Software is math
Mathematical formulas are generally recognised as non-patentable because math is not patentable subject matter.
Since the logic (idea) of software can be reduced to a mathematical formula (idea) with Church-Turing Thesis, and because mathematical formulas (idea) are not patentable, patent applications for software ideas should be rejected.
Respected computer scientist Donald Knuth makes the argument:
To a computer scientist, this makes no sense, because every algorithm is as mathematical as anything could be. An algorithm is an abstract concept unrelated to physical laws of the universe.
 Math is not patentable
 Case law in the USA
In the USA, math is unpatentable because it is a "law of nature", that is to say a "scientific truth", and as such it can never be "invented", only "discovered", and patents are not granted for discoveries.
The non-patentability of math was confirmed in the case Parker v. Flook (1978, USA):
Respondent's method for updating alarm limits during catalytic conversion processes, in which the only novel feature is a mathematical formula, held not patentable under 101 of the Patent Act.
Also, in the 1948 case Funk Bros. v. Kalo Inoculant:
He who discovers a hitherto unknown phenomenon of nature has no claim to a monopoly of it which the law recognizes. If there is to be invention from such a discovery, it must come from the application of the law of nature to a new and useful end.
Ideas which use math can be patentable, but this is not controversial:
While a scientific truth, or the mathematical expression of it, is not patentable invention, a novel and useful structure created with the aid of knowledge of scientific truth may be.
 Some judges say math is patentable
In the 2011 UK High Court decision on the Halliburton case, the judge said that math can be patentable because:
the data on which the mathematics is performed ... represent’s something concrete (a drill bit design).
Also in 2011, the US CAFC Cybersource v. Retail 16 Aug 2011 case, an algorithm was held patentable because:
as a practical matter, the use of a computer is required.
 Church-Turing Thesis or Curry-Howard isomorphism?
There are two mathematical bases that can be used to make this argument.(Can you help? This page was written by a non-specialist. Any help would be appreciated.)
The Church-Turing Thesis is the more commonly used based. It is discussed by some documents linked in the #External links section.
Another approach would be the Curry-Howard isomorphism, which demonstrates that computer programs are equivalent to mathematical proofs. If proofs are unpatentable, then computer programs must be too.
 EPO says software is math
According to the EPO, as written in EPO EBoA referral G3-08 (page 12 of 18):
computer programs were to be understood as a 'mathematical application of a logical series of steps in a process which was no different from a mathematical method
 Related pages on en.swpat.org
- Anti-lock braking example - if the physical car invention is patentable, should an in-computer game-simulation be?
- Software does not make a computer a new machine
- Australia#Case law - patents on math might be valid in Australia
- Pen and paper patents - what if math is so complicated, a pen and paper are required?
- The Rise Of The Information Processing Patent, by Ben Klemens (Church-Turing is discussed on page 8)
- Church-Turing thesis, Wikipedia
- Curry–Howard isomorphism, Wikipedia
- Offshore Software Development
- Does not compute: [US] court says only hard math is patentable, Aug 2011, Timothy B. Lee
- (in German) http://www.users.sbg.ac.at/~jack/legal/swp/tech-turing-lambda.pdf
 PolR's articles on Groklaw
- An Explanation of Computation Theory for Lawyers, 11 Nov 2009
- Physical Aspects of Mathematics (An Open Response to the USPTO), 27 Sep 2010, (submission to USPTO 2010 consultation)
- A Simpler Explanation of Why Software is Mathematics, 8 Sep 2011
- 1 + 1 (pat. pending) — Mathematics, Software and Free Speech, 26 Apr 2011
- What Does "Software Is Mathematics" Mean? Part 1 - Software Is Manipulation of Symbols, 13 Oct 2012
 Counter arguments
- Computer Software is Not Math, 15 Dec 2008, IP Watchdog
- On Abstraction and Equivalence in Software Patent Doctrine: A Response to Bessen, Meurer, and Klemens (challenging, inter alia, Klemens's "repeated mischaracterizations of the Church-Turing Thesis")
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