Motions in the space of knowledge

Forward: What you can speak about ...


It is often appropriate – especially in science – to differentiate between the reality we speak about, and the knowledge we express in doing so. But this differentiation should not disguise that all things available in science – or anywhere else – are data, derived from measurements, observations, and so on. We can call them “real” – but “knowledge” quiet as well.


All that we can explore and speak about is part of our knowledge. Therefore it should make sense to study thoroughly the characteristics of knowing, its implications, and so on. This way we may discover forms and rules present in every part of reality. And, the other way round, the laws of nature should reveal the nature of knowledge, too.


When we talk or think about something, it becomes some thing. This is an effect we cannot avoid; so it is a fact we should not ignore. The world consists of things. But what does this mean? What is the nature of a thing, how do things interact and combine? Questions like these seek for something between the things or beyond them, something different and complementary – even though again our analysis will not be able to discover anything but things.


Speaking or thinking about something are a sort of reflecting: something gets transformed and appears in a new medium where it will be easier to overlook and to handle. This would not work unless the material thing and its mental representation were the same somehow. In a sense they must share one and the same space.


We would not be able to speak about anything if our speaking were not responded by understanding. Somebody must hear our words and know what to do with them. All persons involved have to play the same game – although slight differences and variations should be possible.


Perception and cognition are games of sameness and difference, of overlapping spheres and interacting things. Thus they do not differ from any other real things or events.


Mental representations of material things might be assumed to be much like photos, primarily produced by perception. Compared with the original they seem to be flat and poor, rather dead – but actually the mind is alive, too, and so is everything it contains. The notion of poor flatness arises from lack of congruency of different living spaces.


Here we basically speak about a logical system. All the rest is application – nothing else could found and justify our logic.


To distinguish between reality and the knowledge about it is an important impulse not to stop scrutinizing the consistency of our knowledge. So it drives our search for truth on and on. Still it is not the whole truth – which, of course, does not mean that any other truth would be the whole.

Home: We know


Knowledge is always embedded in a particular context, especially in a community sharing it. Here it finds the form it needs to get conserved and exchanged. This form alone, without the context, would be senseless. In order to really own the knowledge, we have to learn to handle its forms the right way. To acquire knowledge – as well as to proof it – we have to use it.


Knowledge belongs to a community, a “we”. Knowledge is based on a common fundament – and constitutes it. To “know” means to draw a line: there, on the other side, there is the unknown, here, on this side, here is the known, here are we.


Bringing the contextuality and delimitating effect of knowledge to the head, one might feel obliged to say that every particular piece or part of knowledge were associated with a specific we. But this sounds strange, of course, because this term, the “we”, does not really conform to such a splitting up: rather it puts together.


Advance in knowledge means to broaden the horizons. But in order to get there, we should know the limits. First we have to see that there is a horizon – then we may wish and try to look and go beyond it. That is why it is so important to recognize previously invisible demarcations, to draw the line, to differentiate. The first step towards knowledge is feeling the gap.


We go step by step. Relying on what we have for sure, we adopt the new. But our knowledge does not really have to grow. Integrating the new reforms the old. Knowing simplifies, we see clear where impervious thicket has dulled the sight.


Here we do not speak about an “I”, not about “me” as being the actor and focus of knowing, but rather accentuate the community. We do not have to act like an isolated individual, unable to cast its skin. We can grow – without losing our identity. And without forcing out others: for we are able to integrate and to restructure ourselves.


The We may help to abolish the illusion of a bounded individual existence. However, all our traditional enhancements and extensions, the traditional communities we identify ourselves with by habit, are not at all adequate to comprehend the ultimately new perspective of this opuzz.

Way 1: Knowledge persists


Stability and constancy are fundamental for all that can be called “knowledge”. Whenever we want to investigate and judge something, it might be an event of the past, the presence, or the future, something near by or far away: the only things we can rely on are those definite measures and laws that last forever unchanged. Certain facts – that's what we know.


Our exploration tries to go ahead with the concept of knowledge as something definite that does not alter. Although taking effect tacitly in the background all the time, this idea with all its consequences has never been worked out. We do not feel comfortable with conclusions like the one that infinite spaciousness will be required to store every fact for ever unchanged. We have never realized that the space of knowledge – where this takes place – is a real thing. Therefore we have never inspected it.


We can define knowledge in contrast to activity, as its counterpart. All changing is activity, while knowledge is the rest that does not change. This is yet a poor definition, of course, a first approach waiting for being developed and completed.


Knowledge and activity – one defines the other, one causes the other. So they do not appear but together. Knowledge would not persist if there were not any changing around it – and even inside.


Knowledge may be called “frozen activity”. Every piece of knowledge is unchangeable as a whole, but includes kinds of inner changes, namely certain differentiations. Knowledge is constituted by them, conserves them, and will reproduce them as needed.


Nothing could be anything without some constancy. So we might say that everything participates in knowledge, or that knowledge is an essential part of everything, or even that everything is knowledge. But of course, if knowledge were everything and there were nothing else, it would not be anything.


The difference, differentiation, is contituent for knowing. Thus knowledge is somewhat straddling. It generally manifests kind of extension. That makes space the natural product of knowledge. While knowledge turns out to be the ultimate dimension of space.


Knowledge is closely associated with the notion of matter. Not only because it has always some material representation, but also and above all because matter is nothing else than compact knowledge. It is memorized experience that renews itself contantly in acting with perfect regularity.


In science knowledge is often reflected as a mathematical equation. This ascertains that actually distinguished expressions are identical. That is exactly the task that knowledge performs: it manages identity. The calm in swirling transiency.

Way 2: Activity changes


Activity always moves something, induces a change, makes a difference. And every difference is due to activity.


Activity is the necessary complement to knowledge. The permanently constant equality of knowledge is facing its counterpart, the unequality, difference, alteration. Both belong together, neither could be without the other.


Activity is almost nothing, almost no thing. Activity is where the things do not be, between them. It is that what makes the things appear, what makes them real. And after all, activity has to be some thing, too. Everything must be a thing, or it would not work – and we could neither see nor think it. In particular, when we speak of (a plurality of distinct) “activities”, then we treat activity like a concrete thing.


Activity lies between two different states. It differentiates – and so constitutes the states. It belongs to both and neither of them. They would not exist without that activity, which itself does not exist like them. It stands for the separating gap, but serves as direct connection, too. Thus it causes difference as well as sameness.


Activity is usually thought to depend on time. So instead of “activity” we might have introduced the concept of “time” as fundamental, especially in addition to “space”. But here we prefer activity because of its open nonlinearity and discontinuity.


Every definition is much too narrow: activity is always wider. Sometimes activity appears as a distinct action. Every action is coupled with many others. United actions act as one thing ... and so on. Still there is more: activity is more.


Activity cannot be catched. It always reaches out. So it scatters. But on the other hand it is always focused, directed, aiming at a target. The point is that some targets are neither hit nor known yet.


Activity is everlasting genesis. It is never finished, strictly speaking it is nothing. Sometimes it appears as the point that itself has no extension in any dimension, although it holds every dimension. The infinitesimal that generates infinity.

Way 3: Activity creates knowledge


Activity leads to a new state distinct from the other. Activity makes the distinction. We could not speak of “activity”, if it did not result in something distinguished and recognizable.


Activity cannot be detected but by the fact that something is happening, something is changing. Pure activity would be nothing. What actually can be grasped and understood to be activity already has become something fixed, a fact, it has become knowledge.


Steadily occurring transformation of one state into another becomes kind of state of its own. Every steady state is due to constantly recurring activity. Again and again the same is happening: that is what makes the stability and certainty we call “knowledge”, that is a fact.


The flow of activity continues. It does not stop renewing itself – and everything else. But while it pours out, it gets cold, it becomes stiff and concrete. Everything is getting old. And in the end it must die. But beyond death – and even in it – there is living. The source of activity will never run dry because it is always everywhere, in every piece of everything.


In contrast to the passive passing of time the flow of activity is creative. While time keeps associated with the deadly dull tick-tock of clocks, activity is meant to be continuing creation, destruction, recreation, and so on – with the chance of something new.


All activity creates knowledge. And since knowledge endures, this means that all activity is stored permanently. It leaves some track in the space of knowledge. In a sense it is this track.


Knowledge is obtained through activity. This activity manifests itself in forming certain structures in the space of knowledge. If the activity occurs regularly and continually these structures solidify. Until they become real matter. Thus physical matter is kind of frozen activity. It is knowledge not aware of any doubt nor any knowing. Usually no question unsettles and inquires this compact body of certainty.


Activity establishes different states or faces of something as well as their relations to each other. So there arises knowledge.

Way 4: Activity springs from knowledge


Activity makes one stable state pass into another. These states are left unchanged and so may be called “elements of knowledge”. They are defined only by their relations to others. Thus, to constitute themselves, to be what they are, they have to interact. So they cause activity.


Knowledge expresses itself in activity. And all the activity is nothing but expression of something that may not yet be known, but nevertheless has to be some knowledge. Activity must have such a cause – for that reason we are looking for knowledge.


Knowledge persists because of its permanent inner dynamism. All the time it is declaring and stating, solving the problem, crossing the bridge. It does not stop answering and so conserves the question it relates to, tending to raise it again and again. But the immanent activity is not completely canalized and captured this way, it cannot be. It pours out, it reaches for more.


Everything that exists is knowledge. Every region of space is full of it – but might be called “empty”, too, because enduring knowledge cannot exist without its opposite, the all-changing all-negating activity. So every point in space may be regarded as vibrating permanently, between “to be or not to be”. But it is a specific being (and not-being). Thus arises a continous directed flow.


Knowledge results from activity, represents it, and causes it again. Since knowledge does not change, the corresponding activity should be always the same. But the actual effect of applied knowledge depends on a lot of other factors – that are applications of knowledge, too.


Everything has a specific radiation; it is the source of a continuous flow of activity. Everything is all the time sending the information that and how it exists. Every receiver of this message gets its own customized copy of the sender – in creating it anew, in multiplying it. In being active.


Normally the source of perfectly regular and automatically occuring activity is not called “knowledge” but “matter”. Knowledge is considered to have some degree of freedom, of openness. So to say, knowledge is interactive. It is the surface of the bodies of knowledge. But in the end even their interior, although scarcely explicitly known, cannot be anything but knowledge.


Activity is what makes knowledge real. Knowledge would not be knowledge if it did not result in activity. It may be written down in a book or stored elsewhere as precisely as possible: only if someone who knows it acts accordingly, that is the right way, it is knowledge.

Way 5: A thing unites


Things are points of intersection where different lines meet. Activity takes place between these nodes. Every thing is a source of activity - and an objective, a result, where activity leads to. Therefore things are the elements of knowledge, entities that rest unchanged while all the activity pulsates between them.


The world is composed of things. Activity separates them, connects them, and so defines them. Every thing has its own specific way of interacting with its environment. These interactions make its unique existence, of them it consists. And only by them it can be recognized.


A thing is one. But we would not understand the meaning of “one”, if there were not any plurality. “Uniting” implies existence of parts. Sometimes we mainly look at these. Then the thing is not at all simple, but of vast complexity. Nevertheless, the point is that we are able to grasp it altogether – in grasping the one thing.


A thing is a repetitive pattern of activities. It is closed in the sense that it binds activity. Once initialized, the activity follows certain rules, coming back again and again. So activity is imbibed and captured. – But turning around all the time, it induces an outpouring stream, too.


The simplicity of a thing may be described in terms of its (inter)actions: then a thing is that what acts as one. This does not necessarily mean that a thing be one single action – although every distinct action is kind of thing – but that it is considered the same in whatever action it occurs, in whatever relation to other things.


Uniting is a central phenomenon, fundamental for cognition as well as for every material existence. But too often it is ignored or mistaken. It seems to be rather mystic: scientific thinking has to be analytic, thus going in the opposite direction. – Yet no analysis would be possible without prior synthesis of simple things.


In a sense uniting is too simple. It needs no work. It just happens. It is like falling to the ground. We do not make it. So it seldom is worth mentioning.


The united simple thing is kind of energy minimum or ground state. It appears without any effort – in the appropriate environment. Though this may be hard to achieve.


A thing incorporates at least two different strands of activity. Each of them connects and keeps apart both sides of a distinction, what may be represented by two separated points or the line between them. If two such differentiations interact to form a unity, they span a plane and define an area therein. But the resulting thing is not flat. It links to all corners like the top of a pyramid, rising above the base area – into a new dimension, constituting a thing of its own, in its own space, with its own interactions.


Uniting is active, dynamic. A thing is not a static isolated unit, it needs to be fed by underlying diversity. It gathers the components of which it is built, it attracts them, concentrates them - and thereby transforms them. A permanent flow of activity is targeted on the thing, thus allowing it to come into appearance.

Way 6: Things multiply


A thing does not exist but interacting with others, with its environment. Every interaction is a new appearing. Only if these are very much alike, if each is a copy of the other, we can speak of one “thing”.


Multiplying produces always the same thing – yet somehow one copy differs from the other. We might say that multiplying constitutes a minimal difference, one that does not destroy sameness. In a sense it is just this elementary difference that makes sameness possible, for at least two are needed to be the same. Thus they must be distinguishable – though not essentially.


Multiplying involves differentiation between distinct instances – and every differentiation implies reproduction, for two different things might always be discribed as different appearances of one and the same.


Reproduction means moving of the thing to a new situation, a new place, a new point or a new moment. Besides this, there have to be some constancy, too, a constant frame or a constant flow, for example. Both are needed: the new item – and continuous progress, progressing continuity. Both advance in their own way: the thing multiplies, while the continuum – space – extends.


The duplication of a thing is almost something like the inversion of the unification that constitutes the thing. Still these processes are distinguished by more than their direction. “Uniting” assembles and merges things of different kinds, whereas “multiplying” produces identical things.


Every process can be described as continual reproduction of something. It is a matter of defining the appropriate thing. Or, accordingly, a matter of defining a suitable space, so that every change will be kind of motion, a displacement along the dimensions of this space.


Every appearance of a thing may be considered to be a thing of its own. Accordingly different things may be understood to be fundamentally the same, to be different appearances of the one thing that unites all these things.


Sometimes it is probably advisable to distinguish between the thing and its appearances. Then the thing may be kind of a prototype, a pattern, an idea. But on more thoroughgoing sights the thing itself gets lots of facets, its complexity grows. Every sight represents a new appearing – and, at the same time, adds something new to the thing. We can expose either the sameness and oness of the appearances of the thing or the differentness of its effects and views. Both should always be found.

Way 7: Things appear in their spaces


Occurring repeatedly, a thing spans a space. This space is coupled with the thing and may be viewed as its emanation – or, the other way round, as the environment suitable for bringing it about. It vibrates with activity that makes the thing appear by disseminating the message of its being.


In a sense, the space is the thing and the thing is the space. Both are different aspects of the same. They perform different functions. While the thing pools, the space spreads and so is rather intangible. Actually it manifests just when the thing appears. But without this appearing – and hence without the space making it possible – the thing would not be anything.


Space is a product of reproduction and the medium wherein it takes place. So it is the medium of differentiation. It is built of the coming together of different appearances of a thing. It is the sum of their places. These places are somewhat distinguished from each other. That is why we can differentiate between distinct appearances. Some kind of space is the reason or the base for every differentiation.


Space is potentiality. It is the appropriate condition for appearances of a particular thing. Sometimes it may be seen as its reason. Or we think of it as representation or embodiment of all the laws that rule the thing's existence and behaviour.


Space is specific: it lets appear nothing but different instances of one and the same thing. This is an extremely narrow restriction. Yet remains a rest of freedom, expressed in differentiating the thing from its space – and between different items, what, after all, leaves room for variations.


As different views of a thing form a surface, the space between may be regarded as the corresponding interior. While the thing holds the appearances together, space holds them apart. Space extends. So it represents kind of moving – exceeding all appearances, stimulating their (re)generation.

Way 8: Spaces are endless


Space has neither beginning nor end, nor does it own any boundaries. You cannot enter it, and you will not find an exit. Nothing can come in or out. There is no inside and no outside.


Since space has no outside, it must be the whole universe. Everything exists here – or it does not exist. That is why we usually accept only one single overall space.


Different appearances of a thing take different places. The space of a thing is the whole of all its possible places. Thus it is an endless continuum of distinct locations – and nothing else. There is no foreign space between these locations, no gap, no internal border. If there is space between two places, it is filled with others, that may be empty, but must be places nevertheless, ready to be taken by proper appearances.


Being the space between different appearances, space is that what these do not be, their negation. But they need it, for without these breaks no distinct items would be possible. And it is specific, formed by the thing – and giving it its form. Thus results a process of continuing reproduction. It is endless, because it does not produce anything else, especially no limits to stop itself.


A thing is the realized potential of its space – while the space is the thing elevated to pure potentiality. Both raise one another and so alternate. This way the thing multiplies – and space extends to infinity.


The space between the faces of a thing, its inside, may sometimes be treated as a closed chamber, cut off from any contact to the outside. So the heart of everything becomes a black box. But this is ultimately just another face. An endless number of them may be added. So the true heart of every thing is luminous space.

Way 9: Spaces penetrate one another


Coexistence of various endless spaces would be impossible without interpenetration. This is the only way to preserve the diversity of things as well as the richness of knowing – even though knowledge persists.


Mutual penetration means that moving in several spaces at the same time is possible. Every movement implies this, yes, movement can even principially be described as transition from one space to another. Whereby neither is really entered or left. Rather it is a change of perspective, of the point of view.


Penetration allows the mixing of spaces from which arises a new one. This may have much in common with some of the others, so that their associated things resemble – yet there will always be something special, making it a unique space with its own kind of thing.


Every combination of spaces is a space of its own, which then represents the sum of all potential places of a certain thing. Then everything appearing there is nothing but a copy of one and the same. So, in the end, all differences reduce to differentiations between locations.


The perfect sameness of multiplied things and the purity of their spaces need complementing penetration. This way different processes of duplication may interact and lead to variations that otherwise would not be possible. So a new thing can unite differing streams of activity. It becomes obvious and its space gets the focus.


Without penetration there would be no unification. Every mixing or superposing of different spaces generates a new space – and so it corresponds to the uniting of different parts into that one thing which this superposition is the space of.


Without penetration there would be no multiplying. Every appearing of a thing takes place in a new environment, it is bound to interactions with other things, which means that the corresponding spaces penetrate and superpose.


Mixing different spaces by means of penetration may result in a completely new space. In this case a new thing is created. But very often the thing remains the same old one and there is nothing generated but a new instance or state of it. Then the space remains the same, too, although slightly varying, allowing internal differentiations – due to superposition of other spaces.

Way 10: All spaces form the space of knowledge


Every superposition of multiple spaces is a space itself – and so must be the union of all spaces. This overall one we usually call the “space of knowledge”, but it comprises all things and all the activity as well.


Since every space is endless and so the one and only space, every space is a – or the – space of knowledge. Thus every space may be seen as superposition of an infinite number of spaces. But then the latter ones seem partial. The knowlege they are representing has to be imperfect, whereas the knowledge of the space of knowledge is taken for perfect.


The space of knowledge can embrace all other spaces because it generalizes. Thus the thing appearing in it must be the most generalized form of all things, their bare skeleton. But usually it wears clothes; it appears in all possible forms. It manifests in every thing. So it does not really seem to be one single thing multiplying itself.


The all-surrounding and all-penetrating space is a space of knowledge because it does not only comprise the material things, but also their mental representations – as well as the act of knowing, its description, analysis, rules, and so on...


The space of knowledge has many dimensions, it allows and embraces all thinkable differentiations. But in the end all boils down to one. Everything that has some extension, that bridges some distance and so holds it, is fixed activity, is knowledge. Thus knowledge gives measure as well as direction. It is the generalization of the concept of “dimension” – and may be called the “forth”.


We distinguish the space of knowledge from every other, partial, space. Yet we know that if we were focused on any other and then investigated it, we would not assume it to be partial. It would be as complete as our space of knowledge, everything appearing there would be the most general thing (in disguise), and so on – but we would not speak and think about it like we do it here, everything would sound different and look different.


The space of knowledge contains the space of three dimensions where bodies do not penetrate one another but displace and rule out. Its nature eliminates all the others, banishes them to nowhere. From where they launch fear and terror. As well as the redeeming promise. Both have to find their places.

Way 11: The space of knowledge is structured


In the space of knowledge we cannot completely isolate a region or a partial space. But always and everywhere we may find a particular structure. The things are distinctively coupled and arranged by activity. These arrangements express knowledge.


When we recognize the structure of a region of the space of knowledge, this structure becomes a thing. This way we can establish a kind of hierarchy: taking appropriate intermediate steps, we summarize more and more regions in fewer and fewer things. This is a possible structure of selected spheres of knowledge. And if we focus on it, we shall find it everywhere and so be apt to take it for absolute.


The space of knowledge is not homogeneous, for it comprises all other spaces. Each of them is represented by a special kind of structure, capable of receiving or producing or forming a corresponding kind of things.


Different structures in space have different effects on the things: there are all sorts and strengths of attraction and repulsion. On the other hand, space is structured by the things. Thus there is mutual impact, continuous interaction between the space and the things.


Knowledge is structure in space, in an extremely general one, the most general, the one that we call “the space of knowledge”. „Structure“ means that there are differences between neighboring regions, that something moving there performs changes. Everything moves like this, following the lines of knowledge – and drawing new ones.


Of course, in speaking of the “space of knowledge” we imply kind of uniform overall structure, in the end homogeneity. So it has to grow beyond its own limits. For this is its only chance to keep on being the structured space of knowledge.

Way 12: The space of knowledge contains itself


The space of knowledge contains everything. It is the whole universe – but far away from being perfect. There will always be something else, something still unknown. This is not a fault, but the nature of knowledge and activity, of the things and the all-embracing and all-permeating space.


Again and again we shall find out that our prior knowledge will have been inadequate. We shall explore and populate new regions of the space of knowledge; and all that we regard now as being the whole will look like nothing but a limited section.


It is essential for every thing's space to contain itself. This ensures self-identity of the thing despite all changings, developments and growings that it may perform. Remaining the same implies a higher or wider or more general sameness (represented by the space), one that integrates all former identities and every potential modification.


Containing itself makes the space of knowledge expand: wider and wider it gets, again and again the former universe becomes a region in itself. But this makes it contract, too, for the partial region is no longer the unlimited space. All in all, the space of knowledge cannot grow.


The space of knowledge contains itself all the time. Thus all the time it is expanding and contracting – not only as a whole, but also in its smallest region. It does not stop oscillating between dispersing space and massing thing. So it is pure activity.


In the space of knowledge even the smallest region is somewhat the whole. Actually there do not exist separate regions. No part is really isolated. Diving into any tiny detail can reveal the whole story. But each opens a different way – and so differ the stories.


The space of knowledge is the most general physical field. All the activity is completely defined by the structure of the space. Every motion and every mutation derive from the actual state, they are locally determinable and computable, thus predictable – although the whole field exerts influence and is influenced in return. Because every infinitesimal small region is a condensed universe extending to infinity.


In the course of our investigations the space of knowledge becomes more and more concrete, a thing that is understood better and better. Still the space of this thing itself has to be called “space of knowledge” ...

Repose: The meaning of being is love


Although unable to leave or lose the own, nothing is ever allone. Everything is reachable. The knowledge that measures and enlightens the entire space is nothing else than the heart of the things, the activity that creates them, that keeps them existing, and takes them back to the whole. It might be called “love”.


In this sense love cannot be demanded or stipulated, but just discovered. It is (omni)present. It is the essence of life and knowledge. Nothing more is to speak about it. But all our speaking expresses the search for it – that we cannot lose. Even our speaking cannot dispel it.


It is love that holds everything. It gives existence and identity. To be loved means to be known, not forgotten. It means to be created by someone's will and perception. To love means to hold on to the dream that the best is real – and so to realize it.


Everything gets its meaning from interactions with others. Nothing could exist alone. Therefore everything's most vital and supreme concern is to keep interacting.


Everything is nothing but light. It is vibrating activity as well as luminous knowledge. Its deepest sense reveals when turning away from obfuscating ossifying separation – back to forever new arising.


To live means to make experiemces. Every experience shows another side of our own body of knowledge. So we find ourselves. To love means to accept this. To open oneself for oneself.



The Notion of Dimension

The term dimension is used in many ways and therefore does not have one distinct meaning. Often it appears in the context of measuring, for example when we speak of the “dimensions of an object”.

Starting from this definition and abstracting from the particular object, the denotation of “dimension” may shift so that it becomes a unit of measurement – or the total range of all possible values of a measurement.

It is also possible to generalize and thus to say that not only measurings, but all sorts of differentiation take place in a specific dimension. Those distinctions than constitute this dimension which, on its side, provides a suitable frame for differentiations of that kind.

Here in this opuzz we speak of the “space” that allows differentiating between discrete instances of a thing. So the notion of dimension links to our notion of space.


The Dimensions of Space

In geometry and physics we know the three dimensions of space. These equal one another as they derive from the same kind of differentiation: they all are dimensions of length or distance or extension.

Extension in one dimension is often represented by a straight line. Then the other dimensions correspond to lines rectangular to the first and to each of them. The right angle defines the vertical that does not slope to any side. The vertical has no extension in any of the dimensions it is perpendicular to. It is independent of them and thus marks a dimension of its own.

This way exactly three dimensions are to distinguish in physical and mathematical space. Usually they are represented by the three axes of a coordinate system. There would be no room for a fourth axis; it could not rise perpendicular to the three others and so be independent of them; every straight in space advances in at least one of the well known three spacial dimensions.

We just do not need any more: by means of the three dimensions of space all static spacial relations can be described precisely. The abstract space of geometry is nothing else than the total range of all the points or localizations definitely determinable by the corresponding three coordinates.


Time Dimension

In mechanics space appears as a framework for motions. Just because it does not move itself, it can serve as an objective benchmark for everything that moves.

But to describe motions sufficiently some additional gauge is needed. The development of mechanics got vitally affected by the growing availability and comparability of time measuring instruments, the clocks, and their influence on our modern understanding of time. This way time could enter computations and theories as a physical quantity. It became a dimension.

Indeed, in the course of this process time approached space and its dimensions more and more. It too became a kind of framework for the “localization” of events. And it became a line where all possible periods are staked stretches. From the linear scale it was not a long way to the coordinate axis. So time stood on an equal footing vis-a-vis the spacial coordinates and the dimensions represented by them.

Finally it resembled them so much that it became one of them...


Realizing Four Dimensions

In physics time occurs in the context of processes. It adds a dynamic component to the static system of three dimensions. So it blazes the trail for an enhanced model that may be called “four-dimensional”. To get an idea of what this could mean let us consider some analogies:

Plane pictures cannot show but flat reflections of space and its solid objects; linear projections of two-dimensional areas can ease computations but never realize their whole reality. So the same must be true for three-dimensional bodies: they cannot be but poor single-sided images of underlying four-dimensional entities and their arrangement in a corresponding kind of space.

But perhaps we should better say “single-spaced” than “single-sided”. For the crucial fact is that envisaging a fourth dimension requires the assumption of an infinite number of three-dimensional spaces.

This may sound strange because we are used to speak of different “states” of the one space rather than of different “spaces” appearing in the course of time; we would hardly know how to distinguish one space from another.

On the other hand it is not unusual to speak of the “planes” that sum up to the supposedly three-dimensional space – even though these are in no way plainly visible or predefined. But we can define them – by means of the dimensions. Because that is exactly what they are for. A dimension is the essence and totality of all differentiations of a certain kind.

Consequently a dimension is treated as consisting of infinitesimal intervals. While following a spacial dimension we seem to pass through one tiny thin disc after the other. And likewise the everywhere present clocks strike us to keep on walking from one moment to the next – which affects not only us, but at the same time everything, the (or one) whole space.


Knowledge, The Fourth Dimension

The preceding contemplations on time gave us an idea of what a fourth dimension might be. Yet that does not mean that time would be the best casting for that role.

The meaning of time should not be forced too much. If it tries to rank among the real dimensions it has to undergo deformations that will take it out of normal service.

Do we really need a fourth dimension? – Well, in a sense this is actually the concern of every kind of science. Yes, every quest for knowledge is about the recognition of global contexts: how this combines with that, how that results from this. It is about processes, but also about relations which, though extremely regular, cannot sufficiently be described as temporal successivenesses. To make a long story short: it is about four-dimensional entities.

Ever since, while looking for insights, we had to assume the existence of a four-dimensional reality. Although until now that what we are seeking has been called “knowledge”.

There is no need to change it. Nevertheless it should be useful to realize that our search takes place in the four-dimensional space. This may encourage us to study its topology and dynamics and so to optimize our search.

By the way, this is exactly what logic is about ever since – and, just now, this opuzz, the exploration of the space of knowledge.


The Question of Reality

The space of knowlegde is the ultimate extension of our common three-dimensional space because it comprises all possible states of all possible objects.

This may seem to be a too great leap. But after all it is necessary. The notion of space asks for generalization to a very high degree. For it is the frame for everything.

Though mostly more specialized spaces are demanded. As, for example, in mechanics, where we are interested in moving objects. Then other processes and relations are neglected – which is pretty advisable whenever we seek for some detailed understanding. Simplification is indispensable for each kind of knowledge. (Therefore it plays a leading role in this opuzz as unification by means of the thing.)

There is yet another constraint seemingly indispensable for us. Particularly in the context of physical sciences we take little interest in imaginary possibilities, but a lot in reality. Only that matters what is really happening; this is what we want to know, to compute, to predict.

Hence time is so fascinating: because it is not only a measurement or a computable variable, but as well and first of all the very moment; it is always actually now. And likely the solid objects of three-dimensional space: they are real because we can handle them and they may touch us physically. We are among them, we are an essential part of space; we are the here and we are the now.

This is near the heart of reality. Between it and our knowledge there will always remain or arise some distance, even in natural sciences. Still we cannot stop insisting that all theory and all research should have some impact on our life. It must bear on our physical existence. As straightly as possible. It must be tangible, it has to become experience.

But the spectrum of our experiences is wider than we will ever be able to realize. In the end, all our questions have something to do with us – and thus with reality.

And naturally our bodies too are ultimately four-dimensional. Which, besides, is one good reason to speak about “us” (instead of “me”) in this opuzz.


Four-Dimensional Bodies

Let us have a look at four-dimensional bodies. Starting from time we saw what extending into the fourth dimension has to be: a series of states. But to constitute a four-dimensional object these states must have some internal coherence, a kind of identity. That is why we say in the opuzz that they are different appearances of one and the same thing.

But, of course, the different states or appearances do not have to be entirely identical – they actually cannot, for they are distinguishable. They even may differ quite a lot. This may reach the point where any internal cohesion remains totally obscured and we see nothing but different objects, not recognizing that they belong together. It is all a matter of view. Or, to keep the context, a matter of knowledge.

Extension in the fourth dimension allows a degree of connectivity and coherence that otherwise would be impossible. Things can come together and form one thing. And one single thing can branch off and reach out into most distant regions. It is the task of knowledge to describe those hidden connections and identities, to remember them, and to make them handy. We want to get hold of them to make use of them. So what we need is a direct link – and we have got it. We have got knowledge where all this uniting and multiplying takes place, it is part of us. Or maybe we should say that we are part of it: we live in the space of knowledge.

Actually, as there are no points without any extension and no lines without any broadness and no disks without any thickness, there are no bodies without any participation in the fourth dimension.


Inside Knowledge

Trying to look into the thick of solid bodies we seldom see very much. Whatever we see is surface.

So, what about four-dimensional bodies? Do they have visible surfaces – as well as hidden kernels? And if so, can they really be knowledge?

In deed they can. Because knowledge does not have to be always known. We can discover it; we can forget it. Though that seems merely subjective. The objective knowledge should never get lost, it has to last forever.

But on the other hand there is also a fundamental dynamism. So that both is true: knowledge persists – and everything changes. This is far from being a paradoxon. It is what can be seen from different points of view. Different perspectives are essential for knowledge.

To bring it to the point: no dimensionless and therefore extensionless things can be found in the space of knowledge. Every tiny little fact or information or whatever must be a four-dimensional body, extending into the dimension of knowledge. It bridges a gap, there are some internal distances, built-in differences. Without differentiation there would be no knowledge. Every fact states different states – as well as the corresponding movings, the transitions.

Thus in knowing we move. And not only us, every thing moves. Between different states representing different views. If there is some thing, there are different views of it. For it is coated with a surface that looks into different directions and is to be seen from these different directions.

But if we want to look right into a thing, if we want to explore its internal structure, its blueprint, we have to change much more. Analyzing means enlarging the surface. In the course of this process the thing may get porous, it may even fall into pieces. It is only because we know the linking relations that we can still speak of the same but “analyzed” thing. This way we can illuminate obscurities.

The other way round, in heaping knowledge over knowledge we obscure a lot of it. We loose it, cannot find it again, we forget. But sometimes this is not so bad as it may first seem. It might be the natural process of creating new simple four-dimensional objects – with solid kernels that we can rely on, without questions, without endless dispersing arguing.


More Or Less Free

The existence of a fourth dimension misses all mysteries if we reconstruct it from the different states of the universe and its objects. Ever since somehow we know that these states are interconnected somewhere. This is the base of all our researches, of our search for knowledge. However, we never really became aware of the conclusions that have to be drawn therefrom.

The problem is that everything threatens to dissolve. We seem to lose the firm ground. But actually this is nonsense. Much like the idea that, if space were spherical and free rotating in space, we inevitably should drop off. This does not happen. Nor do all our dependencies on real life get lost just by discovering and accepting our participation in the infinite possibilities of the fourth dimension.

By and by we may learn to make use of them. And if so, they will enrich our life matchlessly. But first of all, the dimension of knowledge has to be understood and taken seriously – with all its implications. Astonishing may be, for instance, to recognize that in being bigger and reaching further than we ever thought we lose a lot of the freedoms and chances we seemed to have as long as we supposed to live completely isolated. For this gave us the illusion of being free from the consequences of our doings. The price we were paying for that was our reduction to an individuum tending more and more to zero.


Real Limitations

The new dimension changes everything – although it leaves everything like it is. It opens new horizons we never imagined – but has always been there. We have already settled in. The fact that we did have and do have other possibilities does not break the limitations determining our recent existence.

But we can, for example, learn to understand them. If we take account of alternatives. Only then the actual reality comes out clearly. It has to stand out. And then we may change a few things, naturally. Yet, if it turns out that we are unable to live the wider perspective, the doors just shut again. For the new freedom does not fit into the old narrow mind.

The problem is that we are trained to recognize nothing but closed separated things. Everything else that does not trace back to elements of this kind cannot be real. It does not really matter. So we need the narrow view to remain on the firm ground of reality.

But this is not the whole story. It goes on. The ground reaches further than we thought. This is not at all obscure. It is well known. We are talking about knowledge.


Dynamic Impressions

The space of knwowledge is what we finally find when we search for the wider space, the more capacious one, the enhancement of the concept of three-dimensional space. It is in this context that we are calling knowledge the “fourth dimension”. But this does not mean that we would always be able or actually forced to distinguish exactly four dimensions in the space of knowledge. There may be more or fewer – or perhaps none.

The three-dimensional space finds its completion in the space of knowledge because the latter comprises all possible states of all possible three-dimensional bodies. And because it broadens the static concept of space to a dynamic one. So it may be called “space of activity”, too, or maybe “space of events”.

All changing is motion in this space. A motion that does not stop delivering new impressions, reinterpreting existing things, rearranging them in a new manner. Every new impression is full of overflowing and contagious activity and so it is real. It adds another piece to reality; or rebuilds it. Without demaging the old. Because activity penetrates and revives everything. It is light uniting with other light to form new structures, new knowledge, while it does not displace anything.


A Thing's Knowledge Space

We get impressions of a thing in interacting with it. A thing's interactions with others are what determines it. The thing is all that occurs in these interactions. All its different views or appearings make up the thing - and span its space.

Each time a thing appears it shows a new facet. This will rarely reflect a completely new property, but much more often a certain value within the range of an already known one. We may think of this as of localizing the thing on a scale - or in a space representing the property's possible assignments. Like this a thing should have as many spaces as it has properties - but because it comprises all of them, the different spaces sum up to one: the space of all the determinations of the thing, all that can be said of it, all that can be known about it. Every new appearance of the thing marks a new location in this knowledge space. And each of these locations is given by interactions with other things.

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