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!

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Basically, there are different levels of depth achievable in generalizations.

In very short, yes, that’s a valid summary, though I think your summary is really only the first step. After all, a computer could presumably be programmed to generalize concepts more and more. I would probably summarize as follows:

i) there are different levels of depth achievable in generalizations.

ii) there are infinitely many different directions in which one can start generalizing.

iii) for any given problem, certain directions and certain levels of depth are more interesting/important/useful.

iv) abstract thought is the art of being able to identify those directions and levels of depth mentioned in iii).

Again, I think it’s a hard concept to “summarize”, but I also do think it’s more than simply generalizing to various degrees. I’d love to hear what you think!

Yup, I agree. I took 2 and 3 more as givens. 4 is pretty deep to understand. Is it the process of identifying, or creating? If you say identify, then you’re assuming that they already exist and have yet to be discovered. Do they already exist before we think about them, or only when we have the thought?

I’m trying to think of what directions are possible, i can understand horizontal and vertical for breadth and depth, but are there other directions possible? Funny trying to make abstract ideas about what abstract thought is hahahaha.

Sorry for my lateness on this, and great questions! The question of whether or not ideas exist outside of us or are created by us is as old as our ability to ask it. Thus, my own two cents here will really mean very little in the grand scheme of things, but here goes. I think in general mathematicians are Platonists in this regard, meaning that we in general believe that mathematical ideas “Exist” with a capital E. This has a lot to do with Penrose’s “Three worlds” view, which I wrote a post about in the Blog section. That said, what we’re talking about here with general abstract thought has to do with non-mathematical ideas, which I think a lot fewer people would see as “already existing”. For example, the existence of the integers is a very objective notion, and therefore it might be reasonable to think that in some ideal Platonic world of mathematical forms—or something—the integers “really exist”, whatever that means. However, the notion of “walking the dog” is a very human one, and we could easily suppose that some life form on the other side of the planet is such that domesticated pets don’t exist, and thus neither does the idea of “walking the dog”. In general, we’d like to think that ideas that are somehow dependent on the nature of humans are constructed by humans, whereas ideas that seem to transcend our silly cultural or social norms and habits at least have a chance of “really existing”.

In terms of “directions” of abstract thought, I’m with you in that I generally visualize it is horizontal and vertical, but I think the question is itself ill-defined. Namely, by trying to visualize it we’re giving concrete meaning to notions that are inherently abstract, so the horizontal vs. vertical discussion is really only a tool (and a sometimes counterproductive one) and not really reflecting anything “real”.

Again, though, these are just my thoughts on the matter. Libraries can be (and have been) written on this stuff, so there’s no doubt that my short and rambling post here will do very little justice to the issue, and probably only serve to piss off some philosophers 😉

I love the questions though!