Can you fool a mathematician?

What magic(ians) and math(ematician)s have in common

I took up card magic last year after spending time on YouTube in thrall of Shin Lim (if you’ve never heard the name, now is the time to get acquainted). I’d had fleeting experiences with magic before. As a child I once scuppered a magician’s act by accidentally destroying his prop (two interlocking Polos). In later years I would teach mathematically-inspired tricks to groups of schoolchildren. But this was my first plunge into the craft proper. It was in the early stages of the pandemic, after I had been reliably informed that an effective coping mechanism for lockdown is to keep one’s hands occupied. Card magic seemed like one of the more savoury options. 

My card skills are nothing to behold just yet, but I’ve scoured hundreds of tricks through various books and courses, mastered a handful of sleights and flourishes, and even attended a couple of online card magic conventions. I’ve found a safe sanctuary in the annals of Vanishing Inc’s impressive array of downloads (magicians really do make great marketeers).

Along the way, I’ve been struck by the parallels with my own ‘craft’ of mathematics. I’m not merely referring to mathematical card tricks (of which, it is true, there are many), and I will do well in this post to avoid using that well-worn term ‘mathemagician’. Rather, I’m speaking of the shared cognitive aspects of magic and maths. 

Here are two intellectual disciplines that, on the surface, defy comparison - after all, magicians trade in deception, mathematicians in cold logic. Yet the more I peered into card magic, the more I found resonance with the mental habits of mathematicians (card magic is my reference point but this probably applies just as well to other genres).

In this opening post we’ll see how both magic and mathematics can mislead us, and how a considered approach to either can rescue us from our cognitive biases.

Fool me once

Magicians would make great cognitive scientists. They have an intimate understanding of how humans, as relentless pattern seekers, are easily duped by illusory correlations. We rely on rapid-fire thinking processes (fashionably termed ‘System 1’) to process the world around us with efficiency. Speed is privileged over accuracy, resulting in a raft of cognitive biases that make us a perfect target for magicians. 

A magic trick unfolds as a coherent narrative, each move seamlessly tied to the next. So keen are we to have order in the world that we will infer cause-and-effect mechanisms when they don’t exist. So long as one move precedes another, we’ll automatically assume there’s a logical and inevitable connection between them. 

Even when we’re on the lookout for deceptive cues, we can’t help but direct our eyes and ears at the magician’s behest, lapping up whatever version of reality they are presenting to us and ignoring the deft manoeuvres that come back to bite us when the final effect is revealed. Magicians employ a range of visual and audio cues that are subtle enough to defeat our perceptions. Sleight-of-hand artists like Shin Lim move with such unnerving speed and grace that most of us can barely keep up (even when we replay his act at ½x speed). 

The curious-minded among us are not content with having our reality tampered with. The real ‘wow’ moment comes from figuring out the subtle devilry of the magician’s act. The legendary Penn & Teller (where Shin Lim first earned fame) have premised an entire show on attempts to debunk magic acts. The duo have been fooled several times over by the deft hand of budding magicians, but they usually come out on top, and it’s hard not to be impressed by the perceptiveness of these seasoned campaigners. They see and hear what the vast majority of us miss. The show attracts millions of viewers who all want to play along and search for those fine-grained deceptions that pull the effects together.  

When 2 becomes 1

If you’ve found yourself playing close attention to a magic act, trying to get one up on the performer, you have more in common with mathematicians than you may realise.

A polished magic act bears the hallmarks of a mathematical proof, each step meticulously accounted for and the overarching argument anchored to a coherent narrative. The difference, of course, is that a mathematical proof doesn’t seek to deceive. Mathematics brings scientific rigour to pattern-seeking, distinguishing meaningful connections from fabricated ones.

Mathematicians are among the most transparent of people. They expose their every assumption and every leap in logic to scrutiny, embracing the most uncompromising standards for verifying their claims. 

A mathematician understands that the subtlest logical flaw can have stark consequences further along the chain of an argument. The mathematical analogue of a magic effect is a seemingly plausible argument with an absurd conclusion. Take, for instance, the classic ‘proof’ that 2=1:

It’s easy enough to fall for this argument; each step seems credible enough at first blush . It’s only when we’re faced with an unpalatable conclusion that we take pause to retrace each step and uncover the logical fallacy that escaped our attention the first time.

A visual example: here are the same four shapes arranged in two different ways. A square appears to have gone missing - but where is it?

Examples like these are the fake news of maths. Debunking them involves stepping back and giving space for our slower, more rational thinking processes (‘System 2’) to take hold. Mathematicians have their own fun unpicking arguments and exposing the tamest violations of logic  - even for arguments whose conclusions are highly plausible.

The only difference, perhaps, is that where Penn & Teller have to unravel the magician’s secret on the fly, a mathematician isn’t time-bound (although Mathematicians: Fool Us could make for quite a spin off). Hours, days, weeks are spent combining through the details of an argument. 

This is not to suggest that mathematicians are perfect reasoning agents. We’re hardly immune to cognitive bias, especially when we’re in the real world dealing with thorny concepts that evade strict definition and logic. Mathematical proof is sometimes described as a form of high-altitude training; a space to examine the most abstract arguments with unflinching attention to detail. It is best thought of as conditioning for the mind that perhaps, just perhaps, makes us less prone to being fooled. There’s some Penn & Teller in all of us, if we’re willing to give it a shot. 

So, have you found the missing square?