Umberto Eco, in his essay on the semiotics of theater performances [1], wrote an intriguing episode which beautifully gives us an allegory for our situations today vis-à-vis information technology, computers and networks. It begins with telling a tale told by Jorge Louis Borges, entitled Averroes’ Search. In Borges tale, Eco tells us, the great islamic philosopher Averroes, and a Koran teacher called Farach chat with the trader Abulcasim who has just returned from travels to places in the far East. Abulcasim tells his friends how he had witnessed an obscure scene, on a patio somewhere far away, where fifteen or twenty adult persons wore masks and acted as if they would perform real actions. They were horse riding, yet there was no horse on the patio. They acted as if they were fencing, yet without rapiers. They seemed to die without ending up dead. Abulcasims friends knew nothing about theater, they had never seen a single performance because of the Islamic ban of graphic depictions. They knew nothing of a literary genre comprehending dramatical texts. Abulcasim assured his friends that the people he saw were not insane – they »acted something out« he tried to explain, »they wanted to demonstrate something«. Yet Averroes and Farach did not understand. »Imagine«, Abulcasim tried again, »someone showing you a story rather than telling it to you«. This makes sense neither to Averroes nor to Farach. To them this seems like an incredible waste of efforts. One single narrator would be enough to pass on the tale in a lively manner, they maintain. Borges makes sure that no one might think Abulcasim’s listeners were slow in thinking or even dimwitted  – Averroes is an important islamic philosopher with an oeuvre covering areas like law, medicine, psychology and theology. Averroes work coins the intellectual landscape in Europe at least up to the 16th century. He counts as an expert on Aristotle, on his Logics and his Poetics. He works more than 30 years on translating Aristotle to the Arabic language, and it is only with his help that many of Aristotle’s writings have survived in Europe, where they had been forgotten or lost – among them, the Poetics. We are told by one of our narrators that Averroes was indeed puzzled for a long time about what the two terms  »tragedy« und »comedy« might mean, which feature so centrally in the last part of the book. We learn in our fictitious account that he eventually decided on assigning these terms to the literary genre of praise and ode. The experience of apparentness could only mean revelation to Averroes, so he assigned them to the sphere of spirituality.

If this story should actually have happened, similarly to how Borges and Eco are picturing it, the correct thinker and rationalist Averroes would indeed have been deplorably ignorant of a large part of the semiotic universe at stake. Eco comments accurately how regrettable that were, given that Averroes obviously has had available, in his comparatist approach to the spiritual, a symbolic form for actually developing a theory of demonstrative performance. While the Western culture during the middle ages did not lack an ordinary experience of the dramatical staging, what was severely missing there was a network of comparatistic scope to cast over these experiences of an oeuvre, which literally abounds into situations by enveloping the observer into the scene of the happening.

Working with data analysis, data modeling and simulation, we seem to be little better off today than what Borges (and Eco) imagine here for Averroes – who found his way through making sense of Aristotle’s Poetics in a paradigm (immediacy and revelation) that Aristotle had in fact just left behind, with his start to theorize the figurative dimensionality of speech and language.

This blog is about computability as an emerging literacy. The initial point of fascination for it can be simply put, albeit it may well feel like barely worth mentioning: computing is algebraic. What started out as a rather dry project and felt to me, at first, not particularly exciting but rather something necessary, namely to develop an idea of what we mean, and what assumptions are involved, when we say that “computing is calculating”, or “programming works with logical propositions”, or “networked infrastructures are working logistically” soon developed in a quite unexpected manner. I stumbled across indexes to explanations for many of the questions that I perceived to be insufficiently explicitly, or at least unsatifiably explicitly, thematized in those discourses where I have been educated or socialized in, media science or cultural theory, sociology, cybernetics, and science and technology studies, nor information science or political philosophy . While they al take as their object of theory the computer, IT, electronic media, or expressions of medialization for example in fashion, film, music, social codexes, or in the technical and planning sciences as logistical infrastructures, none of them is debating and considering the very consititutive and uttermost abstract level on what their theories inevitably build: namely a clear separation and identification of how the concepts of magnitude and multitude are thought to relate, measuring and counting, cardinality (what is being ordered) and ordinality (regulating framework), units and metrics, predication and proposition, mathesis and calculus, synthesis and analysis. In short, the symbolic make-up of the very forms of thought and construction they work with, in their theoretical approaches. The more I started to dig, as a non-mathematician, about the actual ‘object’ of algebra – the Unknown Quantity, as John Derbyshire puts it illustratively in his A Real and Imaginary History of Algebra (Plume, 2007) – and about its virtualization into abstraction throughout the 19th century, the more startling appeared the place where I found myself! The problematics of quantification, in a not purely technical (blind) sense, is about existence, reality, and how to conceive of truth.

Once one affirms the algebraic vector of abstraction and does not seek to resolve the problem (of quantification) by arriving at a solution, but by exploring how the formulation of the problem can be articulated in a variety of ways, and what are the strategic moves and moral stances involved in those ways, reconsidering issues so fundamental that it seems impossible to even consider doubting the establish paradigms in “how to” becomes the most exciting theme! For example considering what Quine actually suggests that today, the grandest philosophical topos, Being, was best addressable as “being the value of a variable” (literally “to be is to be the value of a variable”,  Willard Van Orman Quine, “On What there Is” (1948)); or considering what was at stake when Alan Turing and Alonzo Church came up with the definition of a specific class of Computable Numbers, namely to propose a subclass of the real numbers which guarantees the universally uniform applicability of arithmetics (Allen Turing, “On Computable Numbers, with an application to the Entscheidungsproblem” (1936)) – a proposition which clearly takes position within the quarrels considered “metaphysical” that were preoccupying the algebraists at that time (Dedekind’s “conceptualism”, Kronecker’s “algorithmicism”, Hilbert’s “formalism”, Brouwer’s and Weyl’s “intuitionism”).

Small excursion: what is problematic is the “universally uniform” applicability, because this assumes a generalized notion of structure, whereby algebraic structures are algebraic precisely because they are not general but abstract and domain-specific. Algebraic domains have the role to delineate the boundaries of what Michel Serres calls “islands of rationality”, domains where arithmetics can be applied uniformally (yet specific to the domain as which they are conceived). In consequence of this peculiar specificity, algebraic domains are not called “territories”, but “bodies” (Zahlenkörper in German, corp de nombres in French, is translated somewhat awkwardly as fields into English). This goes back to Richard Dedekind, one of the key persons in establishing algebraic number theory. Dedekind was very conscious of the implications it would have when speaking of those domains in a general way, by simply treating them as “sets”, rather then in terms of conception and corporeality. The main issue is that with sets, you ascribe the set-defining properties to the operation that ranges over the elements and makes them “alike”, whereas with the corporeal characterization of “bodies” (Körper), the defining properties are established as mutually operative indexing relations between an “abstract” ideality and a “concrete” collection of instances to be termed conceptually. In distinction to the set-approach, the corporeal approach can be called “conceptual” because it involves, necessarily, reference to an ideality level which needs to be well defined and which is more abstract than the elements it allows to “comprehend”. Many of the posts here will deal with this, but as an introductory overview I would like to recommend especially Leo Corry’s book Modern Algebra and the Rise of Mathematical Structures (1996))

But let’s come back to the problem of quantification more broadly, and to the philosophical excitement it bears as a theme to reconsider, once we affirm the algebraic vector of abstraction and explore how the formulation of a problem can be articulated in a variety of ways, rather than seeking to judge, immediately, the suggested solutions. It also allows to develop in interpretation of Wittgenstein’s seemingly disrespectful distance towards Russell’s own theory of logical atomism, because Russell did not separate, as was important (and troubling!) to Wittgenstein, equations from functions but rather aimed at conflating this very distinction. Wittgenstein’s distance ceases to appear like a manifestation of disrespect, and instead becomes understandable, through the implications both of their stances on the problem of quantification entail. For Wittgenstein, language,and mathematics, are operative tools for a practice, he does not conceive of them a representational framework for a defined metrics that can be purified in order to orientate thought in non-ambiguous ways, once and for all. Russell clearly took side with the representational stance. Different to the view maintained by Russell, logic has a transcendental status in the view of Wittgenstein, it has ultimately (i.e. on Russell’s “foundational” level) the status of tautologies, instantiated variably in different “life forms”. Certainly, this makes things irresolvably complex, with no positivist theory about language, and the quantification problem, in sight. It may seem likely to assume that Russell, aware of Wittgenstein’s problems (In the view of many, Wittgenstein got somewhat stuck regarding his theory of knowledge, in what appears to many as a quasi-mystic position), considered well and chose to abstain and withdraw from explicating his own views on a social theory. Yet far from it, while Wittgenstein grappled with his difficult concept of life-forms, Russell spelled out his views in positivist terms, and in a quite straight forward manner, in his book entitled The Scientific Outlook (1931). Already from a rough glance at both these perspectives it becomes clear that the latter aimed at a social theory for which no matters of personality, and hence of individual capacities and abilities, ought to play a role – Russell’s notion of knowledge is strictly impersonal. Wittgenstein, on the other hand, never affirmed to separate knowledge from the process of learning, and he saw no way how the process of learning could, in any exhaustive way, be generalized. In short, while Wittgenstein seems to have been interested in a non-transitive notion of learning, Russell considered learning only insofar as it can be treated transitively, as the learning of particular skills or the acquisition of information, while issues of personality and authority ought to be treated in strictly aristocratic terms. The Scientific Society, as he viewed it in his social theory, is a dystopia very close to George Orwell’s vision in Animal Farm – but unlike the latter’s, his is not an allegorical novella, a satire, but a positive theory opening up “The Scientific Outlook” for future societies (it is worth to get familiar with this side of Russell’s views by reading the original text (available for free download).

The issue of learning in relation to knowing is, interestingly, a line which makes Richard Dedekind and Ludwig Wittgenstein “allies”, and also relates them to Alfred North Whitehead. For a similar line of conflict spans also between Russell and his own teacher and co-author of his main work, the Principia Mathematica (1910). Before they started their joint project, Whitehead had accomplished, in his Universal Algebra (1891), one of the first systematic treaties on the latest advances in symbolic algebra, both in relation to number theory as well as to geometry; at roughly the same time, Russell had formulated a treaties on how to purify the legacy of geometry from the algebraic complexity introduced through many of the approaches affirmed and systematized by Whitehead (especially Grassmann’s Extension Theory (1862)  and Riemann’s theory of manifolds in “On the hypotheses which lie at the foundation of geometry” (1968)). The two interests, those of Russell and those of Whitehead, could hence not be more at odds with one another: while Russell seeks to purify inherited perspectives (the ultimately cosmologically governed belief in Euclidean Geometry as the basis for ordering knowledge), Whitehead is interested in understanding the implications of these same developments by putting inherited perspectives radically into question (a theme which he obviously picked up in his later work, probably his main contribution to philosophy, namely Process and Reality: An Essay in Cosmology (1927)).

These stories here, involving such major philosophical positions in such a short form, aim at nothing more than making the reader sensitive to the odd situation that throughout the late 19th and early 20th century literature on algebra, there are statements all around which involve the monstrous, the diabolic, the allegorical devil and angel even (Weyl’s famous statement that “In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain”, in Weyl: “Invariants”, Duke Mathematical Journal 5 (3): 489–502, 1939). What is at stake with the philosophical problem of quantization, obviously joins mathematics with morals.

It is the interest of this blog to attend anew to the involved issues around the problem of quantification, and to relate them to the pervasive and ubiquitous role which information technology plays for our contemporary world. Today, it seems to me, it should be possible to do so from a somewhat more relaxed position than when the categorical upheavals were still fresh and tremendously unsettling. Today, we are largely familiar with electricity (a key achievement of the mathematics at stake) – even if there still is no coherent theory as to what exactly we ought to take electricity for (we account for it by referring to the wave-particle dualism, and treat the implied “contradictions” as complementary pairs of “quasi”-elements – which means, in a less sophisticated formulation, that we reduce electricity to some sort of campfire), we trust it enough to build our most basic urban, logistic, infrastructures in its terms. Similarly with information – the way we treat it today is strictly technical, i.e. quantitative and formalist; yet, as a concept in its own right among the categories of science, information is not well understood yet – Norbert Wiener’s unsettling statement that information was reducible neither to energy nor to matter (“information is information not matter or energy”, Norbert Wiener in Cybernetics, 1958), still lacks an interpretation today that would be meaningful with respect to our theme – namely the philosophical problem of quantification. Information, strictly speaking, must be said to be non-existent; and yet it obviously has measurable and producible impacts and effects. Michel Serres goes as far as seeing in the the capacity to store, expand, emit, and receive information the common denominator of all things existent – which means nothing more and nothing less, in terms of the roles ascribed to certain concepts, that information takes over the role that used to be ascribed to materiality. The interesting and unusual move of Serres, in this lecture entitled Les Nouvelles Technologies: Révolution Culturelle et Cognitive (2007) is that he does no longer seek to find a positive determination of an “entity” or “referent” labelled “information” – he inverts the perspective and looks differently at the world within which information features so prominently and irritatingly today. I translated the quote here to give a better sense of Serres own radicality with this stance:

 “I do not know any living being, cell, tissue, organ, individual, or perhaps even species, of which we cannot say that they store information, that they expand information, that they emit it and they receive information. […] I know of no object in the world, atom, crystal, mountain, planet, star, galaxy, of which one could not say again that it stores information, it deals with information, they emit and they receive information. So there’s this quadruple characteristic in common between all the objects of the world, living or inert. So in our ‘hard’ sciences, where previously one only spoke of forces and energy, there has, as of late, been talk of what we generally call ‘soft.’ Hard sciences are engaged also with the ‘soft.’”

In order to help the reader build his expectations about where I am speaking from, with my interest here in this blog, I feel it is adequate to name some of my anchor points explicitly.

• First: my interest in the problem of quantification/quantization in terms of “quantitability” has been raised primarily in the writings of Gilles Deleuze (especially Difference and Repetition, 1968) and in a second step those of Jules Vuillemin (La Philosophie de l’ Algèbre (1962), Leçons sur la première philosophie de Russell (1968), Nécessité ou contingence. L’aporie de Diodore et les systèmes philosophiques (1984), What are Philosophical Systems? (1986), Mathématiques pythagoriciennes et platoniciennes (2001)).
• Second: my interest in the problem of quantification/quantization in terms of “quantitability” aims at understanding better the role of abstraction in learning (in a sense neither strictly generalizable nor psychologizable, but rather close to what C.S.Peirce was aiming at with his notion of abduction). The “heroes” in this line of thought (as far as I have encountered them yet) spans from Joseph-Louis Lagrange, Leonhard Euler, Arthur Cayley, Augustus de Morgan, George Boole, Charles Sanders Peirce, Peter Gustav Lejeune Dirichlet, Richard Dedekind to Emmy Neother, Saunders Mc Lane and William Lawvere. In a somewhat different manner, the economic approach to changes and transformations of language by André Martinet, and especially Louis Hielmslev’s glossematics was equally enlightening, as well as in a yet again somewhat different (yet not necessarily less important) manner, Gertrude Stein and her poetry.
• Third: my interest in the problem of quantification/quantization in terms of “quantitability” is driven by my awareness of contemporary digital tools available for data analysis, modeling, processing. Working with conceptual tools like Eigenvalues, Vectorspaces, Self-Organizing Maps, Principal Component Analysis is performed by code (not numbers or magnitudes) and it likely involves probabilization and parametrization. The latter are themes which directly rise out of the open, troubling, and seemingly unresolved issues around metrical vs symbolical notions of spatiality (and temporality, insofar as time is modeled in discrete, sequential ways, or space-time if mapped in continuous ways) initiated and made available by abstract algebra. It seems that there is great lack of clarity regarding concepts like proportionality, common means, structure, dimensionality, and systematicity in a philosophical sense (not in a mathematical sense).