# Is it possible to generalize without abstracting?

According to Wikipedia,

Abstraction: Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena

Generalization: A generalization is the formulation of general concepts from specific instances by abstracting common properties.

I can think of abstraction without generalization, in the case where there is only one or same number of objects even if you go higher up the abstraction. e.g. if there is only one human and that human is the only mammal that exists.

And while it seems like people talk of abstraction and generalization as two distinct things, I can't think of how one can generalize without abstracting out some property.

• Well let's say you prove a theorem about an equation with a parameter. You first prove it for a single, easy to work with parameter (say $p =1$) and then go on to prove it in the more difficult cases, $p \in \mathbb{R}$ for example. I would say this is generalization, but does not really require any abstraction. – rubikscube09 May 19 at 0:29
• Thanks, @rubikscube09 but isn't this similar to generalizing an equation about squares to a more general one about rectangles via abstraction (removal of property of all sides being equal, or removing p = 1) – csp2018 May 19 at 0:46
• sure, but i don't really think that going from squares to rectangles is really an abstraction, so i guess i dont agree with your premise. – rubikscube09 May 19 at 0:47
• @rubikscube09 Hm, interesting. Would you consider rectangles as a more general case of squares? Doesn't going from squares to rectangles require abstracting out the property that all sides are equal? – csp2018 May 19 at 1:01
• Here is my vague notion of it all. Abstraction is taking an idea, removing certain details and making it sufficiently independent of anything but mathematical concepts. Generalizing is simply enlarging a class of objects. In probability theory and stochastic processes, many different scenarios can be abstracted into the same model: a counting process may model the number of telephone calls, number of insurance claims, number of earthquakes, etc. Generalizing the model from having a constant mean rate of arrivals to a deterministic function of time giving the mean rate enlarges its uses, etc – Nap D. Lover May 19 at 2:14