I am not sure why you are using the phrase "natural science", is there any other kind? Lets revisit with the wikipedia definition:
Science (from Latin scientia 'knowledge') is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
That seems pretty general and all-inclusive to me. Any knowledge about humans; or societies; or the stuff human societies invent is knowledge about the universe. Because we are part of the universe.
Now, you seem to be trying to differentiate the different sciences from one another - tell us why. What's the purpose of drawing a distinction between science and (what you call) "natural science"?
I am not doing so. I am saying that if you want to classify mathematics as a science on the basis of the existence of theorem validation software on which someone may experiment, then it is a science on human constructs, ie, a social science, and not a natural science.
The argument here for math to be a science is very tortured and contrived, but if it has any even technical validity then it makes the argument for math as a social science.
>if you want to classify mathematics as a science on the basis of ... The argument here for math to be a science is very tortured and contrived,
I am not classifying Mathematics as a science on the basis that you claim. I am not even making an argument! All I am doing is empiricism!
I am observing a fact about the universe: What humans call science is any enterprise which builds and systematically organizes knowledge about the universe. I am also observing that Mathematics is one such enterprise. So it's a science.
If you think society's definition of "science" is too broad - that's fine.
If you want to use a narrower definition which renders Mathematics a non-science - that's also fine.
But I am still asking why?
Why does it matter whether Mathematics is classified as a science; or a non-science; or a social science; or a natural science?
Now, you seem to be trying to differentiate the different sciences from one another - tell us why. What's the purpose of drawing a distinction between science and (what you call) "natural science"?