It's funny because Tegmark actually gives an answer of how the physical adheres to math beyond epistemology--which seems very rare in the philosophical literature--and yet he receives incredible criticism by philosophers (who often haven't actually read his book from my experience). Not that I agree with Tegmark, but he at least sees the full horizon.

"The same gravity that is projected on the inner sphere must be spread out over the larger surface of the outer sphere. Since the surface of the larger sphere increases with the square of the radius of the sphere, the gravity must fall with its inverse square." - This makes sense if the universe behaves mathematically ... But why would the universe behave mathematically in the first place?

With regard to math’s being a language, @Philomath, if you mean merely a language, please see my answer to this question

“Is it unreasonable that our physical universe has three spatial dimensions that are orthogonal? What would it be like if these dimensions were not orthogonal?” - I agree with everything except this. Non-orthogonal spatial dimensions are actually physically identical to orthogonal ones by a mere coordinate transformation. +1 though.

"Physics doesn't follow any rules." This is nothing more than speculation. If it doesn't follow rules, why do the equations stay persistent over time?

@Syed Feel free to set a rule for physics and see if it follows it. Most likely not. So we make up 'rules' that arbirtrarily closely describe physics. It is descriptive rather than prescriptive (just like English grammar). And even then we have to adjust constantly, cfr the paradigm shift from classical mechanics to quantum mechanics when nature stubbornly refused to follow Newton's beatiful rules. When we investigate a quantum state and the wave function collapses, nature becomes even indeterminate. Who or what follows a rule here? Rather than that, nature sets the rules.

@user80226 But why would the universe behave mathematically in the first place? If it would behave differently, it would still be mathematically. The math would be different though.

@PaulTanenbaum You can create a link straight to your answer by using the share link in your answer.\ This is a very interesting issue. Just like in mathematics you can create correct sentences in English that have no relation to the real world: Colorless green ideas sleep furiously. This is grammatically correct English. Yet, most English is about the real world. Just like with mathematics, English can be about fictional worlds too. There are just fewer constraints.

@controlgroup Very interesting. I am an amateur when it comes to maths so bear with me. But If you say that these non-orthogonal vector-bases are physically identical then does it follow that we could describe the world in this vector-base but the math would just be unnecessary complicated. Right? But isn't it still true that the orhogonality of the vector-base leads to the inverse square relationship of gravity? You don't have to explain but I would be grateful to a hint where I could learn more about that.

@Philomath how do you know that it’s descriptive rather than prescriptive? That’s just an assumption on your part. Again, if it’s just descriptive, why do many of the rules stay the same the next day?

@Syed Because if it were prescriptive then you (or some other authority) could simply rewrite the rules of physics in your textbook and nature would follow those rules. This is clearly not the case. This is just the meaning of words: we describe the world and not prescribe it. If there is an assumption in my assertion then it is that there is not some rule-following agent in nature and a rule-setting supernatural being. Yes, I will concede that.

@Philomath “ Because if it were prescriptive then you (or some other authority) could simply rewrite the rules of physics in your textbook and nature would follow those rules”. That doesn’t follow. The rules could be inherent in nature

Let us continue this discussion in chat.

I'm putting this as politely as I can, but your examples are complete and utter nonsense. For example: "The presence of pi is linked to the circular symmetry inherent in the normal distribution, where the probability density is evenly distributed around the mean, similar to how points on a circle are distributed around its center." This is at best completely meaningless, but more realistically intentionally misleading. It is in fact a complete word salad.

The same is true for the next one: "Is it unreasonable that our physical universe has three spatial dimensions that are orthogonal? What would it be like if these dimensions were not orthogonal?" Again a completely meaningless word salad. Orthogonality (and dimensions) are not properties of our physical universe, they are properties of our models. And as noted above, any non-orthogonal Euclidean model is equivalent to an orthogonal Euclidean model by a coordinate transform. All your examples sidestep the actual question: Why are all the good models mathematical?

Re: "If it would behave differently, it would still be mathematically." That is a very bold claim. What do you base that on?

@Servaes Thank you for your comments. I'll await your answer in eager anticipation and if it is any better than mine I will duly delete my answer.

"So where does pi come from? The presence of pi is linked to the circular symmetry inherent in the normal distribution, where the probability density is evenly distributed around the mean, similar to how points on a circle are distributed around its center. Not so mysterious after all." << Hi! Have you got a source that expands a little on that mysterious paragraph?

@Stef There is an earlier comment that recommends a video of 3blue1brown I haven't watched it myself but I think it covers this exact topic.

@Servaes Re: I'm putting this as politely as I can, it doesn't matter if you have no manners, you're welcome here. Re: my examples. I am not a mathematician, but I am keen on learning more, so please tell me how I can improve my answer. I am serious.

@Philomath We can continue in chat if you want a deeper explanation, but it’s a fairly simple coordinate transformation between skewed coordinates and orthogonal coordinates. There’s still inverse-square gravity, but the coordinate transformation doesn’t really require very complex math.

@Servaes This might be an ignorant response, but aren't all the good models mathematical simply because all the models, good and bad, are mathematical? I'm not really sure what a model that was both reliably predictive in practice and non-mathematical would look like, beyond broad statements like "gravity makes objects fall". How would you measure its accuracy to determine if it was a better model than alternatives if it wasn't quantitative?