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Spatio-temporally graded causality: a model

In this work a model is proposed, according to which causality can be considered to be spatio-temporally graded. It is outlined how this can be described using the formalism of fuzzy sets theory, with the degree of causality varying between 1, that is no separation between causes and effects, and 0, that is perfect separation between causes and their effects as in classical 'billiard balls' models of physical systems. The model posits that subjective moments of time are like fuzzy sets, with their extension determined by local degrees of causality, resulting from information integration processes extended gradually in space and time. This, we argue, is a way in which a notion of causality could be, to a certain degree, spared and reconciled with a variant of Bergsonian duration theory as formulated in the theory of continuous change as the dimension of consciousness.


Introduction
The very conception of causality 1 in the natural world requires that a cause and its effect be two distinct events (for an overview, including possible alternatives to 'events', see [2]). It assumes, in essence, that causes cause effects, which themselves may cause another effects, and so on, with the identity of subsequent events being different, which determines a chain of separate, distinct events ( Figure 1A). Whether the direction of a causal chain is identical to or can be different than the direction of a time flow is disputed, but either way, it reduces to the notion of an ordered set of distinct events, which thus can be identified with distinct moments of time.
This natural conception of causality is also assumed by the information integration theory of consciousness (IIT), which posits that the integration of information (within temporal windows of certain duration), as determined by a particular structure of the cause-effect chain of an underlying physical system, gives rise to a unified subjective experience [3,4].
According to the theory of consciousness as continuous change (CC) [5], derived from a re-consideration of the Bergson's theory of duration [6][7][8] (developed also, for instance by Whitehead, as process philosophy) and contesting the assumption that nature consits of exact realizations of the model of mathematical point, what is real (and directly accessible to us) is CC. However, practically, CC can be considered as a spatiotemporally 'blurred' blob 2 , with some spatiotemporal extension, but with no clear-cut, sharp boundaries ( Figure 1B, left part). Then, however, on this view, taking it in the spatio-temporal terms, there is essentially no clear-cut spatio-temporal distinction of separate events, and thus no separate moments of time, or its linear flow. On this view causes can be considered, in a sense, to interact and influence themselves, with causes themselves being effects caused by their own effects, and hence 'causes' being their own 'effects'. Effectively, there seems to be no real distinction into cause and effect, which can be illustrated as a loop ( Figure 1B, right part). This leads to the conclusion that on this view there is no notion of causality, as it is naturally defined. This observation is, we posit, what underpins the Bergson's conception (described somewhat vaguely at times) of constantly appearing novel forms in which there is always some 'originality', not determined by or derivable from any preceding (or any other) state (and thus being inherently 'surprising'). This, in turn, has been the main source of (most accurately targeted) objections to the duration theory, stating that since it diminishes (or even eliminates) the role of causality as a foundational and essential element of description of the natural world, which is at the very center of our, scientific or popular, conceptions of it [2], then this approach is of no use and maybe-even worse-not accurate. The question is then, can (and whether there is such a need, in the first place) the notion of causality be spared, in some way, in this Bergsonian view as formulated more strictly by the CC theory?
1 Note that 'cause', and so 'causation'/'causality' as well, are said to be "truly obscure [concepts]" (see [1]). Here, by 'cause' ('effect') we will mean simply an event, that may potentialy be a cause (effect) of another events. And by 'event' (from the lack of clear definition of which this whole "obscurity" stems), we will mean (in an effort to reduce this obscurity in our argumentation to a minimum), when talking about phenomenal consciousness-a particular content of an experience, and when talking more generally about the world-a state of (a part of) the world at a time t [1].
2 Note on terminology: we will be using here the somewhat informal term 'blob', which, however, we see as a relatively accurate illustration of the idea, possibly reflecting our currently limited cognitive abilities, allowing us to think productively only in terms of space and time.

Graded causality
We propose that it can be spared, albeit to a certain degree only, in a form in which it is assumed to be graded, ranging from no causality (with causes not at all discernible from their effects; which we may consider to be pure 'indeterminism'), to perfect causality (that is, perfect separation of causes from their effects, as in classical 'billiard balls' models of physical systems).  Specifically, what we propose is to consider that it is spatio-temporally graded, and that there is no causality when a CC blob is 'seen' from the 'inside', and perfect causality only when a blob is 'seen' in an abstract manner from the 'outside', that is, when we are able to 'see' a blob's perimeter and thus many blobs as (almost) distinct entities (which, perhaps, is afforded to us by faculties of our mind, which is capable of 'extracting' and reflecting on separate fragments of CC-an issue on which we elaborate further below) ( Figure 1C). As we move towards a more abstract view, and consider separate blobs rather than the CC axis as a whole [5], individual blobs start to 'crystalize' gradually, causes tend to move away from their effects, causes become pure causes, effects become pure effects, the loop (as the one in Figure 1B) 'unfolds' itself into a straight line, and an approximation of a succession of separate blobs emerges. It is then equivalent to a (possibly deterministic) sequence of events, and to separate moments of time, giving thus an impression of a linear 'passage' of time (as in Figure 1A). Moving in the opposite direction, blobs gradually 'dissolve' into an unitary whole, and causes again become indiscernible from their effects. This could be described and analyzed using the tools of the fuzzy sets theory [9], with 1 denoting no causality, whereas 0 perfect causality. Then, 1 would be 'inside' the blobs, and 0 'outside'.
The model posits that the spatiotemporal 'diameter' of a blob will correspond to the extension of a subjective moment of time. Specifically, in a brain, a subjective moment of time, with a specific duration, associated with a synaptic clock with a specific time unit (as defined in [5,10]), will be constituted by a fuzzy set with a specific spatio-temporal diameter, encompassing and integrating in a graded manner certain amount of spatio-temporal neural activity. In a brain, many such sets will be overlapping, each 'centered' on a different synaptic clock, and each with a different diameter. Thus, due to the integration processes, each portion of neural activity, spatially and temporally located (in the reference frame of a brain, as recorded from in an experiment), will have a different degree of causality (i.e., membership) with respect to (or, from the perspective of) different subjective moments (i.e., sets). This is illustrated in Figure 1D, which shows only the 'spatial' dimension (corresponding to brain regions), with the spatial diameter of each set (that is, spatial range of information integration, X = v * T ) determined by the time unit of synaptic clock on which it is centered (that is, temporal window of information integration, T ) and 'velocity' of information propagation across regions (v, depending on the anatomical and functional connectivity). Then, some portion of neural activity, integration of which is giving rise to a longer-lasting unitary event (a large 'loop'), will be at the same time 'unfolded' into a more deterministic sequence of cause-effectlike events of shorter duration (small 'loops'), as being located within the spatio-temporal range of different albeit overlapping sets (i.e., moments; proportional to X 1 and X 2 , respectively) ( Figure 1E). For instance, some ac-tivity in a visual area might result in a sequence of visual percepts, e.g., of objects in the environment, while at the same time being integrated into one unitary percept, e.g., of a spatial location, as belonging (possibly with a different degree) also to another (here, 'hippocampal') set (i.e., moment), characterized by a wider spatio-temporal diameter. Perhaps, the experience of such a sequence would be enabled-or rather, ameliorated-by its 'embedding' in the longer-lasting event acting thus as a 'background'.
With separate blobs (that is, events) emerging, which one then is the 'cause' and which is the 'effect' ? One could posit, as an initial guess, that what is fundamental is the very process of (gradual) separation, resulting in a contrast, with the "effect" label being attributed to the one to which the "cause" label is not. The direction may not be inherent to it (but rather be a somehow superimposed description, or impression). In Husserlian terms, such a process of separation might be what is responsible for the appearance of 'retentions', 'impressions', and 'protentions', looking as contents of experience with different "temporal modes of presentation" [11-13], corresponding to the crystalizing 'causes' and 'effects', and thus constitute a basic experience of extension (i.e., separation) and causation as emerging out of the unity (rather than uniformity, or homogeneity) of CC.
The question of whether such a processes-of 'looping in' and 'unfolding' of the cause-effect lines, corresponding to the dissolving and crystalizing of individual blobs, respectively-actually occur, as real phenomena in the world, rather than being only a some kind of artificial construct of the mind (i.e., some sort of 'illusion'), is a central (and in our opinion open) one 3 . According to the theory of duration, the answer to it seems to be in the affirmative [7]. In Bergsonian terms, we would say that the physical world as well as our 'rational' mind (perhaps as a result of their interaction) both tend (or, are 'falling") towards 0 (that is, spatio-temporal separation and perfect causality), whereas our 'intuitive' mind tends (or, is 'falling') towards 1 (that is, pure flow and unity of experience, and what appears to be pure 'indeterminism'), and their interplay and relative strenght will, 'locally', determine the degree to be somewhere in the range between 0 and 1. However, when the CC is considered in its entirety, it is neither exactly 0 nor 1. Importantly, this does not correspond to the position along the CC axis [5], but rather to the two sides of experience, as illustrated using the analogy of the time-frequency uncertainty principle ( Figure 1F).

Discussion
There is a view, according to which, as we read it, there would be no real ordering of events into causes and ef-fects, and thus no notion of real time at all, with all possible events (or actually, configurations of matter) simply existing as a collection of points in an ideal configuration space [14] (this view being thus a form of 'eternalism' [11], with each of the co-existing points representing a separate, 'closed' moment of absolute simultaneity). As the conclusion is that causality in the traditional sense is not something that could be considered real in such a world, it may thus appear as pointing in a direction similar to the one presented here. However, such a lack of causality would be of a radically different nature, one that could be called static, in contrast to our dynamic view. Our objection to this kind of conceptions is obviously that they are based on the assumption that in nature there can be different events, or configurations of matter, representable as individual points, that would be distinct and completely separable (by some 'borders' and 'gaps') from one another. A conception like this, of discontinuous, unordered moments of 'now-ness', we find as an underpinning of the view on subjective experience stating that there is no fact of the matter as to what were the contents of one's experience in previous moments of time (i.e., ones that, assuming their ordered succession, would precede a current moment of experienced 'now-ness') [15]. Our view, in contrast, posits a continuity, and thus reality of the flow of time (see also [16]).
Our blobs may remind the Leibnitzian monades, that is, elementary unitary beings unable to interact, arranged into a spatio-temporal order due to some 'pre-established' harmony. In contrast to that, however, we posit that the natures of conscious 'beings' and of physical reality are not disjoint and incompatible, with the CC being common to them, and perhaps they both having extensional (i.e., 'separational') as well as non-extensional aspects (or 'sides') to them. Different beings can thus interact-the constant, continuous interaction is what would constitute a 'harmony'. CC could be seen as a 'common currency', shared by these two realms, as sought also by the temporospatial theory of consciousness [17].
According to the "unfolding argument", causal structure theories, like IIT, cannot explain consciousness, because for systems with a specific causal, in particular recurrent, architecture, there can always be, in theory, devised equivalent feedforward systems characterized by exactly the same input-output function, with the two variants of a system being not distinguishable experimentally [18]. According to the CC theory (departing from a phenomenological viewpoint), consciousness as such is not neccesarily due to the recurrence, or any other specific architecture of a system. Rather, specific network structure is what will give a structure to an experience, 'channel' it into different contents. The cause-effect structures of various systems (when considered in terms of CC) are perhaps always inherently 'folded', to varied degrees (as we have been arguing here).
When we say that a subjective moment of time, and thus some subjective experience, will be related with some activity in a given nervous system, e.g., some action potentials propagating, or proteins being phosphorylated, that is not to say that it is identical with that activity. When a physical system is looked at in a classical (that is, usual) way, then action potentials are action potentials, protein phosphorylation is protein phosphorylation (and so on), nothing less nothing more (see also [19]). It is then obviously unthinkable that, for instance, a particular propagating wave of sodium ions flowing in and out of a given cell could at the same time constitute, on the one hand, a spatio-temporal sequence of events, and on the other, one unitary event. The point is rather that if subjective experience as well as physical systems are assumed to have CC in common, then different subjective moments will be differently related with (or, constituted by) its different 'portions' (with CC being both 'fluid' and potentially 'complex', having a dynamic structure on different, more fine-or coarse-grained, spatiotemporal scales of integration).
The CC view suggests the 'source' of causality (as well as of other tendencies, or rules, that they display, or follow) to be in the phenomena (thus being 'immanent'), rather than existing 'outside' of them (beyond time and space, or as we should rather say-beyond the CC) as a collection of prescriptions telling how exactly they should occur (see also [20]). It leads us onto the position of Best System Analysis, with laws of nature being on this view ontologically derivative, not primary, and taken simply as "the best system of regularities that systematizes all the events in universal history" [1].
As has been argued in [5], the notions of space and time can be considered abstractions derived from a juxtaposing of blobs of different 'sizes', that is, from CCs of different 'rates', slow and rapid, respectively. Two blobs, when juxtaposed (in an abstract, retrospective manner), can be considered to be relatively more spatial (i.e., appearing to change less rapidly; with the 'blurring' effect more extended; perhaps 'ameliorating' the interactions between different 'temporal' blobs), or more temporal (i.e., appearing to change more rapidly; with almost no blurring), respectively. In other words, when we move along the axis of CC (between different 'rates', or 'sizes', of CC), space appears to be 'transformed' into time, or vice-versa. It is then like we were moving between 'macro' and 'micro' scales. In other words, what we commonly mean by 'space', which allows to see it as an abstraction derived from CC, is how much change needs to occur in the world, for instance, when I am walking a street to some target location, then how much change in the configuration of matter needs to occur (i.e., of my body in relation to the seemingly static environment, e.g., the surrounding buildings, clouds on the sky, etc), so that I find myself, i.e., have an experience of being, in a different 'place' (which I consider to be a different point, with different coordi-nates in my imaginary absolute spatial reference system). That clasically conceived space and time (à la Newton) will in fact be both inextricably linked and malleable, or that there are no objects in the world with definite positions in space and time-all this, according to the theory of duration and now CC, can be seen (but perhaps only post factum) as trivial predictions stemming from a simple reasoning (as outlined in [5], challenging the assumption that nature consits of exact realizations of the model of mathematical point; with CC, however, not to be seen as an exact negation, or a 'mirror image', of the notion of a collection of discontinuous mathematical points, but simply as something different, which is to be characterized in some positive terms), that should rather be treated as a general starting point and basis for building specific physical theories.
In summary, what this proposal suggests, is to simply acknowledge and incorporate into formal analyses the two facts, one from physics and one from biology/psychology, that do not seem to be fully explainable(-away) with currently available methods (but see [1]), namely, on the one hand, the apparent "disappearance of determinism" from the fundamental physics [2], and on the other, the common observation that behavior of individuals is never fully predictable (due to its (inherent) 'messy-ness'; which, however, must not in itself imply any 'free will' of some non-physical nature).