# Rambling Thoughts

## Algebra vs Analysis

Algebra and analysis are two major areas of mathematics, and much of mathematics is divided into these two categories. The algebraic and the analytic way of looking at the mathematical world can be very different, and I’ll explore some of these differences through over-generalizations. A disclaimer: All of these over-generalizations are based on what I’ve seen of these two disciplines; they might have no relationship with reality.

Algebra is the study of collections of objects (sets, groups, rings, fields, etc). In algebra, people care more about the structures of these collections and how these collections interact than about the objects themselves. In fact, with homomorphism and isomorphisms, the original objects become irrelevant. This is in direct contrast with analysis, which primarily studies individual objects; for example, an analyst might study the smoothness of an individual function. In many cases in analysis, studying collections of objects might just be too hard. For instance, the study of PDEs often focuses on individual equations instead of general theory because the general existence and uniqueness problem is simply intractable.

The way mathematicians go about studying the algebraic and analytic parts of the world is also quite different. Just as algebra is the study of structures, algebraic theory is also quite structured. There are endless similarities between algebraic objects, and the goal is often to classify these objects and show when they can be thought of as the same. In this way, seemingly unrelated problems can be linked and solved by the same methods. A good example of this is category theory, which leaves even the details of algebraic objects behind. The analytic world, on the other hand, is full of ad hoc methods designed to circumvent technical difficulties such as convergence issues. In fact, the ad hoc nature of analysis even makes it difficult to over-generalize.

Part of the reason for this difference between algebra and analysis is that they often have different ultimate goals. In algebra, people are always looking for equalities: Are these objects isomorphic? Can we classify all objects of this type? Analysis, on the other hand, deals in inequalities and error terms. This is evident from the very beginning, in the theory of epsilons and deltas. Instead of obtaining precise values, it’s sufficient to show that epsilon and delta are within a certain range. In order to show convergence, we just need to show that the error terms are small. Thus, there’s often no perfect bound or best approximation, and there doesn’t need to be; all that is needed is for the bound or the approximation to be good enough. In that sense, analysis doesn’t require so much structure, and people can get away with being less general.

It appears, then, that analysis deals with details while algebra takes a broader view. Both are important to have in mathematics, and any interesting problem would most likely contain a mix of the two. As much as many mathematicians would like to avoid having to deal with any area that is not their own, everyone has to learn about and understand mathematics as a whole.