As promised, a post about how natural it is for humans to think of infinity (see also My Grandma).
Let me say first that there are some very big and important questions about infinity that mathematicians and philosophers are still trying to answer (and I’m just talking about mathematical infinity here!). Unfortunately I have nothing to say here about those questions. This post is very naive.
As I said, the question that I’m interested in here is: How natural is it for people to think about infinity? I want to consider three cases: my son, myself and my grandmother.
One thing that has me thinking about thinking of infinity is that my son has been talking about infinity lately. He is 6 years old and in kindergarten. He has asked me several times “Dad, is infinity a real number.” This catches me off guard and I usually mumble “umm…no. It’s an extended real.” Of course this is not at all helpful and usually prompts a “Huh?” from him.
I just asked my 4-year-old daughter what infinity is. She said that her teacher hasn’t told her yet.
What I wonder is, where did he even learn the word infinity? Where and when did he learn what it means? I don’t remember ever specifically teaching him what infinity is. Then again, I do remember him asking me questions about what the biggest number is. I told him that if you had a number that you thought was biggest you could always add one to it. . . which leads me to. . .
I don’t remember where or when I learned the words infinity or infinite. On the other hand I remember exactly where and when I learned the word finite. It was in the fifth grade. I immediately understood what it meant: not infinite. The word finite itself seemed very funny to me.
I know that I had started thinking about infinity much sooner. When I was, oh probably 6 or 7, I guess. I remember trying to come to grips with the following paradox: it was absolutely impossible for me to comprehend space going on forever and it was absolutely impossible for me to comprehend space “stopping” at some point (what would be just beyond that point?). It seemed that one of these alternatives had to be true (I hadn’t yet learned that there are compact, boundaryless 3-manifolds).
So how natural is it for us to think about infinity? First of all, from studies that psychologists have done with newborns it appears that we are born thinking about numbers (at least the numbers “one.” “two,” and “three or more.” I don’t think that thinking about infinity comes that early.
So here is my hypothesis (prompted by the above two anecdotes): at some point around 6 years of age it is natural for people to think “do numbers (or space) ever end? If they do, why can’t you just go one step further? But it seems impossible that they could go on forever. The resolution of this paradox, for most of us at least, is to construct for ourselves and accept the concept of infinity.
I could be totally wrong on this. I am no psychologist or philosopher or anything or the sort.
Of course the one monkey-wrench in the works of my hypothesis is my grandma. I can’t be sure whether she understood the concept of infinity or not (see My Grandmother).
So this post is more of a question and request: What are your fist memories of thinking about infinity? How natural do you think it is for us to think about infinity? Please share your ideas.