March 28, 2018 at 07:04How is the state of an object evaluated?
At school, it took me tremendous effort to understand Cantor's concept of the multitude of power continuum. But then I realized that I didn’t understand anything, but only learned the rules for working with such objects. In the realm of understanding, as there was a white spot, so it remained white. Since then, I have repeatedly returned to this issue until I became acquainted with statistical physics and sopromat.
The concepts of matter and the concept of state were very well defined in these sciences. It was said that to determine a substance, a volume of a minimum size is needed, less than which we no longer have a substance, but a set of molecules, and it takes a finite time to evaluate the state in order to register some value related to the state. If we talk about the state using time intervals less than the minimum, we get not an assessment of the state, but something incomprehensible.
I understood that on this basis it is possible to build different mathematics with different axioms. I did not do this, but I remembered that to assess the state, you need to specify the minimum time during which it makes sense to talk about taking a measurement, how to determine the substance you need a minimum volume. Then this time will be considered an instant to assess this condition. This time may be different for different properties and evaluation methods. For example, in order to understand what color the bus is with the eyes, it takes milliseconds, and in order to understand the current state of the Earth’s climate with a thermometer, it takes several years.
The second difficulty is the choice of these time intervals. If we decide that to evaluate the properties 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 estimated the instantaneous state of the object. But there is something missing in this decision. Question: what? You probably guessed that there are two questions that need to be answered: will the minimum amount of time itself depend on time? Is it possible to cut time intervals in another way? Yes and yes. And all these arguments are broken about these questions.
And again, statistical physics, or sopromat, comes to our aid. They consider many partitions. If we are talking about assessing states, then we are talking about many different partitions of the time interval into intervals within which the state can be estimated. Further, any question that worries us about the object whose state we are examining should be addressed not to a specific partition, but to a lot of partitions. For example, if we investigate the state of rotation of the motor shaft, recording its “instantaneous” rotational speeds, we can ask the question: how many revolutions did the engine complete in a finite time interval? To do this, it will be necessary for each partition to make a summation of the products of speed and time. For each partition we get a certain answer and then we need to make the passage to the limit.
This transition is made in statistical physics and compromise, but is poorly described both there and there. The fact is that it is connected with a mathematical model that is far from reality, which states that a property can be measured instantly, not in the sense that we defined earlier, but in the literal sense. This assumption, which is not recognized by many, leads to the proof of the convergence of partial amounts, but it is pure focus. And this trick may not pass. There are properties for which you can build different series of partitions, the partial sums of which will converge to different limits. However, we usually do not model such properties, considering them extravagant. Our warm tube properties are not like that. For them, we can build partitions and make assumptions about convergence. True, there is a difference from the mathematical model, and it is what you need to remember about the accuracy of this kind of transition. The accuracy will be determined by the frequency of the breakdown, below which we cannot go down due to the fact that the instantaneous property will no longer be determined. We will have a spread in the data, which must be able to process. This processing is described in statistical physics in sufficient detail and is called fluctuation of the property. Only not of the property whose values we collected, but of a new, calculated one.
So what is a property of an object in a time span? 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 particular partition and the “instantaneous values” of a property on a particular partition, this is not an object property over time, it doesn’t know what to call it. Will you help?
Perhaps you can give an example of non-warm and non-lamp properties, for which partial sums depend on the partition?
This definition of the assessment of the property of an object leads to the fact that the assessment of the property depends on the measurement method. As a rule, we believe that the value of the property does not depend on the measurement method. Perhaps it is, if we neglect 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.
The concepts of matter and the concept of state were very well defined in these sciences. It was said that to determine a substance, a volume of a minimum size is needed, less than which we no longer have a substance, but a set of molecules, and it takes a finite time to evaluate the state in order to register some value related to the state. If we talk about the state using time intervals less than the minimum, we get not an assessment of the state, but something incomprehensible.
I understood that on this basis it is possible to build different mathematics with different axioms. I did not do this, but I remembered that to assess the state, you need to specify the minimum time during which it makes sense to talk about taking a measurement, how to determine the substance you need a minimum volume. Then this time will be considered an instant to assess this condition. This time may be different for different properties and evaluation methods. For example, in order to understand what color the bus is with the eyes, it takes milliseconds, and in order to understand the current state of the Earth’s climate with a thermometer, it takes several years.
The second difficulty is the choice of these time intervals. If we decide that to evaluate the properties 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 estimated the instantaneous state of the object. But there is something missing in this decision. Question: what? You probably guessed that there are two questions that need to be answered: will the minimum amount of time itself depend on time? Is it possible to cut time intervals in another way? Yes and yes. And all these arguments are broken about these questions.
And again, statistical physics, or sopromat, comes to our aid. They consider many partitions. If we are talking about assessing states, then we are talking about many different partitions of the time interval into intervals within which the state can be estimated. Further, any question that worries us about the object whose state we are examining should be addressed not to a specific partition, but to a lot of partitions. For example, if we investigate the state of rotation of the motor shaft, recording its “instantaneous” rotational speeds, we can ask the question: how many revolutions did the engine complete in a finite time interval? To do this, it will be necessary for each partition to make a summation of the products of speed and time. For each partition we get a certain answer and then we need to make the passage to the limit.
This transition is made in statistical physics and compromise, but is poorly described both there and there. The fact is that it is connected with a mathematical model that is far from reality, which states that a property can be measured instantly, not in the sense that we defined earlier, but in the literal sense. This assumption, which is not recognized by many, leads to the proof of the convergence of partial amounts, but it is pure focus. And this trick may not pass. There are properties for which you can build different series of partitions, the partial sums of which will converge to different limits. However, we usually do not model such properties, considering them extravagant. Our warm tube properties are not like that. For them, we can build partitions and make assumptions about convergence. True, there is a difference from the mathematical model, and it is what you need to remember about the accuracy of this kind of transition. The accuracy will be determined by the frequency of the breakdown, below which we cannot go down due to the fact that the instantaneous property will no longer be determined. We will have a spread in the data, which must be able to process. This processing is described in statistical physics in sufficient detail and is called fluctuation of the property. Only not of the property whose values we collected, but of a new, calculated one.
So what is a property of an object in a time span? 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 particular partition and the “instantaneous values” of a property on a particular partition, this is not an object property over time, it doesn’t know what to call it. Will you help?
Perhaps you can give an example of non-warm and non-lamp properties, for which partial sums depend on the partition?
This definition of the assessment of the property of an object leads to the fact that the assessment of the property depends on the measurement method. As a rule, we believe that the value of the property does not depend on the measurement method. Perhaps it is, if we neglect 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.