will be better off the more philosophy gets involved. The human mind is capable of astonishing feats and wondrous habits of thinking, but it needs material and training to do its best. 

The resources here are meant for parents to share with their kids, for better thinking, self-improvement, and scads of fun all around. "Doing philosophy," as philosophers call it, should be natural and painless, although it can take hard work. Above all, it should give those growing brains - and yours - interesting things to do.