PERHAPS the playwright Alfred de Musset put it best. “I can’t help it,” he declared in 1838. “The idea of the infinite torments me.”
How do humans, who have nothing in their daily lives that even approximates to infinity, conceive of the infinite? Most of us probably don’t like to think about it – it’s the kind of abstract thought that leads to severe headaches. Fortunately for us, there are people who have given infinity, and the question of how we deal with it, a great deal of attention. And their answer, perhaps rather surprisingly, is that we conceive of infinity by using our bodies.
Metaphorically speaking, that is. George Lakoff, a cognitive linguist at the University of California, Berkeley, believes we can only understand infinity based on what we can do with our bodies. More specifically, he says we deal with the headache of infinity by drawing on our familiarity with repetitive and iterative motions – walking, jumping and breathing, for example.
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Lakoff’s research has led him to believe that we handle all kinds of abstract concepts by using metaphors. Consider statements such as “she is a very cold person”, for instance, or “please give her my warmest greetings”. Here, the abstract concept of affection is understood metaphorically via our sensory experiences of temperature. Or when we describe the progress of a task, we link it to our experience of reaching destinations: “I’m not there yet” or “I’m close to it”, or “there’s something standing in my way”.
Of course, we are all familiar with using these kinds of metaphors, and Lakoff says he has found thousands of similar examples across all the world’s cultures. But they are not restricted to people and their actions or feelings. We use similarly physical metaphors when discussing abstract concepts such as economic policy: phrases such as “France fell into a recession” or “India is stumbling in its efforts to liberalise”, for example. So if our minds grasp abstract concepts of economics in terms of what our bodies can experience, are our bodies also the way we can understand the infinite?
That is the contention of Lakoff and Rafael Núñez, a cognitive scientist at the University of California at San Diego. In the early 1990s, Núñez was wondering about how the human mind grasped the concept of an “actual” infinity – such as a point that exists at infinity – in contrast to a “potential” infinity, the sort that is merely implied but never actually reached. For example, the infinity of the counting numbers is a potential infinity, since the numbers never actually get there. Núñez reasoned that since no human has ever directly experienced an actual infinity, we must use some kind of metaphor to think about it. But what?
In 1993, Núñez went to Berkeley to work on the problem with Lakoff, who at that time was leading research into the metaphorical nature of language and human thought. The duo soon became a trio when they enlisted the help of then Berkeley doctoral student Srini Narayanan, now a researcher at the International Computer Science Institute in Berkeley.
Narayanan was developing computer models of how the brain controls bodily movements. He noticed that all his motor-control programs, which aimed to simulate the brain’s control mechanisms, had the same basic structure, involving states such as “get ready”, “initiate movement”, “check if purpose has been achieved”, and so on.
When Lakoff came across Narayanan’s work, he realised he had seen this same structure elsewhere. Languages the world over encode it in their grammar to allow people to describe and think about actions and events. For instance, linguists have shown that when someone says “John jumped”, we visualise a series of states that starts with John preparing to jump, and ends with John landing somewhere.
Lakoff, Núñez and Narayanan speculated that the structures in the brain that control body movements might also be used to handle all abstract concepts. So France “falling into” a recession is our best stab at representing the country’s economic plight. “The logical underpinning of problem-solving or reasoning about complex events may have some connection to the structure of movement and structure of perception,” Narayanan says.
With this idea in place, Lakoff and Núñez built an entirely new understanding of mathematics. We conceptualise numbers as metaphors, such as a series of points on a line, Lakoff says. And if any reasoning that invokes metaphors is constrained by the neural processes that govern movement, mathematics is likely to be similarly constrained: we will only be able to understand mathematics in terms of parallels with what our bodies can do or experience. In fact, the researchers go so far as to say that mathematics is actually born out of our physical experience of the world. “We have created mathematics; it’s a product of our bodies, our brains, and our functioning in the world, and it’s uniquely human,” Lakoff says. “Other beings that have different bodies might have totally different notions of mathematics, if they had mathematics at all.”
So where does this leave our understanding of infinity? It depends on which kind of infinity we are talking about. Cases of potential infinity, such as creating polygons with more and more sides, can be easily conceptualised as a never-ending process. Each iteration still produces a comprehensible result: a polygon with one more side. However big a polygon we are given, we can always cope with adding an extra side to it.
The real challenge is conceptualising actual infinity – infinity as a thing in itself, such as the point that exists at infinity. But remember, in Lakoff and Núñez’s view, we are in control of infinity – it is a concept we have invented to deal with certain aspects of what we see and experience in the physical world. And so, to conceptualise it, we have come up with a clever trick. They call it the Basic Metaphor of Infinity (BMI), in which we simply impose a metaphorical end to cap off whatever process is repeating.
The researchers believe the BMI appears in pre-Socratic Greek philosophy, in which every object is an instance of a higher category. A cow is an instance of the category Cow, a goat is an instance of the category Goat. Since the categories Cow and Goat are things in themselves, they are instances of some even higher category. This iterative process of categorisation goes on until you come to an end – the final category of Being. “We claim that probably most of the monotheistic systems somehow deal with the BMI, in which you have to put together all kinds of frames of reference in one line, to give you one last bang – some sort of an almighty being that is the end of the chain of relationships,” Núñez says.
Using the BMI in mathematics is a bit more complicated. “If you want to apply the BMI to mathematics in specific domains, then you need training,” Núñez says. But he and Lakoff have analysed many of the well known instances of actual infinity in mathematics, such as how two parallel lines can meet at infinity (see Graphic), and they hypothesise that these are all special cases of the BMI.
It is worth pointing out that Lakoff and Núñez have dedicated an entire book to explaining the reasoning behind their ideas on mathematics, not just two pages of a magazine. Their work is contentious – after all, it challenges the age-old idea that infinity and mathematics exist as universal truths that we have simply stumbled upon – but it has been very well-received. The acid test is, perhaps, whether it feels right – is this really how mathematicians visualise actual infinity? John Allen Paulos, a mathematician at Temple University in Philadelphia, Pennsylvania, thinks so. “The BMI is more or less the way I think about infinity,” he says.
But if Lakoff and Núñez are right, what does this do for the romance of infinity and mathematics? “Realising that the notion of actual infinity is metaphorical and made up by people changes your whole idea of the relationship of mathematics to the world,” Lakoff says. “It’s much more interesting to understand mathematics as a product of human beings.” On the other hand, you might see it as cold comfort. If, like Alfred de Musset, you are tortured by the concept of infinity, you have only yourself – or at least your body – to blame.