However, there is pretty conclusive evidence that there were quadratic relations among various physical quantities in the earlier universe, notably the inverse square laws for electrostatics and gravitation. It seems extreme to blame that on humans...

I will repeat one of my comment from yesterday [1] as we argued about the same topic.

I want to argue that whenever you think that there is some math already contained in the universe, it is actually you accidentally introducing it without noticing it. Look, trees! One, two, three, ... See? There are numbers in the universe. And look, I can add the number of trees in this wood and that wood.

It is of course tempting to conclude that the math was already there. But I argue that it was not, that it is you inventing it. It is your concept of object identity that defines what a tree is and when two trees are the same or different trees, you divide the world into different objects. It is your concept of grouping objects into sets. It is your concept of pairing the elements of two different sets to compare their cardinalities. It is your concept of including one more element in a set that defines what counting means. It is your concept of combining two sets that defines what addition is.

That are all things you can do with objects in the universe, but nothing of that is inherent. If you look carefully enough, all the things that look like math already exists in the universe actually require you making use of a lot of abstract man-made concepts. Many of them, like object identity and counting, are just so deeply ingrained in us that you don't notice that you are using them if you don't pay careful attention.

That is why I asked for the empty universe, I wanted to take away all the objects on which you could accidentally impose structure. If math were truly fundamental, if Platonism were true, then seven should still exist in an empty universe. I don't see how this would be true in any meaningful way.

If we added a mathematician to our empty universe, he could invent math. He could come up with the idea of objects and identity and sets of objects and he could think about operations with those objects and consequences of the definitions he made. But it would all have no relation to the otherwise empty universe, it would be a game of formal rules, mostly manipulating strings of symbols. If the mathematician had great imagination, maybe he could imagine actual things existing in the universe, how he could label them with those numbers he invented, and how he could visualize addition by moving things around.

In another comment I argued the same for your example of the inverse-square law. It is you who decided how to measure distance. It is you who decided to look at spherical shells around a charge. Nothing of this is inherent. I could easily make it an inverse-cube law by adopting a different measure of distance. I could decide to look at boxes instead of spheres and ruin the symmetry. It is certainly correct that there are more and less natural choices, but I don't think that is a good enough argument to conclude that some preferred math is intrinsic.

Particles flowing outwards from an isotropic source with constant velocity in straight lines in three-dimensional Euclidean space will generate a density field following an inverse square-law. But that is not intrinsic, it is just a useful description of the situations, by itself there are only particles traveling at constant velocity in straight lines. Systems with simple rules, especially all systems with chaotic behavior, can produce behavior with staggering complexity requiring some serious math to describe, but, by itself, there are just many things following simple rules and interacting with each other according to simple rules.

Give me a large enough pile of sand and I will use it to implement set theory on top of it. And then on top of set theory a huge part of math. Does that mean that grains of sand include most of mathematics? That does not sound reasonable to me. I believe people are confusing the ability of describing things with math or implementing math on top of something with the thing already containing the math. Also note that there often many ways to treat the same thing, are they all equally intrinsic?

That all seems very similar to the general question of existence. Do all books and pictures already exist, including those that will be written or taken in the future? An empty SD card certainly has the ability to contain them, I just have to flip the correct bits. To me it seem to simply stretch the definition of existence way to much to declare that everything I can imagine or everything I could create already exists. So being able to do arbitrary math with grains of sand or being able to describe what is going on in the universe with math does similarly not imply that math already exists.

That became way longer and repetitive than intended. On the other hand I repeatedly failed yesterday to convey my argument, so I will just leave it as it is.

[1] https://news.ycombinator.com/item?id=13794255