The subject of today is quite elementary, and not that remarkable from a mathematical point of view. It is really a special case of my earlier, unpublished, work on Communication and Behavioural Theory. Still it might be of some interest for the uninitiated, so I will devote today to speaking of how to write good texts.
To define a way to talk, in an intuitive manner, about how to make people understand and be interested in what you are telling them about, we need to find a way to measure "the distance" between what you are telling them and what they understand out of it. This is of course done by providing a metric topology, and we must consider the pieces of information that you are sending and them recieving as discrete points. By providing a constant for what the maximum distance between these points are, we can determine how "close" the information you are sending them and the information that they percieve are, and thus determining if the idea is actually transferred.
Different types of texts have different values of the constant (often denoted in topology as the "delta"), for instance Mathematical texts have relatively small deltas compared to other types of texts because of the exact precision of the theorems and logical arguments. Poetry, especially of the abstract kind can have very big deltas, and it is in fact this that can allow the reader for a subjective interpretation.
Even though mathematical texts have relatively small deltas, these still allow for very different interpretations and the restriction of a certain delta can still allow the reader to view things from many different perspectives. Though the delta is specified, the amount of dimensions that the metric topology have is not, and if every property could be represented as a one-dimensional independent component, we could easily add more dimensions and adjust the metric accordingly.
Different types of texts, especially books have different ways that they should be written in as well. We might want to define some quantity measure how intriguing the plot is as a function of how many pages you have read, to provide some examples of writing styles.
Above is shown have a good thriller is written, or rather, how it is planned. Your goal is to have a story that reaches its peak at the end (this is generally a good way for most stories), which this one clearly does. Different rules apply for dramas though, as well as biographies, where you would rather have an even distribution of excitement.
The thriller is characterized by this discontinuity near the end, which represents a shocking twist in the story such that you realize that the storyline really went the more smooth path of the dashed line. This is often near the end, which is marked with the filled black dot at the end of the graph.
Having a continuous graph is preferable, because it helps the reader to follow what is happening. If there were to many discontinuities, the story would be impossible to follow, and it would seem more like a dadaistic poem. Textbooks are preferred to be as smooth as possible, even though it wouldn't be a measure of the storyline, but instead the trail of thought leading to the ideas, such that they are understandable. Consider as a theoretical experiment taking a Calculus book, and randomly rearranging all the texts and theorems in it, and then try to read it. Even for a devoted mathematician it would be almost impossible to comprehend it, and even less enjoyable.
Also, another important concept for writing books is the initial conditions, telling the reader what level of knowledge is needed to understand it. This applies both to textbooks as well as for novels(eg. series such as Lord of the Rings). If the initial condition for the book is not met, the reader will have a hard time understanding it because of the discontinuity that will arise of the readers progress.
Those are some important things to consider when writing an arbitrary coherent text.
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