Scalar is a basic type of mathematical object that we have all dealt with in our first maths lessons. Its value is a number. There are two basic operations that can be performed on scalars: adding and multiplying. Subtraction, division and more complicated operations, such as power, square root and factorial are derived from these two primary operations. Vector is another type of mathematical object, slightly more complicated than a scalar. Its value consists not of a single number, but two non-interchangable ones. The vectors are also submitted to basic operations, but because they have a more complicated structure, there are more primary actions: adding vectors, multiplication by scalar, scalar product and vector product. Scalars and vectors are not the end of mathematics. New types of mathematical objects have been created when needed and, as you can see, they are more complicated, leading to more complicated operations. Vectors are followed by matrices whose values are defined by sequences of ordered numbers. There are more operations performed on matrices: addition, multiplication, external product, internal product, and besides, they can be subjected to rotation or divergence... The following types of mathematical objects are functions, and so on.
There are many types of mathematical objects, and you can define your own operations on each of them. In mathematics, calculus studies the impact of operations on objects. Do only mathematicians dealing with calculus understand it in such a general way? In fact, what we did with numbers in mathematics lessons does not deviate too far from what we do in our everyday life with objects of other types. I will give you a simple example: mathematicians add numbers and receive results, electronic engineers combine various elements to make a radio, chemists synthesize chemical compounds, and my mother, combining various components in the right order and then applying thermal factors, makes a delicious meatball. Lawyers, on the other hand, check the credibility of the accusation and then pass it through the function they call the penal code, and as a result they get the length and type of punishment for the criminal. After all, length and type are nothing but a vector. It is enough to look properly to find that mathematics, in its most general form, permeates our whole life.
Scalar and vector calculus were originally used by the Ancient Greeks, the tensor calculus was introduced in 1890 by the Italian mathematician Gregorio Ricci-Curbastro (1853-1925). Both of these calculi contributed to a better understanding and better analysis of physical phenomena. One form of calculus is not universal, specific calculi are best suited to explaining and analyzing specific issues. In theoretical mechanics it is best to use vector calculus, in fluid mechanics a differential calculus, and in the strength of materials, a tensor calculus. In The Physics of Life, a population calculus would be very helpful, which would allow us to look at living objects through the prism of population. This is understood mathematically as a set of objects characterized by at least one similar property, the state of which can be studied. Using this would allow us to better understand issues related to the development of life on Earth (and the development of ourselves), and with it we will be able to transfer knowledge to each other in a much faster, simpler and more practical way.
3.24.1 Definition of popsorThe basic object in population calculus is the popsor - a mathematical object similar to a vector or a matrix, but more complicated. As the introduction to the issue, it should be explained where the name came from. Since we have to name the basic population calculus object, it does not seem to be possible to find anything better than the prefix "pop" from the word "population", with the suffix "or" taken from vector and tensor. However, the connecting "s" was added because "popsor" sounds better than "popor".
Mathematically speaking, a popsor is a self-contained collector of different types of information that is used to record the states of the property of a given group of objects. It is a set of elements, sorted out in a characteristic way, of which a detailed definition is given in the following table.
No. | Symbol | Name | Comment |
---|---|---|---|
1. | ^{P}height | Popsor identifier | The letter P informs us that we are dealing with a popsor, the further sequence of characters (height) is the identifying name. |
2. | Object | Investigated object type | Information that precisely defines a group of objects that belong to the popsor. For example: (human) or (person not younger than 17 and not older than 37) or (women); (men) |
3. | Definition of a property | Unambiguous determination of the tested property, methods of its measurement, applied accuracy and units. | Human height: length given in centimetres, determining the distance between the heel bump and the top of the head. The measurement is performed when the object is standing straight, its result is given in centimetres, and the millimetres are rounded according to the mathematical rounding rules. |
4. | (Inf,Sup) | The range of the property values. Infimum - lower limit (name and minimum value), supremum - upper limit (name and maximum value) | From a mathematical point of view, these elements are not necessary to determine the popsor, nevertheless they are often very useful from an illustrative point of view. These are names given to the extreme values assumed by the examined property. For example, for popsor ^{P}height it will be: ("Low": 150 cm; "High": 250 cm). |
5. | N | Size of popsor | A number that specifies the number of objects belonging to the popsor. If, for example, we study one hundred people, then N = 100. |
6. | Popsors matrix | This is an N-dimension matrix containing the values of the tested property of each object belonging to the popsor. In other words, it is a set of all the results of height measurement, obtained during the study of this hundred-person population. |
When we are dealing with the scalar "height" of, for example, John, we measure his height and say: "John's height is one hundred and seventy-nine centimetres.". We treat the popsor in exactly the same way. In the case of a scalar, we said "John's height" and we gave the number and the unit, and in the case of popsor we will say:
Someone may ask, "And why do we need that? After all, this is probably the distribution we talked about in one of the previous chapters." It it is similar but not precisely. The difference lies in the fact that popsor is subjected to various operations, such as combining popsors or transforming them through a function, for example, selection. Some other operations that have no counterparts in scalar nor in vector calculus, will be also defined. Some of them have interesting names such as: *****;
3.24.2 Popsor matrix distribution3.24.3 Traf***
Despite the fact that we have not yet defined the concept of good or evil, by using our initial knowledge of popsors, we can try to answer the age-old philosophical question "Are humans inherently good or evil?".
No one is trying to answer the question "Are men born tall or short?". The right answer is: "The average person is of medium height.". It is puzzling, however, that in the case of good and evil, we constantly hear statements that a person is fundamentally evil or fundamentally good. This origin of this illogical question has several causes. The first one is the fact that good is not a property of human object, but the state of a property. Identically, "175 cm" is the state of the "height" property. The second reason is the lack of an unambiguous definition of "good" and "evil", and the methods of its measurement. "Height" is a well-defined property, the state of which is measured in well-defined units (eg in centimetres), and no measurement units of good or evil have been invented yet.
So how can we try to answer the age-old philosophical question "Are humans inherently good or evil?" First, the question is illogical, because there is no unambiguous definition of a property whose state could be good or evil! However, if we assume that evil are those who steal and murder, and good are those who give help and assistance, then we have created the range of the state of a property: (good, bad). Now we have to name this property, for example: "Saintness". In the next step, we should determine how this property should be measured, enabling us to measure it even in a newborn baby. Only after examining all the people living on Earth, could we save the result in the form of a popsor:
3.24.4