Contemporary technologies operative on the symbols of universal algebra and their code-based characteristics, namely information, have meanwhile spelled out three fields that begin to appear within the emerging logistically based order. I will attempt to characterize these fields (as the beginning of a to-be-continued reformulation and articulation):
Quantum Activity as a kind of actuality that abstracts from the stasis/movement distinction and instead of beginnings and terminating introduces an aspect of doping, of investing points with dimensionalities, and the systems they are considered to form are invested with axialities – axialities allowing to orientate the virtually empowered points along any common axes that can be actualized, by using any unit in any way. The dynamical system becomes a system-in-act. Invested with axialities, it is possible to take measure across the dimensionalities. This means, proportionality can be rendered. Figuratively put, in addition to the potential a point has within an order of seriality, where it can potentially occupy a position within various series – the points now acquire virtuality. It helps (arguably) to see a quasi-aristotelian dynamics of privation/deprivation at work within the potentiality that comprehends all potentials, as the potential to occupy places in different series cannot be regarded in uniform manner anymore. Because the points are equipped with dimensionalities, and the serial order under consideration is strictly dispositional (equipped with axialities). In German, the technical procedure of doping is called Ausstatten, because it abstracts from place-value logics and instead works with introducing arbitrarily and decidedly ‘equipped’ points into semiconductor grids. All Electrical Systems are Systems-in-Act in that sense – any of them is inadequately characterized in thermodynamic terms.
Data and the Curse of Dimensionality
What we take as given today, for working with, is objects in their symbolical, algebraically encoded form, as data. There is a phenomenon to which data engineers refer to as the Curse of Dimensionality. It labels various phenomena that arise when analyzing and organizing data in high-dimensional spaces, with hundreds or thousands of dimensions. Why should one do this?
An answer to this question has to do with the status we ascribe to objects. It makes sense if we take an evolutionary perspective of either ‘optimizing’ the general capacities proper to a certain object-class, or of ‘breeding’ capacities that can be appropriated by individuals within certain milieus or environment. The status we ascribe to objects depends upon the way we decide to deal with the disparity which is constitutive, in every evolutionary perspective, for a population. Put in more formal terms, do we treat our data-objects in a Cartesian way by positing them before a generally coordinated container space, where each can be related to each all others according to a position-value logic or do we seek to find, in a Riemannian manner, the topology of a space such that it is capable of accommodating the elected objects under consideration not in a general, but in a unique way? While the benefit of the prior decision allows for efficiency in terms of class performance and productivity, the latter offers benefits in terms of sophisticated articulation in which the results can be produced. We either sacrifice the generality of the process to problem-specific optimization, or reversely. What underlies these stances is the crucial distinction between abstraction and generalization: while generalization works by finitary methods which proceed by measuring a population according to the norm of a set standard, abstraction inverses the focus. It applies infinitary methods and looks at the individual within a population – without settling first on a generalizing class notion. Rather, it makes the individual potentially commensurate in all possible respects according to all possible criteria that can be extracted from a given population, infinitarily so. Thus we can say that while generalization condenses the disparity of a population into one magnitude, abstraction elevates each component of the disparity such that it may accommodate potentially all the others (with which it ‘unites’ as the disparity of a population).
To express it more technically, the common theme meant by data engineers when they speak of the curse of dimensionality is that when the dimensionality increases within which you work with your data, the volume of the space you have to master increases so fast that the available data dissolves its representational links within a vast combinatorial space. In high dimensional data, all objects appear to be sparse and dissimilar in potentially so many ways, that detecting areas where objects form groups with similar properties can hardly be achieved; they are statistically not significant and sound, because of an insufficient amount of data worked with. In consequence, the objects appear as singularized representatives, claiming to speak in the name of a population, each demanding attention and consent, and this in ways that not only conflict between the different objects, but that are also brought forward on very weak grounds.
Insistence. The distinction between intensity and extensity seems insufficient for characterizing a notion of order capable to express quantum-activity. The qualification of intensity refers to a definite quantity being strained, stretched, spanned, in tension, while the qualification of extensity refers to a relaxed state, to the amount of voluminosity a definite quantity occupied. Yet the power-at-work in order emerging from quantum activity can hardly be referred back to an interplay of continuous forms in rotation and constellation, nor to the dynamic balances of discrete distributions in contained compartments. Systems-in-act cannot be properly studied if we apply linear schemata in our analysis. Whether we synthesize these schemata geometrically, by exploiting the mechanical impacts of extensive quantities, or arithmetically, by exploiting the variability of intensive quantities in series, systems-in-act demand consideration of their symbolic constitution as data. Logical quantification in terms of existence and universality is insufficient. The insufficiency which I would like to point out consists mainly in the distinction’s departure, on both sides, from an assumed definiteness of quantities. The rise of abstract or symbolic algebra concerns this now challenged assumption of a quantity as being definitely given – if not necessarily as a finite quantity then at least as a positively definite one. But symbolic algebra is capable of expressing quantities as terms, such that quantities became the subject of conceptual definition. Quantities can no longer be conceived as partitioning one continuous order of magnitude; rather they now accommodate an abounding magnitude within an architectonics of terms. The fact of the conceptual definability of quantities, rather than their positive givenness in definite form, eventually split the camps between the intuitionists (Brouwer, Weyl), the formalists (Hilbert), as well as (within realist philosophy) different groups of conceptualists (Russell, Wittgenstein, Deleuze, Badiou) in 20th century philosophy. How to think this wholeness which abounds without tying it back to inherited notions of the One?
Algebraic forms are not representatives – this would be to idolize the abounding magnitude they conserve – they are contracts. Contracts make arrangements, and of the arrangements they put in act, we can speak about in terms of insistence. What insists has value without being necessary or privative. The notion of existence captures what can persist in terms of extensive presence and intensive potential through sequences of time and sections of space, and we can proceed similarly with the notion of insistence. We can assume that it captures acts of abstraction that are being mastered and legitimized – made public – such that they may persist in terms. That is, practices that are verbalized, formalized, mathematized. Complementary to what exists, what insists can be thought as that which is ‘conserved’ in the literacies and proficiencies of practices which are formalized and mathematized.