The philosophy of dividing by ... or confession of a madman

Introduction

It should immediately be pointed out that in this article there will be no deep mathematics. There will only be a discussion on the topic indicated in the title. All further described only the opinion of the author. No more. Nearly.

Mathematics feature

Mathematics, as a science, finally went deep into the systematization and abstraction, thereby creating for itself a position in which it fell into a crisis state. What is meant by this? The great philosopher and mathematician Kurt Gödel proved with his excellent theorems that some mathematical foundations cannot be proved or disproved by the means of mathematics itself.



And although it is obvious to many that axiomatization is always based on observations of physical reality (that is, on experience), for some reason these many concentrate exclusively on mathematics itself, that is, structure (form) without content. Because they sometimes do not imagine what they are doing, but they know how. Most of those who tried to approach the described problem, like a cat that chases its tail, stubbornly walks in a circle. Here, most likely, the very professional ossification that Lorentz wrote about in his excellent work is manifested.

Comparison as the most important tool of cognition

Everything is relative.



Rene Descartes

To begin with, it should immediately be noted that all mathematical operations occur in people due to the possibility of identifying common signs. That is, due to the statement of the condition and the relation of the condition to the objects, the calculation itself occurs. From here arithmetic operations are derived. Simply put, through the comparison, the initial calculation took place. Many physical quantities are accepted standards (standards), examples of which are carefully stored in Paris. This implies that an initial unit is established on the basis of which numerical representations ( concepts ) of physical phenomena are derived. Simply put, the same calculation is done. “Things in oneself” (a term proposed by the great philosopher - Immanuel Kant) seem to us such objects of being that we cannot comprehend with the mind, due to the imperfection of human capabilities. An elementary comparison of things and compilation of categories on this basis enables us to systematize objects of knowledge, which leads to some processed knowledge (“things in ourselves” become incomplete “phenomena”, because we may not know all the properties of something) . If we could not determine the differences between bodies (form, color, taste, size, and so on), then for us all objects would remain “things in themselves”. Kant established that the selection of categories is the basis of inexperienced (a priori ) thinking, which is directly related to the mathematical variety of cognition, that is, we can immediately indicate that by highlighting equality or inequality (similarity), we only establish (produce) the possibility of counting. Of course, the absence of inexperienced thinking excludes the possibility of carrying it out (the exclusion of the human “function” of comparison makes counting impossible). Many, by the way, are categories of objects in which there are some conditions for the presence of elements.



Take as an object of consideration a bar of silver (a popular object of thought experiment). We can distinguish its mass on the basis of experimental comparison with the accepted unit in the SI system (kilogram). We can also distinguish its length and width based on experimental comparison with the accepted unit in the SI system (meter). If we mentally discard the measures taken and all known objects, except the ingot itself, then we will only have the object of cognition given to us in our subjective, sensual representations (which will still be part of the inexperienced knowledge of the object, because thinking cannot be completely turned off (as well as past experience, the definition of which is very difficult)). We cannot compare any number with it simply because we cannot compare it with anything. Based on all this, it is easy to come to the conclusion that the numerical representation of a physical quantity has a connection with elementary comparison (comparison) in general (this is obvious, but it is necessary to indicate for clarity of judgment).



If we change our unit of measurement (standard), then we can get any numerical representation (within the framework of real numbers) of the same ingot, based on the rule (theorem) of scale change. Imagine that an ingot weighed one kilogram, that is, it was completely compared with the accepted unit of measure by weight. But if we do not use the traditional standard (kilogram), but replace it with half of the previously adopted unit of measurement (kilogram), we get that our bar weighs two “accepted units”. Of course, in this case, the scale will extend to all the objects being compared (within the framework of the consideration) for the possibility of comparing them, but this does not negate the possibility of changing the numerical representations of the compared quantities in the adopted framework (action of the rule (theorem) of scales). Thus, I single out separately the numerical representation (value) obtained by comparing it with the accepted unit of measure (measure). We can compare a silver ingot of two hundred kilograms and another silver ingot of four hundred and half kilograms, which implies the use of different numerical representations and different accepted units of measure (measures). Of course they will be equal, with the same measure. Accounting for units of measure plays an important role in physics, which helps to avoid errors (and paradoxes) in the calculations. But mathematics allows itself to ignore this approach, despite the fact that it is possible to derive any numerical representations based on the choice of “accepted unit”.

The most important problem of the mathematical approach

Mathematics can be defined as a doctrine in which we never know what we are talking about, nor whether what we say is true.



Bertrand Russell

When we have to work with quantities, we, by default, consider them one-dimensional (deduced through one, general measure). This applies to the numerous educational calculations of mathematicians who do not take measures into account in decisions. This approach immediately creates a certain orientation. “Ideal representations”, worked out mathematically, do not fully correlate the phenomena of reality, due to the complexity of the phenomena themselves (there are too many factors that cannot be taken into account immediately). A problem arises in which the “ideal representation” may itself not be complete from the very beginning, and its verification becomes completely impossible (experience cannot unambiguously confirm this). All this is sufficiently confirmed by the existence of such a wonderful parameter as accuracy (“ideal representation” (some universal law) is derived from experience and is itself determined by experience, which is quite funny). Humor may still consist in the fact that the initial conditions for obtaining “ideal ideas” may no longer be correlated with current reality (the universe then and now). Based on the same biology (the example of which is easier to see), our reality is constantly changing, as we are changing. Centuries ago, laws worked out may cease to fulfill their role after some time due to changes in reality itself (without already informing about scientific revolutions). Apparently, due to these problems, the approach to measurements and standardization is gradually changing (it becomes more suspicious skeptical ). But why all this?



The author of this article will refer to the second section of the book by Konrad Lorenz (“The emergence of new system properties”), in which the scientist points out not obvious changes in the parameters during the formation (combination) of systems of individual elements, where each individual element demonstrates its own characteristics, but, when combined with others, these features are distorted - that is, a linear sequence of causes is eliminated. Thus, I want to draw attention to the fact that the observed phenomena cannot always follow the mathematical approach (and even the same arithmetic in the simplest cases) as some people know. And if we take into account that mathematics itself arises in the processing of experience by our mind (with other functions of the human body), then solving mathematical problems through experimental testing is not something criminal.

Zero as a number

There are already quite a few analyzes regarding the representation of zero, and therefore the section will be brief.

The problem of zero and division by it

For some reason, people finally made sure that as a result of multiplying the number by zero, the result will be zero itself. Of course, this conclusion has reason. The author agrees with them, but still it is necessary to understand a little. Take the same ill-fated ingot of silver and multiply by five. Get five ingots. The value increased, but the measure remained the same. Take an ingot and multiply by zero. We get the number of ingots - 0. The measure is still the same. Take a boring ingot and divide by two. The result will be half an ingot. The measure is the same. The value has changed. Or not? What prevents us from reporting that the measure has changed? Meaninglessness. Dividing the ingot by one, we get the same ingot. Dividing the ingot itself, we get the net value without measure (quantity). You can easily remember that the default denominator of all real numbers is one. The same unit, which, in fact, is a measure (standard) of calculating our value. It is worth changing the unit (changing the measure, changing the denominator, dividing) and we are changing the numerical value. So what happens when dividing by zero?

Division by zero as a cognitive process

The measure is being destroyed. Our idea, due to which the quantity is developed (calculated), is destroyed. Each time a person who performed mathematical operations on indefinite quantities, all the time he did this unconsciously. He took a sign (condition) by which he developed a value in his imagination and, after performing the calculations necessary for himself, he got rid of this idea (from short-term memory). Having an observable example in which we allow ourselves to divide some term into zero, we simply eliminate the example from this term (it turns out that it is simply ignored, because the measure, in this case, no longer coincides when comparing the terms themselves - it simply does not exist for of this). If we draw an analogy with programming languages, then dividing a variable of some type by zero, we should actually delete the memory (allocated even for a “wrapper” (named pointer)) allocated for this variable (this is too radical for the reasons described below) .



The author developed a notion in which this operation is closely connected with the theory of information, cognitive ( cognitive ) psychology and all other “exact” ones (the author cannot afford to call sciences exact, in which there are no exact calculations, for which it is enough to recall representations of limits with infinitely small (large) quantities, not to mention the discriminators ( differentials ) and unreasonable ( irrational ) numbers) sciences.

Systematization problem

At first, most likely, they received the multiplication operation (through the sum), and only then the division operation was deduced for it as the opposite. The peculiarity is that multiplication is commutative, associative, distributive, and so on, in contrast to division. That is, by properties there is no longer the same comparison as when adding and subtracting. No logical symmetry is already observed here, so to speak. When multiplying and dividing by zero, the famous dilemma arises, because any number multiplied by zero will always be zero, not to mention division, for now. What to do in this case?

Sentence

Just as at some time people decided to introduce complex numbers to solve cubic equations, you can introduce a special kind of numbers to solve the problem of returning values ​​when dividing and multiplying by zero. At first glance, all this is meaningless. On the second, the meaninglessness is still obvious, but the author has not just touched upon the natural sciences and cognitive psychology. Provided that the calculation measures can be compared with each other in a wide variety of mathematical calculations, there should be a need when taking into account various measures and features of the calculation of quantities. Accounting itself will be the necessary information that forms the return value when dividing and multiplying in various complicated problems of physics and standardization (not to mention the calculation of systems with connected subsystems and elements).



When multiplied by zero of any quantity, the quantity itself becomes zero, but the measure itself remains. This approach prevents the creation of a reverse operation. You can enter "memorable numbers", which in the examples themselves will cease to be perceived after dividing or multiplying the value by zero, but after the reverse operation, the previous value (value) will be returned taking into account the (previous) measure. This approach opens up new spaces for comparing measures and quantities in calculations. Moreover, this approach can allow you to compare not only numbers, but also other, non-mathematical objects with each other, but all this is already a fantasy that alludes to category theory.

$$

$$\ frac {X * 0} 0 = \ frac X 0 * 0 = X;$$

The consideration of the returned parameters when multiplying and dividing by zero should depend on the application and justification, but already at this step it can be deduced that the operation to destroy the representation (measure) is the reverse for the destruction of the value. These operations themselves, of course, in the framework of conventional calculations do not make sense (although this will show the future and experience).

$$

$$\ frac 0 0 = 1;$$

Based on this, information about the previous value and measure of the number multiplied or divisible by zero remains in square brackets.

$$

$$X * 0 = 0 [/ 1 - specific measure by default; * X - past-value];$$

$$

$$\ frac X 0 = 0 [* X - past-value; / 1 - specific measure by default];$$

Of course, you need to add notes ([*; /] or [/; *]), specifying in which places the previous value and measure, because when multiplied by zero it is necessary to put the measure that remains in the first place. When dividing, it is necessary to put in the first place the previous value, and only then the measure that is destroyed. The resulting "memorable numbers" cannot interact with other numbers through arithmetic, although they must interact with each other, due to the presence of the same measures, but this is already established by the calculator. Folding meters with liters you can’t bring everything to the same value. That is the reality. Another thing is that numbers are one-dimensional, when used on their own.

$$

$$1 + X * 0 = 1 + 0 [/ 1; * X];$$

$$

$$1 + \ frac X 0 = 1 + 0 [* X; /one];$$

$$

$$\ frac 1 1 = 1 [* 1; /one];$$

$$

$$\ frac 1 2 = 1 [* 1; / (\ frac 1 2)];$$

Entering arithmetic rules is pretty straightforward. It is enough to compare the values ​​of the same measures. For fitting, simply follow the scale reduction rule using the available measures, that is, multiply the existing value by a measure.

$$

$$1 [* 1; / 1] + \ frac 1 2 [* 1; / (1/2)] = 1 [* 1; / 1] + \ frac 1 2 * \ frac 1 2 [* 1; / 1] = 1 [* 1; / 1] + \ frac 1 4 [* 1; / 1] = 1 \ frac 1 4 [* 1; /one];$$

$$

$$\ frac 1 2 [* 1; / 1] * 0 = 0 [/ 1; * \ frac 1 2]$$

$$

$$\ frac {\ frac 1 2 [* 1; / 1]} 0 = 0 [* \ frac 1 2; /one];$$

The author did not think much about which symbols to use for the remaining measures and quantities, and he took the square brackets just like that, but if by some miracle his ideas would be used and meaningful by other people, then please use Russian symbols for uniqueness images and introductions of Russian, cultural tradition.

Some thoughts on the subject of “uncertainties”

The indicated division by zero problem has given rise to several well-known uncertainties. But with the conclusion of previous ideas, they do not seem so unsolvable.

The author of the article strongly opposes the use of limits (indefinite functions for which the method of achieving a given value is not indicated) for this review, because to achieve many quantities, even if they are uncertain, you can always try to approach their estimation, otherwise the values ​​themselves would be we are not perceived (to the question of comparison).



In this formula, the last result is easily displayed by substitutions:

$$

$$0 ^ 0 = (x - x) ^ {x - x} => \ frac {(x - x) ^ x} {(x - x) ^ x} => \ frac 0 0 = 1;$$

As you can see, limits are absolutely not necessary for the perception of established, (for example) real numbers. When it comes to emptiness, it implies a definite state (the absence of something by condition, as a result), although in reality no one has ever seen absolutely empty states (though the condition is blurry with this approach).



An important problem arises when comparing infinities with zero, but the whole point is that the infinities themselves are indefinite. It is only necessary to give them a functional look and many conclusions in the evaluation themselves suggest themselves through induction. I recall George Cantor’s excellent speculations about “capacities,” thanks to which many appeared.



Suppose we have functions F (x) and G (x):

$$

$$F (x) = \ sum ^ {\ infty} _ {X = x} {X} = + \ infty;$$

$$

$$G (x) = \ sum ^ {\ infty} _ {X = x} {X * 2} = + \ infty;$$

Can we not get an explicit answer when dividing these functions?

$$

$$\ frac {F (x)} {G (x)} = \ frac 1 2;$$

Moreover, what prevents us from assessing the speed of reaching various infinities, given the same Cantor “powers”? Yes, nothing.



The division of one infinity into another should be equal to unity, if only because the designation is the same. Otherwise, the introduction of a functional representation of infinities is a necessary need that will help determine their difference even in symbolic representation. Based on the adoption of infinity as a result, it is easy to come to the conclusions:

$$

$$\ infty ^ 0 = 1;$$

$$

$$1 ^ \ infty = 1.$$

Have the courage to use your own mind.



Immanuel Kant

This is another attempt to touch the unknown (something like this was written by a colleague on the concern of the problem of division by zero), which has been carried out here (on the resource) more than once. It’s just that the author believes that it is necessary to use other methods, rather than mathematical ones, in determining various grounds. For example, self-awareness ( reflection ) is enough.

I don’t imagine what happens to people: they do not learn by understanding. They learn in some other way - by rote memorization or otherwise. Their knowledge is so fragile!



Richard Feynman

Literature:



Conrad Lorenz: “The Back of the Mirror”;

Rene Descartes: “A discourse on a method to correctly direct your mind and seek truth in the sciences”;

Immanuel Kant: “Critique of Pure Reason”;

Aleksandrov Alexander Danilovich: "Geometry".