One of the nice things I get out of this blogging gig is to get e-mails from some truly fabulous people either commenting on something they saw here, or sending me a link that might be of interest to us. Sometimes, I even get to meet them in person when they visit our Institute or, more generally, Bangalore. Recently, I had the privilege of meeting one such person: Niranjan Srinivas who graduated with an integrated M. Sc. in Mathematics & Scientific Computing from IIT Kanpur in May 2008, and is currently a Ph.D. candidate in the Computation & Neural Systems program at Caltech.
During our short meeting, Niranjan happened to mention an article he wrote sometime ago; when I asked him for permission to post it here, he readily agreed.
* * *
What I learned from an undergraduate education in mathematics
Typically, one’s undergraduate years involve several different kinds of learning: academic growth in the chosen field of study, social development arising from meeting and interacting with a fairly diverse set of people, and personal and emotional growth from teenage adolescence into full-fledged adulthood. In this article, my focus is on academic growth. Specifically, I shall try to articulate what I learned from my undergraduate education in mathematics, with its own particular virtues and weaknesses, and why I think such an education is valuable in the “real world”. I use “mathematics” in a broad sense (which includes theoretical computer science and statistics.)
Although the philosophy of mathematics and its relationship to science and logic is the subject of active debate, much mathematics may be viewed as an exercise in “pure reasoning”. An education in mathematics hence provides training in precise and careful logical reasoning. This logical thought process is a fundamental tool for solving complex problems, whether they arise in mathematics, scientific research, or industry, and is therefore a valuable skill to acquire.
Mathematics is as much a language as it is a field of enquiry. Indeed, it is a powerful language for expressing and investigating complex quantitative relationships between objects of interest. Therefore, it is the language of choice for the physical sciences and engineering. Facility with this language, which is essential to formulate and solve quantitative problems, is another advantage of studying mathematics.
Abstraction is an essential feature of mathematics, as it deals with the properties of and relationships between abstract entities. Although the entities are often inspired by the “real” world, mathematics is concerned with the abstract entities rather than the real world “instances” of those abstract entities. Even the number two is an abstract entity; one may find two birds or two stones or two mountains in the real world, but never just “two”. The ability to reason about abstract entities and formulate general questions is another skill one develops while learning mathematics. This skill is important because generalizing from the specific to the abstract and articulating a precise question that captures the heart of a complex problem is often a formidable challenge in itself.
In my personal experience, my training in mathematics has proven very valuable even though I have been working in very different fields. After I graduated, I spent one year working in the financial industry in a quantitative role. After that, I returned to academia to pursue my Ph. D. My current research interests lie in the intersection of computer science, bioengineering and nanotechnology; my work involves engineering smart molecular systems using synthetic nucleic acids. My colleagues include biochemists, physicists, mathematicians, and theoretical computer scientists, among others. Being trained in mathematics is very useful as it is often a common language between scientists from different backgrounds.
Apart from these specific skills and ways of thinking, an education in mathematics, which is one of the oldest intellectual endeavors, provides a historical perspective on scientific thought. A typical first course in calculus would consist almost entirely of mathematics that would be considered “well-developed” by the nineteenth century. Contrast that with a first course in molecular biology, which would consist almost entirely of ideas developed in the last fifty years. For an academic, this historical perspective is important because it provides context for his or her work. For the non-academic, the perspective provides one possible framework for thinking about modern issues and challenges.
However, majoring in mathematics as an undergraduate is not without its challenges. Although the mathematical education helps one acquire several important skills useful in the “real world”, one could conceivably graduate without any domain-specific knowledge about anything else. Leveraging the skills one acquires to actually solve problems outside mathematics would therefore entail acquiring knowledge about the particular field or problem of interest. However, this is usually not difficult if one has an open mind and is actually motivated to learn about the concerned subject. In my personal experience, acquiring new knowledge about a particular subject is significantly easier and faster than learning to think carefully and precisely in a mathematical way.
Related to this is what I like to think of as the “purist” trap. I think it is frightfully easy, at least if one is interested in “pure” mathematics as an undergraduate, to care only about mathematics for its own sake and not be interested in any “applications”. This is a perfectly admirable attitude after one has learnt a lot of both “pure” and “applied” mathematics, but is probably not the most helpful attitude for an undergraduate since it might bias the student against a lot of beautiful mathematics due to an arbitrary distinction between “pure” and “applied” mathematics. Indeed, historically much “pure” mathematics owes its development to particularly important and interesting real world problems. Personally, I fell into this trap as an undergraduate and it took me about three years of study to realize that a lot of mathematics I had labeled “uninteresting” due to my pre-conceived notions was actually quite fascinating.
Lastly, I think the single most important thing I got out of my undergraduate education is learning how to learn. Nearly everything I did after I graduated required me to learn a variety of new skills, and very few things I learned during my undergraduate education were directly relevant to my work. Indeed, almost every opportunity in an individual’s career would likely be based on what he or she can learn, rather than what he or she already knows. Learning how to learn is more valuable than the actual subject or field of study one chooses to focus on, because it empowers the individual to learn other subjects at will.