In various posts throughout this site we have discussed the notion of abstract thought. Abstract thought is the primary tool that mathematicians use in practicing their art form. In fact, one could reasonably say that mathematics is the art of pure abstract thought. The only problem with this, however, is the potential circularity in the reasoning: math is abstract thought, and abstract thought is that which mathematicians do. Thus, we need to try to attack abstract thought directly. This is no easy task.
Generalizing: Driving your car = skydiving
Abstract thought is very closely related to the mental process of generalizing. Another way to think of this is that abstract thought is that which explores what something really “is”.
For example, I am currently drinking a glass of water. I can generalize this in an infinite number of different ways, but here are a few: I am drinking water, I am consuming water in some way, I am nourishing my body, and I am doing something. In each of these cases, the statement “I am drinking water” is only a special case. Namely, if I am drinking water, then I am certainly drinking, I am nourishing my body, I am consuming water in some way, and I am doing something. The converse is not necessarily true. For example, I could be drinking orange juice, in which case “I am drinking” is true, but “I am drinking water” is not. Similar counter examples can be found for the other generalizations.
This generalization is nice because anything that I can say about drinking, or nourishing my body, or consuming water in some way, or doing something, will also be true in the case of drinking water. For example, if I say “drinking is good,” then it will also be true that “drinking water is good”, because drinking water is a special case of drinking. You can think up several different examples, and it’s usually pretty fun to do so.
More than just generalizing
Abstract thought also includes the act of appraising the value of a certain generalization. In other words, it is possible to “over-generalize” and reach a point of generalization that is no longer fruitful. In the above examples, “I am doing something” would be a point of over-generalization in my opinion. This is because if I want to make a meaningful generalization of “I am drinking water,” then I don’t want to generalize to the point that “drinking water” and “fighting a gorilla” are both special cases of the same thing.
This is, of course, a matter of taste in this instance. In mathematics, however, the extent to which an idea is generalized is immensely important for making meaningful progress. For example, if I took an object that could be generalized to a group (which is a very special type of set, with some added structure) and “over-generalized” it to a generic set (because a group is a set, but a lot of sets are not groups), then I will have lost a lot of meaningful information about the object. Yes, it is true that anything that I prove to be true about a set is true about a group, but there are likely many important things that I can prove about a group that I can’t prove about a set, and I therefore might not want to generalize everything to a set.
This is how abstract thought is more than mere generalization. It is also the intangible knowledge of when to stop generalizing. We will often see the power of this type of thought, and indeed it is abstract thought that makes all of math “go”. For now, however, it might be fun to try to generalize everything in your life to an almost comical degree. For example, driving your car is a special case of driving a vehicle, which is a special case of driving, which is a special case of transporting yourself. Sky-diving is also a special case of transporting yourself (transporting yourself from a plane to the ground, quickly). Thus, driving and skydiving are, in precisely this way, the same!