For most of the history of mathematics, there was really no such thing as infinity.

It was an idea, but not a reality.

Many thought that only God could be infinite, so having it be a real mathematical thing would be like heresy.

It wasn't until the 1870s, through the work of Geyor Cantor and others that

infinity was established as a true mathematical object, and we talked about that pivotal moment in

math history in a recent two-part episode. It was a key idea in laying the foundation for modern

research mathematics, but that doesn't make the idea of infinity and all its different

sizes and species any easier to get your head around.

Now, while it's best described as a fringe position, there's a non-zero population of mathematicians,

philosophers, who think that infinity shouldn't really be a thing. Why, they ask, should we lean on an idea of endlessness when our world is finite?

Welcome to the Quanta podcast where we explore the frontiers of fundamental science and math.

I'm Samir Patel, editor-in-chief of Quanta magazine.

As part of our recent special series, the evolving foundations of math,

science and technology journalist and regular quantum contributor Greg Barber wrote about these folks,

a small group of mathematical dissidents, heretics, even, called

ultrafinitists who think that we should excise infinity from math itself.

Greg's here with us to talk about the story.

Welcome back to the show, Greg.

Thanks so much for having me back.

So, Greg, what's the big idea?

So words like heretics and dissidents, and I've even heard this described as a crackpot

theory of mathematics, these are all honestly pretty appropriate ways to describe

...