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I don't agree that math is in nature. Math, and numbers themselves, were invented by man. Numbers are abstract symbols, sometimes denoting amounts of something ("3 peanuts"), sometimes standing as symbols of themselves ("3"). And math itself is an abstract concept, sometimes applied to a concrete situation ("2 peanuts and 3 peanuts gives you 5 peanuts") and sometimes remaining purely in the abstract realm "2 + 3 = 5").
There's nothing in nature about "math". Sure, a water molecule consists of 2 hydrogen atoms and 1 oxygen atom, due to the way physics works, but it was man that assigned the symbol "2" to denote the amount of hydrogen atoms involved; nature couldn't care less about that. And it was man that created math and created equations that describe the laws of physics, but nature doesn't care about those equations. The laws of physics existed in nature before any intelligent race created equations to describe them.
So I disagree with the notion that "math is in nature", unless one goes down the road of "man is in nature, therefore anything man does is in nature, and man created math, therefore math is in nature - Q.E.D.", at which point every idea, thought, dream, or fantasy is in nature, including every invention and every invention idea, whether those inventions are made out of atoms or made out of bits.
However that doesn't rule out the "discovered" concept. Isaac Newton invented calculus, and I'd argue that calculus didn't "exist in nature", but one could argue that it did already exist in the "abstract", waiting to be "discovered" by a member of an intelligent species probing the abstract realm. But that could be said of any invention, be it software or physical.
As for "ideas shouldn't be patented", I don't agree with that as a general principle (though I agree with it in particular instances). If the idea isn't obvious (and particularly, if the idea took a large amount of resources to come up with and develop), then why not?
And if we say, "only implementations (not ideas) can be protected (via patents, copyright, or other mechanism)", then how similar does implementation B have to be to the original implementation A in order to say that B infringes on A? Does changing the order of program instructions in B free it from infringment charges on A, even if changing the order of those instructions doesn't affect the "output" of the program? Does B's using a linked list whereas A used a binary tree free B from infringemnent charges on A? Or in the physical (non-software) world, if an invented device is made from brass and someone makes a copy of it using steel, is the copy freed from infringement charges because it's a different "implementation"?
At one extreme is "B infringes A only if B is an exact copy of A" and at the other extreme is "B infringes A if B implements the same idea as A, even if the implementation is totally different". I'd disagree with both of those extremes, generally speaking. I'd judge on a case-by-case basis whether B infringes on A, depending on the "idea" itself. Most software patented "ideas" describe, not just a general concept, but the implementation as well - not the exact source code, but an "algorithm" (that could be implemented by an infinite number of source codes), or more broadly, a fairly well-defined method, that could be implemented by numerous "algorithms", but those algorithms would all look quite similar to one another. The "idea" that is patented is typically the concept PLUS the described method (and sometimes the two are intertwined and unseparable). And if implementor B alters the "method" enough (that is, enough to be considered of some significance (not something like using a hash table rather than a tree)), then B can be free of infringement charges on A.
But I don't think these things are black-and-white. Apple does push the envelope with its "look-and-feel" claims.
Really? Math isn't nature? What we write down is abstract. The letters we use is an abstract. The symbols we use is an abstract. But math is not symbols.