Why is it useful to represent, how is formed, where does the tree of mathematical concepts grow from?

The current state of mathematics and physics, who uses it, such is, that so many concepts appear as if from nowhere, simple declaration — this is what we define so and so, we accept such and such axioms. Why, why — understand for yourself. And we will go further and build on these axioms a beautiful theory. As a result, a veil of mysticism penetrates into science, its presentation loses clarity, support in our daily experience. Often gets so clever, that even a person, those who are educated in this particular area have to take many things simply on faith. Common place in this regard are, eg, concepts of quantum mechanics. Its very foundations are presented as something initially contradictory.. Don't try to understand. Take formalism on faith and do not ponder the meaning of symbols, just calculate. It's all, what is available to you. About a hundred years have passed since the beginning of its creation., And nothing has changed. All this has extremely sad consequences for science itself., and for the general public, deprived of any opportunity to form a worldview appropriate to the time. That's why it often happens, that some scientific terms are taken out of context, endowed with a completely inherent meaning to them and then are widely used to the place and, more often, out of place. for instance, word “energy” — what is bioenergy? Or mental?  Or “chakras” — “vortices of energy”? And the concept of a field in such a context — “biofield”? Or the concept of the number of dimensions, which is easily used without any measurements?  The right word, new middle ages. And this is primarily due to the fact, that the scientists themselves, which these concepts use, do not take the trouble to state them clearly and clearly. Most often because, that they themselves do not achieve the required degree of clarity and clarity in their worldview. As they say, clearly understood — clearly stated. And when it is not clear to yourself, then the presentation leaves much to be desired…

Chapter “Thinking out loud” of this site just aims to convey, if possible, to the widest audience that understanding of the meaning, origin, role and place in our knowledge of mathematical constructions, without the use of which the description of the world cannot do,  which I formed in the process of building this description. Maybe this will help someone. It doesn't mean, that I am going to revise the axioms themselves and the concepts they define. Not at all. I'm going to describe those reasons, those visual examples of the real world, which are summarized, turn into ideas, and only then these ideas are formulated as sets of axioms. I emphasize. Real world examples are the foundation, generating idea. But ideas, as such, implemented by such examples, ultimately they are not the same. Moreover, In most cases, especially when it comes to infinities, potential or actual, any implementation looks like real world examples only approaching to the ideal, to that, which would correspond to the formulated idea in its entirety.  Of course, cover the whole variety of mathematical concepts (ideas) not completely possible, it is too big. But touch on the main, critical moments, which are usually presented in an unfortunate light in my opinion, you can try. And this article itself is intended to collect these concepts, or at least some of them into a single system, to the tree of mathematical structures, growing naturally from the needs of describing certain properties of our world and ways of such a description. Wherein, of necessity, I will have to lead two parallel lines of narration. One line — this is a sequence of ideas, collected in mathematical concepts. Ideas are born from the real world, then formulated as part of the description of the properties of one phenomenon or something common to many phenomena. But then they get a life of their own, combining into new ideas or generating others through change (negation of properties, characterizing the idea, adding new ones, etc.) original definition of the idea. Moreover, the self-consistency and consistency of the new idea, generally speaking, optional. Formulate (describe) ideas as absurd as you like. but, when it comes to mathematical concepts, then for them self-consistency and consistency are mandatory. Another line — this is a sequence of examples of the implementation of these ideas by objects and phenomena, that we see in the real world. The presentation here will largely overlap with other articles., dedicated to one or another concept mentioned here. I see no reason to avoid it. Better to repeat several times, than to miss something useful.

A few words about, why do I focus on geometry, not mathematics at all, as a holistic science. The reason is, that for me the whole world of mathematical ideas is easier to comprehend, using a constructive way, where each concept is introduced as a definition of an idea, through the description of its properties. Ultimately it's easy enough to see, that any axioms are such definitions, often veiled. But with an emphasis on pure axiomatics, the integrity of mathematics is lost., her vision breaks down into a set of different, loosely coupled theoretical frameworks. Upon realizing that fact, what axioms are just definitions of words through the description of what is meant by these words, properties attributed to the idea (requirements, restrictions, available operations, etc.. etc.) the big picture is oversimplified, becomes a tree with different branches and common roots. And merging and / or crossing of different branches (dialects of mathematics) it is also much easier to notice. As you can see below, for me all math starts with two basic ideas, whole idea and ideas of its part, continuous and discrete, with a description of their unity and differences. Moreover, it is easy and simple to get the second from the first idea., but the way back, just generates that very tree of mathematical concepts, in question. AND continuous for us naturally associates precisely with the geometry. This is why the link to geometry was included in the title..

IN basis and mathematics, and physics lies the idea of ​​some elementary, inseparable element — point. The only property,   which is associated initially with the idea of ​​the point — this existence. Existence, in no way related to other concepts, such, eg, as time and place. Just some element there is, we think as a single and indivisible whole. And that's it. Time and place as concepts will appear later. Why do you need this concept in mathematics? It `s that, where does set theory begin, and geometry, and number theory, etc.. and so on. — shorter, all mathematics.

Its origins in the real world are all sorts of things.. Precisely things, individual items, parts of the world are easy to visualize element, as a subject of set theory, allow us to understand this idea. But with a visual representation of point, as the main element of geometry more difficult. The best, what can be suggested for clarity, this process, a set of objects smaller and smaller sizes, vanishing in the limitNS. A little later, I will suggest something different as an image of a geometric point., much more natural and, you can say, quite ordinary, habitual. And now I want to focus on that, what a visual image of a point like object leaves something very essential for geometry overboard, what is its essence. Then, what makes it different from other branches of mathematics. This image allows you to link geometry with set theory., consider the concepts of geometry as concepts, describing a set of points, many elements with some specific properties. But these specific properties of the point themselves are not emphasized by such an image., does not make it clear. It's about, what geometry is a description of continuities, the aggregate points, as elements internally related, not disunited. Sets of points as separate and, in the same time, as holistic objects. That's why, the first axiomatic theory in geometry, Euclidean geometry, relies (before formulating the well-known five axioms!) not only on the concept of a point, but also on the concept of a continuous line (straight), and even on the concept of a surface (plane). It means, that there should be more axioms, because these concepts also need to be formalized. Later, this internal connectedness of points in mathematics began to be described by a set of axioms, which fix the properties of the set of points, as a whole (which means, and the elements themselves, its components). And not so small a set. I will not list them. Who is interested, see axiomatics of topological spaces and manifolds. From these, already quite complex concepts, geometry begins. But all these axioms were created precisely for this, to formulate idea of ​​continuous, using terminology, well understood by discrete. And one more idea. Limit ideas — process, endless process, which relies (sometimes) completed. The possibility of reaching such a limit must be taken on faith, how definition, that he exists. And the basis for faith is such examples, like threads, fabrics and other things, which can be divided into parts, remaining examples of the same things. But to divide which into these parts, you need to make some special efforts.. Generally speaking, these are bad examples, because. we already know well, what any such thing after some the final divisions are no longer the same, what we started to share. But imagine, what is something could have taken place at endless continuation we may well. And in this sense, such examples are enough satisfactory for formulating the idea itself. Later I will give a different image of an indissoluble whole, based on our experience. And now I will only note, that the root of all problems is in the understanding of phenomena, which are united by the name “quantum mechanics” is located right here. The idea of ​​the integrity of our world, the inextricable connection between all its parts seems incompatible with the experimentally found limit of the possibility of unlimited division of this integrity into parts.

By and large, to form geometry, as a system of concepts, in addition to a point in the form of a discrete element, component of the set and leading to the concept of a natural number, all that is needed is the concept of a continuous line together with the concept of a real number. In this case, we can already talk about geometry of space one measurements. I will emphasize again. Concept of a line as a continuity on its own, apart from the concept of a point. It is in a sense more fundamental, because. easily allows you to formulate the concept of a point. for instance, divide the line into two pieces and call each one a point. Divide these pieces again, and again we can call the new parts dots. Etc. These will be different implementations of the discrete with the help of the continuous. Of course, this example is limited and allows further talking only about sets of pieces, not about the geometry of the line. On the other hand, it gives an idea of ​​the reverse process, about forming a line from dots. However, this reverse process is not very simple., it requires a description many special properties, required points, so that the result is exactly a continuous line. This complexity just stems from the independence of the concept of continuity, its initial independence from the concept of a separate element, points. Therefore, it is necessary to isolate and describe all the necessary properties of the points, which make them elements of a continuous set. And that, that such properties it should be possible to formulate follows from the ability to define points as part of a whole. These properties are described by a certain set of axioms (definitions of properties of points as elements of sets). As I said above,  axioms create a bridge from discrete to continuous. But from the point of view of physics, all this complex verbal construction can be conveyed in a rather simple way, highlighting the very essence of the idea of ​​continuity. A continuous line is a set pieces linked by the ends (concatenated means that the endpoints of the pieces belong to two pieces at once, segments),  always having some the size. If we imagine, what pieces can you endlessly divide, so size resulting completing this endless process becomes equal to zero strictly, This last piece is what we call a point. Note, that since all fission processes available to us are finite, then when applying these ideas to describe the continuous real world, we necessarily associate a variety of objects with points and, in this way, form approximate descriptions of the real world. But into math, as the language of pure ideas, we put exactly imaginary perfect concept about a point as an element of continuity, no size. Moreover, in mathematics such points, continuous line, are by definition the same. This sameness can be broken, if some points are assigned special, additional labels, not destroying continuity as such. This idea is also quite clear to us. — eg, thread with knots. Such singular points can also be marked with integers.. In this sense, line with special dots implements embedding of integers into real numbers. Besides, if the idea of ​​the point, as a segment with zero size requires a certain voltage, then idea of ​​some special points, indivisible further for one reason or another, but staying always connected, chained in sequence, much easier for our minds. In the end, exactly this idea, the idea of ​​many elementary events, collected in sequence of cause and effect, and became for me the basis of understanding as physics, and mathematicians. Event, which we cannot imagine by any means as the combination of other events is for us the ultimate, indivisible, elementary. We associate such an event with the mathematical concept of a point that does not have dimensions.. And this is the very last approximation available to us. Yes, such an image of a point is by no means suitable for all points of continuity, which we call “our world”. But this is all that is available to us and we have to reckon with it.

Now I want to focus on one of the most important properties of the above examples., which is always overlooked in the presentation of ideas in mathematics. note, for the presence of a word “the size“. This word is not defined in any way, its meaning is assumed to be obvious. So there is something important behind the scenes. Namely, behind the scenes there is a concept “dimension“. And it is no coincidence that the speech has already gone about geometry of space of one dimension. But then this concept also needs to be clearly defined.. However, mathematics does not yet offer any explicit definitions for the measurement., although he uses it with might and main. Actually, as soon as it came to numbers, then behind them always worth measuring. Even if the numbers are natural.

I will not try to come up with an axiom system for the concept “dimension”.  Any sets of axioms can be considered as an enumeration of the properties of the concept being defined. therefore, same way, as for the concept “element”, “point”, i will list properties, united in this concept:

• Measurement — this procedure, act (action set, operations), and not the subject. This procedure is undoubtedly the subject of physics., as an image of the real world. In mathematics, in most cases, it remains outside of its system of concepts., at least today. On the one hand, this is useful, since it allows you to operate with already formulated ideas without caring about their foundations. On the other hand, forgetting the foundations impoverishes mathematics itself, and it complicates its application in science.
• This procedure includes several components., some of which are always present, and the other part may not apply. Moreover, you can add new properties to the previously listed ones., which specify the measurement procedure. In this way this concept also represents a certain hierarchy of concepts, not necessarily sequentially nested (this hierarchy can branch out like a tree at certain stages). Common to the hierarchy are the components, available always :

1. Any measurement procedure starts with unit selection. Examples of: stone, human, any subject (ie. element, point already as a mathematical concept) or object, with some specific properties, eg, “the size”, “size”.
2. Another required element in the measurement procedure — comparison Togo, what is measured with unit of measure, which is also called the scale.

Comparison results are displayed by elements of a special set, sets of numbers. In the simplest case of measurement, comparison is made according to the characteristic that determines the scale — matches / does not match. Result — natural number. 1, 2, 3, 4 stone. 10 items. Just 10 or simply 1 these are just symbols, not numbers. These symbols become numbers then, when, at least implicitly, we consider them to be results counting anything. Score — already measuring procedure. The simplest, Yes. But quite understandable to us, common in the real world. but, mathematics tried to leave the origin of numbers behind the scenes and focused on operations with them. For operations with numbers, it is not essential, what are the numbers of stones or arbitrary objects. The results will be correct for any scales used to form the numbers., if only they were obtained by the same measurement procedure.  And when we then operate with numbers we we tacitly assume it. If you want to know, how many guests will you have in the house, you will count exactly the people who came to you, and not mix in a heap for the account of what is horrible — of people, umbrellas, hats… In this detachment from specifics, “universal” the applicability of the mathematics of numbers, its enormous power is concentrated. But her weakness is concentrated in her,  if they forget about,  that operations with numbers must still take into account their origin.

By the way, we are not inclined to consider the counting procedure as a measurement, not only in science, but in everyday life too. Measurement for us is, first of all, length measurement (width, heights), after, may be, weighing and volume measurement. Well and measuring time — day, night, day, year. The latter stands a little apart, because. comparison with scale is not entirely obvious here. Besides, many other dimensions are now common enough — pressure, mains voltage, sound power and more. Nonetheless, the counting procedure is always the measurement procedure in its most basic form. So such sections of mathematics, like number theory, set theory and others related to them have their roots in this simplest, but still the measurement procedure. but, geometry as a way of describing continuities is generated by more complex measurement procedures.

The most important property of numbers, resulting measurement procedures is their dimension. Dimension is the name of the scale, that unit of measure, as a result of comparison with which these numbers were obtained. I want to emphasize — this fact is completely ignored by modern mathematics, which only works with “naked” numbers. But physics cannot do this in principle.. And many of the problems of today's physics, if not all, generated by this completely unjustified abstractness of mathematics. Below I will return to the problem of the dimensions of mathematical objects in more detail.. Now we will pay a little more attention to the description of the measurement procedures themselves..

Let's dwell on measuring the length. This is one of the main dimensions, to which all the rest can be reduced (except for measuring time and counting the facts of events, e.g. Geiger counter; the last type of measurement is reduced to the basic counting procedure). Really, all other measurements in one way or another are reduced to reading readings from a scale with divisions applied to it. What is this, if not length measurement?

In the simplest version, the measurement of length is also limited by the first two components of the measurement procedure. In this case, only the unit of measurement must have one specific property. — “the size”, “magnitude” or “the length”. It is by this property that all other parts of the world are compared with the unit of measurement itself.. All other properties of parts of the world are ignored. unit of measurement (scale) in this case it is always considered as a piece, part of continuity, a set of two non-coincident but connected points. For geometry, length measurement, or distance between the points became the basis of one of the fundamental ideas, allowing describe continuities as sets of individual points. We are talking about coordinate system idea. This idea is final in the definition of the concept “manifold“, it is the addition of this idea that turns the concept “topological space” into the concept “manifold”. Actually, and the concept of a topological space to a certain extent (through the concept of a neighborhood of a point) already related to the concept of measurement. I do not want to discuss in detail all these axioms.. I can only say, that everything is brought together in them, which distinguishes the continuous from the discrete and isolated these concepts from the concept of the number line. Word “straight” it is not important here, could be said about continuous line, the points of which are assigned real numbers. The standard way of writing math stops here. “Some set are assigned real numbers”. One set or several. How is it done — mathematicians don't care. This is an axiom, accepted without proof and without a constructive description of the way to implement such a correspondence. Clear, that such an approach has a right to exist. Moreover, it is very effective at playing the mind with different ideas. But in order to then benefit from this game, it is required for each of the ideas laid down in the foundation of the construction to find at least an approximate implementation of it in the real world.. AND in the real world this correspondence can be achieved only with a measurement procedure. Even if you just place marks on the line at arbitrary distances from each other and name them “0”, “1”,”2″ etc., you are already implementing a certain measurement procedure. In which the unit of measurement can change from point to point. And the real numbers are nothing else, as a coordinate system. Mathematics section “topology” not interested in markup, but only by maintaining order, continuity itself. Therefore, a direct emphasis on the coordinate system is already made in the following concept, diversity. Usually they say, that the manifold is topological (continuous) space, which behaves like Euclidean space near the point in terms of the presence of a coordinate system. The reliance on Euclidean space is done for generality, to allow an arbitrary number of measurements. And the concept itself can begin to form from one dimension. (not from scratch measurements!), with number line. At the same time, mathematics prefers not to consider the number line itself as a subject of geometry., but highlights these issues in a separate section, number theory. At school, geometry begins to be taught in two dimensions., with plane geometry. The reason is simple — it is impossible to formulate an idea without going into two dimensions straight lines, which is one of the basic concepts of Euclidean geometry. However, in that hierarchy of ideas, which is introduced into the geometry by the description of the properties of the measurement procedures, most of it can be easily seen already for the one-dimensional case, for just a line, as a single set of points. The same case underlies the implementation of geometric ideas with examples from physics, from the real world. Moreover, this is essential for physics.. Therefore, for now, I will focus on discussing the properties of the one-dimensional coordinate system (and measurement procedures, its generators).

• When organizing compliance “line point — real number” using the measurement procedure, four more elements need to be added to its description, four properties , not obligatory for the invoice procedure:

1. Point selection, from which distances will be measured, ie. choice countdown, he points, to which the number will be assigned (distance) “zero”.
2. Selecting the unit property, scale, attributed to the size, magnitude equal to one. For the line, the scale is chosen set of two points, necessarily separable from each other (ie. having disjoint neighborhoods), which are also called a line segment. One point (with zero size) cannot be scale.
3. The scale is assumed to be available, existing at every point. There is already a certain contradiction here. — one side, scale is the distance between two different points, on the other, it is concentrated in a single point. At the stage of diversity as a certain set of ideas, the mathematician (and physics) this contradiction can only be resolved recognition of the strangeness of the scale of the line itself. He is external, an additional element to the line. Its two points, defining unit, are not line elements. Ruler, which we measure something is always separated from that, what do we measure. Such a property of scale can be (and need!) consider as an axiom. but, if we want to determine the scale only from the internal properties of the line (and this seems very natural), then it is intolerable to have a contradiction in the fundamentals. This contradiction can be resolved, but for this you need to expand the set of ideas, move from the concept of diversity to another concept, the concept of space with connectivity and even further, to the concept of a layered space. What are these concepts, we will discuss later.
4. Direction is chosen, in which the points are assigned positive numbers as coordinates (naturally, in this case, the negative direction). This direction turns out to be inherent not only to the points of the line, which are described (digitized, listed, arithmetic). It also becomes the second (after magnitude) scale property, because. and the two scaling points become ordered — beginning and the end, zero and one. In this way, already in the one-dimensional case, scale has two properties — size and direction. Although the direction of scale has been reduced to a ratio of the order of, and the direction in one-dimensional space is reduced to the sign of the resulting coordinate.

To distinguish the concept of a manifold from the concept that follows it in the hierarchy of geometric constructions space, equipped with connectivity (in common jargon, only such geometric sets are called spaces) important, that to each point of the manifold an existing (in a mathematical sense, out of time, but just “there is, there is”) it contains a unit of measurement, scale. And it is by definition equal to 1 (its magnitude) everywhere. AND, by definition, no other comparison is made between scales at different points. They are, and measurement results, obtained with their help, at different points are not connected in any way.

I emphasize, this is a description of ideas. And how these ideas are realized with real examples, which is very important, I will discuss below. It is also useful to talk about the meaning of the word “coordinate”. It comes from the idea of ​​order. “Order” in Latin. Ie. points are assumed to be ordered. “NS” this “general”, “joint” order. Points are marked (described) such symbols (numbers), which allow them to be kept in order for any such description. The points are distinguishable, but are described by their labels not separately from each other, and all together, as a collection, system. This is the meaning of the concept “coordinate system”.

It is also useful to note, that the ideas already described can be considered as root, generating many branches of mathematics, intertwined with geometry, but also independent from it, allowing independent development. It doesn't mean, that these sections only have these roots, in that sense, what their (sections of mathematics) can be grown (although not fully!) and based on the ideas of pure discrete. But the roots that appear in the formulation of geometric concepts are enough for growth, the formulation of all such sections already in their entirety.

for instance, these are mapping theories, transformations of coordinates and functions. Continuous groups and their representations also arise here. Matrix algebra. And much more. How does this happen?

Points can be assigned numbers in different ways.. What are the degrees of freedom in choosing these methods? One obvious degree of freedom is the choice of the origin.. Zero offset along the line. Shift. Any distance. Every number, corresponding to any point in this case changes. Namely, increases or decreases by the amount of shift.

Here 2 display — point to another point and numbers, corresponding to a point to another number, associated with the same point.

There is already and function (constant in the simplest case), the amount of change in the coordinates of a point when switching to a new way of assigning coordinates to them.

Here coordinate transformations, as functions of new coordinates from old, and old coordinates from new.

There are several groups (defined as a set of many elements and one operation, to which you can expose these elements so, that they remain in the same set and satisfying some additional conditions), in which the group operation is addition. Base group, group of measurement procedures, differing in the choice of the origin. This group, as a concept belongs more to physics, not mathematics in its purest form (mathematics tries to keep this concept implicit, out of consideration). However, here I explicitly admit some inconsistency of my jargon with the exact mathematical concept of a group. but, for “groups” there is no suitable measurement procedure in mathematics accurate concepts, because, that mathematics has never really dealt with measurement procedures. So I ask you to forgive me for this jargon. “Group” coordinate systems corresponding to these procedures. Group of transformations from one coordinate to another. But this is already a real group, without quotes. Matrix group (here degenerate into a singular number in the table, due to the limitation of consideration to a single dimension), describing these transformations. But this group can be considered and forgetting about, what are these transformations. Just a group of real numbers (or matrices) by addition. Last property, the ability to abstract from the origin, makes matrix transformation groups (generally continuous) a key tool for group theory and their representations. AND group introduction any of the above groups can be called. It is the connection between all such groups that forms the idea of ​​different representations of the same groups. Etc.

Another extremely important concept is immediately defined here. — concept vector. After all, the scale, existing in the only (any) point, its additional property, this is a very specific object, requiring separate consideration and precise formulation as an idea. Remember, because the scale, assigned to a point already has two properties at once — magnitude (the size) and direction. With an increase in the number of measurements at a given point, (defined as existing) additional scales, differing in direction, and, may be, and size too. Besides, from here lies the path to such a concept, as metrics. Although the origins of this concept are already laid here (remember — the unit of measure for a line is defined as the distance between its two points), its full-fledged formulation is possible only for spaces, in which the connection is defined — relationship between scales at different points. For the simplest reason, that the concept of metric is about two points (even in the case of their endless proximity). We will return to these two directions in the development of the hierarchy of geometric concepts., for now, let's dwell a little more on the concept of a group.

Group idea, which is born from the ability to choose the measurement procedure in different ways is one of the central for physics (and for mathematics too, of course). And she is born here, at the beginning, when creating a description of continuity with labels, allowing and distinguish between its elements, points, and keep them together,  orderliness, links between them. Coordinate systems are such shortcuts (sets, many shortcuts). And measurement procedures are the means, which allows you to create these shortcuts, numbers. And so many concepts, in physics, as describing the world (almost all)  originate from the properties of measurement procedures. But don't forget, that the measurement procedures themselves in physics (and in mathematics too) must have real world examples, and not just be a made up idea. Actually, the idea of ​​a group can be formulated after defining the concept of a set — set with one given operation, does not deduce elements from it and some additional properties of elements (binary operation, existence of elements neutral and inverse to the operation).  But at this level only discrete groups can be considered. Here, the concept of a group is extended to geometry (continuity theory), with its help, it becomes possible to consider the geometric structures themselves, space in terms of those properties, which are covered by the concept of a group. But the opposite is also true., it becomes possible to consider group properties by methods, designed for geometry, function theory, mappings, etc..  In this way, the theory of continuous Lie groups was developed, as spaces of a certain number of dimensions (number of group parameters, which are used as coordinates on the group, viewed as integrity, continuity, ie. as a geometric space). As a matter of fact, when describing group properties, the method of describing group elements using coordinates was again used. And this possibility is based on the understanding that, that the transition from one measurement procedure (recalculation of coordinates) feasible only measuring the same one scale another. Almost all groups belong to continuous groups. “groups” measurement procedures and their presentation, because. almost all parameters, characterizing the measurement procedure (these are parameters, characterizing the position of the origin and describing the scale, ie. their sizes and directions) can, by definition, change continuously. The only exceptions are groups for changing the sign of coordinates, changing the direction of their positive counting to the opposite. These are discrete groups. but, sometimes, under certain conditions, and these groups can be included in continuous ones as a special case. In the sense, that the transition between the elements of the group, having opposite directions can sometimes be carried out with the help of a continuous change of directions, and not just by changing the sign of the numbers.

For fundamental physics, the concept of a group is fundamental, that's why. Measurement procedure groups (further i omit the quotes, primarily because, that almost always we will talk about possible transformations, transitions between measurement procedures, which are exactly the groups) fall naturally into subgroups, which stand out in, that some set of procedure parameters is fixed, and only the remaining ones change (one or more). for instance, the group of shifts of the origin is completely independent, although it is organically included in the complete group of measurement procedures transformations (and coordinate transformations too). This group of transformations is available to us in its entirety., without any restrictions. So can the description we create depend on the choice of the origin and its shifts? Maybe, of course. But! This dependence is of this kind, that we are obliged to consider it as a description that does not change the essence, not changing the identity of the, what we describe. Any differences in the numerical description of a point property (or more difficult part of the world), due to the difference in the choice of the origin, we need to interpret as a single representation of this property (from different points of view).

And now the question. We discussed ideas here. Including, idea of ​​the measurement procedure. Are you sure, that all our ideas are realizable without any restrictions by objects or actions in the real world?  First of all, exactly are all the desired measurement procedures? Should make you sad, if you are sure. Many times I have paid attention to the fact, that the very concept of continuity, in the form we described it (as an idea), includes the idea of ​​actual infinity, the reached limit of the infinite fission process. Yes, the real world gives us tons of examples of proof, that such a limit exists (at least sometimes). But we cannot implement an almost endless procedure., and we can never. This is a natural limitation. For what? At first, on our ability to describe the world accurately and one-to-one (so that each element in the description corresponds to one and only one element of the real world and exhausts the world completely). Secondly, on our ability to implement all those measurement procedures, necessary for such a description of the world, the ideas of which we formulated (as holistic, continuous object). Which means, among the group of all these measurement procedures (coordinate systems, coordinate transformations) naturally stand out those subgroups, which we can implement in full and they,  which we can implement only approximately or not at all. Therefore, that the representations of those subgroups, which we can fully implement, we should consider as a look at some one property of the point of the world from different points of view. But if, to move from one such representation to another, we have to use a transformation that is not available to us, then these two views will look like different objects for us. Many have probably heard, that elementary particles are today described as representations of different groups — Poincaré groups, different unitary groups. This is where this description comes from. No matter what exotic structures we would attribute to the elements (points) the world, the points themselves, as elements of integrity, continuity, we describe using coordinates, which we get, applying measurement procedures to the world. All our measuring instruments, of necessity, tied to massive bodies, existing in time (at least to ourselves). We can change the parameters of these devices. But there are often objective limitations. (and some of these restrictions are always present), we cannot cancel, although our ideas easily go beyond these limitations. therefore any structures we use break down into representations of those transformation groups, which in each situation we describe correspond to the measurement procedures available for this situation. Although from the point of view of broader groups of all conceivable transformations, these different views, different objects can be combined into one representation of a single object.

Examples of restrictions.

  Some of the restrictions are artificial, owes its existence to our choice, convenience reasons. English speaking countries use feet, and the rest of the meters. Because of these preferences, different representations of the descriptions of the world arise. — in feet or meters. But you can measure one by the other and recalculate. This is a transition to a wider group of transformations, to a wider group of measurement procedures. Different representations of the size of bodies or distances can be correlated and understood, that we are talking about the same. The same is the case with measures of weight and other things..

Limitations can also be quite objective., caused by certain circumstances, however, under other circumstances they can be avoided. So, in the case of cartography, all measurement procedures are, if necessary, on two scales, necessarily located in the described surface. And when navigating flights, all three directions in space are practically equal.

There are also absolute limits., through which it is impossible for us to step. The idea of ​​changing the direction of the flow of time (sequences of events, some of which are reasons for others) available to us quite. Just implement such measurement procedures, in which causes change places with effects none of us can. About those restrictions, which lead to the need to describe our world locally by pseudo-Euclidean space can be read in the corresponding article.

Now let's move on to discussing the concept vector, which lies at the heart of a huge branch of mathematics. As well as for the concepts of set and group,  clarification of certain specific properties in the idea of ​​a vector leads to its different applications, placing different accents in these applications, but at the same time the very concept, its basic features become unifying for very different branches of mathematics, let you see their unity. What are the features in the concept of a vector can be defined as basic? The initial premise in the concept of a vector is the absolutization of some properties of one of the basic components of the measurement procedure, scale properties. More precisely, focusing attention only on certain of its properties.

To serve “good” unit of measurement in everyday life, any scale should always be at hand (ie. exist everywhere, where is it needed) and be unchanged, the same, identical with oneself, again everywhere. These properties in everyday life (ie. in the practical implementation of any measurements in the real world) taken for granted. In mathematics, this given forms the idea of ​​such an independent, absolute scale. More precisely, the idea of ​​many such scales, since understanding is also a given, what to choose as a scale, we have the opportunity to many different (but similar in required qualities) objects. With one-dimensional geometry, when only one scale is chosen, its basic properties are not yet fully displayed. This completeness is achieved already in the geometry of two dimensions.. The development of the idea continues with an increase in the number of dimensions., but quantitatively. All quality scale properties, as an absolutized entity (ideas), can be seen well already for two dimensions, two scales used simultaneously. In this sense, two is already a lot, enough for, to formulate the right ideas in a clear way. The number two is not accidental. It is in one-to-one correspondence with two interrelated scale properties — size and direction. Both of these properties are inseparable from the very concept of scale., as an aggregate two related ordered  elements, points, ends of scale. When such scales are used to describe continuity exactly two, both of these properties are fully manifested. Besides, and with an increase in the number of dimensions of the considered spaces, their two-dimensional subspaces and two-dimensional subgroups of transformations turn out to be the most effective means of describing all properties of spaces.

The concept of a vector as such, in its entirety appears then, when the scale becomes explicitly directional, when it ends (and the scale in geometry always has the size, which means that even in the one-dimensional case, this is a line segment and has two ends) become unequal. It would seem, determination of direction, carried out by establishing order among pairs of points does not correspond to the scale itself, but that continuity, which is digitized (acquires tags, coordinates) using this scale. But! Every scale in geometry is a piece of that very continuity (well, or some other, but continuity), coherent integrity. Therefore, adding the concept of direction precisely to the definition of scale is extremely easy and natural for us.. However, not completely. And that's why. The most obvious examples of scales for us are rulers.. And we are used to handling them when measuring by turning them as required.. In this case, the direction looks like a property external to the scale, not inherent in him. I will note however, that even here it implicitly remains. After all, by turning the ruler, we still combine the zero mark on it with one point., and then read the measurement results. So much for the direction, internal to the ruler as a scale. And in the case of measuring time, there is no question at all. Set of events only, rigidly organized in the direction, the sequence from the past to the future is available to us as the basis of our scale time. Direction is natively embedded in the time scale.  And since in reality all our measurements are in geometry, including measuring with rulers, based on time measurement (cm. article on pseudo-euclidean), then the scale in geometry always has two properties — size and direction. And it is he who is the main example of the vector. Not for nothing, when initially studying this concept, they rely on the idea of ​​a directed segment, specially named, radius vector.

Now let's look at the properties of the scales., ways of describing these properties from a slightly different point of view. Let's remember, that the whole set of measurement procedures forms a group. Operation, forming a multitude of all (or parts) possible measurement procedures is the actual measurement, comparison of the scale of one procedure with the scale of another procedure. There are exactly as many comparisons of such for each of the scales., how many scales are included in the measurement procedure. Let be this amount is indicated by a number n. it nothing else, as the number of measurements of that continuity, to describe which these procedures serve. Let's emphasize: beyond the continuity itself, we can measure and the scales assigned to it. Each scale is also described using n numbers, scale component. These components are also sometimes called coordinates. But no longer the coordinates of the points of continuity, and object coordinates (scale), associated with this (and often with each) point of this continuity. When measuring each of the scales of a given measurement procedure with the scales of the same procedure, component kits, describing them turn out to be very simple — these are all zeros, except for the result of measuring the scale by itself. And this result, obviously always equal to one. But when measured with scales from another procedure, the components can be quite arbitrary. Consider the situation, when the scales of one fixed measurement procedure are measured by two other, generally different, scale sets. Two sets of components can be built, generate two sets of coefficients for converting these components into each other. These conversion factors are usually conveniently organized in tabular form., in which the lines have the scale number from the first measurement procedure, and the columns — scale number from the second procedure. Such tables are called matrices. And the sets of scale coordinates themselves are conveniently organized according to the same principle. — as columns or rows, where the component number is the same as the scale number, comparison with which gave its meaning. Columns and rows are also called matrices, like any tables in general, organized like this. Our conversion factors are matrices of one special kind. They always have the same number of rows and columns., if the number of scales in different measurement procedures is the same. From studying (descriptions, transfers) possible properties of matrices, a huge number of branches of mathematics grow, her concepts, which unite different such branches by a common root. One such concept is linear independence concept (or dependencies) between columns and rows. Which has its own representation in the theory of finding solutions to equations and systems of equations, and in transformation theory, and in the theory of the matrices themselves, and in the theory of vectors and vector spaces. AND, of course, in the theory of continuous spaces as such. The very concept of quantity is inextricably linked with this concept., number of dimensions of any space. This concept grows from the analysis of the possibility or impossibility of measuring one set of scales by another. So far, to organize the measurement procedure, a single scale is enough, everything seems to be simple — all tables, connecting different scales, are singular. This number for a group of scales characterizes pairwise ratios of their values. When the group is reduced to a single vector, then it's just its magnitude. Is there a limit on this number? Yes, there is! By its very meaning, it cannot be zero.. With an increase in the number of required scales (n, the number of dimensions of the mathematical space) this constraint is moved to demand (axiom!) linear independence of coordinates exactly n scale, collected in a square matrix, accompanied by the requirement of a mandatory linear dependence of any n+1 scale. And the same requirement for matrices, connecting any pair of such different sets of scales. The wording of this requirement leads to the consideration determinants of coordinate transformation matrices and volumes (in its general sense, including the magnitude of a single vector and the area as a property of a pair of vectors), as necessary, natural properties of the scale set, taken as a whole.

Easy to see, that introduced in this way vector concept rests on the idea of ​​combining several numbers into a single whole (vector coordinates), Well and for two operations with these numbers — componentwise addition and multiplication of all components at once by a number. Plus the idea of ​​linear independence, as a fixer for a specific number of coordinates (component) vector, fixer of the number of vectors necessary and sufficient to describe the given continuity (basis or benchmark in space)  and the fixator of admissible non-uniqueness of such bases. It is these properties that are formulated in the form of axioms in the mathematics of vector spaces (often specified — linear vector spacesin, since only linear operations with vectors are used as a basis — addition and multiplication by a number). Only usually the connection of these axioms with the properties of measurement procedures remains behind the scenes in mathematical courses.. but, you can often find such vector algebra courses (otherwise very good!), in which the presentation begins with the definition of a vector as an expansion in the basis. In this case, the definition of the basis is taken for granted. Moreover, this is the approach that is usually given first., and in a regular school, and in the highest. This poses a colossal problem for mathematics., and for physics.

In itself, such a presentation would not create problems., if as a basis element, separate ort,  considered just a special case of the vector, having a single unit component and all other components equal to zero. Ie. first the given number is determined (actually, number of measurements) vectors of a particular type and then with their help, vectors of a general form are redefined. But the standard is a completely different way, across scalar product any vector with not clear how certain orts, which is called the projection onto the given unit vector (for this direction). For, to figure it out, what is wrong here, need to discuss tWhat are the concepts, as invariant quantities, specific (geometric) magnitudes, conversion factors, linking coordinates, obtained by different measurement procedures (coordinate transformations), conjugate vectors and the resulting dimension relations.

Let's start with the last. I already mentioned, what numbers, obtained as a result of measurements naturally have the dimensions — need to indicate, which unit was used to measure. for instance, the x-coordinate of the point is 1. What does it mean? Distance from point, marked as zero, origin, up to this equals 1. But why? Meters, inches, yardstick, centimeters? Or maybe feet? It doesn't matter for mathematical relationships? Most people think so today. And wrong. So that for mathematical relations (between numbers) it didn't matter, what units of measurement are adopted for arithmetic some integrity, continuity superimposed on these ratios the requirement of their covariance. Ie. only covariant relations have the right to exist and no others. The meaning of this term is very simple. Co-variant ≡ co-converting. Since integrity descriptions are possible in different ways, using different scales, then only such ratios between the numbers will be valid for all possible descriptions, that connect numbers, obtained using the same scales. All other ratios turn out to be meaningless. In physics, these same requirements are formulated as the inadmissibility of mixing in arithmetic operations (addition, subtraction, equality) quantities of different dimensions — meters with kilograms are not added or compared.

We call the transition from one description to another coordinate transformation, which accompanies the replacement (transformation) scale. Consider two line descriptions — using scale e with the value 1 m (coordinate x)  and using the scale e’ with magnitude 1 cm (coordinate x’). The origin is at the same point. Another point will have coordinates in different systems, eg, x=1 [m] and x’=100[cm]. Quite a common and obvious situation. What are the coefficients for converting one coordinate to another? Obviously, there are two such coefficients — from NS To NS’ and vice versa. Namely, these are numbers 100 and 0.01. All coordinates NS must be multiplied by 100, to get coordinates NS’. Question, these numbers have dimensions? Of course. Number 100 has the dimension [cm / m], and the number 0.01 has the dimension [m / cm]. These numbers are ratios of scales, measurement results of the old scale with the new one and vice versa. These numbers do not refer to any old, nor to new line descriptions. They are intermediate, describing not a line, and the scale, used to describe the line. AND, in the same time, these coefficients, which in the most general case can depend on a point on the line (after all or old, or new scales, or both sets together, can in principle change when moving along the line, it is not prohibited by anything, except for reasons of convenience), are additional characteristics of each point in any line description. Especially, if we take into account not two possible descriptions of the line, and all such possible descriptions without exception. Well, or some of them, something more convenient for us. for instance, description group, all scales of which are the same for all points of the line. With every point of the line, in addition to coordinates in a specific system, the entire set of transformations possible at this point into other coordinate systems is associated. I will note, that there is already a path to the concept stratified space. Base — line, first level layers — the scale (vectors) in different coordinate systems, more layers — matrix spaces of admissible transformations between coordinate systems, etc.

Note also, that the coordinates conversion factors are, in addition to the results of measuring one scale by another, also in some way specific quantities. And they are always present conjugate pairs. There are two of them — one says, how many new scales are there on one old, and second, on the contrary, how many old are the new.

It can give the impression, what, since all numbers without exception are obtained only as measurement results, then all numbers, lines appearing in the description (or the continuity of more measurements) will definitely depend on, what scales are chosen to build the description. Will depend on the coordinate system used and will be dimensional, with dimensions, determined by the scales used for the description. This is the wrong impression, and that's why.

Consider the line description with highlighted on it special dots. for instance, thread with knots. These special points can be marked with labels. By different. Including, and integers. Choosing some point beyond the zero, we will assign number one to the neighboring one, and so on. Or we can just count them. And get a certain number of such points (eg, knots) in a given area (piece) lines. Yes, and to these numbers we must assign the dimension. This dimension will have the name “singular point”, or “knot”, or something else similar. But will this dimension be related to the dimension of the coordinate along the line? In general, no. After all, the scales for describing the line, assigning labels to its points, coordinates, have nothing to do with, there are such special points, or not, and when describing the singular points themselves are not used at all.

The number of points in a certain selected area of ​​the line will remain unchanged, whatever coordinate systems we choose on the line. As well as their numbers, if they were attributed to points. Such magnitudes, which do not depend on the choice of the coordinate system (scale) are called invariant (unchanging) or scalars. Scalars in terms of the choice of scales for describing continuity have no dimension. These are dimensionless numbers in the description of continuity. Their dimensions external for description and description are not taken into account, ignored.

Scalar values ​​can be assigned not only to individual points, but also to all points of continuity. Their meanings are then considered as scalar field, defined on the line (or in the continuity of more measurements). Or how scalar coordinate function. This function can be continuous and differentiable (it may not be). For the one-dimensional case, for the line, a derivative of such a function is associated with each point. And for several coordinates, this will be the gradient of the function (in the one-dimensional case, of course, this is also a gradient). What is this design? What is its dimension and how does it transform during transitions between coordinate systems?

The dimension of the gradient of a scalar function is quite obvious — each component will have the inverse dimension of the corresponding scale. After all, this is the limit of the ratio of the function increment to the argument increment. Dimensionless function, the argument has the dimension of scale. The value in its meaning is specific — how much scalar (eg, singular points) per unit of measurement. And gradient transformations are easy to get. They are the same, like the vector, only the conversion factors need to be taken conjugate — not from old coordinates to new ones, and vice versa. In this way, in geometry, there are initially two types of conjugate vectors — those geometric objects, which are the idea of ​​scale, and those geometric objects, which are the idea of ​​their conjugate specific values. Conjugated by product operation, which results in the invariant, scalar. In that part of math, which deals exclusively with vector spaces, this idea of ​​conjugacy is also introduced from the very beginning, using the appropriate set of axioms. Here I tried to describe its genesis, the reasons and necessity for the existence of this concept (pairing) and that, how its examples are implemented, what are its origins in the real world. And in mathematics, the idea of ​​conjugation is very widespread., and not only in the theory of vector spaces. for instance, Fourier transform, forward and backward. Important note — vector conjugation does not require the introduction of a metric or scalar product of vectors. Although very similar to the dot product operation. The point is, that conjugation is a product followed by summation different kinds of vectors, rather than vectors of the same kind.

Let's go back to the two types of vectors. And the similarities and differences between them are concentrated in the law of transformation., recalculation of components during transitions between coordinate systems. The similarity is, that the recalculation is carried out by multiplying one single matrix, linking coordinates in different systems (scale in these systems)  per column or row vector component. And the difference between these vectors is, that in one case, the transition matrix from old coordinates to new ones is used, and in the other — her reverse. And this difference is indicated by adding a special pointer to the word “vector” — contravariant (counter-transforming) for vectors, similar to the scale vector, and covariant (co-transforming) for conjugate vectors, similar to the vector of the gradient of the scalar function. When writing mathematical formulas, instead of names, different positions of the component indices are used. For a contravariant vector, they are written in the upper right — qⁱ, and for covariant bottom right — p_i. At the same time, a different arrangement is also tacitly assumed. (above and below) row and column indices when writing the coordinate transformation matrix and the correspondence of the superscript to the row number of the matrix, and the bottom — column number. In connection with this rule, the matrices of contravariant vectors look like columns, and the matrices of covariant — like strings. Considering covariant vectors as strings, then their transformation is carried out using matrix multiplication on the right, so that the matrix product rules are fulfilled, row per column. I must notice, that not all authors of geometry textbooks adhere to these conventions, which then badly affects the understanding of the subject.

I will also draw your attention to the obvious illogicality of the names. It would seem, co-transforming (together with the scale) should be called those vectors, which were called contravariant. Well, there's nothing you can do about it, names to remake yourself more expensive. I will only explain , why did it happen. The point is exactly, what originally (and even now it is done all the time) vectors were introduced as expansions in basis vectors, and with the help of the dot product. Ie. the components of the vector are defined as projections onto the corresponding unit vectors of the basis (as dot products with each of the unit vectors). This approach automatically makes the vector and the base vector as conjugate vectors, rather than vectors of the same type. Yes, all this is usually done in Euclidean space, in which the difference between covariant and contravariant vectors is leveled by the presence of the metric, allowing you to define the dot product for two vectors of the same type (as covariant, and contravariant). But such a vector space is a very special case. The common vector space allows you to get a scalar only when convolution of two vectors of different types. And that, that in order to define the concept of a vector, you need to already have a basis of vectors and completely leads to a dead end — and the vectors of this basis, what are they? Nonetheless, this has been done and is done often. And after discovering the difference between the two types of vectors, the choice of the name played a role that, which, due to the conjugacy noted above,, the vectors thus defined are transformed using the matrix, the inverse matrix of transformations of the basis unit vectors. So they called them “contravariant”…

Another side effect is the above-mentioned way of introducing vectors.. I have seen many mathematicians have an idea of ​​the tensor in general. (and a vector is the first example of such a multicomponent mathematical object as tensor) how about a scalar, obtained by the convolution of all tensor indices with basis vectors. It's enough to understand, that because, that the basis vectors themselves can be folded with themselves only in a very the specific case of the presence in the vector space of the dot product, ie. concepts complementary to the concept of a vector (what before, vector or scalar product of vectors?), to see all the groundlessness of such an idea of ​​the nature of the tensor (and vectors, in particular).

I want to dwell a little on the above example of a scalar on a line, based on the assignment of numbers to special points, integer. Nothing prevents the use of such, whose scales give these singular points exactly integer values. It turns out, these numbers can be simultaneously considered as a scalar function, and as a coordinate along the line. Here such a fundamental concept for geometry arises as a scalar parameter. This concept is used everywhere in geometry., and, in my opinion, also without the necessary explanations. A scalar parameter is usually used to select a single line in multidimensional space., the way, or family of paths. Of such parameters, naturally, maybe more than one, then we are talking about surfaces (two parameters) and subspaces of more dimensions. It is important to understand, that scalar geometry parameters are also the results of some measurement procedures. But! These procedures are removed from the group of all possible points in space.. Even if the group of possible includes them too (and this, usually, always takes place). This means, that the measurement results, which we use as scalar parameters are fixed when changing the general description, they stand on the sidelines. Cutlets separately, flies separately…

Everything, what was said above, in terms of describing structures, arising in geometry, did not go beyond the concept of diversity, beyond its axioms. Of course, in a number of remarks, further directions of enrichment of geometric concepts were outlined, but only. Now the question —  maybe this set of concepts is already enough to describe wholes, continuities? After all, the goal was just that — create concept construction, allowing to describe all properties of whole objects, which can additionally be considered as a set of separate parts. We need more than just a description, and the description necessary and sufficient, adequate to the world as an integral object.

The answer to this question is as follows. — set of concepts, united by one word “manifold” not enough for our purposes. The reason is, that the measurement results at different points of continuity, no matter what geometric objects they are described, with one single exception — scalar — compare, which means that you cannot add or subtract. The result won't make sense, until a connection is established between the scales at these points. The scalar is an exception because, what its measurement procedure is external to describe continuity, fixed and assuming by definition the identity of its scales at all points.  This assumption is also true for each individual measurement procedure., generating a specific coordinate system (description) for a given continuity. But for the whole set of descriptions, if admissible transformations include such, which correspond to the change in scale from point to point (when viewed from any other coordinate system), there is no structure in the construction of the manifold, which would allow for such a change. You can solve this problem by prohibiting all such, problematic measurement procedures. Actually, in the presentation of the geometry of Euclid, this is done. About all sorts of curvilinear, etc.. coordinates are simply not mentioned. Exceptions are polar coordinates,  which are not at all completely adequate to Euclidean geometry, but this is due to the fact, that Euclidean spaces also actually go far beyond the axiomatics of manifold. Yes, you can do this. By the way, exactly this path was outlined in the Erlangen program of Klein, which drew attention to the fact, that the concepts of geometry are directly related to those constraints, which are imposed on the group of admissible transformations (not only on this, of course). But we can definitely say, that such a description would not be necessary, nor sufficient. The first is because, that there is no reason to ignore any possible measurement procedures and there is no priori (for this it is necessary to indicate, why are they excluded from consideration). And the second is practically because of the same — sufficient means exhaustion of all possibilities. And here not a small set of possibilities is simply thrown out of consideration.. So we cannot avoid defining an additional structure..

This the structure should bind the values ​​of the scales (ie. basis vectors) at least at infinitely close points. Diversity already has a tool, allowing to describe the proximity of points. This is the concept of an infinitesimal displacement vector. So you need to add structure, describing the, what happens to the scale with such a displacement and we will be able to link all the structures, obtained by measurements for different points, even remote (therefore, we are talking about infinitely close points, because. the case of the remote turns out to be special; but on the contrary it will not be so). At least, in relation to addition operations (subtraction) and comparisons. This structure is for geometry, as a set of ideas, is, in fact, another axiom (more precisely, some of them) and it is called affine connection. And the corresponding descriptions of continuities are called spaces with affine connection. Depending on specific properties, required from connectivity, these spaces fall into different classes — with and without torsion, allowing the introduction metric (Riemannian) or not, etc. Among Riemannian spaces, affine spaces occupy a special place, and among them — Euclidean. Connectivity allows you to define many different additional structures, parts of which in the section “Thinking out loud” dedicated articles. You can say, what if connectivity is given, known as a feature set for all integrity, then everything is known about this integrity as continuity.

It is useful to clarify the meaning of the name of this structure., so important for geometry. Connectivity — quite obvious, what are we talking about communication scales at two infinitely close points. Explanation “affine“, what does it mean in Russian “linear” means, that this relationship is postulated with certain conditions. The first of these conditions, which is marked with the word “affine” He speaks, what describes that part of the change in scale, which is proportional to the infinitesimal displacement, ie. depends on it in the first degree. This limitation in reality is insignificant., because, that the coefficients, describing this dependence as a function of the point themselves may well be nonlinear functions of the point. This means that there is no real limitation on the relationship between scales in this sense.. The second connection condition is actually more restrictive, although attention is not focused on it at all. Affine connectivity, as it is defined, keeps track of relative changes in scale, ie. point dependence of the ratio of change in scale to scale itself. There are good reasons for this approach.. However, in terms of constructing ideas,  both of these conditions can in principle be modified somehow. What will come of it — the question is completely different. You can just do it. Physically, it is affine connectivity that is generated by the real world in terms of describing it using measurements.

so, if the connectivity is known, then everything is known about space. In geometry, in addition to the tensor algebra at each point, tensor analysis appears, allowing to consider changes in tensors (measurement results) when going from point to point, describe areas, parts of space as a whole, etc.. but, the development of the mathematical concept of integrity does not end there. I have already noted, that among all imaginable measurement procedures, by no means everything can be implemented by us. There are many reasons, but the main thing is, that we cannot even realize the process of potential infinity, not to mention the actual. What does it say? About that, what our claim to construct a completely adequate description of at least part of the world as a whole is excessive. We do not have the means for this (the ability to implement the ideal measurement procedures required). But I still want to describe the world as accurately as possible. This is where the following construction in the geometry hierarchy can help — layered space. In this concept, it is possible to unite more than one separate space of affine connection, and an infinite number of them, somewhat different from each other, but coinciding in descriptions that are critical for us. What are these critical traits? Pure mathematics can consider as such any available or introduce (redefine) new. But for physics, connecting these ideas to the real world (using these ideas to describe it) this is serious, critical question. Linking (construction of description) carried out on the basis of and through experiments, using that set of facts, which are already available or can be guaranteed to be obtained using measurements available to us and their consistent interpretation. You can find a description of this problem from the point of view of physics in my books. And here I will only outline a general idea of ​​a layered space..

By and large, the space taken separately with an affine connection itself is an example of a fibered space. That's in what sense. Scales in mathematics are objects, alien to the described (arithmetic) their continuity. For physics, this situation is unacceptable., physics needs internal scaling, in their obligatory belonging to the described real world. Just because, that we are part of this world and cannot be outside of it. Nonetheless, scale as an idea, as the ultimate transition from a piece of the world to an ideal structure, tied to one single point in the world, we completely understand. And not as an idea of ​​the only vector, and as the idea of ​​the space of all such possible vectors. This space is usually called tangent space. Remembering the existence of vectors, conjugate scales, we can talk about the second natural vector space, associated with each point of continuity. He's called cotangent space. These two vector  space, available at each point of continuity, but not clearly belonging to her, and give examples of layers over this continuity. Affine connectivity is that tool, which allows you to operate with these specific, vector layers, even if they are assigned to different points of the described continuity. If you think about it, it will become clear, that in this sense the fields of any quantities over continuity can be regarded as layers. In particular, and matrix spaces, containing all possible admissible transformations. And the connected field too. If desired, you can hang as many and such layers., as soon as you want. Here's just physics, and math too, should try to be limited only to natural structures, justified for their introduction. Nonetheless, from the most general reasons it is clear, that the quantities, forming such layers in the most general case do not have to transform during coordinate transformations also, like tensors or related tensor densities. And I would like to work with them. This problem is solved by introducing a connection already in a stratified space individually for each layer.. The basis for this introduction is the mechanism, tested on the special case of vector layers, ie. affine connection. Only the reliance is not on the vector transformation law, and for that group of transformations, which works in the studied layer. In this case, the description of such a group itself is used in an explicit form as a certain parameterized space. I will not go into further details., for a general idea of ​​the subject, what has already been said, in my opinion, is enough.

In conclusion about, why our physical description of the world as space-time requires the use of stratified space. And what a layer, in addition to natural tensor, the resulting layers, need to be taken into consideration. The essence of the matter is precisely, that we do not and cannot have the scales we need to describe the world at each point, because we ourselves belong to this world, describe it from the inside, not outside. Our experiments, no matter how much we make them, will not give us information about all points of the world at once. Information, available to us, in a sense is a discrete component of all possible information, which could be obtained with an ideal description of the world as a whole with the help of the mathematical concepts developed by us. The way out of this situation is, to temper ambition and try to take into account all possible descriptions of the world, highlighting those, which contain all the experimental data we have accumulated (well, or at least some of them). This is where the layer comes in., which can be called a state space. States of scale, that one, which we can realistically implement, time scale. These states can be described by the matrix, and quite a certain kind. So we're talking about the state space, how about the space of such matrices, representing a well-defined group. To determine states at other points of continuity (the world) based on known conditions in some, introduced (naturally generated) connectivity in this layer, which is also described by matrices of the same kind, by the same group. That's all.. Such, maybe you know the terms of physics, as gauge fields it is just about the connectivity in this layer. And the wave function, spinors, etc.. — it's about the representatives of the state in the layer itself.

Today's mathematics is a huge collection of very different concepts., most often systematized only partially, in its special area. Cross-links between such areas are rarely described.. Clear, that I could only touch on a fairly small fraction of all these concepts, only those who were of interest in my own work. And even then it is very superficial. But, In my opinion, better at least something, than nothing at all. Maybe this article will help someone in his work or just to satisfy interest. Finally, I want to draw your attention to the importance of extremely careful use of words., terms. For each such term you need to see a well-defined meaning., otherwise misery…

 

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