“All algebraic inquiries, sooner or later, end at the Capitol of modern algebra over whose shining portal is inscribed the Theory of Invariants.”

This writes Arthur Cayley in a letter, around 1850, to his friend James Joseph Sylvester [1]. It strangely resonates, as a statement, with another very famous saying, namely every road leads to Rome. And isn’t this, indeed, what so many people are frightened of with regard to algebra – its “imperialistic” strivings?

Yet what algebraic inquiries lead to, as envisioned by Cayley, is not the Capital of an Empire, but the Capitol of Algebra, and as such, of Modern Science. The term capitol as well can be traced back to Roman antiquity, where it was the name of the smallest of the seven hills surrounding the ancient city. The capitol was the domicile of the Roman sainthood, allegorically considered as the home of Jupiter, the God of the Sky and father of all the gods. To any roman, before christianization, Jupiter was a divine witness to oaths, those political contracts about future “contingencies”, and the sacred trust on which justice and good government depended in the Republic.

So Cayley, in his parable, pictures modern science as a republic and algebra as the republics capitol. Does it express then, straightforwardly, an aspiration for Invariant Theory to make claims as a theory of the forms of the intelligible? Indeed, the options to determine what the algebraists at that time meant by Invariance are highly abstract (to say the least), invariants are that which cannot be determined as a value governed by constants and variables. Invariants have neither location (in time or space) nor physical manifestation. Today, they mainly mean the abstract quantity conserved by Noether’s Conservation Laws. A handy wording puts it as a quantity of energy that remains invariant throughout a transformation process. This may well describe invariances from the point of view of their observable effects, yet it doesn’t help to dis-obscure the peculiar character of such an abstract symbolic quantity. (cf. for a levelheaded and matter-of-factly introduction to the topic the article by E.T.Bell on Invariances [.pdf] in his The Development of Mathematics (1945), and for a more recent discussion about the relation between Invariances and Emmy Noether’s conservation laws: Nina Bayers, “E. Noether’s Discovery of the Deep Connection Between Symmetries and Conservation Laws” (1996) [.pdf]).

Perhaps, instead of celebrating the orchestration of alarm bells ringing “I knew it, algebra brings theosophy (metaphysics, theology, whatever you like) back into science!” it might be worthwhile to take a breath first, and get rid of any agitational excitement. We want to consider the problem, after all.

There are now many interesting directions to go. The one I would like to consider  is Cayleys characterization, in his parable, of Science as a Republic. (In order to demarcate when I use a concept as a dramatized mask rather than as a “plain” concept, I will capitalize their spelling). A republic, literally from L. res publica for public things, is foremost a political model countering that of monarchic forms of governance, literally from Gk. monos for one, and archein for beginning. Hanna Arendt considered very insightfully, throughout her Oeuvre, that to begin also means to lead (uncomfortably perhaps, but irreducibly so). A monarch is one who begins and leads, classically such a figure was understood as the son of a god. The republican model of governance now puts at its basis not the immediacy of a One which enters the scene, appears, is born, or whatever, but characterizations of this Oneness. The Republicans have installed these determined characterizations into a System of Representatives. The republican model thus started to rationalize the autarchic concentration of power proper to model of monarchy for governance. They distributed this characterized and rationalized Oneness into many locations with addresses in space and time. Republican Grid Power presumes and generalizes Monarchic Beginnings as having been set. It concentrates its attention on settling constellations for what has already begun. It develops theory around the administration and cultivation of a legacy. In contrast, the monarchic Central Power knew theory as well – but in a much more immediate role. It meant for them a means to establish ties. Before the inception of philosophy in Ancient Greece, Theoreticians were those people who brought the rites (from L. ritus, for religious observance or ceremony, custom, usage), and culture (from L. cultus for care, labor, cultivation, culture, worship, reverence) from the city center to the rural areas. Theoreticians were like a cultural clock, they spread around the year from the center to he periphery, to orchestrate and synchronize the members of the governed people. Republican theoreticians were much less involved and more formal. They were ambassadors within the System of Representatives. Their role was not to establish economical conditions for a people to be governed as a people, their role had changed towards establishing political conditions for governing varieties of diverse peoples, with diverse rites, cults, customs.

So what does that help us for considering Cayley’s statement, or more adequately, for considering the strange status ascribed to abstract algebra in it? Cayley clearly ascribes to Algebra, within the Republic of Science, the status of what people in antiquity liked to call a science of divination, or forecasting. And he ascribes this status in a not entirely positivizable sense by placing the Theory of Invariances as the Capitol within the City of his Republic called Algebra. But this does not automatically turn algebra into theosophy, and certainly not in one of these much feared versions of scholastic or baroquish-absolutist forms of theology. Finding ourselves in a Roman like environment, in Cayleys parable, indeed would not suggest to understand Theory as being in the service of a God, but as being in the service of the worldly affairs proper to a Res Publica.

Theory in the Republic, we have argued, is political because it takes the economical practices as its object under consideration within a System of Representatives, not as its form of expression like the enacted synchronization achieved by Monarchistic Theoreticians in order to establish (not to consider and rationalize, like the Romans) the conditions for economic governance. Now in Cayleys parabola, it seems likely to assume, the System of Representatives is Science with its Disciplines and Faculties. The communities represented by them would be the peoples, with different “cults”, “rites”, and “customs” to be integrated by the Ambassadors, by the Theoreticians acting in a mediating role between the Compartments. And is not the Community of a Discipline today defined and identified according to norms and standards of instruction commonly considered legitimate and trusted ?

Yet there is an obstacle to the smoothness of this analogically constructed scenario. Theory, so the common sense throughout Academia today, is expected to exhaustively understand itself through its being instrumental to concrete practices of the Disciplines. There is, even within philosophy itself, a pervasive mistrust against theory which claims to be in any sense more abstract, and more generally applicable beyond a very restricted and specific set of heuristically familiarized cases, administrated by the Disciplinary Communities. Isn’t this mistrust an expression of refusal to acknowledge Theoreticians as Ambassadors within the Res Publica of Science? As if they wanted to impose an orchestration, yet one that is perceived, by the Disciplines, as somehow illegitimately claimed?

Different from the rural members of Persian, Egyptian or Greek communities, the members of disciplinary communities unite under the democratic understanding that is possible for communities already living within urbanized areas. Having abolished their city-rurality stratification and along with that, their center-periphery dynamics, the Peoples of Disciplines consider their “Customs” as Methods rather than Rites.

Yet wouldn’t this mean, effectively, that they are, as Scientific Domains, underway to define themselves in relation to an Absolute instead of a Republic? Are they not, thereby, transforming into Monarchically Reigned Domains, nourished and supplied by a Republic (of science at large), yet increasingly unaware about this? Regarding “Customs” as “Methods” – despite the absence of Ambassador Theoreticians which would stabilize Correspondance between the Disciplines – means regarding Systems of Representatives as Nature. The Dream of Urbanized Domains, with their striving for a Democratically Monarchic Governance, is that of a Politics which does not depend upon Cities and Citizenship.

Yet how could some sort of paradoxical constitution in terms of democratic monarchies within the Common Wealth of Nature be political, if they don’t consider themselves as States, Actors in a Political Game whose relation to an Absolute (instead of a Res Publica) would leave, for the Disciplinary Domains, nothing but hegemonic competition? Wouldn’t an understanding of Democratic Monarchies, decoupled from their embedding within a Republic, necessarily abolish the dimension of politics through Corruption? After all, next to Economical Governance which maintains and organizes Wealth and Supply for a People, Political Governance consists in establishing and maintaining the conditions for free action and free speech of Individuals within Societies.

Within traditional monarchies, the monas at power is an Individual (a Son of a God, or a Heroic Man). How could that possibly be reconceived in terms of a Democratic Form of Governance? In other words, how can there be Heroicness without the Singularity of a New Beginning?

By considering the Individual as the Generic. Within such an Archein of a Democratized Monas, within a Generic Beginning and Leading of the Collective, there can be no internally political individuality anymore. The democratic character of such a Monas would depend upon the reification of the generic monas into a Collectively-Singular Individual, made up as A People with Economic Power to Domesticate the Outside of its domain, but with no political freedom on the level of the members of this People – in our Analogy the members of disciplinary communities. All any “one” member can ever be, in our allegorical figure of a Democratic Monarchy, is A Particle,  a “One” that is fully determined from the Domain it happens to participate in. A Particle is a One without a proper Personality, this would be another way to express the same. Now, considering itself democratic, such a Generically Monarchistic Governance would claim “freedom” for its particular members. If we conceive of freedom as the contrary to being deprived of ones potential, it means, if applied to the Members of the Monas as Particular Individuals, a pathological reduction of Personal Integrity to the granted Right to fulfill ones duties. To live one’s Individuality, to develop one’s Personality and Integrity – what Political Governance aims at securing – would mean, within such a Democratic Monarchy raised to oppose against the Republic of Science by claiming to Go Back To The Roots, to Turn Exclusively Towards Nature, comes to grant and sanction the Pride of Civility in proportion to the degree of Devotion Maintained for the Urban and the Generic.

The image that comes to mind, for such a form of social organization, is not Gilgamesch’s City Wall, but the Erection of the Tower of Babylon. What if it collapsed, after all, not because the people building it were speaking different languages and living different customs, but because they were driven by a vision of uniting, as a People, through building an Emblem of Power in the name of an Absolute?

I don’t mean to drive this reading, inoffensively inspired by Cayley’s figurative remarks (which were probably much more casually meant than how I have made it appear here), much further. But in many regards, Modernity has meant to instigate such a Democratically Monarchistic beginning, the beginning of something which will lead collectively, what it begins. The scenery articulated in Cayley’s parable appears like a calm and generous offer to me, because it makes a suggestion of how to involve an Outside to decouple the unhallowed complicity between the Democratic with the Absolute which theories of Socialism are trying to penetrate for many decades now.

Cayley’s idea articulated in this parable was the vision of a Scipublic (science as a republic). He saw in algebraic inquiries not a tool for imperial expansion, but a formal authority for the foresight needed whenever political governance is at stake that aims at characterizing the figure of the Absolute, just like the figure of the One had been characterized and installed into a System of Representative in the Roman Republic. Cayley’s parable suggests a Capitol within the center of Science as the New Republic. Yet its capitol is not a temple of Jupiter as an allegorical God, but a philosophical school which extracts – from the Theory of Invariances –  the methods for dealing with one of the main pillars of modern science: the study of population dynamics through analyzing the probable.

Cayley has adapted the Platonic inscription on the portal of the ancient Academy, which said that only those are welcome who master Geometry, to the state-of-the-art of science in the 19th century. Not Geometry can act in the role of a general method, instead he puts Algebra in the position of the Forms. Indeed, the basic interest of the algebraists in the 19th century was a method for discovering  invariants of various “symbolic forms” (polynomial equations). Cayley and his friend Sylvester were both important protagonists in this field. Cayley himself demonstrated as a first systematical study the finite number of invariants proper to binary quartic forms (i.e., forms of degree four in two variables); about a decade later, Gordon proved that every binary form, of any degree, has a finite basis (a finite number of invariants). He proved this by algorithmic means – basically, by a method of trial and error. But the developments went clearly in the direction of generalizing the applied methods for dealing with symbolic forms, and when Hilbert astonished the mathematical world in 1888 by announcing a new, conceptual approach to the problem of invariants, Gordon declared: “this is not mathematics; it is theology.” [2]

The object of theory which Cayley considered for his School at the Capitol of the Republic of Science, namely Invariants, do not concern the positivity of anything given, but the representability of what is intelligible. By altering the inscription from Geometry to Algebra, he seems to consider turning the Platonic Academy into some sort of realist idealism school of philosophy. Realism because of its service to the Res Publica, but Idealism because invariants would instantly loose all their integrative capacities if they were ever determined, figured or delineated in any concrete way. They incarnate infinities as qualifiable capacities within the abstract, and they are, as Cayley conceived of them, of purely instrumental interest, and not, in themselves, an Authority. Perhaps this all is not so different, after all, from what Plato had conceived for the Forms, within his Pedagogical understanding of Dialectics. In any case, this declared interest in the “republican” instrumentality of Invariances (Algebraic Forms of Conserved Quantities) is also what Gordon eventually realized about Hilbert’s conceptual approach. Once Hilbert’s conceptual approach could easily be reproduced as a constructive proof  (a fact which Hilbert himself did not consider significant at all), it aroused the following response from Gordan: “I have convinced myself that theology also has its advantages.” [3]

Perhaps the main problem with the complicity between Democracy and the Absolute is that it leaves no room for abstraction. We ought to introduce hallowed spaces into our Urban Territories, thus the suggestion of Cayley. Instead of raising temples, we could consider: What if the Roman Capitol were transformable into a Philosophical School within the Republic of Science? What would be an architectonic form for such a capitol? A Garden (Epicur’s school), a Portico (Stoics), a Lyceum (Aristotle), an Academy (Plato)? A Cloister, or another Form of an Monastery (Monotheistic Religions)?

If we look at the twentieth century, the most likely candidate would perhaps be the Conservatory. Was this not the idea of the Logical Idealists, Atomists and Empiricists, from Frege to Russell, Carnap and Popper, and on to the Cyberneticists like Wiener, Buckminster Fuller, the Green and Sustainable City movements suggested today? The conservatory as the architectonic form of the Roman Capitol today would seem like the ultimate affirmation of what I have allegorically called here the Politics of a Generic Monas! Philosophers, in such a Conservatory, are sitting – allegorically speaking – in the Cockpit of Space Ship underway throughout a Cosmos, trying to find their ways around through the Spacetime by Ecological Maps plotted from a General Theory of Relativity. Under the enormous pressure  they are exposed to, in their Cockpit of the Ship called Human Future, they consider it their Political Duty to train and send out their Ambassadors in order to economically govern the Representatives of an Urbanized Globe. Like the Theoreticians in the Ancient Monarchies, Theoreticians are sent out from the Conservatory – yet not in the name of a Leader, but in the name of an Absolutized Res Publica.

What Architectonic Form would provide philosophy in its own right a place within the Republic of Science in the 21st century? Perhaps the Observatory would be a likely candidate to set Philosophy free again and keep it from being exhausted by the Shortcircuits it must perform in its Duty to Political Economy. The Architectonic Form of the Observatory would provide it a place for studying Constellations and Events in the Sky of Theory. As such, it provides the place of seclusion necessary to keep the distinction between economic and political governance in place within the Res Publica – and hence, it keeps the Democratic Monas from identifying itself in relation to an Absolute. Within such an Observatory, Philosophers could learn to interpret the Constellations and Events in the Sky of Theory, by studying them as Starring and Distant Quantums of Cityness.

Studying the Theory of Invariances, coupled with the Probabilistic Empiricism, philosophy might be able to better and better understand how to quantize the genericness of the Democratic Monas algebraically – not unlike, perhaps, as Cayley dared to consider, from how the Ancient Schools have been successful in learning how to elementarize and proportionalize the genericness of the Tyrannical Monas.

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[1] cited in: Israel Kleiner,”Emmy Noether and the Advent of Abstract Algebra” in A History of Abstract Algebra, Birkhäuser 2007, p. 92.

[2] ibid. p. 93.

[3] ibid. p. 93.