Rational Dimensions

Basically, the world is quite simple. It is cumbrous specifications that make it complex, as we try to discern ever finer details. Everything would seem equally valuable, and one would struggle to keep on a slightest bit of the whole. This spurious hope leads from one specter to another, so that, in the end, it's becoming hard to believe in anything but apparitions. Science gets bogged down in the speculations of scientific method, art is to incessantly savor a fashionable caprice, philosophy tends to force nature into a single all-embracing scheme. In the moments like that, it is especially important to hold back for a while, to drop a far-away look at the world and our worldly destiny. It often happens that more detail does not automatically imply higher accuracy, and popular explanations may be far from vulgarity. Hundreds of formulas cannot eclipse the effect of an apt figure, while a rigorous theory may never go farther than an extended metaphor. A few naive observations to please a curious amateur may well come as imminent in science as thorough calculations or laborious experimenting. Well, those seeking for provocative originality are free to skip the rest of the reading.

In real life, we are accustomed to move in any direction, combine numerous experiences, discover unexpected faces of the quite ordinary things. On the other hand, to cope with an intricate undertaking, one may need to split it into distinct stages, fix intermediate goals, or run branch projects. In either case, there is a wide field for cooperation, with the whole activity distributed among several (sometimes many) individual participants. Putting themselves to science, people would build it reproducing the very same logic: any research is aimed at the enhancement of either our ability of combining the building blocks or the techniques of disassembling things into potentially reusable parts. No need to explain that the requirements of production always dominate over inquisitive analysis, so that any decomposition would serve the interests of some future construction.

As soon as it comes to connecting separate things or breaking the whole into elemental pieces, there is a question of the possibility of universal methods, or, so to say, master technologies. Philosophy says: yes, they do exist; however, there are no pre-defined patterns, or assembly charts, but rather specialized implementations of the same principles depending on the current instant and need. Such fundamental ideas are known as philosophical categories. Of course categories may be built into various categorical schemes, or represented by a hierarchy of schemes.

The antithesis and mutual penetration of quality and quantity is probably the most famous scheme in the cultural history of humanity. So, let us take it for the starting point of our review of the mathematics of dimensionality. Once again, without comprehensive elaboration, in general, in an intuitive manner.

Admit that we keep producing a socially valuable something. Provided the product satisfies the corresponding cultural need, we find its quality quite acceptable. Any other product that might satisfy the same need (within the same activity) will do as well, being of the same quality. In this sense all such products are equivalent (interchangeable). Still, the very "otherness" assumes distinction; this difference of the qualitatively identical things is called quantitative.

In this context, quality and quantity are interrelated entirely within the embracing activity, in respect to its product. Certainly, similar relations also exist in inanimate or living nature, regardless of human intervention. Still, in a note devoted to formal space construction, we'll stick to the activity prototype, with other object areas to be discussed in a different wording.

Obviously, mere distinctness of equal things is not yet a kind of quantity. Thus, a Chukchi reindeer breeder may know each of his animals by name and aspect, so that, for him, they are all qualitatively different, though one reindeer is as good in a sledge team as any other. In this manner, quality can easily grow into a hierarchy; for humans, this is related to the interdependence of different activities. Similarly, in mathematics, the elements of a set are in no way dependable on each other, they are just different, but any single element belongs to the same set. On the top level of the hierarchy, speaking of sets, we do not pay much attention to how one element differs from another; this is a lower-level quality.

To make the difference quantitative, one needs, beside production and consumption, to bring in some order. That is, all thing satisfying the same need are to sorted out, with establishing a gradation in priority and preference. In the limit case, when one thing is absolutely no better than another, such an ordering may develop from some outer influences: this thing was made before that, or maybe just came to sight first... However arbitrary the enumeration may seem, some practical considerations can always be found. The position of a thing in the row represents its quantity, in the narrow sense. Of course, quantity can also be hierarchically organized: at some level of preference, one finds a group of things, so that additional effort is needed to establish order within the group. From the higher-level viewpoint, such inner quantitative differences will look as negligible, or even infinitesimal corrections. Additionally, the cost of separation of one level of adequacy from another may vary; thence yet another quantitative hierarchy.

In the theory of dimensionality, quality and quantity thus interrelated in the context of a common activity represent space and time respectively; together, they form what we call a space on unit dimensionality (one-dimensional space, or spatial dimension).

To put it plain, the term "space" expresses the possibility of motion, going from one possibility to another "in a natural way", following a once established order. To ensure that our abstract space represent a portion of a real world, we need to order things in a right (objective) way. Space and time are just one of the aspects of motion; they are only definable in respect to a certain class of changes.

Qualities are never comparable as such. Two qualitatively different thing are just different. However, the presence of order makes thing comparable within the same quality. We speak of one as closer, another as farther away... As for equality, it is not as straightforward; basically, it is an indication of an inner hierarchy.

Let us adopt, for a while, the traditional "plain" digitalization of a one-dimensional space: there is the origin of the system (a reference point related, for instance, to the subjectively ideal product of activity), and any other point is characterized with a number generally meaning the time needed for the transition from the origin with some "standard" speed; in physics, following the ancient tradition, the speed of light is taken for the standard, and that is why ("by construction") this speed is the same in any frame of reference.

One can easily guess that there are no negative quantities in nature. Any hierarchy unfolds itself from the top down. Still, comparison of positive numbers provides a hint to the direction that should be taken to come to the destination point from some initial position; this direction we conventionally designate with the sign + or – . In more complex (multidimensional) spaces, the direction will be represented by the phase (angle), or something like that. In general, the transition will follow one of the possible trajectories. In everyday life, this corresponds to the choice of a particular mode of production, unfolding the activity into a hierarchy of actions and operations. For instance, to get a warehouse, we can build the edifice from scratch; alternatively, we might choose to rebuild some church to serve the purpose.

Depending on what and how we enumerate, all kinds of space-time will arise. In the background, there is certain activity, whose organization our mathematics is to reproduce. Discreetness and continuity, finiteness and infinity, no borders or some limitations... These are the examples of qualitatively different quantities. The conversion of quantity into quality is the other side of the same. Indeed, chairs may vary in size; however, a very small chair is no longer a piece of furniture, but rather a toy; similarly, a giant chair is alright as a sample of monumental art, but nobody is going to sit there the common (regular) way.

To be sure, nature does not know any zeros or infinities. Everything is good to certain extent; lack of moderation often leads to regrettable consequences. As long as we merely toss abstractions, there is practically no risk: well, yet another paradox, or an intrinsic contradiction... Just get into real-world production, and the abstractions will need to become workable tools, their robustness to be proved by the typical applications. In many cases, within each quality, there is vast zone of admissible quantitative variations, far from the limits of applicability; this is a kind of operating range where any border effects are of no importance. The commonly known idea of a spatial dimension as the possibility of infinite motion in any direction is an example of such a local structure. Centuries ago, this extensibility was meant by default. Today, everybody heard about quantum mechanics, which is primarily focused on the study of boundaries, admitting any inner (local) motion as virtual, where appropriate. Still, a certain degree of moderation would be welcome, there too.

We have already seen that quantitative and qualitative hierarchies can be unfolded in many ways. On the preliminary (planning) stage, all kinds of weird ideas are allowed, all the visions of the product, and we intentionally loosen the hold of our fantasy in order to grope for the right direction of development. Finally, it is time to start: at a certain moment we decide that the construction is acceptable as it is, and stop any further elaboration. From now on, one can deal with the structures in the science-like manner, leaving aside the philosophical questions. It is only encountering a nonsense that will revive the interest to the foundations.

Admit that we are given two one-dimensional spaces. Is it always possible to paste them together into a two-dimensional something? From the formal viewpoint, nothing prevents us from considering a pair of numbers, each one for the corresponding dimension. Two pounds of pork, and a stone block. Since all aspects of human activity are heavily intertwined, there is always a possibility to discover (or intentionally produce) the situation where such a fusion would not seem a bit surrealistic. Still, even there, we intuitively feel that mere concoction is not enough for the feel of spatiality. Two qualities could only be considered as the different dimensions of the common space if there were a qualitative homogeneity, a kinship: though each dimension represents a separate quality, each of them also represents the same a higher-level quality. In other words, there is an activity that assumes a combination of the two other activities, each of the two being equally necessary for the final product. The mode of aggregation does not much matter. Thus, we can first set up a fence, and then paint it red; alternatively, we can paint the parts of the future fence beforehand, just supplying them with the rigs for assembling the whole in-place; any intermediate options are also possible (for instance, involving additional paintwork on the seams). The product of such a compound activity can be logically represented with as point in a two-dimensional space, while the process of production corresponds to one of the possible trajectories from the initial to final state.

This is exactly what we do in linear algebra: there are basis vectors, and a sequence of their linear combinations arranged in time. After all, the construction of a basis is mainly nothing but the matter of taste, so that the destination point remains reachable with any choice of the axes and scales. Here, once again, we get an example of the numerous modes of unfolding in any hierarchy, which reveals its different positions (hierarchical structures). The integrity and qualitative definiteness of the product is in the core of that mutability. It is the possibility of rearrangement, the symmetries of the whole that distinguish a multidimensional space from a mere list of assorted qualities ("a tuple"). The traditional notation of the linear algebra id to stress that circumstance: we speak of the addition of vectors rather than component-wise processing.

The omission of the requirement of hierarchical integrity would lead to a mathematical object of a different kind, a graded set, a collection: each element of the collection (a special quality) is associated with a number, which could be interpreted as the prominence of the corresponding quality, or the specific weight of the corresponding component. For instance, a traveler's bag may contain two pairs of pants, a few shirts, several pairs of socks, plus books, medicines, and personal hygienic articles. Taken together, all these things, too, constitute a kind of integrity (the baggage); but this does not need to have anything in common with the characteristic integrity of space-time.

Surely, there is no rigid delimitation. Collections may grow into spaces, as we reduce the independent qualities to a single one (just like market economy is only concerned with the exchange value of the product); conversely, the so called phase spaces (with heterogeneous quantities mapped along different axes) essentially resemble weighed sets.

A multidimensional space represents a quality as expressed trough some specific, or "partial" qualities. In this hierarchical structure quantity becomes hierarchical as well: the quantity of the whole is derived from the quantities for individual dimensions. Practically, this means that the components of the whole will no longer be independent; that is, there is a physical constraint on the system's motion favoring a particular class of displacements. Thus, our habitual Euclidean space relates the square of a vector's length to the squared lengths of the vector's projections to the axes of a coordinate system. Similarly, an estimate of the bank capital will depend on currency exchange rates or share quotations. Besides, the quality of multidimensionality is complemented with a peculiar quantity expressed by the number and sequential order of unit dimensions (from a trivial enumeration of the basis vectors to an outline of economic priorities). The both characteristics are related to the local (metrical) and global (topological) properties of the space. In the following, let us focus on the number of units, the dimensionality.

Now, to construct a multidimensional space, one needs to list the unit dimensions in a certain order, and then relate the quantitative parameters of any element of the space with its unit measures, the quantities pertaining to each component space. A symmetrical constraint will induce the corresponding symmetry in the whole space. In particular, the Euclidean metric invariant in respect to any reflections, shifts, and rotations. The dimensionality of the whole space can be quantitatively characterized by the number of dimensions (the size of any acceptable "basis"). It should be stressed once again, that this picture is only valid in a local sense, within the operating range of the space-time model; for very large or very small "distances", the same constraints are no longer applicable, with quantity becoming quality, and the other way round.

Well, supposing that there are no more significant problems in constructing spaces of any positive integer dimension, can we apply this constructive ability to the unit dimensions of a multidimensional space, representing them with multidimensional spaces of a different kind? Why not?

From this angle, it is the right time to observe that, in any two-level structure (like the whole space vs. individual unit spaces), it is not only the top-level quality that becomes hierarchical (and "split" into component qualities), but also the quality of each unit dimension will no longer be the same as if we were only interested in the production of that partial thing as such. The imprint of the whole on its parts and elements tends to distort their original nature, up to radically changing the very their essence. Thus, trying to satisfy the people's need in a specific thing is almost incompatible with market-oriented production, with the same thing to be just monetized; in the latter case, market economy opens much room for half-baked rubbish, all kinds of fake, and even irreparable harm. In mathematics, it rarely comes to the tragic heights; nevertheless, we readily discover an additional level in the quality of each unit dimension, namely, its capacity of playing the role of the dimension number such-and-such in the space of this-or-that dimensionality. In the same manner, the unit quantity becomes hierarchical, with the coordinate value linked to its position in the dimension list. Hierarchical structures of individual dimensions and the hierarchical structure of the whole space are the two complementary representations of the same, the different positions of the hierarchy.

We agree that the relation of the whole space to its unit dimensions is characterized by dimensionality N, the number of components. Conversely, the relation of each component to the whole can logically be expressed with the number 1 / N representing the dimensionality of the corresponding subspace. An isolated space may have integer dimensionality M ; taken as a subspace of a space of N dimensions, it will be characterized by dimensionality M / N . This is how we come to the idea of rational dimensions.

Let us look at it from opposite direction. Admit that there is an N-dimensional space. Every unit dimension (in any basis) taken as a subspace will have dimensionality 1 / N . Now, let each unit dimension is unfolded into a space of dimensionality M . Then its quality related to both the embedding space and the component spaces is characterized by the quantitative dimensionality M / N . However different, all modes of construction will result in the same rational dimensionality. This closely resembles the original geometrical picture: it does not matter whether we divide a line segment of the length M into N parts, or prefer to put together M unit lengths 1 / N . In either approach, representing a point of the original space with a k-dimensional space (where k may well be rational), we get a space of the dimensionality kM / kN ~ M / N .

Naturally, formal constructions will only be intelligible in the context of some activity permitting that particular hierarchy. Traditionally, mathematics does not care for the origin of its objects and the inter-level relations; the dimensionality of a space is deemed to be an inherent feature independent of any other spaces. This works fine as long as we deal with the systems that do not change their behavior when included in the context of a different system. On the contrary, one would have to account for contextual dependence if the inclusion modified the constraints, introducing new interactions. Rational numbers can be reduced to integers while we can unfold new dimensions with appropriate constraints. For real-valued dimensionalities, commensurability would no longer hold, and we'll speak about hierarchical embedding, with the processes of one dimensionality developing in a space with a different number of dimensions.

The numerous "theories of everything" will readily come to mind, with their multiple "folded" dimensions. The resemblance is unlikely to be just accidental. However, nature would hardly ever prefer one dimensionality to another, and all kinds of hierarchical structures might be physically possible. To discover them, we'll need to rearrange our activities, to involve non-integer dimensionalities. As a hint, recall the above two modes of the construction of rational spaces: either inner or outer approach. We cannot immediately see certain effects due to the spatial (and temporal) organization of our bodies; still, there is a trick of making such features an inner property of observable systems. Quantum physics provides many well-known examples of such intertwined hierarchies. Here, it should be quite clear that interference of virtual processes is possible regardless of any quantization, in an entirely classical system, as well non-integer dimensionalities enter the game. This might be considered as an indication of the relative nature of the difference between quantum and classical physics.