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.

The reason for making software patentable is just one:

Big corporations see an advantage in it.

That is all about it. That's the reason, they started with this and that is the reason, why nobody changes this system, although anybody with a clear mind can see what harm is being done. The corporations pay hundreds of lobbyists and lawyers to ensure, they can protect their support-system.

All the other reasons told, are just gibberish.

So what distinction, exactly, do you propose? Somehow a definition has to be offered that can be consistently applied by the courts. I don't see what that could be.

Algorithms are mathematical structures, and they're also machines. Both of these claims are true in the plainest, most common meanings of the words possible. An attempt to create a distinction between classes of algorithms has to be based on something. Again: what could that be?

EDITED to add: just so it's clear where I'm coming from -- I think algorithms are clearly patentable subject matter. The problem we have with software patents is that the obviousness bar is far, far too low.

> Algorithms are mathematical structures, and they're also machines.

Sorry, but you can kick a machine. You can not kick an algorithm, exactly like can you can not kick any kind of mathematical construction.

Any program you can write, you could build a physical machine to execute.

Now consider a physical general-purpose computer with your program loaded into it. Can you really argue that this is fundamentally a different kind of object than the special-purpose machine? They behave, let's suppose, identically.

Okay, maybe you would say, the program considered abstractly is still an object I can't kick, so it's not patentable; the only thing that's patentable is a physical computer with the program loaded into it. But that doesn't help you, because no one can actually use the program without loading it into a computer anyway. So if you get a patent on a computer with your program loaded, that has the same practical effect as a patent on the program itself. No?

Any algorithm has an infinite set of different physical machines that could be built that will execute it. A machine expressing an algorithm is not the same as the algorithm itself.