How is the assessment of the state of the object?

At school, I needed tremendous efforts to understand Cantor's concept of a multitude of power continuum. But then I realized that I did not understand anything, but only memorized the rules for working with such objects. In the area of ​​understanding how there was a white spot, it remained white. Since then, I have repeatedly returned to this issue until I became acquainted with statistical physics and co-production.

The concepts of substance and the concept of state were very well defined in these sciences. It was said that to determine a substance a volume of the minimum size is needed, the smaller of which we already have is not a substance, but a set of molecules, and to assess the state it takes a finite time to register some value associated with the state. If we talk about the state, using time intervals less than the minimum, we get not an estimate of the state, but something incomprehensible.

I understood that on this basis it is possible to construct another mathematics with other axioms. I did not do this, but I remembered that to assess the state, it is necessary to specify the minimum time, during which it makes sense to talk about the measurement, how a minimum volume is needed to determine a substance. Then this time will be considered an instant to assess this state. This time may be different for different properties and evaluation methods. For example, in order to understand what color a bus is with its eyes, milliseconds are needed, and in order to understand the state of the Earth’s climate with a thermometer, it takes several years.

The second difficulty is associated with the choice of these time intervals. If we decide that to evaluate the property of an object, we need a finite time, for example, a day, then we can divide the time axis into daily segments, determine the values ​​for each such segment and say that we have estimated the instantaneous states of the object. But in this decision something is missing. Question: what? You probably guessed that there are two questions that need to be answered: will the minimum amount of time be independent of time? Is it possible to cut the time intervals in another way? Yes and yes. And all these arguments are broken about these questions.
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And again, statistical physics, or iron oxide, comes to the rescue. They consider sets of partitions. If we are talking about the assessment of states, then we are talking about the set of different partitions of the time interval into intervals within which the state can be evaluated. Further, any question that concerns us about an object, whose state we are examining, should be addressed not to a specific partition, but to a set of partitions. For example, if we examine the state of rotation of the motor shaft, registering its “instantaneous” rotational speeds, then we can ask the question: how many revolutions did the engine make in the final time interval? To do this, it will be necessary for each splitting to do the summation of the products of speed and time. For each partition we get a certain answer, and then we need to make the limit.

This transition is made in statistical physics and strength of materials, but is poorly described both there and there. The fact is that it is associated with a mathematical model, far from reality, in which it is asserted that a property can be measured instantly, not in the sense we defined earlier, but in the literal sense. This assumption, which many do not realize, leads to the proof of the convergence of partial sums, but is pure focus. And this trick may not pass. There are properties for which it is possible to construct different series of partitions, whose partial sums will converge to different limits. However, we usually do not model such properties, considering them to be extravagant. Our warm lamp properties are not. For them, we can build partitions and make assumptions about convergence. True, there is a difference from the mathematical model, and it is that one should remember about the accuracy of this kind of transition. Accuracy will be determined by the frequency of the partition, below which we cannot sink due to the fact that the instantaneous property will no longer be defined. We will have a variation in the data, which must be able to handle. This treatment is described in statistical physics in some detail and is called a property fluctuation. Only not the property, the values ​​of which we collected, but a new, calculated one.

So what is a property of an object on a time interval? This is the set of all partitions of this segment into intervals, within which the “instantaneous value” of this property is defined. When we fix a specific partition and “instantaneous values” of a property on a specific partition, it is not an object property over time, it’s not sure what to call it. Will you help?

Perhaps you can give an example of non-warm and non-tube properties, for which partial sums depend on the partition?

This definition of the evaluation of the property of an object leads to the fact that the evaluation of the property depends on the method of measurement. As a rule, we consider that the value of the property does not depend on the method of measurement. Perhaps it is, if we neglect the accuracy. But in fact, different measurement methods give different values. And this, at times, leads to fundamental limitations on the accuracy of our measurements, as, for example, happened in quantum mechanics.