3 Models and Systems
3.1 Models
Finally, before reaching the primary topic of this site, there is one last topic I would like to explain. In mathematics, models have a formal definition, but it isn’t really necessary to use it here. The important thing is that a model uses assumptions and works within a given universe.
For example, consider the band Ghost. In 2013 they released a cover of Roky Erickson’s song “If You Have Ghosts” (Ghost 2013). One of the lines in the lyrics reads almost like an implication: “If you have ghosts, you have everything.” Normally you would read this metaphorically, but if we change the model through definitions, we can change the meaning.
Let ghosts mean specifically the band, and let everything be defined as music. These definitions will make up our model. Then consider that if you listen to Ghost’s cover, you have ghosts. Thus, in our model, by modus ponens we have everything, or in other words, we have music.
For a more formal version of this quasi-proof:
Proof.
Definitions for the model, followed by modus ponens:
\[ \begin{array}{l} \text{ghosts} := \text{the band Ghost} \\ \text{everything} := \text{music} \\[1em] \text{ghosts} \implies \text{everything} \\ \text{ghosts} \\ \hline \therefore \text{everything} \end{array} \]
Models, however, are bound by their definitions and constraints. This means that if you change the assumptions of the model, you can change the output of the model.
With our previous example, if we define ghosts as something always true, such as the tautology “it is raining or it is not raining”, and define everything as the contradiction “it is cold and it is not cold”, then we have changed the model:
Proof.
\[ \begin{array}{l} \text{ghosts} := \text{it is raining or it is not raining} \\ \text{everything} := \text{it is cold and it is not cold} \\[1em] \text{ghosts} \implies \text{everything} \\ \text{ghosts} \\ \hline \therefore \text{everything} \end{array} \]
This shows how an argument can be logically valid but not sound because of the model it is in. Expanding a model to work with a large number of things means introducing a universe.
For this example, I will use all dogs as our universe. We can say that every dog in our universe must be a mammal.
A formalized version of that would be:
Let \(S\) be the set of all dogs and \(M\) be the set of all mammals.
\[ \forall e \in S,\ e \in M \]
This is a complicated way of saying that if we put all the dogs in one group and all the mammals in another, the group of dogs is fully contained in the group of mammals.
3.2 Systems
Generally, a system is a collection of interacting parts. Reading this website requires several systems working together. Consider the nervous system: a group of cells that work together for sensation, among other things. As an organism, you have a myriad of systems keeping you functional and alive. More importantly, notice how the cardiovascular system has different parts that depend on each other, such as how the heart moves the blood but depends on the lungs to oxygenate it. Apart from the organic systems powering you and me, in order for this website to be accessible, you need some sort of computer system. This computer system could be a phone, a laptop, or something else. A phone seems like a single self-contained thing, but inside is a system of parts and circuits working together to produce a response to each input. Systems can be social, too: a club or a church is a system of people.