What is Math Really About? part 2- models

When we try to describe a situation in math, we come up with some collection of objects that give all relevant information to understand it. To describe a family tree, we might want a set of people (the family) and a set of triples (an ordered group of three things) where the first in the triple is the mother, the second the father, and the third the child. The individuals in the set of triples are from the set of people, of course. There are some other constraints, such as that a person can't be their own parent, but lets not go into too much detail. In the definition (the theory) we don't want a particular set of people, or particular set of triples, but rather the notion that a family tree consists of people and parents. There are other ways to do this, but if we chose this way then a description might be "A family tree consists of a set F (called the family), and a multi-relation FxFxF. A family is given by (F, FxFxF) such that for all (a, b, c) in the set of triple, a, b, and c are distinct.".

While studying these objects, we will look at their properties. This may include ways of ordering them, combining them, and describing parts of their structure. That one of the most important things to study is how to translate the objects into each other, especially when the transformations preserve some aspect of the structures. One of the reasons such transformations are important is because they can be used to describe properties (often by some single objects that embodies the property in some way, and then the transformation to and from that object and others). Because of this, a theory will generally have some objects it is concerned with, and some notion of transformation that is appropriate and interesting for that object. Remember that in math we take situations we find interesting and encode them as theories, so there should be theory that describe this situation of having objects and transformations and combinations of these objects. This theory would be pretty fundamental and would have far reaching consequences- recall that the theory of symbols is logic, and we are talking about the models of these theories (the objects they describe). This theory is called Category Theory. A category is exactly what we just saw- some type of object, and some appropriate way to translate or transform between objects. In Category theory we tend look not at the internal structure of the object (which is based on definition or interpretation) but at an objects relation to other objects, so we often give properties as objects and some class of transformation between them. Category theory has pretty clear place in math from this perspective- it generalizes the notion of model. This makes it less surprising (although I personally found this immensely surprising) that categories have something call an "internal language" that is the logic they describe, and logics have an associated category.

I am not a mathematician, so take my generalizations about the importance of Category Theory with a grain of salt- I'm not sure its the amazing paradise I've made it out to be. However, it does describe an important situation, it is extremely abstract, and it is a great deal of fun.

So thats all for part 2. The last thing I'm going to go over is the manipulation of symbols in math, which, as a computer scientist, is really my place in all this.