A Circular Aesthetics


If the world really is to be ‘everything that is the case’,[1] it must also contain within it, as an integral part, our conception of the world. Our perception and intuition is grounded in the assumption of two fundamental conditions. First, the relativistic condition that all its objects are not given to us in themselves, but only in relation to each other. And, secondly, the reflexive condition that there is a subject of perception: an observer, who is himself such an object—that is to say, he too is part of the world he observes.


The first, relativistic assumption—identification—means that something is a given by virtue of being differentiated from all other things by the observer. If no difference is apparent, then nothing at all is apparent in the first place. Thus, the plurality of given objects and events is an interrelation fixed by perception itself. The individual things in the world differ above all in their localisation, so that it may be said that their individuation is their localisation. Put another way, individuation and localisation fundamentally mean the same thing. Both terms also imply a context in which the given objects and events are located—and that context is solely and simply that they are objects of a single and consistent act of perception.


In general and thus in mathematical terms, such a relativistic world, as an accumulation of different interrelated objects and events, is a two-dimensional manifold. That means nothing more than that we are dealing with a set of interrelated elements and subsets. But why two-dimensional? Could there not be an indefinite number of modes of differentiating objects, just as there is an indefinite number of objects themselves? After all, it is obvious that our actual perceived environment is a vast manifold of things with the most diverse properties and peculiarities! This is but an everyday observation, seemingly running counter to the principle of perception, which is to establish coherence. Apparently, however, perception has a way of managing that. This is about the general form of the content of perception being made coherent—not how this is done for any individual percepts. It is about how their representations are coded.


This is best illustrated by the example of a video file. The video shows a complex series of events—a crime drama or a romance, perhaps—in which various people in different environments, such as rooms or vehicles or the outdoors, go about in a wide variety of clothes, circumstances, activities, using props of all kinds, in colour or in black-and-white, talking or silently, with or without music, and possibly, as technology progresses, even with olfactory or haptic effects. But on a purely technical level, this immense variety of things, situations, and qualities experienced by us as the audience is represented by nothing more than a sequence of the simplest binary-coded signals—bits. Using suitable recording and playback devices with a read/write head, it can be made available at any time, its wealth of information restored to its full breadth. That is to say, the one-dimensional sequence of bits is written or read by a processor defining and comparing its individual components, just as someone reading a newspaper combines a mass of letters into words and sentences until some meaning can be abstracted from them. The plethora of possibilities inherent in such a simple one-dimensional set—a sequence of symbols—is thus a function of the relations between its subsets. And those relations, which we may now identify as the actual objects of perception in the manner described, only manifest themselves through an entity that is doing the comparison—the reader or the processor, or whoever or whatever.


The stage on which this is played out is no longer simply the one-dimensional line of symbols, but a two-dimensional manifold of relativistically interrelated percepts, objects, and events of all kinds. Mathematically speaking, instead of the simple set, we are now dealing with its power set, that is the huge set of all its subsets and, in particular, the web of their interrelations. The general basis of topology is the two-dimensional plane. With all objects given to us relativistically, that is in relation to each other, the world now no longer presents itself as a mere linear accumulation of objects, but as a complex web of relations, literally on a higher plane.


In the early modern period, the one-dimensional line of real numbers that had been common algebra until then was extended by a second version of it, initially for purely mathematical purposes.[2] Around a common point of origin—zero—both lines of numbers now open up a plane: the plane of complex numbers. These thus each consist of two components, the customary real part and, in explicit contradistinction, the so-called imaginary part. This extension of algebra soon proved itself extremely productive and useful in all areas of science, especially in physics—surely for no other reason than by providing unlimited scope for formal thinking in a now adequate, because two-dimensional, model of the objectively given world. There was now sufficient space for freely manipulating subsets. Today, all this may now also be represented as a vector space or matrix.


Disregarding, for the moment, my second fundamental assumption posited at the beginning—that of the subject of perception—we can say: in its general form, independent of the observer, the objectively perceptible world appears as a two-dimensional manifold, as a plane, not to say as an interface. The familiar dynamic of the content of our perception, that is, the differences of place, size, and all manner of other properties, their divisibility and mutability, all of our colourful, turbulent and protean reality, when viewed in the light of complete objectivity, is simply flat.


How so? What we consider our entire being is seriously supposed to reduce to such a modest general form? The example of the video recording suggests as much. So does the scientific finding that, at the lowest physiological level, that is, neurologically, our perceptual activity, all our sensory inputs and motor outputs also reduce to a particular distribution of a single basic binary contrast of either actively firing or passive neurons.


The subject

The second fundamental assumption (which I have called reflexive) regarding our perception and intuition is that there is a subject. But this subject does not view the objects and events in the world from offstage: it forms an integral part of these same objects and events. It is itself contained and present in the world, as a part, a player, and an internal observer of the web of interrelations. The subject is given to itself, in a circular way, through its self-identification. Self-identification is its essence.


Whereas, in the absence of an ‘outside world in itself’, all objects, events, indeed anything at all is given to us only in and by the subject’s perception, the subject, among all the other objects and events, does play a special, crucial part. This is not a solipsistic point of view to the extent that I, the subject in this particular case, acknowledge that there are other subjects besides me among the objects in this world: fellow human beings, cats, dogs, birds and insects, trees and fungi, pebbles and so on, to which the same applies, irrespective of how much consciousness I am prepared to attribute to them. But this in no way alters the fact that here and now, I am at the centre of the world as the particular subject, the special case of ‘everything that is the case’. Naturally, the same applies to each and every one of us.


It must be emphasised again that the given objects and events of this world do not exist ‘in themselves’, but only in relation to each other. And that obviously also applies to subjects and, in particular, ‘to myself’ as the immediately given subject. I, too, only exist as a part of and in relation to the web of objects and events in my surroundings.


Relations between things may be more or less immediate or more or less indirect. Any given object or event, including in the shape of the subject, thereby establishes something akin to a hierarchy or classification among other given objects and events, roughly structures the local environment into near and far, and takes up space (in the broadest sense), at the centre of which it locates itself. If the given object or event is the subject, with its competence for observation, this ‘here and now’ will locate itself at the centre of this space, seeing no reason to question this. It is situated and experiences itself in so-called egocentric space, as described by Gareth Evans.[3] Egocentric space is the environment of a given subject. It is the most primitive—and therefore fundamental—topology and orientation. Its most important property is consistence, that is, the coherence of its content (objects and events) given solely by perception. In our everyday surroundings, we locate ourselves, starting from the ‘here and now’ as a local point of origin, by distinguishing the conceptional pairs of up/down, front/back, left/right, and before/after. The four-dimensional spatio-temporal perspective they express, however, is a special structure that remains unexplained for now.


The crisis of egocentric space is found, in the first place, in communication. This is much more than different subjects communicating with each other. By giving validity, publicity, and connectivity to our orientation, communication is also the basis of our reason itself. Without this practical application, we would lack judgement, and orientation would remain purely arbitrary and delusional. Egocentric space, by a kind of evolutionary cultural achievement, is generalised intersubjectively—not only between human beings but also, for instance, between a buzzard and a mouse, even between a brook and its pebbles—into a public and objective space. The egocentric spaces of individual subjects are properly embedded within that public space, which by its essentially practical use acquires something like a superior truth value—even though, as an abstraction from egocentric space, it is perceived much less directly than the latter. Objectivity and immediacy of perception are two different things.


These reflections notwithstanding, I still do not question my earlier assertion regarding the general two-dimensionality of objectively given space—irrespective of our everyday experience of a four-dimensional, asymmetric environment with three spatial dimensions and one temporal dimension.


The crisis of both egocentric and now also public space is, secondly, its incompleteness in principle. At short sight, as it were, that is not a problem. The idea we have of our surroundings is open in principle and not necessarily complete—that is certain. Where it might end is not a question we initially ask ourselves. Somewhere, far out, everything may dissolve. Perhaps into nothingness—but then, something cannot be dissolved in nothing. Somewhere far away, everything may have started some time, perhaps in a Big Bang or in whatever completely inexplicable singularity. Unfortunately, however, we cannot avoid this problem even in the here and now. For even in objectively given space, the subject remains an essential part of the observed world with which it must interact through feedback.


The orientation explicitly discussed here is grounded in the continuous coherence of the contents of perception. In principle, it consists of a perfectly unbroken and complete idea of the environment, only on the basis of which is it possible to react and act ‘sensibly’. This necessarily requires a comprehensive, consistent, thus formally unambiguously definable and, as I have shown, two-dimensional image. Any part of this will also only have validity if it is complete and consistent in itself. A valid idea is not given merely by an arbitrary amount of percepts, but only by their consistent, combined, and unambiguous form. As a whole and in relation to each other, individual objects are not addends, but factors.


But the completeness and consistence of our idea of the objective world remain elusive: the problem here is the inescapable self-referentiality of the subject’s perception. As an observer and as an agent, it is itself an essential component of what is being observed. This means that our conception of the world is circular from the outset. Formally, it rests on feet of clay—and, to make matters worse, there is no ground beneath them. We are dealing throughout with a logical loop—a ‘strange loop’, in Douglas Hofstadter’s words.[4]


The paradox of self-inclusion inherent in this is known, among other names, as Russell’s antinomy. It concerns the set of all sets that are not members of themselves. This set would only contain itself if it did not contain itself. The contradiction inherent in this negates the very idea of such a relativistic context, that is, the objectively assumed relativistic context itself.[5] Against this backdrop, total annihilation beckons. It is not surprising, therefore, that in everyday life, we never seem to encounter this problem and know of no situation in which it actually makes any difference. We can bring it to mind only descriptively, as in the self-referential proposition ‘This sentence is false’ along with the question whether it really is false or perhaps true by virtue of its falsehood. Another well-known example is the Cretan village barber who shaves precisely all the men in the village who do not shave themselves. And the question then is: Who shaves the barber?


An actual instance of something like that happening seems inconceivable. But that is all the more peculiar when considering that we, subject and at the same time object of our perception, are constantly and from the outset in that very self-referential position where the unthinkable and the absolutely impossible is an inevitable given. Our valid idea of the world, our entire orientation is always and everywhere doomed to fail completely from the outset. We may consistently block out that relation—which is not really a relation—of our orientation and content ourselves with the superficial coherence of our objective conception of the world that is both vital and comfortable. We may ignore and overlook this systematic ‘blind spot’[6] of self-reference as a matter of course. It’s as easy as that! But, in fact, the ever-present collapse of our orientation cannot be eliminated. Nor can the vital necessity of then redesigning that orientation over and over. And just as surely as all our designs lack permanence from the outset, so does the actual moment of transition, of crossing over between planes, incontrovertibly assert itself: the ‘here and now’, the dynamic present, where all this occurs incessantly. In contrast, there literally is no past and no future: there was a past, more or less, and only its traces remain; there will be a future or not, as the case may be.


The subject consists of nothing but its self-identification. And the paradox of self-inclusion is the ultimate dynamic principle. By observing a world as subjects that are themselves integral parts of that world, our every observation is ultimately introspective and recursive, that is, circular and anti-formalist. Any form-finding by the subject is stopped at once by the contradiction of paradoxical self-inclusion. This precludes a consistent and complete orientation, without contradictions, at all times and in principle. For the power of orientation consists only in its formal coherence: its completeness and consistence. As Roger Penrose puts it, ‘Reflection principles provide the very antithesis of formalist reasoning.’[7]


But with orientation being the general and necessary objective of all perception and also its basis, it is imperative to block out the inevitable circle of self-inclusion as far as possible. Blocking it out, however, does not mean doing away with it, and so our conception—however far it may reach—negates itself at once and requires a restart. Time and time again. For that is what we call time.


An actual encounter with the paradox, however inconceivable that may have seemed earlier, is thus nonetheless directly and demonstratively perceived in its effect; by no means is it the case that it can only be described awkwardly, as by the barber or the philosopher. Rather, we have this encounter in every moment, always and everywhere. But this amounts to the same thing as never directly perceiving it, only its effect: continuous change.


The given objects and events of a relativistic world, as already mentioned, relate to each other like factors forming products, not as mere summations of addends. While, in arithmetics, an addend of zero does not change the sum, a factor of zero resets the whole product to zero. Similarly, the paradox of self-inclusion functions as ‘event zero’ in a relativistic context. Amid all the other objects and events in the causal interdependency of our environment, perceived in a circular way, it resembles a short circuit: the fuse has blown and the circuit breaks. It at once negates the whole context in which it is situated. It may be objected again here that we can only encounter this situation on a symbolic level, never actually in everyday reality—that the paradox only represents and does not ‘embody’ nothingness. No doubt that is so, but it should be remembered that what we assume to be objective reality is, in turn, the product of our perceptual feedback and thus itself nothing but a representation. And the observer’s self-inclusion is indeed part and parcel of wholly concrete given events. The paradox in relativistic reality, like zero as an addend in arithmetic, does not alter the result. But its effect as a factor is equally destructive in both cases. It permanently causes our conception to fail and immediately forces a reorientation—a formal update.


As everyday experience shows, however, it is not the complete conception along with its content, the many objects and events perceived, that is extinguished from one moment to the next. Otherwise, only pure chaos would remain, and orientation would be wholly impossible. Objects and events, however, are apparently of varying stability and permanence. Apparently, change and permanence belong together and are interdependent. How could it be otherwise? What must be renewed completely and from the ground up, again and again, is merely the special relational context in which our perception presents things. What collapses in every moment is not the content of perception as such, nor its general external form (to be derived in full shortly), space and time,[8] but its particular spatiotemporal distribution and configuration. It is the orientation of the individual that must be accomplished anew, again and again, not its general form of perspective.


Representation and perspective

To contradict a common saying: time does not fly. The ‘here and now’ is that most persistent position where the conception of an objective world fails at once, making way—literally giving up its space—for another such conception. It is the permanent link between a level of orientation that is always extinguished and one that must be established anew. It is the moment of crossing from one level to another and thus the interface between two more or less distinctive and, as discussed above, flat environments. It is the only really stable topos of perception. Since the environment, in relation to that position, is doubly two-dimensional, it is the origin of a four-dimensional perspective.


Now, however, there is no way in which a meta-position is available, such as the one I seem to be taking with my claim. There is no cutting table offstage—in a fifth dimension, perhaps—from where such a four-dimensional space could be observed and represented. All real relations exist only within the world of which the subject itself is objectively a part. In its general form, this world is not objectively given anywhere in global space but, precisely because of its circular constitution, only as a local perspective of the subject. From the inside, it cannot be grasped in its entirety. The representation of our perceived environment only seems to be modelling an ‘outside world in itself’: for such a world does not exist. Rather, there is a fundamentally introspective representation, a self-image; perception is entirely endomorphic: the self-referential view of its world by the subject is both internal and external. In this feedback system, subject and object ultimately cannot be distinguished; they only differ at short sight, as it were. They ‘differ’ in the same way as the ‘two sides’ of the Möbius strip: globally, they do not differ at all; they do so only locally. It would be wholly adequate—if more difficult to imagine—to make a comparison not with a Möbius strip, but with a closely related shape, the Klein bottle,[9] a surface closed in on itself (like a sphere) which, however, (unlike a sphere) does not separate an inside from an outside and does not contain any volume. I would like to claim that the interface between subject and object, our true interface, in principle is like a Klein bottle—a shape, incidentally, that can only be represented without overlap in four-dimensional space.


Mathematics, unlike our everyday imagination, readily provides a representation of four-dimensional space or indeed any n-dimensional space. It can lay claim to an ideal meta-position that is unavailable to our actual perception, given that we see the world from the inside. Actual perception of a world with four spatial dimensions, however, is impossible in principle, irrespective of its mathematical representability. This is not only the result of a lack of ability, but a first principle. If by some magic trick all four dimensions of our perceptible world that I have shown were available to us each in the same way,[10] we would find ‘being’ to be singular, complete, infinite, indivisible, and immutable, with all the change, all the differentiation, indeed all the dynamics of our existence nothing but mere opinion (doxa),[11] not to say ‘pure illusion’. We would have a metaphysical view of the world ‘as it ultimately is’,[12] a finding of absolute truth and of no practical value whatsoever.


But within the four introspectively given dimensions, the position of ‘here and now’, from moment to moment, is really a trajectory—a track of the subject through its environment, a linear, one-dimensional sequence of positions within the three remaining dimensions of its environment. The subject is absolutely bound to this track; there is no metaphysical basis from which it might dispose of its history, as at the cutting table already mentioned. However, it is able to review the traces of its immovable history and, in the other direction, to assume an open and as yet indefinite continuation of its trajectory as the future. Thus, that aspect of its position which we must call ‘temporal’, the ‘now’, faces the remainder of its three-dimensional ‘here’ from a kind of introspective meta-position. Time and space of its perception together form the asymmetric physical continuum of its perspective. Or should I say its ‘introspective’? The objects and events within it appear as a physical interrelation which, mathematically, is a doubly two-dimensional, that is, a four-dimensional manifold, a matrix with the signature (- + + +).[13] This signature expresses the opposing orientation (contravariance) of temporal and spatial dimensions as given by self-referentially circular introspection, in other words: through the endomorphism of self-representation.



The essential idea of my reflections concerns the paradox of self-inclusion of our perception, which I believe is—as much for good reason as by force of habit—now too hastily ignored by current science. For the principal concern of natural science in particular is, rightly, formalistic regularity, that is the completeness, consistency, and unrestricted verifiability of its propositions. This, however, according to Roger Penrose, is incompatible with reflection principles, which nonetheless play a far from negligible part in our reality—in particular, the principle of self-reflection. Physics, above all, is constantly faced with the problem that the observer cannot ultimately be separated from the observation. From different angles, it arrives at formally sound, correct, highly verified, tried and tested propositions and theories, but when taken together, these are inconsistent and completely incompatible and will doubtless remain so—like quantum theory and the theory of relativity. Both these theories have the same ‘blind spot’, having blocked out self-referentiality and circularity, because physics must be strictly objective. The paradox here appears, unavoidably, in its practical form: the contradiction between microscopic and macroscopic approaches to observation. Between looking in and looking out—into a chasm that has no existence in fact. It appears more digestible in the abstraction of mathematics, as in Bertrand Russell’s set theory or Kurt Gödel’s incompleteness theorem.


The paradox of self-inclusion, in my opinion, is the foundational paradox of our perception. We block it out for the sake of our orientation. But only by acknowledging it as a real and inescapable given is it possible to derive the special topology of the observed world in a satisfactory way. For its asymmetrical, four-dimensional spatiotemporal structure is due to our introspective, self-referentially recursive perception. In this, similar to the circular ‘chicken-and-egg problem’, I see the ultimate principle of dynamics, which blinds us to Parmenides’ singular, complete, eternal, and immutable being.



Peter Angermann

Candlemas 2022-02-02


Translated from the German by Philipp Rau


[1]    Ludwig Wittgenstein, Tractatus Logico-Philosophicus.

[2]    To make it possible to extract a root from a negative number.

[3]    Gareth Evans, The Varieties of Reference.

[4]    Douglas Hofstadter, Gödel, Escher, Bach.

[5]    As Russell’s precursor Gottlob Frege realised to his despair…

[6]    Niklas Luhmann, Die Wissenschaft der Gesellschaft.

[7]    Roger Penrose, The Emperor’s New Mind.

[8]    Immanuel Kant, Critique of Pure Reason.

[9]    If one sewed together the edges of two Möbius strips, one would end up with a Klein bottle. If, however, one were to cut the edgeless (that is, closed) surface of a Klein bottle, one would end up with two interlocking Möbius strips.

[10]  Such as in four-dimensional quaternion space.

[11]  Hermann Diels, Parmenides Lehrgedicht.

[12]  Hermann Weyl, Raum. Zeit. Materie.

[13]  Hermann Minkowski, Space and Time.