What Is Pure Mathematics?

Mathematicians develop abstract structures to study complex realities. For example, it is often convenient to group like items together, as when the apples in the market are placed separately from the oranges and the pears, and mathematicians developed the notion of “set” to capture this. It sometimes happens that members of one set can be paired up with members of another set, and we use the concept of “number” to describe this tendency. In our daily lives we encounter numerous examples of related quantities, such as the relationship between temperature and the time of day, or between the distance we have travelled and the time during which we have been travelling. The notion of “function” is an abstract way of describing such relationships. Sometimes it is necessary to organize and manipulate large collections of data, and “matrices” are often useful for that purpose.

The pure mathematician takes the attitude that if abstractions like sets, numbers, functions and matrices are routinely useful for studying so many aspects of our daily lives, then they are also worth studying for their own sake. History records many instances of the usefulness of this approach. When Isaac Newton sought to understand the trajectories of projectiles, he found success not by studying projectiles, but by studying continuous functions. When Einstein was working out his theory of relativity, non-Euclidean geometry, a branch of mathematics developed for reasons having nothing to do with physics, proved to be indispensable. These are just two of many possible examples.

The usefulness of pure mathematics is only part of the story, however. There is also the tremendous beauty of the subject. It is hard to imagine an object more banal than the counting numbers, yet their structure is so complex that mathematicians routinely discover novel facts about them. Right triangles are all around us, but who would suspect that the square on the hypotenuse is equal to the sum of the squares on the other two sides?

It is this combination of beauty and usefulness that explains the importance and appeal of pure mathematics.

Jobs in Pure Mathematics

Training in pure mathematics can lead to job opportunities in a variety of fields. In addition to teaching and research in mathematics, there are also fields like technical writing and investment banking to consider. Mathematics is also good preparation for those entering law and medicine. The Mathematical Association of America maintains a list of possibilities.

What Do We Offer?

Students interested in studying pure mathematics should take MATH 245 as early as possible. This course provides an introduction to proofs and to topics in discrete mathematics. From there, our offerings can roughly be grouped into those covering discrete topics on the one hand, and those covering continuous topics on the other. For those interested in discrete topics, we offer elementary number theory (MATH 310), graph theory (MATH 353), abstract and linear algebra (MATH 430, 431, 434) and geometry (MATH 475). Continuous topics include real and complex analysis (MATH 410, 411, 360), and topology (MATH 435). We also offer a course in the history of mathematics (MATH 415).

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