How the Human Brain Contends With the Strangeness of Zero
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Welcome to the Quantiscience Podcast. Each episode we bring you stories about developments in science and mathematics. I'm Susan Vallett.

Zero, which was invented late in history, is special among numbers. New studies are uncovering how the brain creates something out of nothing. That's next. Explore the world of science in the quantum book Alice and Bob Meet the Wall of Fire, published by the MIT Press.

Available now at Amazon.com, BarnesandNoble.com, or your local bookstore. Around 2,500 years ago, Babylonian traders in Mesopotamia impressed two slanted wedges into clay tablets. The shapes represented a placeholder digit squeezed between others to distinguish numbers such as 50, 505, and 5005.

An elementary version of the concept of zero was born. Hundreds of years later, in 7th century India, zero took on a new identity. No longer a placeholder, the digit acquired a value and found its place on the number line before one. Its invention went on to spark historic advances in science and technology. From zero sprang the laws of the universe, number theory, and modern mathematics.

Andreas Nieder is a neuroscientist who studies animal and human intelligence at the University of Tübingen in Germany. Zero is by many people, mathematicians definitely, considered one of the greatest or maybe the greatest achievement of mankind.

because for a long time in human history, there was no zero. It was absent. It took a long time, an eternity, until mathematicians finally invented zero as a number. Perhaps that's no surprise, given the concept can be difficult for the brain to grasp. It takes children longer to understand and use zero than other numbers, and it takes adults longer to read it than other small numbers.

That's because to understand zero, our mind must create something out of nothing. It must recognize absence as a mathematical object.

Benji Barnett is completing graduate work on consciousness at University College London. Zero is kind of just unlike any other number that we know. So other non-zero numbers, they directly map onto countable objects in the environment. Like we can see there's one elephant or three elephants in front of us and it's easy to map onto those things. But thinking about zero requires an extra kind of cognitive step. It's not about...

tagging something that's present in our environment, we have to go one step further and say, "Okay, there wasn't anything there, therefore there must be zero of them." Rather than it just being a simple, you detect something and then you can count how many things that is. It's like an extra level of abstraction away from the world around you. In recent years, research started to uncover how the human brain represents numbers, but no one examined how it handles zero.

Now, two independent studies, led by Nieder and Barnett respectively, have shown that the brain codes for zero much as it does for other numbers, on a mental number line. But one of the studies found zero holds a special status in the brain.

Carlo Semenza, a professor emeritus of neuroscience at the University of Padua in Italy, wasn't involved in either study. Zero is a very peculiar number and it's very interesting in itself because the fact that it represents nothing is a contradiction in itself. It looks like it is concrete because people put it on the number line, but then it doesn't exist.

And that is fascinating, absolutely fascinating. The new studies are the first to reveal what goes on in the brain when a person thinks about zero. And they bring up broader questions about how the mind handles absence, a pursuit that would have pleased Jean-Paul Sartre, the 20th century existentialist who claimed that nothingness carries being in its heart.

The idea of zero, originally called sunya for empty in Sanskrit, first made its way out of India to the Arab world. Then, in the 13th century, a humble traveler named Fibonacci picked up the idea in North Africa. He brought it back to medieval Europe along with a base 10 number system and Indo-Arabic numerals.

At first, zero caused confusion. Nieder says its ability to represent nothing and enable complex mathematical operations challenged deeply ingrained theological and philosophical ideas. Particularly due to the influence of the church, philosophers and theologians associated nothing with chaos and disorder and were disinclined to accept it.

Barnett says at one point, many people feared zero. So the kind of intellectual Europeans at the time really rejected it. And because of all its weird numerical properties, it was kind of thought of as the devil's number. But he says soon, merchants recognized that zero was valuable for business. It only started to evolve into a more kind of quantitative notion through the use of transactions.

tradespeople. So it was actually a very anti-intellectual concept at first because tradespeople really needed it to balance their books. So they needed to know when their incomings were equal to their outgoings, which was obviously needed to be balanced at zero. And that's why it became quite popular. By the 15th century, zero had become widespread in commerce, finance and mathematics across Europe, but never shed its aura of mystery.

Here's Semenza again. Representing zero is something that is extremely hard for the human mind. When you multiply times zero, it disappears. And that is very, very hard to understand. Although zero is now pervasive and seemingly simple, math students and mathematicians alike continue to wrestle with it. Neuroscientist Andreas Nieder calls it the eccentric uncle in the family of numbers.

To use zero in calculations, mathematicians had to establish all sorts of rules. You can't divide any other number by zero, but you can divide zero by any other number. A non-zero number to the power of zero gives you one. Zero to a power of a non-zero number gives you zero. But zero to a power of zero gives you a calculator error. And a headache.

Neil Barton is a philosopher of math at the National University of Singapore. The idea of zero or something that plays the role of a zero somehow appears all over math. Without zero, modern mathematics wouldn't exist. You wouldn't be able to solve a function, do calculus, or distinguish between one and one million. However you look at it, zero is unique.

For researchers interested in how the brain handles numbers, Nieder says zero is the most fascinating number of them all. He suspects that if zero is special in history and math, then the brain must process it specially too.

Nieder's group has previously shown that some neurons in the brain have favorite numbers. Some favor three and will fire more rapidly when presented with three apples, for example, than they would with two or four, and much more than with five or seven.

The more rapidly neurons fire, the more interested they are in a specific number. This is true not only in humans, but also in other animals. While non-human animals don't understand numbers when they're represented as digits, an entirely human construct, they can estimate quantities, also known as numerosities. Researchers previously found that monkeys and crows have neurons that are specifically tuned to the numerosity zero.

But until very recently, no one ever probed for the neural basis of zero in humans. Barnett was interested in absence before he was interested in zero. The majority of perception science and visual science over the last hundred years or even before that has focused on when we detect the presence of something in our environment. The brain is wired up to detect those things. You know, neurons fire when the cells in your retina are activated because there's something there.

But this really ignores the whole other side of things, which is that you can often have experiences of something not being there. If I try to get you to see a stimulus that I flash really quickly on the screen, half the time, maybe you're going to say, actually, I didn't see something. And that's a really fundamental component of our conscious experience. Like if you go down to get your keys to leave the house and you realize your keys aren't where you left them.

that experience of an absence is really quite core and something that we will share. Previously, researchers assumed that absence was represented in the brain by neurons not firing. But recent studies have shown that the brain encodes absence with unique neural patterns. To push this work further, Barnett turned to zero.

A few years ago, he recruited 24 participants to perform tasks related to Xero as they sat in a magnetoencephalography scanner. So basically you just have people sitting in a scanner. It kind of looks like these old school hairdressers with like a big kind of machine over their head.

And it's measuring the magnetic fields associated with populations of neurons in your brain. So when they're firing, obviously they're firing electrical impulses, which has these associated magnetic fields. So these incredibly sensitive sensors in the MEG system can pick up these tiny disturbances in the magnetic field. And then you can capture which parts of the brain are firing at which points and in what kind of patterns they're doing that.

So researchers can learn how populations of neurons respond when prompted to think about specific topics, such as zero. Barnett and his advisor at University College London, Stephen Fleming, were looking for evidence of the numerical distance effect. That's a phenomenon that occurs when the brain processes non-zero numbers.

Basically, the brain can more easily distinguish between two numbers if they are far apart on the number line than if they are close together. So it confuses 6 and 7 more often than 6 and 9. The researchers figured that if the brain processes zero similarly to other numbers, it should also show the numerical distance effect.

Indeed, in results published in Current Biology in August of last year, Barnett and Fleming concluded that the brain treats zero both as a digit and as a quantity, in much the same way that it treats the other numbers. So basically, Barnett says: Zero is represented on this neural number line. Nieder, for his part, has been obsessed with zero and absence for the better part of a decade.

In 2016, he proposed that the neurological mechanisms that encode absence may be shared with those that encode zero. He hypothesized that zero must have evolved from more fundamental representations of perceptual absence. First, the brain had to understand the absence of a stimulus, like a light being off.

Only then could it recognize nothing as a category akin to something, but representing everything that isn't something. Finally, it had to turn nothing into a quantitative concept. Nieder believed by understanding how the brain encodes zero, we might be able to understand how the brain deals with absence.

Since 2015, Nieder has collaborated with Florian Mormann, the head of the Cognitive and Clinical Neurophysiology group at the University of Bonn, who treats epilepsy patients. For their treatment, these patients have electrodes implanted in their brains, which Mormann can use to collect neuroscience research data with patient consent. Mormann recorded brain activity from single neurons as the patients performed number-related tasks.

In the analysis, he and Nieder first focused on non-zero numbers and found that the brain processes large numbers differently than smaller ones. Then they went back to their data to focus only on zero. As they reported in Current Biology in September of 2024, Nieder and Mormont found the same thing Barnett had: that zero exhibited a numerical distance effect in the brain. Nieder says that means to the brain, zero is just another number.

Or is it? He says when you look closely, zero is still an outlier among the other numbers. Nieder's team found some differences between the way the brain represents zero and the way it handles other numbers. For one thing, more neurons had zero as their preferred number than any other small number. Because there are more neurons that code for zero, they found the brain can represent the empty set with more accuracy than it can represent other small quantities.

Nieder says zero is encoded together with other small numbers, but is represented more distinctly compared to them. He says the brain represents it as an eccentric outlier. However, they didn't find any difference in accuracy in the way the brain represented the digit version of zero compared to the other digits. Nieder says zero is treated like any other formal number.

Nieder's findings are slightly different than Barnett's. Barnett didn't observe any distinctions, even slight ones, in the way zero was represented compared to other numbers. What could explain the discrepancy? After being asked to look at each other's papers, Barnett and Nieder agreed that their findings are complementary. They say any differences most likely have to do with scale. Barnett's study looked at large populations of neurons, while Nieder's looked at single neurons.

Brian Butterworth, a cognitive neuroscientist at University College London, says the study's differences could instead be due to the studied brain area. Barnett's group was able to examine the entire brain, including the parietal cortex, the region that's considered the hub for processing numbers. But Nieder and Mormann were limited by where patients' electrodes were placed.

They closely examined single neurons in the medial temporal lobe, which is known for processing memory. Here's Butterworth. It's like looking for your lost keys under the lamppost, because that's where you can see what's going on. You know, they happen to have access to the temporal lobe, but not to the parietal lobe. I think that's why it appears to be a discrepancy.

Butterworth says the pair might have accidentally surveyed memories for zero rather than the digit or idea itself. Nieder disagrees. He says that area of the brain, as well as many others, has also been found to be able to process numbers.

Despite the differences, the results have excited Carlos Samenza. He had previously hypothesized that zero would have a place on the mental number line and not be a distinct concept from other numbers. I wish I'd done those experiments myself.

All the researchers agreed that these studies are only a start in uncovering how our brains process zero and how to reconcile its different modes. For example, neither of the research groups looked at how the brain processes zero as a written word. Philosopher of math Neil Barton wonders how many different zero concepts there could be, and how could we unify these ideas under one umbrella?

He says it's well understood mathematically, but he'd love to see more from neurobiologists.

While Nieder hopes to continue dabbling in the world of number neuroscience, Barnett has his sights set on the concept of absence. He says if he can find similarities in how the mind represents zero and absence, then maybe Nieder's theory is right, that zero might have evolved from a more fundamental ability to grasp the idea that nothing really is something. Music

Arlene Santana helped with this episode. I'm Susan Vallett. For more on this story, read Yasmin Sapakoglu's full article, How the Human Brain Contends with the Strangeness of Zero, on our website, quantummagazine.org. Make sure to tell your friends about the Quanta Science Podcast and give us a positive review or follow where you listen. It helps people find this podcast.