Tufts Blogs

I am a double major in Philosophy & Mathematics. Yes, I know – these two fields at first could not seem more different. One of them is a model of precision & exactitude, and has only built upwards across millennia, and the other is notorious for being the token field that has not solved any of its core problems. However, I also cannot shake that in my mind, these two fields are the most fundamental for any human inquiry, and the basis to ask questions in any other field.

This semester, I will be focusing some of my blogs around the classes I am taking within both of these majors, and sharing what appeals to me about them. My first post will be focused on Math 145, or Abstract Algebra I. It is my favorite class this semester! This class is centered around the notion of a group. Put in simple terms, it is not too different from the colloquial notion: a group is basically a collection of objects that can be manipulated in specific ways: every group has an operation that you can use to “combine” objects inside it, it has an “identity” that each object can be combined with to keep it the same, and every object has an “inverse” within the group that it can be combined with to result in the identity. It turns out that groups can be used to construct many mathematical objects we know and love, including the integers and the real numbers, and are also the foundation of linear algebra and have applications as wide-ranging as quantum computing, cryptography and fundamental physics.

However, what I like the most about this class is not just the content but the method. Unlike the more elementary mathematics classes of high school or below, a true college math course like Math 145 focuses on deductive reasoning and a true intuitive understanding of the material. This is because the dominant type of question in such classes is not one where you are asked to “calculate” something: you are expected to rigorously prove your answers to questions. The questions too, are not simply about applying some principle to solve a calculation problem, but rather, to rigorously say something definite about an object. For example, it might be required to show that a group of objects consisting of the remainder of the integers divided by n has subgroups divisible by some integer m that is a factor of n, to help you better understand the structure of what you are learning about. At least for me, the process of truly immersing myself in mathematics and absorbing knowledge regarding the beautiful structures underlying it via osmosis is one of the most intellectually and satisfying activities possible.

At heart, mathematics and philosophy both involve the same core skill: argumentation. In philosophy, the argumentation is often more negativistic (at least at the undergraduate level), which is to say, given some author, the objective might be to examine the flaws in their reasoning and tentatively propose certain alternatives. I find the argumentation in mathematics generally more intuitive: in the sense that maybe the argumentative strategy involves rejecting a certain claim, but ultimately one is trying to arrange the nuts and bolts of a system that is presupposed to exist rather than trying to see if it could exist. I tend to find this certitude comforting, and I would recommend anyone else who likes it to take this class.