Patents and mathematics

>>"Surely nobody could apply mathematics if it were necessary to pay a license fee whenever the theorem of Pythagoras is employed."

I've seen this argument why algorithms and mathematics shouldn't be patentable many times. A fact that is always overlooked is that this hypothetical patent would have expired sometime in the 5th century BC. To make this argument one would have to find more recent examples. How about anything created within the last 20 years?

In my view, the greatest things in mathematics aren't practical to be patentable and the practical ones aren't anywhere close in magnitude to the Pythagorean Theorem. So, where is the problem?

It's interesting, the unpatentability of mathematics is used to justify unpatentability of such things as algorithms, software etc. They'd say "It's just math!" How about "it's just physics"?

Without patents we'd never know what PageRank is and we'd still think this is best we can do...

• U.S. Patent No. 6,285,999 "Method for node ranking in a linked database".

There are a few "pure math" patent applications published by the Patent and Trademark Office.

• US Patent Application No. 20050232511, Image model based on n-pixels and defined in algebraic topology: patent. The inventors are three Canadian topologists. The subject is related to Discrete exterior calculus.
• US Patent Application No. 20060013505, Analysis of geometric surfaces by conformal structure. One of the inventors is a Field medalist from Harvard.
• US Patent Application No. 20070036434, Topology based method of partition, analysis, and simplification of dynamical images. This is about computational topology to image analysis.
• US Patent Application No. 2010/023771, Systems and methods for visualization of data analysis, by Ayasdi. This is an application of computational topology and topological data analysis.