You don't get to patent arbitrary methods.
Yes you do. Arbitrary: subject to individual will or judgment without restriction; contingent solely upon one's discretion. An 'arbitrary method physically instantiated' is commonly known as an "invention"; a time-limited monopoly on its control is called a "patent'.
Furthermore, OP is wrong, as well. An algorithm is an *arbitrary* expression of a *creative* method of solving a problem. Your choice of the *expression* of that method is what is arbitrary; it's only necessary that it be comprehensible to the patent examiner, so that he or she may then understand the originality of your method. It is the method which is original. That method *is* an invention and you *did* build it, out of singular mathematical primitives instantiated as specific constructs of physical information directives, also known as software codes. That is why algorithms should be patentable -- NOT merely copyrightable.
To make an analogy with copyright, let's put it like this: letters representing specific sounds are our common property. Arbitrary strings of those letters -- commonly known as words -- are generally considered the common property of all of us -- but not *necessarily*: thus the rise of trademarks (which *may*, like the now-common concept 'xerox' as a synonym for "copy" pass into the common language. Note /.'s notation at the bottom of this list: "Trademarks property of their respective owners."). Arbitrary conjunctions of those strings are considered copyrightable, and, since 1976 [in U.S. law, at least], are considered "instantly" copyrighted by their creator upon formation -- thus /.'s notation at the bottom of this list: "Comments owned by the poster". However, there is nothing to stop people from designating their personal "creative expression" as 'in the public domain': free for unlimited use by other humans, with neither reference nor payment to the original creator.
Now let's look at software patents: original methods which *do* things to information *utilizing* information in the form of software codes. Parent makes the salient point: Mathematics is the birthright of every human. I would state it as "mathematical operations are our common property". Operations *do* things: mapping one thing to another, or transforming one thing into another. Arbitrary strings of those operations -- commonly known as functions -- are generally considered the common property of all of us. (Example: the Newton-Raphson method to find the zeros of an arbitrary equation.) Functions *do* things, using information to operate on information. A "mathematical operation" can be represented as a single software code: say, 0x40 == "add" ; arbitrary combinations of *those* operations -- software functions -- *could be* considered the common property of all of us -- but not *necessarily*: consider the JPEG and MPEG algorithms (proprietary). [But see here for the furor surrounding patent issues about JPEG.] Arbitrary conjunctions of those functions *could* be considered eligible for software patenting, if their operations *do* "non-obvious things. (However, there is nothing to stop people from designating their personal "methodical expressions of operations" as 'in the public domain': free for unlimited use by other humans, with neither reference nor payment to the original creator (think *some forms of* 'open source software').)
The essential difference between copyrightable property and patentable property is that patents cover methods which *do* things; copyrights cover expressions which *say* things. Methods which merely attempt to cover the transformation of codes from one form to another (say, a patent which covers ASCII-to-UNICODE: too obvious) or are clearly invalidated by prior art (like JPEG eventually was) are not eligible for patent coverage. But what if someone applied for a patent which, as one of its methods, asserted the solution to an as-yet unproved mathematical question, such as the zeroes of the Zeta function, as necessary to the legitimacy of the patent -- AND included the proof. Since mathematicians have been trying for decades to solve that problem -- its solution is non-obvious -- *that* would certainly be considered an invention, would it not? The $64,000 question is: where do we set the bar for "non-obvious"?
I think that this is where Parent goes astray: asserting that mathematics as our common heritage mistakenly implies the *unpatentability* of anything based on mathematics. As I have elucidated above, extending his analogy to "creative expression" -- substituting semantics for mathematics -- would invalidate the meaningfulness of copyright as well -- a right long established in common law, reflecting the right of creators to a monopoly over their own work. But surely Parent would agree that mathematics *does* things; otherwise, it would have no utility. Therefore, it seems that software patents have a right -- indeed, a necessity -- to exist; the only question is where does software stop being obvious and start being non-obvious?