(Reposted from my comment on a Groklaw article)
The SFLC Bilski brief argues that (1) the US Supreme Court has essentially held that mathematical algorithms cannot be patented, and that (2) if mathematics were to be patented, this would be bad for innovation:
… This Court has repeatedly held that subject matter which would have the practical effect of preempting laws of nature, abstract ideas or mathematical algorithms is ineligible for patent protection.
…
II. Excluding Software From Patentable Subject Matter Maximizes Innovation In Software
If mathematics were patentable, there would be less mathematical innovation. Only those who were rich enough to pay royalties, or who benefited from subsidization by government, or who were willing to sign over the value of their ideas to someone richer and more powerful than themselves, would be permitted access to the world of abstract mathematical ideas. Theorems build upon theorems, and so the contributions of those who could not pay rent— and all the further improvements based upon those contributions — would be lost. …
The second point of the SFLC could have been made stronger by reference to (e.g.) the 1991 Report of the Committee on Algorithms and the Law of the Mathematical Programming Society.
It seems clear from the previous discussion that the nature of work on algorithms is quite different from that in other fields where the principles of patents apply more readily. This in itself is a strong argument against patenting algorithms. In addition, we believe that the patenting of algorithms would have an extremely damaging effect on our research and on our teaching, particularly at the graduate level, far outweighing any imaginable commercial benefit. Here is a partial list of reasons for this view:

Patents provide a protection which is not warranted given the nature of our work.

Patents are filed secretly and would likely slow down the flow of information and the development of results in the field.

Patents necessarily impose a longterm monopoly over inventions. This would likely restrict rather than enhance the availability of algorithms and software for optimization.

Patents introduce tremendous uncertainty and add a large cost and risk factor to our work. This is unwarranted since our work does not generate large amounts of capital.

Patents would not provide any additional source of public information about algorithms.

Patents would largely be concentrated within large institutions as universities and industrial labs would likely become the owners of patents on algorithms produced by their researchers.

Once granted, even a patent with obviously invalid claims would be difficult to overturn by persons in our profession due to high legal costs.

If patents on algorithms were to become commonplace, it is likely that nearly all algorithms, new or old, would be patented to provide a defense against future lawsuits and as a potential revenue stream for future royalties. Such a situation would have a very negative effect on our profession.
The first point of the SFLC brief could have been stronger as well. It may not be entirely clear that the Supreme Court has ruled that mathematical algorithms as such are unpatentable. The FSF Bilksi brief argues that algorithms with insufficient “postsolution activity” are unpatentable:
A. THIS COURT HAS RULED THAT INFORMATION PROCESSING ALGORITHMS WITH “INSIGNIFICANT POSTSOLUTION ACTIVITY” ARE BARRED FROM PATENTELIGIBILITY.
There is little controversy that information processing algorithms in their pure, ethereal forms, with no physical component or manifestation of any sort, are excluded from patentability. [ Novel mathematics, for example, is outside the scope of patenteligibility. “Whether the [mathematical] algorithm was in fact known or unknown at the time of the claimed invention … it is treated as though it were a familiar part of the prior art.” Parker v Flook, 437 U.S. 584, 59192 (1978) (internal citation omitted). ] However, what is under debate is how much of a physical manifestation an information processing
algorithm must have before it is patentable.
In a trio of opinions issued over the span of nine years, this Court clearly rejected the patentability of an information processing algorithm with “insignificant postsolution activity” appended. First, in Gottschalk v. Benson, 409 U.S. 63 (1972), the Court quoted approvingly the 1966 President’s Commission on the Patent System: …
Second, in Parker v. Flook, 437 U.S. 584 (1978), this Court made a more general statement, reiterating the position that loading an algorithm onto a standard computer is merely an attempt to circumvent recognized limitations: …
Third, in Diamond v. Diehr, 450 U.S. 175 (1981), the Court directly reiterated its two previous holdings, while also acknowledging that bona fide, patenteligible inventions may include a software component: …
To me, the crux of the matter seems to lie in the part of judgement of the US Federal Circuit in re Alappat 33 F.3d at 1543 n.19, 31 USPQ2d at 1556 n.19 (1994) which essentially states that the US Supreme Court has been confused, inconsistent or at best unclear in its understanding of mathematics and how it relates to patentable subject matter:
19 The Supreme Court has not been clear, however, as to whether such subject matter is excluded from the scope of Section 101 because it represents laws of nature, natural phenomena, or abstract ideas. See Diehr, 450 U.S. at 186 (viewed mathematical algorithm as a law of nature); Benson, 409 U.S. at 7172 (treated mathematical algorithm as an ‘idea’). The Supreme Court also has not been clear as to exactly what kind of mathematical subject matter may not be patented. The Supreme Court has used, among others, the terms ‘mathematical algorithm, ‘mathematical formula,’ and ‘mathematical equation’ to describe types of mathematical subject matter not entitled to patent protection standing alone. The Supreme Court has not set forth, however, any consistent or clear explanation of what it intended by such terms or how these terms are related, if at all.
20 The Supreme Court’s use of such varying language as ‘algorithm,’ ‘formula,’ and ‘equation’ merely illustrates the understandable struggle that the Court was having in articulating a rule for mathematical subject matter, given the esoteric nature of such subject matter and the various definitions that are attributed to such terms as ‘algorithm,’ ‘formula,’ and ‘equation,’ and not an attempt to create a broad fourth category of excluded subject matter.
The Manual of Patent Examining Procedure, 2106.02 Mathematical Algorithms [R5] – 2100 Patentability says this ruling “recognized the confusion”.
In practical terms, claims define nonstatutory processes if they:

consist solely of mathematical operations without some claimed practical application (i.e., executing a “mathematical algorithm”); or

simply manipulate abstract ideas, e.g., a bid (Schrader, 22 F.3d at 29394, 30 USPQ2d at 145859) or a bubble hierarchy (Warmerdam, 33 F.3d at 1360, 31 USPQ2d at 1759), without some claimed practical application.
Professor Lee A. Hollaar (whose Bilski amicus brief has been previously discussed on Groklaw) holds in his treatise that the Alappat decision affirmed the US Patent Office’s practice on software related inventions, including the idea that a programmed general purpose computer is a specialized machine.
Alappat allowed the Federal Circuit to restate and clarify its past decisions on whether softwarerelated inventions are patentable. In particular, it is clear that a programmed general purpose computer must be regarded as a specialized piece of hardware both for determining whether a claim is drawn to statutory subject matter and when determining whether the invention is novel and nonobvious. It is also clear that the ‘mathematical algorithm’ exception to statutory subject matter first discussed by the Supreme Court in Benson is limited to abstract mathematical concepts, not mathematics applied to a practical application. Machines, even though they carry out mathematical operations, are patentable.
This really did not differ substantially from the Patent Office’s practice. The time was long past when the Office rejected an application just because it was a softwarerelated invention. There were over 10,000 patents that could be considered softwarerelated at the time of Alappat. But the Office position had swung back and forth on the patentability of softwarerelated inventions. Alappat restricts the Patent Office from treating softwarerelated inventions more strictly under Section 101 than other inventions.
Would it have been wise for the SFLC to have called attention to the Alappat ruling and its consequences, and asked the US Supreme Court for further clarification of its rulings on mathematical algorithms, in light of the First Amendment? Hasn’t the US Supreme Court ruled on the idea that a general purpose computer when programmed with a mathematical algorithm is a special purpose machine for the purposes of patentability, even if the machine does nothing other than execute the algorithm faster than can be done by hand? Hasn’t the US Supreme Court ruled on what is a “practical application” in relation to the claims of a patent, in particular, whether an idea for a practical application is patentable if the only novel part of the idea is the execution of a mathematical algorithm on a general purpose computer? Should the SFLC have reminded the US Supreme Court more strongly about such rulings and what their bearing should have been on Alappat, let alone Bilski?