# Studying a Mathematical Subject: An Oxford Physics Master's Graduate's View

## Wondering whether to study a mathematic subject over a humanity? Sanjay (1st Class BA Physics graduate & Master's in Mathematical and Theoretical Physics with a Distinction, University of Oxford) gives his view on the Pros of Mathematical study & essential study skills!

In some ways, Maths and Physics, being the most mathematical of the sciences, are the easiest subjects to learn. Ideas are neatly and logically defined through the language of mathematics, and the scope of a course is also clearly laid out. To see what I mean, let's compare with a humanity.

Suppose you are learning about the First World War in History. There are countless things you can learn that will help you do well in the exam, and the relationships between these things are often subtle and up for debate. For instance, a syllabus may not mention any need to know about the Russian Revolution in detail, being a huge topic in its own right, but this would surely help you understand the War more thoroughly. Similarly, the question of whether certain events before 1914 contributed more or less to the outbreak of War can be answered in many different, but equally valid ways.

On the other hand, if you are studying calculus in Maths, you need to know relatively few particular things, such as what a derivative means, and how to differentiate certain functions. If you are studying mechanics in Physics, you need to know Newton's laws of motion, conservation of momentum, and know how to apply these to standard types of problem. Compared to huge and messy topics like the First World War, this is, I think, neat and nice.

It's worth taking a moment to ask why this is. It's slightly different between Maths and Physics, but the underlying point is that the language is mathematical.

In Maths, we study theories, such as calculus or algebra. These theories have particular starting points (unlike the First World War) such as axioms or definitions, and proceed logically (also unlike the First World War) through a series of well-defined steps from these to obtain results. Knowing the series of steps between the starting points and the results, and understanding why and how each step is taken, is basically all you need, together with some experience doing problems to solidify this understanding.

In Physics, objects of study are perhaps better described as models than as theories. For instance, in a course on mechanics, we might consider what happens when two perfectly spherical billiard balls on a frictionless table collide. Frictionless tables don't exist - or rather, they exist only in our model. Similarly, when you learn about nuclear physics, you are learning a particular model. There are many subtleties in more accurate models that are completely missed out. But in the model you are taught, there are certain rules, which you can learn, as well as a rationale behind them. The nucleus as you understand it only exists in this model. Models are what saves us from the hugely messy nature of reality, which is often extremely difficult to handle. It is only really in the second half of a degree in Physics that you start to look at the models that are actually used in research - or even later - I had to wait until my Masters to learn quantum field theory, the framework of particle physics.

What this all means is that as long as you know a certain prescribed and relatively short list of facts, and really understand the relationships between them, you will probably do well in the course. These two aspects, knowing facts and understanding their relations, are in fact symbiotic. On the one hand, you obviously can't understand relations between facts if you don't know the facts in the first place. On the other, understanding relations helps you understand why certain facts are important, or why we make certain definitions; the better you understand how a particular fact fits into the whole framework the more firmly it will be committed to your memory.

Of course, much of this is true of any subject, but again, it is because by nature of mathematical subjects the required facts and the relations between them are clear, rather than nebulous. Understanding that mathematical subjects are, in this sense, 'easy', is the key to changing your approach to studying them so as to exploit this feature.