You know about the problem of distance in mathematics, don’t you? In a strange way it’s probably the most metaphysical question of science, or at least it seems to be extremely metaphysical to me and here is why.
    The mental construct of distance is based on our vision and our sense of continuous time. Our brain or, better to say, the way it ‘perceives’ space-time is playing a trick on us by performing an inverse transform in the imaginary space of possibilities. I mean that what really exist is the possibility for a physical object located roughly near the point A to move to a proverbial point B. It is especially ‘obvious’ when there is ‘nothing’ in between and we call this visible absence of material obstacles, separating A and B - ‘space’. Watch once again what your brain just did: it presumed that if there is ‘nothing’ between A and B then the object that is in A now can move to B. This in its turn is based on presumption that the ‘dimension’ of time exists and is changing continuously.
    Another assumption that has snuck into our observation is the cause-effect relationship between the present state of an object ‘in point A’ as a cause of what can be the ‘same’ object but ‘in point B’. This is the main reason why the brain performs a backward calculation from that visibly possible state to the present state of the object and says that the difference between these two is ‘space’ or even ‘distance’ - a more quantitative feeling that the brain invokes.

Note aside: This is one of the mental traps of graphic depictions of all sorts and kinds. It seems to be ‘obvious’ that by means of some kind of ‘nudging’ one point can be ‘moved’ (continuously) from its present position to another one. Not in the slightest! In most real-life cases there is no way to ‘nudge’ a data point, in this sense all the points between A and B don’t even exist, but they are present on your graph, n.

As to the fact that once it has been moved from A to B the object is not necessarily the same every student of Thermodynamics who saw the P/V diagrams of the ‘Carno Cycle’ or other similar ones knows that the other (hidden, not depicted on the diagram) variables may change dramatically if you change the path or even the direction (hi ‘hysteresis loop’) of movement along the path between these symbolic ‘points’ A and B.

    At the same time we are clearly missing something here, namely the opportunity to create an ‘obvious’ representation usable in our most abstract reasoning, for instance a way to quantify the dissimilarity of meanings or optimality of models… The traditional deliberation about the ‘replaceability’ of different changes in dimensions is too narrow and most importantly is lacking the apparent dependence on the ‘previous history’ of the phenomenon under observation.

The Universal Method.

    Is there a ‘Universal’ method that would be so intuitive that it would apply in any and every case? Of course there is! We’ve just discussed it, it is called “getting from here to there”! The difference in position or, even broader, state can be described constructively as a path from point A to point B built of a sequence of steps. The steps can be large or small or even infinitesimally small or infinitely large, but they are a continuous sequence.

The Universal Dimension.

    And from what we’ve said in the beginning and our human ‘intuition’ we immediately know how to ‘parametrize’ this sequence: by the time spent on going down the path! Notice that we can include anything and everything in the definition of ‘time spent’, all the information gathering and (hard) decision making can be included. That is very useful for our synthesis, it is very ‘intuitive’, probably even ‘obvious’. And it’s a scalar too; we know how to deal with scalars, because their composition is just an arithmetical sum that we know how to calculate.

    Notice that our ‘invention’ covers pretty much everything else: the ‘distance’ measured along the straight line, the ‘Euclidian distance’ defined by the Pythagoras theorem, all sorts of ‘information distances’ (because we have a decision algorithm ‘metrics’ as a factor in the time duration). Looks nice. I like it. Maupertuis would like it too, I guess, he was ‘forward-thinking’, this is a ‘history-thinking’ version of it.

    The ‘scalar’ is not a necessarily useful quality though. It is probably better to assign a direction to this dimension, namely the direction from the present location towards the immediately preceding (past) location. What immediately comes to my mind are: Leibniz’s ‘differential’, tangent bundles-shmundles etc., speed as a hidden dimension (with a direction), speed as a second part of the exact definition of motion for all times (he-he) as mathematicians view it, etc. etc.

A full Cycle, two Half-cycles and linked Complementary Cycles.

    There is a main subtlety here though, because when the object goes down the second and further parts of the composite path it is happening after the first part which might have changed the ‘hidden variables’, intrinsic to the object. And this is exactly the prompt that we were looking for! Let’s form two structures: a full Cycle and two Half-cycles. The full Cycle being an intuitively understandable return to initial position (only ‘position’, in the observed variables, not the ‘state’ in all variables) and the two Half-cycles construct being a situation when the object goes from one point A to a distinct another point B down two disctinctely different paths but the intitial point and the end point are exactly the same in terms of the observed variables. The visual metaphor (and example) of/for this are two half-circles, one travelled clockwise and another - counter-clockwise; each of them ‘can be’ completed and in this way become a full cycle.

What is this ‘return to initial position’ you may ask? Simple! Remember thouse Elie Cartan’s ‘Null vectors’? Same thing. As I understand the first mentioning of ‘spinor-like’ constructs is in the Euclid’s book X of “Elements”. I think I will be calling them ‘cycles’ from now. More about it later.

    We’ve just made a major logical leap here: we passed from exactly factual reasoning of paths to hypotheticals of multiple connected events and equivalences. The construct of two Half-cycles is necessary because in many cases you can not even pretend that you are returning to the initial state, you can only pretend that you can make one more experiment with the ‘same’ initial conditions. Notice how many hypotheticals are involved in this line of reasoning.
    Also, don’t forget that when one thing ‘rotates around’ another the other thing ‘rotates around’ the first, which means that these ‘Complementary’ cycles are an object of investigation. I have a picture in my head of two cirles with a common radus (one center on one end of it, another center on the other) drawn in two perpendicular planes. I will explain it later somewhere.
    I will expand on this (main) subject in full detail later, because it gives us exactly the toolset of techniques for our analysis that we are looking for.

What variables are hidden then?

    By refusing to assign any value to the paths of movements that haven’t happened yet we declare pretty much everything to be a hidden variable. For instance in the simplest case of a mechanical system not only the velocity, acceleration and mass are hidden variables for a particular position of the object, the distances between the objects are hidden too until they start moving and reach each other. In a sense what we are thinking about here is a reincarnation of the signaling with light in the traditional metaphor of the Special Relativity Theory. We are not using any ‘light’ though, we are using the natural transformations of the system itself that are possible and nothing else. By the way, notice how many hypotheticals are in the metaphor of signalling with light! Now we’ve shed some lite on that subject too, along the way.

What observable variables look like from the space of hidden variables?

    I have a feeling that in the same way as with complex variables the real part is imaginary from the point of view of imaginary variables, but we’ll see later.

GitBook

Later.

P.S. http://approximatelycorrect.com/2019/02/17/openai-trains-language-model-mass-hysteria-ensues/

P.P.S. deliberate - late Middle English (as an adjective): from Latin deliberatus, ‘considered carefully’, past participle of deliberare, from de- ‘down’ + librare ‘weigh’ (from libra ‘scales’).