“Young man, in mathematics you don’t understand things. You just get used to them.”

– John von Neumann

This is a line:

I was introduced to this mathematical object when I was in elementary school. The teacher drew one on the blackboard, and then said “a line is composed of an infinite number of points. The dashes at the ends indicate that it continues on forever, in both directions”. At first, I was surprised by the mention of infinity. It’s hard to visualize something that is infinite; our poor mortal brains are not used to think about things that untangible. So, as you’d expect, I couldn’t see the line going on forever. But logically, it made complete sense. Since it never ends, you must need an infinite number of points to trace it completely. It’s kind of like two immortal creatures forever walking in opposite directions. Nobody will ever see all their footprints, but we know for sure that there will be an infinite amount of them.

If you are wondering “how do we know that lines are infinite?”, I want to make it clear that lines don’t exist in the real world. They only exist in our minds, and they are perfect mathematical objects that are exactly what we imagine them to be.

At the top of this page there is a drawing of a line; but that drawing is not at all like what I had in mind when I drew it. I was thinking of a perfectly straight, infinitely thin and infinitely long sequence of points, and all I drew is just an approximation of it. Like Magritte would say, “this is not a line”.

Put it simply, a line goes on forever because we want it to. It doesn’t need a reason to be what it is, it simply is.

The next idea is the one of half-lines. The teacher then picked up a red-colored chalk and drew a dot on the line, like this:

They said “this is a half line. It goes on forever in one direction, but not in the other. This,” and pointed to the red dot, “is the origin of the half-line. Just like a line, a half-line has an infinite amount of points”.

If you understood lines, this one is easy. Instead of two creatures, it’s only one that takes the neverending walk, leaving all of its infinite footprints.

I hope you’re still with me, because now is when my mind exploded.

“If you pick two points on a line, you get a segment”:

“A segment has infinitely many points too.”

Huh? A segment, infinitely many points? It has a start and an end, and in no direction keeps on going forever. I can draw a whole segment, without putting dashes at the endpoints. How can this have anything to do with infinity?

The answer lies in the fact that, as I stated before, geometrical shapes are exactly what we want them to be. More specifically, we want our points to be infinitely small - dimensionless. We want them to be so small that no matter how close you can put two of them, there will always be some space in the middle. We want the immortal being to take steps so tiny that any finite number of them will not bring it any closer to its destination. Actually, we want it to be unable to take any steps, because the smallest step you can think of is still too big. That’s how small we want our points.

Of course, such a thing can never exist in the real universe. By merely existing, an object will occupy a 3-dimensional space, and have a size. But in our minds, we can imagine a thing that exists, but it doesn’t occupy space.

So, infinitely many infinitely small points. I’m still confused and fascinated by it to this day.

With this knowledge, I want you to try and answer this question: given one of the endpoints of a segment, what is the closest distinct point that stands on that segment?

What is the point that is so close, no other point will fit between it and the extreme of the segment?

Maybe it’s obvious, maybe it isn’t, but the only possible answer is that there isn’t one. No matter which point you pick, there will always be room for other points. In fact, there will be room for infinitely many more points.

The real number line

One thing that mathematicians like to do is to associate points with numbers. A number is like a name: it makes it clear that you’re talking about that specific thing.

If you take a line (a series of infinitely many, infinitely small points) and somehow manage to associate every one of them with a number, you get what mathematicians call the real number line. The numbers associated to each point are called real numbers.

The assertions that we made earlier about half-lines and segments directly translate to subsets of the real numbers. For example, while reals are infinite, all positive reals are too, just like all reals bigger than 7, and all reals bigger than four billions.

There are also an infinite amount of numbers that are bigger than -1 and smaller than 1, or bigger than 2 and smaller than 2.05.

And, just like before, if I told you to find the biggest real number that is smaller than 5, you’d tell me that it doesn’t exist.

The last implication that I’d like to explore is the fact that, following from all this, we can deduce that not all sets of numbers have a maximum value or a minimum value. This was casually presented in my analysis textbook and was given no apparent importance, yet I had to stop and think about it for quite some time, because it is not obvious at all.

Consider the set of all real numbers between 2 and 4:

What is crucial here is what I mean when I say “between”: are we talking about all numbers in that range including 2 and 4, or excluding them? Do we want to only include one, but not the other? This sounds like a detail, but it is the heart of the problem.

If we include 2 and 4 in the range, then 2 is the minimum and 4 is the maximum, and that’s all well and good. But if, say, 2 is excluded from the range, then we can no longer identify a smaller value, because that would be 2’s next real number - and as we said before, there is no such a thing.

We can still talk about the range being inferiorly limited by 2, as it does not contain any number smaller than 2; but since 2 itself is not in the set, it can’t be called the minimum of that set.

Why is this important? Well, first of all, there are so many advanced theorems that depend on the existence or the absence of minimums and maximums of certain sets; laws by which scientists derive algorithms to make computers work faster and better.

Second… it’s just cool. How often do you find something that is so out of reach, yet still makes so much sense?