So you think you know 2+2? Try again.

ON PAPER, IT’S ONE OF THE SIMPLEST maths problems in the world: 2+2. If you’re counting something, such as screws at the hardware store, it’s pretty straightforward. But the lines blur in other contexts. If you add two cups of vinegar to two cups of baking soda, and the reaction produces five cups of a fizzy mess, does that mean 2+2=5? We bring assumptions into the world of mathematics. In this case, the simple ‘counting numbers’ – the whole integers 1, 2, 3, and so on – signify a gulf between math’s abstraction and application. Using ‘2+2=4’ as food for thought, mathematicians are exploring the circumstances in which 2+2 doesn’t actually equal 4, at least not neatly, and we can extend those interpretations to larger questions in epistemology – how we know what we know.

Kareem Carr, a biostatistics PhD student at Harvard University, ignited a ‘Does 2+2 ever equal 5?’ debate on Twitter. On 30 July 2020, he wrote, ‘I don’t know who needs to hear this, but if someone says 2+2=5, the correct response is, “What are your definitions and axioms?”, not a rant about the decline of Western civilisation.’

In his Twitter thread, Carr pointed out that counting numbers ‘are abstractions of real underlying things in the universe’, so we should be mindful of how those abstractions distort truth when introduced to real-world scenarios. Arithmetic works well in a textbook, but practically, it often runs into contextual questions that don’t account for parts of a whole, approximations, or more relevant vectors.

For example, if you’re adding whole degrees to an angle, eventually you’ll circle around to an angle that measures 360°. But a 360° angle has the same orientation as a 0° angle, so whether the angle measures 0° or 360° depends on context. Likewise, if you drilled a screw five full rotations (1 800°) instead of four (1 440°), the screw’s orientation remains the same, but in one case, it’s deeper inside the wood.

Carr’s tweet received some replies displaying other examples of arithmetic’s real-world limitations. Many people pointed out that two animals can become three through reproduction (1+1=3, or 1+1=1, depending on your parameters), or that two machines could become three machines if you had some spare parts from each machine and a little elbow grease. Others pointed out that

2.3 rounds down to 2, but 2.3+2.3 rounds up to 5, making it possible through a certain filter that 2+2=5.

In general, the idea that we innately learn counting numbers – whole values only, no fractions or decimals – is a common misconception among people who aren’t trained in maths or human development. Young children learn numbers one at a time, by counting, but only begin to learn more sophisticated counting – higher numbers – once they can recognise quantities quickly, an ability called subitising. It becomes easier for us to count to seven, for example, when we can recognise a group of four things and then count the fifth, sixth, and seventh things. Counting is an unnatural, learned skill – even the non-human animals who can ‘count’ to four or five, such as dogs and chimps, are considered exceptional – so imposing abstract counting numbers on to the real world creates an innate tension.

There are more problems with the abstraction of on-paper mathematics. Carr grounds his ‘2+2=5’ concept in the ways statistical models can cause harm to marginalised groups across certain parameters. ‘Whenever you create a numerical construct such as IQ, or an aggression score, or a sentiment score, it’s important to remember that properties of this score might not mirror the real things being measured,’ he says.

Sentiment scoring is the primary way companies analyse reviews and customer service replies for positive or negative ‘feeling’, while aggression scales are used in assessing psychiatric patients. In each model, people must assign arbitrary number values (on a scale of 1 to 10, for example) to a criterion that isn’t tangibly measurable – how ‘pleasant’ a transaction was or how ‘violently’ a patient behaved. ‘When you’re trying to create a statistical construct of some mental phenomenon, my sentiment could be changing from moment to moment,’ Carr explains. ‘You’re not really sure how concrete this thing is.’ It’s hard to rate your feelings when they change so much, or when the minimum or maximum of the scale – is your pain level really a 10, as bad as it could possibly be? – isn’t easily conceived by our experience.

Some bad-faith critics have flooded Carr’s mentions, saying the value of maths is its reliability and rigidity. But Carr’s response points to the distinction between using maths as a tool to find answers, and maths as a tool to learn. ‘There are a lot of people who seek out maths and statistics for a sense of certainty: “This is the answer,”’ he says. ‘And there are people who close their minds. I’m more on the other side: Is there something else I could discover in this complex of ideas? It’s a thrill of discovery, like when people do metal-detecting.’

Ultimately, Carr says expanding people’s conception of the pros and cons of various mathematical applications will lead to deeper critical thinking about the way maths intersects with our lives. ‘There’s a need for this sort of thinking, because we’re basically turning everything into data,’ he says. Movies have Tomato-meters, podcasts have star ratings, and social media is rife with ratios. ‘If we’re going to be a world that’s just in apps, we need to be sure these things are working how we think they work.’