By humans? Do circles not exist without humans to point them out?

They do. http://www.hemsleyandhemsley.com/assets/Screen-Shot-2014-04-...

Not according to the author:

> According to intuitionism, mathematical objects are products of our mind, like characters in a novel. I agree, as far as it goes...

> Fictionalism holds that the mathematical universe is a collective fiction, like Star Trek or Game of Thrones.

Well, different characters behave differently in different novels. How is this like an idea of a circle?

> the Cantor set or the Borel Hierarchy is like the USS Enterprise or the Iron Throne. In fact Star Trek fans talk of the Star Trek universe and GOT fans of Westeros and they talk like these places really exist.

You're right, it feels like quite a loose analogy, and only holds when discussing a "canonical" fictional universe. Still, I'm sympathetic to the idea that when we're talking about a circle we're talking about a shared abstract understanding rather than something which objectively exists.

The pupil in the eye. The sun through the clouds. The craters after an impact. The waves when you drop a rock in water. The section of a bamboo stick... And it gets more evident when you go to microscopic world. Some eggs are perfect spheres, as small bubbles in water. I think circles are a real natural shape. Not an invention. Nature brings us all kind of shapes, some of them are discovered studying living creatures, like the scutoid.

It's not a very heavy argument anyway. Circles are in swirls, tornadoes and even the Sun looks like a bright circle looking from the Earth. Water waves draw circles when you throw a rock in it. Circles are everywhere. Some people says "nature hates perfect shapes" but it's not true, nature creates perfect cubes in some minerals, some fish eggs are perfect spheres... And so on.

None of those things are literally circles, though. If you look close enough there will be a deviation.

Water waves are perfect circles as waves move at same speed in every direction, so its points are always equidistant. Let a soap bubble settle in the table and take some measurements with a ruler. They are almost always perfect circles. You can deny they're circles, but you know it's not true.

They're not actually perfect circles. They're very close, but at an atomic scale there are clear imperfections.

At atomic scale sometimes there are perfect cirlces. The argumentum a silentio fallacy is bidirectional in this case.

Even at atomic scales there are always minor distortions due to external influences. For example, although the shape of an electron s-orbital is said to be a sphere, the only way this could theoretically be a perfect sphere is if the atom is and always has been completely isolated from all external fields and forces.

Circles exist in nature. You can deny it, but it isn't true.

Now go with your atomic imperfections and external influences and take a nap, because it must be exhausting to be so repellent.

I'm sorry for being dismissive but you threw a barrage of contentious examples at me without really trying to engage with my point.

The way I see it, the circle is an abstract idea which resides within the language that we use to share our understanding about the patterns we observe in the external world. Those patterns may conform to the abstraction to varying degrees but they are fundamentally not the same thing. Mathematics is a very precise and formal way to communicate this type of knowledge, but that doesn't mean the ideas therein exist objectively. This contrasts with the views of someone like Max Tegmark, who has the impression that the universe is inherently mathematical.

Except that they're not.