I think that you're responding to an extreme interpretation of the "unreasonable effectiveness of mathematics". I don't think you have to believe that nature follows perfectly simple mathematical rules to be surprised to the extent that mathematics is effective at describing physical reality.
You make a good point, that mathematics is often designed and invented to describe reality. You need to take into account the fact that a lot of what you "get out" of mathematics has already been "put in". But it is my opinion that even accounting for this, it is quite frequent that you "get out" more than you "put in" to mathematics.
I think a strong set of examples of this sort of behavior is when attempts to address one problem in a mathematical theory often end up addressing other seemingly unrelated problem. The introduction of quantized energy states to explain the blackbody radiation distribution then goes on to explain the photoelectric effect. (I'm not too familiar with what physical phenomena were observed before QM but needed QM to explain them to give more examples).
Now you might say "well of course this theory solves both issues, because they are essentially the same phenomenon, and so there's nothing surprising about your theory explaining what it was designed to explain". But that's precisely the point, that mathematics gives us the tools to see relations between things that on the face of them are different. And that there's no reason that this /has/ to be true (without appealing to mathematics), makes the fact that it is true seem surprising.
Of course, being surprising, or unreasonably effective, etc is a relative statement. Certainly to someone who expected the universe to be described mathematically to precisely the degree that it is in reality, nothing would be surprising. But I think to most people who haven't been introduced to mathematics (kids), the power it gives you certainly is surprising.
You make a good point, that mathematics is often designed and invented to describe reality. You need to take into account the fact that a lot of what you "get out" of mathematics has already been "put in". But it is my opinion that even accounting for this, it is quite frequent that you "get out" more than you "put in" to mathematics.
I think a strong set of examples of this sort of behavior is when attempts to address one problem in a mathematical theory often end up addressing other seemingly unrelated problem. The introduction of quantized energy states to explain the blackbody radiation distribution then goes on to explain the photoelectric effect. (I'm not too familiar with what physical phenomena were observed before QM but needed QM to explain them to give more examples).
Now you might say "well of course this theory solves both issues, because they are essentially the same phenomenon, and so there's nothing surprising about your theory explaining what it was designed to explain". But that's precisely the point, that mathematics gives us the tools to see relations between things that on the face of them are different. And that there's no reason that this /has/ to be true (without appealing to mathematics), makes the fact that it is true seem surprising.
Of course, being surprising, or unreasonably effective, etc is a relative statement. Certainly to someone who expected the universe to be described mathematically to precisely the degree that it is in reality, nothing would be surprising. But I think to most people who haven't been introduced to mathematics (kids), the power it gives you certainly is surprising.