Musical scores include elements like sequences and stacks (chords) of notes, tempo, time signature, tonality, and dynamics. Yet, the performer is only faced with a draft. Every notation captures only some features of a composition, leaving a vast horizon open for interpretation. Phenotype and genotype operate similarly. The genetic code is not a set of instructions like a computer program. DNA (genotype) is interpreted and subjected to a process of ontogenesis—one that involves the environment—where the result is not just an instantiation of a genre, not merely an individual, but a singular being. Every musical performance of the same piece is unique. Every individual of the same species is also unique. No two leaves, rocks, clouds, mountains, or bodies of water are identical.
Yet, all these phenomena involve mathematics. Mathematics, understood as a “science of patterns,” as Godfrey Harold Hardy claimed, underpins them. Musical notation captures mathematical patterns of time, intervallic relationships, and pitch sequences. Genetic information constitutes a code. It can be written as a sequence of four nucleotide bases: A, C, G, T. The same applies to language: a system made of elements, rules for constructing formulas, and connectors. Linguists distinguish between the general human capacity for articulated elements that convey meaning (called language), the particular languages spoken around the world (such as Spanish, English, Hindi, or Japanese), and the unique emissions produced by specific people in specific times and places (in French: langage, langue, and parole).
Consider a simple beehive. The mathematical spirit will marvel at the hexagonal tessellation of space. The materialist will be struck by the fact that no single hexagon is perfect and that no two are identical.
We write down nature in the language of mathematics. But it is not true that “nature speaks the language of mathematics,” as Galileo famously said. It is rather as if nature relies on mathematical diagrams, and yet must interpret them in order to give life to real and concrete individuals. The signature of the real is singularity—but on the basis of some pattern. To this day, it remains undecidable whether mathematics was invented or discovered. Its astonishing applicability to nature challenges the view that it is a mere product of the human mind. Mathematics does not merely explain natural phenomena; it also makes possible the creation of machines—from cannons to computers. It even allows us to predict the existence of empirical objects, as in the case of black holes. Still, all applications of mathematics cover only a relatively small portion of its most abstract corpus. It remains a mystery how the human mind alone could penetrate the abstract interior of nature when our senses are our only channel of information.
This strange status of mathematics mirrors the scheme of score and interpretation. Scores map only certain features of music. They are outlines or, if we prefer, diagrams of sound in time and space. Storyboards, musical scores…
via Wikimedia Commons
via Wikimedia Commons
Diagrams capture individuals, relationships, local and global structures, strata, vand elocities.
Different systems of notation capture different patterns of the real. In a sense, every notation is a form of memory. It is writing. But there is another movement—an inverse arrow—flowing now from notation and diagram to effectiveness. This is called actualization or interpretation. In diagrams, we may find iconic, indexical, and symbolic elements, to use Charles S. Peirce’s terminology. These elements are involved in every process of information. The movement from the real to the diagram can also be seen as codification or virtualization. Interpretation or actualization is not the same as materialization. Matter and form are never separate. Actualization is individuation. This involves the work of expressing virtualities and potentialities in partially formed matter. It is thus a process of morphogenesis. Every piece of information is recorded in formed matter. Matter is capable of interpreting the information it carries.
This language aptly describes the work of mathematics in science. Modelling means mapping a phenomenon. It requires a system of coordinates within which a phenomenon can be described. We can model static structures or dynamic processes; the combination of both is called a dynamical system. In physics, a phase space is a diagram that allows us to represent the set of all possible states of a system. A pendulum is a dynamical system, whose possible states correspond to its possible positions. We can express this through waves in time, capturing its oscillatory movement. Since the pendulum’s movement is periodic, we can also diagram its phase space as a circle.
In more general terms, this is what mathematics enables us to do: to create spaces, diagrams, or structures that describe all possible formal objects—even those that contradict common sense and our conventional understanding of space and order. Riemann surfaces, for example, are constructions that allow for the representation of functions that “break” in standard spaces. Continuous functions are differentiable at every point; however, some complex functions exhibit singularities or disruptions in continuity.
To restore continuity, Riemann gathered all the pieces of the complex plane where the function behaved well. He then glued these patches together through local intersections of the function. This process is known as analytic continuation. The outcome is a new space: it behaves locally like the complex plane, but not globally.
DNA is a real thing—a molecule. But it is also a piece of writing, a system of notation. Nature remembers. Nature writes its history. But, as with every story, it must choose what to record. As we have already pointed out, there is not only a movement of notation or idealization. Genes must be expressed in new individuals, in new generations. This is the transit from genotype to phenotype. This expression is no longer mathematical; it is delivered by art, in the sense of performance.
For Hegel, logic is the system of notation of the absolute. This particular rose will die, like every other. Spirit begins with the death of immediate being—the corruption of flesh and the drying of blood. The concept or notion brings resurrection in the ether of spirit. After the disappearance of all particular roses, the concept of “rose” will survive and redeem them from oblivion. Concrete life continues, sheltered by ideality.
But this is only one half of the whole movement. Concepts are not safe from time and space; they are required to descend again and again to earth in order to be actualized. There is a second coming. And a third, and a fourth…
Concepts are continuously actualized and interpreted anew. The same holds for genes, books, and all other systems that carry information. The entire reign of the concept must be actualized again and again; it must descend to the levels it believed were already sublated (aufgehoben). Sensibility, for instance, gives rise to more abstract levels such as perception; perception gives rise to empirical concepts, and so on. But this is not a pure progression—there is also regression: from concepts back to perception, and from perception back to sensibility. The artist trains themselves through a manifold of experiences. These exist virtually in several maps, constituting an atlas or a sheaf of diagrams. But as a creator, sensations, ideas, and concepts are actualized in a work of art—a sculpture, a book, a painting.
Let us return to mathematics. Mathematics allows humans to map the world. Mapping means reflecting structural elements of one being into another. If mathematics captures something of nature, it is because nature has real relationships. This is evident in how we use mathematics in science: mathematics wraps around objects like a climbing plant. We perceive contours, shadows, and patterns. It is not the thing in itself, but it belongs to it.
Extending this argument further, we might say that nature not only follows but creates mathematical patterns. Nature is a working mathematician. We can easily detect mathematical patterns at all levels of natural reality. Yet, patterns do not exhaust being. It is, as we have said, as if nature uses mathematics to create diagrams that give rise to beings. But—as with DNA—diagrams and notation are only guides: they must enter the real world to give rise to concrete individuals, who are never identical to one another.
Philosophy is situated near both art and mathematics. Mathematics enables us to map the world. Art allows for the performance of maps. Philosophy sits between the two, thinking the actual and the virtual. We might call this diagrammatology.
