The Games Mathematicians Play

I.

One thing I always found frustrating is that no one properly explained to me what modern mathematicians do.

A few hundred years ago it was obvious. People did experiments. Other people (or often the same people doing the experiments) invented new mathematical tools to describe what they saw.

Advancements in physics and mathematics went hand in hand.

But at some point both disciplines became increasingly decoupled.

Nowadays most mathematical research activity is largely detached from physics.

One reason is certainly that progress in physics has significantly slowed down. All mathematical tools needed to describe experiments was already invented and well understood.

There was also a vibe shift pushed by guys like David Hilbert towards more abstract mathematics. Abstract mathematics is “more pure“. Hence this is what the smartest minds should focus on.

It took me far longer than I’m willing to admit to understand what games mathematicians are playing now instead.

II.

Here’s an example.

Take an intuitively familiar notion like “symmetry” or “size”.

Then ask: “What even defines a symmetry?” or “What even defines a size?”.

Let’s start with symmetry.

Imagine I’m holding a ball in front of you. You close your eyes. I do something to the ball. You open your eyes. If you can’t tell I did anything, that thing I did was a symmetry.

Rotating the ball? That’s a symmetry since you can’t tell.

Moving it three feet to the left? Not a symmetry since you can definitely tell something changed.

Now we can’t just think about the symmetry of a ball, but also of, say, a square, or a set of balls, or even something more abstract like an equation.

The idea remains the same. When a transformation leaves something unchanged we intuitively can imagine it as a symmetry.

Thinking about this gives you a list of common-sense criteria that all intuitive examples of a symmetry fulfil.

(1) Doing nothing counts as a symmetry.

(2) If you do one symmetry and then another, that’s also a symmetry.

(3) Every symmetry can be undone by another symmetry.

These three rules capture the essence of symmetry. Any transformation that follows these rules counts as a symmetry, even if we can’t easily visualize it.

Now let’s look at “size” the same way.

Let’s take a normal number like 3.

What is its size? “Uh, 3?” you say, looking at me like I’m an idiot.

Okay, what about -5? Is its size -5, or is it 5?

Most people would say 5 since we care about how far from zero it is, not which direction.

This is what your 7th grade teacher called “absolute value.”

Just as for symmetries we can now come up with a list of criteria that captures the essence of what “size” means:

(1) The size of the number 0 is 0, and no other number has size 0.

(2) If you multiply two numbers together, the size of their product is the product of their sizes.

(3) If you add them together, the size of their sum is less than or equal to the sum of their sizes.

Take a moment to verify the familiar absolute value follows these rules. The size of 0? Zero, check. Multiply 3 × 4? Size is 12, which equals 3 × 4, check. Add 3 + 4? Size is 7, which is less or equal than 3 + 4, check. It works.

So far, so boring.

But once we did the boring work of rigorously defining the essence of a common sense notion, we can start to think about what other examples fulfil it besides the intuitive ones we used as our starting points.

Take symmetry first. This is how we discover plenty of strange symmetries beyond the familiar ones that are easily visualized as, say, “rotational symmetry of a ball”.

Like the symmetries described by groups like SU(3) that show up in particle physics, or weirder ones like the exceptional group G2 that has 14 dimensions and no simple geometric picture at all.

Same deal with size.

Here’s one: pick a prime number, p.

Now declare that the “size” of a number is based on the highest power of p that cleanly divides it. So if p is 3, then 18 has “size” related to 9 (which is 3 squared), because 18 is divisible by 9 but not by 27.

Throw in some logarithms to make the rules work exactly, extend it to fractions, and you get the p-adic numbers.

A completely different number system with a completely different notion of what “size” means. Numbers that are close together in the normal sense can be far apart in the p-adic sense and vice versa.

III.

Once we found a few examples we can try to systematize our search.

In the case of symmetries it turns out there is only a finite set.

For notions of size on the familiar numbers, there’s the usual absolute value, the boring one where everything except 0 has size 1, and one p-adic version for each prime. That’s it.

The same pattern shows up everywhere.

For example, once you formalize what it even means to be a “number system” (something where you can add, multiply, and divide (except by zero), and where multiplication behaves nicely with respect to size) you find there are exactly four: real numbers, complex numbers, quaternions, and octonions. This is Hurwitz’s theorem.

So this is one game modern mathematicians play. Take a common sense notion. Formalize it rigorously. Find ALL the strange beasts that technically satisfy your definition. Classify them completely.

That’s what we did with symmetry and size. We wrote down the rules, then asked: what are all the possible things that follow these rules? Sometimes you get a messy zoo. Sometimes you get a shockingly short list.

A secret hope is always that some (all?) of these strange mathematical beasts somehow play a role in nature.

After all, why would nature only make use of some specific narrow slice of mathematical technology?

If it uses real numbers, why not complex numbers too?

Oh wait, it actually does in quantum mechanics.

So what about quaternions and octonions?

Maybe a strange symmetry like E6 or E8 (related to octonions) describe some undiscovered fundamental symmetry of nature.

Maybe p-adic numbers is exactly the mathematical technology we need to finally make sense of quantum mechanics?

Why would the universe be weirdly conservative, only employing specific mathematical structures, when all these other wild possibilities exist?

Maybe it wouldn’t be. Maybe it uses all of them and we just haven’t found where yet.

IV.

There is a second related game mathematicians play.

Once we’ve written down a list of criteria that define a “thing” we can also ask: what if we relax one of these assumptions?

The historical precedent that motivates why this makes sense is Non-Euclidean geometry.

For thousands of years, Euclidean geometry was just common sense. Parallel lines never meet. The angles in a triangle add up to 180 degrees. The shortest distance between two points is a straight line.

Then mathematicians started asking: what if we drop the parallel postulate? What if parallel lines could meet?

Turns out when you do this you get hyperbolic geometry, spherical geometry, and all sorts of other weird spaces.

Then Einstein shows up decades later and realizes these weird structures is exactly what we need to describe gravity.

Riemann’s abstract mathematical game from the 1850s became the language of general relativity in 1915.

With this in mind it seems natural to push further.

What if we relax the assumptions built into Riemannian geometry

You can study geometric frameworks that have torsion, or that aren’t metric-compatible, or both.

What if spacetime has torsion? What if it has non-metricity?

So in this optimistic view, mathematicians are systematically exploring the full landscape of mathematical possibility.

Classifying everything that can be classified. Relaxing every assumption that can be relaxed. Building the complete atlas before physicists show up needing a map.

V.

Do these mathematical games ever end?

Can we classify all the things that can be classified? Is there a structured way to think through all the ways assumptions can be dropped?

Well, there’s category theory.

Once you’ve classified all the symmetries, you can study the structure of how symmetries relate to each other.

Once you’ve found all the number systems, you can study how they map into each other.

The relationships between mathematical objects are themselves mathematical objects you can formalize and classify.

And once you understand those patterns, you can classify entire “mathematical universes” with different internal rules.

This is topos theory, where even basic logical laws might work differently.

In other words, you can study what happens when you relax the assumptions at the very heart of mathematics.

At what point do these games stop being useful and start being silly?

No clue.

The only thing that for sure seems silly is trying to predict what mathematical tool will end up being useful to describe nature.

Doesn’t mean all of it will be useful.

But a ton that seemed silly ended up making sense. So mathematics will most likely remain unreasonably effective.

Moreover, leaving physics aside, there is also the hope that if you keep pushing the frontier of abstraction, you’ll eventuell reach at the “heart of hearts“.

Some fundamental layer where everything just clicks. Where all the strange coincidences and mysterious connections between different areas of math finally make sense because you’re seeing things the right way.

Whether that exists, nobody knows.

But it definitely feels like it should, just like a more unified framework of fundamental physics.

And I can totally see how someone might go crazy by repeatedly banging their head against that final abstraction ceiling for too long.