## Manifoldness: Spacetime as Hypernetwork with Leibniz, Whitehead, and Riemann

by Christopher Vitale (crossposted at *Networkologies)*

What might it mean to look at space and time as networks? Historically, a networkological approach to time and space naturally finds one of its forefathers in Leibniz, but first its worth getting a sense of what he is reacting against. The early modern period is one of enormous flux as far as theories of space are concerned, both in terms of the establishment of space as a metric, measurable, and absolute domain (primarilly via Descartes invention of analytic/coordinate geometry), but also in terms of the parallel and complementary notion that this metrically determined sense of space and time governs the realm of things (*res extensa*, radically distinct from the non-metric realm of *res cogitans*). The Cartesian mind-body split and the notion of space (and with it the moving metric of time) as a metric is indissociable from the assumptions which make the Newtonian spatial project possible – a distinct observer, the progress of a spatialized, linear, uni-directional time, etc. This tradition, which starts with Descartes, is further formalized by Netwon, reaches its apotheosis in Kant, and is based upon a split of the world into subjective and objective, one which continues to haunt the pretensions to ‘objectivity’ in much of contemporary empirical science.

In contrast, the counter-tradition which Leibniz represents does not separate mind and body in the radical sense pursued by the Cartesians. For Leibniz, mind and matter are, to use a term employed by Deleuze, ‘enfolded’ in each other, two sides of the same great origami-like structure we call existence. Furthermore, perception is not something limited to rational beings – all parts of the universe, due to their possession of a ‘perspective’ on the universe, present an opening onto the entirety of all that exists, in both space and time, in nuce, so to speak, that is, in miniature. For in fact, were not all space and time the way it is, any given spacetime location or entity wouldn’t be what it is either. The whole is present, virtually and differently, in the parts.

This is why the universe for Leibniz is, to use a modern term, holographic – take any part of the universe, and you can deduce the space-time structure of the whole, so long as you know the manner in which the perspective of the object in question has ‘taken in’ and ‘recorded’ this history of which it is the result. Each particle in the universe indicates a worldline, in the most literal sense of the term.

The fact that so many contemporary terms are applicable to Leibniz should indicate the fact that Leibniz is in many ways more our contemporary than is the Cartesian tradition. And Leibniz is in fact an enormous influence on Whitehead, the first thinker to attempt to bring metaphysics into conversation with the theory of relativity and quantum mechanics. The fact that Leibniz provides a roadmap for Whitehead to follow perhaps says as much about Leibniz’s epistemo-ethico-aesthetics, and the manner in which epistemo-ethico-aesthetics affects one’s ontology, than anything else. For in fact, Leibniz’s relational approach to the universe and his non-dualistic approach to matter/mind is in fact more of an emanation of his core aesthetic approach to the universe than the other way around. Leibniz’s reltionalism is in fact precisely that which makes him quite so contemporary. Leibniz is in fact the first philosopher to be fully fractal, relational, ‘fuzzy’ (as in non-binary logics), and holographic in his picture of the universe.

This is all taken a step further by Whitehead. For Whitehead, the very notion of the primary ‘unit’ of experience, that is, the spacetime ‘monad’ of sorts that he interchangeably calls the ‘actual entity/occasion’, is in fact of variable size. Any event can fit into this category, from the Russian Revolution to a quantum flicker in an accelerator. While some might have greater internal structure than other, some of which may go beyond the ‘clarity’ of our knowledge (and here again we see Leibniz as a thinker of gradation rather than ‘the cut’, in that knowledge of what is beyond us is unclear rather than non-existent), the point is that Whitehead’s approach to the world finds its ultimate unit as one of both space and time. The event is the foundation of Whitehead’s mechanics.

This is something which he builds up from a Leibnizian approach to the cosmos, and which also provides a foundation for the often maligned ‘mereological’ aspects of the penultimate sections of *Process and Reality*. These sections, in which he shows the manner in which points, lines, and planes are in fact abstractions from the more complex entities we see in the world, works to show how the very notions of geometry, which many view as primary and foundational to any approach to spacetime, are in fact abstract derivatives thereof. This section, almost inpenetrable without having read Whitehead’s works leading up to *Process and Reality* (such as *On the Concept of Nature* and *An Enquiry Concerning the Conditions of Natural Knowledge*), are a reworking of his notion, developed in these earlier works, of what he calls ‘extensive abstraction’. Here we see his argument, sketched in much clearer terms, that the mathematical models of space and time that we inherit from Newton and Descartes are in fact convenient and socially useful abstractions from the primary way in which we experience the world, which is that of moving, flexible, continually reconfiguring spacetime. And in fact, we all know this anyway – we often say things, when driving a car, for example, such as ‘oh, I need to make a left in like 5 minutes’. Space and time are naturally cojoined, and it is the abstractions of our modes of knowledge that separate them, leading to a situation which then needs to be overcome if we are to understand existence a bit more clearly.

While working to dissolve the distinctions between space and time, Leibniz and Whitehead also dissolve the related boundaries between mind and matter within an ontology that knows only events. If mind and matter are not fully separate, but rather two aspects of the same thing (namely, events), then the need to firmly separate the objective world of rigid, geometrical extension from the subjective world of potentially erring impressions, ceases to be a pressing matter for the establishment of scientific enquiry. And we have seen this with the rise of the ‘subjective’ seeming elements in quantum physics and relativity. What matters is not what ‘really’ happens to some impossible objective observer, but rather, what appears to happen from the perspective of the observer in question. And rather than these appearances be deemed ‘less real’, in line with a Newtonian/Cartesian approach to the world, rather, all these appearances are equally real. This is precisely why Bergson, another descendent of Leibniz, refers to all that exists as ‘an image’ – the universe has many images of events, each themselves events, none of which are more or less real than others.

That said, within the theory of relativity, one’s own ‘intertial frame’ is still privileged, giving some point of reference, and there is no question of this. This is why, for example, when you fly in an airplane, you don’t feel like you are moving at 500 some-odd miles an hour, because ‘you’ and the airplane you are in, your ‘frame of reference’, move together. But when we really get down to it, ‘our’ inertial frame is even itself a subjective judgment, one that gets smaller and smaller the more ‘our’ different parts are in motion relative to each other.

The result, particularly when we take into account the manner in which time and space stretch and bend according to the theory of relativity, is a universe which is distinctly not at one with itself. Depending on where you are in the universe and how fast you are going, you may see different spatial arrangments of the world around you, different distances between the objects in question, and a different order between the temporal relation of these events.

The world is then, shall we say – *manifold*. This term, taken from the German term *Mannigfaltig*, one which in French is most often translated as *multiplicite*, or in English, ‘multiplicity’. But we must avoid confusion here – simply calling the world multiple does not mean it doesn’t change, it is a self-altering multiple.

But this creates some difficulties, for how can we describe the unfolding of this change, if the order of events – that which is usually used to ‘tell’ time – is itself no longer stable? Whitehead’s term to describe change is thus not linked to time (which for him presupposes perspective and its limitations), but rather, what he calls the ‘creative advance’ of the universe. This advance is the sum totals of the changes and/in the ‘times’ of the events of which the universe is composed. Such a notion spatializes time as much as it temporalizes space, while providing a spur to the thought of change beyond standard conceptions of space and time.

This becomes a bit clearer by getting a sense of what is really at stake, mathematically, with the notion of spacetime. The events of which the creatively advancing manifold is composed are thus times as much as spaces, in that the only distinction between time and space is, if we are being mathematical, nothing more than the speed of light squared. That is, space takes a certain amount of time to cross at this ‘top speed’, while time going at top speed indicates a certain amount of space. Time and space are strictly convertible, at least numerically, with only a minus sign and a conversion factor, a multiplier, between the numbers that describe one, and those that describe another. But does that mean they are two different ways of describing the ‘same’ thing?

Basically, yes, so long as we understand that time is the more compressed of the two, and that there are a set of rules that govern how we convert one to the other. That is, if we are moving at the same speed, we only know we are moving because of the changes in the world around us. We only know distances by means of comparison of objects that seem to be the ‘same’ in space, and we only know time by comparing these as some objects change and others don’t, or via rhythmic oscillations between these spatial invariants. Time can be unfolded as space, and space scrunched together into time, but ultimately, at long scales, we cannot tell the difference between them.

Or, take objects which are very far away from planet earth, the picture of which tells us of events which are far away *and* far back in time. The relation between the two can only be judged in reference to other such objects. Objects more distant from us versus further back in time are ultimately only differentiated by their relation to each other, and the constant of the square of the speed of light.

What then is spacetime, and how do we begin to think of it as a network? As Einstein has shown us, objects going towards heavy objects tend to find their space expanded and time dilated, while objects going at high speeds experience the opposites. Its as if matter ‘opens’ up space by making more of it for things to travel in around them, just as high-speed objects do the opposite, recollapsing space with it. And with space goes time. But if spacetime is different, in fact, at each perspectival point in the universe (in varying degrees, of course), then what sort of network are we dealing with?

Mathematician Bernhard Riemann is the one who provided the answer back in his inaugural lecture back in 1850, and it was the math he announced that provided Einstein with the tools for turning the universe into a mass of interconnected, 3D rubber ‘sheets’, or perhaps better yet, stretchy sponges. Riemann essentially extend’s Leibniz and Newton’s approach to the calculus beyond 2D, into 3 and multi-dimensional spaces. For example, in calculus when we take the derivative of a term, we can view it as the increasingly close tangent line to a curve. That is, as a tangent line gets closer and closer to a curve, it begins to approximate it. Riemann attempts to take this from a 2D picture to a 3D one with his notion of the manifold.

A manifold is a portion of 3 (or more) dimensional space. It may be Euclidean, or it may be scrunched or expanded (what scientists generally call ‘curved’, but this term often confuses non-scientists because we can easily think of curves in 2D, but when transposed into 3D, scrunching/expanding is a more intuitive description of what really happens).

Here’s what we mean by ‘curved’ spacetime. Imagine a 3D cube. If it is scrunched, a particle flying through it on a straight trajectory will take more space and time to go through the middle sections, while if it is expanded, the opposite. This is what we mean by a section of space that isn’t extrinsincly (exteriorly), but rather, intrinsicly (interiorly) curved. If you put a bunch of these cubes together, the scrunched ones will add up to a space in which particles move differently than in ones made up of the same number of expanded cubes. The result is that an observer moving in spaces made up of these cubes, sewn together, will always think it is going straight, but will find the objects in space around it rearrange in odd ways. And from an outside perspective, it will look like the observer is curving around. This is what we mean by curved space.

Riemann’s insight was that he came up with a way to do calculus on these cubes. Just as we continually shrink a tangent line to a curve in 2D calculus, the calculus of manifolds continually shrinks a 3D cube down to a single point. As it approaches a point to the limit, the 3D cube will approximate non-curved Euclidean space, no matter what the curvature. This is just like any curve will be approximated by a tangent line in the limit. Thus, Riemann provides the mechanics to take calculus beyond 2D. Spacetime, understood via Riemannian calculus, is what he calls a manifold, an assemblage of potentially stretched and warped sections, sewed together.

And as an observer moves around in this manifold, the light or other media which convey data to our senses warp and sway, expand and contract, just like a person walking in front of a fun-house mirror. That is, as we move in this manifold, we see spacetime unfold and contract in front of us in warped foldings and unfoldings in 4D.

How then is this a network? For Reimann, the only way to understand a manifold is by the calculus of limits, but taken into 4D. That is, the whole can only be understood as the limit of the additions of sections of spacetime, swatches of events, decomposed into limit points and then resewed together. This is the analytical network whereby a manifold can be understood.

But spacetime is more than this, for it is also the network of perspectives which link together to form the incompossible world which is the manifold of the known universe. Riemann gives us the math to comprehend this, even if the sum total of the interconnected perspectives indicate a sort of hypernetwork, a network of networks which are incompossible except in terms of virtual superposition.

And this shows us the manner in which the universe is perhaps superposed on multiple levels of scale. Quantum phenomenon are superpositioned in time and space at multiple levels of scale. Likewise, the human mind can, in its own way, be in multiple spaces and times at once, via memories and associations, at multiple levels of scale, all interpenetrating. And as we have seen in our discussion of how spacetime warps over large distances, the largest entities in the universe have a similar relationship with each other.

In a sense then, actualization in a particular location in spacetime is perhaps something which only happens at medium levels of scale, and at local levels but not at great distances in time or space. For we know not if the universe has a boundary or is in fact curved in upon itself – and what we would see in either case might not be all that much different.

The result is that the universe can be thought of as composed of fuzzy quantum superpositions, actualized locations in spacetime, the superpositions of these within the echolike images they produce as they refract off each other, and the ‘prehension’ or ‘perception’ of these images as they impact events. The universe as hypernetwork.