Heads Properly Placed

If hands were feet, 't would be in vain to have the head still host the brain.

Meraïlih

Every normal person intuitively grasps the essential difference of time from the space. We cannot clearly explain the feeling, we just need them to differ, to prevent our life from turning into a sheer mess, with no order at all. It takes a really dogged theoretician, far from the need of earning the daily bread by digging the ditch from here to then, to keep on with the frivolous geometrical play in some perpetual god-made scenery. Still, even the enlightened ones occasionally have to pace something out and consult the watch.

Just here, we say "Hold hard!", suggesting to drop a couple of thoughts to the physical sense. Which, as every (normal) physicist perfectly knows, is determined by the way we measure the physical characteristics; this is exactly what we treat as their (practical) definition. One could proceed with comparing different technologies to each other and spot the possibilities of substituting one for another. For instance, having no nails, or no hammer to drive then in, or just facing a damn-concrete surface, we still have the option of using an assembly adhesive, so that the mirror would keep the proper position anyway. Alright, we happen to drop in a bar without an alcohol meter; nothing disastrous, we still can estimate the overall hardness of the drinks by the percentage of the passed out. Similarly, the physical laws connect certain values to each other, which allows us to judge about something we cannot (or do not want to) directly reach.

Blunt logic concludes, that, to distinguish this from that, we only need to compare the modes of practical treatment... Sheer miss. In many cases things exhibit just no apparent differences. In Ancient Greece, this circumstance gave rise to quite an industry of coiners; fortunately, some Archimedes could dive deep in the essence of things, and managed to flush the dirty tricks off. In the same way, to take temperature for something different from the length of a liquid column (or the volume of a solid body), one needs a real piece of work: first invent analytical mechanics and thermodynamics, spice them up with classical and quantum kinetics, and it is only within that hierarchy of sciences that temperature could be treated as an essential parameter of the theory irreducible to anything else. Of course, no warranty that some smart engineer would not, one day, bring about a gadget from far beyond the nice theories, which would drive us to dumping our comfortable notions in a dusty corner to get more room for a newer conceptualization.

Why, isn’t it what Mr. Einstein and a whole lot of the later promoters of relativism, on all the levels of vulgarity, keep telling us for a century, or so? That is, the intuitively felt distinction of time from space is nothing but spurious, while reality (whatever is meant) would never care, and there is no time, and no space, but solely a four-dimensional geometry with a proper signature. Consequently, things never move (god forbid!), but rather stand (or hangs?) transfixed (by whom?) like a huge piece of abstract art. However hard we try, we can nothing change in this ideal geometrical world; well, and there is no longer any need to... Some people (namely, the money-bags) would certainly find that flavor of science really handy!

To speak of fundamental differences, blunt logic is not enough. We have to engage much stronger abstractions, so to say, abstracting from the very abstractness. Thus, instead of a plain scientific theory, we’ll get a general principle, a guiding idea. Which is most helpful in revealing the true ideological face of anybody, and hence follow the line of Archimedes as to ruling the tricksters out. For instance, the advocates of the bourgeois principle of social equality are never really favor the attempts to compare them with any rabble, cutting short any encroachment on their personal and real estates, access to the pork, or their bank accounts. Similarly, just try to explain a high-ranked physicist that he does not really understand what he does, and, the next moment, the scientific community will spare science your further presence. A professional does not need to talk to dabblers, as the laymen are not permitted in the refined world of self-contained professionalism.

The dictate of geometry is epidemically ubiquitous. Well, it may be a kind of fun, to watch a philosopher who, just to share his intimate vision of the spatiality and temporality with the world, has to waste dozens of tedious pages taking an oath of absolute love for Einstein and indignantly denying any suspect in plotting against the foundations of modern physics. In the modern fashion, physics runs the show, and any philosophy must make friends with physics, consult it for each trifle issue and ask for the royal permission at every turn. This may closely resemble the similar apologies to Aristotle or the Pope, about a thousand years ago.

Still, let us dare to part two classes of measurements that cannot be reduced to each other, so that their common basis is to be sought for in something much more sublime.

Roughly, there are things we can observe together as co-existing within the same activity; however, there also exist "incompatible" things that can only show up one by one, as the appearance of one means the disappearance of another. In the former case, a thing is given immediately and at once, so to say, in the same time, and in the entirety of its "spatial" scope. The latter case assumes a mere sequence of simultaneities, and we just guess about the whole from these snapshots, ideally establishing their commonality within yet another integrity, "time".

Of course, we often express parallelism through sequencing, and the other way round. Does that remove their principal difference? Not at all. It may only change its form, flow from one level to another. Thus, space is readily represented by the time of some "standard" motion. However, you still need to simultaneously consider the starting point and the result of the motion as belonging to the same thing (space); otherwise, there will be nothing to express. Attaching some "unit length" segment to a spatial thing several times, we are to (mentally) retain the whole sequence of applications, counting them. When it comes to the distances much greater than the size of the unit, we run off the count, losing the original motivation, and all we get is the process of measurement as it is, pure time. Conversely, any spatial representation of time by the spatial range of some standard process (like the wavelength stands for the oscillation period) means that we can distinguish the starting point from the destination: that is, mere phase picture, or a standing wave, is not enough; we have to reconstruct the process of propagation, shifting the position of the same phase.

In the popular literature, they like drawing the diagrams of motion, setting up one axis (say, horizontal) for space and another one (vertical) for time. Isn’t it a perfect static vision, in a complete agreement with the idea of geometrical time? Well. Just draw there a circle. Is it a mere geometrical figure or a process? What, in one context, is but a diffraction picture may well come out in another as a working cycle of a thermodynamic machine. The feeble objection that, in a diagram of spatial motion, no curve cannot be closed (since time does not go back) dies at the spot: first, some physical object just like turning time back (e.g. positrons); second, if an unclosed curve is to represent a process, we still need to specify the order of its points, thus mentally adding time to sheer geometry. Moreover, in real life, most different scale may apply to the same. For instance, let a point oscillate with a constant period; the graphically, we get a sine curve along the time axis (say, left-to-right). Now, switch to a scale with unit much greater than the oscillation period. What will you see? A horizontal band. Is that virtual up-and-down motion of a point, or is it a stationary state of a continuum of points (a segment of a line)? Why not take that for a reason to identify the two physical systems that are indistinguishable in this scale (though they may well get apart on a different level)?

Let us come back to the procedures. We measure space comparing one length with another; this implies a (virtual) displacement along the measured thing, as well as counting the number of the steps (the total time). Conversely, to measure time we stick to the same point and run a clock, which would (hopefully) march with the same pace; however, we need to somehow distinguish a "tick" from a "tock", and the only way to do that is to attach spatial markers: a clock hand jumps from one mark on the dial plate to another, the sand runs from the upper compartment to the lower, and so on. The fancy electronics does not really change it, as it only serves to visualize spatial distinctions (take the proof for a home exercise if you wish).

Rescaling time, we risk to run into a resonance, effectively eliminating time, with the clock hand stuck to the same grade. Similarly, spatial measurement can produce the impression of no progress: we apply the ruler time after time without no visible change. This is what we call symmetry. Should that cause any real problem? Just take another ruler and a different clock, and everything will wake up and run on. Unfortunately, life would not always leave us much choice, and even what we have may be far from what we need. No, it is not just mean. There is a very solid reason that we call hierarchy.

The difference of space and time is related to the organization of the world in general; for humans, it is primarily revealed in the forms of human activity. Hierarchy is a way of universally connecting things to each other, so that a push to some element is bound to influence all the other elements, though this influence is never immediate, but rather propagates from one element to another in a definite order, which is called hierarchical structure: the elements of the same level respond "simultaneously", they belong to the same "space". What if we push a different element? Well, a different wave of responses will propagate through the whole, and this new ordering will show a sequence of level of its own. That is, the same hierarchy unfolds into many hierarchical structures (the positions of hierarchy). Obviously, one does not need to resort to brute force; thus, developing a theory of that hierarchy, we can vary the basic notions, with the corresponding change in the structure of the theory.

Of course, this does not exhaust all we know about hierarchies; still, the overall idea is clear enough to bring more light to the nature of (physical) space and time. Here, it is important that a hierarchy cannot be observed but through one of the possible unfoldings. A mathematician would prematurely identify a hierarchy with the set of the possible hierarchical structures; but this is wrong, since a hierarchy can also be treated as many hierarchical systems, and besides, there is a history of the hierarchy’s growth, which adds a specifically hierarchical aspect.

All the positions of a hierarchy are equally valid (since they represent the same in different ways); this is the core of relativity principle. However the equivalence is also relative, referring to a specific level of the hierarchy of hierarchical conversion; in a wider context, the paradigms will mutate.

Human activity looks like a sequence of actions, so that every action is to start with something certain and to produce a specific result, thus spanning a definite volume of the "cultural space" (psychologically represented as a motivation space). The beginning and the end of an action are fixed, with something very diffuse in between, which implements the very process of transition. In respect to its actions, an activity is a higher level of hierarchy: we can indicate the previous and subsequent actions for every particular action, but the whole activity does not imply a beginning, nor end, being an abstraction of succession, pure time. For each action, we have just enough time to construct its inner space; all the rest is beyond it, at "infinity".

However, what serves as a higher level (activity) in respect to an action will also play the role of an action in a more global context. Looking to an action from above, from the level of an envelope action, we can no longer discern the starting point and the end, they merge into a single entity, entirely hiding the details of the transition. Thus an action folds into an operation: a point, an instant. This does not remove the inner complexity, and we can unfold an operation into a full-fledged action whenever needed. An operation is infinitesimal; but it does not equal zero, sheer nothing. The problems of formal theories mainly arise from too literal approach to a point (an instant) as an entirely unstructured abstract indivisibility, utterly impenetrable. Mathematically, this leads to all kinds of singularity.

Now, the distinction of space and time is related to unfolding a specific hierarchical structure; in a different position we’ll get a different picture, which may look like a mixture of space and time : in the new space one find a portion of both former space and former time, while new time gets formally represented with a combination of the old-cut space and time. Why? Just because the formal approach flattens the hierarchy, removing the distinction of its levels in a savage manner, just identifying one with another. For example, consider a conventional picture of 3-dimensional body (tetrahedron) in the plane:

Most people would easily grasp the idea and mentally add the missing dimension, putting the vertex corners at different levels. From the formal viewpoint, all the nodes and links lie in the same plane, and there is no reason to prefer one order to another. Similarly, some "physicists" do not distinguish space from time, since their formulas treat them on the same footing. True physics would never confuse the ways of expression with natural phenomena; to get to a meaningful result we have to somehow introduce physical time, even at the price of neglecting a bit of mathematical "rigor".

Alright, but why the all-embracing geometrization still happens to properly work? Because the positions of a hierarchy, in their turn, form a hierarchy, which may (and must) be unfolded in all the directions. In particular, in some hierarchical structures, the distinction of space from time is of little importance; it goes to the lower levels of hierarchy and we can dismiss it, to a certain extent, within a particular approximation. On the other hand, as long as we employ the traditional means and methods of measurement, there is no need in a more profound theory. Still, if, some day, we manage to implement transition between the reference frames moving in respect to each other faster than light, our models of space and time will have to take a more appropriate form.

Every act of measurement compares one thing with another, mentally (or practically) dividing the whole to portions that we deem to be equal (by definition). The latter condition is not as easy to satisfy in real life: thus, evaluating distances by the number of steps (or any other "physiological" measure like feet, inches or furlongs), we have to admit a certain degree of variability in the unit size during the very process of measuring; still, this minor circumstance may make no practical difference (never influencing our decisions), and we can boldly promise to account for it in the next approximation (if ever). However careful our choice of the units might seem, their constancy is an entirely practical issue, and nobody can guarantee that the "fundamental" constants (including mathematical ones) would never start to drift. Nevertheless, there is the same basic commensurability of a physical parameter and its unit, as any quantity can only sprout from a definite (and hence relatively stable) quality. Can we immediately measure distances with a clock? No, we first need to somehow convert the clock readout into spatial units, or the other way round. There is a traditional trick, multiplying time by a dimensional constant, the speed of light. Insofar, the fundamental nature of that speed (and its constancy across all the reference frames) is to be directly postulated, just on the grounds of that, with the present technologies, there is no other choice. That is, the speed of light is certainly constant because we evaluate velocities in the unit of that very speed, no alternative in stock.

Similarly, one can express time in terms of distance as long as there is a practical procedure converting one into another. That is, the geometrically posed physics will work within the areas of such (at least plausible) feasibility. Exactly the same could be said about the measurement of intensive characteristics, like electric current or temperature: where the available technologies allow associating one feature with another, the physical theory is free to exploit such conceptual links.

How is that reflected in the organization of human activity? Well, if you can split an action into several actions of the same kind, this will produce a quantitative estimate. For example, to march from point 1 to point 4 in three steps, we first go to point 2, then to point 3, and yet another step will fetch us to point 4. In a different scale, we can might need 10 or 1000 steps; this does not change the overall procedure of repeating the same operation time after time, always remaining on the way to the destination point. The possibility of such a sequential transition from the starting point to the end is related to continuity.

On the contrary, when an action gets unfolded into a combination of qualitatively different operations, no quantitative estimate is assumed. Thus, one could directly go to point 3, and get to point 4 in one more step; however, the two operations are incomparable, as we are not sure to reduce the first leap to sequential unit steps (just admit that point 2 is physically inaccessible, or forbidden by some "selection rules"). In some situations, one may need to first jump up to point 5, and then step back to point 4. This is the principle of laser construction. Thus structured activity is called discrete. Note that the "unit" operations do not need to be equal in any physical sense: in the C-major scale (and other musical modes), the intervals between the adjacent notes differ, still referring to the neighboring grades of the scale.

In general, any activity unfolds into a diversity of operations, which are only compatible in being arranged for the same purpose. To build a house, one is to dig a foundation pit, lead in the service lines, lay the foundation, mount the walls and floors, and so on, up to outside and inside finishing. In some cases, we could find a common measure even here: thus, if we are not specifically interested in the house but rather in the overall construction time, any operation will lose its individual quality and become a sheer duration. Similarly, the market transforms qualitatively different articles into abstract values that can be exchanged in any combinations. Recalling that, in K. Marx’ economic theory, the value is nothing but a quantitative expression of the socially necessary production time, we can conjecture that any quantity at all is primarily related to time, while the qualitative distinctions are akin to the spatial organization. In particular, natural numbers sum up a specific activity, enumeration (counting), which puts some objects in a definite order, thus making than sequential time marks. In this context, it is clear why physical "theories of everything" mainly add spatial dimensions (including any inner spaces), while the one-dimensional nature of time is always preserved.

Let us look closer at the construction of the clock. A physically perceptible thing necessarily has a definite quality, and hence it is spatially organized. This means, in particular, that we can compare different things, taking then all together, at once, "now". Thus, whenever mechanics is to characterize the position of a material point by a coordinate x, this implies comparison with some reference point, the origin of the coordinate system. It is important that the two points are intentionally treated as different: if they happened to be the instances of the same point, their distinction would be quantitative, as of the stages of motion, and one would have to seek for some qualitative characteristics to be able to speak about material points as such. That is, the very possibility of geometrical representation (including the construction of the frame of reference) implies instant (at the current level of hierarchy) transition from one point to another, and back. In the common terms, we just look at one thing, then look at another, and then tell how they differ. If so, whence the idea of a speed limit?

To produce an impression of motion the same thing must be taken in different respects. Is it, in real life, that we need to do several things in a time? Every now and then. However, a direct comparison of the different aspects of a thing will just add one more spatial dimension, producing the same static picture. Playing as much notes as you wish together, you make yet another (albeit utterly dissonant) chord, which is normally perceived as a peculiar timbre of a single note rather than a melody, a succession of notes; yes, cool professionals can mentally decompose a chord into melodic movements, but the formation of such an ability takes years of training, tuning one’s perception to a definite musical system. In other words, to get the "time coordinate", we need something very special, which is not comparable to the rest of the system’s parameters and hence cannot be placed in the configuration space. This odd thing (or activity) must be unrelated to the original activity; thus we return once again to the idea of hierarchy.

So, where is it, time? Let us recall the simple fact that a description of a thing does not coincide with that very thing; a name is unlike what it names; a formula merely represents a physical law, having nothing to do with the physics thus meant. Sometimes, one can take a thing’s image in a mirror for a real thing, but we perfectly know that such deceptions are transitory, as any illusion. Just try to pay attention to what you do, and you’ll stop doing it, switching to a quite different activity, known as reflection. Returning to the prototype activity (enriched with the experience of self-observation) requires a special effort.

That is, in the mechanics of a material point, the activities of two different levels meet: first, the determination ("measurement") of the current state of the system (formally represented by the spatial coordinates), and second, repeated distraction to yet another activity, comparing the results of different measurements (and hence a kind of self-reflection); the latter (within its proper level) is much like comparing the points of the configuration space, which may lead to a formal similarity, an illusion of time coordinate.

Obviously, the separation of the levels of physical motion and its reflection is only possible under rather strict conditions. Any measurement in the configuration space is assumed to be practically instant, entirely kept within a single act. On the other hand, the cycle of reflection should not interfere with any inner symmetries, the characteristic times of inner motion; otherwise, we could either miss motion, or find there inappropriate (artefact) structures. In a very simple case, one measurement immediately follows another, so that the system’s state is not likely to significantly change in between. The opposite limit of very scarce measurements will produce an impression of random motion, possibly obeying a definite statistics. When the motion of the system cannot be separated from the acts of measurement, we speak about "quantum interference", or, at least, a combination of quantum and classical traits.

Of course, the anthropomorphic slang is not to be mystically interpreted: the same physics is produced in the interactions of natural things, without any human snooping, so that one thing becomes a mirror for another, or the other way round, depending on the character of interaction. People are certain to intrude with the nature’s affairs; they are bound to adapt things to human needs. However, nobody can impose anything to nature that would not comply with its inner predispositions; we only use what we can get.

Provided the reflection cycle is very short (while remaining much longer than the characteristic time of measurement), another mechanical system of the same type (a material point moving in a space) can serve as a clock. In this case, time labels for the acts of measurements will compare the positions of two points in their corresponding spaces. Thus time takes on even more spatial look, prompting us to boldly draw diagrams with the position of the point of interest on one axis (say, X) and the position of the reference point on the other axis (space Y). In particular, a photon could be taken for the "clock hand", to get the dimension of time dividing the photon’s coordinate by the speed of light. Similarly, in two-dimensional configuration space, we can, under certain conditions, establish a correspondence between the displacements along the two orthogonal axes (or, alternatively, the radial motion and rotation). However, doing that, we compare the motion of two entirely different systems, or the incompatible aspects of motion of the same system. Such pictures (and the corresponding formal "spaces") are called phase diagrams; they are very unlike configuration spaces, where physical motion is to proceed. The geometry of phase spaces has nothing to do with the geometry of the physical space, since it essentially depends, in addition to the methods of determining the state of the system, on its dynamics and the modes of observation. On the same grounds one might take the momentum p of the particle in the point x for a phase coordinate, obtaining a phase diagram of the well-known type as an alternative representation of the system’s motion. The (x, t) and (x, p) diagrams are dynamically interrelated, and they may even look alike. For the motion with a constant speed, we get a straight line on both diagrams, though this line will be parallel to the X axis in the (x, p) space. For a constantly accelerated particle with a constant mass, the (x, p) diagram will exhibit a straight line slanted in respect to the axis X, while the (x, t) space will seem "distorted".

In general, the state of a physical system is a hierarchical structure, the way of unfolding a hierarchy. Any transition to another state (provided the integrity of the system is preserved) assumes folding one structure and unfolding another, that is, hierarchical conversion. Physical motion is mainly an example of such conversion, a cycle of folding and unfolding. That repetition serves as a natural, or "inherent" clock for physical time. Of course, time is hierarchical as well, and one find very different "characteristic times" on the levels of that hierarchy. It is only in certain approximations that we can reduce time to a single number, a time coordinate. To absolutize the odd cases like that is far from the true method of physics, since the artificial character of such a formalization will manifest itself sooner or later, requiring a separate consideration of the effects of different scale.