What If There’s No Infinity?

I’ve long been an infinity skeptic, and I didn’t even know it. Today, I read What Can We Gain by Losing Infinity? | Quanta Magazine by Gregory Barber, and it opened my eyes. It’s an article about a fringe group of mathematicians called the ultrafinitists. They believe that there’s no such thing as infinity and they’re trying to create the math to prove it. It’s a hard project for a lot of reasons. It turns out that a lot of our familiar math, even basic arithmetic, has the assumption of infinity built in. Most mainstream mathematicians are outright hostile to the project. And young mathematicians don’t want to risk their up-and-coming careers on such a controversial topic.

The article, and the ultrafinitists, really resonated with me, though. I firmly believe that math is made up. It’s a tool we use to describe the world we live in. So, any math is only as good as its ability to accurately describe the world. If a math depends on infinity, but there’s no actual infinity in the real world, then that math is flawed.

The question then becomes, are there any infinities out there? My instincts say no. People talk about there being an infinite number of grains of sand on the beach or an infinite number of stars in the sky, but that’s clearly figurative language. Just because it’s impossible or impractical to count them, there is a specific, finite number of grains of sand and stars.

When people talk literally about infinities, they say things like, “you can always add one and get a bigger number,” and “you can always cut it in half to get a smaller piece.” But if there is only a finite number of things in the universe, does it even make sense to talk about numbers that are bigger than that number? And what if we can’t always break something into smaller pieces? Atoms are made of electrons, neutrons, and protons. There are things like quarks and bosons. But what if it doesn’t get any smaller than that? If there is no way to divide an electron, does it make sense to talk about something smaller?

The other place you hear of infinities a lot is from people who espouse multiversal theories. It seems to be assumed that if there is a multiverse, then there are an infinite number of universes where everything that can happen does happen. Now, I’m skeptical of there being a multiverse at all, but why infinite universes if there is one? Why not two or three universes? Or why not, like the stars in the sky, an unimaginably huge number that’s not infinite? If there are new universes being created all the time, you can stop at any given moment and count how many universes there are right at that moment.

Anyway, that’s my two cents about a subject I know very little about. I’ve always been a little suspicious of infinities. My confirmation bias found Barber’s article very satisfying.