Min-Max Math, ep. 01: What is mathematics?

Falling into the rabbit's hole

I have spent at least 50% of my life studying, practicing, and teaching mathematics. There is a wide range of misconceptions I encounter when people find out that I’m a mathematician.

“You must be good with numbers.”
“You must be talented.”
“You must be without a job, because mathematics is useless.”

These are all wrong. I am terrible with numbers, but that’s fine; mathematics is about structures, not numbers. I am also not that talented; understanding math takes hard work, not talent. (Talent makes it easier though.)

And finally, mathematics is extremely useful. All our technological advances are deeply rooted in math. Without understanding the mathematical foundations, an engineer is rendered to a mere end-user of technology. A user whose job will be automated by AI.

Thus, I am starting a new course to provide a min-max introduction to the wonderful world of mathematics. Minimally technical, maximally useful.

Returning from my summer break, I want to take The Palindrome to the next level. So, this course will be my first truly premium content, exclusive for paid subscribers. (This first part is available for everyone. There will be a free post around each month.) Instead of convoluted results and technical details, we’ll have weekly lessons about

• the tools of (mathematical) thinking,

• the fundamental mathematical objects,

• and how they relate to life, science, and technology.

Let’s get started by clearing up the essentials: what is mathematics?

Walks and graphs

There’s a reason why the best ideas come during a shower. A shower allows the mind to wander freely, unchained from the restraints of strained focus.

This is true for long walks as well, a trick well-known by mathematicians for a few millennia. These very lines are inspired by regularly interspaced physical activities, followed by short bursts of intensive work.

To prove the point, the regular daily walks of the legendary Leonhard Euler gave birth to graph theory. (Of course, Euler’s genius also played a massive role.)

Let’s take a look at the map of Königsberg (now known as Kaliningrad, Russia), where these famous walks took place.

The seven bridges of Königsberg

This particular part of the city is interrupted by several rivers and bridges. (I cheated a little, and drew the bridges that were there in Euler's time, but not now.) Euler was a playful person, so he came up with a challenge: can he take a walk that touches every bridge exactly once? This is now remembered as the problem of the seven bridges of Königsberg.

Take a moment to look at the map and try to figure out the solution. This is important, as I want to show the gears in your thinking process. Pick up a pen and paper, and manifest your thoughts. In sketches, doodles, and words. Whatever works. I’ll wait.

Welcome back.

Now it’s time to disassemble the process. Upon riding your train of thought, you might have realized that the exact route doesn’t matter. Only the key parts: the bridges and the islands.

Consider these two routes, highlighted with yellow and red.

Are the yellow and red routes that different?

From our point of view, these are the same. Both touch the same bridges in the same order, and they start/end on the same islands.

Shall we complicate things by taking the exact routes into account? Of course not.

Thus, we proceed to mentally strip the geographical and urban details. This way, we are left with a “map” that represents only the key information, and nothing else. The very skeleton of the problem.

The reduction of the problem of the seven bridges into nodes and connections, a.k.a. graphs

In this “map”, islands are represented by nodes, and bridges are represented by edges between corresponding nodes. This is called a graph, an abstraction of nodes and their connections. Thus, the question becomes: can you traverse the graph by touching each edge exactly once?

Graphs are not just for representing urban maps: they are used in electrical engineering, sociology, biology, and practically every other field of science. For instance, we can model social networks by representing people with nodes and social relationships with edges. Here is a graph modeling the structure of a small university karate club, described by Wayne W. Zachary in his famous paper An Information Flow Model for Conflict and Fission in Small Groups.

Zachary's karate club. (Source: Wikipedia.)

From concrete to abstract

The process you’ve seen here is called abstraction, the primary driving force of mathematics. Flowing from concrete to abstract, mathematics translates real-life phenomena into structures, then tools.

For instance, we translate urban layouts into graphs, then apply optimization theory to find the fastest routes between two locations.

The map and the graph

Still not convinced about the power of abstraction? I’ll give you more examples. For one, we can translate the transmission dynamics of infectious diseases such as COVID into a set of abstract rules. Rules whose analysis yields a prediction that yields a vaccination strategy.

Or, we can translate the motion of an object into a set of differential equations that give a real function. Like the motion of celestial objects. Like the Earth orbiting around the Sun.

The orbit of Earth around the Sun

These mathematical structures are defined by a formal language, a language unlike any other. Mathematical results are recorded via definitions, theorems, and proofs. This is why math is difficult. The learning curve is steep, and it often seems like magic.

To illustrate, recall the problem of the seven bridges. The question is: can you take a walk in Königsberg by touching each bridge exactly once?

The seven bridges of Königsberg, graph edition

Such a traversal is called an Eulerian path. This is what its formal definition looks like.

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Definition. (Eulerian paths.)

Let G be an arbitrary graph. A finite walk on the edges of G is an Eulerian path if each edge is visited exactly once.

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But what is a graph, let alone a finite walk? We'll talk about them in detail way down the line, but the point is: mathematics is hierarchical. Definitions are followed by theorems, and theorems are followed by proofs. For instance, the following theorem answers the problem of the seven bridges.

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Theorem. (The existence of Eulerian paths.)

Let G be an arbitrary graph. An Eulerian path exists on G if and only if there are at most two nodes that have an odd number of edges.

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So, can you cross all seven bridges of Königsberg exactly once in a single walk? I'll leave the solution for you to figure out.

We'll see the proof of this particular theorem later, but ponder on it for a while. Anyway, what is a proof? Aren't we verifying the result via experiments? No, and this is why mathematics is different from all other fields.

Fear not: we’ll build a solid mathematical vocabulary in no time. By the end of this course, you’ll be able to speak it at a conversational level.

In the next lesson, we'll talk about the definition-theorem-proof structure.

You can find the previous episodes of the Min-Max Math series here.