This article combines the results we obtained in previous articles and brings the theoretical reasoning made in them to a practical level. I introduced enough terms to look at the concept of a property and explain how to build a property model. This article can be read independently of others, therefore I will repeat a part of the reasoning made earlier, a part I will skip, and I will add a part.

Introduction

Those mathematicians or physicists who are starting to study business analysis have a hard time. There is a huge difference between basic science and those practices that are set out in different standards for business analysis. Attempts are being made periodically to familiarize the business analyst community with the point of view of contemporary philosophers, but such attempts have so far failed. Because of this, a mathematician or physicist, plunging into the study of business analysis standards, is experiencing a light shock. I will try to bridge the gap between what the physicist or mathematician used to work with and the models that analysts are building.

For this, I formulated a body of knowledge, which I called projection modeling, because the method outlined in it resembles plotting. In drawing lessons, we learn to model spaces. At the same time, the space model is separated from the interpretation of this space. Depending on the problem being solved, the simulated space can be interpreted both as a piece of aluminum, and as part of the water, and as an airplane wing: the space model does not depend on its interpretation. In projection modeling we do the same: first we create a model of space, but already in time, because our world is four-dimensional, if we consider time as a separate dimension, and then we treat this space-time in one way or another. Just as in drawing, the simulated 3-D volume can be interpreted in different ways, and in projection modeling, the interpretation of 4-D volume is separated from the space-time model.

For example, one subject may treat 4-D volume as a car, another subject may interpret this same 4-D volume as a piece of iron, the other as a function of transporting passengers. The only difference from drawing is that the model of space and time is more complicated than the model of space. Therefore, modeling tools must also be more difficult. As a result, the model of our ideas turns into a two-level model:

1. At the first level, a model of the space-time parts and the relations between them is built
2. At the second level, a model of subjective atomic representations and the relations between them is built.

Why do you need to simulate space-time?

Case 1

Suppose that two different people were asked to tell about one event. One said: a hammer hit the nail head, another said: a nail hit the hammer. They told about the same event, but from different points of view.

However, what is an event if you can look at it from different sides? Hammer hit the nail head - is this an event? No, because this is a story about an event from one point of view, or, more simply, an interpretation of an event. If this is not an event, but an interpretation of an event, then what then is an event?

Case 2

Suppose that two different people were asked to describe the same object. One said: this is a car, another said: this is a boat. They told about the same object, but from different points of view.

However, what is an object if you can look at it from different sides? Is the machine an object? No, because this is a story about an object from one point of view, or, more simply, an interpretation of an object. If this is not an object, but an interpretation of an object, then what then is an object?

Explanation

Both of these cases are united by one thing: the impossibility to express a thought correctly in words Hammer hit the cap - this event. A machine is an object. And this is hard to argue. But what, then, do the subjects interpret as an event and what exactly do the subjects interpret as an object? What exactly do they look at when they make their interpretations? They perceive the same space-time volume, and in this they agree. In the movie, actors often ask the question: are we seeing the same thing? This question means: are we now looking at the same space and interpreting it in the same way?

It is correct to say that there are two different interpretations of the space-time volume. This explanation is correct and accurate. If we do not understand this, our reasoning will be like a snake biting its tail. That is why, if we want to build a model of interpretations, we must begin it with a model of what we see - with a model of space and time, and only then endow this model with different interpretations.

Examples of two-level models

Example 1

Pick up a ball of aluminum. You see a rough surface, you feel the weight and see the shape of the ball. To create a model of such a view, you need:

1. Build a model of space, which can then be interpreted as a matte surface
2. Give the interpretation of this space as a matte surface.
3. Build a model of space, which can then be interpreted as a piece of aluminum
4. Give the interpretation of this space as a piece of aluminum
5. Build a space model, which can then be interpreted as a ball shape
6. Give the interpretation of this space as a form of a ball
7. Specify the relationship between the three spaces, treated as a rough surface, as a piece of aluminum and as a form of a ball. I would suggest this:
   
   1. The space treated as a rough surface is the boundary of the surface treated as a piece of aluminum.
   2. The space treated as a sphere is the idealized boundary of a surface treated as a piece of aluminum.
8. Specify the relationship between the three interpretations of three different spaces. I would suggest such:
   
   1. A piece of aluminum has a surface, an idealized representation of which looks like a sphere
   2. A piece of aluminum has a rough surface.
   3. A piece of aluminum has a weight

Example 2

You look at the stage and see a dancer dancing a dance. To create a model of such a view, you need:

1. Build a model of space-time, which can then be interpreted as a dancer
2. Give the interpretation of this space-time as a dancer
3. Build a model of space-time, which can then be interpreted as a dance
4. Give the interpretation of this space-time as a dance
5. Identify the relationship between two space-times, interpreted as a dancer and as a dance. I would suggest this:
   
   1. The space-time interpreted as a dancer coincides with the space-time interpreted as a dance.
6. Specify the relationship between two interpretations of two different space-times. I would suggest this:
   
   1. Dancer dancing dance

The relationship between the space-time model and its interpretation

The space-time model depends on how it will be interpreted later. Such a model is created for its specific interpretation, or, in other words, for a specific type of property. Two different types of properties will produce different models of space-time volumes, even when it may seem that these volumes are the same. For example, the sphere in the case of a piece of aluminum is an idealization of the real form and differs from its real surface. Therefore, building a model of space based on the statement that the piece has the shape of a ball, we obtain a surface that is different from the real surface of the piece.

The main thing is not to confuse the property and the type of properties. For example, a white car and a white steamer are different properties, different "white" ones. For one property there will be one model of space-time, for another - another. Combines their property type "white". As a rule, we are not able to distinguish a property from a property type. This is one of the problems of the language: the language does not allow us to do this. But in projection modeling, this differentiation of meanings must be realized very clearly by the analyst. Do not confuse the property and its type. This means that the white steamer and the white car will have in common not the properties, as we used to think, but the type of properties. Properties will be different. This means that one white is not at all the same as another white! These whites are different shades, shapes, position in space and time.

Therefore, the space-time model that we will build will be related to the type of properties that generated it. We call this type of properties generic for both space-time and its model.

The generic type of space-time properties is such a type of properties, on the basis of which the given space-time was selected from the total space-time volume.

The generic type of properties of a space-time model is such a type of properties, on the basis of which a given space-time was selected from the total space-time volume and its model was built.

Property model

We came to the conclusion that any type of properties can be generic for a space-time volume and its model. The property model is the space-time volume model, for which the type of the modeled property acted as a generic one. Therefore, if there is a white property, the type of this white property acts as a generic for some space-time volume and its model.

Name of space-time volumes

To refer to spatial and temporal volumes, we use the name of the generic property. And, since all properties are grouped into property types, the name of the type of generic properties becomes the name of the volume. For example, the type of generic properties "white" becomes the name for the "white" property, which we associated with the steamer, and for the other property "white", which we associated with the machine. These are different properties and therefore must have different names, for example, "white # 123", or "white # 124". The analogy with machines: the machine with the number # 123 and the machine with the number # 234 are different space-time parts, which we treat in the same way as machines. Similarly, "white # 123" and "white # 124" are different whites, which we treat in the same way as white. The same applies to the property "length of 10 meters." This is not a property, but a property type. The full name of the property should be: "length 10 meters # 123".

Representation of space and time

To build a model of space-time, you first need to figure out what space-time is. Let us briefly repeat the theses of previous articles and formulate them in a formal way. At the same time, I apologize for the mistakes I made. This is especially true of the spikes, which led me to the indistinguishability of the property and the type of property.

Usually, a story about space and time begins with a story about space, and then they say that time is a change in that space. Why is space considered out of time? Because we can easily imagine a frozen space in time: this is a slice of the space-time volume across time and the consideration of this slice. But we do not understand what is frozen time in space. If we want to cut across the space in order to study time, we must select a point, line, or surface and consider its dynamics over time. Logically, if the cut across time is called space, then the cut across space should be called time. Agree unusual?

Both space and its changes are different points of view on the same section of the studied space-time, but according to the rules of the language, the changes should be tied to the space, but the space to changes should not be. We cannot tell about changes without space, but about space without changes, supposedly, we can. In fact, we always look at changes in space, even when we think that nothing changes. Just sometimes we believe that these changes can be neglected. In order not to be confused, I will speak about space, having in view of its changes, which were insignificant in the framework of the task we are solving.

We introduce the term "dance of space", or simply "dance" as a synonym for space-time. If I say simply "space", I will mean the dance of space, the changes in which are minor.

In any dance chosen for modeling, there is a minimum spatial resolution (atomic point), a maximum spatial volume (volume of the studied space), a minimum temporal resolution (atomic instant), and a maximum time interval (the amount of time studied).

What dance can be given meaning?

Suppose that you have lost the ability to see part of the space. This can be imagined, because each of us has a blind spot. You can become aware of it through special exercises, but then you adapt again and cease to realize it. This is because our consciousness is able to smooth the visible image. Our consciousness does not work with a picture, but with a spline - functions that smooth it out. The same thing with time. If you show the 25th frame, you will not notice it. Therefore, we make the following statement:

The property can be given only the dance that is continuous, or, equivalently, homogeneous.

The question arises: how to build a model of a uniform dance, so that later it can be given meaning, or interpreted?

First we need to formally define the concept of continuity for dance. The first thing that comes to mind is to remember the definition of continuity from mathematical analysis: continuity is when the attribute values ​​differ slightly at two close points. Everything seems logical and beautiful, but the question arises, what is the point?

For example, you are holding a crystal. What is a point on its surface? You can say that a dot is an atom. But, if you argue about the color of the crystal, the atom does not possess color. Color has a surface of a huge number of atoms. For example, for this they must be a million. This means that the point on the surface of the crystal, which has color, contains a million atoms. From this it follows that the points may intersect, because the neighboring points may have common atoms. It turns out that the definition of mathematical analysis does not suit us.

Formal definition of homogeneous space

Take a homogeneous space endowed with a property. Divide it into parts. The properties of each of the parts of this space will be similar to the property of the entire homogeneous space (any part of the crystal surface is similar to the entire surface). Whatever part of a homogeneous space we take, the properties of this part are similar to the properties of another part of this space and to the properties of the entire space as a whole. This will become the basis for the formal definition of a homogeneous space.

The homogeneous space for a given type of property is the set of all possible parts of the space, for each of which a property of a given type is defined.

The model of a homogeneous space looks quite impressive: for this you need to consider all possible parts of it, and there may be a lot of them.

If we consider parts of a homogeneous space, directing their size to zero, at some point we will come to the limit beyond which the obtained parts can no longer be endowed with generic properties. This means that there is a limit to the partitioning of space. This limit determines the size of the points of homogeneity of the space endowed with the property. The resolution of the instruments may allow us to see the structure of the points of uniformity. The same point of homogeneity can not be seen, because its borders intersect with the boundaries of other points of homogeneity. It can only be imagined.

If the size of the point of homogeneity is less than the size of the atomic point of the studied space, we observe an absolutely smooth space. If the size of the point of homogeneity is greater than the size of the atomic point of the studied space, we see it as a rough space.

Let me explain by example. Consider the surface of the carpet. He is fleecy, we see every flee. Any part of the carpet is also fleecy and looks like any other part of it. We will reduce the size of the parts. At some point in one part of the carpet there will be only one pile. Is it possible to call such a part fleecy? No, because one pile does not have the property of fleecy. Therefore, the size of the point of homogeneity for a carpet is greater than the size of the atomic point of the studied space and therefore the surface of the carpet looks rough.

An example of a homogeneous space can be found in the drawing. The shaded area in the drawing models a homogeneous space that can be treated as a substance. It is modeled using all the possible parts that can be obtained from this space. There are incredibly many such parts, they intersect, the size of the point of homogeneity is comparable to the size of a group of a billion atoms.

Knowing the generic type of space properties, one can introduce the concept of continuity: for closely spaced points of homogeneity, the attribute values ​​should be similarly close.

Those who are familiar with functional analysis can see that this definition of homogeneity can be interpreted differently: by means of Fourier expansion. Then the definition of homogeneity will be:

A space that is homogeneous for a given type of property is a space that has a pronounced burst (or sets of bursts) in the spatial spectral decomposition of properties of a given type.

Depending on convenience, you can use either one or another definition of spatial homogeneity.

For example, we talked about an absolutely smooth space. To determine it, we can know nothing about the points of homogeneity. It is enough to know that the peak in the spectrum of properties of a given type on a given space corresponds to the size of the atomic point of the studied space. Peak blur gives us an idea of ​​the spread of property values. If the peak in the spectrum corresponds to a size larger than the size of the atomic point, we can see several bursts at different frequencies, proportional to the base frequency. This will give us an idea of ​​the second and third harmonics, and if we combine this knowledge with knowledge of the phases of spectral decomposition, we can conclude about the shape of the periodic structure. If there are several peaks at frequencies that are very different from each other, for example, dozens of times, this tells us that there are periodic structures, which, in turn, consist of periodic structures. That is, the stone wall consists of blocks, each of which consists of bricks. Today there are plenty of ways to analyze such spatial structures. As a result, our entire understanding of space can move into the area of ​​spectral analysis, thereby fundamentally changing our understanding of reality. Imagine the AI, which will be flashed the idea of ​​space, based on spectral analysis. What will he see? Only waves, no space in our understanding! Interestingly, in real projects and done? By the way, can such an interpretation help us understand the essence of quantum physics?

There is a feature in our reasoning. As soon as we started talking about decomposition into a spectrum, we began to write about the mathematical apparatus of functional analysis. But the trick is that we do not have the right to do this, because the points of homogeneity have completely different properties than the points in the functional analysis. Therefore, I gave a high-quality picture without a claim for a mathematical justification.

The idea of ​​a homogeneous dance

In order to imagine a homogeneous dance, for each atomic instant of the dance under study, we assign the studied space to correspond. A stack of such spaces for each atomic instant of the dance under study will give us a dance model. This way of presenting is similar to stills from the movie.

Why dance is considered homogeneous in time

So that we can talk about the homogeneity of dance in time, the space for each pair of consecutive atomic moments should be similar to each other. If spaces from instant to instant rapidly change their size, shape or position, we cannot speak of such a series of spaces as a uniform dance.

The following statement seems obvious: if for each pair of consecutive atomic moments we can perform a comparison of two spaces, then we will assume that the dance is uniform in time. But in fact, we are now appealing to the atomic moment. We have defined the absolutely smooth dance of space. But the dance may have rough edges. How to formulate the homogeneity of a rough dance without appealing to the moment?

The uniformity of dance in time

Consider the rotation of a ballerina. It consists of movements of the same type: one turn, the second turn, and so on. If we observe this rotation for a long time, we see a uniform dance. If homogeneity in space was determined by the concept of space properties, then the concept of dance homogeneity should be based on the concept of dance properties.

We formulate the sign of temporal homogeneity. If we observe some kind of uniform dance of space, we can divide this dance into temporary pieces (make cuts across time), and see that they resemble the entire dance as a whole. This similarity of parts makes the dance uniform in time. The set of temporal parts of a homogeneous dance, similar to each other, is a model of a homogeneous dance.

A homogeneous dance of space in time for a given type of property is the set of all possible temporary parts of a dance, for each of which a property of a given type is defined.

The star for us is a bright point, it is not uniform in space. But her light is uniform in time. Uniformity in time allows us to see this point and assign it some meaning. If we encounter something that does not have homogeneity either in space or in time, we simply will not notice it.

The model of a dance that is homogeneous in time looks quite impressive: for this you need to consider all of its possible temporary pieces, and there may be a lot of them.

If we reduce the temporal parts of a dance that is uniform in time, at some point we will come to the limit beyond which the parts obtained can no longer be endowed with properties similar to the property of a uniform dance. This means that there is a limit to such a partition. This limit determines the duration of instants of homogeneity of a dance endowed with a property. The resolution of the device may allow us to see the structure of the moments of homogeneity. The instant of homogeneity itself cannot be seen, because its borders intersect with the boundaries of other instants of homogeneity.

If the duration of the instant of homogeneity is less than the atomic instant of the dance studied, we observe an absolutely smooth dance. If the duration of the instant of homogeneity is more than the atomic instant of the dance under study, we see it as a grungy dance.

Let me explain by example. Imagine a wave of sea with standing waves and try to describe the dance of its surface. If someone does not believe that this is possible, I can assure you that such a dance exists: it is formed due to the interference of the waves reflected from different piers. This dance can be called fermenting. Any temporary part of this dance is also worrying. We will reduce the duration of observations. At some point, only one period of oscillation of the wave will fit into one temporary part. Is it possible to call such a part worried? No, because one period of oscillation does not have the property of a wave surface. Therefore, the duration of the instant of homogeneity for the waves of the sea is longer than the duration of the atomic instant of the dance under study, and therefore the dance of the waves looks rough.

An example of a homogeneous dance in time can be found in music. The note on the stave simulates a time-uniform dance of a certain duration, which can be interpreted as the sound of a note. This dance is modeled using all possible lengths that can be obtained from this sound. There are many such parts, they intersect, the size of the instant of homogeneity is comparable to fractions of a second and contains up to twenty sound vibrations.

Knowing the generic type of properties of a dance homogeneous in time, one can introduce the concept of continuity of this property: for closely spaced instants of homogeneity, the values ​​of the attribute must also be close.

Those familiar with mathematics can see that this definition of dance homogeneity can be interpreted differently: with the help of Fourier expansion. Then the definition of homogeneity will be:

A homogeneous dance of space in time for a given type of property is a dance that has a pronounced splash (or multitude of bursts) in the temporal spectral decomposition of a property of a given type in time.

Depending on convenience, you can use either one or another definition of temporal homogeneity.

Uniform idea of ​​a homogeneous dance

We saw dance as a stack of spaces arranged in time. The time has come to abandon this idea and learn to see dance as a single semantic block.

Let us return to the rolling surface of the sea. The dance with which we model this surface in time is homogeneous in time and in space. Let us try to formulate it without breaking the presentation into temporal and spatial.

We can divide the surface of the waves of the sea into parts both along time and across.

The division along gives us the result of observing some part of the sea. Imagine a helicopter hovering above sea level at some altitude. He has a sector of the review, which snatches under him some area for observation. This is an observation of what can be called a part of the dance of space, carved along time. We put a lot of helicopters at different heights. The results of observations - there are parts of the dance, cut along the time.

The division across time looks like the surface of the whole sea, observed by a single helicopter, the observation time of which is limited by the helicopter's fuel supply. We put a lot of helicopters. They will fly away and fly back. The results of the observations are parts of the dance cut across time.

Now combine these two divisions. Let's do the dance division both along time and across. This means that helicopters now begin to observe when they want, and finish when they want, have the opportunity to rise and fall above the surface, changing the field of view. But at the same time, the rule is observed: the minimum observation time lasts longer than the instant of dance uniformity, the minimum viewing sector is greater than the homogeneity point. The results of such observations will give us many pieces that are similar to each other.

We formulate the concept of a homogeneous space-time

Homogeneous spacetime

Homogeneous space-time for a given type of property is the set of all possible temporal and spatial parts of this space-time, for each of which a property of a given type is defined.

We can interpret this homogeneous space-time in different ways. For example, as a wave of the sea.

Separately, it must be remembered that in such structures it is necessary to learn how to correctly measure properties other than generic ones. Only those whose duration is longer than the instant of uniformity can be performed, and the sensitivity is greater than the point of uniformity. No measurement with a duration of less than a moment of homogeneity in a given space makes sense. This means that it is impossible to measure the height of the wave, because its time of existence is much less than the time of homogeneity. No measurement of an area smaller than the size of spatial uniformity also makes sense. This means that you cannot measure the wavelength. But we can measure the thickness of space and understand that it is non-zero: it is as thick as the height of the waves. This boundary belongs both to the ocean and the atmosphere! Each thesis strongly beats on intuition, we see the waves! But everything should be formal. If you see waves and want to describe them, use other minimal spatial and temporal homogeneities, and you will see waves.

Fourier dance analysis

Fourier analysis of dance can be performed not only on spatial or temporal homogeneity separately, but also combining them together. Such an analysis will show us the structure of the production line.

Homogeneous dance anisotropy

We talked about the points of homogeneity of the dance. But these points of uniformity may not be just cubes. They may have elongated shapes. Take velveteen as an example. Its point of uniformity has a pronounced elongated shape. Or the light of a star, for which homogeneity in space is not defined at all. But the space-time anisotropy is much more interesting. It allows us to represent the business function as a set of scenarios of the same type!

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The impact of additional information on the interpretation of homogeneity

If we look at the dance, the limits of the homogeneity of which go beyond the limits of our observation window, we may have discrepancies in the interpretation of what we have seen. For example, waves in the ocean - is it homogeneity in time, or homogeneity in space? I will explain the difference. Suppose we have a string. Suppose that it has the form of a sinusoid. The question is: does it have such a form, or are we in the frame of reference associated with the traveling wave? If we do not have additional data, we cannot say which of the answers is correct. But, if we know the boundary conditions - the boundaries where the string is fixed, or we know the structure of the string and see that we are moving relative to this structure (relative to the string substance), then we can say for sure: whether it is homogeneity in space, or in time. If the string has the shape of a sinusoid, and this sinusoid is fixed relative to the edges of the string or its substance, then we have spatial homogeneity. If the string runs, then we have temporary homogeneity. In some cases, we cannot say for sure what it is: spatial, or temporal homogeneity. And only experience can give us the right answer. And then the waves in the ocean will be temporary homogeneity, and the waves in the picture - spatial.

Two properties of any regular dance

Since a regular dance in space or time can be divided into parts in two ways, we can interpret them in two ways. If we endow parts of a space with properties of the same type as the entire space as a whole, we are talking about one property. If the elements of this dance have properties different from those of the dance, we have a different property. In other words, if I say that the flow of oil consists of flows of oil, it will be one property, if I say that every part of the flow of oil consists of molecules, it will be a different property of the same flow. Often it is the second type of properties - the composition becomes generic to build a uniform dance!

Property classification

Any dance that is uniform in space and (or) time can simulate a property. And vice versa: any property requires a model in the form of a uniform in space and (or) dance time.

At first I wanted to classify all the properties. This was supposed to be a rather interesting story, but so far I have no time for a full account of this issue. I will only give one example that allows you to understand how the representations constructed above help to conduct business analysis.

Business function

The function definition is based on the flow definition. But when the spatial observation area is small, we cannot say what became the basis for the flow formation: spatial homogeneity, or temporal? For example, when a stream of parts flows past, we cannot say for sure: it flows through a space filled with details, or parts are created before entering the perception window and are destroyed upon leaving it. Maybe so and so. Therefore, speaking of flows in the definition of a business function, we can proceed from both the temporal uniformity of the dance: the uniformity of the events that occur (customers regularly come in) and the spatial homogeneity of the dance: the regularity of the flowing flow (the oil flow). Both methods fit the definition of a function; however, temporary homogeneity is often forgotten, meaning only spatial.

When a business function is endowed with many regular flows in time, they forget to say that each flow has a different structure and different moments of homogeneity. By definition, this leads to different models of homogeneous dances. Homogeneous dances, corresponding to different streams, occupy the same volume of explored space, but have completely different structure. These are different dances! Combining them together should be accompanied by a model of relations between them and their interpretations. I have not seen anything like this and this greatly impoverishes the model of enterprise activity. The problem occurs due to the violation of the condition of uniform uniformity for all flows. For example, the instant of homogeneity for multiple streams should certainly be greater than the longest instant of homogeneity among all streams. I see part of how this condition is grossly violated in those or other models built by business analysts who do not feel this restriction. Now you don't have to feel it, you just need to know it.

Thanks

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