As someone who deeply enjoys mathematics, both on an intellectual and esthetic level, I have inevitably been confronted many times with some remark along the lines of

I never used any of the math I learned at school later in life.

or

Mathematics is just intellectual masturbation, no better than philosophy.

or even

Well this all nice and well, but what really matters are the practical applications.

Such remarks are usually expressed in a slightly condescending tone, with the implied signal that my enjoyment of mathematics is not something that Serious People™ who do Serious Things™ do for a living, or even at all. Serious people, so the meme goes, only care about math insofar as it has concrete applications.

This view comes in many flavors, and is pervasive throughout society at large, in particular among the people who decide how math should be taught at schools. As a result, Math curricula end up being designed around “Core skills” and “Learning Outcomes” which in practice involves making students learn how to blindly apply algorithms in a rigid manner.

Such “result-focused” approach to education has already been lamented before as a tragic smothering of children’s creativity to make obedient drones out of them, so I won’t dwell too much on it here.

What I wish to discuss in this post is something I see as a fundamental mistake in thinking about the usefulness of mathematics, which is one of the generators of the thinking I described above. That mistake is viewing mathematics as a set of tools that one uses to solve different types of problems like one uses a hammer or a screwdriver, and then stores in its box.

The reason I think this is mistaken is that, even though mathematics can be used as a box of tools, it is the wrong way to think about it, especially when trying to learn mathematics, as it makes the math you learn less useful to you than it could be. This is counterintuitive, and boils down to a difference in mindset, which I will attempt to convey.

Given that I am trying to transmit an antimeme, I am not optimistic about my chances of success.

Mathematics is a language

“Philosophy is written in that great book which ever lies before our eyes — I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.” – Galileo Galilei

Let me provide a definition of what mathematics are with regard to their unreasonable usefulness in science and society¹. Mathematics is a language. A way to describe the empirical world via abstract symbols. Moreover, this language is formal and abstract. Formal because contrary to natural language, mathematical statements are precisely formulated so that there is no ambiguity in their interpretation². Abstract because these statements never refer directly to objects from the empirical world, but to purely symbolic objects.

I should qualify what I mean by “empirical world”. I use it to mean the set of all possible experiences that an observer (e.g. a human) can have. This includes the “Natural World” that hard sciences concern themselves with, but also thoughts and emotions, anything that can be felt or known³.

When the language of mathematics is used to describe a situation from the empirical world, for example “one apple plus one apple equals two apples”, we are making the assertion that a purely mathematical statement (“1+1=2”) adequately describes this empirical situation. In this case, we are asserting that the language of arithmetic is appropriate for counting apples.

Sometimes, new mathematical language is created for the specific purpose of describing certain aspects of the empirical world. When Newton had his epiphany next to an apple tree⁴, he was trying to find a mathematical theory to describe the movements of bodies, from an apple falling from a tree to planets revolving around the sun.

Sometimes, mathematical theories find applications decades after it was initially formulated. When Einstein revolutionized physics with his theory of General Relativity⁵, he used the mathematical language of Riemannian Geometry, which had been developed roughly forty years prior⁶.

The point of these examples is to insist on the fact that applying mathematics is a two-step process. The first is recognizing an abstract pattern that can be described using mathematical language. The second is using the tools from that language to gain insight into what you’re trying to model.

The first step is often neglected, but it’s arguably the most important part. Einstein is considered a genius in part simply because of the idea to use Riemannian Geometry to describe the universe⁷. The most successful applications of mathematics start with someone recognizing in the wild a pattern they’ve encountered before in another context.

Tools vs. Patterns

I have just used the words “tools” and “patterns” to describe mathematical ideas. Because my use of these words may be idiosyncratic, let me define them.

• A Tool is something one uses on few occasions to perform specific tasks.
• A Pattern is something one sees. They can be recognized at any moment.

The main difference between the two is really a matter of stance. To use a tool, one first needs a problem to apply it to, and to muster the intent to apply it. To see a pattern, one need only adopt a mindset of constantly looking out for them.

Tools are only as useful as the number of problems you use them on, whereas patterns allow you to find new problems. I think the reason so many people complain that the math they learned was useless to them is that they thought they were handed an Abstract Hammer 5000™ that can only be used on Patented Abstract Nails 3000™, without realizing that this is literally just a hammer that can be used for other things than just hitting nails.

Granted, seeing patterns is more difficult than using tools. It often takes some imagination and creativity to realize that one problem is just the same as another in disguise. That’s because mathematical patterns are abstract. The exact form in which they are instantiated is irrelevant, what matters is the behavior of those instantiations.

How to make tools pay rent

While some people are naturally more gifted at discerning abstract patterns than others, I believe this is a skill that can be trained. A good way to do that is to constantly try to describe the world around you using the mathematical language you know. Like, at any moment, focus on whatever you’re doing, or the place you’re currently at and think a bit about how you would go about describing it mathematically⁸.

Maybe you’re in public transport looking at the route map, and think of applying graph theory to it. Maybe you’re trying to solve a Rubik’s Cube and you try to formulate it in the language of group theory.

Maybe you’re taking a walk in a park, and you start comparing plant leaves to surfaces. Some of them look “elliptic”, others “hyperbolic” or even “fractal”.

It won’t necessarily work super well, and you’ll end up with something a bit contrived. That’s okay, your abstract muscles are still developing. Every once in a while, though, you’ll land on something that just fits somehow, and you’ll try to use the mathematical language you just applied to draw a conclusion about what you’re trying to model, and it will work!

When that happens, you will have tasted the joy of mathematicians.