I saw an opinion piece by Stephen Ornes, in the March 16 issue of New Scientist which ties the ongoing debate about the nature of mathematical ideas, to a modern one about money and ownership. Ornes argues that patentability is one of the most hotly contested issues in software development. The problem, as many see it, is that not all software is patentable because of its dependence on mathematics. Mathematics is understood as the exploration of abstract ideas, not the invention of new products. Ornes referred to an essay by David Edwards (University of Georgia) in the April 2013 issue of Notices of the American Mathematical Society. In the end, Edwards is calling for an update of the patent laws because the current laws do not promote the development of technological innovation. I wasn’t very inspired by the discussion. However, when I went to find the Edwards piece in the AMS Notices, I stumbled upon an essay, written in a completely different spirit, and published in January 2012. Jason Scott Nicholson, then a Ph.D. candidate in mathematics at the University of Calgary, addressed Eugene Wigner’s consistently cited query into the “unreasonable effectiveness of mathematics.” But Nicholson explores the puzzle of mathematics’ effectiveness using the structure of the ideas brought to life in the book Lila by Robert M. Persig, author of the widely read Zen and the Art of Motorcycle Maintenance.
Nicholson explains that, in Lila, reality is dual-aspected. One of these aspects is what Persig calls Static Quality, and the other, what he calls Dynamic Quality. A very brief explanation of these ideas is this:
Dynamic Quality is understood as the creative urge, the constant stimulus to move, perhaps to something ‘better.’ Static Quality is what is given in the patterns reflecting the “realization” of the undefined Quality that is the world. Static Quality is created in response to Dynamic Quality. It exists on 4 discrete but related levels:
Inorganic Biological Social and Intellectual
In this system, the biological builds on the inorganic, the social on the biological, and the intellectual on the social. Nicholson tells us that Perig uses a computer analogy to illustrate this idea:
He describes the relationship between these levels as being analogous to the relationship of computer hardware to computer software—the software is run on the hardware, but has nothing, really, to do with it. The program that you run on your computer and write your article with has nothing to do with the computer hardware itself. Furthermore, the content of your article has nothing to do with the program you write it in. In this way the levels of static quality are related to each other: Biological is built on Inorganic, Social is built on Biological, and Intellectual is built on Social, but each level is independent of the other.
Persig’s Static Quality creates a relationship among manifold patterns – from the bonding of atoms, to the mating of animals, to the formation of nations, to the dogma of religions, and the intellectual patterns of art and science. And this relatedness becomes the crux of Nicholson’s argument:
…since nature is simply inorganic and biological patterns of value that follow Dynamic Quality, it is not surprising that mathematics, a static intellectual pattern of quality that also follows Dynamic Quality, should arrive at the same conclusions. That is the reason that mathematics that is done in isolation ends up explaining nature so well—both are patterns of static quality created by following Dynamic Quality!
This configuration of Quality, Dynamic Quality and Static Quality is also used by Nicholson to describe the art/science character of mathematics:
Art is the realization of Dynamic Quality in a given medium—that is, Art is following Dynamic Quality, and the pattern of static quality which is a “work of art” is left in its wake, in whatever medium the artist chose. In this sense, mathematics, especially pure mathematics, is an art, as it is the realization of Dynamic Quality in the medium of mathematical definitions and their logical consequences.
But mathematics is also a science. It is commonly classified as such, being in the science faculty of most universities. More to the point, though, it is also generally seen as similar to empirical sciences in that it involves an objective, careful, and systematic study of an area of knowledge. It is, however, different because it verifies its knowledge using a priori rather than empirical methods. But, within the Metaphysics of Quality, its methods are totally empirical. In fact, it may be argued that from this perspective, it is even more empirical than the other sciences. Mathematics is following empirical reality (Quality) directly, whereas other sciences are one step removed from empirical reality (Quality): they follow nature, which, in turn, follows Quality. Thus mathematics is really both an art and a science and, in fact, can act as something of a bridge between the two.
The nature of Pirsig’s ‘Quality,’ and the use that Nicholson makes of it, reminded me of Leibniz again. For both Pirsig and Leibniz, our perceived reality is the consequence of structure being brought to something we cannot see, something that isn’t even material in the way we understand material. For Leibniz, this fundamental reality is the harmonious existence of monads. Leibniz’s monad is:
Something that has no parts can’t be extended, can’t have a shape, and can’t be split up. So monads are the true atoms of Nature—the elements out of which everything is made.
The text of Leibniz’s Monadology is not easy reading. It is a heavily logic-based analysis. The Internet Encyclopedia of Philosophy is one of many philosophy sites that discusses the document. There the point is clarified that:
Leibniz thus distinguishes four types of monads: humans, animals, plants, and matter. All have perceptions, in the sense that they have internal properties that “express” external relations; the first three have substantial forms, and thus appetition; the first two have memory; but only the first has reason (see Monadology §§18-19 & 29).
There is no formal correspondence between the Persig and Leibniz. But there are most certainly parallels. Leibniz’s appetitions, for example, as explained by the Stanford Encyclopedia of Philosophy are:
“tendencies from one perception to another” (Principles of Nature and Grace, sec.2 (1714)). Thus, we represent the world in our perceptions, and these representations are linked with an internal principle of activity and change (Monadology, sec.15 (1714)) which, in its expression in appetitions, urges us ever onward in the constantly changing flow of mental life. More technically explained, the principle of action, that is, the primitive force which is our essence, expresses itself in momentary derivative forces involving two aspects: on the one hand, there is a representative aspect (perception), by which that the many without are expressed within the one, the simple substance; on the other, there is a dynamical aspect, a tendency or striving towards new perceptions, which inclines us to change our representative state, to move towards new perceptions. (emphasis added)
I’ve been intrigued for some time by the view of reality Leibniz gave us and, to a large extent, because of its unmistakeable mathematical character. But I’ve also been captivated by how non-materialistic it is. Also from The Stanford Encyclopedia of Philosophy is this about Leibniz’s philosophy of mind.
In short, Leibniz stands in a special position with respect to the history of views concerning thought and its relationship to matter. He rejects the materialist position that thought and consciousness can be captured by purely mechanical principles. But he also rejects the dualist position that the universe must therefore be bifurcated into two different kinds of substance, thinking substance, and material substance. Rather, it is his view that the world consists solely of one type of substance, though there are infinitely many substances of that type. These substances are partless, unextended entities, some of which are endowed with thought and consciousness, and others of which found the phenomenality of the corporeal world. The sum of these views secures Leibniz a distinctive position in the history of the philosophy of mind.
I thought it worthwhile to bring these ideas up again in the context of Jason Scott Nicholson’s response to Wigner.