> The whole argument to support this point would be too long to write it in a single post, but the general idea is that mathematics is a system that can easily describe counterfactual worlds. We use it to also describe our physical world because, of course, it can do that. But then asking the question about the "surprising effectiveness" is moot: we deliberately made it to be as effective as possible, so how is it so surprising that it is?
The "surprising effectiveness" is, I think, one of three things. First, the surprise is that we can create mathematics that describes our physical world.
The second surprise is that, when we find out something new, it often takes the form of existing mathematics that we didn't design to describe the physical world. (Though, from your point of view, I suppose you could say that we created mathematics to describe everything that could be described by mathematics, and so it's not surprising that something was there to describe reality.)
The third surprise (maybe this is just a restatement of the first one) is that mathematics really describes the physical world. It's not that we find some math that describes it, and then we change the situation a little bit and we need to find some new math. The surprise is that the math describes what is going on so well that it applies to situations that we didn't know about when we devised the math that applied. That is, it's predictive, not just descriptive.
Well, the last one is a kind of surprise like... if you believe in real numbers, then there are some nonrepeating real numbers (pi is believed to be one of them) that embed all other real numbers. I mean, to someone unfamiliar with the concept it sound amazing and improbable that some number embeds all other numbers (all of which are infinite). But this is because we aren't used to dealing with infinite things.
Similarly, if mathematics is so powerful as to be able to describe any physical reality, it shouldn't be surprising that it can describe ours, no matter how complex and detailed.
The "surprising effectiveness" is, I think, one of three things. First, the surprise is that we can create mathematics that describes our physical world.
The second surprise is that, when we find out something new, it often takes the form of existing mathematics that we didn't design to describe the physical world. (Though, from your point of view, I suppose you could say that we created mathematics to describe everything that could be described by mathematics, and so it's not surprising that something was there to describe reality.)
The third surprise (maybe this is just a restatement of the first one) is that mathematics really describes the physical world. It's not that we find some math that describes it, and then we change the situation a little bit and we need to find some new math. The surprise is that the math describes what is going on so well that it applies to situations that we didn't know about when we devised the math that applied. That is, it's predictive, not just descriptive.