Infinity

What is infinity?

A few weeks ago, a nine-year-old girl looked up at me and asked, without warning, “What is infinity?”

She asked the question in Vietnamese, but her aunt translated it for me into English.

I said, “Infinity is without boundary.” Her aunt kept translating. The girl tilted her head. “What’s a boundary?”

So, I tried again. “It’s without limit, without border or outline. Infinity has no end – but also no beginning.”

She smiled, turned away, and went back to the little card game she had made herself. I looked at her aunt and parents and said, almost in awe, “What an amazing question.”

Later, I kept thinking about my answer. I had only described infinity through negation – what it isn’t: no boundary, no limit, no end, no start. Yet infinity is so rich. It’s vastness beyond vastness. I failed to pass on the immeasurable richness of infinity to that little girl.

I remembered reading, as a teenager, a text by a philosopher who was struggling to imagine infinity. In German, vorstellen – “to imagine” – literally means “to place before.” But how could one place before oneself that which has no outline, no front, no back, no form? How to contain in an area that which cannot be contained? Even in English, to “imagine” means to form an image – to copy something into the mind’s eye. But boundlessness can’t be framed. Limitlessness exists untouchable by any containment whatsoever.

I thought of my math classes too, where infinity was always something we reached for by counting: start somewhere, keep going, never stop. But that kind of infinity is one-sided – it has a beginning. And if there’s a beginning, there’s already a limit. This means that a-symmetrical infinity isn’t truly infinite.

So how can we understand infinity without defining it by what it’s not? The best word I found thus far is “open.” Infinity is openness itself. Infinity is pure openness – an opening that cannot be contained by anything.