Some reasons for studying mathematics

A different sort of post today.

I've never had much interest in doing maths. I'm fundamentally incurious about mathematical objects, I don't feel joy or excitement when I prove a theorem, and I rarely enjoy the process of solving mathematical problems. The typical questions that I encounter in my course of mathematical study don't appeal to my intellectual hunger in the same way that, say, a computer science problem can take over a corner of my brain for weeks. This is how I know that mathematics is not for me. So why did I choose to study it?

When I studied algorithms in high school I learned a bunch of graph search techniques and dynamic programming and computational geometry etc, but more importantly I developed a sort of mental toolbox for reasoning about systems and processes. Items in the toolbox could also be described as "problem-solving techniques" or "approaches" or "paradigms" or "meta-algorithms"; they can feel abstract and fuzzy and difficult to describe, but certainly seem to exist as distinct concepts in my brain so that I can manifest them at will. They're obviously useful in the context of algorithmic problem solving, but they're also applicable to problems in totally unrelated fields. There are hundreds of tools in my toolbox of various levels of abstraction and usefulness. I'll have a go at identifying some of the tools in my algorithms toolbox:

• Thinking in terms of growth rates
• The notion of "greediness"
• "Sliding window" methods: we can solve a sequence of overlapping problems by solving the first problem and efficiently transforming that solution into a solution for the subsequent problem, and so on.
• Divide and conquer
• Laziness vs eagerness
• Invariants
• "X reduces to Y", meaning in a precise sense "X is at least as hard as Y, and we can't solve X without solving Y"

This mental toolbox represents the greatest benefit I received from studying algorithmic problem solving.

The study of mathematics has given me a whole new set of tools to add to my toolbox. As in the examples above, these tools tools are used constantly in the study of maths, but they have little to do with mathematical problems themselves. These mathematical tools are often far more insightful and valuable to me than the problems that they help to solve, and I can apply them to problems outside of mathematics. On its own, this is a good enough reason for me to study maths (it's not the only reason – see below). In this entry I will list and briefly describe some of the tools I've added to my toolbox in the past few years of maths education. Most of these tools can be expressed as statements about systems or structures. I may update this list over time as new tools occur to me. All tools are stated informally.

• Action to structure: we can study actions on a structure by treating the actions as first-class objects themselves. (for instance studying symmetries via the symmetry group. Reverse of "Structure to action".)
• Basis: in some cases, the behaviour of a system is wholly described by its behaviour on a small set of representative inputs.
• Continuity: the property of being able to induce arbitrarily small changes in the output of a system via correspondingly small changes in the input.
• Contradiction (proof by): one can assert a proposition by supposing that it doesn't hold and subsequently taking a logical sequence of steps to arrive at an impossibility.
• Convexity: the property of "not having dents"; not necessarily in a geometric setting.
• Density arguments: if a property holds for a subset $A$ of a set $X$ and any element of $X$ is the limit of elements in $A$, then we may be able to extend the property to all elements of $X$. (This is useful when $A$ consists of "simple" examples where the property is easily explored. The obvious example is the treatment of simple functions in integration theory.)
• Diagonalisation: if we have a countable number of instances from some class, we can generate a new instance of that class simply by ensuring that it differs from each previous instance at exactly one point.
• Differentiation: we can learn about a system by observing its instantaneous change.
• Duality: Two seemingly disparate domains sometimes pair up conveniently, so that we can study items in one domain by looking at the corresponding item in the other domain.
• Eigen-: we can learn about a system by studying the inputs on which it is particularly well-behaved.
• Exchange arguments: we can show that a state is optimal by taking any other state and showing that it doesn't get any worse if we bring it a step closer to the proposed optimal state.
• Graphs: complex systems and structures often have natural representations as networks.
• Integration: we can learn about a system by observing its cumulative effect.
• Isomorphism: rather than speaking about the equality of two objects, it's often more meaningful to consider equivalence with respect to some structural property.
• Lifts: methods that transform solutions for a problem in one domain to solutions in a broader domain, hence "lifting" them.
• Limits: properties that hold as we get close to something may or may not hold once we get there, depending on the attributes of the surrounding space.
• Locality vs globality: properties that hold at a micro level often don't hold at the macro level and vice versa.
• Orthogonality: a stronger or more precise form of independence. If A, B are orthogonal then we can change A in any way without affecting B and vice versa.
• Pigeonhole principle: if you're placing balls into boxes with more balls than boxes, at least one box will have more than one ball. (And in reverse: if there are less balls than boxes, at least one box will be empty.)
• Probabilistic method: informally, if a property has non-zero chance of holding for a randomly-selected element of a set, then the property holds for at least one element of the set. If the mean of the values in a set is $x$ then at least one of the values is at least $x$ and at least one of the values is at most $x$.
• Projection: we can learn about a complex structure by studying what it looks from a fixed perspective. I'm stunned at how often the notion of projection appears in mathematics and how abstract it can become. We can project in the literal geometric sense – as in projective geometry, or projecting a vector onto another – but we can also project the integers onto the set {0,1} via the equivalence relation of parity, or we can project a vector space to another vector space via a linear map, or we can project a sigma-algebra to a sub-sigma-algebra via conditional expectation. Projection is closely linked to the notion of homomorphism – so much so that I wish our study of homomorphisms had begun with an explicit acknowledgement that homomorphisms are precisely the maps that behave "projectively".
• Representatives: we can sometimes study the interactions of complex structures (often equivalence classes) by instead considering the interactions of individual representatives from those complex structures. (e.g. dealing with cosets via coset representatives.)
• Structure to action: we can learn about a structure by constructing an action of that structure on some class of objects and studying the action instead. (for instance studying a group via its group action on a set. The reverse of "Action to structure".)
• Substructure: we can learn about a structure by studying the smaller structures embedded within it. (see subgroup, subspace, submodule, subgraph, etc.)
• Symmetry: we can learn about a structure by studying the ways in which we can transform the structure without changing the way it looks.
• Two-player game: some notions can be understood in terms of a two-player game oppositional game. For instance epsilon-delta proofs interpreted as a game between an epsilon-picker and a delta-picker.

Why else do I study maths if I fundamentally don't care for it? I find maths to be an exceptional form of brain training. Most of this training happens via two mechanisms:

1. Structuring and summarising information: as part of my maths education I'm expected to be able to regurgitate in exam conditions something like fifty theorems and proofs per course. Each proof can consist of pages of dense equation manipulation or tricky argumentation, which is far too much to learn by rote. Instead, I need to memorise the core ideas of each theorem and proof in summarised form, but in a way where I can elaborate on those summaries as much as required until I arrive at their full unsummarised forms. I also need to develop a rich mental map of the relationships between the objects of study. Both of these processes develop my reasoning abilities.
2. Coping with abstraction: the objects of study in pure mathematics become increasingly abstract throughout a course of study, but we still need to reason about them. So we develop skills and techniques for coping with abstraction. For instance, we can fall back to geometric intuition (vector space = "space with arrows that can be added to each other and scaled"; orbit-stabiliser theorem = "sending a fixed vertex of a polyhedron to all other vertices then counting symmetries which fix that vertex in place"), as long as we keep in mind the limitations of that geometric intuition. Or we can comprehend abstraction in terms of sufficiently distinct representative examples whose generalisation "spans" the full abstraction (for instance it's difficult to understand abstract measure theory in full generality, so we develop most of our intuition by comparing and contrasting canonical examples: Lebesgue measure, counting measure, Hausdorff measure, Dirac measure, ...). If we think of an abstract concept as a space of instances of examples ("measure theory" as the study of the set of measures), this technique corresponds to choosing a set of examples at different points on the boundary of the space that help us to reason about any instance inside the boundary. I suspect that this skill is exercised more in mathematical study than in any other area of study.

Studying maths for me feels a lot like going to the gym. My interest in number theory or complex analysis is about as deep as my interest in push ups or squats. To me their value is as means of self-improvement rather than as ends in themselves. I deeply respect my lecturers and mathematically-minded peers, but as my study has progressed and the mathematical objects in consideration have become more contrived, I increasingly struggle to understand the motivation behind a career in research mathematics. But of course many academic mathematicians say the exact same thing about careers in tech or finance. Different strokes for different folks...