U.S. PATENT LAW DOES NOT ADD UP

Being derived from medieval concepts about land, U.S. patent law lacks an objective basis for awarding ownership rights in intellectual property. 

Land is excludable: you can keep someone from it with a fence. Land is rivalrous: no two people can stand in the same place at the same time.  According to the economic theory of public goods, knowledge is non-rivalous and non-excludable: sharing knowledge does not diminish its quantity; and you cannot prevent someone from knowing what they know. Thus, it is generally true that you cannot get a patent on a mathematical theorem.

See 35 U.S.C. §101: Subject Matter Eligibility• The four statutory categories of invention:– Process, Machine, Manufacture, or Composition of Matter and Improvements Thereof• The courts have interpreted the categories to exclude:– “Laws of nature, natural phenomena, and abstract ideas”• These three terms are typically used by the courts to cover the basic tools of scientific and technological work, such as scientific principles, naturally occurring phenomena, mental processes, and mathematical algorithms.Evaluating Subject Matter Eligibility Under 35 USC § 101: August 2012 Update  Office of Patent Legal Administration United States Patent and Trademark Office here.

However, patents have been awarded for mathematical ideas. Of course a plethora of patents exists for computer software, even though a program is only an algorithm, a method of calculation.

US Patent 4133152 A was awarded to Roger Penrose for “Set of tiles for covering a surface.”  (Patent here.)

http://mathworld.wolfram.com/PenroseTiles.html

“A set of tiles for covering a surface is composed of two types of tile. Each type is basically quadrilateral in shape and the respective shapes are such that if a multiplicity of tiles are juxtaposed in a matching configuration, which may be prescribed by matching markings or shapings, the pattern which they form is necessarily non-repetitive, giving a considerable esthetic appeal to the eye. The tiles of the invention may be used to form an instructive game or as a visually attractive floor or wall covering or the like.” – US Patent 4133152A

A Penrose tile is a “non-periodic tiling generated by an aperiodic set of prototiles. – Wikipedia.

The Penrose tiles are a pair of shapes that tile the plane only aperiodically (when the markings are constrained to match at borders). These two tiles, illustrated above, are called the "kite" and "dart," respectively. In strict Penrose tiling, the tiles must be placed in such a way that the colored markings agree; in particular, the two tiles may not be combined into a rhombus (Hurd). – from Wolfram here. 

Penrose Tiles and Trapdoor Ciphers … and the Return of Dr. Matrix 

by Martin Gardner. 

Washington DC: Mathematical Association of America, 1989. 

"Because the tiles lend themselves to commercial puzzles, [Roger] Penrose was reluctant to disclose them until he had applied for patents in the United Kingdom, United States, and Japan.  The patents are now in force." Penrose Tiles and Trapdoor Ciphers … and the Return of Dr. Matrix by Martin Gardner. Washington DC: Mathematical Association of America, 1989, Page 6.

So, mathematical ideas can be patented if you claim that the theorem is a puzzle (or has some other physical instantiation).  

Mathematician David A. Edwards asserts that no economic distinction can be made between a discovery and an invention. 

From an economic point of view, there is no rationale for distinguishing between discovery and invention, and we would advocate dropping entirely any subject matter restrictions whatsoever on what can be patented. One should be able to patent anything not previously known to man. In fact, a good economic case can be made2 for allowing the patenting of many things that are well known but are not being commercially produced.
 “If we're going to have a general patent system, then algorithms should be as patentable as lasers. For example, general relativity is used in GPS…  My colleague Carl Pomerance developed fast primality testing algorithms in the late 1970s but couldn't patent them. My colleague Victor Wickerhauser developed the fast wavelet transform in the early 1990s and was able to patent it as a software patent.  … If we want these things to be patentable, then Congress needs to change the law.”  (“Platonism Is the Law of the Land,” David A. Edwards
April 2013 Notices of the AMS Volume 60, Number 4 pp475- 478 here. 

Intersecting the discursive plane from another angle, mathematician Robert Palais points to the low esteem in which the USPTO holds mathematics. 

 …  an applicant to practice before USPTO must demonstrate, in accordance with the USPTO’s requirements, that he or she possesses scientific and technical proficiency sufficient to address issues that arise in patent law. Notably, however, mathematics is explicitly excluded as a subject for this purpose. …   I downloaded the “General Requirements Bulletin for Admission to the Examination for Registration to Practice in Patent Cases Before the United States Patent and Trademark Office” to see for myself. It lists 32 subjects in which bachelor’s degrees exhibit adequate proof of the necessary scientific and technical training, as well as 2 1/2 pages of acceptable alternates. Then it states the “Typical Non-Acceptable Course Work: The following typify courses that are not accepted as demonstrating the necessary scientific and technical training:” and in the middle of this paragraph, there appears: “…machine operation (wiring, soldering, etc.), courses taken on a pass/fail basis, correspondence courses, …home or personal independent study courses, high school level courses, mathematics courses, one day conferences, …” 
 … Ironically, USPTO requires mathematics coursework for prospective examiners in the computer arts (employees) that it doesn’t recognize as qualifying for practitioners.
 This is not a debate regarding the appropriateness of patenting mathematics. There have been many such conversations elsewhere. But in these times that mathematics is becoming increasingly visible in valuable patents (e.g., Google’s Page Rank, a linear algebra algorithm, was licensed by Stanford for US$336 million) it seems that the USPTO should be encouraging, not discouraging, the mathematical fluency of the lawyers whose work it recognizes. 
—Bob Palais, Math Dept., Pathology Dept. University of Utah, Notices of the AMS, January 2010, page 9,  here. 

Euclid's Proof of the Pythagorean Theorem
Proposition 47 Book 1.
(Not Socrates's Demonstration,
which Plato and the USPTO say that you were born with.
Too bad you were not born with this one.)

Plato believed that mathematics exists within you to be discovered by introspection.  In the Dialog of Meno, Socrates draws out from a slave an application of the Pythagorean theorem to show the doubling of an isosceles right triangle.   The so-called “Socratic method” favored by many teachers assumes that this knowledge is inherent within you.  That is the epistemology of the US Patent and Trademark Office.  Mathematics need not be invented – or even discovered by external exploration – because we all have it in our heads.   That theory just does not add up.

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