Infinity doesn’t come in just one size.

Since the beginning of history, scientists have been fascinated by infinity – whether it be the idea of endless space, endless time, or endless numbers. Actually, it’s not just scientists who’ve been fascinated by the subject. For many of us, the notion of infinity holds an intriguing, almost magical power. 

What many people consider infinity comes in two versions. Of course, there’s infinitely big. When scientists explore this type of infinity, their work often approaches mysticism. Some Russian mathematicians of the 1920s, for instance, conflated their mathematical explorations of infinity with theological explorations of God.

But there’s also the idea of the infinitesimally small, or infinitesimal – and it’s no less puzzling than the infinitely big. In mathematics, for example, each number is divisible into infinitely many fractions

Since the time of the ancient Greeks, mathematicians have used infinitely small fractions to calculate certain geometric forms and volumes. But for a long time, scientists regarded the idea of the infinitesimal with suspicion. That’s because most of them didn’t believe that infinitely small fractions actually existed in the real world. They were convinced that the smallest unit of the physical world was an atom, which couldn’t be split up any further. 

In the seventeenth century, French mathematician Blaise Pascal shortly revived the idea of the infinitesimal, using it to calculate the areas under curvilinear forms – that is, forms characterized by curved lines and shapes. Isaac Newton, too, made use of the infinitesimal when he calculated the exact orbit of a planet around the sun. But Newton agreed with his contemporary Gottfried Wilhelm Leibniz that infinitely small quantities were nothing but “well-founded fictions” in mathematics – useful, but not really real. 

Over the next two centuries, the infinitesimal fell out of fashion. Most mathematicians didn’t see the point of basing calculations on quantities that didn’t really exist. They feared that somewhere down the line, this approach would turn up huge contradictions.

It was only in the 1960s that mathematician Abraham Robinson was able to rehabilitate the image of the infinitesimal. Robinson was able to draw on Gödel’s completeness theorem – the counterpart to his incompleteness theorems – to show that even if we can’t be sure of the reality of the infinitesimal, we can still be sure of the logical consistency of the calculations that involve it. That, Robinson argued, should be enough to satisfy the standards of modern mathematics.

But perhaps the infinitely small isn’t just a useful fiction. Today, we know that atoms can indeed be divided – specifically, into electrons, protons, and neutrons. The latter two can be divided into even smaller particles called quarks. And there’s some evidence that even quarks can be divided up further. So maybe the universe is not just infinitely big – maybe it’s also infinitely small.