[throwawaykf03]

The "software is mathematics" argument has a few flaws:

1) It is reductio ad absurdum, like saying "machines are metals". It is not the naturally occurring metals that are patentable in physical inventions, it is how you configure and use them. Similarly it's not the mathematics that is patentable, it is the application thereof to a practical problem.

2) More importantly, it misunderstands what abstract math is from a legal perspective. People will invoke the Church-Turing thesis and several related theorems to prove that software being executed is math, but they miss the point.

An abstraction, by definition, cannot affect or effect anything in the real world. An idea in your head is abstract. The moment you act on it in the physical world, it is real. Similarly, if you can run some piece of software and get useful, practical, real world results, it is not abstract.

[harshreality]

You're splitting math into separate categories that are not adequately defined by court precedent.

The legal profession has at best an inaccurate and non-uniform concept of what math is. Relying on what the legal profession views as math therefore is not a valid argument and depends on which part of the legal profession you survey.

If software by itself can be a patentable device/machine for the purposes of patent law, that needs to be made explicit, and I happen to think it will cause serious problems; it already is causing major problems, even without explicit supporting precedent.

Almost every software patent you can find mentions network communications, memory storage, disk storage, display, or other artifacts of general purpose computing. If software by itself is, or should be, patentable, why is everyone trying so hard to patent complete machines rather than only software algorithms? Maybe it's that they don't agree with you that pure software implementations of solutions to real world problems are patentable?

According to your paradigm of allowing patents for application of math to a practical problem, you're fine with someone patenting a moon slingshot maneuver to get back to Earth? That's an application of pure math.

We agree you can't patent using equations to predict motion, right?

Can you patent using equations to predict motion and reversing that to generate a solution to a specific orbital problem?

Can you patent running that algorithm on a general purpose computing machine?

Can you patent an embedded platform that is built out of a general purpose computing machine, but is sold as a black box running only that algorithm with suitable inputs and outputs?

[throwawaykf03]

I'm not sure which categories you mean I'm splitting mathematics into. As for precedent, even the Supreme Court cannot define abstractness other than "we know it when we see it." Jurists may certainly have a loose grasp of mathematics, but it makes no difference at a general level because saying "software is mathematics" makes no sense when faced with the fact that executing software achieves tangible, practical results, and above all law is about practicalities. You won't convince any judge, jury or any practical person that the software they use is abstract, because they derive practical benefit from it on a daily basis.

A moonshot is certainly patent-eligible. You said it yourself: it is an application of pure math, and it achieves tangible results, namely, transporting an object through space. The pure mathematical equations are not patented. You can use those equations for other uses, say calculating asteroid trajectories.

As you said, equations on predicting motion are not patentable by themselves because they don't do anything by themselves. They simply describe aspects of the physical world. Any patent hoping to claim anything as such is supposed to be, and usually is, rejected out of hand.

But, a computer running those equations for a practical purpose, such as say, predicting impact with a car on the road and warning you, is patent eligible. Or maybe a method of calculating those equations more efficiently or faster using e.g. a GPU.

Do you see the difference? Neither of those examples is a patent on the equation itself, but rather an application thereof or a particular way of implementing it. None of those preempt you from using the equation somewhere else, e.g. to control a robot that can catch a ball you throw at it.

That leads to why most software patents include language for hardware. The reasons are manifold:

1. To be properly enabled, they must describe the invention in as much detail as possible. Ironically, after adding all that boilerplate about the hardware, they then skimp on the description of the invention itself.

2. The software is useless without the hardware. No machine will infringe without executing that software on hardware.

3. The term software itself is not well defined in the context of patent law. The constitution certainly seems to allow it, because it is undeniably a "science and useful art". But when the laws were drafted there was no such thing, and so there is no statute allowing it or denying it. So it is claimed as both a method or process (a way of doing things) or a system (a machine that does things) or both. And really, software is both: when executed it does things, and the machine that executes it is a machine that does those things.