Yes, what is patented is some commercial application of some knowledge, and that application is the invention. It doesn't matter whether the knowledge itself is invented or discovered. When an algorithm is patented, we're no more patenting math than we're patenting Newton's laws when a car brake is invented. We're patenting a commercial application of either, and that must be the invention.
Commercial application is irrelevant to a patent law. Patent applications need only demonstrate "eligibility, utility, novelty, and non-obviousness." Math and other basic research are ruled out on eligibility grounds, not for being non-commercial, but rather for being "abstract ideas."
Also, algorithms are a bad example, as they are just mathematical functions, i.e. "abstract ideas." They should not be patentable. Although I realize patent law is not actually logically consistent.
Commercial application is the motivation behind patent law, and logical consistency is not the point, but rather legal consistency. When you patent an algorithm you do not patent the idea any more than you patent thermodynamics when you patent an engine. You patent a particular application of an idea that performs some function or functions. The algorithm -- or thermodynamics -- stay completely free for anyone to know, study and disseminate. In fact, patents are designed to make the knowledge public so that they could be studied and improved upon.
Whether or not it works is a separate question (and I completely agree that software patents do not perform their role), but patenting a device to predict stock prices is not patenting math, just as patenting a drug is not patenting chemistry.
