The other day, my four-year-old son asked me, point blank:
—Dad, can you count to infinity?
—No son, you can't, it would take forever.
—But what's infinity minus three?"
—It's infinity, too.
—But how do I write the number infinity?
—It's like an eight lying down.
—Is that even a number, like one or two?
The point is, infinity is more an idea than a number. It's a concept we came up with to represent endless sequences of numbers, or a point in space or in time infinitely distant from our position or from the present moment. It's not something you get to; it's something you think about. It's a representation of our own limitations, finite beings that we are. (But for this reason, also a representation of our amazing creativity.)
And yet, we use the notion of infinity all the time. In cosmology, present data indicates that our universe is infinite. This means that if you start moving in one direction you will never come back to your starting point. Had the universe been finite, like the 3d version of a surface of a sphere, you could circle around and come back to your destination point.
Can we be absolutely certain of the infinity of the universe? No, we can't. All we can say is that the portion of space that we can measure, what is called our horizon — the distance traveled by light since the beginning of time some 13.7 billion years ago — is flat or very nearly so. And a flat geometry, like the surface of a table, would go on forever. But that's where our certainty ends. It's possible that our flat patch of space is really part of a huge curved universe. If we have no access to information beyond our horizon we can't really tell what's there. All we can do is infer.
And what about points and lines? Strange concepts, too. A point marks a location in space. However, it occupies no space at all, as a point has zero volume. And a line, linking two points in space, has no thickness. If we say that a line is a sequence of adjacent points it gets even weirder. How can something with no volume be adjacent to something else? So, we represent our spatial reality by things that don't really exist in space, more ideas than real things. Given out sensorial limitations (small things become invisible below the accuracy of our vision), approximations such as representing points by dots on a page are very efficient. The wonderful thing about mathematical representations is that they work, even though they are intangible.
Things get even weirder when we go to physics and look for the smallest material constituents of reality, the building blocks of matter. What happens if you cut a chunk of matter down to its smallest pieces?
You get to atoms, aggregates of protons, neutrons, and electrons. Protons and neutrons are not indivisible, being made of smaller particles called quarks. So, as far as we can tell at present, matter is made of quarks and electrons. Can quarks and electrons be cut down to smaller pieces? Here, things get tricky. An electron is not a simple little ball of energy with a negative electric charge. It's an entity that can either look like a little ball of energy with a negative electric charge, or it could look like a wave carrying electric charge. If you were to ask a high energy particle physicist, she would say that an electron has no inner structure, meaning that we don't think there is anything in there. But can we be sure?
Just as with the flatness of the universe, we can only base our arguments on what we can measure. In the world of particle physics, we study structure by colliding particles at each other at speeds near the speed of light, somewhat like throwing an orange against a wall (or another orange) to see what's inside. So, all that we can say is that current measurements and theories don't indicate that the electron has an inner structure. We may treat it like a point particle with negative charge, but should keep in mind that that's only an approximation to the real thing. And what is the real thing? We may never know. As with the ideas of infinity and points and lines, electrons and quarks are constructions of our intellect to represent what we see of the world. Their reality is contingent on how we see them.