John W.P. Phillips on Fractals
As a supplement to Couze Venn’s commentary on Benoit Mandelbrot, John Phillips writes about fractals and the border between mathematics and the humanities, focusing on the work of Mandelbrot, MacFarlane Burnet and Jacques Derrida, and arguing for a view of distinct disciplines that focuses more on general principles.
Image: Mandelbrot set. Created by Wolfgang Beyer with the program Ultra Fractal 3 / CC BY-SA 3.0.
The problem
It is true that much of the work of the humanities and the social sciences acknowledges debts to mathematics and physics, or at least betrays influences from the mathematical sciences, including the mathematical world of fractal geometry. But to a great extent this latter work owes much to analogical modes of application. Famously N. Katherine Hayles identifies “parallels between poststructuralist philosophy and scientific attitudes towards chaos.” Chaos, uncertainty and predictability are “deemed more fecund” than order, predictability and “arbitrary definitions of closure” (176). This supposed “inversion” of traditional priorities has often set the tone for social science and the humanities in the various treatments of their different topics. The point, though, seems not--on closer inspection--to be that of inversion, but of the complicity of determinate and indeterminate processes, privileging neither one nor the other, but in locating their shared constituting ground.
The question of what happens on or within the border between mathematical disciplines and humanities disciplines therefore still needs to be explored.
The works of certain scientists, philosophers, artists and writers share with mathematics a precise yet nonetheless independent understanding not only of stochastic processes, multi-dimensionality, formlessness and fragmentation, but also (and more to the point) of the complicity of indeterminacy and determinateness, and of randomness and order (Determinate chaos).
So a greater awareness should be gained of philosophical and mathematical confluences, and of conditions that disciplines share with their declared objects (The feedback effect).
What Mandelbrot calls the “flight from formlessness” was always more obvious in the progress of mathematics and the natural sciences than in at least certain idioms of philosophy, art and literature.
An awareness of the deterministic element of stochastic processes has been a feature of several slowly progressing yet influential disciplines, emerging more or less independently but as elements of a supposedly shared historicity.
The so called mathematical crisis between 1875 and 1925 (Mandelbrot 1977: 21), out of which most of the mathematical innovations of the last century have arisen, coincides with crises in philosophy, art and science from more or less exactly the same era. Some have been tempted to connect the many crises of the same era besetting world history. Although the connections between these critical spheres do not seem rooted in a single catalyst or an event as such, they nonetheless form the particularities of what today might perhaps be regarded as an event. But what is an event?
Separate fields: the destructive philosophies of Nietzsche, Freud, Heidegger; the revolutionary mathematics of Cantor, Peano, Frege; the stylings of Mallarmè, Woolf, Duchamp; the biological science of Ehrlich, Bordet; and so on. Is there a common ground? One possible--perhaps likely--answer to this question lies in developments roughly half a century later, still more or less independently of each other: in philosophy, mathematics/geometry, experimental poetry, and cell biology. There are other examples but in these spheres particularly claims are made that would justify careful scrutiny and comparison, not merely of a loose analogical kind.
And not merely the thematics but more to the point the technicalities of infinity, chance, randomness, permutation, and the enigma of stochastic processes are decisive in the development of the idioms in question. But so also is the problem of mediation, which is at last acknowledged as insuperable. And in every case some interest in the functioning of language as a tele-technology plays a central role. What formerly were regarded as problems and limitations to science now begin to be recognized as resources for its continued development. (Though this is still largely ignored in mainstream science--even in linguistics.)
We would probably need to consider the question of method, and the question of the relationship between method and the heritage that a writer, scientist, inventor, might wish to cultivate. The significant connection is that between the technical specifics of the idiom in question and the heritage, which the technical specifics help to transform.
Three Cases
1. Benoit Mandelbrot:
Mandelbrot’s earliest published writings address the obscurely related fields of structural linguistics and economics. Working with several precedents he reveals fallacies based on power laws in economics (concerning trading schemes and wealth) that require different interpretations than those normally (and disastrously) made by economists (Mandelbrot 1963). Even before this, he had shown how the law discovered by George Zipf (“Zipf’s Law”) governing the relation between word frequency and word rank in language use was not, as Zipf had supposed, a function of human behaviour (and “the principle of least effort”), but a function of a law that operates quite independently of human behaviour. His calculations, exemplified by the (already classical) hypothesis of a monkey with a typewriter, show that the animal hitting keys randomly will produce a language also obeying Zipf’s law (Mandelbrot 1961). Despite this such a language would not operate quite like the so-called natural languages, which can only be used meaningfully in finite and relatively orderly formations (so that only a small and repeatable minority of possible combinations at any time apply). The “monkey language” forms what in mathematics is regarded as an un-denumerable infinite set (a Cantor Set): a set of potential combinations following no specifiable order. The texts of such a language would be impossible to catalogue or organize. No dictionaries or glossaries would be conceivable. Jorge Luis Borges had already given fictional form to the idea in a lyrical and evocative short story, “The Library of Babel” from Ficciones (1956). Yet, not even the economically geared down natural languages are free of the effects of this infinite play, as can be seen with instances of homographic resemblance (words in different languages spelled in the same way, like the German Gift, meaning “poison”). These provide only one example of the ways in which the general economy of the Cantor Set infiltrates the restricted economies of communicative interaction; but the impact of this kind of infiltration can also be glimpsed with literary works that exploit them, like James Joyce’s Finnegans Wake or David Melnick’s Men in Aida (a phonetic reproduction of Homer’s Greek written entirely in existing English words). Furthermore, the distinction between a restricted and general economy had already been established, independently of mathematical disciplines, in readings of G. W. F. Hegel, notably by Georges Bataille and Jean Hyppolyte in the 1940s. The indispensable reference would be to the key distinction in Hegel’s Logic between a spurious or “bad infinity” and the “good infinity” of conceptualization.
If languages belonging to the general economy, as well as those belonging to the restricted economy, can be shown to have fractal dimensions, then clearly the fractal dimension operates as a kind of boundary object that connects otherwise distinct spheres. It is in this that the emergence of continental, especially French, philosophy (and so also literature and literary theory) reveals consonances with the contemporaneous emergence of fractal geometry. In each case the roles of formlessness, random indeterminacy and chance are increasingly subject to scrutiny and to question.
2. Macfarlane Burnet: Somatic Mutation
The leading breakthrough in the transition to molecular biology in the 1960s has also to do with the growing recognition of the role of stochastic processes in the functioning of immune systems. In 1969 Burnet had identified five levels on which stochastic processes play a determinate role in biology. On the first level errors are introduced in the process of replication and these provide in the form of somatic mutation the raw material for all biological change. A role for error therefore needs to be explored as an essential and ineradicable element of biological development. This applies at the level both of the species and of the individual. On the second level stochastic processes inform the way selection is made for some of the many available patterns of information in, for instance, sexual recombination. On the third level, environmental factors allowing proliferation of certain types is enabled at the cellular level as well as that of the individual. Fourth, viral and bacterial models reveal stochastic processes in the storage, replication and expression of biological information. And, fifth, the determinate aspects of protein structure depend on stochastic processes. At this time Burnet could claim that immunology, “with its unique blend of stochastic and determinate processes, is better fitted to exemplify the realities of biology than any other major field” (309-310). And at this stage the merging of determinate with stochastic processes in biology can be regarded as genuinely revolutionary.
In the further development of cell biology this becomes increasingly clear. Burnet’s theory of somatic mutation was established as scientific fact in 1976, by Susumu Tonegawa, in the discovery of gene fragments. An antibody is produced through the random combination of three different kinds of gene fragment (in humans more than 8000 possible combinations are possible with innumerable further insertions available at any time). To abduct the mathematical idiom, it looks like a restricted economy is produced in connection with the random production of further uncountable somatic resources. The forms of combination are approximate to, rather than merely analogous with, those that inform written and spoken languages.
Furthermore a classical distinction between nature (phusis) and ethics (habit, ethos, character, etc.) is located as already operating within the bios itself. Aristotle had distinguished the natural capabilities from those that must be learned through training (ethics). He identifies the latter as produced through repetition and the deliberate avoidance of error (initial conditions lead to major differences in outcome). The possibility of error therefore lies at the heart not only of ethics (as is well known) but also of biology.
3. Jacques Derrida: Iterability
When Jacques Derrida coined words like iterability, différance, dissemination and deconstruction in the 1960s he did so with an eye on how these words themselves instantiated the conditions they were coined to describe. Iterability is coined from what seems to be a homograph: two words in languages far apart, meaning either “other” or “different” (Sanskrit itara) or “again” or “repeat” (Latin iterum). The term in this way instantiates its designation: the ability of a mark or word to signify differently in repetition. This “ability” inscribes into a mark both a regularity and the possibility of mutation.
Derrida’s chief methodological principle, as is well known today, involves the structure of repeatable marks or traces, which in their repeatability can give rise to systems that claim a “curious tendency” as their property (language would be just one of many such systems): “they simultaneously incline towards increasing the reserves of random indetermination as well as the capacity for coding and overcoding or, in other words, for control and self regulation” (“My Chances”: 2). Because a mark can be repeated it functions both as condition for institutional stability and yet also as a potentially destabilizing force, in principle separated from the idealized guidance of a subject of address or a regulating context. In this principle the infinite can no longer be opposed to the finite.
In the article “Différance,” Derrida uses the adjective “scientific” (in quotation marks) to describe possible ways in which different modalities of economy could be put into relation. For this idea of relating he refers to existing attempts to indicate how it might operate (previous texts by Derrida) and towards a time when this practice might indeed deserve the designation science, which would amount to something like a collaborative undertaking (the coming community of deconstructive scientists).
Referring to his 1965 article on Bataille, (“De l’économie restrainte à l’économie générale: un hegelianisme sans reserve”), he writes:
I have attempted to indicate what might come as a rigorous and, in a new sense, “scientific” relating [une mise en rapport] of the “restricted economy” that takes no part in expenditure without reserve, death, opening itself to non-meaning, etc., to a general economy that takes into account the non-reserve, that keeps in reserve the non-reserve, if it can be put thus. (20/19)
The “new sense” that qualifies the word “science” is implicit in the quality of the relation, which implies combining in a single scene or drama otherwise incompatible or at least structurally distinct modalities. In the unqualified sense “science” possibly suggests a “restricted economy” of knowledge, in which a movement towards finite and thorough calculations promises both transparency and completion. To say of such an economy that it “takes no part” in effects that would otherwise disturb the movement towards transparency and completion suggests that the exclusion of such effects helps to produce the difference between these structurally distinct modalities. Later in his career, Derrida refers to the pathology of autoimmunity, which implies that systems supposedly produced for defense are built in such a way that they target the conditions of their own emergence (i.e., repeatability, somatic mutation, etc.).
On the other side, the reference to a “general economy” implies that a kind of economic operation exists that “takes into account” these effects (gift, death, sex, style, chance, non-sense, etc.). Evident throughout the history of philosophy, but particularly in the last 100 years or so, both kinds of economy operate if not necessarily as absolutely distinct then as strongly incompatible tendencies.
So the “new sense” of the adjective “scientific” ought not to be attached to one or other side of structurally distinct modalities but rather it should be regarded as describing the sphere of operations that constitute the relation, i.e., the relating (the “mise en rapport”). It is therefore difficult to think of this as anything other than a third modality.
The interest in fractals therefore would coincide with an interest in philosophy (including perhaps especially political philosophy) in the sphere of a possibly paradoxical “third modality.” I propose a more sustained and technically precise reading of distinct fields (mathematics and physics, art and poetry, biological science, philosophy) with an eye on general principles that serve not only as their conditions of development (in a constitutive sense) but also as the conditions on which their respective objects, aliments, or topics operate too.
References
Burnet, Macfarlane. Cellular Immunology Books I and II. Melbourne: Melbourne University Press, 1969.
Derrida, Jacques. “My Chances/Mes Chances: A Rendezvous with Some Epicurean Stereophonies.” Taking Chances: Derrida, Psychoanalysis, and Literature. Ed. Joseph H. Smith and William Kerrigan. Baltimore: Johns Hopkins University Press, 1984.
---. De la grammatologie. Paris: Minuit, 1967 (Of Grammatology. Trans. Gayatri Chakravorty Spivak. Baltimore: Johns Hopkins UP, 1974).
---. La carte postale de Socrate à Freud et au-delà. Paris: Flammarion, 1980.
Hayles, N. Katherine. Chaos Bound: Orderly Disorder in Contemporary Literature and Science. Ithaca: Cornell University Press, 1990.
Mandelbrot, B. The Fractal Geometry of Nature. New York: Freeman, 1977.
---. “On the Theory of Word Frequencies and on Related Markovian Models of Discourse.” In R. Jakobson, ed. Structures of Language and its Mathematical Aspects. NewYork: American Mathematical Society, 1961.
---. “Stable Paretian Income Distribution when the Apparent Exponent is Near Zero.” International Economic Review 4 (1963) 111-115.
---. “The Variation of Certain Speculative Stock Prices.” Journal of Business 36 (1963) 394-419).
---. “New Methods in Statistical Analysis.” Journal of Political Economics 71 (1963) 421-440.
John W.P. Phillips is Associate Professor in the Department of English Language and Literature at the National University of Singapore, and an editorial board member of TCS. He writes on philosophy, literature, critical theory, aesthetics, psychoanalysis, urbanism and military technology. He is co-author with Ryan Bishop of Modernist Avant-garde Aesthetics and Contemporary Military Technology: Technicities of Perception (Edinburgh University Press, 2010), and he is currently researching a project on autoimmunity in biotechnology and political philosophy.
