Why does the site about unified field theory raise this issue?? What is the current generally accepted view of the role, the place and organization of mathematics does not suit me? AND, finally, what place should be given to mathematics, her concepts, constructions and methods in our picture of the world?

First the answer to the first question. One of the goals books “Dimension and Properties of Space-Time”, explicitly stated in the Introduction, was finding out those reasons, which make us involve certain mathematical constructions to describe the real world.  The result was the formulation of a belief system, which can also be called the unified field theory. Naturally, in the language of mathematics, some of its sections. By and large, this is set theory, algebra (complex numbers), geometry, theory of groups and functions (as regular, and generalized).  In the book, for all used mathematical concepts, justifications for the need for their application are given.. These rationales are often very brief., because. implied, that the reader already owns the material, at least, in the volume of the university course. Here I want to give at least the most general outline of those considerations, which explain the naturalness of all these mathematical constructions for describing the real world.

Second question. In the process of creating this frame of reference, I had to very carefully revise the definitions, properties and origins of all these mathematical concepts. In a number of cases it turned out, that they are introduced in this way, that their application in physics becomes a little natural, accents usually given by mathematicians obscure their physical meaning. This situation is quite understandable. After all, mathematics took shape a long time ago as an independent, self-sufficient science.. Its development very often came from internal formal needs or simply from the love of art.. The connection of mathematical concepts with the real world is now extremely weakened even in those cases, when she seems to be self-evident. And for the concepts that grew on this tree far from the roots, the situation is completely depressing. Nevertheless, I managed to restore these connections. (or install, since in some cases no mention of such relationships in the literature could be found). In the site section “Thinking out loud” most of the articles are devoted precisely to the presentation of my point of view on the origin and properties (which means, for physical meaning) a number of mathematical concepts. In this article I want to describe, at least briefly, how a tree of mathematical concepts grows from our experience in the real world, of course not everything entirely, but only the roots of this tree. I would also like to draw attention to those moments, presentation of which in textbooks or monographs, formally correct, turns out to be a clear brake for those, who wants to apply such concepts to describe the real world. And if they are not used for this, what are they for??

How did I start building my belief system?? Strange as it may seem, had to start from the end. From the formulation of the goal of physics as a science: “The task of physics as a science is to create the most complete and accurate (adequate) real world image“. And if you think about it, there is nothing strange about it. To reach the goal, you need to be aware of it, this technique is applicable to any area of ​​human activity. Can, of course, and just go somewhere. You will come somewhere too. AND, may be, even to something useful. Or maybe not. This method is much more widespread in the activities of mankind.. Including in scientific activity. But this is not my method.

It is not enough to formulate a goal. Questions arise about the meaning of all words in the wording. Form. What is it? Complete and accurate. And what's that? 

Form. The photo is good? Or a picture? Or a movie? In general, yes. But only as part of a whole, illustration, and nothing more. The image of the world must be formulated in words. What's the use, that I or someone else was able to create it in my head. Until it became available to everyone, his, it could be considered, simply no. Consequently, we are talking about language, common to all or part of people (well, or other reasonable).

And the words “complete and accurate” represent the requirements for this language, or, it is better, to that part of the language, which can and should be used to describe this image. Because language as such does not have to be complete, accurate and consistent. But for that part of him, which claims to have a one-to-one correspondence with the real world as a whole or some of its parts, this requirement is completely justified. No Consistency, completeness and accuracy — there is no desired image of the world, no theory. But there is also a problem here.. The thing is, what could be, what we are not we will able to meet these requirements in their entirety. And it really is. The completeness and accuracy of the description of the real world available to us has limitations.. So you need to abandon the declared goal? Not at all. You just have to be honest — this is the degree of completeness and accuracy we can achieve, and more — not, for such and such reasons.

The part of the language we need, trying to meet these requirements (let a long time and not formulated) mankind has given a special name “science“. And the name of the most advanced part of science on this path — mathematics.

Today's mathematics has practically forgotten its origins and tries to deal only with ideological component of words, which denote its concepts. but,  in physics, such a use of mathematical concepts can lead to, usually, leads to very serious problems.. Up to the complete loss of connection of some fashionable theories with reality. And for mathematics itself, this approach is by no means harmless.. Along the way, I will try to illustrate some of these points.. With the help of any language, it is possible to formulate both true and false statements.. And it's not always obvious (like the first; if it wasn't so, then there would be no need to prove theorems). Especially, if we are talking only about the ideological component of the language. If an incorrect statement of the general form does not cause special worries — well wrong, so what? Then wrong definition, accepted without proof, on the basis of which some construction is further built, should not appear in the language, claiming to be a complete and accurate description of the real measure. The only criterion, which i can suggest, to avoid such definitions, this the need to give for each definition an example of its implementation in the real world. And a strict description of those situations, when such implementation is possible, and when not. In the same time, may well be (and has been repeatedly), that when some definitions were first introduced in mathematics, it was difficult to find such examples, or they simply did not even try to look for them. And later, the constructions built on these definitions proved their usefulness when applied to the description of some phenomena in the real world.. And here you need to understand, that just this success is the very realization, for the need of which I advocate. therefore “free” in this sense, creativity in mathematics is quite acceptable. Perhaps, that ideas that are worthless today will someday come in handy, and find their justification by such application. But in physics or any other science, claiming to describe real phenomena, all mathematical constructions must have clearly described examples of implementation. Actually, this possibility makes mathematics a separate science., self-sufficient, not purely utilitarian, secondary to other branches of science. Internal development for application prospects. If the direction of development liked, felt right, then this branch of mathematics will succeed.

The division of the world into distinct parts and the designation of these parts with words is the basis of all languages ​​and any attempts to describe this world.. Just because, that this division is a property of the real world itself. It has rocks and sand, water and trees, different animals and different explorers of this world. This is where the mathematical concept comes from. “lots of“. Although it was formalized only in the 19th century. What was done in the process of formalization? The basic concept of “element“, part of any set. This concept is reduced to a single property of any such element: property “exist”, “to be”, “be available”. All the rest — never mind. I'd like to note, what is a mathematical concept “Existence” significantly different from the physical concept of existence. Here's the difference. The physical concept of existence in most cases means existence in time, existence has a duration. Mathematical “Existence” has nothing to do with duration., but is related only to the fact of the presence or absence of something. The duration of this presence is already the next property, at this stage of formalization, not taken into account.

Here it is useful to digress, dedicated to the method of formalization, creating definitions, which, in my opinion, should be used in the construction and presentation of mathematics as a science. There has been and still is a lot of controversy on this subject., there are philosophical and mathematical directions for substantiating mathematics, which are often grouped under such names as logicism, intuitionism, constructivism, formalism, axiomatic and set-theoretic directions. Maybe there are others, I don’t want to pretend to be a complete classification. Everyone knows about Euclid's axioms from school. But these axioms, in turn are tied to such intuitive concepts as the dot, straight, plane, right angle and, may be, other. It's already bad, that it is necessary to start with a noticeable number of intuitive concepts. And even worse, when, eg, the concept of a point is defined as something, having no length, no width, no height. Even before, how these concepts (length, width and height) appear in geometry. In my opinion, much better to start with a single (and the most basic) intuitive concept — properties of anything “exist” or “not to exist”. And construct new concepts, adding one new explicitly declared property to this property, by explicitly defining this property. Well, or at least, adding a sufficiently small number of such properties, if you can't limit yourself to one. In the end, All mathematicians do this, giving definitions and formulating axioms. Only after all, the axioms without loss of generality are the same definitions of properties, like any other. And it makes little sense to single them out in a separate category.. It is this path that I will follow when introducing certain mathematical concepts..

so, at the heart of the whole concept “element“, its only property — Existence. I will use one more word as the name of this concept, as a synonym. This word “point“. There is no need to look for examples of the implementation of this concept in the real world.. Table, stone, animal, tree, etc.. and so on. These are all good examples..

Next concept — “set of elements” or “set of points“. The added property can be called the word “number” distinguishable elements in a set or “cardinality of the set“. It is by this single property that sets themselves differ from each other as sets. “amount” this is the second property, distinguishing this concept from the basic concept element, point. Any set also has the very basic property of existence or non-existence. And no other properties yet. Examples sets, existing in the real world anyone can bring many. sorry for the pun. So this is a formalization of some property of the real world, ideas already in it regardless of, do we describe the world or not.

At this stage, it becomes possible to define another set of mathematical concepts, different from and from the concept of a point, and of concepts multitude. These concepts are combined with the words “operation“, “act“. The basis for their introduction is also purely intuitive., but, naturally, based on direct experience, drawn from the real world. Operations are carried out with sets (and with their elements too, but every single element is a set, consisting of it). Operations are different and are determined by the end result, after that, how sets change after operations are applied to them. for instance, union operation (additions) sets. Or selection operation one set within another, allocation inclusions. And another operation, seizing it (subtraction). Implementation examples: there are a lot of stones; you add stones to it or remove it; or allocate other heaps in the common heap. I am not going to build and describe here all the necessary mathematical concepts., I will limit myself to only a few of them.. that part, which is useful for illustrating the connections that exist in mathematics between its seemingly unrelated branches, between the branches and roots of this huge tree. And this part of the concepts is vital for me as part of the language of physical theory. Their precise formulations will greatly facilitate the understanding of my system of views on the unified field theory.. It is important, because. often behind the same words someone can see a different meaning.

At the same level, one can already define other concepts that are very important for mathematics.. And for all other sciences too, because the mathematics with its consistent development on the basis of the properties of the real world becomes universal, the most appropriate language of any science; or, at least, part of such a language, his spine. It's about the concept “display” and a very close concept “function“. Both of these concepts link elements of two or more arbitrary sets., put in correspondence with each or part of the elements of one (or more) of a set one or more elements of another set. To some extent all operations on sets can be considered mappings or functions, just different properties, but traditionally in mathematics a distinction is made between operations and functions. Implementation examples: Do you have several animals on the farm?; you tie a ribbon around the neck of each animal; some, especially capricious, also a bell on this ribbon. Animals — lots of; ribbons — lots of; bells — lots of. And they're connected to each other.

The concept of a function makes it easy to introduce ways to distinguish elements in the same set, Associating with each of them some label. Obviously, that the labels themselves will be nothing more than elements of some set. Or elements of a set of sets. Here lies a direct path to the concept a natural number. This concept has already appeared, because. the second property after the existence, we introduced the quantity, with which the concept of number is closely related. But this concept has not yet been formalized properly., not all of its properties are described. And not described because, that the concept of number — comprehensive, includes the concept of operations with them, and the notion of mapping, and the concept of order. These concepts are all interconnected.. Their association begins with definitions of special, special set, which has only one element. More than one such set is allowed.. Implementation examples you already have. The union operation allows you to build on their basis sequence, orderly in count. Mapping allows you to put a set of sets, realized by this sequence, many of their powers (quantities) and it will basis of the set of natural numbers. Implementation example: heaps of stones — only one stone in one; take the second pile of one stone and add to the first; implemented the second set in the sequence; etc. And on paper, when implementing each set, we draw a new icon, unique to him. The set of these signs is a mapping of the set of sets. You can enter signs for addition operations, subtraction and equality. Equality is an operation that establishes a one-to-one correspondence between two sets, and, in this way, asserting the coincidence of their powers. Let it be signs +,- and =. And the signs of the initial sets in the set of natural numbers will be 1,2,3,4,5. Then, using these signs, we can write the connections between these sets, formulated in the definition. Write formulas: 1+1=2, 2+1=3, 5-1=4 etc. Here we also introduce new concepts in mathematics, and they should also be strictly defined and examples of implementation should be given. But all this has already passed into the category of well-known truths and cannot cause any difficulties for people., who can count. I will not try to chew everything in the world. I will dwell only on the moments, who are able to create a false impression of themselves.

One of these moments is the extension of the set of natural numbers to the set of positive integers. This is done by defining the concept “empty set“. Of course, this concept is introduced independently of the concept of a natural number. But it's more convenient for me to stop on it right now.. We have already identified one special set among all possible — lots of, containing one element. It is a brick, from which all the rest are built. Everything, except for one single, another special — empty set.

The empty set is the set, in which there is no element. That is, this definition clearly denies the very essence of the definition of the concept of set. The usefulness of this concept in the formalization of operations with sets is very great.. What remains, when we remove element by element from some finite set, and, finally, delete the last? Nothing. This “nothing” and has a special name “empty set”. To add it to another set means “don't add anything”. If you add an empty set to an empty set what happens? Nothing, empty set again. Important point: if you take “more than one” empty set, then the result will not be a set of these sets, but again just an empty set. The empty set is unique in nature.  The empty set is the symbol for the absence of any other kind of set.. And it is only in this sense that it must be understood.. But in the literature you can find a description of the procedure for constructing natural numbers only on its basis: one empty set is equivalent to 0 (that's its meaning) and 1 (because it seems that there is already one set), two — deuce, etc.. And from nothing they get something. Notice, already at the very beginning two characters are assigned to the empty set. This is a direct path to absurdity, and it is possible only as a result of forgetting the special meaning of this concept.. In the set of numbers, the empty set is associated with the symbol 0. Only 0. And as an example of implementation in the real world, you can consider an empty box, banning, but, explicitly consider multiple empty boxes. The box is a container for a lot, and not just a set. A few empty boxes are just a few boxes, not several empty sets and they form a set of boxes, not empty sets. For all its limitations, an intuitive idea of ​​the concept of emptiness, such an example still gives. If there was an item in the box, this box separated from the rest, and then they took it out of there and there was nothing left. Isolation, a generalization of the idea of ​​an empty set is obtained using arbitrary empty containers, and at the end of formalization, with due imagination, you can do without them at all.

Set of natural numbers, with one of his operations — adding a new element — allows us to formalize another important mathematical concept, idea potential infinity. The idea of ​​the absence of a limit in the implementation of the operation of addition. And then any other operation. There are sometimes two different ideas mixed up here.. Herself unrestricted possibility of repeating the operation and the result of this sequence of actions. For the most part, they should be distinguished. For the second idea, as for the idea of ​​an empty set,  special symbol in mathematics — ∞. Yes, there is a name too. — actual infinity. Since mathematics is very much (although not completely freed from ordinary language) uses special characters for its concepts, then very often the same symbol is also used to indicate the idea of ​​a never-ending sequence of operations. In most cases this does not lead to problems., but still be careful, and it is not critical to mix both of these ideas. Problems may appear, when they begin to deal very freely with sets of sets, having themselves this infinitely large number of elements. I don't want to discuss this kind of problem here.. But the second idea brings us very close to another concept, essential for physics (for mathematics, of course, also) — to the concept of actual infinity in the real world, realized in intuitive notions of continuity, integrity and continuity of ties, first of all, causal.

Before discussing the idea of ​​continuity, useful to complete the discussion of the concept of number. As we have already said, it comes from the needs of the account, and its formalization begins first with a description of the concept of a natural number, which, with the addition of the concept of zero, is extended to the concept of the set of positive integers. But everyone knows, that the matter does not end there and the practice of the real world required the introduction negative and rational numbers. Introduction irrational numbers is already closely connected precisely with the concept of continuity.

Negative numbers can be implemented with real world examples in many ways, in particular, as symbols of duty. I prefer to introduce them as an extension of the ability to perform a subtraction operation., inverse addition, ie. adding many new elements. After all, the possibilities of how to add, so take away (subtract) an element from a set have the same intuitive basis in the real world. But for addition, this possibility has no restrictions, and for subtraction it seems like there is such a limit — if the resulting set is empty, what you want to remove from it? A clear disparity between operations, seemingly having exactly the same origins as ideas. This equality is easy to return, if you use the concept of containers as a crutch, to which empty spaces are added, where to put something. Of course, these props, as in the case of the idea of ​​zero, empty set, optional. But to get an intuition about the idea helps. Like other similar examples of the implementation of a negative number. This particular implementation example is useful for those, what emphasizes the removal of the limit from the operation of subtraction, it also gains the ability to be executed potentially an infinite number of times. This happens by expanding the concept of number. Lots of numbers, which are now simply called whole, includes natural numbers, zero and negative integers. This example also allows us to note the fact, that the ideas of the operation and that, What is this operation for? (in the operand) although different, but inextricably linked. One is not defined without the other. And the definition is, as I noted before, listing properties and presenting at least one real-world implementation example.

Rational numbers appear with the definition division operations and, indirectly, comparisons one element (subject, or his formalized idea) with another. The division operation as an idea has a huge number of examples in the real world.. One class of examples collects the division of a certain number of objects (multitudes) for a number of subsets, without affecting the integrity of the elements, it's a lot of ingredients. In the set of natural numbers, this class corresponds to the concept of integer division. I want to note, that when defining this operation in the set of numbers, its properties are specified, complemented by a comparison operation. for instance, there are many ways to divide a set of 4 objects into two subsets. But in the division of natural numbers, the only such way is chosen as a definition — necessary more, so that the resulting subsets are equivalent in cardinality (number of elements). In numbers it sounds like both halves are equal to each other. Also when dividing into any number of subsets (parts) — parts must be equal by definition. This operation differs “division” in the set of numbers from the operation “division” in any arbitrary set. It is this additional property that leads to the concept of a rational number.. It is forbidden, keeping the requirement of equality of result sets, divide 3 item for two. And one too. And if you treat objects as sets, then you can. for instance, all alone, and another (others) nothing (empty set).  The concept of a rational number is at first easily introduced when dividing, even without the need to expand the set of natural numbers itself.. for instance, no problem to share 6 on 3. Get it 3 equal numbers, equal 2. The result remains in the same set. But as soon as we ask, so that for all numbers the result of division remains in the original set, it is immediately found that it is impossible to satisfy this requirement. One or two can't be divided by three, so that the result is a natural number. The results are new numbers, not whole. They get a new name, rational. And natural are strictly included in the new concept. The division operation can be extended to negative numbers without problems.. But with zero, the problem arises. If we divide zero, “nothing”, can be any other number — the result is obvious, by definition of the empty set “nothing”, that is zero, and it will turn out. That's what it means to divide by zero? This operation is undefined in the set of rational numbers, she is forbidden. In the set of rational numbers, not all elements are equal for all certain operations. One of the numbers is special with respect to the division operation. I didn't say anything about the multiplication operation. The way it is introduced as a symbol for some special addition suboperation is well known.. There is also a comparison operation — add up equal numbers. And that, that the operations of multiplication and division are mutually inverse (define) simply. Zero has no special properties with respect to multiplication.. How many empty sets do not add, so there will be an empty set, zero. And if you don't do this “never”, what don't you apply it to, so you get nothing, ie. zero. Intuitively clear. Note, that in the set of rational numbers, the operations of multiplication and division are not equal for one of the numbers. This disparity can also be removed., extension of the very concept of number. You just need to choose an intuitive (ie. implemented by some real world example) concept (idea) for the result of division by zero.

It is useful to dwell in more detail on the above-mentioned comparison operations, which acquires a special role in physics, as the basis of any description of the world that allows verification by experiment. And other descriptions (not allowing verification by experiment) I'm not interested here. The comparison operation is performed by us quite often.. And in everyday life, and in science. However, in science it is formalized to the limit and is reduced only to relations of the form: something could be more, equal to or less than something else. Besides, it is possible to refine these simple relations in the absence of equality — how many times one more (smaller) than another. As a matter of fact, the same idea-forming concept of a set — its power, amount of elements (points) in multitude — and appears as the result of this comparison operation. And the concept of number, naturally, also. One moment, obvious to everyday language and lost by mathematicians, I would like to emphasize in this regard. By and large we have the right to compare only homogeneous entities. For example rams with rams, and tables with tables. Only in this case the result will make sense. but, in everyday language, we can place accents differently and comparison of seemingly incomparable entities can make sense. for instance, if we are only interested in the property of the object, object to be, exist, have (the same, which we have identified as the most basic, initial property-concept), then by this property you can compare any objects. Say, question — “how many items are in the box?” — quite meaningful for a completely arbitrary set of a wide variety of objects. Unfortunately, today mathematics believes, that she relies only on this, extremely cleared of all other shades, possible in the real world, concept and therefore does not attach any importance to the above restriction on the applicability of the comparison operation. But all numbers in mathematics, when it is used as a language to describe the real world, without any exception is the result of applying this particular operation. And when using such numerical descriptions in relation to various phenomena of the real world, forgetting the origin of the numbers themselves can become fatal.. In experimental physics, the number appears as a result of procedures for counting objects of the world or as a result of measurement, ie. comparison of one object with another. If desired and counting objects can be included in the concept of measurement as a special case, as should be clear from what has been said on this topic above. Measurement as a complete procedure (when can it be finished) gives a rational number, and whole numbers, counting result, are a subset, a special case of rational numbers.

Let's take a closer look at rational numbers., considered precisely as the result of the measurement of one object by another. For a very long time there was a belief, what the ratio of any two homogeneous objects can always be expressed precisely by a rational number. Ie. believed, what there is such a pair of integers, that this relationship can be written using them. for instance, 1:2, 2:3 etc. In this sense, the measurement procedure in terms of determining such a ratio is always completed, the correct choice of the unit of measure allows you to complete the comparison procedure in a finite number of steps. I will explain, what I mean. Let the denominator of the desired ratio (fractions) equal n, and the numerator k. Let's choose instead of the original subject, with which we compared the other (let's call this item scale or unit of measure) his n-share. The ratio of the old scale to this can be written whole number n. But the attitude of that subject, which we measured with the old scale to the new unit of measure will also be recorded whole number, equal k*n. Ie. the belief stated above can be reformulated as the following statement: for any two homogeneous objects (parts of the real world) you can find such a share, that the ratio of both objects to it will be expressed in whole numbers. Note, that the concept of a rational number is genetically related to the union of integers in pairs and the operation of their comparison.

This belief turned out to be false.. As I know, discovered it first Pythagoras. It turned out, what do this for the hypotenuse and any of the legs in a right triangle, legs of which are equal to each other,  impossible. Today this discovery is regarded as the first crisis in mathematics. It allowed me to see, that in the real world there are examples of the realization of the idea of ​​actual infinity, realized potential infinity, endless process, completed. Not by us. real world, its parts.

Concept, which allows us to come to the realization of the actual infinity is a formalization, idealization of a set of experimental data, which we combine in the representation of continuous objects, all parts of which connected together. It does not mean, that such objects cannot be divided into parts at all. It is possible, and, Furthermore, it is possible to continue for any parts resulting from such a division. Then, what the fission process can be continued potentially indefinitely and is the main, defining property of the very concept of continuous. The simplest examples of the implementation of continuity are such objects of the real world as a rope, string, thread and the like. I want to emphasize the second side of continuity, which is usually overlooked. The idea of ​​continuity, if it is maximally formalized and cleared of its particular manifestations, it can still be framed as connection idea, connectivity of parts of the real world. That's in what sense. The idea of ​​continuity as a property of an object to be infinitely divisible without losing this property may seem inconsistent with the real measure., if you take into account, that all massive bodies with a mass of rest are parts clearly separated from the rest of the world. Although here it is quite possible to appeal to the idea of ​​the line of existence of any such body. True, it will be necessary to deal with the meaning of the statements of quantum mechanics about the absence of a well-defined trajectory for the smallest such bodies, such, like an electron. But there are also causal relationships between events.. In order not to go into these details here, I will draw your attention to something else (unrelated to the presence of a rest mass) manifestation of connections between parts of the real world. I mean those manifestations of such links, which is called the field. for instance, electromagnetic field. Yes, visually demonstrate relationships such as linear, superficial or volumetric formations directly impossible. It is possible only with the help of their secondary manifestations (eg, sawdust lined up along “power” lines). But demonstrations of these links between, eg, two magnets, great multitude. And these connections are also the realization of the idea of ​​continuity.. Continuity as an idea of ​​the universal connectedness of the world also acts as a formalization of the idea of ​​the world as a whole.. Overall, containing various parts. Parts are different, separable from each other in a certain way, but at the same time interconnected.   What is not connected with the world (not associated with any part of the world), that in the world and not. Our world cannot consist of several unconnected pieces.. A purely discrete set of parts cannot be regarded as a satisfactory description of the world., because it does not even contain a hint of the presence of connections between the parts. To describe the world we need a different concept, different idea, which we called continuity. This idea necessarily also includes the discrete, at least as a collection of arbitrarily chosen parts. New, contiguous set is also called continuum. Mathematicians have developed many different ways to work with continuum means, naturally defined for discrete sets. Naturally, they all rely on the concept of realized potential infinity, ie. actual infinity. First of all, such a concept in mathematics is the concept limit.

The concept of a limit, sequence limit, probably first took shape precisely as a formalization of the idea of ​​the possibility measurements some line segments, the ratio of which with the chosen unit of measure cannot be expressed by a natural number. Indeed, in this case it is easy to see, what if we split the unit of measure, say ten smaller equal units, and repeat this procedure an infinite number of times, then we can always get two infinite sequences (two potential infinities), the sums of which will always be one less than the measured segment, and the second more. Moreover, the difference between these two amounts is always, at each step will be equal to the unit of length of the segment chosen for the step. And this unit will be less and less, compared to the initial. In this procedure, we consider four segments: One, the length of which is taken as a unit. Second, which we want to measure with this unit, ie. compare the lengths of these segments and match the resulting result with a certain number. And two more service segments, at each step of the procedure for comparing different. One, having a length less than the measurable, and at the same time expressed rational number. And second, per current unit of measure (current fraction of initial unit) more than that, and also just more than the measured segment, and also having a length, expressed rational number. This relationship between the four segments clearly convinces us, what if we were able to continue this procedure an infinite number of times, then the end result would be, because it is between two, arbitrarily close rational numbers. This belief has a basis in our experience and does not require other evidence.. Result, as a number, is the image of the actual infinity and is called the irrational number. Corresponding place in a continuous line (in this case, this is the end of the measured segment) can be called a dot, that essence, of which the line itself consists. A side effect of this procedure is the understanding that, what in an arbitrarily small segment of a continuous line, there is always an infinite (actually infinite) number of such points. Another side effect is the understanding, that with the help of such a measurement procedure, on any segment of a straight line, one can place (match its points) some part of the set of rational numbers. And if you measure an unbounded line, not a segment, then the entire set of rational numbers can be put in correspondence with a part of the set of points of the line. The union of all points of an unbounded line will correspond to the union of the sets of rational and irrational numbers, which is called a set valid numbers. set of points, constituting continuity, using the measurement procedure, we associate the set of real numbers. It is important to understand, that these concepts simply describe some properties of the real world, rather than derived from the applied procedure. Here, two ideas are inextricably intertwined into a single whole., very important for mathematics, same for physics. one idea — this is a representation of a set of a different type, than those sets, that emerge from the simplest idea of ​​existence. All properties of the previously defined sets are present in it.. New properties added. According to these properties of the set, one can distinguish: discrete and continuous, Continuity. These properties are not formulated in one word, they formalize a set of special procedure properties, which implements examples of new types of sets in the real world. In physics, this procedure is called measurement procedure. This is a rather complex set of operations, which includes detailing the selection operations, comparisons, division and can itself be considered as an implementation of a complex operation of creating a mapping of one set to others. One of these images, one such set is, as a matter of fact, the backbone of mathematics, and physics too. This is the set of real numbers.

The attentive reader must have noticed, that the measurement procedure in its simplest version was also used by us in formalizing the concept of discrete sets. Something new, what appeared, was the result of refinement of the properties of this procedure, its enrichment. This process is not completed.. For a new type of sets, additional features appear. for instance, in addition to choosing the base unit of measurement, scale, you can choose that element of the continuum, which will be assigned such an element of the set of real numbers as zero. Choose start countdown on a continuous set. You can also choose direction on a continuous set, which will be considered positive. There are also opportunities to consider continuums of the number of dimensions of a larger, than one. And a lot, much more. Colossal branches of geometry grow from here, group theory, algebra and many more. Ideas, filling them, separated somewhere, intersect somewhere up to complete alignment, becoming different dialects of the language, talking about the same.

I will not go on discussing all these important concepts further.. I will make only a few remarks that I think are important..

Modern mathematics has consigned to oblivion those origins of the origin of the concept of number, which I mentioned above. In math we just work with numbers. Choice of scales, their implementation by objects of the real world does not interest mathematics. But in vain. Many concepts would be much easier to grasp, if mathematicians did not fall into snobbery “world of pure ideas”. And for those, who uses the achievements of mathematics to describe certain aspects of the real world, such snobbery is not at all acceptable. Perhaps, it is worth briefly listing here those moments, which I consider important to study, and for the further development of mathematics, as part of science, Well, of course, for its successful application to describe the real world:

• World of ideas exists. But not as something different from the rest of the real world, separate from him. It is an integral part of both the Universe itself, any part of it, and any description. Just because, that the description is the isolation, concept formation, ie. “ideas”. Any “good”, adequate to some property of the world (or some part of it) idea, being complete is at the same time limited. In the sense, which does not describe the whole, and part, one single property or a finite set of properties.
• These ideas are very diverse.. They are combined into languages. Including, in such a variant language, as science and its components (dialects) mathematics, physics, biology, etc..
• Then, that science and, in particular, mathematics claims the adequacy of their concepts (ideas) the real world is justified only then, when there is at least one example of the implementation of an idea in the real world. This is especially true for basic, elementary ideas, which form the basis of new directions in the same mathematics. And not only, of course. Absolutely clear, that all languages, and mathematics as well, allow to formulate as true, and false statements (ideas). But one can also formulate such ideas, fidelity or infidelity (matching them to the real world, any examples in it available) can (and should) be accepted as an axiom, as a possible option. This is what Gödel's theorems say.. All our direct experience speaks of this.. Binary logic, when there are only two options to choose from, only one of the logic options, no more. We all know examples of situations, when a choice is possible from an arbitrary number of possibilities, up to potential infinity, and examples of the continuum make this infinity relevant. Considering the elementary idea (axiom, definition) as an enumeration of some set of properties, then Gödel's theorems only say that, what is a potentially infinite enum. From the point of view of such a new idea, adding a new property can neither be proven, neither rejected on the basis of only the entire previous set of described properties. The only criterion “fidelity” such a new definition for us can only be the discovery of at least one example of the implementation of a new idea in the real world. One single implementation example is already enough.
• In this regard, the presentation of mathematics would be useful to build on a constructive basis., creating new concepts (ideas, axioms) or on the basis of existing, their various combinations, either based on those phenomena of the real world, who have not yet received the formalization of such ideas. And always examples of the implementation of each such idea.
• It does not forbid the so-called “free flight of thought”. He can be very successful.. Moreover, our history provides a huge number of examples of this kind. But it's important to understand, that the success of such creativity becomes clear precisely then, when the implementation of such “free” ideas real world examples. That's just for the development of such ideas by the broad masses of at least scientists, I'm not talking about people far from science, these examples are vital.
• Due to this, the only justification for meaningfulness and consistency as mathematics, and other branches of science can only be the correspondence of their ideas to the real world.

In the end, I want to emphasize again, that I consider the idea of ​​the existence of the world of ideas, especially, collected under the name “mathematics”, as a world completely independent of the real world, vicious and dead-end, “wrong idea”. Not sure, who was the originator of this idea, possibly Plato. But even now it dominates so many minds. for instance, she is one of the leading ideas of R. Penrose, which is well stated in his book “Path to reality, or laws governing the universe. Complete guide.”

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