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.

]]>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.

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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.

]]>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.

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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.

]]>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.

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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 o*puzz* 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 o*puzz*.

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 o*puzz*, the exploration of the space of knowledge.

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.

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...

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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.

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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 o*puzz* 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.