A child thinking about infinity
Introduction
Young children's thinking about infinity can be fascinating stories of extrapolation and imagination. To capture the development of an individual's thinking requires being in the right place at the right time. When my youngest son Nic (then aged seven) spoke to me for the first time about infinity, I was fortunate to be able to tape-record the conversation for later reflection on what was happening. It proved to be a fascinating document in which he first treated infinity as a very large number and used his intuitions to think about various arithmetic operations on infinity. He also happened to know about “minus numbers” from earlier experiences with temperatures in centigrade. It was thus possible to ask him not only about arithmetic with infinity, but also about “minus infinity.” The responses were thought-provoking and amazing in their coherent relationships to his other knowledge.
My research in studying infinite concepts in older students showed me that their ideas were influenced by their prior experiences. Almost always, the notion of “limit” in some dynamic sense was met before the notion of one-to-one correspondences between infinite sets. Thus, notions of “variable size” had become part of their intuition that clashed with the notion of infinite cardinals. For instance, Tall (1980) reported a student who considered that the limit of n2/n! was zero because the top is a “smaller infinity” than the bottom. It suddenly occurred to me that perhaps I could introduce Nic to the concept of cardinal infinity to see what this did to his intuitions. My aim was to show him the correspondence between the set of even numbers, the set of odd numbers, and the set of all (whole) numbers, and to explore related ideas. He showed a great versatility in thinking, producing some surprising insights of his own.
Later, he discussed the notion of infinity with his friends. He returned with a new view of infinity as a single large entity that is bigger than anything else and has no bigger number. The way in which he rationalized this with his earlier ideas involves a fascinating personal idea of the number line. The whole episode shows the amazing ability of a young child to deal with various infinite concepts and attempt to make the ideas fit together in coherent way — a task that reveals the endless fascination of the workings of the human brain.
Section snippets
Early experience with number
Nic began his regular schooling at the age of four years and three months, because he was born at a time when the birthrate suddenly diminished and younger children went to school early to fill up the classes. At home for lunch in the winter, aged four years and six months, he heard the weatherman on television mention that the temperature would be down to “minus two degrees centigrade.” He asked what “minus two” meant. Fortunately, we had an outdoor thermometer and I took him outside and
Talking about infinity
At the age of seven years and one month, Nic came out with a comment that took me totally by surprise
“I've invented a number bigger than infinity.”
We had never talked about infinity. I was so flabbergasted that I asked him if he minded my making a tape of what he said. This is part of the conversation that followed with Nic's comments in heavy type.
“A little bit earlier today you told me about a new number. What's it called?”
“Infinity.”
“Who told you about infinity?”
“Chris.” [His nine-year-old
Comments
Certain factors seem clear in Nic's remarks. He regarded infinity as a large number that can be added, subtracted, and multiplied like any other number. His system includes positive and negative infinite numbers that are in different places. On a subsequent occasion, I asked him about “one over infinity,” and he was quite convinced that this was a “very, very small number.” In effect, he imagined a total arithmetic system including infinitely large and infinitely small numbers, though the
Cardinal infinities
We have already seen that Nic had a sense of “infinity” being a “very high number” and that he did not think an infinite set of children and an infinite set of sweets could be arranged to give two sweets to each child. To begin talking about infinite correspondences, I started by attempting to establish the notion of the infinite set of numbers (which, to him, meant the whole numbers, 1, 2, 3, …), as follows.
“Now let me write down the numbers, one, two, three, four, five, dot, dot, dot” [Writes
Reflections
Did I do right to press on with infinite correspondences with a seven-year-old? I don't know. He clearly enjoyed the experience, as he wanted to continue with the conversation long after the two episodes described had finished. One thing is certain. I took a child with a consistent personal view of infinity as an arithmetic entity whereby
and gave him a conflicting idea where“infinity plus infinity is two infinity,”
“aleph plus aleph is aleph.”
In each case, he used previous experiences to come to
Interlude
The following day, we talked about infinity again. Nic remained confident. But in him, there were the seeds of conflict. A week later, he came to me and said
“I don't believe infinity plus one is bigger than infinity any more.”
“What is it then?” I asked.
“Infinity,” he replied. “I've been talking with my pals and we all think that you can't have bigger than infinity.”
He was also under some pressure from his older brother Chris who regarded infinity as “going on forever” and so “you can't get past
References (1)
Mathematical intuition, with special reference to limiting processes
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