In the early 1990s the U.S. Patent Office issued several patents that reawakened interest in the patentability of “pure” algorithms. The first, U.S. Patent No. 4,744,028, issued to one Dr. Karmarkar and was assigned to AT&T Bell Labs.

This patent covers a new linear algebra technique for allocating scarce resources in a large system such as a telephone network (AT&T’s obvious application of the invention). The Karmarkar algorithm describes an improvement on the well-known (to mathematicians) “simplex method” for solving a very large series of equations, which is how these resource allocation problems are set up mathematically.

The second patent issues on a “pure algorithm” covers a mathematical technique known as the Discrete Bracewell Transform in the field of signal processing. Bracewell’s advance was to create an algorithm that handles sophisticated signal processing without using what are known as “complex” numbers. (These are number which are based on the square root of negative one.)

These patents, which are expected to lead to applications by other mathematicians, raise anew the problems hinted at in the Benson and Diehr cases: what is the nature of mathematics? How do algorithms relate to laws of nature and natural products? Should patents be allowed on “this type” of subject matter?

In comparing computer algorithms to natural products and laws of nature, Justice Douglas states:

Phenomena of nature, though just discovered, mental processes, and abstract intellectual concepts are not patentable, as they are the basic tools of scientific and technological work

Benson, 409 U.S. at 67.

What view of algorithms, and mathematics as a whole is implicit in this statement?

The debate amongst mathematicians on the exact nature of what they do has taken many forms. However, it is possible to simplify the various positions by marshalling them into two main groups.

First are platonists, who believe that mathematics is a real phenomenon which is discovered by mathematicians in the course of their research. On this view, mathematicians simply discover the ordered relationships that nature has laid down.

The alternative view is that mathematics is simply a formal game, which mathematicians “make up” in accordance with strict rules. According to this “formalist” theory, mathematics does not describe any underlying reality. One must simply be careful to state mathematical assertions according to the accepted “rules of the game”. This view comes closer to the theory that math is “invented” by mathematicians.

One overview of the field states:

Most writers on the subject seem to agree that the typical working mathematician is a Platonist on weekdays and a formalist on Sundays. That is, when he is doing mathematics he is convinced that he is dealing with an objective reality whose properties he is attempting to determine. But then, when challenged to give a philosophical account of this reality, he finds it easiest to pretend that he does not believe in it after all.

P. Davis & R. Hersh, The Mathematical Experience 321 (1981).

But the view that math is invented is more starkly stated in the philosophy of Imre Lakatos. Lakatos, whose Proofs and Refutations was published in 1976, sets out a theory of mathematics which places it more properly within modern traditions of the history of science. That is, Lakatos believed that mathematics grows by the criticism and corrections of theories which are never entirely free of ambiguity or the possibility of error. According to Davis and Hersh:

Starting from a problem or a conjecture, there is a simultaneous search for proofs and counterexamples. New proofs explain old counterexamples, new counterexamples undermine old proofs. To Lakatos, “proof” in this context of informal mathematics does not mean a mechanical procedure which carries truth in an unbreakable chain from assumptions to conclusions. Rather, it means explanations, justifications, elaborations which make the conjecture more plausible, more convincing, while it is being made more detailed and accurate under the pressure of counterexamples.

P. Davis & R. Hersh, The Mathematical Experience, supra, at 347 (1981).

Note that in this passage, the authors are discussing Lakatos’ view of that part of mathematics which is in the process of growth and discovery, rather that “settled” mathematics. However, the authors point out that “informal” or unsettled mathematics “is of course mathematics as it is known to mathematicians and students of mathematics” – i.e., the most significant part of the field.

These two authors conclude that neither the Platonist nor the Formalist philosophy of mathematics is ultimately satisfying. They propose instead a view of mathematics that combines the objectivity of the Platonist view with the reliance on social consensus of the Formalist view:

Mathematics is not the study of an ideal, pre-existing non-temporal reality. Neither is it a chess-like game with made-up symbols and formulas. Rather, is is the part of human studies which is capable of achieving science-like consensus, capable of establishing reproducible results. The existence of the subject called mathematics is a fact, not a question. This fact means no more and no less than the existence of modes of reasoning and argument about ideas which are compelling and conclusive, “noncontroversial” when once understood.

P. Davis & R. Hersh, The Mathematical Experience, supra, at 410 (1981).

Mathematics, the authors conclude, has “conclusions [which] are compelling like the conclusions of natural science. They are not simply products of opinion, and not subject to permanent disagreement like the ideas of literary criticism.”

That is, while admitting that at any given time certain propositions at the frontiers of mathematics may be fallible or correctable, they deny that this makes mathematics a meaningless battle of symbols.

What does all this mean for the patent system? First of all, it sheds some light on the naive Platonism of the early Supreme Court opinions on algorithms. As Davis and Hersh point out, there is no consensus among mathematicians that they are in fact discovering a preexisting reality. Thus, the Supreme Court’s treatment of algorithms – as akin to other “found” natural objects, such as products of nature – conflicts with the views that many sophisticated mathematicians see to have of their field. Of course, these views are normally expressed only when “frontier” or pioneer mathematics is at issue; much of the applied mathematics which is the subject matter of algorithm claims would probably be considered outside the discussion of mathematical philosophy anyway. However, even these applied algorithms raise the same philosophical problems. It must be noted that since applied mathematics strives to emulate underlying physical relationships, there is much stronger pull toward the Platonist position when this branch of mathematics is under investigation.

Perhaps this explains some of the cases we have examined. For instance, the use of the Arrhenius Equation in the rubber-curing process at issue in the Diehr case is well within the realm of applied mathematics. That is, this equation tries to capture a physical relationship and state it as a “law”. For the variables stated in this equation, the relationship which it sets fort will always hold. On the other hand, consider the algorithm at issue in the Benson case. This was a “pure” mathematical algorithm which converts binary coded decimal numerals into their binary equivalents. Since numbers of a given base (e.g., base 2 or base 10, the decimal system) do not really correspond to any physical objects, this is an algorithm which states only an abstract relationship. (Compare this to the variables in the Arrhenius equation, which stand for physical properties – pressure, heat and so on.) Perhaps the differences between the Arrhenius equation in the Diehr case and the “pure” number conversion algorithm in the Benson case go a long way toward explaining the different outcomes of the two cases. In any event, the statements made in Benson about the nature of mathematics surely conflict both with the offhand treatment of the mathematical aspects of the Diehr process and the way in which mathematicians themselves view their field, or at least that part of it which deals with purely abstract matters.

The underlying view of mathematics contained in the Benson case may one day be tested when the new generation of mathematical algorithm patents – such as the Karmarkar patent discussed above – come under review.

In the meantime, the debate over the nature of mathematical algorithms is very much alive. Consider some recent comments on the Bracewell and Karmarkar patents, discussed above:

Unlike an industrial technology, an algorithm, the step-by-step recipe for carrying out a mathematical calculation, might seem more like something that is discovered than invented. But in the last few years, corporations have been patenting these abstract procedures, leading many mathematicians to complain that the free flow of ideas is in danger of being interrupted.

“The tradition in algorithms has been that they should be free,” said Ronald Rivest, a mathematician at the Massachusetts Institute of Technology, who said he had mixed feelings on the subject. “Research generally has proceeded on that basis.”

Michael Ian Shamos, a mathematician and computer scientist at Carnegie Mellon University in Pittsburgh and a lawyer in private practice, said that the patenting of important algorithms is contrary to the best interests of science.

“Mathematical facts are the building blocks of research,” he said. “I’m an intellectual property attorney. I like patents. But the patent law was never designed to apply to algorithms. The argument that you spent lots of money developing an algorithm and therefore you should be able to protect it is nonsense.”

G. Kolata, Mathematicians Are Troubled by Claims on Their Recipes, N.Y. Times, March 12 1989

For an argument that the entire software patent issue should turn on the invention/discovery distinction, see John A. Burtis, Comment, Towards a Rational Jurisprudence of Computer-related Patentability in Light of In re Alappat, 79, Minn. Law Review, 1995.

Burtis observes that “Mathematical expressions may be used to describe both discovered and invented subject matter and are therefore imperfect proxies for mathematical truths and other laws of nature.” He concludes by arguing that a “tightly-defined test built on a robust discovery and invention distinction” would improve on Alappat. He then tries to enunciate a test to identify whether an algorithm claim essentially encompasses a “natural truth,” in which case it is an unpatentable discovery, or whether it contains “an implicit, but real, use limitation,” i.e., is tied to a specific application or field of use. Id., at 1165.

In the end, the analysis is helpful because it focuses on the scope of software claims. Recall that in many ways this was the underlying concern in Benson – the case that caused most of the headaches that now plague the law in this area. This approach can be seen as implicitly arguing that software patent doctrine went awry when it rejected “field of use” limitation as a way of preserving patentability.