Modernism, Postmodernism, and Digitality
Cultural System Explained, Linear Algebra
Cultural System Explained
The idea of this book is that various different cultural systems are important to understanding the history and that the history that we have can illuminate what Cultural Systems are in play. Because we are at a particular point in time the Cultural Systems that elucidate the history will be most important though we shall show that other Cultural Systems are important at different periods of time. We also need to set out a mathematical system for looking at Cultural Systems because they are more understandable if they are set up in this form.
The first thing that we need to explain is why a Linear Algebra system is the correct form for producing historical mathematics. At the present time a certain amount of economics is used in history, but we shall show that it is better to think of history as a branch of biology. Basically, history starts out as the biology of human beings but then makes significant changes so that the biology of human beings is not enough to explain history. In biology, the most prevalent form is taking an ecology and following through on reproduction, predation, and caring for the members of the species that are being looked at. This means that we can map onto a mathematical arrangement based on the habits of the species, that is, who reproduces and how, what care they give their young, and how they die. We are most interested in predation as a signal event but remember that even death does not end the cycle for a given member of the population because even the dead become food for the living bacteria and scavenge for other animals.
This level of detail is enough to see how there is a boom in prey animals which will often then lead to a boom in the animals that prey upon them. It also means that we can see that there are bifurcations in the population which give rise to diseases that spread out. A classic example is Ludwig et al paper on the spreading of spruce budworms and its effects on the forest. These functions exist in history, and we can point out numerous times when a member of the human race died and the effects that it had on society. For example, JFK was assassinated on 22 November 1963, and it became the touchstone far beyond the local members of his immediate family. But this is only the start of mathematics in history. A simple recount of births and deaths is important for the structure of population pyramids, but the math that they give us is only a small fraction of what can be deduced from math.
This is because if one thinks about it, conception and death can be modeled as a Dirac Delta function which changes the population.[i] While this is useful if is only the dipping of the toe into what can be determined by mathematization of the data that history gives us access to. This is because there are numerous forms of Dirac Delta functions in history. While consumption is one such place homo sapiens go much farther in their use. Other forms of biology do so as well but it is a much smaller aspect of reproduction even though it forms the basis of Darwin and Wallace’s theory of evolution and several of its key points. But no one other than a human being on Terra has written a book, and yet one can model the writing of a book as a direct delta function that changes the lives of other people. This means that biology becomes different when it is put in the hands of homo sapiens who run with it to the point of needing a differing form.
This is because while there are a few Dirac Delta functions that dominate biology: reproduction, fitness, predation, and dying; there are many others that dominate history. Partially because human beings can communicate with each other far more than other animals. We structure our society more than most other creatures. All of this means that the Dirac delta function shows up in a large number of other places and has a more far-reaching effect than biology is capable of explaining. Thus, history is a descendant of biology but has enough features that are unique unto itself that it needs a new mathematization to explain the uses of various themes that are present in history but only in smaller ways than biology. A human being takes biology to another level entirely.
The first step is to explain how the Dirac Delta function works in the form of history, but that means we need to set parameters. And this means delving into vector algebra, because Linear Algebra is the best way of bringing math to history, or if you like, bringing history to math. When one does this one is immediately struck by one fact of human biology which it shares with a small number of other species: most species are programmed by their genetic information to behave in certain ways. This is not to say that they are merely automata, but their configuration at the level of the brain, should they have one, is mostly determined by genetics. But human beings as with other species have discarded a great deal of the genetic programming and allowed the individual to absorb information and thus make their behavior far richer and more complex. This means that we need to model not only genetic information but the cultural information. This means that genetics allows the animal to create different forms of behavior that are wider than can be included in genetics. This means that epigenetics and culture need to be included in the mathematics of human behavior.
Now we turn to the means of mathematical information which is the subject of a great deal of the primer. This allows us to model the behavior and find moments that are important in the current subset.
Linear Algebra
The first concept to understand is that every individual at any particular moment in time can be represented as a tensor, and any given state on that tensor is a vector. This tensor changes with time so time is a variable in each tensor.
What this means is that there are many possible states some of which are physical, such as the location of the individual, but others are theoretical. Later on, we talk about the ownership of color television sets, if you think about it this is a vector with the owning of one or more television sets as the points. This means that there are many possible sets of vectors.
A tensor is a set of vectors each one assigned to an individual. This means of course that many vectors will have no meaning at a given point in time. For example, in 400 CE no one owned a color television set. Later we will find out that only a limited number of vectors can be used on a given individual for a given vector. For example, living in a particular location is a vector and we can find out which other vectors will be necessary. To take again the vector of owning a color television set, the location is quite likely to be an important variable, but it may not be as important as just the country that the individual is living in. Beyond that Kansas City may be the only slightly more likely than St. Louis. In other words, we can limit the number of vectors from any individual which are important to the matter at hand.
This means that a limited number of vectors can be used instead of all available vectors. A simple example is whether or not a single individual is near a particular battle, and the results of this vector will tell you whether or not the individual was a casualty of this battle. This is already done in history but it becomes more important when one invokes mathematics in history.
This means that linear algebra will be an important branch of mathematics because it deals with vectors and tensors. Most of what follows is relatively straightforward linear algebra with the addition of how this linear algebra can be used for Cultural Systems in a historical context. What this means is that the linear algebra will be used as proof as to why individuals of a particular time thought as they did, and how this model changes through history.
This means that history will join other disciplines in using the techniques of linear algebra with certain features that will be noted later: history must use both Fourier and Laplace transformations given the way in which these are both defined for history, we can look at alternative views in history, but not in the way which it is normally done, and it shows how one Cultural System replaces another because of objects, ideas, and techniques. These will be explained later but realize that the mathematics is relatively much as it would be for physics, biology, or any other field.
The primary difference is that human beings are capable of making assumptions that may or may not happen and can communicate to others those assumptions and have others alter their choices. An example would be the American Civil War: when Lincoln became the president after the election of 1860, the slave-owning population realized that they needed to be able to expand and that Lincoln was probably going to inhibit this expansion. They then gathered state by state to secede from the “Union” based on this assumption. This will be a Laplace transform rather than a Fourier transform because there was no actual act or threat of an act. However, dimensional this Laplace transformation was at the root of a coming Civil War. This means that imaginary constraints can be squared and located on the real plane in history.
In both physics and biology Linear Algebra is used to manipulate data. It is also accused of political science and demographics but has often been without a deep underlying structure. This will be rectified by building up towards things such as population pyramids from the basic vector algebra.
The first point is to understand Linear Algebra which can be done by reading through a basic textbook such as Linear Algebra by Shilov or more completely Linear Algebra by Strang as well as by lecture at MIT. Because this is an entire semester of content this book will only touch on the specific points rather than being a textbook on Linear Algebra. The will do more for Xaotic systems, Single Value Decomposition, and primary component analysis as these are essential to the mathematical structure of history and Cultural Systems within that subject.
The first point is that linear algebra will represent a number of values based on experiment and measurement just as with medical science. The number of measurements will then be reduced by primary component analysis so that most of the information will be maintained. We shall then move on to xaotic analysis and show that in history and Cultural Systems analysis the limits of Cultural Systems will grow to an unmeasurably large distance because of the fact that even trivial differences in initial conditions will not be the same.
To have a measurement in a vector space certain conditions must apply:
For vectors x, y, and z and scalar values a and b:
1. The vector 0 exists.
2. ax is a vector.
3. x + y is a vector.
4. -x exists. (Inverse)
5. 1x = x (Identity)
6. x + (y + z) = (x + y) + z (Associative)
7. a(bx) = (ab)x (Associative)
8. x + y = y + x (Commutative)
9. a(x + y) = ax + ay (Left Distributive)
10. (a + b)x = ax + bu (Right Distributive)
This means that all measurements in history must conform to these requirements. Since there is very little mathematization, this is easy enough to do. Since these are logical extensions of algebra and the requirements for vectors, little explanation will be offered.
With the toolkit of linear algebra, Fourier and Laplace transformations, xaotic systems, and Dirac delta transitions a means of looking at Cultural Systems in a historical context will be presented.[ii]
The first step is to describe how Dirac delta functions are the basis of the discipline.
Key Points of the linear algebra:
1. Linear algebra forms the basis for a mathematics of Cultural Systems in history.
2. Fourier and Laplace transforms are useful in this vector space.
[i] The equivalence of a Dirac delta function is not equivalent to a legal discussion of when a member of homo sapiens is entitled to various legal forms of protection. I merely note this because this is foremost in many people’s minds and a disclaimer that this book is not about those protections.
[ii] While the common use of the word “chaotic” is the standard the problem with this is that the word “chaos” means both deterministic and random in different contexts. Therefore, I will use Xaotic s