The more vague the laws we allow, the more prone they are to abuse. Also, it's not clear that there is any meaningful distinction that can be drawn between types of algorithms. Any process which has numbers (or symbols, bits, data, etc.) as input and output is mathematical. The fact that all types of data are equivalent may only go back to Godel or so, but algorithms themselves have been known as mathematical since the very beginning.
> "all algorithms are math" is an irrelevant reason for making software not patentable
Not so, the statutes themselves are a bit unclear, but they say that processes are patentable but math is not. Unfortunately, the courts have not fully realized the fact that algorithms are mathematical processes nor are they overly clear about what is and is not math, as they have clever lawyers crafting arguments that go against what mathematicians have considered as math for millennia.
al-Khwārizmī may be responsible for the world algorithm, but they've been used by mathematicians since at least Euclid in ~300 BC.
Your mistake is assuming that the criterion for distinguishing between unpatentable math and patentable algorithms must come from within the domain of computer science. There is no reason that has to be the case. The distinguishing criterion can be wholly external, rooted in policy concerns.
Laws can be made in any way they like. The argument is not in how laws can be made, but in whether they are any good. This is an argument about quality and usefulness of law. As Knuth puts it, "Man-made laws can be significantly helpful but not when they contradict fundamental truths.".
You are right that the method for deciding on what parts of math are subject to patent can be wholly external to maths and not pay any attention to mathematics whatsoever, however what Knuth is saying, and I would tend to agree with him, is that this would be folly.
That is a truism that I did not find useful to call out. It's true that they can redefine words, even after millennia of usage, to suit their purposes. But, for example, defining that pi in legal contexts is exactly three serves only to make communication impractical, as if we are trying to talk to Humpty Dumpty in Alice & Wonderland whose words mean exactly what he wants them to and nothing else.
So it's true, they can create an arbitrary distinction. The problem with that is that it can be made to mean anything and, for that reason, is inherently meaningless.
I understand this to be the sort of vague law they, quite reasonably, normally attempt to avoid. I cannot see any reasonable justification for adopting that here, though I suppose that those in the patent bar have a monetary interest in saying otherwise.
> "all algorithms are math" is an irrelevant reason for making software not patentable
Not so, the statutes themselves are a bit unclear, but they say that processes are patentable but math is not. Unfortunately, the courts have not fully realized the fact that algorithms are mathematical processes nor are they overly clear about what is and is not math, as they have clever lawyers crafting arguments that go against what mathematicians have considered as math for millennia.
al-Khwārizmī may be responsible for the world algorithm, but they've been used by mathematicians since at least Euclid in ~300 BC.