TO THE Iranian mathematician Maryam Mirzakhani, the first woman to win the Fields medal, mathematics often felt like âbeing lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricksâ.
âWith some luck,â she added, âyou might find a way out.â
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Mirzakhani, who died on 14 July at the age of 40, ventured deeper into the mathematical jungle than most. Nonetheless, most of us have spent enough time on its periphery to have a sense of what the terrain looks like.
Increasingly, it seems as if humans are the only animals with the cognitive ability to hack their way through the undergrowth. But where does this ability come from? Why did we develop it? And what is it for? Answering these questions involves diving into one of the hottest debates in neuroscience, and reimagining what mathematics really is.
The natural world is a complex and unpredictable place. Habitats change, predators strike, food runs out. An organismâs survival depends on its ability to make sense of its surroundings, whether by counting down to nightfall, figuring out the quickest way to escape danger or weighing up the spots most likely to have food. And that, says Karl Friston, a computational neuroscientist and physicist at University College London, means doing mathematics.
âThere is a simplicity and parsimony and symmetry to mathematics,â says Friston, âwhich, if you were treating it as a language, wins hands down over all other ways of describing the world.â From dolphins to slime moulds, organisms throughout the evolutionary tree seem to make sense of the world mathematically, deciphering its patterns and regularities in order to survive.
Friston argues that any self-organising system â and so any form of life â that interacts with its environment needs an implicit model of that environment to function. The idea goes back to the 1970s and the âgood regulatorâ theorem, co-developed by Ross Ashby, who pioneered the field of cybernetics. To provide effective control, the theorem says, a robotâs brain must have an internal model of its mechanical body and its environment. âThat insight is becoming increasingly formalised now in machine learning and artificial intelligence,â says Friston. The corollary being that an animalâs brain, too, must model its body and the world in which it moves.
No thought required
The remarkable thing is that none of these creature modellers are aware of what theyâre doing. Even we human beings, when we run to catch a ball or dart through heavy traffic, are unconsciously doing some pretty complex mathematics. Each of our brains is constantly using its models to predict what weâll encounter, says the theory, and these models are kept updated by checking the predictions against actual sensations.
Those mathematical functions are undoubtedly being computed by particular bits of the brain, says Andy Clark, a cognitive philosopher at the University of Edinburgh, UK. But this is not to say that there are specialised modules in the brain similar to buttons on a calculator that we can call up on demand: one to perform multiplication and another to work out cosines. âWe donât have access to that,â he says.
Although these models try to ensure our survival in a complex world that follows the laws of physics, their insistence on keeping us alive means they sometimes have to compromise on correctness. Take the gamblerâs fallacy: the mistaken belief that, if the roulette ball keeps landing on red, a bet on black is the best one to make. In reality, of course, both results are equally likely, but the models our brains have built of the world, perhaps to tell our ancestors when to move on from an unsuccessful foraging area, blind us to that simple statistical observation.
âWe could have a sense of number as strong as our sense for colourâ
Or take the Weber-Fechner effect, which governs our response to external stimuli. Found to hold true across all our senses, it states that our ability to discriminate between sensations of a similar magnitude diminishes as their magnitudes increase together. So while a 1-kilogram weight can easily be distinguished from a 2-kilogram one, for example, weights of 21 kilograms and 22 kilograms are harder to tell apart. The same applies to the brightness of lights, the volume of sounds and even the number of objects you can see.
Even though human brains share such aberrations with those of other animals, we have developed the ability to identify and overcome some of these flaws. Most obviously, we invented numbers: a system of notation that allows us to instantly deduce that 21 and 22 are as far apart as are 1 and 2. The creation of this complex, symbolic language for mathematics not only allows us to overcome certain such limitations of our subconscious mind, but also to explore abstract concepts in depth and communicate them to others. But how did we develop the tools to consciously understand what our bodies do instinctively?
Innate numeracy
One long-standing idea says we are born with a conscious sense of numbers in the same way we are conscious of colours. In his 1997 book The Number Sense, Stanislas Dehaene of the INSERM-CEA unit for cognitive neuroimaging in Gif-sur-Yvette, France, hypothesised that evolution endowed humans and other animals with numerosity, an ability to immediately perceive the number of objects in a pile. In other words, three red marbles would produce a sense of the number 3 just as they would produce a sense of the colour red. Dehaene proposed that this numerosity was exact for numbers below 4 and fuzzier thereafter, but nonetheless represented a hardwired ability. Armed with such an instinct, our paths through the mathematical jungle would quickly start to clear.
Evidence to support this ânativistâ view soon started to accumulate. Elizabeth Spelke at the Massachusetts Institute of Technology and her colleagues showed that 6-month-old children could distinguish between an array of eight dots and one with 16 dots. Then Dehaene and his colleagues reported that the Munduruku Indians in the Brazilian Amazon, who donât have words for numbers larger than 5, could approximately discriminate between much larger quantities, suggesting that this ability was independent of culture.
Other studies indicated that humans instinctively represent numbers spatially on an imaginary ânumber lineâ, their values increasing from left to right. There was even evidence of numerosity in animals. This all pointed to an innate number sense that millennia of culture had helped expand.
But before long, some researchers grew uncomfortable with the conclusions of these studies. Might subjects, for example, be distinguishing two arrays of dots based not on the number of dots, but on other attributes such as their spatial distribution or area of coverage? âThese are cues that are usually correlated with number, so it would be unwise not to use them,â says Tali Leibovich at the University of Haifa in Israel. âIf you are an animal in nature and you need to hunt something and need to do it very quickly, you want to use all available cues.â
Indeed, on further examination, it seems that people also rely on these non-numerical cues. Soon, a different hypothesis emerged. Perhaps, instead of having an innate sense of number, we are born with a sense for quantities â such as size and density â that correlate with the numbers of things. âIt takes time and experience to develop and understand this correlation,â says Leibovich.
More-refined cognitive tests in children tend to support this view. For example, children younger than about 4 years of age cannot understand that five oranges and five watermelons have something in common: the number 5. To them, a bunch of watermelons simply represents more âstuffâ than the same number of oranges.
Even teaching young children to identify the order of numbers â going through the motions of counting â doesnât immediately impart their meaning, says developmental psychologist Daniel Ansari at the University of Western Ontario in Canada. That occurs informally through long-term exposure to parents and siblings. âThis points to the strong influence of cultural practices on the learning of exact representations of number,â he says.
Study of the cultural aspects of numerical cognition has suffered from bias, says Ansari, in that not enough attention has been paid to data collected from non-industrialised cultures. These findings, he believes, cast serious doubts on the nativist hypothesis.
Take the Yupno people of Papua New Guinea. Rafael NĂșñez at the University of California at San Diego has learned, for example, that they donât use the supposedly universal mental number line. Also, they have no comparatives in their language to say that one thing is bigger or smaller than another.
This is not to say that the Yupno language is primitive. Far from it. Take demonstratives. In English, there are only four: this, that, these and those, to specify the proximal or distal nature of things. The Yupno, on the other hand, have words to indicate whether something is higher or lower than them in elevation (in keeping with their mountainous terrain), and they have nuanced words to capture not only how proximal or distal something is, but also by how much.
The Yupno are not alone in having a language that doesnât emphasise numbers. NĂșñez points to a study of 189 Aboriginal Australian languages, of which three-quarters were found to have no words for numbers above 3 or 4, while a further 21 went no further than 5. To NĂșñez, this suggests that exact numerosity is a cultural trait that emerges when circumstances, such as agriculture and trading, demand it. âHundreds of thousands of humans who have language, sometimes very complicated and sophisticated language, donât have exact quantification,â he says.
Even languages that do, such as English or French, can only take you so far. Last year, Dehaene and his student Marie Amalric reported the results of scanning the brains of 15 professional mathematicians and 15 non-mathematicians of the same academic standing. They found a network of brain regions involved in mathematical thought that was activated when mathematicians reflected on problems in algebra, geometry and topology, but not when they were thinking about non-mathsy things. No such distinction was visible in the other academics. Crucially, this âmaths networkâ doesnât overlap with brain regions involved in language.
This suggests that once mathematicians have learned their symbolic language, they start thinking in ways that donât involve normal language. âIt sounds strange, but itâs almost like being able to download an intuition into another world, the world of mathematics, stand back, and let it talk back to you again,â says Friston.
Some of this sophisticated mathematical language certainly develops out of our inbuilt sense for numbers or magnitudes, however imprecise it might be at birth. But it probably also leans on many other abilities: language to communicate ideas, working memory to hold and manipulate concepts, and even cognitive control to overcome the kinds of biases apparent in the gamblerâs fallacy.
Counter-revolution
The exact moment when culture transformed whatever instincts we may have had into a recognisable mathematical ability is unclear. One of the earliest pieces of evidence of humans dealing with numbers comes from the Border cave in the Lebombo Mountains in South Africa. There archaeologists found 44,000-year-old bones with notches, including the fibula of a baboon etched with 29 such marks. Anthropologists think that such âtally sticksâ were an aid to counting, and represent evidence for an emerging symbolic understanding related to consciously representing and manipulating numbers.
Counting and measuring hit new heights sometime around the 4th millennium BC, in the sophisticated Mesopotamian culture of the Tigris-Euphrates valley, a region in modern-day Iraq. Eleanor Robson at the University of Oxford has argued that mathematics in Mesopotamia was a cultural invention needed to keep track of days, months and years, to measure areas of land and amounts of grain, and maybe even to record weights. And as humans took to the seas, or studied the skies, we began developing the mathematics required for navigation and for tracking celestial objects. But it was always, in the beginning, a product of cultural necessity (and if you think trading-driven mathematics is a thing of the past, think again: some of the most sophisticated mathematics is being developed for trading stocks and bonds on Wall Street).
âSome ask if mathematics is invented or discovered. Itâs not an either-orâ
With the help of fundamental mathematical tools, humans have built an immense pyramid of mathematical knowledge. Over the past 5000 years or so, mathematics has expanded into ever more abstract domains, seemingly further removed from the processes that govern the world around us. And yet, the more we learn about the universeâs hidden workings, the more such mathematical innovation seems to describe the things we see. When David Hilbert developed a highly abstract algebra that worked in an infinite number of dimensions rather than the familiar three dimensions of space, for example, nobody could have foreseen its use in the emerging field of quantum mechanics. But soon after, it turned out that the state of a quantum system could best be described using such a Hilbert space â with the underlying mathematics remaining key to our attempts to make sense of the quantum world.
The ubiquity of such connections between mathematics and physics led the physicist Eugene Wigner to comment on the âunreasonable effectiveness of mathematicsâ at describing the natural world. To many physicists today, the success of mathematics as a language speaks to its primacy in the organisation of the universe.
Max Tegmark of MIT is one of these. He thinks the universe is a mathematical structure in that it has only mathematical properties â and we are slowly uncovering this structure, brushing away the dust to reveal the theorems and proofs that underpin reality. âIt used to be that it was very easy to list the small number of things in nature that you could describe with maths. Now itâs very easy to list the small number of things you cannot,â says Tegmark. Even biology, which long resisted mathematical rigour, is slowly succumbing: witness the proliferation of mathematics in genomics or computational neuroscience.
From this perspective, mathematics is a discovery rather than an invention. For researchers like NĂșñez, however, that is an overly simplistic distinction. âWhen the question is asked â is mathematics invented or discovered?â he says, âthere is a supposition that itâs exclusive. If you invented it, you donât discover it, and so on.â But it is not an either-or situation, he says.
Think of the âElementsâ, compiled by the ancient Greek mathematician Euclid, which unified all of Greek mathematical knowledge of the time and codified the laws of geometry. Euclid based his work on a series of rules or axioms, one of the most famous being that parallel lines never meet. Over time, the patterns, regularities and relationships that emerged from these âinventedâ axioms were explored by other mathematicians and proved as theorems. In a sense, they were âdiscoveringâ the landscape of Euclidean geometry. But then, thousands of years later, other mathematicians decided to start with axioms that contradicted the ones Euclid set out.
Riemannian geometry, for example, which owes its name to the German mathematician Bernhard Riemann, explicitly relies on the idea that parallel lines can in fact meet. This unorthodox starting point led to the discovery of a rich vein of mathematics that Einstein would use to formulate his general theory of relativity and describe the curvature of space-time. âThe world out there has all kinds of patterns and regularities and ways of behaving, and any creature that is going to build a mathematics is going to have to build it on top of regularities that are constraining the behaviour of the stuff that they encounter,â says Clark.
But no matter which axioms we start off with, mathematics might not be as complete a system of thought as we like to believe. We owe that insight to Austrian logician Kurt Gödelâs incompleteness theorem. Gödel showed that within the bounds of any formal system of axioms and theorems, you can make statements that can be neither proved nor disproved. In other words, there are some questions that mathematics can ask, but it will never have the tools to answer.
In which case, perhaps it is too early for us to make any sweeping statements about mathematics being a universal truth. After all, whoâs to say that our little corner of the jungle is in any way representative of the whole? But physicists like Tegmark have hope. For him, the biggest hurdle to a mathematical theory of everything is a description of consciousness, the crucible of our own numerical ability. Getting maths to explain its own origins? âThatâs going to be the final test of the hypothesis that itâs all mathematics,â he says.
This article appeared in print under the headline âThe origin of mathematicsâ
