Algebraic Concepts Characterized / Lectures / Pre-specificity / Thinking as an Algebraic Mechanist

The Soul of the Universe, Fabulous Discretion, and Dis-cyphering the Being of Chance

* this is the manuscript to the paper “Serious stories around the fantastic dream of a mathesis universalis and the inception of Universal Algebra in the late 19th century” delivered at the The Common Denominator Conference, Universität Leipzig, March 21st 2014.

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The title of my paper announces “serious stories” around the inception of universal algebra, so let me perhaps begin by elaborating on that strange term. For are we not either dealing with stories, or seriously with issues that are real?

– when I prepared the abstract for this talk and imagined what I would be speaking about, what I had in mind was a compositional body of anecdotal stories around the major protagonists in the late 19th century scene of mathematics. There was something happening at that time to how we might think about number and calculation which I am trying to understand for quite some time now, because it seems to lurk behind any understanding of electricity, computers, code, and programming languages. In essence, what is at stake in these developments regards the question of what we might consider to be the transcendent referent of mathematical “necessities”. Herman Weyl has famously expressed the sentiments that surround these developments: with the symbolic numbers, he held, “the angel of topology and the devil of abstract algebra are caught up in a fight about the soul of mathematics”.

Thus, I was thinking of giving a few accounts of how this drama actually did take place – I was thinking of speaking about how it might have come that what we today call “Boolean Algebra” has very little to do with the Algebra George Boole envisioned in his Laws of Thought [1], Or I was thinking of speaking about Richard Dedekind’s   brilliant procedure of how to give categorial conceptions of numbers by instrumenting and orchestrating pairs of them such that they may create bodies of local rationality within the continuous ocean of real numerosity,  and how violently it was corrupted   in its genuinely acrobatic character when later generations have rather shamelessly linked up Dedekind’s Cut with the axioms (Peano) of how to establish a hierarchical class-order of numbers within the transfinite cosmos of ordinality and cardinality. Or I was thinking of JamesJoseph Sylvester, who is together with Arthur Cayley perhaps the eminent figure in the theory of invariants, and who wrote a rather idiosyncratic book entitled The Laws of Verse, Principles of Versification – a book which I profoundly misunderstood   while studying long ago for my degree in literature  as a formalistic and somewhat soul-less attempt to disenchant even the art of poetry. Sylvesters’ emphasis in The Laws of Verse on that which never changes (just as Boole’s in Laws of Thought)  stand odd in the landscape of the time marked by the Romantic desire to dissolve all form into one collective unity from which an infinite amount of apparent Gestalten can be yielded – by the strictly balanced transformations of purely generic properties. I was also thinking of Robert Musil who expresses these Romantic sentiments perhaps most drastically in his epos, the Man without Quality, which he begins with, quote “A sort of introduction from which interestingly nothing follows.” He continues by giving an account
of the protagonist’s world as a logistic and strictly balanced place: “Over the Atlantic, there was a barometric minimum; it wandered eastward, towards a maximum over Russia, not yet showing any tendency of bypassing it to the north. The isotherms and isotheres were doing their job. Air temperature was in orderly proportion to the mean…”.

In short, the seriousness of the event that marks the birth of universal algebra regards not algebra and its formulaic reasoning about unknown quantities itself, but the attribute “universal”[2] – can its computations count as real despite their symbolicness? or as ideal due to it?  What I am interested in   in the following is not so much the disputes around the truth status of computations, but the role which the nature of numbers plays for learning. Thus the remainder of this paper will be about two serious stories   that relate the nature of numbers to learning: Plato’s Timeaus and Quentin Meillassoux’ Le Nombre et la sirène : Un déchiffrage du Coup de dès de Mallarmé, a book on Stéphane Mallarmé’s The Throw of the Dice.

So let us turn to Platos Timeaus, the first text, arguably, which introduces a concept of the universal together with a procedure of how to make sense of the world in a general manner. It is the first account which does not tell the genesis of the world in purely mythical terms, that is by generations of Gods, but that aspires to tell this tale in a manner that allows for decoupling a method of understanding from the binding obligations to remain faithful to the traditions of linearly branching progeny. In all brevity, into the miracle of the genesis of the world   Plato has introduced a governing principle    with which anyone can identify   if she learns to understand it. The cosmological account of the Timeaus is one which desires to empower all people who care to deal with it. There are no heroes anymore in this space governed by the logics of the cosmos’ fabrication, instead now there are masters. People who are sophisticated in the craft of ordering things, and this sophistication can be shared through teaching and learning, it is not restricted to the exclusiveness of being gifted by talents. Thus, let us see how, in the Timaeus, conceptual symmetries are established that allow for the exchange of craftmanship, artisanry, and technical skill according to such a universal method.

The Demiurg begins by creating the Soul of the Universe as the being of wholeness which he intends to distribute equally between Cosmos and Earth. For this, we are told, the Demiurg takes „the three elements of the same, the other, and the essence“ (the citations are from Benjamin Jowett’s translation). He mingles them in a cup, and out of it he creates numericalness. This, he sets out to partition into measured blocks of pure ratios. Plato tells us: „When he had mingled them [the same and the other, VB] with the essence and out of three made one, he again divided this whole into as many portions as was fitting, each portion being a compound of the same, the other, and the essence.“ Thus, the Demiurg proceeds in his partitioning of wholeness as follows: „First of all, he took away one part of the whole (1), and then he separated a second part which was double the first (2), and then he took away a third part which was half as much again as the second and three times as much as the first (3), and then he took a fourth part which was twice as much as the second (4), and a fifth part which was three times the third (9), and a sixth part which was eight times the first (8), and a seventh part which was twenty- seven times the first
(27).“

This account on how the Demiurg composes the blocks of ratios sounds rather mysterious, but it’s logics has a lot to do with Pythagoras so-called Circle of Fifths. In this, Pythagoras famously created his notion of numbers out of sounding harmony – he was capable of representing visually the lofty body of a sound produced by one vibrating cord in such a manner that it became systematically measurable into the relationships among the 12 tones of the chromatic scale, their corresponding key signatures, and the associated major and minor keys. I know too little about this, unfortunately, for explaining how Plato’s adaptation works in detail. But Plato has, it seems, the Demiurg create the Soul of the Universe (1) by considering the conflicting natures of the same, the other, and essence according to musical harmony, and (2) by engendering it in the character of blocks of ratio cut out of being of wholeness. Plato is strictly Pythagorean, by taking from him the idea of the circle in vibrating quickness – rather than in its geometric and timeless comprehensiveness.

Ok, lets see now how the Cosmos is being created after this Universal Soul has been engendered through fabulous discretion. In a manner strikingly similar to Richard Dedekind’s procedure of the cut, according to which Dedekind grew capable of giving exact conceptual definitions of numbers, Plato’s Demiurg takes all the numbers thus produced, he takes all of this partitioned and formed wholeness, which is now (after the preparations discussed) made up entirely of blocks of ratios, and he “took and stretched it and cut it into two”, and “crossed, and bent
[it] such that the ends meet with ends” (Timeaus xx). Like this, two intertwined circles are created, an inner and an outer, and we are told that they revolve around the same axis. The motion of the outer circle is called the Same, and the motion of the inner circle that of the Diverse. To the outer circle belong the intelligible forms, and to the inner belong the sensible and corporeal bodies. But what both of these circles are made out of, namely the soul of the universe, is dispersed throughout all of it. This is the precondition for Plato’s Atomism to work: the triangle, this geometric atom which is the operable representant of a similarity notion that can claim to be universal, can only mediate between the sphere of the intelligible and that of the sensible because the principle that animates both is distributed discreteness, universal, and a visual representation – a snapshot – of an articulatory trembling that stems from the restlessly circular swinging of a primordial chord (between same, other, and essense). What links the diversity of how the elements of fire, water, earth, and air, appear and decay in the sensible world   to the solids in perfect form and regularity in the cosmic sphere of the intelligible  is blocks of universal wholeness, numerosity cut apart into ratios each of which encapsulates a split unit. The universal does not exist, neither ideally nor in pure elementarity. It insists in all things being, in the distributed manner of the disparse and heterogenous – or shall we rather say, by foregrounding its operational character: the disparse and the genital? In any case, it is the soul of the cosmos distributed in blocks of ratios, units that embody a split, which allows for the figure of the Master to enter the stage of methodical thought and science. And the master is master for no other reasons than because she has learnt to be one.[4] Her authority is decoupled from a logics of progeny and God-given priviledge, and from now on entwined with artistry and craft that is pursued methodically.

Let me dramatize a bit tendentiously why this appears so very interesting to me, with regard to the EVENT that is, according to Husserl and many others, supposed to mark the CRISIS OF SCIENCE at large: universal algebra and its logistical principles of ordering. In the Timeaus, quite obviously, it is discreetness (and its nature)  that grant the possibility of learning within the openness of the Universe. Forms may well grant the possibility of recognition, and hence of knowledge, – but the split units, the blocks of ratios, play a role more profound in that they provide the possibility for our individual souls to remember what had been intuitive to them before they had been born. There is, it seems to me, an entirely different approach to the notion of intuition which, supposedly and according to many, has been dethroned (if not degenerated) and is being misappropriated on the basis of purely algebraic renderings. The accusation being that algebraic symbols are amphibolic, which means according to Kant, that they “confound an object of pure understanding with appearance”. Yet precisely this is the very character of Plato’s geometric atomism! It wouldn’t work, if it were otherwise! Kant was well aware that amphiboly in the concepts of reflection introduces a peculiar kind of activity that he could not accommodate in his system of critique: Transcendental Deliberation[5] turns into the necessary intermediate between the two spheres of correlation in Kant’s theory.

So let us turn to our second protagonist, Quentin Meillassoux. His reading of Mallarmé’s poem The Throw of the Dice as well presents us a particular understanding of learning that is enabled through the distributive nature of numbers. I must perhaps point out right away that Meillassoux himself does not see in what he describes an understanding of learning. Rather he wants to see in the poem the liturgical text of a secular religion, as he calls it. My own reading will not follow him in this direction.

All right, to the poem then. Its main protagonist is, significantly, the generic persona of the Master. We encounter him on a boat in the midst of a stormy and wild sea, holding dice in his fist and pointing his hand into the air. The poem never resolves what the Master actually does or intends to do with the dice, whether he wants to throw them in order to learn about his near destiny, whether he believes that he can intervene in the “fulfillment” of what appears to be his “predicament.” Are the dice a sign of the Master’s despondence, his impotence to continue being what he is, a master, vis-à-vis the powers of cosmic chance that science has just began to affirm in the stochastic methods introduced by Laplace and others? Does the calculation with probability mark the ultimate end to any form of mastership, and instead enforce a more humble stance for man in a cosmos whose nature is determined indirectly, on the level of a second derivative, as a paradoxical determination of being undetermined?

Most of the interpretations somehow unfold along these lines.[6] Yet the brilliance of Quentin Meillassoux’s reading lies in presenting nothing less than an understanding of Mastership in an entirely original way, which relies neither on annihilating chance nor on desiring to control it – but in engaging with the Being of Chance. His claim is to see in Mallarmé a true symbolist master, because he sees him as having engendered his own numerical corpus—i.e., a symbolic nature of numbers, from “placing” in the manner of a distribution (hidden in the seemingly arbitrary meter of the poem) the one number that cannot be another: 707.[7] The entire analysis of Meillassoux revolves around determining the “identity” of this number—as the Being of Chance that consists in making itself infinite.[8]

With this, we have a true affirmation of the idea that numbers allow for learning. If they allow for learning, then it must be their nature to allow for being engendered in their kind – and with this, we step out of the mimesis tradition where understanding meant only recognizing, and where no creativity is to be involved with regard to what we can learn to understand. On the other hand, if we can engender the nature of number in its kind, then we can engender acts of thinking within thought.[9] And this, indeed, is what Meillassoux admires in Mallarmé. For his poem does not only make one think, in all vagueness and subjectivity such a statement ordinarily implies. Rather, I would like to postulate after reading Meillassoux’ acrobatic performance in decyphering Mallarmés nature of number which can be extracted and formalized, and thus generalized, communicated, taught and learnt – a method about how to counter what I would call the improbability of learning what cannot possibly be anticipated, as against recognizing everywhere what we expect.

Meillassoux’ reading departs from gathering a mixture involving all that he knows of and that might be relevant to  making sense of the poem: the semantics, the conventions of harmonic and graphical meter, the broader historical-political-cultural context as well as the history of the legacy Mallarme continues (poetry), and all hermeneutic aspects one can think of. We don’t have a natural tendency towards learning, quite the opposite, he too seems to think. Our natural tendency is to recognize what we expect, thus we must trick our own nature and dissolve our expectations by method such that we can allow for the unlikeliness of learning to happen: and he creates one particular formal inventory out of all the distinctions he can think of.

So how then does Meillassoux find this number, or rather, find that his number holds the clue, from within such a “confused” and “oversaturated” situation?

He takes the noninitiate reader through a fabulous journey to how he ends up with the number 707, which—in the finale of this speculative trip through possible codes—turns out to be, and I am sorry for the prosaicness in putting it this way, the chance variable we know from ordinary statistics, the sum of all the counted words. The number-that-cannot-be-another facilitates to carry out probabilistic analysis on Mallarmé’s text. Even in statistics, a random variable is not a variable strictly speaking, for it has no fixed value. In other words, it is not a magnitude of which we could ask metrical questions like how much? What it does is label a number that counts a magnitude that is unknown. As such, a chance number (I would prefer to call it an “indexical magnitude”) can incorporate a possibility space, and allow to experiment with it in probabilistic terms, by partitioning it into a set of events that can be combined in their interplay.

Meillassoux’ procedure might best be called “hypothetico-inductive.” He experiments with adjoining (encoded metaphorical, nonmathematical) “domains of rationality” for his hypotheses, and thus creates a hypothetical modeling space for studying distributions, patterns, and regularities that might fit into a certain metrical scheme. It manifests a brilliant example of how the “triangles” – Platos geometric atoms which allow for measuring – must be symbolically contracted: it must be opened up to articulation, and the scope of imaginativeness provides the alphabets of coding according to which a work, comprehended as a cypher, may be dis-cyphered (rather than deciphered). Because the “key” does not decipher something that would have been there before the key and the particular reading it opens up has been engendered –

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for a more detailed version of this reading of Meillassoux’ book cf my article Articulating a thing entirely in its own terms, or what can we understand by the notion of engendering?


[1] cf. the article by Theodore Hailperin, “Boole’s Algebra isn’t Boolean Algebra, a description, using modern Algebra, of what Boole really did create” (online at Jstor)

[2] rather than simply “symbolic” or “abstract”

[3] Verrätselung

[4] (and, we must add with regard to Plato, because he happens to be speaking and reading Greek, male and a free citizen of Athens, i.e. not a woman or a slave). With regard to the change I want to stress out, between cosmogony and cosmology, these restrictions are not of principle relevance.

[6] The “death of the author,” which was proclaimed by Roland Barthes, Maurice Blanchot, and Jacques Derrida, among others, was decidedly rooted in particular readings of Mallarmé’s great character of our poem, the Master.

[7] The whole argument is summarized in the chapter entitled “Sommes” (Summations) in Meillassoux, Le nombre et la sirène, 47ff.

[8] Significantly, in the subtitle of the German translation of Meillassoux’s book, déchiffrage is translated as Verrätselung, not as Entzifferung, as with the English translation (decipherment). In English, Verrätselung could perhaps best be expressed as “dis-ciphering.”  It strikingly makes Meillassoux’s point explicit: that Mallarmé’s oeuvre seeks to dissolve, rather than to represent or even resolve, the nature of the divine.

[9]This is the formula not only of Deleuze, but also of other Algebraists like Boole, Dedekind, Peirce, Hjelmslev:  “Penser c’est créer. Il n’y a pas d’autre creation. Mais créer, c’est d’abors engendrer “penser” dans la pensée.”

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