Mathematics is more than the memorization and application of various rules. Although the language of mathematics can be intimidating, the concepts themselves are built into everyday life. In the following excerpt from A Brief History of Mathematical Thought, Luke Heaton examines the concepts behind mathematics and the language we use to describe them.
There is strong empirical evidence that before they learn to speak, and long before they learn mathematics, children start to structure their perceptual world. For example, a child might play with some eggs by putting them in a bowl, and they have some sense that this collection of eggs is in a different spatial region to the things that are outside the bowl. This kind of spatial understanding is a basic cognitive ability, and we do not need symbols to begin to appreciate the sense that we can make of moving something into or out of a container. Furthermore, we can see in an instant the difference between collections containing one, two, three or four eggs. These cognitive capacities enable us to see that when we add an egg to our bowl (moving it from outside to inside), the collection somehow changes, and likewise, taking an egg out of the bowl changes the collection. Even when we have a bowl of sugar, where we cannot see how many grains there might be, small children have some kind of understanding of the process of adding sugar to a bowl, or taking some sugar away. That is to say, we can recognize particular acts of adding sugar to a bowl as being examples of someone ‘adding something to a bowl’, so the word ‘adding’ has some grounding in physical experience.
Of course, adding sugar to my cup of tea is not an example of mathematical addition. My point is that our innate cognitive capabilities provide a foundation for our notions of containers, of collections of things, and of adding or taking away from those collections. Furthermore, when we teach the more sophisticated, abstract concepts of addition and subtraction (which are certainly not innate), we do so by referring to those more basic, physically grounded forms of understanding. When we use pen and paper to do some sums we do not literally add objects to a collection, but it is no coincidence that we use the same words for both mathematical addition and the physical case where we literally move some objects. After all, even the greatest of mathematicians first understood mathematical addition by hearing things like ‘If you have two apples in a basket and you add three more, how many do you have?’
As the cognitive scientists George Lakoff and Rafael Núñez argue in their thought-provoking and controversial book Where Mathematics Comes From, our understanding of mathematical symbols is rooted in our cognitive capabilities. In particular, we have some innate understanding of spatial relations, and we have the ability to construct ‘conceptual metaphors’, where we understand an idea or conceptual domain by employing the language and patterns of thought that were first developed in some other domain. The use of conceptual metaphor is something that is common to all forms of understanding, and as such it is not characteristic of mathematics in particular. That is simply to say, I take it for granted that new ideas do not descend from on high: they must relate to what we already know, as physically embodied human beings, and we explain new concepts by talking about how they are akin to some other, familiar concept.
Conceptual mappings from one thing to another are fundamental to human understanding, not least because they allow us to reason about unfamiliar or abstract things by using the inferential structure of things that are deeply familiar. For example, when we are asked to think about adding the numbers two and three, we know that this operation is like adding three apples to a basket that already contains two apples, and it is also like taking two steps followed by three steps. Of course, whether we are imagining moving apples into a basket or thinking about an abstract form of addition, we don’t actually need to move any objects. Furthermore, we understand that the touch and smell of apples are not part of the facts of addition, as the concepts involved are very general, and can be applied to all manner of situations. Nevertheless, we understand that when we are adding two numbers, the meaning of the symbols entitles us to think in terms of concrete, physical cases, though we are not obliged to do so. Indeed, it may well be true to say that our minds and brains are capable of forming abstract number concepts because we are capable of thinking about particular, concrete cases.
Mathematical reasoning involves rules and definitions, and the fact that computers can add correctly demonstrates that you don’t even need to have a brain to correctly employ a specific, notational system. In other words, in a very limited way we can ‘do mathematics’ without needing to reflect on the significance or meaning of our symbols. However, mathematics isn’t only about the proper, rule-governed use of symbols: it is about ideas that can be expressed by the rule-governed use of symbols, and it seems that many mathematical ideas are deeply rooted in the structure of the world that we perceive.
Featured image credit: “mental-human-experience-mindset” by johnhain. CC0 via Pixabay.