Complex Analysis Reveals Maths is Just Squiggles

By Professor Lord Lord – Chief Science Writer

The mathematical world is in crisis.

Far from being static and eternal, it is in fact no stranger to controversy. From mathematicians publically challenging each other over the solutions to cubic equations (it was the 16th century and reality TV wouldn’t be around for another couple of centuries) to 17th-century clerics in Rome trying to ban infinitesimals, mathematics has had catastrophe after catastrophe. Every time it’s survived, but every time it’s become even less comprehensible. Now mathematics faces one of its greatest challenges yet.

“All this time we thought we were doing maths and discovering new truths about our fields, but it seems like most of the time we’re just drawing diagrams, or even squiggles,” says Abigail Koss of the Upton Snodsbury Institute of Mathematics.

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This is Pi. Pie is food. Pi is mathematics. Squiggles.

To the general populace, most mathematics beyond calculus seems like squiggles, so your correspondent asked Dr Koss to elaborate: “Take working with complex numbers, a field we call complex analysis. A lot of applied calculations involve, basically, drawing lots of lines around points where you end up dividing by zero. It all works out in the end – that’s why it’s used so widely – but I feel like I’m just doodling sometimes,” she laughs. “On the other hand, we get pretty graphs,” she says as she presents your correspondent with colourful pictures of functions and fractals.

Complex analysis is one of the older and more well-established branches of mathematics, and your correspondent wondered if Dr Koss’ worries about the foundations of mathematics extended to other, more abstract fields. A short walk down the corridor answers his questions as he meets with some topologists – mathematicians who study shapes and space in different dimensions.

Crocheted models of higher-dimensional objects adorn Doug Hole’s desk. On sabbatical from the University of Nork Rise, he originally intended to keep his head down and finish a research project, but was quickly caught up in the crisis sweeping mathematics.

“I always thought that while we do tend to do a lot of visualisation in topology, that was just an aid to help us,” he says, deftly sketching a diagram. “Nowadays it feels like most of my job is visualisation.”

His colleague Owen Borous agrees. “Back in the 70s, two mathematicians proved that you need a minimum of four colours for any map, but they did it with a computer and most people didn’t accept it. During my PhD I spent a lot of time – and I mean a lot – colouring in maps trying to check that proof. I think that’s when we started to realise maths might be more about squiggles and pictures than we had thought.”

As we speak, Dr Hole disappears and then reappears, accompanied by the scent of fresh doughnuts. These aren’t your normal shop-bought doughnuts, though: these are twisted and knotted into fiendish patterns, models of the low-level topology Drs Hole and Borous are working on. Here low-level refers not to the simplicity of the shapes – some of them made your correspondent’s eyes hurt – but to the number of dimensions.

“I did originally try making five- or six-dimensional doughnuts, but it didn’t go very well,” says Dr Hole. “Most of the time they didn’t work. The one time they did, they disappeared and I found a message in binary which said ‘your six-dimensional doughnuts are trivial and insignificant, but extremely tasty’ on the counter. Not sure what’s up with that. Anyway, we use them as models.” In the middle of our conversation, he and Dr Borous are drawing on the doughnuts with edible pens, marking out regions and scribbling equations. “And once you’re done with them, you can always eat them. I do feel like a lot of my maths is now done through baking, though.”

As your correspondent munches through a freshly-made topological doughnut, a mess of knots and holes, he has to agree with the author of the mysterious note: they really are extremely tasty. Finally, he heads over to the other side of the Institute.

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This is a Mandelbrot Set. This is mathematics. It’s not even numbers! HOW DO I!?

A hot new proof has swept the world of mathematics: two brilliant mathematicians, Keita Yokoyama at the Japan Advanced Institute of Science and Technology and Ludovic Patey at Paris Diderot University, recently proved that a certain theorem may be a statement about infinite objects but can be reduced to using finite objects alone. This has far-reaching implications for scientists, who work in a finite world but who often use infinite quantities in their work.

Unfortunately, your correspondent was not able to contact Professor Yokoyama or Dr Patey for comment, but he was able to speak to some combinatorists at the Institute – broadly speaking, mathematicians who work on graphs, combinations and order.

“This theorem – Ramsey’s theorem – is really foundational in combinatorics,” says Alina Graf, who was also working on Ramsey’s theorem at the time. “We can use it to study the conditions where order appears. It’s quite beautiful, really.” A small common room is covered in spidery lines and diagrams your correspondent couldn’t make out. “These diagrams – these are really good ways to explain Ramsey’s theorem. Well, except Ramsey’s theorem usually works with infinite sets, and it’s not like we have an infinite amount of space here…I have graduate students working on combinatorics right now, and honestly, most of it is squiggles. I think we can reduce maths to squiggles, really.”

So where does this leave other disciplines? “I called it!” exclaims Vera Careless, a physicist working on the Parallel Universe Project. “But really, I don’t think maths being on shaky foundations changes the fact that the maths does work in practice. I think we should be fine. I’m cautiously optimistic. And if it doesn’t work out, we’ll just break it a bit, right? That’s what we were doing before.”

Beyond traditional disciplines, science-based artists are very excited about this new development. “I can’t say I knew much about this, but it just seems like something that could happen, you know?” says Art Tiste, currently working on his solo sciart exhibition Xthlzzzzz’q5ft. “I mean, it’s been a while since I touched any maths, but so much of what I do remember was drawing contours. So many contours. Anyway, I really hope this gets mathematicians and artists to talk to each other more. This could potentially be really good, do you get me?”

Whether you think of maths as an objective reality waiting to be discovered, arbitrary squiggles, or totally and utterly incomprehensible, it still describes everything around us and underpins everyday life. And if it does turn out that mathematics as we know it is just drawing lines and shapes in fancy ways, maybe that will help to demystify it.

Whatever happens, your correspondent maintains that those topological doughnuts were delicious.

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