It’s the height of wanton hubris to claim that “mathematics is all around us.” Mathematics isn’t all around us, mathematics is only in our heads. To claim that “functions are everywhere” or that “numbers are everywhere” is only to claim that what is going on inside of our heads is actually going on outside of our heads. Belief in mathematical Platonism is anthropocentrism at its worst and is completely absurd.

For example.

Let’s look at $$\sqrt{2}$$. $$\sqrt{2}$$ must exist, right? Like… it has some sort of an existence. What is $$\sqrt{2}$$? One of the things it is, how $$\sqrt{2}$$ arguably came about historically, it’s the length of the hypotenuse of an isosceles right triangle with leg length 1. Okay. How do we know an isosceles right triangle with leg length of 1 has a hypotenuse of length $$\sqrt{2}$$ and not 1.4147? How do we know that? The Pythagorean theorem. How do we know the Pythagorean theorem is true? We know that the Pythagorean theorem is true because of the Euclidean Parallel Postulate. How do we know the Euclidean Parallel Postulate is true? WE DON’T KNOW ITS TRUE! IT’S STIPULATED. We ASSUME it to be true. We take it to be true. Why do we take it to be true? Because it kinda seems like it might be true. When we’re dealing with flat surfaces.

The only reason that we know the hypotenuse of the triangle is $$\sqrt{2}$$ is because of math. Not because we can measure the length with a ruler. Not because a measurement conforms to our sense experience. Not because $$\sqrt{2}$$ exists as an object outside of mathematics. $$\sqrt{2}$$ is just a figment of our imaginations! In order to have that isosceles right triangle, you HAVE to have the Euclidean Parallel Postulate. We HAVE to be working on a flat surface. Otherwise… we don’t have the triangle, we don’t have $$\sqrt{2}$$.

We have to define distance, a figment of our imaginations. We have to define area, also a figment of our imaginations. And then later comes $$\sqrt{2}$$. In the above sense, $$\sqrt{2}$$ doesn’t exist outside of the Pythagorean theorem. (I mean, it does exist outside of the Pythagorean theorem, we can find $$\sqrt{2}$$ other ways… but they’re all math ways.) It’s not like an empirical, “oh. THIS thing right here that I am pointing at is the square root of 2.” It’s all just made up. Number itself has only some sort of quasi-empirical existence. The idea of “number” itself is a map that humans lay on top of the territory of our sense experience. The map is not the territory.

P.S. And to ask students to “discover” mathematical objects as though they exist outside of the minds of mathematicians is completely absurd and unfair.