Algebraic Concepts Characterized / Distinguishing the General from the Generic / Metalithikum / Pre-specificity / Thinking as an Algebraic Mechanist

Characteristica Designata IV: algebra as an undiscovered continent, and attempts to appropriate it as the symbolic positivity of ‘pure instrumentality’

In his classic textbook The Development of Mathematics (1950), E.T. Bell describes how abstract, symbolical Algebra appeared like an ‘undiscovered continent’ on the horizon.  Those who pushed the application of the symbolic method without dedicated political or economical commitment were ‘adventurers’, whom Bell calls ‘illegitimate Kings’ striving for ‘profit’: masses of young mathematicians were recruited, he writes, who mistake the ‘kingdom of quantics’ for the ‘democracy of mathematics’. Quantics was the name of the branch which studied algebraic forms, before it turned into a general theory of invariances. Leading Algebraists were accused of mobilizing ‘totalitarian’ regimes of calculation, by recruiting mathematicians for theory without applications and use. If we recall the decision communicated by the French Academy of Science shortly after the French revolution, that the classical mathematical problems like squaring a circle should no longer be credited within institutional science because they consume the workforce of mathematicians for metaphysical interests, it seems understandable that the attractivity of studying algebraic forms for its own sake, was perceived as an offense or even deceit of enlightenment values. In effect, it was stigmatized a political threat.

Cayley’s numerous successes, quickly followed by those of the prolific Sylvester, unleashed one of the most ruthless campaigns of totalitarian calculation in mathematical history. […] Such misdirected foresight was not peculiar to the algebra of quantics in mathematics since 1850. In the accompanying theory of groups, for example, especially permutation groups, there was a similar panic. Once the means for raising unlimited supplies of a certain crop are available, it would seem to be an excess of caution to keep on producing it till the storehouses burst, unless, of course, the crop is to be consumed by somebody. There have been but few consumers for the calculations mentioned, and none for any but the most easily digested. Nevertheless, the campaign of calculation for the sake, apparently, of mere calculation did at least hint at undiscovered provinces in algebra, geometry, and analysis that were to retain their freshness for decades after the modern higher algebra of the 1870’s had been relegated to the dustier classics.“ (E.T.Bell, The Development of Mathematics p. 429/30)

The political struggle about the appropriation of the power, and the capacities set free by understanding the algebraic forms in symbolic, rather than geometrical (continuous) or arithmetical (discrete) terms, is no less virulent today than in those former times. The conceptual instrumentarium algebra provides for reclaiming the lands of the new continent, for cultivating its fertility and for populating and domesticating on its soil, undoubtedly sets the actuality operative which we today refer to as globalization. The peculiar kind of actuality proper to logistical networks is manifest in the generic manner of a place-holder-structure. This genericness, this place-holder structure for what will be likely to happen, is virulent without being existent – without being there, in a positive manner.

Marx had been very receptive and quick in understanding the power of this emerging new order. For him, this peculiar non-present and not-manifestly existant modality of the logistical order was clear – he named Bureaucracy the Spiritualism of the State. He writes:

“Bureaucracy is the imaginary state beside the real state, the spiritualism of the state. Hence everything has a double meaning, a real and a bureaucratic meaning, just as knowledge and also the will are something double, real, and bureaucratic. What is real is dealt with in its bureaucratic nature, in its otherworldly spiritual essence. Bureaucracy possesses the state’s essence, the spiritual essence of society, as its private property. The universal spirit of bureaucracy is the secret, the mystery sustained within bureaucracy itself by hierarchy and maintained on the outside as a closed corporation. The open spirit and sentiment of patriotism, hence, appear to bureaucracy as a betrayal of this mystery. So authority is the principle of its knowledge, and the deification of authority is its sentiments. But within bureaucracy spiritualism becomes a crass materialism, the materialism of passive obedience, of faith in authority, of the mechanism of fixedly formal activity, fixed principles, views, and traditions. For the individual bureaucrat the state’s purpose becomes his private purpose of hunting for higher positions and making a career for himself. In one respect he views actual life as something material, for the spirit of this life has its separate existence in bureaucracy. Bureaucray, therefore, must aim to make life as material as possible. In another respect, life insofar as it becomes the object of bureaucratic treatment is material for him, for his spirit is not his own, his purpose lies outside, his particular existence is the existence of the bureau. The state then only exists in various fixed bureau-spirits whose connection is subordination and passive obedience. Actual knowledge seems lacking in content, just as actual life seems dead, since this imaginary knowledge and this imaginary life pass for real.” (Karl Marx, “Hegel’s philosophy of the state” in Writings of the Young Karl Marx on Society and Philosophy, ed. by D. Easton and Kurt H. Guddat, Doubleday & Company 1967, p. 186)

This is a passage which not only expresses Marx’ lucidity in a way that could almost not be put more clearly – and which is being referred to and re-instituted by political philosophers today like Giorgio Agamben or Paolo Virno, who flirt with affirming a quasi-angelic order for an Authority of the Collective, which they name, with explicit recourse to Marx, the General Intellect (it is clear that the General Intellect in Marx is a concept from his later works called The Grundrisse, and not the same as his notion of Spiritualism cited above – but they cannot, it seems to me, be treated hygienically as unrelated and distinct topoi in Marx). An interesting PhD thesis by Alexandru Cistelecan entitled The Discrete Charm of Bureaucracy traces its role through the work of Hegel, Marx and Lacan – here.

Alain Badiou, one of the few contemporary philosopher who forcefully claims recognition for the capacities of algebra and its powers for philosophy today, goes one step further and claims a peculiar positivity for this bureaucratic governance, this non-present Being of what Counts. In his book  On Numbers Badiou dedicatedly considers the complex numbers not as ‘numbers’ but as ‘operators’.

After introducing the central role he wants to ascribe to the computational numbers better known as surreal numbers or cellular automata, invented by Conway a.o. as the building blocks for a physics of the symbolic, he elaborates in a footnote:

“One might object at this point that our Numbers do not authorize the representation either of complex numbers or of quaternions, upon which physics relies to a considerable extent.
But are complex numbers and quaternions numbers? I think it can be reasonably maintained that, from the moment we take leave of all ‘linearity’ when we abandon dimension 1, we are dealing with constructions based on Numbers rather than with Numbers per se. Basically, the innermost essence of complex numbers is geometrical, it is the ‘complex plane’ which delivers the truth of these ‘numbers’. Around the complex numbers is organized the profound link between pure algebra (the extension of fields) and the ontological scheme of space as topological concept. I am tempted to call complex numbers operators, operators whose function in thought is to articulate algebra and topology. Hence the simultaneously combinatorial (a complex number being a pair of real numbers) and geometrical character of these ‘numbers’. They are in fact numbers which do not number, but suggest schemes of representation and inscription which are already, in effect, something very close to a conceptual ‘physics’. Moreover, it seems to me unreasonable to speak of ‘numbers’ when it is not even possible, in terms of the operational field considered, to say that one ‘number’ is larger or smaller than another. In short: a field of numbers must in my view be an ordered field, which neither complex numbers nor quaternions are. Finally, I restrict the concept of Number, in so far as it is thought of as a form of being, to that which can be deployed according to the intuition of a line. This is made clear by the decisive part played in the definition of the being of Number by that fundamental ‘line of being’ constituted by the ordinals.” (Badiou, On Numbers, p. 228)

Just like Russell, Whitehead, Husserl etc, also Badiou seeks to respond to the striking implications of algebraic number theory: Measuring and counting had been subjected to conceptual definition.

But why are these implications so striking? In the late19th century, algebraists began to consider the universal – common property – in terms of that which remains invariant rather than that which might be considered constant.  In novel fields called The Calculus of Variations or Theory of Determinants, the grand philosophical theme of essentiality was suddenly considered in an inverse way: Invariance is not what is constant, invariance is what can be conserved.

The algebraists developed means and instruments for tilling it and cultivating its wealth which had been incomparably more successful than the administrative strivings could cope with – E.T.Bells account gives a vivid illustration of just how much this was perceived as a political and societal threat. Their insight that we need not state positively that of which we can say it is a property of all things – that we could instead politicize it. They learnt how invariances can be computed by applying what George Boole – who admired Leibniz as much as Kant – described as ‘inverse operation’:

“It has been remarked that beside considering the result of direct operations we can propose to determine the nature of a subject upon which the performance of a given operation shall produce a given known result. It would appear that question is in its very nature indefinite. If we obtain a solution we cannot a priori prove that it is the only one – only by considerations proper to the cases.”

The political dynamics of such inverse operations was that that which is truly substantial, from a societal point of view, namely that of which we can say that it is a property of all things, cannot be stated without declaration. It needs to be declared as ‘that which remains invariant across all possible transformations’ (possible in the sense of computability). Such an inversion entails neither negating nor disbanding the idea of positivity; the best I can say for now is that it tackles the idea of positivity subtractively from outside-in, rather than additively from inside-out. It liberates us from the necessity to communicate acts of abstraction and instead allows to conserve them and make them available.

It is clear that this is nothing less than an appropriation by science of politics! The algebraists appropriated the political idea of a general method, displayed that all statements characterize that which they state as public common wealth by declared designation. We could go as far as postulating, from observation as well as from principle, that political science works with characteristica designata.

Symbolic algebra expresses quantities as terms. These terms, we must acknowledge despite Kant’s cautions, live an amphibolic life by being rooted in different symbolical grounds. Dedekind’s publications on the algebraic kinship of numbers had offered a synthetical procedure for giving rigorous conceptual definitions of number classes and corpusses. His procedure of the Cut is still the standard method applied today. It is a generally applicable mechanism because it works entirely decoupled from any assumption regarding an absoluteness or relativity of the magnitudinal values termed by numbers.

With this I would like to come to my concluding part which relates these number theoretical issues to the institutionalization of eduction – if not to the political role of something like the spiritual ‘substance’ of societies.

What troubles Badiou as it did Russell (we will see in a moment) is radical modernization – they want a new beginning, from scratch, and they subject their philosophical thoughts entirely to their political agenda.

“This is why I have a concept of absolute beginnings (that which necessitates a theory of the empty) and of singularities of thought which are incomparable in their constitutive gestes (that which necessitates a Cantorian theory of the plurality of types of infinitude). Deleuze has always maintained that by doing that, I fell back into transcendence and into the equivocity of analogy. But if it is in fact necessary to sacrifice immanence and the univocity of Being (which I do not believe, but it is not important here), for a political revolution, for an amorous meeting, for an invention of the sciences, or for a creation of art to be thought as distinct infinities, under the condition of dividing and incomparable events, I will do it. […] If, against the ascesis of the fold it is necessary to maintain that the fidelity to an event is the militant recollection, the transition of which remains obscure, and to reduce it to its actuality as a generic multiplicity having no virtuality beneath it, I will do it. I do it.”

We can see a similiar commitment in a text Russell wrote in 1931, The Scientific Outlook (for download here), in the introduction of which he articulated the dilemma around notions of learning and literacy I wish to accentuate – I therefore insert all of the 4 pages of the introduction before citing from a later chapter in the same book on how Russell thought about the role of education in the “Scientific Society”:


The passages which I cite on the following slide are from a later chapter in the book, and I would like to clarify that my intention is not at all to polemically raise any sort of accusation – indeed, as many people hold, it might well be that Russell wrote it as a parody. While this would certainly be a great relief, the context of these passages (to which he introduced in the 5 pages I cited before) does not seem to frame liberties that we would ascribe to literary texts. I do think that the issue of the role of education is an unresolved and an urgent one, and I raise this difficult historical document to underline the vast scope of any engagement with the symbolicness of algebra.


I think every philosophical position must be related to the political views they manifest, and every philosophical position ought to be read, criticized and continued or discarded according to how they seeked to relax and resolve the urgencies of the times in which they had been articulated. In the case of Russell’s document, my point is that we tend to forget today about the sheer singularity of the situation he wished to address. Indeed, when the symbols which are operatively-technically dominant at a certain time lack a philosophical-ethical concept to embed it, that specific moment in time will suffer from insufficient sagacity and inability to think appropriately. Such a crisis marks the entire 20th century, and still our present today.

I sympathize with thinking that problems of this order, of crisis, may be ill-posed if we seek to coerce decisions. We can pose such problems in terms of continueing intellectual legacies in our philosophical positions. This not only prevents the exploit of history for the sake of authority claims which instrumentalize one position or another, it also abstracts from the level of individual intention onto an impersonal yet not a-subjective level which might be called a ‘body of thinking’.

It is with such an interest that I would like to contrast two vectors in the legacy of realist philosophy on how to place the social role of eduction, a platonic one and an aristotelian one. Yet I do not regard this marked out contrast as a dichotomy which ought to be decided in order to overcome the blockage it manifests, quite contrarily to that I would like to see in them mutually implicative strands that seem, in their motoric interplay, generative to the entire history of philosophical-political thought.

I will post separately my first attempts to this, please click here.

This post in context:

Characteristica Designata VI: emerging fields in the synthesis and analysis of data

Characteristica Designata V: legacies of philosophical realism

Characteristica Designata III: an existential ‘genitality’ proper to symbolic numericalness

Characteristica Designata II: Polynominality, and the question of structural amphiboly

Characteristica Designata I

5 thoughts on “Characteristica Designata IV: algebra as an undiscovered continent, and attempts to appropriate it as the symbolic positivity of ‘pure instrumentality’

  1. Pingback: »popularizing insistence« | monas oikos nomos

  2. Pingback: Characteristica Designata VI: emerging fields in the synthesis and analysis of data | monas oikos nomos

  3. Pingback: Characteristica Designata V: legacies of philosophical realism | monas oikos nomos

  4. Pingback: Characteristica Designata III: an existential ‘genitality’ proper to symbolic numericalness | monas oikos nomos

  5. Pingback: Characteristica Designata I | monas oikos nomos

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